Three-Dimensional Reconstruction and Seepage Simulation of Vermiculite Materials Based on CT Technology

Abstract

The purpose of this paper is to investigate the 3D reconstruction and seepage simulation of vermiculite-based sealing materials in order to clarify the relationship between material structure and fluid leakage in order to improve their performance and reliability and to provide a foundation for accurate leakage rate calculation under various conditions. CT scanning and 3D reconstruction techniques were used to examine the pore structure and fractal properties of vermiculite-based sealing materials. The seepage mechanism under compression was examined using a porous medium model. The findings demonstrate that the vermiculite-based sealing materials have self-similarity and scale-invariant fractal properties. The porous media model in compression exhibits a low leakage rate, opening up a new avenue for realizing microscopic leakage studies with a leakage rate of 1 × 10⁹ in compression.

1. Introduction

High-performance non-metallic sealing materials have become increasingly important in the sealing sector as a result of advancements in industrial technology. Asbestos sheets, ceramic/glass fiber, mica, and graphite are currently frequently used materials for static sealing [1]. These materials possess exceptional mechanical properties and processability. However, gaskets made from these materials may experience extensive wear and a notable rise in leakage rate when subjected to prolonged high temperatures and pressures. Vermiculite possesses a typical molecular structure of a 2:1 layered silicate. It can expand at high temperatures and through chemical reactions under specific conditions. After losing water and expanding, vermiculite has a structure similar to expanded graphite, resembling that of a worm. As a result, it maintains the ability to return to its original shape effectively when compressed. Additionally, vermiculite is structurally stable and exhibits excellent resistance to high temperatures, being able to withstand up to 1000 °C without oxidation. China has an abundant supply of vermiculite, and the expanded vermiculite produced through chemical and high-temperature expansion processes has found significant applications in various industries, including construction, sealing, agriculture, metallurgy, and chemical production [2]. In the field of sealing, sealability is a significant indicator of gasket performance, and microleakage has always been a research hotspot. The vermiculite-based sealing material is a new type of sealing material suitable for high-temperature environments with excellent sealing performance. However, relatively few studies have been conducted on the microstructure and seepage patterns of vermiculite-based sealing materials. The objective of this paper is to investigate the three-dimensional reconstruction and seepage simulation of vermiculite-based sealing materials to clarify the relationship between material structure and fluid leakage and to provide a basis for the accurate calculation of leakage rates under different conditions.

There are two types of gasket leaks: interface leakage and penetration leakage. Interfacial leakage occurs between the gasket and the flange sealing surface, while penetration leakage mainly occurs within the non-metallic gasket material. This is because non-metallic gaskets, made up of various fibers and binders, are not densely compacted, resulting in sparse internal tissue. Under the pressure of the medium, the material’s internal voids are more likely to be permeated, leading to an increase in the leakage rate. With the increasing human requirements for the environment and safety, petrochemical, nuclear power, aerospace, and other industrial fields are becoming more and more stringent in the control of fluid leakage. This context necessitates advanced analysis of sealing materials, specifically through simulation of microscopic seepage characteristics and the establishment of models reflecting the internal pore structure of gaskets.

