# Estimating Permeability of Porous Media from 2D Digital Images

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## Abstract

**:**

## 1. Introduction

_{2}sequestration, geothermal energy extraction, and unconventional onshore and offshore oil and gas recovery processes [10,11,12,13,14].

## 2. Methodology

_{2D}will be derived based on the tube bundle model and fractal geometry [29,39,40]. In the capillary bundle model, the capillary bundles are perpendicular to the images (i.e., capillary bundles run along y direction). Then, by combining the KC equation and fractal geometry, 3D pore structure of porous media in flow equivalence will be estimated, and the permeability of natural samples (in 3D) K

_{3D}will be derived. Finally, a bridge function connecting K

_{2D}and K

_{3D}will be proposed.

**Permeability of 2D cross-section:**In general, based on 2D imaging (thin section, back scatter electron, micro-CT scanning, scanning electron microscope, scanning electron microscopy, and electron probes, etc.) and digital image processing techniques, pore structure parameters (areal porosity, pore size distribution, average pore radius, pore perimeter, specific surface area, pore fractal dimension, coordination number, and shape factor, etc.) of 2D cross-sections can be determined [26,31,50,51]. For instance, pore fractal dimension of 2D cross-section can be obtained by using the box-counting algorithm [38]. In addition, areal porosity of 2D cross-section can be obtained by thin section analysis or digital image processing techniques. Moreover, by using digital image processing techniques, various pore structure parameters (pore size distribution, the maximum pore size, minimum pore size, average pore radius, pore perimeter, and specific surface area) of 2D cross-sectional images can be easily determined [27,36,46,52,53]. For example, for a given 2D digital image, parameters (r

_{max}, r

_{min}, and φ

_{2D}) can be determined by using the Avizo software, which can effectively determine pore space morphology and extract pore networks [25].

_{2D}, pore fractal dimension d

_{f}) from 2D imaging and fractal theory, permeability of 2D cross-section (i.e., 2D composite permeated with tortuous capillary tubes that are perpendicular to the 2D cross-section, Figure 1) K

_{2D}is given by [12,29,37,40,55]:

_{t}is tortuosity fractal dimension in 2D space, which is [42]:

_{t}ranges from 1 to 2 in 2D space. When the matrix plane is filled with so highly tortuous capillary tubes, d

_{t}is equal to 2; however, for straight capillary tubes, d

_{t}is equal to 1, and on this condition, Equation (1) can be simplified as:

_{f}in Equation (1) is the pore fractal dimension, which ranges from unity to 2 (1< d

_{f}< 2) in 2D pore space. Mathematically, d

_{f}can be determined by box-counting algorithm [38] or by the following equation [29,35]:

_{e}is 2 in 2D pore space. Equation (5) reveals that, under a given ratio r

_{min}/r

_{max}, d

_{f}increases with an increasing of areal porosity φ

_{2D}. In addition, by fitting the experimental data, Chen et al. [56] also found a positive correlation between d

_{f}and φ

_{2D}and suggested d

_{f}could be estimated by ${d}_{\mathrm{f}}=0.974{\phi}_{2\mathrm{D}}^{0.173}$. Equations (1) through (5) form the basis for analysis of K

_{2D}, which were utilized to estimate permeability of 3D parent porous media in flow equivalence.

**3D pore structure in flow equivalence:**As 2D cross-sections come from 3D parent samples, various studies show that pore structure information (pore morphology) of 2D slice is identical to that of 3D matrix system [17,30,57]. For example, Chen et al. [17] suggested that areal porosity of 2D cross-sections φ

_{2D}should be similar to bulk porosity of 3D matrix φ

_{3D}. In addition, to estimate permeability in 3D from 2D images, Saxena et al. [26] also assumed that φ

_{2D}was similar to φ

_{3D}. Figure 2a compares bulk porosity of 3D parent samples versus areal porosity of 2D cross-sections [17,26,31,36,46,53]. Taking the data from Saxena et al. [26], for example, Saxena et al. conducted modeling on 4 samples (i.e., a Fontainebleau sandstone, a Berea sandstone, a Bituminous sand sample, and a Grosmont carbonate). Based on digital rock technology, the bulk porosity φ

_{3D}for each sample was determined. Then, multiple 2D cross-sections (lying perpendicular to the flow direction) of a parent cube sample were sampled, and the corresponding areal porosity of each slice was measured using digital image process technology. Similarly, for 3D samples with bulk porosity determined, Wu et al. [36,53] divided the original sample into various layers along vertical direction using low- and high-resolution X-ray CT scanning technologies. Then, areal porosities in each layer of low- and high-resolution digital rock images were measured. Besides digital rock technology, thin section analysis was used by Berryman and Blair [31] and Chen et al. [17] to measure areal porosity of 2D thin. As shown in Figure 2a, porosity of 3D sample φ

_{3D}is approximately in the middle between the maximum and minimum values of areal porosities of 2D cross-sections. Specifically, for a 3D porous media with bulk porosity φ

_{3D}, most of the areal porosity data φ

_{2D}for 2D sectional sections are in the range of 0.9φ

_{3D}and 1.1φ

_{3D}. Figure 2b compares bulk porosity of 3D samples versus average porosity of 2D cross-sections. Results suggest that porosity of 3D sample is quite consistent with the average areal porosity of 2D slices. Specifically, the slope of the fitting line is 1.009, which is close to unity. And the intercept of the fitting line is close to 3.38 × 10

^{−4}, which is quite close to 0. As a result, for simplicity of the model, areal porosity of 2D cross-sections φ

_{2D}is assumed to be identical to bulk porosity of 3D matrix (parent samples) φ

_{3D}, i.e.,

_{T}) is larger than unity. Based on KC equation, Chen et al. [17] developed a bridge function connecting capillary radius in 2D space to that in 3D space. As stated by Chen et al. [17], for a given capillary with pore radius r in 2D space, the corresponding pore radius r

_{3D}in 3D space is:

_{3D}in 3D space is less than that in 2D space, and the ratio of r to r

_{3D}is tortuosity to the fourth power, i.e., τ

^{4}. With the same method as Chen et al. [17], but assuming that the ratio of specific surface in 3D space to that in 2D space is 3/2 and not the tortuosity, Saxena et al. [26] suggested that the ratio of r to r

_{3D}is $9{\tau}^{2}/4$. To some extent, $9{\tau}^{2}/4$ can be considered a special case of τ

^{4}(e.g., when τ is assigned as 3/2, $9{\tau}^{2}/4$ equals τ

^{4}), so, in this paper, to make our model more general, Equation (7) was used to correlate r and r

_{3D}.

