Investigation of the Impact of Particle Shape on Pore Structures and Clogging Properties of Filter Layers

Abstract

This study posits that soil particles in the filter layer are ellipsoidal. The effective pore radius of the filter material was calculated for various particle-shape parameters and distributions. The relationship between the porosity of the filter material and the ratio of the long axis to the short axis of ellipsoidal particles in a loose arrangement was also examined. The results indicate that the porosity of the filter material initially decreases and subsequently increases with increases in the ratio of the long axis to the short axis; however, the rate of increase progressively slows. A method for transforming irregularly shaped particles into ellipsoidal forms is proposed. The particle-shape parameter, S, is introduced to characterize the shape of irregular particles. The relationship between particle-shape parameters and the ratio of the long axis to the short axis was investigated specifically for ellipsoidal particles. It was found that the particle-shape parameters exhibit an approximately linear relationship with the ratio of the long axis to the short axis within a specific range. The discrete element method was employed to investigate the impact of particle shape on the filtration characteristics of the filter layer and was complemented by comparative experimental analysis. By analyzing the pore structures of spherical and ellipsoidal particles, this study predicts the relationship between pore structure and particle-shape parameters for any irregularly shaped natural-particle filter layer.

1. Introduction

According to incomplete statistics, there are 12,655 tailings reservoirs in China, including 3032 disease-prone reservoirs, 1265 hazardous reservoirs, and 613 critically hazardous reservoirs. The risk associated with tailings pond accidents ranks 18th among 93 types of accident hazards globally, making it a significant hazard that cannot be overlooked.

In tailings accidents, dam failures and tailings leakage account for as much as 96% of incidents. Leakages from dams and drainage facilities are the primary causes of these failures and leakages. Therefore, the study of seepage and stability in tailings reservoirs has always been a critical issue in tailings design and research. The study of seepage in tailings reservoirs differs from that in general reservoir dams in two key aspects. First, tailings seepage involves muddy water carrying tailings slurry, rather than clear water. Second, tailings sand is a byproduct of ore processing, which involves separating valuable components from waste. If tailings are discharged with water, they can cause severe pollution. Therefore, tailings-seepage management must not only ensure effective drainage but also prevent the discharge of tailings sand with the water. This requires the installation of a filter layer in the drainage facilities.

In the study of the filtration characteristics of filter layers, David C. Mays [1] analyzed the clogging mechanisms of granular filter layers from the perspective of fractal dimensions. Dang et al. [2] addressed muddy-water seepage as an issue of unstable seepage through variable infiltration paths and proposed an effective theoretical method for calculating such seepage. Indraratna and Raut [3] examined the interaction between the filter layer and the protected soil within the filtration system and developed a method for evaluating the filter layer’s performance when it is infiltrated by protected soil. Peter To et al. [4] demonstrated that the erosion resistance of granular filter layers is correlated with their relative density. Delgado-Ramos et al. [5] investigated the mechanisms and principles of internal erosion in filter layers where the protected soil is clay. Srivastava and Babu [6] proposed a novel method for the design of granular filter layers and assessed their performance using probabilistic analysis. Li et al. [7] identified three modes of clogging through laboratory tests and proposed corresponding determination methods. The principles and guidelines for filter-layer design have been discussed by Liu and other researchers [8,9,10]. John Szczap [11] investigated both experimentally and from a theoretical perspective the clogging of two types of filtration media by the slurry under varying operational conditions.

The aforementioned literature on tailings filter layers predominantly focuses on the effects of gradation and particle arrangement. In these studies, soil particles are typically assumed to be spherical, and the studies do not account for the impact of particle shape. However, as a granular material, tailings sand features complex and variable particle shapes, and these significantly affect the clogging characteristics of the filter layer.