In recent years, porous structures have been widely used in fields such as mechanics, aerospace, material chemistry, and medicine. Lattice structures, honeycomb structures, and minimal surfaces are common porous structures. The Three-Period Minimal Surface (TPMS) structure has attracted more and more scholars’ attention due to its controllable parameters, interconnected holes in the overall structure, and the ability to be accurately described by mathematical expressions. Mathematically, a minimal surface needs to be described by area and curvature [3], that is, a minimal surface that satisfies all external constraints, or a surface with zero average curvature is theoretically a minimal surface [4]. There are many different types of TPMSs, including P surfaces, G surfaces, D surfaces, and I-WP surfaces. Rajagopalan and Rob [5] were the first to apply P-surface-type TPMSs to establish a biomimetic skeleton model. Afterward, Melchers et al. [6] modeled the target object using G and D surfaces as unit bodies. To better integrate the advantages of various types of TPMSs, a method of mixing multiple cell bodies was adopted to establish the model. Yoo [7], Nan Yang [8], and Wang Qinghui [9] et al. used a more complex TPMS scheme to describe the changes in the microstructure of a scaffold by controlling the position and parameters of the unit body. Compared to the reconstruction model, the three-period minimum curved porous media model has all the pores interconnected and does not contain islands, so it is not as accurate as the three-dimensional reconstruction model containing islands to reflect the pore characteristics of vermiculite porous media. At the same time, the complexity of the pore model generated by the arrangement and combination of three-period minimal surfaces is still not as high as that of the three-dimensional reconstruction model, and it lacks a certain degree of randomness. Therefore, in this study, microscale 3D reconstruction of vermiculite-based sealing materials was performed using advanced digital image processing techniques based on CT scanning experiments with Avizo software 2021.1.

CT scanning technology has become increasingly prevalent in the analysis of the microstructure of geotechnical bodies. E. Rosenberg [10] and Ams [11] utilized CT scanning to create a three-dimensional digital model of sandstone, enabling them to analyze various characteristic parameters of sandstone samples. Suna [12] employed the CT scanning method to construct a digital model of rock samples, which was then used to establish a pore network model. The equivalence of the digital core and pore network models was verified by calculating and analyzing parameters of the pore network model, such as spatial topology, pore throat size features, and shape features. He Kaikai [13] utilized CT scanning experiments, as well as Matlab2021 and Avizo software2021.1, to establish reconstruction models of coal samples in different directions and sizes. The characteristic parameters of the reconstruction models were compared with mercuric compression and penetration test results for validation. Ni et al. [14] used X-ray tomography together with multiphysics field simulation software to analyze the flow of CBM in a large-pore coal medium and obtained the absolute permeability pressures in the X, Y, and Z directions, and Ramandi HL et al. [15] combined the CT technique with a scanning electron microscope and further accurately calculated the permeability of the rock samples. Zhong Jiangcheng [16] performed a numerical simulation of uniaxial compression on the three-dimensional structure of coal samples reconstructed by CT scanning technology and compared it with actual test results to verify the accuracy of the model. Chenghao Zhang [17] used CT scanning to reconstruct a real three-dimensional pore and fracture model of mudstone samples and combined it with COMSOL Multiphysics software to perform a finite-element simulation and revealed the silica sol grouting slurry diffusion law. Sayan G et al. [18] (μ-CT) analyzed the porosity of nanocomposite scaffold-like gel for dual-response adjustable drug delivery.

In this paper, the 3D reconstruction technique is introduced in the study of vermiculite-based sealing materials, and the real 3D pore model of vermiculite-based sealing materials is established by CT scanning and 3D reconstruction techniques. Compared with the traditional 2D characterization method, the 3D reconstruction model can more accurately describe the pore structure and fractal features of the material. In addition, the compression state of the vermiculite-based sealing material was considered, the porous medium model under compression was developed, and the effect of the compression state on the seepage mechanism was investigated. These improvements enable the text to reveal the microstructure and seepage law of vermiculite-based sealing materials more comprehensively and provide a new research method and theoretical basis for improving their performance and reliability.

2. Sample Description and Test Principle

In order to model the porous media of vermiculite materials, the pore structure of vermiculite-based porous media was characterized using various experimental methods. Firstly, the elemental composition of the vermiculite-based sealing materials was investigated using energy spectroscopy. Secondly, the surface and cross-section micromorphology of the vermiculite-based sealing materials were qualitatively analyzed by scanning electron microscopy for fractal features. The pore parameters of the vermiculite-based porous materials were measured by a high-pressure mercury pressure test to quantitatively analyze the spatial distribution of the pores. Finally, a three-dimensional model of vermiculite-based porous media was constructed using CT imaging technology and 3D visualization software.