_{3D}is [29]:

_{f}in 3D space as:

_{3D,max}should be larger than r

_{3D,min}, which means D

_{T}in Equation (10) should be less than 1.25. Based on fractal theory, in 3D pore space, D

_{T}is [42]:

_{f}is pore fractal dimension in 3D space, which can be also determined by [29,35]:

_{f}can be rewritten as:

_{T}can be determined. Then, based on Equation (13), D

_{f}can be determined. By substituting Equation (10) into Equation (9), we have:

_{3D,max}and r

_{3D,min}can be determined. Then, according to Equation (A11), 3D permeability K

_{3D}of porous media in flow equivalence can be given by:

_{max}, r

_{min}, φ

_{2D}, and d

_{f}) from 2D digital image. Specifically, for a given 2D digital image, parameters (r

_{max}, r

_{min}, and φ

_{2D}) can be determined by using the Avizo software [25]. In addition, d

_{f}can be determined with box-counting algorithm [38] or with Equation (5). Subsequently, parameter φ

_{3D}can be calculated using Equation (6).

_{T}by solving Equation (14). Then, determine D

_{f}by using Equation (13). In addition, determine parameter V by using Equation (15). Then, r

_{3D,max}and r

_{3D,min}can be determined by using Equation (10).

_{3D}and τ

_{av}of the corresponding parent sample using Equations (16) and (A12). In addition, the bridge function (i.e., the ratio of K

_{2D}to K

_{3D}) can be determined by combining Equations (1) and (16). Moreover, other 3D pore structure parameters of the parent sample in flow equivalence can be further estimated.

## 3. Model Validation

_{3D}of 23.5% and a total volume of 1.64

^{3}mm

^{3}, the volume (voxel) is 648

^{3}, and the voxel volume is 2.53

^{3}µm

^{3}. With the D3Q19 model stated in Appendix B, the determined permeability in 3D is 27.8 × 10

^{−3}µm

^{2}. In addition, for site B with a bulk porosity φ

_{3D}of 23.4%, the volume (voxel) is 681

^{3}, the voxel volume is 1.55

^{3}µm

^{3}, and the total volume is about 1.06

^{3}mm

^{3}. Then, based on LBM simulation, the 3D permeability of site B was determined, with the value of 69.4 × 10

^{−3}µm

^{2}. More detailed information about the D3Q19 model can be found in Appendix B. For the sake of simplification, site J (J = A, B, C, ⋅⋅⋅) from the sample I (I = 1, 2, 3) is labeled as sample I–J.

_{max}, r

_{min}, φ

_{2D}, and d

_{f}) from the 2D digital image were determined; then, based on our derived model, the 3D permeability of the carbonates was predicted. In addition, our predicted 3D permeability was compared to that from D3Q19 LBM to validate the feasibility and effectiveness of our derived model. The experimental data and modeling results are summarized in Table 1. As shown in Table 1 and Figure 4a, the areal porosity (2D porosity) of the cross-section consists of the bulk porosity (3D porosity) of the parent sample. The results (Figure 4a) suggest the slope and the intercept of the fitting line are 1.048 (close to unity) and 7.56 × 10

^{−3}(close to 0), respectively. Similar findings are shown in Figure 2. The results shown in Table 1 suggest that the predicted 3D permeability of each parent sample from the derived model exhibits excellent agreement with the experimental data (the results from the LBM simulations). Taking parent sample 1-A (Figure 4a), for example, in light of the image processing technology, the φ

_{2D}of the selected 2D cross-section (Figure 4b) is 31.24%. In addition, with the box-counting algorithm [38], the double logarithmic (Lg) plot of the box size (pixel) versus the total pore number is presented in Figure 4b; then, the pore fractal dimension d

_{f}(the negative of the slope) was determined to be 1.686. As shown in Figure 4b, the pore structure of the 2D cross-section of sample 1-A has a typical fractal scale. In our modeling, d

_{f}and φ

_{2D}were assigned as 1.686 and 31.24%, respectively. r

_{max}was assigned as 535.06 µm, and φ

_{3D}was assigned as 23.5%. Moreover, r

_{3D,max}and D

_{f}were assigned as 41.18 µm and 2.81, respectively. Then, based on the derived model, the predicted permeability K

_{3D}is 27.8 × 10

^{−3}µm

^{2}, which is in agreement with that determined by the LBM simulation. Figure 4c presents the pore structure parameters (e.g., τ

_{av}, r

_{max}/r

_{3D,max}, and D

_{T}) of the samples. As we can see from Figure 4c, the parameter r

_{max}is larger than r

_{3D,max}, which is consistent with the results stated by Chen et al. [17]. For these 9 samples, the ratio r

_{max}/r

_{3D,max}ranges from 3.18 to 10.58. In addition, D

_{T}ranges from 1.10 to 1.15, and τ

_{av}ranges from 2.19 to 4.69. The results (Figure 4c) also suggest that even from the same parent sample, the different sites behave strongly in heterogeneity. For instance, sample 1-A and sample 1-B come from the same parent sample 1; however, their physical properties vary greatly. Figure 4d presents the average tortuosity versus the ratio r

_{max}/r

_{3D,max}. The results (Figure 4c) show that there exists an approximately linear relationship between the ratio of the maximum pore radius in 2D space to that in 3D space (r

_{max}/r

_{3D,max}) and the average tortuosity of porous media. Physically, when the average tortuosity of porous media is unity, the corresponding ratio r

_{max}/r

_{3D,max}approximately equals unity. As we can see from the fitting equation, when τ

_{av}is assigned as unity, r

_{max}/r

_{3D,max}is determined to be 0.82, which is close to unity, which verifies our model. The reason for the deviation may be caused by the computational error.

**Dataset of Saxena et al. [26]**

**:**To further verify our derived model, the predicted K

_{2D}and K

_{3D}from the new model were compared with available test data [26]. Saxena et al. conducted numerical experiments on 12 samples from various geologic formations [26]. For these 12 samples, samples 1 to 3 come from the same parent Berea sandstone and represent Berea sandstones A to C, respectively. In addition, samples 4 to 6 represent Bituminous sands A to C, respectively. Samples 7 to 9 represent Fontainebleau (Font) sandstones A to C, respectively. And samples 10 to 12 represent Grosmont carbonates A to C, respectively. During the numerical experiments, Saxena et al. first calculated the K

_{3D}of the parent samples based on 3D digital rocks using an LBM simulation; then, they sliced 2D thin sections along the flow direction from the parent samples and performed an LBM simulation to estimate the K

_{2D}of the thin section [26]. It should be noted that to calculate the K

_{2D}of the thin section, the pore structure was restructured by permeating the pores in the thin section with straight tubes along the flow direction. That is, for the restructured porous media used to determine K

_{2D}, the tortuosity fractal dimension d

_{t}was equal to 1. As a result, during our modeling, d

_{t}was assigned as 1, and Equation (3) was used to validate our derived model.

