The Lattice Boltzmann Method and Image Processing Techniques for Effective Parameter Estimation of Digital Rock

Abstract

Several numerical simulations of fluid flow were performed using the Lattice Boltzmann method and image processing techniques to estimate the effective properties of 2-D porous rocks. The effective properties evaluated were the physical characteristics that allow fluid flow including the effective porosity, permeability, tortuosity, and average throat size to determine the storage and transport of fluids in porous rocks. The permeability was compared using the Darcy model simulation and the empirical Kozeny–Carman Equation. The results showed that the Lattice Boltzmann method and image processing techniques can estimate the effective parameters of porous rocks. Furthermore, there was a good correlation between permeability and parameters such as effective porosity, tortuosity, and average throat size. The Darcy model simulation revealed a gamma distribution in the permeability, while the empirical Kozeny–Carman Equation showed a log-normal distribution.

1. Introduction

Porous media are composed of solid mineral grains and empty spaces, or pores. An important parameter of this pore structure is the spaces between these grains and their shape. The spaces in porous media serve a dual purpose, serving as channels for fluid transportation, forming interconnected pores, and as reservoirs for fluid storage. During geologic time, when sediments were deposited and rocks were formed, some void areas became isolated due to excessive cementation. As a result, many void spaces are interconnected, while others remain isolated [1]. There are several types of porous media, one of which is rock. Rocks are porous media because there are empty spaces in them. The spaces in the rock can be empty spaces or filled by fluid. In most rocks, reservoir rocks generally have a higher pore volume so that the empty spaces in reservoir rocks are generally the places where hydrocarbons accumulate [2].

The porosity of the rock is the ratio of the volume of pore cavities to the overall volume, and can be classified as the absolute or effective porosity. Effective porosity is when the pores are interconnected, allowing fluid flow, while absolute porosity is calculated without connections between the pores. Therefore, effective porosity is the most important parameter influencing permeability [3,4]. Porosity measurement in reservoirs is important in evaluating and making decisions in petroleum field development. Physical rock parameters are measured in the laboratory but this is time-consuming and can destroy rock samples. Another method to determine rock parameters is by estimation based on matching parameters from rock lithology, grain size, and others, but this method is limited to rocks from specific locations [5].

Permeability is an essential petrophysical parameter to determine the flow characteristics of hydrocarbons [6] and can be calculated by fluid flow simulation, nuclear magnetic resonance, core analysis, and digital rock physics [7,8,9]. However, laboratory measurements are difficult to perform for fluid flows that are not gas because some clays attach to the porous media, causing the fluid to be contaminated and mixed [10]. Computational Fluid Dynamics (CFD) serves as an alternative method for determining effective parameter values in rocks through fluid flow simulation. This approach offers advantages such as the elimination of fluid contamination and reduced measurement time [8]. In this study, the fluid flow simulation is conducted using the Lattice Boltzmann method (LBM) for single-phase flow [11]. The LBM is a numerical technique that employs a mesoscopic approach to model fluid dynamics, particularly effective in handling complex boundary conditions like those found in porous media. Previous research has utilized the LBM for various applications, including investigating the flow of non-Newtonian fluids in fracture networks [12], simulating gas flow, and predicting permeability in porous media by comparing single relaxation time (SRT) and multiple relaxation time (MRT) schemes [13], and modeling heat transfer processes [14,15]. These studies have demonstrated the validity of the LBM in simulating fluid flow in complex geometries. The present study focuses on simulating single-phase flow in 2-D porous rock media using the LBM to obtain velocity and pressure profiles, which are then used to calculate effective fluid parameters. The permeability of the rock was determined through Darcy’s law, comparing fluid flow simulation results with digital image analysis [8]. The rock sample used in this study was sandstone, which had undergone prior pre-processing [16]. The physical parameters were calculated using the OpenLB software to simulate fluid flow [17].

In this paper, we employed a 2-D model to investigate the interrelations among effective porosity, permeability, average throat size, and tortuosity due to its ability to effectively capture essential behaviors and trends while being computationally efficient. The 2-D model allows for extensive parameter sweeps, higher resolution, and more iterations within a feasible timeframe and resource, providing a practical approach for our study. Our findings from the 2-D model offer valuable insights and establish a solid foundation for understanding fluid flow in porous media. We recognize that future research can extend these findings using 3-D models to further enhance our understanding of these phenomena.