The vermiculite-based sealing plates used in the CT scanning experiments were obtained from Zhejiang Guotai Xiaoxing Sealing Material Co., Ltd. (Hangzhou, China) (Figure 1). These plates were transformed into small cubes measuring 1.5 mm on each side. The basic physical parameters of the samples are presented in

Table 1. Basic parameters of vermiculite-based sealing material.

Sample	Density (g/cm3)	Specific Surface Area (m2/g)	Elemental Analysis (%)
			O	Si	Mg	C
Vermiculite sheet	1.1–1.13	0.82	35.80	23.71	15.89	6.80

.

By observing the microstructure of vermiculite-based sealing materials through scanning electron microscopy, it can be determined whether the two typical characteristics of fractal porous media—“self-similarity” and “scale invariance”—are reflected in them. Figure 2 shows the surface image of the sample, with magnifications of 200×, 500×, and 1000×.

Comparing the microscopic morphology maps of the samples at 200×, 500×, and 1000×, the morphology and distribution of the matrix and pores in the samples meet two obvious characteristics: the first is scale invariance, which means that images magnified at different magnifications all exhibit the same structural characteristics; and the second is self-similarity, which refers to the similarity of local and global structural features. Therefore, the microstructure characteristics of vermiculite-based sealing materials are consistent with fractal features, and fractal theory can be used to describe the complexity and tortuosity of their structures.

Computed tomography, or CT scanning technology, is a non-destructive imaging technique that provides clear images with fast scanning times. The principle behind it involves the penetration of high-energy X-rays through an object, causing collision with atomic electrons outside the nucleus and resulting in energy loss. The absorption capacity of an object can be determined by examining the difference in energy before and after the X-rays penetrate the object. If the internal structure of the object is not homogeneous, different parts of the same object will absorb X-rays differently, thus providing information about the distribution of the internal structure of the object.

The experiment utilized a General Electric-manufactured Phoenix V|tome|x S CT scanner as the primary instrument. The imaging parameters are detailed in

Table 2. CT scanner working parameters.

Parameters	Name	Maximum Tube Voltage (kV)	Maximum Tube Power (W)	Minimum Distance from Focal Point to Workpiece (mm)
Norm	Phoenix V|tome|x S	240	320	4.5

. A total of 910 2D CT slices were obtained during the experiment, as depicted in Figure 3. The image resolution was 1478 px ∗ 1009 px, with voxel measurements of 0.7 μm. The grayish white color represents the high-density specimen, and the black color represents the pore space of the low-density specimen. A reconstructed 3D grayscale model of the sample was obtained from the scanned effective grayscale image, as shown in Figure 3.

3. Three-Dimensional Reconstruction of Pore Structure of Vermiculite Material

3.1. Image Noise Reduction

The CT scanning procedure reconstructs pictures with non-quantum and impulsive noise. These sounds interfere with the distribution of the image’s gray value, causing the model’s accuracy to degrade. As a result, there is a demand to minimize the noise in the image. The most prevalent noise reduction methods are Gaussian, median, and mean filtering. Zhu Honglin [19] and Lv Bangmin [20] utilized the median filtering algorithm to process grayscale images of porous carbon and multi-low-permeability sandstone. This approach considerably mitigated the effects of pretzel noise on the images. In this paper, median filtering was applied to the images, as shown in Figure 4, depicting the photos before and after undergoing the noise reduction process. The results show the removal of small noise points and the filling of small pores, which improves image smoothness. This result is a useful prerequisite for proper segmentation and quantitative analysis in future studies.