^{3}and a pixel size of 0.74 µm, for example, the bulk porosity φ

_{3D}is 4%, and the areal porosity φ

_{2D}of the thin sections ranges from 2% to 6% with an average of 4%. Based on simulations, the LBM 3D permeability of the parent sample was determined as 3 × 10

^{−3}µm

^{2}, and the average LBM 2D permeability of the thin sections is 624 × 10

^{−3}µm

^{2}. In our modeling, both φ

_{3D}and φ

_{2D}were assigned as 4%. Based on Equations (3) and (13), K

_{2D}and K

_{3D}can be calculated. The results (Figure 5a–d) suggest that the predictions (K

_{2D}and K

_{3D}) provide a good match over the test data. Besides K

_{2D}and K

_{3D}, other parameters (e.g., d

_{f}, τ

_{av}, r

_{max}/r

_{3D,max}, D

_{T}, and D

_{f}) of each sample are presented in Figure 5e. As we can see from Figure 5e for these 12 samples, τ

_{av}ranges from 1.10 to 1.50, which is smaller than that of the carbonates in Figure 4. This reveals that the carbonates in Figure 4 have stronger heterogeneity than the rocks in Figure 5. The reason may be that the rocks in Figure 4 are carbonate core plugs; however, most of the rocks in Figure 5 are sandstones, which are more homogeneous. In addition, Figure 5e shows that the ratio r

_{max}/r

_{3D,max}ranges from 1.15 to 1.74, D

_{T}ranges from 1.01 to 1.03, d

_{f}ranges from 1.64 to 1.82, and D

_{f}ranges from 2.69 to 2.83. The results (Figure 5f) also suggest that τ

_{av}and the ratio r

_{max}/r

_{3D,max}are linearly dependent. When τ

_{av}is assigned as unity, the ratio r

_{max}/r

_{3D,max}is estimated to be 0.93, which is approximately equal to unity. Similar findings are shown in Figure 4e.

**Dataset of Berryman and Blair [31]**

**:**Based on digital image analysis, Berryman and Blair first studied the two-point correlation functions, porosity, and specific surface area of glass beads (55 µm), Ironton Galesville (IG) sandstones, and Berea sandstones [31]. Then, they predicted the permeability of these samples with the following equation:

^{−1}, an areal porosity φ

_{2D}of 17%, and an image permeability K

_{2D}of 0.312 µm

^{2}. For example, Berryman and Blair found the bulk porosity (laboratory porosity) φ

_{3D}ranged from 15% to 18% with an average value of 16.5%. In addition, based on Equation (14), the laboratory permeability of K

_{3D}was determined to be 0.023 µm

^{2}. In our modeling, φ

_{3D}and φ

_{2D}were assigned as 16.5% and 17%, respectively. The results (Figure 6a) suggest that the predicted results (K

_{2D}and K

_{3D}) of these samples (i.e., two glass beads, four Berea samples, IG-775 and IG-785) provide a good match over the test data. In addition, the pore structure parameters (e.g., d

_{f}, τ

_{av}, r

_{max}/r

_{3D,max}, D

_{T}, and D

_{f}) of each sample are presented in Figure 6b. As pore structures (d

_{f}, τ

_{av}, r

_{max}/r

_{3D,max}, D

_{T}, and D

_{f}) of these samples are different, the curves show different behaviors. In addition, the results (Figure 6c) also suggest that τ

_{av}is remarkably correlated linearly with the ratio r

_{max}/r

_{3D,max}.

_{3D}) composed of identical spherical grains, the average particle radius R

_{ap}of the porous medium could be written as [26,31,58]:

_{3D}is the specific surface area. Based on the derivation in Appendix C, R

_{ap}can be also expressed as:

_{avs}is the average spherical particle radius, and R

_{avc}is the average circular particle radius. More details about R

_{avs}and R

_{avc}can be found in Equations (A22) and (A24). By combining Equations (15) and (16), the weight coefficient of the average spherical particle radius can be determined. Figure 6d compares the calculated average particle radius for spherical particles and the calculated average particle radius for circular particles with that from the former model. The results from Figure 6d demonstrate that the calculated average particle radius from Equation (15) is approximately in the middle between the calculated values from Equations (A22) and (A24). This indicates that the model gives predicted values that are quite consistent with the results from the former model. In addition, by combining the results from Equations (15) and (16), the weight coefficients for different samples have been determined in Figure 6d. Taking sample 3 for example, when the weight coefficient is unity, the average particle radius can be effectively determined by Equation (A22).

## 4. Results and Discussions

_{f}and D

_{T}) on the 3D permeability of porous media. Figure 7 shows the influence of pore fractal dimension D

_{f}on the average tortuosity τ

_{av}and K

_{3D}of porous media in 3D. In this case, the maximum and minimum pore radii in 2D were 3 µm and 0.13 µm, respectively. The parameters φ

_{2D}and φ

_{3D}were both 0.15, and parameter D

_{T}was 1.1. During the calculation, the parameter D

_{f}ranged from 2.2 to 2.8. Based on our derived model, the maximum pore radius in 3D space r

_{3D,max}was determined, which ranged from 1.4 µm to 0.82 µm. In addition, the determined parameter r

_{3D,min}ranged from 0.0077 µm to 0.0045 µm. Specifically, when parameter D

_{f}was assigned as 2.2, the corresponding values of r

_{3D,max}and r

_{3D,min}were 1.4 µm and 0.0077 µm, respectively. As one can see from Figure 7a, there exists a positive relationship between D

_{f}and the average tortuosity τ

_{av}. The main reason is that a larger value of D

_{f}means a more complex pore structure of porous materials, leading to a larger value of τ

_{av}. Furthermore, the permeability of porous media in 3D decreases with an increase in D

_{f}, which is anticipated. Figure 7b demonstrates that the linear correlativity between τ

_{av}and the ratio r

_{max}/r

_{3D,max}is very prominent. Moreover, when τ

_{av}is assigned as unity, r

_{max}/r

_{3D,max}is determined to be 0.9951, which is close to unity.

_{T}on the τ

_{av}and K

_{3D}are shown in Figure 8. In this case, the basic parameters applied in the model are summarized in the corresponding figures. During the calculation, the parameter D

_{f}was assigned as 2.2, and the parameter D

_{T}ranged from 1.05 to 1.2. Based on our derived model, the parameter r

_{3D,max}ranges from 2.2 µm to 0.07 µm. The results (Figure 8a) suggest that a larger τ

_{av}corresponds to a larger D

_{T}. However, K

_{3D}decreases as D

_{T}increases. Specifically, for this case, when D

_{T}increases up to a certain value (e.g., D

_{T}≥ 1.2), the value of K

_{3D}is extremely small. The main reason is that a larger value of D

_{T}means larger seepage resistance, resulting in a small value of K

_{3D}. Figure 8b also reveals that τ

_{av}has distinct linear correlations with the ratio r

_{max}/r

_{3D,max}. Similar findings have been also demonstrated in Figure 6.