2. Materials and Methods

2.1. Materials

The 2-D sandstone rock data from CT scans were downloaded from https://www.digitalrocksportal.org/projects, accessed on 20 August 2024. Several types of sandstone, including Bandera, Berea, Bentheimer, Castlegate, and Leopard, are predominantly composed of quartz, with variations in feldspar content, clay minerals, and lithic fragments. Bandera and Bentheimer sandstones exhibit low anisotropy and a low degree of alteration, characterized by exceptionally clean quartz grains. Berea sandstone shows moderate anisotropy and low-to-moderate alteration, while Castlegate sandstone tends to be anisotropic with a moderate degree of alteration. Leopard sandstone is distinguished by its characteristic iron oxide staining and visual anisotropy, although its mechanical properties and fluid flow characteristics are generally isotropic. The rock data samples were originally 3-D, and were segmented and sliced into 2-D because sandstone rocks have homogeneous porosity so the porosity distribution is the same for each rock segment. The eight selected rocks were categorized based on their porosity and grain shapes. Digital Image Analysis was used to calculate the total number of rock pores against the total rock volume. The porosity range of the rock samples used was in the range of 0.21–0.29. This range corresponds to the porosity range of sandstone reservoir rocks. Figure 1 Illustrate the microstructural differences among the samples, with black areas representing the pore spaces and white areas representing the solid matrix. The visual comparison highlights the variation in porosity and pore connectivity across different sandstone types, which is essential for understanding their fluid flow properties. This set of images serves as the basis for analyzing the interrelations among effective porosity, permeability, average throat size, and tortuosity in our 2-D study.

The porous rock parameters were obtained in several stages, starting from data collection to the visualization of the data obtained, as shown in Figure 2.

Figure 2 illustrates the research methodology employed in this study, starting with the collection of 3-D X-ray micro-computed tomography sandstone data. The data undergo segmentation to identify different phases, resulting in a $1000\times 1000\times 1000$ pixels of 3-D rock data segmentation. These 3-D data are then resampled and sliced to produce $256\times 256$ pixel of 2-D binarized images for simulation input. The Lattice Boltzmann method (LBM) is used to simulate fluid velocity and pressure profiles, which are then segmented to produce average throat size and effective porosity. Permeability is calculated using the velocity and pressure data through Darcy’s equation. Additionally, the Kozeny–Carman permeability is obtained from image processing techniques, utilizing tortuosity, specific surface area, and absolute porosity. The data visualization and analysis stage includes comparisons and correlations among the various calculated parameters, culminating in a comprehensive understanding of fluid flow properties in the sandstone samples.

2.2. Lattice Boltzmann

The LBM is a numerical method that uses a mesoscopic approach to describe fluid dynamics with complex boundary conditions such as geometry in porous media. It has been used extensively to characterize the transport mechanism in porous media, and has several advantages including having a simple, efficient algorithm in parallel systems suitable for overcoming complex geometries and boundary conditions [18]. In addition, it is a powerful technique for computational modeling of a wide variety of complex fluid flow problems, including single-phase [19,20] and multi-phase flows in complex geometries [21]. This method naturally accommodates various boundary conditions and can simulate solute transport and solve advection–dispersion equations [22]. Figure 3 shows the schematic of the 2-D Lattice Boltzmann method.

The Lattice Boltzmann method employs a particle movement scheme with discrete velocities represented as DnQm, where Dn corresponds to the dimension (n) and Qm denotes the direction of particle movement (m). One widely utilized model is D2Q9 for 2-D simulations, featuring 9 directions of particle movement. This research utilizes the D2Q9 model for Lattice Boltzmann simulation. While higher DnQm values enhance simulation quality, they also increase computational complexity and time requirements, demanding higher computer performance, but the use of such models to improve simulation accuracy has been supported [23]. Equation (1) is the form of the generalized Lattice Boltzmann Equation.

$$f_i\left(\mathit{x}+{\mathit{c}}_i\Delta t,t+\Delta t)=f_i(\mathit{x},t)+{\Omega }_i\left(f\right)\right.$$