In a study of particle shape, Dong et al. [12] investigated the actual shape and internal porosity of carbonate particles and discussed how particle shape influences the crushing behavior of individual particles. Johanson [13] found that the number of contact points between soil particles significantly affects soil strength in work based on studies of particle accumulation with varying shapes. Kock and Huhn’s study [14] shows that particle shape significantly affects friction strength, shear-band direction, and particle rotation. Khan and Latha [15] studied the effect of particle shape on multi-scale shear at the sand–geomembrane interface. Baten and Garg [16] investigated the effect of particle shape on the sand-accumulation process. Shahid and Usman [17] examined the motion of particles with various shapes and their interactions in a fluid. Xiong et al. [18] investigated the influence of aspect ratio on the seepage performance of both coarse and fine particles. Wu et al. [19] examined the effect of particle shape on the triaxial mechanical properties of soil samples. Abou-Chakra et al. [20] developed a three-dimensional analysis model of particle shape by projecting particles in various directions and analyzed particle shape across different particle sizes. The results indicate that there is no clear relationship between the shape of particles and their size within the same material. Tu and Wang [21] compiled commonly used parameters in particle-shape analysis and compared various particle-shape parameters. Behzad Majidi [22] used DEM to investigate the effects of particle shape and friction coefficient on the VBD. Alireza [23] investigated the effect of particle size distribution through three-dimensional DEM simulations and included rolling resistance as a parameter to account for particle behavior.

Despite this, most of the current research on particle shape is conducted at a qualitative level and primarily analyzes the shape of individual particles. Furthermore, there is a lack of studies examining the effect of various particle shapes on the pore structure of soil.

In this study, filter soil particles were modeled as ellipsoidal particles and the pore radius of the filter soil was calculated based on varying particle-shape parameters and distributions. The filtration characteristics of the filter-layer soil directly reflect the soil’s pore structure. Therefore, this study employed ellipsoidal parameters to model filter-layer soil particles using particle-flow software PFC 5.0. A filtration-permeability test was conducted to investigate the impact of these ellipsoidal parameters on soil filtration characteristics and the variation in the permeability coefficient of the filter layer. Additionally, indoor tests were performed for comparison and validation.

2. Analysis of the Pore-Structure Characteristics of the Filter Layer

Soil pores are classified as effective or ineffective [24,25]. In the seepage process, effective pores are those that are connected to the external water flow and allow both water and soil particles to pass through, while ineffective pores are those that cannot participate in the seepage process, such as closed pores. Seepage can occur only through effective pores. In this study, the “effective pore radius” refers to the maximum pore radius through which tailings sand can pass during seepage through the filter layer. This radius reflects the clogging characteristics of the filter layer. This parameter is essential for analyzing seepage clogging within the filter layer and will hereafter be referred to as the effective pore radius of the filter.

2.1. Ideal Spherical-Particle Filter Layer

Firstly, it was assumed that the filter layer consists of spherical particles. The plane projections for a loose arrangement and a tight arrangement are shown in Figure 1 and Figure 2, respectively.

Assuming that the radius of the uniform spherical particles is R and that the effective pore radii of the filter are r₁ and r₂ for the loose and close-packed arrangements, respectively, the values can be calculated from Figure 1 and Figure 2, as presented in Equations (1) and (2).

$$r_1=\sqrt{2}R-R=0.414R$$

(1)

$$r_2=2({\displaystyle \frac{\sqrt{3}}{3}}R-{\displaystyle \frac{R}{2}})=0.155R$$

(2)

Thus, for ideal uniform spherical particles, the effective pore radius of the filter ranges from 0.155 R to 0.414 R.

2.2. Ideal Ellipsoidal-Particle Filter Layer

Secondly, it is assumed that the filter layer consists of ellipsoidal particles.

2.2.1. Theoretical Maximum Pore Radius

Under ideal conditions, the particles in the filter layer that form the theoretical maximize the effective pore radius are arranged as depicted in Figure 3. The analysis was conducted using the upper-left portion of the axisymmetric diagram. The long-axis radius, OA, is denoted as a, and the short-axis radius, OB, is denoted as b. The contact point between the two ellipsoidal particles is denoted as C, with OC = c. The effective pore radius of the filter for this particle arrangement is BD = r_max.