_{bridge}connecting K

_{2D}and K

_{3D}(i.e., f

_{bridge}= K

_{2D}/K

_{3D}), we studied the influences of D

_{f}and D

_{T}on this bridge function f

_{bridge}. Figure 9 presents the influences of D

_{f}and D

_{T}on f

_{bridge}. For the calculation, the parameters applied in the model are summarized in the corresponding figures, which are identical to those for Figure 7 and Figure 8, respectively. As one can see from Figure 9, the bridge function increases as D

_{f}(or D

_{T}) increases. The main reason is that for a given 2D pore structure, a larger value of D

_{f}(or D

_{T}) corresponds to a smaller value of K

_{3D}. Similar findings can be found in Figure 7a and Figure 8a. Moreover, Figure 9 reveals that with an increase of D

_{f}(or D

_{T}), the increase rate of f

_{bridge}increases. The main reason is that with an increase of D

_{f}(or D

_{T}), the increase rate of the seepage resistance in the porous media increases.

**Model advantages and limitations:**The developed model provides a theoretical basis for predicting the 3D permeability of porous media from 2D digital image analysis without the need for 3D reconstruction, achieving high accuracy even with only 2D information. Compared to DRP, our model not only reduces the computational cost of high-resolution scanning, 3D pore network reconstruction, and numerical simulation but also captures the effect of realistic pore shapes on permeability. Furthermore, our model can be used for inverse modeling to estimate relevant parameters, such as tortuosity and the pore fractal dimension in 3D, making it highly practical for estimating permeability in heterogeneous porous media. Overall, our derived model is of great practical significance, as 2D images are easier and cheaper to access than 3D images and reconstructions of 3D pore networks.

_{2D}is assumed to be identical to the bulk porosity of the 3D matrix. However, φ

_{2D}varies with the slices, and sometimes the difference between the maximum and the minimum 2D porosity is relatively large. Although φ

_{3D}is approximately in the middle between the maximum and minimum values of φ

_{2D}, 2D porosity is not enough to represent φ

_{3D}. Moreover, our developed model is limited to intact porous media and ignores the effect of micro-fractures on the 3D permeability of porous media. Thus, further research is required to reduce the uncertainty in estimating the 3D permeability of porous media from 2D images without reconstruction. Furthermore, in general, the pore surface of porous materials is rough, and the surface fractal dimension is crucial to characterize the fluid flow in porous media [41,59,60]. For example, Lei et al. [59] and Xiao et al. [41] derived theoretical models to study the fluid flow in porous media, and they concluded that the effects of surface fractal dimension on permeability and relative permeability in porous media were significant. Thus, to make our model more reasonable, in our future work, a rough pore surface and surface fractal dimensions will be taken into account. Moreover, as mentioned above, the interaction between solid minerals and fluids will significantly affect the permeability of porous media. To improve the applicability of our derived model, the interaction between solid minerals and fluids will be taken into account. What is more, as the information on fracture systems may not be obtained from 2D cross-sections, in this paper, our derived model focuses on predicting the 3D permeability of intact porous media, and the fractures are ignored. In our future work, we will try to extend our model to study the 3D permeability of fractured porous media from 2D cross-sections of parent samples without 3D reconstruction.

## 5. Conclusions

_{T}, pore fractal dimension in 3D space D

_{f}) on the 3D permeability K

_{3D}of porous media. The results indicate that the average tortuosity τ

_{av}decreases with an increase of D

_{f}(or D

_{T}). In addition, the average tortuosity τ

_{av}is remarkably correlated linearly with the ratio of r

_{max}(the maximum pore radius of 2D images) to r

_{3D,max}(the maximum pore radius of 3D porous media). Moreover, the permeability K

_{3D}decreases as D

_{f}(or D

_{T}) increases. With an increase of D

_{f}(or D

_{T}), the decrease rate of the permeability K

_{3D}increases.

_{2}geology storage, unconventional onshore and offshore oil/gas development, and groundwater seepage.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Latin symbols | |

A | The cross-sectional area of the representative element (µm^{2}) |

A_{av} | The average pore area (µm^{2}) |

b | A constant that depends on the cross-section of the tubes (dimensionless) |

c | Lattice sound speed, which is determined by Δx/Δt (m/s) |

d_{e} | The Euclidean dimension is 2 in 2D space (dimensionless) |

d_{f} | The pore fractal dimension in 2D space (dimensionless) |

d_{t} | The tortuosity fractal dimension in 2D space (dimensionless) |

D_{f} | The pore fractal dimension in 3D space (dimensionless) |

D_{T} | The tortuosity fractal dimension in 3D space (dimensionless) |

e_{α} | Lattice velocities (m/s) |

e_{y} | The unit vector along the y-axis (m/s) |

F | The formation factor (dimensionless) |

f | The pore size distribution in 2D space (dimensionless) |

f_{3D} | The pore size distribution in 3D space (dimensionless) |

f_{α}(y, t) | The evolution of the density distribution function (kg/m^{3}) |

K_{2D} | Permeability of the 2D cross-section (10^{−3} µm^{2}) |

K_{3D} | 3D permeability of porous media in flow equivalence (10^{−3} µm^{2}) |

${K}_{ij}$ | Permeability in the $i$ direction when the flow is driven in the $j$ direction (10^{−3} µm^{2}) |

L | The actual streamlined length of a tortuous capillary (µm) |

${L}_{j}$ | The size of the computational domain in the $j$ direction (µm) |

L_{u} | The edge length of the cubical unit cell (µm) |

N | The total number of pores in the representative element of a 2D cross-section (dimensionless) |

N_{3D} | The total number of pores in a representative elementary volume (dimensionless) |

N_{total} | The total lattice number (dimensionless) |

p | The transient pressure (Pa) |

p_{in} | The inlet transient pressure (Pa) |

p_{out} | The outlet transient pressure (Pa) |

r | The pore radius in 2D space (µm) |

r_{av} | The average pore radius in 2D space (µm) |

r_{3D} | The pore radius in 3D space (µm) |

r_{3D,av} | The average pore radius in 3D space (µm) |

r_{max} | The maximum pore radius in 2D space (µm) |

r_{min} | The minimum pore radius in 2D space (µm) |

r_{3D,max} | The maximum pore radius in 3D space (µm) |

r_{3D,min} | The minimum pore radius in 3D space (µm) |

R | The particle radius of the porous media (µm) |

R_{ap} | The average particle radius of the porous media (µm) |

R_{avc} | The average circular particle radius (µm) |

R_{avs} | The average spherical particle radius (µm) |

R_{max} | The maximum particle radius (µm) |

S_{2D} | The specific surface area of the 2D cross-section (µm^{−1}) |

S_{3D} | The specific surface area of the 3D cross-section (µm^{−1}) |

u_{j} | The velocity at the void point j (m/s) |

u | The flow velocity (m/s) |

V | The pore volume of representative elementary volume (µm^{3}) |

V_{av} | The average pore volume (µm^{3}) |

V_{u} | The volume of the cubical unit cell (µm^{3}) |

Greek symbols | |

τ | Tortuosity (dimensionless) |

τ(r_{3D}) | Tortuosity of pore radius r_{3D} (dimensionless) |

τ_{av} | The average tortuosity (dimensionless) |

τ_{0} | The relaxation time (dimensionless) |

φ_{2D} | The areal porosity of 2D cross-sections (dimensionless) |

φ_{3D} | The bulk porosity of a 3D matrix (dimensionless) |

µ | The dynamic viscosity (mPa·s) |

Δy | The lattice distance or the voxel size (µm) |

y | The grid location (µm) |

Δt | The time step (s) |

${u}_{i}$ | The component in the i direction of the volumetric average velocity (m/s) |