3.2. Threshold Segmentation

Image segmentation involves dividing an image into regions based on similar values of features, such as grayscale, color, texture, or geometry. In this experiment, the regions of interest are the skeleton region and the pore space region. Commonly used segmentation thresholding methods include the maximum class spacing variance, maximum entropy, and porosity-based search thresholding. The maximum class spacing variance method and the maximum entropy method are suitable when the porosity is unknown. However, when the porosity is known, the pore characteristic parameters of the porous media model can be extracted based on the porosity searching threshold. These parameters can then be verified by comparing them with the results of the mercury compression experiment [13]. The selection of different feature thresholds determines different categories of image pixels, and choosing an appropriate threshold can improve the accuracy of image segmentation and analysis [21,22].

The AutoPoreV 9600 fully automated core porosity measurement system was used in this study, and the porosity of the vermiculite-based seal was measured to be 52.28%. This measured value served as a constraint for threshold segmentation. The approximate range of thresholds was determined by iteratively adjusting the thresholds, and the corresponding porosity of the vermiculite-based porous media model was calculated for each threshold value. The threshold value that yielded the smallest difference between the calculated porosity and the measured porosity was selected as the optimal threshold for segmenting the vermiculite-based sealing material. After selecting a gray value of 102, the average porosity of 500 images following threshold segmentation was found to be 50.579%, which closely aligned with the measured porosity of 52.28% obtained from the mercury compression experiment. The error fell within 3%, confirming the selection of a gray value of 102 as the final threshold value.

3.3. Three-Dimensional Reconstruction of Pores

The 2D slices were imported into Avizo software sequentially. The scanning parameters, such as the number of slices and voxel size, were set. The 3D reconstruction model of the vermiculite sealing plate was then created using the volume rendering module (Figure 5). In Figure 5, (b) is the 3D reconstructed model of vermiculite-based sealing material built by ct scanning, (a) gray is the skeleton model, and (c) blue is the 3D microporous model. This reconstructed model provides a more realistic and intuitive representation of the spatial morphology of the skeleton and pores of the vermiculite-based sealing materials. Additionally, it preserves a greater amount of information about the 3D data field.

In order to study the internal pore structure of the samples in more detail, the corresponding pore network model was extracted using the maximum sphere method based on the reconstruction model [23,24,25]. The pore network model was obtained using the “Separate Objects” command and the “Generate Pore Network Model” module in the Avizo software2021.1 (Figure 6). In Figure 6, the red sphere represents the hole with the largest volume, followed by the yellow color, and the blue-green color represents the hole with the smallest volume. In this model, spheres represent internal pores, with larger spheres indicating larger pore diameters. Columns, on the other hand, represent internal throats, with thicker columns indicating larger throat radii. The length of the columns represents the length of the throats, with longer columns indicating longer throat lengths. The pore network model, also known as the ball-and-stick model, provides a visual representation of the connectivity characteristics of the sample pores.

3.4. Construction of Finite-Element Models

The following steps were taken to produce a finite-element model in compression: 1. testing and optimizing the mesh quality; 2. obtaining the compression resilience curve of the vermiculite gasket; 3. writing the velocity UDF; 4. constructing the compression-state model.

Avizo’s Meshing module can generate meshes for the reconstructed sample model. However, a mesh quality setting of “High” may still generate some low-quality meshes. Avizo can detect and repair unreasonable meshes. Mesh irregularities can be identified and resolved by translating and merging vertices. As depicted in Figure 7, the arrowed portion refers to a triangular mesh with an excessive aspect ratio, i.e. a malformed mesh, and the circle shows the modified mesh node.

Vermiculite-based sealing materials are available in thicknesses ranging from 0.8 mm to 1.5 mm. Due to their thinness, it is not feasible to directly perform compression tests and CT scan analysis. Instead, the porous media in compression need to be modeled by shifting the boundaries of the original-state porous media model. The moving grid method is employed to handle boundary movement, allowing for the changing shape of the flow field over time. In this study, a vermiculite-based porous media model in compression was constructed using this method. In the dynamic mesh method, there are several iterative methods for repairing the mesh, including the dynamic layer method, the mesh smoothing method, and the mesh reconstruction method. Among the mesh smoothing methods, the diffusion smoothing method was chosen because of its moderate computational volume and superior mesh quality. However, it becomes necessary to utilize the mesh reconstruction method when the mesh deformation reaches a certain extent. Therefore, both the diffusion smooth method and the mesh reconstruction method were used in this paper.