(1)

where f represent the distibution function, while $f_i$ is the spesific distribution function represents the density values moving in the i-th direction with velocity ${\mathit{c}}_i$ towards its nearest point, $\mathit{x}+{\mathit{c}}_i\Delta t$, at time $t+\Delta t$, $c_i$ is the discrete velocity vector, and the particles also experienced collisions with each other, affected by the collision operator, ${\Omega }_i\left(f\right)$. Using the Bhatnagar–Gross–Krook (BGK) approximation, the calculation of the particle distribution function at each time iteration can be divided into two main steps: the collision step and the streaming step. The collision operator is expressed by Equation (2).

$${\Omega }_i\left(f\right)=-{\displaystyle \frac{f_i-f_i^{eq}}{\tau }}\Delta t$$

(2)

where $f_i^{eq}$ is the equilibrium distribution function and $\tau$ is relaxation time. The distribution function that changes due to the collision operator will then be streamed to its nearest point. The equilibrium distribution function is expressed by Equation (3):

$$f_i^{eq}(\mathbf{x},t)=w_i\rho \left(1+{\displaystyle \frac{\mathbf{u}·{\mathbf{c}}_i}{c_s^2}}+{\displaystyle \frac{{(\mathbf{u}·{\mathbf{c}}_i)}^2}{2c_s^4}}-{\displaystyle \frac{\mathbf{u}·\mathbf{u}}{2c_s^2}}\right)$$

(3)

where $c_s$ is the lattice speed of sound, u is the macroscopic velocity, $c_i$ is the discrete velocity vector, $\rho$ is the density, and $w_i$ is the velocity direction weight for the D2Q9 lattice model. The weight factors for each speed link, as shown in Figure 3, are expressed in

Table 1. The weighting coefficients for D2Q9 model.

Model	${\mathit{w}}_{\mathit{i}}$
D2Q9	${\displaystyle \frac{4}{9}}(i=0)$
	${\displaystyle \frac{1}{9}}(i=1,2,3,4)$
	${\displaystyle \frac{1}{36}}(i=5,6,7,8)$

.

The macroscopic values such as density, $\rho$, and fluid momentum, $\rho$u, are derived by summing the distribution functions, as shown in Equation (4).

$$\begin{array}{cc}\hfill & \rho =\sum _i{f}_i\hfill \\ \hfill & \rho u=\sum _i{c}_i{f}_i\hfill \end{array}$$

(4)

The value of pressure P is affected by $c_s$ and $\rho$, and the relaxation time, $\tau$, is affected by $c_s$ and the kinematic viscosity, v, defined by Equations (5) and (6).

$$\begin{array}{cc}\hfill & c_s^2={\displaystyle \frac{1}{3}}{\displaystyle \frac{\Delta x^2}{\Delta t^2}}\hfill \\ \hfill & P=c_s^2\rho \hfill \end{array}$$

(5)

$$\tau ={\displaystyle \frac{\Delta t}{2}}+{\displaystyle \frac{v}{c_s^2}}$$

(6)

The simulation involved streaming and collision steps, with collisions further categorized into single relaxation time (SRT) models. The widely adopted collision model is the Bhatnagar–Gross–Krook (BGK) approach, characterized by a SRT. This model has been extensively employed in various studies [24,25] to enhance numerical simulation stability, minimize spurious velocities, and eliminate dependence on viscosity, showcasing its efficacy in refining the overall simulation performance. A maximum of 360,000 iterations was used so that each simulation was in a convergent state using upper and lower bounce-back boundary conditions. The permeability was then calculated using the velocity profile data and pressure profiles.

2.3. Porosity

The absolute porosity and effective porosity were calculated using digital image processing. The total porosity was calculated as follows:

$$\begin{array}{c}\hfill \varphi ={\displaystyle \frac{V_p}{V_t}}\end{array}$$

(7)

where $V_p$ is the pore volume pore and $V_t$ is the total rock volume. The effective porosity was calculated using velocity distribution data, as shown in Figure 4. Fluid always flows in the effective pores of the rock. Fluid flow in effective pores will produce flow velocity while ineffective pores will not show any fluid flow velocity. Therefore, velocity profile segmentation can be used to calculate effective pores characterized by the presence of flow velocity in the rock.