v | The kinematic viscosity (m^{2}/s) |

ρ | The fluid density in the D3Q19 model (kg/m^{3}) |

ω_{α} | Fixed weighting factors (dimensionless) |

π | Circular constant, which is approximately equal to 3.1415926 (dimensionless) |

λ | The weight coefficient of the average spherical particle radius (dimensionless) |

Subscript symbols | |

α | The direction in the D3Q19 model (dimensionless) |

2D | Two-dimensional space |

3D | Three-dimensional space |

max | Maximum |

min | Minimum |

## Appendix A. Fractal Theory of Porous Media in 2D and 3D Space

_{f}can be calculated with the box-counting algorithm [38]. Moreover, with thin section analysis or digital image processing techniques, areal porosity φ

_{2D}and pore size distribution can be easily determined. As the pore structures of most sedimentary porous media follow fractal characteristics, fractal theory has been widely applied to describe the pore structures of porous media [29,34,35]. Based on fractal geometry, with the determined pore fractal dimension d

_{f}, the maximum pore radius r

_{max}, and the minimum pore radius r

_{min}, the pore size distribution f and the total number of pores in a representative element of 2D cross-section N can be written as [29,35]:

_{av}and specific surface area S

_{2D}of the 2D cross-section are:

_{2D}can be obtained as [12,29,40,54,55]:

_{t}is the tortuosity fractal dimension in 2D space, which can be determined with [42]:

_{t}is assigned as unity, K

_{2D}can be simplified as:

_{f}, the maximum pore radius r

_{3D,max}, and the minimum pore radius r

_{3D,min}, the pore size distribution f

_{3D}and the total number of pores in a representative elementary volume (SRV) N

_{3D}can be written as:

_{T}can be determined as [29,42]:

_{3D}can be obtained as [12,55,61,62]:

_{av}is:

_{3D}in 3D space, we have:

## Appendix B. The Algorithm of the D3Q19 Lattice Boltzmann Method

**y**is the grid location, c = Δy/Δt is computed with the grid distance Δy (namely the voxel size here), and τ

_{0}is the dimensionless relaxation time, which is determined by the kinematic viscosity v, namely ${\tau}_{0}=3v/\left(c\Delta y\right)+0.5$. Additionally, the lattice velocity

**e**

_{α}and weight ω

_{α}in the moving direction $\alpha \in \left[0,Q-1\right]$ were selected according to the D3Q19 model (3-dimension and 19-velocity) [64]. For an arbitrary grid at

**y**, its neighboring grids are exactly located at

**y**+

**e**

_{α}Δt since the magnitude of

**e**

_{α}depends on c. The equilibrium distribution function ${f}_{\alpha}^{\mathrm{eq}}$ was determined from the local fluid density $\rho $ and flow velocity

**u**that were computed in the absence of internal/external force as follows:

**u**. The bounce-back scheme was used at the complicated solid surface for the non-slip boundary condition and was naturally implemented as the half-way bounce-back scheme with a higher accuracy because the actual solid–fluid interface was exactly located in the middle between the solid and fluid grids in the digital rock simulations. The flows in the current study were driven by a pressure difference ${p}_{\mathrm{in}}-{p}_{\mathrm{out}}$ between the inlet and outlet (equivalent to a density difference due to $p=\rho {c}^{2}/3$), and the density constraint was imposed by the robust non-equilibrium extrapolation scheme [67]. The permeability was computed after convergence:

## Appendix C. Determination of the Average Particle Size

_{3D}was estimated with the idealized geometrical model shown in Figure A1, assuming that the unit cell was composed of spherical particles of the same size. Based on Figure A1, the volume of the cubical unit cell V

_{u}and its edge length L

_{u}are:

**Figure A1.**The arrangement of spherical rock particles for the average pore radius of porous media: (

**a**) the cubical unit cell; (

**b**) front view of the unit cell.

_{av}is:

_{av}can be also estimated with:

_{3D,av}is the average pore radius in 3D space, which is:

_{avs}is

_{max}, which is [12]:

_{avc}can be also determined as:

_{ap}of the porous media can be expressed as:

## References

- Millington, R.J.; Quirk, J.P. Permeability of porous solids. Trans. Faraday Soc.
**1961**, 57, 1200–1207. [Google Scholar] [CrossRef] - Whitaker, S. Flow in porous media I: A theoretical derivation of Darcy’s law. Transp. Porous Media
**1986**, 1, 3–25. [Google Scholar] [CrossRef] - Renard, P.; De Marsily, G. Calculating equivalent permeability: A review. Adv. Water Eesources
**1997**, 20, 253–278. [Google Scholar] [CrossRef] - Meng, M.; Ge, H.; Ji, W.; Wang, X.; Chen, L. Investigation on the variation of shale permeability with spontaneous imbibition time: Sand-stones and volcanic rocks as comparative study. J. Nat. Gas Sci. Eng.
**2015**, 27, 1546–1554. [Google Scholar] [CrossRef] - Guo, D.; Lv, P.; Zhao, J.; Zhang, C. Research progress on permeability improvement mechanisms and technologies of coalbed deep-hole cumulative blasting. Int. J. Coal Sci. Technol.
**2020**, 7, 329–336. [Google Scholar] [CrossRef] - Walsh, J.B.; Brace, W.F. The effect of pressure on porosity and the transport properties of rock. J. Geophys. Res. Solid Earth
**1984**, 89, 9425–9431. [Google Scholar] [CrossRef] - Lock, P.A.; Jing, X.; Zimmerman, R.W.; Schlueter, E.M. Predicting the permeability of sandstone from image analysis of pore structure. J. Appl. Phys.
**2002**, 92, 6311–6319. [Google Scholar] [CrossRef] - Costa, A. Permeability-porosity relationship: A reexamination of the Kozeny-Carman equation based on a fractal pore-space geometry assumption. Geophys. Res. Lett.
**2006**, 33, L02318. [Google Scholar] [CrossRef] - Pan, J.; Zhang, Z.; Li, M.; Wu, Y.; Wang, K. Characteristics of multi-scale pore structure of coal and its influence on permeability. Nat. Gas Ind. B
**2019**, 6, 357–365. [Google Scholar] [CrossRef] - Kozeny, J. Uber kapillare leitung der wasser in boden. R. Acad. Sci. Vienna Proc. Class I
**1927**, 136, 271–306. [Google Scholar] - Carman, P.C. Fluid flow through granular beds. Trans. Inst. Chem. Eng.
**1937**, 15, 150–166. [Google Scholar] [CrossRef] - Xu, P.; Yu, B. Developing a new form of permeability and Kozeny-Carman constant for homogeneous porous media by means of fractal geometry. Adv. Water Resour.
**2008**, 31, 74–81. [Google Scholar] [CrossRef] - Nomura, S.; Yamamoto, Y.; Sakaguchi, H. Modified expression of Kozeny-Carman equation based on semilog-sigmoid function. Soils Found.
**2018**, 58, 1350–1357. [Google Scholar] [CrossRef] - Liu, C.; Zhang, L.; Li, Y.; Liu, F.; Martyushev, D.A.; Yang, Y. Effects of microfractures on permeability in carbonate rocks based on digital core technology. Adv. Geo-Energy Res.
**2022**, 6, 86–90. [Google Scholar] [CrossRef] - Panda, M.N.; Lake, L.W. Estimation of single-phase permeability from parameters of particle-size distribution. AAPG Bull.
**1994**, 78, 1028–1039. [Google Scholar] - Rodriguez, E.; Giacomelli, F.; Vazquez, A. Permeability-porosity relationship in RTM for different fiberglass and natural reinforcements. J. Compos. Mater.
**2004**, 38, 259–268. [Google Scholar] [CrossRef] - Chen, X.; Yao, G.; Luo, C.; Jiang, P.; Cai, J.; Zhou, K.; Herrero-Bervera, E. Capillary pressure curve determination based on a 2-D cross section analysis via fractal geometry: A bridge between 2D and 3D pore structure of porous media. J. Geophys. Res. Solid Earth
**2019**, 124, 2352–2367. [Google Scholar] [CrossRef] - Chen, X.; Yao, G. An improved model for permeability estimation in low permeable porous media based on fractal geometry and modified Hagen-Poiseuille flow. Fuel
**2017**, 210, 748–757. [Google Scholar] [CrossRef] - Adler, P.M.; Jacquin, C.G.; Quiblier, J.A. Flow in simulated porous media. Int. J. Multiph. Flow
**1990**, 16, 691–712. [Google Scholar] [CrossRef] - Yeong, C.L.Y.; Torquato, S. Reconstructing random media. II. Three-dimensional media from two-dimensional cuts. Phys. Rev. E
**1998**, 58, 224–238. [Google Scholar] [CrossRef] - Hilfer, R.; Manwart, C. Permeability and conductivity for reconstruction models of porous media. Phys. Rev. E
**2001**, 64, 021304. [Google Scholar] [CrossRef] - Keehm, Y.; Mukerji, T.; Nur, A. Permeability prediction from thin sections: 3D reconstruction and Lattice-Boltzmann flow simulation. Geophys. Res. Lett.
**2004**, 31, L04606. [Google Scholar] [CrossRef] - Andrä, H.; Combaret, N.; Dvorkin, J.; Glatt, E.; Han, J.; Kabel, M.; Keehm, Y.; Krzikalla, F.; Lee, M.; Madonna, C.; et al. Digital rock physics benchmarks-part II: Computing effective properties. Comput. Geosci.
**2013**, 50, 33–43. [Google Scholar] [CrossRef] - Karimpouli, S.; Tahmasebi, P. Conditional reconstruction: An alternative strategy in digital rock physics. Geophysics
**2016**, 81, D465–D477. [Google Scholar] [CrossRef] - Saxena, N.; Mavko, G. Estimating elastic moduli of rocks from thin sections: Digital rock study of 3D properties from 2D images. Comput. Geosci.
**2016**, 88, 9–21. [Google Scholar] [CrossRef] - Saxena, N.; Mavko, G.; Hofmann, R.; Srisutthiyakorn, N. Estimating permeability from thin sections without reconstruction: Digital rock study of 3D properties from 2D images. Comput. Geosci.
**2017**, 102, 79–99. [Google Scholar] [CrossRef] - Saxena, N.; Hofmann, R.; Alpak, F.O.; Berg, S.; Dietderich, J.; Agarwal, U.; Tandon, K.; Hunter, S.; Freeman, J.; Wilson, O.B. References and benchmarks for pore-scale flow simulated using micro-CT images of porous media and digital rocks. Adv. Water Resour.
**2017**, 109, 211–235. [Google Scholar] [CrossRef] - Tahmasebi, P.; Sahimi, M.; Andrade, J.E. Image-based modeling of granular porous media. Geophys. Res. Lett.
**2017**, 44, 4738–4746. [Google Scholar] [CrossRef] - Yu, B.; Cheng, P. A fractal permeability model for bi-dispersed porous media. Int. J. Heat Mass Transf.
**2002**, 45, 2983–2993. [Google Scholar] [CrossRef] - Rabbani, A.; Ayatollahi, S.; Kharrat, R.; Dashti, N. Estimation of 3-D pore network coordination number of rocks from watershed segmentation of a single 2-D image. Adv. Water Resour.
**2016**, 94, 264–277. [Google Scholar] [CrossRef] - Berryman, J.G.; Blair, S.C. Use of digital image analysis to estimate fluid permeability of porous materials: Application of two-point correlation functions. J. Appl. Phys.
**1986**, 60, 1930–1938. [Google Scholar] [CrossRef] - Sisavath, S.; Jing, X.; Zimmerman, R.W. Creeping flow through a pipe of varying radius. Phys. Fluids
**2001**, 13, 2762–2772. [Google Scholar] [CrossRef] - Mandelbrot, B.B. The Fractal Geometry of Nature; WH freeman: New York, NY, USA, 1982. [Google Scholar]
- Katz, A.J.; Thompson, A.H. Fractal sandstone pores: Implications for conductivity and pore formation. Phys. Rev. Lett.