To solve the boundary motion problem using a dynamic mesh, a velocity UDF (User-Defined Function) must be written and compiled. This paper achieves the velocity curve by conducting a gasket compression rebound experiment using a 304-punched vermiculite-reinforced gasket as the research object. The gasket compression rate, rebound rate, and compression rebound curve refer to the compression rebound experimental test standard GB/T 12622-2008 [26].

Table 3. Dimension parameters of 304-punched vermiculite-reinforced gasket.

Name of Sample	Nominal Diameter	Nominal Pressure	Gasket Type	Gasket Standard	Size (mm)
304-punched vermiculite-reinforced gasket	DN80	PN63	Convex surface (RF)	HG/T 20606-2009 [27]	89 ∗ 148 ∗ 3

and

Table 4. Results of compression and rebound by 304 sprint vermiculite-reinforced gasket.

Name of Sample	Load (MPa)	No.	Compression Ratio (%)	Rebound Rate (%)
304-punched vermiculite-reinforced gasket	35	1	33.33	8.57
		2	33.18	9.86
		3	33.33	10.61
		Average value	33.28	9.68

provide information on the gasket parameters and experimental results.

The velocity versus time equation and velocity UDF were derived by scaling the compression resilience curves (Figure 8) of vermiculite-reinforced shims at a pressure of 35 MPa. The velocity UDF was implemented in Fluent, and the compression process of the shims was performed using the diffusion smoothness method and the mesh reconstruction method of the dynamic mesh technique. By adjusting the time-step length and the number of time steps, the problem of error reporting that may occur during compression was solved. The final compressed model was obtained (Figure 9). The compression rate initially increases and then slows down to about half of the thickness of the original porous medium model, which correlates with the actual compression of the gasket.

4. Quantification and Characterization of Microscopic Pore Structure

A 3D reconstruction model of the vermiculite-based sealing material and a similar pore network model were constructed using porosity as a constraint. The structural parameters of the two models were calculated and compared with the results of mercury compression experiments to verify their accuracy. The seepage simulation results were qualitatively analyzed to assess the reasonableness of the reconstructed models [28,29,30]. Additionally, the study explored the size effect of the characteristic parameters of the reconstructed model and identified the minimum model size capable of representing the overall properties.

4.1. Analysis of Three-Dimensional Pore Space Structure

The porosity of a porous material refers to the ratio of the total volume of pores to the total volume of the material. To determine the surface porosity of the material, the area of each slice pore is divided by the total area of the slice. By counting the surface porosity of each section sequentially, it is evident from Figure 10 that the porosity of each section of the reconstructed model fluctuated. However, the overall cross-section porosity ranged from 47.60 to 53.87%. The average porosity of the samples was calculated to be 50.579%, which is slightly lower than the value of 52.28% obtained from the mercury compression test. The reason for this discrepancy is the limited accuracy of the scanning equipment when scanning apertures that were too small in size.

As shown in Figure 11, the majority of pore diameters were between 2 and 4 μm, accounting for 70% of the total. The highest number of pores, accounting for 60% of the total, had pore areas ranging from 10 μm² to 20 μm². Similarly, pore volumes ranging from 5 μm³ to 15 μm³ were observed in 60% of the total.

In addition to using parameters, such as porosity, pore diameter, and pore volume, to characterize the reconstructed model, meander and volume fractal dimension can be used as characteristic parameters to evaluate reconstructive models. Tortuosity measures the curvature of fluid flow in the pore space, while the volume fractal dimension measures the irregularity of the complex object and the efficiency of space occupation. These eigenvalues closely resembled the results of the mercury compression experiments (

Table 5. Comparison of reconstructed model parameters and mercury intrusion experiment results.