By geometric derivation, $r_{\max }$ can be determined using Equation (3), as follows:

$$r_{\max }=\sqrt{a^2+b^2}-b$$

(3)

The relationship between $\tan {\theta }_1$ and the radii a and b of the long and short axes of the ellipsoid can be derived mathematically. The final solution is given by Equation (4), as follows:

$$\tan {\theta }_1={\displaystyle \frac{a^2}{b^2}}$$

(4)

When a = b, the ellipsoid degenerates into a sphere and ${\theta }_1$ is 45°, which is consistent with the previously obtained calculation results.

2.2.2. Theoretical Minimum Pore Radius

Under ideal conditions, the particles in the filter layer that form the minimum effective pore radius are arranged as shown in Figure 4. The long-axis length of the filter particles OA is represented by a; the short-axis radius OB is represented by b. The effective pore radius of the filter in this particle arrangement is denoted as AD = $r_{\min }$.

Through geometric derivation, $r_{\min }$ can be expressed by Equation (5), as follows:

$$r_{\min }=\sqrt{a^2+{\displaystyle \frac{1}{3}}{b}^2}-a$$

(5)

2.2.3. Particles in a Loose Arrangement

Figure 3 and Figure 4 illustrate the maximum and minimum effective pore sizes that can occur in the filter layer under ideal conditions; these are seldom observed in practice and represent extreme scenarios. When the overall particle distribution of the filter layer is considered, the stability of the particle arrangement should also be taken into account. An unstable particle arrangement (Figure 3) is unlikely to be prevalent within the filter layer. Therefore, the loose arrangement and tight arrangement of the particles in the filter layer are depicted as illustrated in Figure 5 and Figure 6, respectively.

Figure 5 illustrates the loose arrangement of particles in the filter layer. In an approach similar to the solution process described in Section 2.2.1, the maximum effective pore radius and tanθ₂ value of the filter can be derived as in Equations (6) and (7), as follows:

$$r=\sqrt{b^2+{\displaystyle \frac{1}{3}}{a}^2}-b$$

(6)

$$\tan {\theta }_2={\displaystyle \frac{\sqrt{3}}{3}}{\displaystyle \frac{a^2}{b^2}}$$

(7)

The three-dimensional spatial structure of the particles in the filter layer under this particle arrangement is expected to resemble the structure of a regular tetrahedron. If all the particle sizes in the filter layer are the same, the porosity of the filter layer in three-dimensional space is given by Equation (8), as follows:

$$n=1-{\displaystyle \frac{\pi a{b}^2}{2\sqrt{6}{\left(\sqrt{b^2+\frac{1}{3}{a}^2}\right)}^3}}{\displaystyle \frac{6\arccos \frac{1}{3}}{2\pi }}$$

(8)

Assuming that the ratio of the long axis to the short axis is a/b = λ, the porosity n of the soil in a loose arrangement for different values of λ is calculated. The results of these calculations are presented in

Table 1. The relationship between λ and n in a loose arrangement.

Ratio of the Long Axis to the Short Axis, λ	1	1.1	1.2	1.4	1.5	1.7	2	2.5	3	3.5	4	5
Porosity, n	0.510	0.501	0.498	0.503	0.512	0.534	0.577	0.652	0.717	0.770	0.811	0.868

. As shown in Table 1, the porosity of the soil in a loose arrangement is minimized at a λ value of 1.2, with a porosity of 0.498. For λ < 1.5, the porosity of ellipsoidal particles is less than that of spherical particles. When λ > 1.5, the porosity of ellipsoidal particles exceeds that of spherical particles. Additionally, the porosity increases with λ, with the growth rate initially rising and then decreasing, as shown in Figure 7.

However, it should be pointed out that the porosity calculated here is based on a single particle size. Due to the presence of particles with varying sizes in the actual filter layer, small particles will fill the pores of larger particles and the particle arrangement may not be loose. Consequently, the porosity is expected to be lower than that calculated here.