**1985**, 54, 1325. [Google Scholar] [CrossRef] - Yu, B.; Li, J. Some fractal characters of porous media. Fractals
**2001**, 9, 365–372. [Google Scholar] [CrossRef] - Wu, Y.; Tahmasebi, P.; Lin, C.; Zahid, M.A.; Dong, C.; Golab, A.N.; Ren, L. A comprehensive study on geometric, topological and fractal characterizations of pore systems in low-permeability reservoirs based on SEM, MICP, NMR, and X-ray CT experiments. Mar. Pet. Geol.
**2019**, 103, 12–28. [Google Scholar] [CrossRef] - Lei, G.; Liao, Q.; Patil, S.; Zhao, Y. A new permeability model for argillaceous porous media under stress dependence with clay swelling. Int. J. Eng. Sci.
**2021**, 160, 103452. [Google Scholar] [CrossRef] - Ghiasi-Freez, J.; Kadkhodaie-Ilkhchi, A.; Ziaii, M. Improving the accuracy of flow units prediction through two committee machine models: An example from the South Pars Gas Field, Persian Gulf Basin, Iran. Comput. Geosci.
**2012**, 46, 10–23. [Google Scholar] [CrossRef] - Cai, J.; Hu, X.; Standnes, D.C.; You, L. An analytical model for spontaneous imbibition in fractal porous media including gravity. Colloids Surf. A Physicochem. Eng. Asp.
**2012**, 414, 228–233. [Google Scholar] [CrossRef] - Lei, G.; Liao, Q.; Patil, S.; Zhao, Y. Effect of clay content on permeability behavior of argillaceous porous media under stress dependence: A theoretical and experimental work. J. Pet. Sci. Eng.
**2019**, 179, 787–795. [Google Scholar] [CrossRef] - Xiao, B.; Li, Y.; Long, G.; Yu, B. Fractal permeability model for power-law fluids in fractured porous media with rough surfaces. Fractals
**2022**, 30, 2250115. [Google Scholar] [CrossRef] - Wei, W.; Cai, J.; Hu, X.; Han, Q. An electrical conductivity model for fractal porous media. Geophys. Res. Lett.
**2015**, 42, 4833–4840. [Google Scholar] [CrossRef] - Liu, L.; Sun, Q.; Wu, N.; Liu, C.; Ning, F.; Cai, J. Fractal analyses of the shape factor in Kozeny-Carman equation for hydraulic permeability in hydrate-bearing sediments. Fractals
**2021**, 29, 2150217. [Google Scholar] [CrossRef] - Lei, G.; Qu, J.; Wu, Q.; Pang, J.; Su, D.; Guan, J.; Lu, C. Theoretical analysis of threshold pressure in tight porous media under stress. Phys. Fluids
**2023**, 35, 073313. [Google Scholar] - Norouzi, S.; Soleimani, R.; Farahani, M.V.; Rasaei, M.R. Pore-scale simulation of capillary force effect in water-oil immiscible dis-placement process in porous media. In Proceedings of the 81st EAGE Conference and Exhibition, London, UK, 3 June 2019. [Google Scholar]
- Khodja, M.R.; Li, J.; Hussaini, S.R.; Ali, A.Z.; Al-Mukainah, H.S.; Jangda, Z.Z. Consistent prediction of absolute permeability in carbonates without upscaling. Oil Gas Sci. Technol. Rev. D’ifp Energ. Nouv.
**2020**, 75, 44. [Google Scholar] [CrossRef] - Soleimani, R.; Azaiez, J.; Zargartalebi, M.; Gates, I.D. Analysis of marangoni effects on the Non-isothermal immiscible Ray-leigh-Taylor instability. Int. J. Multiph. Flow
**2022**, 156, 104231. [Google Scholar] [CrossRef] - Vasheghani, F.M.; Mousavi, N.M. On the effect of flow regime and pore structure on the flow signatures in porous media. Phys. Fluids
**2022**, 34, 115139. [Google Scholar] [CrossRef] - Lei, G.; Xue, L.; Liao, Q.; Li, J.; Zhao, Y.; Zhou, X.; Lu, C. A novel analytical model for porosity-permeability relations of argillaceous porous media under stress conditions. Geoenergy Sci. Eng.
**2023**, 225, 211659. [Google Scholar] [CrossRef] - Karimpouli, S.; Tahmasebi, P.; Saenger, E.H. Estimating 3D elastic moduli of rock from 2D thin-section images using dif-ferential effective medium theory3D elastic moduli using 2D images. Geophysics
**2018**, 83, MR211–MR219. [Google Scholar] [CrossRef] - Srisutthiyakorn, N.; Hunter, S.; Sarker, R.; Hofmann, R.; Espejo, I. Predicting elastic properties and permeability of rocks from 2D thin sections. Lead. Edge
**2018**, 37, 421–427. [Google Scholar] [CrossRef] - Blunt, M.J.; Bijeljic, B.; Dong, H.; Gharbi, O.; Iglauer, S.; Mostaghimi, P.; Paluszny, A.; Pentland, C. Pore-scale imaging and modelling. Adv. Water Resour.
**2013**, 51, 197–216. [Google Scholar] [CrossRef] - Wu, Y.; Tahmasebi, P.; Lin, C.; Munawar, M.J.; Cnudde, V. Effects of micropores on geometric, topological and transport properties of pore systems for low-permeability porous media. J. Hydrol.
**2019**, 575, 327–342. [Google Scholar] [CrossRef] - Lei, G.; Liao, Q.; Lin, Q.; Zhang, L.; Xue, L.; Chen, W. Stress dependent gas-water relative permeability in gas hydrates: A theoretical model. Adv. Geo-Energy Res.
**2020**, 4, 326–338. [Google Scholar] [CrossRef] - Yu, B. Analysis of flow in fractal porous media. Appl. Mech. Rev.
**2008**, 61, 050801. [Google Scholar] [CrossRef] - Chen, X.; Yao, G.; Herrero-Bervera, E.; Cai, J.; Zhou, K.; Luo, C.; Jiang, P.; Lu, J. A new model of pore structure typing based on fractal geometry. Mar. Pet. Geol.
**2018**, 98, 291–305. [Google Scholar] [CrossRef] - Rabbani, A.; Assadi, A.; Kharrat, R.; Dashti, N.; Ayatollahi, S. Estimation of carbonates permeability using pore network parameters extracted from thin section images and comparison with experimental data. J. Nat. Gas Sci. Eng.
**2017**, 42, 85–98. [Google Scholar] [CrossRef] - Comiti, J.; Renaud, M. A new model for determining mean structure parameters of fixed beds from pressure drop meas-urements: Application to beds packed with parallelepipedal particles. Chem. Eng. Sci.
**1989**, 44, 1539–1545. [Google Scholar] [CrossRef] - Lei, G.; Wang, C.; Wu, Z.; Wang, H.; Li, W. Theory study of gas-water relative permeability in roughened fractures. Proc. Inst. Mech. Eng. Part C. J. Mech. Eng. Sci.
**2018**, 232, 4615–4625. [Google Scholar] [CrossRef] - Suarez-Dominguez, E.J.; Perez-Rivao, A.; Sanchez-Medrano, M.T.; Perez-Sanchez, J.F.; Izquierdo-Kulich, E. Mesoscopic model for the surface fractal dimension estimation of solid-solid and gas-solid dispersed systems. Surf. Interfaces
**2020**, 18, 100407. [Google Scholar] [CrossRef] - Cai, J.; Lin, D.; Singh, H.; Wei, W.; Zhou, S. Shale gas transport model in 3D fractal porous media with variable pore sizes. Mar. Pet. Geol.
**2018**, 98, 437–447. [Google Scholar] [CrossRef] - Lei, G.; Dong, P.; Wu, Z.; Mo, S.; Gai, S.; Zhao, C.; Liu, Z. A fractal model for the stress-dependent permeability and relative permeability in tight sandstones. J. Can. Pet. Technol.
**2015**, 54, 36–48. [Google Scholar] [CrossRef] - Qian, Y.H.; d’Humières, D.; Lallemand, P. Lattice BGK models for Navier-Stokes equation. EPL
**1992**, 17, 479–484. [Google Scholar] [CrossRef] - Sukop, M.C.; Thorne, D.T. Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers; Springer: Berlin, Germany, 2006. [Google Scholar]
- Li, J. Multiscale and Multiphysics Flow Simulations of Using the Boltzmann Equation; Springer: Cham, Switzerland, 2020. [Google Scholar]
- Li, J.; Hussaini, S.R.; Dvorkin, J. Permeability-porosity relations from single image of natural rock: Subsampling approach. J. Pet. Sci. Eng.
**2020**, 194, 107541. [Google Scholar] [CrossRef] - Guo, Z.; Zheng, C.; Shi, B. An extrapolation method for boundary conditions in lattice Boltzmann method. Phys. Fluids
**2002**, 14, 2007–2010. [Google Scholar] [CrossRef] - Li, J.; Ho, M.T.; Wu, L.; Zhang, Y. On the unintentional rarefaction effect in LBM modeling of intrinsic permeability. Adv. Geo-Energy Res.
**2018**, 2, 404–409. [Google Scholar] [CrossRef] - Zhang, Y.; Qin, R.; Emerson, D.R. Lattice Boltzmann simulation of rarefied gas flows in microchannels. Phys. Rev. E
**2005**, 71, 047702. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of transforming the 2D pore information into 3D pore structure and predicting 3D permeability from 2D cross-sections.