2.4. Digital Image Analysis

Digital image analysis of the preprocessed images was performed to calculate the absolute porosity, tortuosity, and average throat size. A segmentation method was applied to separate the pores from the solid matrix in the digital image to calculate the absolute porosity ($\varphi$) [26,27]. The image was converted into binary format so that the pixels representing the pores (usually black pixels) and the solid matrix (white pixels) could be separated. This thresholding process can be performed manually or automatically using the Otsu algorithm to determine the optimal threshold value [28]. Once the binary image is obtained, the absolute porosity is calculated as the ratio of the number of pixels representing pores ($N_p$) to the total pixels in the image ($N_t$):

$$\begin{array}{c}\hfill \varphi ={\displaystyle \frac{N_p}{N_t}}\end{array}$$

(8)

where $N_p$ is the total pores and $N_t$ is the total elements. The absolute porosity is obtained from the ratio between total pores and total elements. The fluid flow in porous media will generally take the shortest path, as shown in Figure 5.

The flow lengths of the porous media are $l_1$ and $l_2$, and can be used to calculate the tortuosity ($\tau$) using the A* (A-star) algorithm. This algorithm is known as the efficient shortest path-finding method and is used to navigate through the pore structure in a binary image. The process starts by selecting two points in the image, usually from one side of the image to the other, and then evaluating potential paths based on distance costs and heuristics that account for the path complexity. The shortest path describes the route traveled within the pore structure, and the length of this path is compared with the length of a straight path (L) between those two points [29,30,31]. The tortuosity is calculated as the ratio between the actual path length ($l_a$) and the straight path length:

$$\begin{array}{c}\hfill \tau ={\displaystyle \frac{l_a}{L}}\end{array}$$

(9)

where $l_a$ is the length of the flow through the fluid and L is the length of the porous media geometry. The next step involves calculating the average throat size using watershed segmentation, an algorithm designed to separate dominant watersheds and thereby determine the width of the connecting line. The mathematical model for watershed segmentation is based on the concept of height fields in image processing:

$$D(x,y)=\underset{(x^{\prime },y^{\prime })\in W_m}{min}\sqrt{{(x-x^{\prime })}^2+{(y-y^{\prime })}^2}$$

(10)

where W is the set of pixels forming the detected object edges, m is a marker, and the distance transform, $D(x,y)$, is the Euclidean distance from each pixel $(x,y)$ to the nearest edge pixel. Markers are pixels chosen as the starting points for the flooding process and they can be determined manually or automatically by finding peaks in the distance transform or based on additional information. If m is the set of marker pixels, the flooding process can be viewed as adding water from the marker, m, and inundating the height field, $I(x,y)$. This process can be formalized using the level set function, $L\left(t\right)$:

$$L\left(t\right)=\left\{\right(x,y\left)\right|I(x,y)\le t\}$$

(11)

where $I(x,y)$ is the intensity function of a two-dimensional image where $(x,y)$ represents the pixel coordinates, I is a height field, with the pixel intensity, $I(x,y)$, representing the height at point $(x,y)$, and t is the rising water level. During the flooding process, a watershed line is formed when water from two different markers meets ($m_1$,$m_2$), which can represent the following equation:

$$\underset{(x^{\prime },y^{\prime })\in W_{m_1}}{min}\sqrt{{(x-x^{\prime })}^2+{(y-y^{\prime })}^2}=\underset{(x^{\prime },y^{\prime })\in W_{m_2}}{min}\sqrt{{(x-x^{\prime })}^2+{(y-y^{\prime })}^2}$$

(12)

where $W_m$ is the set of pixels belonging to the connected component derived from the marker, m, once these throats are isolated, and measurements are taken to calculate the average diameter or width of each throat ($T_i$). This measurement is usually repeated for multiple throats in the image to obtain a representative average value:

$$\begin{array}{c}\hfill \overline{T}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{T}_i\end{array}$$

(13)

where n is the number of throats measured.