Parameters	Reconstructed Model	Mercury Compression Experiment
Aperture/μm3	2.17 × 107	2.48 × 107
Totality/μm3	4.29 × 107	4.75 × 107
Body porosity/%	50.579	52.28
Face porosity/%	47.60–53.87	52.28
Average pore size/nm	2336	1211.49
Tortuosity	1–10.9058	4.30
Dimensionality of a volume fractal	2.75	2.999

), suggesting that the spatial geometry and topology of the reconstructed model closely resembled the actual pore structure.

4.2. Analysis of Hole Throat Dimensions

The pore network model was created using the maximum sphere method. In order to determine the parameters that characterize the pore and throat, we measured the internal tangential diameter of the largest sphere that could fit into the pore and throat, thus determining the pore diameter and throat diameter. Additionally, the throat length was measured as the distance between two connected pore spaces. The statistical results of extracting the characteristic parameters of the pore network model, including the throat diameter, throat length, and coordination number, are displayed in Figure 12.

Combined with Figure 12 and

Table 6. Statistics of quantitative parameters of pore network model.

Parametrics	Throat Diameter (μm)	Length of Throat (μm)	Coordination Number
Minimum value	0.41	2.28	0
Maximum value	34.54	82.04	52
Average value	2.84	10.39	7

, it can be seen that pipes with diameters between 1 and 5 μm accounted for 50% of the total number of pipes, and those with lengths between 7 and 12 μm accounted for 55% of the total number of pipes, which suggests that the distribution of pipe sizes was relatively homogeneous. The coordination number represents the number of throat channels connected to a pore, which reflects the pore space topology of the material. About 50% of the pores had a coordination number between 3 and 14, and about 35% had a coordination number of 0, also known as freestanding or “island” pores. A higher coordination number signifies improved pore connectivity. The coordination number influences the porosity, compressibility, and permeability of a sample.

4.3. Qualitative Analysis of Seepage Processes

The Avizo software was used to perform seepage simulation on the vermiculite 3D reconstruction model. This simulation aimed to qualitatively analyze the seepage pattern and verify the model’s reasonableness. The model assumed that a single-phase fluid moves from the left side to the right side along the blue Z-axis. The left and right sides of the model were designated as the pressure inlet and outlet, with pressures of 1.1 MPa and 0.1 MPa, respectively. The hydrodynamic viscosity was set to 0.001 Pa·s. The flow lines’ continuity and distribution indicate the connection status of the orifice throat. The simulation results of the microseepage in the three-dimensional reconstructed model are shown in Figure 13 and Figure 14.

The arrangement of flow lines illustrates the path of fluid movement within the porous material. In Figure 13, consistent flow lines suggest well-connected pores, while intermittent flow lines suggest poor pore connectivity. The color of the streamlines corresponds to the flow rate, with red indicating a fast flow rate and blue indicating a slow flow rate [31,32]. In the reconstructed model, the central region shows fully red flow lines, indicating the presence of pores with numerous connections, forming a comprehensive seepage channel. On the other hand, the blue flow line at the edge is fragmented, representing the flow condition within the silos.

The ball-and-stick model illustrates the pressure and flow distribution in different regions of the sample. In Figure 14, the larger the sphere and the redder the color, the greater the pressure exerted near the pore; the thicker the column and the redder the color, the greater the flow in the throat. Upon analyzing Figure 14, it becomes evident that the pressure within the pore gradually diminishes in the direction of water flow. The pressure is maximal at the inlet and minimal at the outlet. Similarly, the flow rate in the throat is minimal at the boundary and increases as we move closer to the center, mirroring the flow rate distribution.

4.4. Table Cell Extraction

The Representative Element Volume (REV) is carefully chosen to effectively represent the overall properties and to meet the requirements of devices such as computers [33,34].