2.2.4. Particles in a Close-Packed Arrangement

Figure 6 illustrates the close arrangement of particles in the filter layer. In this case, let $\angle \mathrm{EDF}={\theta }_3$; then, the maximum effective pore radius of the filter can be obtained by Equation (9), as follows:

$$r=\left(\sqrt{3}-1\right)\cdot b-{\displaystyle \frac{a}{\tan {\theta }_3}}$$

(9)

3. Ellipsoidization of Particles of Arbitrary Shape

3.1. Calculation of Particle Shape Ellipsoidization

The shape of the actual filter-layer particles varies, and it is almost impossible for them to have a regular spherical or ellipsoidal form. Therefore, it is necessary to characterize the particle shapes using specific shape parameters and attempt to approximate the irregular shapes of the filter-layer soil particles as ellipsoids.

Here, the plane projection area of the filter-layer soil particles is denoted as A and the perimeter as P. The projection is approximated as an ellipse with a constant area and perimeter, and the radii of its long and short axes are defined as a and b, respectively. From the formulas for the area and perimeter of an ellipse, the following relationships can be derived from Equation (10), as follows:

$$\begin{array}{l}\pi ab=A\\ 2\pi b+4\left(a-b\right)=P\end{array}$$

(10)

The value of the radius b of the short axis can be obtained by Equation (11), as follows:

$$b={\displaystyle \frac{P-\sqrt{P^2-4\left(2\pi -4\right)\frac{4A}{\pi }}}{2\left(2\pi -4\right)}}={\displaystyle \frac{P-\sqrt{P^2-32\frac{\left(\pi -2\right)}{\pi }A}}{2\left(2\pi -4\right)}}$$

(11)

When the particle shape is spherical with a radius r, the formulas for the area and perimeter of a projection circle can be substituted into Equation (11) to obtain b = r, which confirms the validity of the formula.

At this time, the long-axis radius a can be obtained by Equation (12), as follows:

$$a={\displaystyle \frac{A}{\pi b}}={\displaystyle \frac{2A\left(2\pi -4\right)}{\pi P-\pi \sqrt{P^2-32\frac{\left(\pi -2\right)}{\pi }A}}}$$

(12)

The prerequisite for deriving Equation (11) is Equation (13), as follows:

$$P^2-32{\displaystyle \frac{\left(\pi -2\right)}{\pi }}A\ge 0$$

(13)

This can also be expressed by Equation (14), as follows:

$${\displaystyle \frac{32\left(\pi -2\right)A}{\pi P^2}}\le 1$$

(14)

For a given area, a circle has the smallest circumference, resulting in the maximum value of Equation (14). Therefore, we can obtain Equation (15), as follows:

$${\displaystyle \frac{32\left(\pi -2\right)\pi r^2}{\pi {\left(2\pi r\right)}^2}}={\displaystyle \frac{8\left(\pi -2\right)}{{\pi }^2}}\le 1$$

(15)

Specifically, for the plane projection of any particle shape, Equation (11) can be applied. Consequently, the plane projection of any particle shape can be approximated as an ellipse using the semi-major and semi-minor axes defined by Equations (11) and (12).

3.2. Ellipsoidization Method Associated with Particle-Analysis Results

The grading curve of particle analysis is determined by the screening method. Assuming that the particle shape is a sphere, the diameter of the sphere represents the screening particle size. If the particle shape is assumed to be an ellipsoid, the screening particle size should be defined as the short-axis diameter of the ellipsoid particle, specifically 2b. At this time, the radii of the long and short axes of the ellipsoid can be determined using the following two methods:

(1) By comparing the particle size obtained from the analysis with the short-axis radius previously determined by Equation (11); if the difference is minimal, the short-axis radius can be approximated using the particle-size results, while the long-axis radius can still be calculated using Equation (12);

(2) When the particle size obtained from the analysis is treated as the short-axis radius, the long-axis radius can then be determined by calculating the projection area. Using this method to determine the radii of the long and short axes, the area of the figure before and after conversion to an ellipse remains constant, while the perimeter changes.