**Figure 4.**Modeling results of the samples: (

**a**) the curves of φ

_{2D}versus φ

_{3D}; (

**b**) d

_{f}of the 2D image of sample 1-A; (

**c**) the determined pore structure parameters for various samples; (

**d**) τ

_{av}versus the ratio r

_{max}/r

_{3D,max}for various samples.

**Figure 5.**Modeling results of the samples [26]: (

**a**) the predicted permeabilities from the derived model versus the data for Berea samples; (

**b**) the predicted permeabilities from the derived model versus the data for Bitumen samples; (

**c**) the predicted permeabilities from the derived model versus the data for Font samples; (

**d**) the predicted permeabilities from the derived model versus the data for Grosmont samples; (

**e**) the determined pore structure parameters for various samples; (

**f**) τ

_{av}versus the ratio r

_{max}/r

_{3D,max}for various samples.

**Figure 6.**Modeling results of the samples: (

**a**) predicted permeability (K

_{2D}and K

_{3D}) and experimental data; (

**b**) the determined pore structure parameters (d

_{f}, d

_{t}, τ

_{av}, D

_{T}, and D

_{f}) of various samples; (

**c**) τ

_{av}versus the ratio r

_{max}/r

_{3D,max}for various samples; (

**d**) the predicted average particle radius for spherical particles and circular particles versus that from Equation (15).

**Figure 7.**Effect of pore fractal dimension D

_{f}on properties of porous media in 3D: (

**a**) curves of D

_{f}versus τ

_{av}and K

_{3D}; (

**b**) τ

_{av}versus the ratio r

_{max}/r

_{3D,max}.

**Figure 8.**Effect of tortuosity fractal dimension D

_{T}on properties of porous media in 3D: (

**a**) curves of D

_{T}versus τ

_{av}and K

_{3D}; (

**b**) τ

_{av}versus the ratio r

_{max}/r

_{3D,max}.

**Figure 9.**Effects of D

_{f}and D

_{T}on the bridge function f

_{bridge}: (

**a**) pore fractal dimension D

_{f}versus the bridge function K

_{2D}/K

_{3D}; (

**b**) tortuosity fractal dimension D

_{T}versus the bridge function K

_{2D}/K

_{3D}.

No. | Samples ^{a} | LBM | Proposed Model | |||||
---|---|---|---|---|---|---|---|---|

φ_{3D}(%) | K_{3D} (10^{−3} µm^{2}) | φ_{2D}(%) | d_{f} | R_{3D,max} (µm) | D_{f} | K_{3D} (10^{−3} µm^{2}) | ||

1 | 1-A | 23.5 | 27.8 | 31.24 | 1.686 | 41.18 | 2.81 | 27.8 |

2 | 1-B | 23.4 | 69.4 | 21.35 | 1.450 | 43.15 | 2.79 | 69.4 |

3 | 2-A | 31.5 | 1450 | 35.71 | 1.576 | 64.04 | 2.78 | 1450 |

4 | 2-B | 21.5 | 216.7 | 26.44 | 1.683 | 70.31 | 2.79 | 216.7 |

5 | 2-C | 34.9 | 1743.1 | 38.55 | 1.611 | 60.24 | 2.79 | 1743.1 |

6 | 3-A | 30.7 | 906.9 | 34.97 | 1.735 | 89.07 | 2.82 | 906.9 |

7 | 3-B | 16.2 | 103.1 | 10.81 | 1.415 | 49.04 | 2.74 | 103.1 |

8 | 3-C ^{b} | 31.3 | 3049.4 | 36.97 | 1.631 | 61.83 | 2.76 | 3049.4 |

9 | 3-D ^{b} | 40.7 | 2660 | 36.74 | 1.636 | 60.20 | 2.82 | 2660 |

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**MDPI and ACS Style**

Lei, G.; Liu, T.; Liao, Q.; He, X.
Estimating Permeability of Porous Media from 2D Digital Images. *J. Mar. Sci. Eng.* **2023**, *11*, 1614.
https://doi.org/10.3390/jmse11081614

**AMA Style**

Lei G, Liu T, Liao Q, He X.
Estimating Permeability of Porous Media from 2D Digital Images. *Journal of Marine Science and Engineering*. 2023; 11(8):1614.
https://doi.org/10.3390/jmse11081614

**Chicago/Turabian Style**

Lei, Gang, Tianle Liu, Qinzhuo Liao, and Xupeng He.
2023. "Estimating Permeability of Porous Media from 2D Digital Images" *Journal of Marine Science and Engineering* 11, no. 8: 1614.
https://doi.org/10.3390/jmse11081614