By analyzing the porosity variation curves of models with different shapes (Figure 15), it can be observed that the porosity is unstable when the thickness of the model is less than 500 pixels. However, the porosity becomes more stable when the model thickness is equal to or greater than 500 pixels, and the increase in porosity is small as the representative volume cell size increases. The effect of thickness on pore diameter is similar to its effect on porosity (Figure 16). When the model thickness is less than 500 pixels, the pore diameter value is unstable. On the other hand, when the model thickness is equal to or greater than 500 pixels, the pore diameter becomes stable and very close to the average diameter value of 1.211 nm derived from the mercury pressure experiments. Consequently, a square with a side length of 50 pixels was chosen as the representative volume cell.

As the volume of the 3D reconstructed model varies, parameters like porosity and mean diameter also change. This means that the 3D reconstructed model exhibits a scaling effect. Consequently, a square reconstruction model with a side length of 50 pixels was examined to draw conclusions applicable to models of large size.

5. Simulation of Seepage in Vermiculite-Based Porous Media

5.1. Control Equations and Boundary Conditions

The simulation was based on a three-dimensional reconstructed model that replicates the real pore structure and includes seepage simulation. Two types of research methods were used in this study: traditional computational fluid dynamics (CFD) and the lattice Boltzmann method (LBM). CFD is a method of discretizing the model and control equations by solving the equations of conservation of mass, momentum and energy. On the other hand, the LBM is based on the theory of molecular motion and simulates a fluid as particles with mass but no volume. By simulating the collision and motion of these particles in a lattice, a simplified kinematic model was developed to reflect the fluid seepage behavior in the pore space. Ma Kun, M E Nimvari, et al. [35,36,37] concluded that turbulent flow is the dominant state of a fluid in a porous medium. In this section, the Reynolds number of fluid flow in vermiculite-based porous media is calculated with the available data and it is determined that turbulence is the flow state of the medium based on its Reynolds number greater than 2300. In this paper, the seepage flow in porous media is numerically modeled using the incompressible Navier–Stokes equations module in CFD. The equations used in the simulation are as follows:

$$\left\{\begin{array}{c}{\rho }_w \left(u·\nabla \right)u=\nabla ·\left[-pI+\mu \left(\nabla u+{\left(\nabla u\right)}^T\right)\right]+{\rho }_{w}g+F\\ {\rho }_w \nabla ·u=0\end{array}\right.$$

where ${\rho }_w$ represents the fluid density in kg·m⁻³; $p$ represents the pressure in Pa; $I$ is the unit matrix; $\mu$ is the hydrodynamic viscosity in Pa·s; $F$ represents the volumetric force in N·m⁻³; $u$ is the flow velocity in m·s⁻¹; and $g$ is the gravitational acceleration in m·s⁻².

The fluid properties were assigned values based on the corresponding parameters at room temperature. The leftmost side of the porous medium model was designated as the inlet boundary, the rightmost side as the outlet boundary, and the four sides were set as free-slip wall surfaces, while the rest of the walls were regarded as no-sliding wall surfaces. The inlet pressure was set to 2.1 MPa, 4.1 MPa, 6.1 MPa, and 8.1 MPa, while the outlet pressure was set to 0.1 MPa. Seepage simulation was carried out for the porous media model in both the original and compressed states.

To simplify the simulation and analysis process, the following assumptions were made: (1) the fluid flows only within the pores of the material and does not penetrate into the skeleton matrix or react with the skeleton particles; (2) the outlet flow rate is monitored using the steady-state method, and the simulation is completed when the flow rate shows regular periodic changes or remains essentially constant; and (3) the fluid is affected only by gravity and pressure.

5.2. Seepage in Porous Media with Different Pressure Gradients

The seepage simulation consisted of the original-state model and the compression-state model. The seepage of porous media was demonstrated under different inlet pressures of 2.1 MPa, 4.1 MPa, 6.1 MPa, and 8.1 MPa and an outlet pressure of 0.1 MPa, as shown in Figure 17 and Figure 18.