3.3. The Particle-Shape Parameter

It can be seen from the previous article that the value on the left side of Equation (14) is minimized when the particle shape is spherical, thus allowing for the evaluation of particle shape using this formula. From Equation (15), the value of Equation (14) can be derived when the particle shape is circular. Equations (14) and (15) are combined to define a particle-shape parameter S, such that the particle shape is circular when S = 1. The parameters are defined by Equation (16), as follows:

$$S={\displaystyle \frac{32\left(\pi -2\right)A}{\pi P^2}}\cdot {\displaystyle \frac{{\pi }^2}{8\left(\pi -2\right)}}={\displaystyle \frac{4\pi A}{P^2}}$$

(16)

The value range of S is (0,1]. The closer S is to 1, the more closely the particle projection shape approximates a circle.

The shape parameter S presented here is the square of the shape parameter S₁₁ in ref. [21], though the methods for obtaining the two parameters differ. In ref. [21], S₁₁ is defined as the ratio of the circumference of a circle with an area equal to that of the circumference of the projection.

The particle-shape parameter S was calculated for different values of λ, and the results are presented in

Table 2. The particle-shape parameter S corresponding to the ellipsoid for different values of λ.

Ratio of the Long Axis to the Short Axis (λ)	1	1.2	1.5	1.8	2	2.3	2.5	3	4	5
Particle-shape parameters (S)	1	0.944	0.863	0.790	0.747	0.689	0.654	0.581	0.472	0.398

. Additionally, the relationship between S and λ is illustrated in Figure 8. From Figure 8, it can be observed that before λ reaches 2.5, the relationship between S and λ can be considered approximately linear. However, as λ increases further, the rate of change in S gradually decreases.

In ref. [26], the shape parameters of soil particles with varying particle sizes were computed. In this paper, the parameter was converted to the particle-shape parameter S; the mean value is 0.764, and the corresponding λ value is approximately 1.9. This indicates that, generally, the particle-shape parameter S of soil particles can be regarded as linear in relation to the ratio of the long axis to the short axis, λ. Therefore, the data with λ values less than 2.5 in Table 2 were selected for linear regression analysis. When λ values are less than 2.5, the relationship between S values and λ values can be approximately represented by Equation (17), as follows:

$$S=1.217-0.232\lambda$$

(17)

For particles within a specific size range, a sample of representative particles can be analyzed to determine the S value. The average of these shape parameters can then serve as the reference value for the particle-shape parameters within this size range [20,27].

4. Construction of Ellipsoidal Particle Unit in Particle-Flow Software

The following model was constructed using the particle-flow software PFC 5.0 to verify the influence of particle shape on the seepage characteristics of the filter layer. Because ellipsoid particles cannot be directly created within PFC, it was necessary to utilize the clump unit within PFC to create an approximate ellipsoid by combining spheres and then employ circular splicing to create an approximate ellipse in the plane projection.

4.1. Basis for the Construction of Ellipsoid Particles

When the ellipsoidal model is compared with the pre-ellipsoidal model, the volume of the ellipsoidal pattern must be equal to the volume of the spherical particles to ensure that the model’s porosity remains unchanged during the transformation of the particles into ellipsoids. As shown in Figure 9, a particle with radius r in the original model was transformed into a model characterized by a long axis a and a short axis b.

Let the ratio of the long axis to the short axis be a/b = λ. This method for constructing the ellipsoid unit must satisfy 1 ≤ λ ≤ 2. If λ exceeds 2, the ellipsoid unit must be reconstructed. Otherwise, errors will occur in the calculation of particle volume in PFC, which will alter the porosity of the model.

The radii of the large and small circles required to construct the ellipse are determined by the values of a and b. Consequently, an elliptical particle model with constant porosity can be approximately constructed in PFC.

To construct ellipsoidal particles in PFC, the ratio of the long axis to the short axis, λ, must be determined. In this study, tailings-sand particles with a size range of 2–5 mm were scanned to obtain their projection images. Particle shapes were analyzed using software ImageJ (https://ij.imjoy.io/, accessed on 17 April 2025) to obtain size information for each particle. Subsequently, the corresponding λ parameters were determined by calculating the ovalization of each particle. The mean value of λ for each particle was used as the basis for the λ parameter in the construction of ellipsoidal particles in PFC. The tailings used for scanning were screened from the Lixigou tailings reservoir.