2.5. Permeability

The permeability was calculated using Darcy’s law of the relationship between the pressure difference and flow rate from the simulation results using the LBM. Darcy’s law describes the flow in porous media as follows:

$$\begin{array}{c}\hfill q=-KA{\displaystyle \frac{\Delta P}{\mu L}}\end{array}$$

(14)

where q is the flow rate in the x direction, K is the permeability, A is the area of the porous medium, $\Delta P$ is the pressure drop, $\mu$ is the dynamic viscosity of the fluid, and L is the flow length of the fluid. The LBM simulates Darcy’s experiment by applying a pressure gradient to the inlet–outlet of a saturated porous medium using the OpenLB Open Source Lattice Boltzmann. OpenLB utilizes the D2Q9 scheme, which describes motion in two dimensions and 9 associated velocity vectors. Based on Equation (14), the permeability can be calculated as follows:

$$\begin{array}{c}\hfill K=-{\displaystyle \frac{q\mu L}{A\Delta P}}\end{array}$$

(15)

The flow rate in a certain direction (in this case, x direction) can be expressed as the rate volume of fluid passing through the cross-sectional area, A, perpendicular to the flow direction as follows:

$$\begin{array}{c}\hfill K={\displaystyle \frac{\langle v\rangle \mu L}{\Delta P}}\end{array}$$

(16)

where $\langle v\rangle$ is the average velocity. Permeability has units of area (${\mathrm{m}}^2$ in SI units), but sometimes, in terms of the geophysical aspect, it is more common to use the Darcy, where 1 Darcy $=0.987\times {10}^{-12}$ ${\mathrm{m}}^2$ or the millidarcy ($mD$) where 1 $mD$ = 0.001 Darcy. In this study, permeability was also calculated using the Kozeny–Carman permeability:

$$\begin{array}{c}\hfill K={\displaystyle \frac{{\varphi }_{\mathrm{eff}}^3}{c{\tau }^2{S}^2}}\end{array}$$

(17)

where ${\varphi }_{\mathrm{eff}}$ is the effective porosity, $\tau$ is the tortuosity, c is a constant, and S is the specific surface area. Kozeny–Carman permeability is used to calculate the permeability of a porous medium based on its pore structure. This theoretical model relates permeability to the geometric characteristics of the pores and fluid flow through the medium.

3. Results

A total of 2000 data points were used to calculate the effective parameters of eight types of 2-D rock samples: Bandera gray, Bentheimer, Berea, Berea upper gray, Berea sister gray, Castlegate, Leopard, and Buff Berea. Each rock sample had 250 2-D rock images. The absolute porosity distribution in Figure 6 shows the range of absolute porosity values and indicates the homogeneity of the rocks. The quartiles were $Q1=0.23$, $Q2=0.26$, and $Q3=0.29$ within the porosity range of sandstone. The effective porosity distribution is shown in Figure 7.

The effective porosity quartiles of $Q1=0.12$, $Q2=0.15$, and $Q3=0.18$ were smaller compared with the quartile values for absolute porosity. The absolute porosity (y-axis) was plotted against the effective porosity (x-axis) in a scatter plot in Figure 8, showing that the relationship between these two porosities is directly proportional with a positive gradient, indicating that higher absolute porosity generally corresponds to higher effective porosity. However, the data points are below the blue line that represents the ${45}^{\circ }$ line, the equilibrium line where absolute porosity equals effective porosity; therefore, the effective porosity in this case, is always smaller than the absolute porosity.

The tortuosity results are shown in Figure 9 for each rock sample. The x-axis represents tortuosity values, and the y-axis represents the frequency or number of distributed data points. The red color represents the average value, the blue color represents the lowest value, and the green color represents the highest value of tortuosity.

The Bandera gray rock shows the lowest tortuosity range, with a log-normal distribution evident from the rightward elongation indicating an asymmetrical distribution. The skewness value of this distribution is also positive (right-skewed), meaning the data is more distributed in the lower range. All distributions for the Bentheimer rock show a log-normal shape and are asymmetrical with positive skewness (right-skewed). All distributions are asymmetrical with positive skewness (right-skewed) for the Berea rock, indicating the data are distributed more in the lower range. The Berea sister gray rock shows the lowest tortuosity range, with each skewness value being positively greater than 1 (right-skewed). The Berea upper gray rock’s lowest tortuosity exhibits a gamma distribution with positive skewness. The tortuosity results for the Buff Berea rock are positively skewed with an asymmetrical distribution. Likewise, the Leopard rock values are positively skewed (right-skewed). The Castlegate rock exhibits the lowest range distribution, approaching normal distribution, indicated by a skewness value of 0.09.