Figure 17 and Figure 18 demonstrate that altering the inlet pressure minimally impacts the overall pressure gradient pattern within the porous medium. Increasing the inlet pressure results in a rapid expansion of the high-pressure zone and a rapid narrowing of the low-pressure zone. Subsequently, the increasing inlet pressure leads to a slower expansion of the high-pressure zone and a slower narrowing of the low-pressure zone. This overall pressure trend suggests that the spatial distribution of the pore structure in the porous medium is non-uniform.

The original porous media model and the compressed porous media model were divided into five equal sections from the inlet to the outlet, parallel to the inlet surface. Figure 19 and Figure 20 display the pressure distributions from section 1 to section 6 of the original porous media model and the compressed porous media model at 2.1 MPa, respectively. In the porous media model, the pore size decreases proportionally as the compression state increases, leading to a decrease in the differential pressure for the same cross-section. The flow rate, mass flow rate, and leakage rate all decreased slightly. However, the overall pressure change follows a pattern of decreasing pressure from the inlet to the outlet.

Table 7. Leakage rate of original porous media model.

Leakage Rate (m3/s)	Strains (MPa)
	2.1	4.1	6.1	8.1
He	3.958529 × 107	5.958315 × 107	7.554999 × 107	8.935514 × 107

and

Table 8. Leakage rate of porous media model under compression.

Leakage Rate (m3/s)	Strains (MPa)
	2.1	4.1	6.1	8.1
He	2.306919 × 107	3.502845 × 107	4.450683 × 107	5.268679 × 107

further validate this observation.

As can be seen from Figure 21 and Figure 22, the velocity distribution is consistent across various sections, both in the original and compressed states. The middle section exhibits high flow rates due to incomplete development at the beginning of the fluid medium. Once the middle layer is fully developed, the flow rate reaches its maximum. The relationship between outlet flow rate and inlet pressure is non-linear, with the former increasing as the latter rises. The rate of increase slows down after an initial rapid increase.

The leakage rates in 10⁻⁷ m³/s are given in Table 7 and Table 8 for the porous media model in the original and compression states, respectively. Comparison of the two shows that the leakage rate of the compression-state porous media model is lower than that of the original-state porous media model. This indicates that the compression state significantly affects the leakage rate of vermiculite gaskets.

6. Discussion and Conclusions

This article aimed to investigate the three-dimensional reconstruction and seepage simulation of vermiculite-based sealing materials, and the following conclusions can be drawn:

(1)

A realistic 3D pore model of vermiculite-based sealing material was successfully established using CT scanning and 3D reconstruction techniques, and the accuracy and representativeness of the model were verified.

(2)

The porous media model under compression shows a small leakage rate, with a compression leakage rate of 1 × 10⁻⁹. This provides a new method for realizing microleakage research.

(3)

The flow simulation results indicate that the seepage process of vermiculite-based sealing materials is influenced by pressure gradients, and the inhomogeneous distribution of pore structure leads to differences in seepage rate.

This study provides a new research method and theoretical basis for the three-dimensional reconstruction and seepage simulation of vermiculite-based sealing materials and lays an important foundation for further research on the microstructure and seepage law of sealing materials. To provide a basis for calculating leakage rates under different conditions, clarifying the relationship between material structure and fluid leakage can help optimize the material design and preparation processes, which is of great significance for improving the performance and reliability of sealing materials in industries, such as petrochemical, nuclear power, aerospace, and other industries.

This study only focuses on vermiculite-based gasket materials and applies to materials that meet fractal theory. The research on other materials is not yet complete. In addition, only single-phase fluid flow within the pores was considered in the seepage simulation, without considering the interaction with the skeleton matrix. Therefore, in further developments, it will be necessary to expand the research scope, conduct in-depth research on the performance and reliability of other non-metallic sealing materials, and consider more complex multiphase flow models.