4.2. Image Acquisition and Information Extraction

Figure 10 shows the shapes of tailings-sand particles ranging from 2 to 5 mm. The parameters of particle projection were analyzed using image analysis software ImageJ. Details of the specific process are outlined below.

Figure 11 shows the black and white binary image obtained by processing Figure 10, This image clearly displays the projection shape of each particle, with the lighter areas representing the particles. ImageJ software can automatically detect the boundaries of each individual particle and assign the particles unique labels, facilitating the subsequent extraction of their shape parameters.

Table 3. Graphical information processing.

Number	Particle Areas and Perimeters in ImageJ Software		The Real Areas and Perimeters of the Particles
	Area	Perimeter	Area (mm2)	Perimeter (mm)
1	7645	356.617	10.333	13.111
4	6855	335.765	9.266	12.344
8	5486	287.279	7.415	10.562
13	4659	274.392	6.297	10.088
average	7137.69	331.552	9.648	12.189

shows the area and perimeter of selected particles analyzed using ImageJ software. The parameters in the software were measured in image pixels. To determine the specific size, a reference size had to be established in the image to determine the proportional relationship between pixel dimensions and actual dimensions, allowing for the calculation of the true shape parameters of the particles. The average value in Table 3 represents the mean of each parameter for all 13 particles.

The reference size in Figure 10 was determined by the width of the slide, which measured 25.4 mm. By comparing the actual size of the slide with the pixel width of the picture, the particle pixel sizes in Table 3 could be converted into actual sizes. The average value in Table 3 represents the mean of each parameter for all 13 particles.

4.3. Shape Parameters of Ellipsoidal Particles

The area and perimeter of each particle are presented in Table 3. The radii of the long and short axes of each particle after ellipsoidization were determined using Equations (11) and (12); from these values, the corresponding ratio of the long axis to the short axis λ and the particle-shape parameter S were derived, as shown in

Table 4. Particle-shape parameters after ellipsoidization of each particle.

Number	Equivalent Elliptic Diameter (mm)		Ratio of the Long Axis to the Short Axis λ	Particle-Shape Parameters S
	Long Axis	Short Axis
1	6.883	2.949	2.334	0.682
4	4.756	2.480	1.918	0.764
8	3.899	2.421	1.610	0.835
13	5.076	2.592	1.958	0.755
average	4.690	2.461	1.906	0.769

.

The scanned particles had a size range of 2–5 mm, and the shape parameters in Table 4 correspond to this particle-size range. (The average values presented in Table 4 represent the means of the parameters for all 13 particles.) Figure 12 shows that the ratio of the long axis to the short axis λ for each particle is approximately linearly related to the corresponding particle-shape parameter S, which is consistent with the conclusion drawn in Section 3.3.

Based on the calculation results presented in Table 4, λ = 1.9 was used as the basis for constructing ellipsoidal particles in PFC, with the corresponding particle-shape parameter being 0.769.

5. Analysis of Numerical-Simulation Results

The spherical particles and ellipsoidal particles were utilized to construct a three-dimensional filter-layer soil column model with a height of 20 cm and a diameter of 10 cm in PFC. Tailings-sand particles of varying sizes were then added above the soil column for a muddy-water-seepage simulation test. The particle size of the filter layer was 2–5 mm, with spherical and ellipsoidal particle shapes, respectively, used for modeling. The ratio of the long axis to the short axis of the ellipsoidal particles, λ, was 1.9. The tailings-sand muddy-water particles were assumed to be uniformly spherical, with particle sizes of 0.5–1 mm, 0.25–0.5 mm, and 0.075–0.25 mm. The walls were designated as impermeable. The model contained a total of 5 × 5 × 10 fluid units, with each fluid unit measuring 2 cm × 2 cm × 5 cm. The initial porosity of the model was set at 0.4, with a water pressure of 20 kPa applied at the top. The seepage direction was from top to bottom. A layer of filter with a pore size of 1.25 mm was placed at the bottom of the model.