The plots in Figure 10 illustrate the relationships between effective porosity, permeability, and tortuosity. In Figure 10a, a strong inverse correlation ($R^2=0.81$) is observed between effective porosity and tortuosity, indicating that, as effective porosity increases, the pathways for fluid flow become less complex, reducing tortuosity. Figure 10b shows a positive correlation ($R^2=0.7$) between effective porosity and permeability, suggesting that higher effective porosity results in better fluid transmission through the rock. In Figure 10c, a moderate inverse correlation ($R^2=0.50$) is found between tortuosity and permeability, implying that more convoluted pathways (higher tortuosity) hinder fluid flow, resulting in lower permeability. Overall, these plots highlight the interrelated nature of these parameters, where higher effective porosity generally leads to lower tortuosity and higher permeability, facilitating easier fluid movement through the rock.

The plots in Figure 11 indicate weak correlations between average throat size and the effective parameters of porosity, permeability, and tortuosity. Specifically, a weak positive correlation ($R^2=0.28$) exists between average throat size and effective porosity, suggesting that larger throats may slightly increase the overall void space of the rock. Similarly, the weak positive correlation ($R^2=0.19$) between average throat size and permeability implies that larger throats may modestly enhance fluid flow through the rock. Conversely, the weak inverse correlation ($R^2=0.27)$ between average throat size and tortuosity indicates that larger throats may lead to more straightforward fluid flow paths, thereby reducing the complexity of these pathways. Overall, while the relationships are present, they are not particularly strong, indicating that other factors likely also play significant roles in determining these effective parameters.

The permeability values were also calculated using the Kozeny–Carman Equation and compared with the permeability values obtained from simulations using Darcy’s Equation. Figure 12 is a plot of permeability versus effective porosity. In Figure 12a, the correlation between permeability and the effective porosity is positive, with the coefficient of determination ($R^2$) indicating that $0.7$ of the variation in permeability can be explained by the variation in effective porosity. This demonstrates a relatively strong relationship between the two variables. The data points are scattered with some deviations but generally follow the overall trend according to the applied equation. In Figure 12b, the relationship between permeability and the effective porosity shows a similarly strong positive relationship, with $R^2=0.73$. The Darcy and the Kozeny–Carman models have a similar range of permeability data but the Kozeny–Carman model permeability values tend to be lower compared with the values obtained using the Darcy model.

Figure 13 and Figure 14 illustrate the permeability distribution results for both the Darcy (or LBM) model and the Kozeny–Carman model.

The distribution of the permeability model using the Kozeny–Carman Equation exhibits a log-normal form with a positive skewness value (right-skewed), whereas the distribution of the permeability data using the LBM is closer to a gamma distribution with positive skewness but is lower compared with the skewness value in the permeability distribution of the LBM.

4. Discussion

The absolute porosity distribution was normal with negative skewness, and thus relatively symmetrical, whereas the effective porosity distribution exhibited a gamma distribution, indicating that the data tend to cluster in the lower distribution range and are positively skewed. Therefore, the effective porosity values are smaller than the absolute porosity.

Not all pores in the rock can be used for fluid flow, as some pores may be isolated or too small to function effectively. This is important because, by definition, effective porosity is the result of subtracting ineffective porosity from absolute porosity, so the effective porosity value will always be smaller than absolute porosity. In other words, absolute porosity includes all pore spaces in the rock, including those that do not contribute to fluid flow, while effective porosity includes only those pores that function in fluid transport. Variations in effective and absolute porosity can reflect different geological processes such as sedimentation, diagenesis, and recrystallization [32]. For example, rocks that have undergone recrystallization, such as Castlegate and Leopard, have high absolute porosity but low effective porosity because many pores are isolated by newly formed crystals [33].

The tortuosity results for all rock samples were within the lower range because the range of tortuosity commonly found in soils and sedimentary rocks is between 1.2 and 3. In reality, it is challenging to find tortuosity values above 2, as this would affect low porosity. Extremely low porosity in some rocks causes the lack of pathways for fluid flow, making tortuosity values impossible to calculate.

There was a clear negative correlation between tortuosity and effective porosity, whereby the tortuosity decreased as the effective porosity increased, indicating an inverse relationship. The high $R^2$ value (0.81) indicates that the Kozeny–Carman Equation provides a good representation of the experimental data. The correlation can be considered representative because higher porosity usually means more direct and less tortuous flow paths, thus reducing tortuosity. There was a positive correlation between effective porosity and permeability, whereby the permeability increased as the effective porosity increased. Curve fitting using the Kozeny–Carman Equation model shows that the data tend to follow the research data well, despite some significant data scatter. The high $R^2$ value (0.70) indicates that the regression model represents the data relatively well. There was also a negative correlation between tortuosity and permeability, whereby the permeability decreased as the tortuosity increased. The curve fitting demonstrated the data trend but the lower correlation coefficient (0.50) indicates that this model does not represent the data, so other factors may affect permeability besides tortuosity. Overall, these results are representative because the higher the tortuosity is, the longer the fluid path is, resulting in lower permeability.