5.1. Comparison of Seepage-Simulation Test Results for Filters with Different Particle Shapes

Figure 13 shows the muddy-water seepage of 0.5–1 mm tailings-sand particles through the spherical-particle filter layer and the ellipsoidal-particle filter layer, respectively. In the Figure 13, Figure 14 and Figure 15, green particles represent filter layer particles and red particles represent tailing sand particles. Comparing the seepage and clogging of tailings-sand muddy-water particles under both working conditions, it can be seen that 2–5 mm filter layers with different particle shapes effectively filtered out 0.5–1 mm tailings-sand particles. Most of the tailings-sand muddy-water particles accumulated on the upper surface of the filter layer or were located at a depth of 15–20 cm within the filter layer, and no particles penetrated the filter layer. However, the muddy-water particles of tailings sand in the spherical-particle filter layer were located significantly deeper than those in the ellipsoidal-particle filter layer.

Figure 14 shows the muddy-water seepage of 0.25–0.5 mm tailings-sand particles through the spherical-particle filter layer and the ellipsoidal-particle filter layer, respectively. Unlike 0.5–1 mm tailings-sand particles, 0.25–0.5 mm particles almost all entered the filter layer during seepage, with most particles remaining at a depth of 10–20 cm within the filter layer and a small number of particles penetrating the filter layer (the particles located below the filter layer indicate those that have seeped through the filter layer). The intrusion depth of tailings-sand muddy-water particles was significantly greater in the spherical-particle filter layer than in the ellipsoidal-particle filter layer, and the number of seepage particles that penetrated the filter layer was also considerably higher than that in the ellipsoidal-particle filter layer.

Figure 15 shows the muddy-water seepage and clogging of 0.075–0.25 mm tailings-sand particles through the spherical-particle filter layer and the ellipsoidal-particle filter layer, respectively. The intrusion depth of 0.075–0.25 mm tailings particles through both the spherical and ellipsoidal-particle filter layers is nearly identical, indicating that for these tailings particles, the pores of the 2–5 mm filter layer were too large to provide effective protection. Furthermore, the different shapes of filter-layer particles have no significant impact on their behavior.

The filter layer was divided into four layers, and the number of muddy-water particles within each of the four filter layers, as well as those above and those that penetrated the filter layer during the seepage process, was recorded to further analyze the filtering characteristics of layers with different particle shapes.

Figure 16, Figure 17 and Figure 18 present the statistical results for tailings-sand particles of various sizes (specific proportions are omitted in the figure when they are less than 1%). It can be observed that as the particle size of tailings sand decreased, the proportion of particles infiltrating the filter layer and the depth of infiltration both increased. With the exception of 0.075–0.25 mm tailings-sand particles, for other particle sizes, the proportion of infiltrating particles and the depth of infiltration for the spherical-particle filter layer were greater than those for the ellipsoidal-particle filter layer.

Figure 16, Figure 17 and Figure 18 illustrate various forms and the evolution of clogging in the filter layer during muddy-water seepage. For the uniform filter layer ranging from 2 to 5 mm, 0.5–1 mm tailings-sand particles were concentrated above the filter layer and at a depth of 15–20 cm within the first layer of the filter layer, indicating surface-internal deposition. Most of the 0.25–0.5 mm tailings-sand particles were retained at a depth of 15–20 cm within the first layer of the filter layer, while a small amount continued to migrate to 10–15 cm within the second layer, indicating internal clogging. For the 0.075–0.25 mm tailings particles, although Figure 18 suggests that the clogging morphology was similar to that of the 0.25–0.5 mm tailings particles, this similarity arises from the significant size difference between the 0.075–0.25 mm particles and the filter layer. The finite element software processes data very slowly, making it challenging to reach an equilibrium state, and observations and calculations indicate that the infiltration of tailings particles of this size exerts minimal influence on the filter layer’s permeability coefficient, which is why only a portion of the calculation results are presented. In fact, because the tailings-sand particles were significantly smaller than the pore diameter of the filter layer, the filter layer cannot provide effective filtration of the tailings-sand particles; the result is the eventual passage of tailings-sand particles through the filter layer, a form of seepage failure.