There was a positive correlation between throat size and effective porosity, with the effective porosity increasing as the throat size increased. The $R^2$ value of 0.28 indicates that about 28% of the variation in effective porosity can be explained by variations in average throat size. This is consistent with the basic theory that larger throat sizes allow for more effective pore space within the rock. There was a weak positive correlation between average throat size and permeability. Throat size is the radius of the cavity within the rock, so the larger the value is, the higher the permeability is. In contrast, there was a negative correlation between average throat size and tortuosity, with decreasing tortuosity as the throat size increased. The $R^2$ value of 0.27 indicates that about 27% of the variation in tortuosity can be correlated with average throat size. Larger throat sizes can reduce the complexity of fluid flow paths, thereby reducing the flow length and resulting in lower tortuosity values. The average throat size has a significant impact on effective porosity, permeability, and tortuosity, with larger throat sizes tending to increase the effective porosity and permeability but reduce tortuosity. Although the $R^2$ values were not very high (0.19–0.28), they indicate that average throat size influences these rock properties. Additionally, the relationship between the average throat size and permeability, effective porosity, and tortuosity shows a non-linear relationship.

The consistent observation that permeability values from Darcy’s law are higher than those from the Kozeny–Carman model can be explained by the differences in their underlying assumptions and how they account for pore structure complexity. The Kozeny–Carman model assumes a homogeneous, isotropic pore structure with uniform grain sizes, leading to an underestimation of permeability in real-world systems where pore networks are heterogeneous and irregular. This model simplifies the geometry and connectivity of the pores, neglecting the complex flow paths and channeling that occur in natural porous media. In contrast, Darcy’s law is an empirical relationship that directly relates flow rate to permeability, without assuming specific pore geometries. As a result, it better captures the effects of varying pore sizes, tortuosity, and connectivity, which are common in heterogeneous systems. This flexibility allows Darcy’s law to reflect the actual flow behavior more accurately, leading to higher permeability estimates compared to the more idealized Kozeny–Carman model.

The difference in the level of asymmetry in the distribution indicates that the asymmetry of the Kozeny–Carman Equation is higher compared with the asymmetry level in the LBM permeability data. However, although the asymmetry level in the LBM permeability distribution is lower, it does not mean that the LBM permeability is better. In some previous studies on permeability [34], the resulting distribution is log-normal, so the permeability distribution of the Kozeny–Carman model is more representative compared with the LBM distribution. Nevertheless, this does not mean that the permeability measurement using the LBM in this study is incorrect, as other studies have mentioned that alternative permeability distribution forms a gamma distribution. Moreover, the calculation using the Kozeny–Carman Equation requires the porous rock conditions to be ideal since it is performed through mathematical calculations, unlike the actual conditions of porous media, which have more complex parameters.

Overall, the relationships between porosity, permeability, and tortuosity derived from our simulations can be directly applied to enhanced oil recovery (EOR) processes, where optimizing fluid injection and extraction strategies is crucial. Additionally, these results have applications in groundwater hydrology, adding in the prediction of aquifer behavior and contaminant transport. Moreover, the insights gained from our study can contribute to the design of advanced porous materials, such as filters and catalysts, where precise control over flow properties is essential.

5. Conclusions

The effective properties of 2-D porous rock including the tortuosity, permeability, effective porosity, and average throat size were estimated using the Lattice Boltzmann method (LBM) and image processing methods. The permeability was strongly correlated to the effective porosity, while tortuosity showed a moderately strong correlation using the Kozeny–Carman Equation. Furthermore, comparing the permeability results from the LBM simulations with those from the empirical Kozeny–Carman model indicated that both methods are representative of permeability calculations. The ideal form of the permeability distribution was log-normal, although the actual conditions can be influenced by various factors affecting the permeability values. Therefore, using simulations for permeability calculations remains a viable alternative due to their representative results.