Through Equations (1)–(3) and (5), the maximum and minimum effective pore radii for both the spherical-particle filter layer and the ellipsoidal-particle filter layer were calculated. The theoretical effective pore radius for the spherical-particle filter layer was 0.31–2.07 mm, whereas that for the ellipsoidal-particle filter layer was 0.11–4.12 mm. It can be observed that the effective pore radius of the ellipsoidal-particle filter layer exhibited a wider range of variation and that the maximum effective pore radius of the ellipsoidal-particle filter layer exceeded that of the spherical-particle filter layer. The observations from Figure 16, Figure 17 and Figure 18 indicate that the effective pore radius of the ellipsoidal-particle filter layer was greater than that of the spherical-particle filter layer. This suggests that the permeability of the ellipsoidal-particle filter layer was likely greater than that of the spherical-particle filter layer, a conclusion that was validated through the analysis of the permeability coefficient of the filter layer.

5.2. The Process of Changing the Permeability Coefficient of Filter Layers with Different Particle Shapes

The permeability coefficient of the model cannot be directly obtained using PFC and must be calculated by employing appropriate theoretical formulas. In this paper, the Darcy permeability coefficient Equation (18) was employed to estimate the permeability coefficient, and the results of these calculations were subsequently analyzed.

$$k=v/i$$

(18)

In the equation, v represents the seepage velocity, while i denotes the hydraulic gradient.

The calculated permeability coefficients for filters with varying particle shapes are presented in Figure 19, Figure 20 and Figure 21. In the figures, SP represents a spherical-particle filter layer, whereas EL denotes an ellipsoidal-particle filter layer. Analysis of Figure 19, Figure 20 and Figure 21 indicates that, based on the results derived from Darcy’s formula, the permeability coefficient of the spherical particle model is lower than that of the ellipsoidal particle model.

5.3. Indoor Test Verification

In this paper, three sets of indoor muddy-water seepage tests were conducted under the same conditions used for the prior numerical simulation using a custom-designed muddy-water seepage apparatus. The test apparatus is illustrated in Figure 22.

The preparation material for the muddy water consisted of tailings-sand particles sourced from the dry beach of the Lixigou tailings dam, while the filter layer was composed of spherical transparent glass beads. The particle sizes of the tailings sand were 0.5–1 mm for Test 1, 0.25–0.5 mm for Test 2, and 0.075–0.25 mm for Test 3.

The test procedure was as follows:

(1) Filling of the filter layer: Before the filter layer was loaded, a 10 cm buffer layer of coarse particles was laid, and water is added to a depth of 10 cm to saturate the buffer layer. Then, 20 cm of filter layer was laid on top of the buffer layer and filled in four layers. Each layer was saturated before the next was added.

(2) Instrument assembly: The main body of the instrument was connected to the water tank and the pressure-measuring tube. Fresh water was added to the water tank, and the valve was opened. The test is formally started after the head of the pressure measuring tube was stabilized.

(3) The test began: At the start of the muddy-water percolation test, the anti-filtration layer was saturated before the first vacuum, and this step was followed by 30 min of clear-water percolation. During this period, the stability of the head of the pressure-measuring tube was observed and the flow rate was recorded every 5 min. Once a stable seepage field was formed in the anti-filtration layer, tailings-sand particles were added to the water tank to start the muddy-water percolation test. Throughout the muddy-water percolation test, continuous stirring of the tailings water in the tank was required to maintain a constant concentration of the test water.

(4) Data collection: After the start of the turbid-water seepage test, pressure-tube readings were taken every 5 min and the flow rate was measured. Subsequently, the time intervals were gradually increased to 30–60 min.

(5) At the end of the test, the filter layers tested at different locations were sampled in layers and dried separately. The particles of the filter layer were then separated from the tailings-sand particles and weighed individually.

The results of the indoor tests are presented in Figure 23.

The results of the indoor tests aligned with those of the numerical simulations. In terms of their influence on the clogging of the filter layer, the variation in 0.075–0.25 mm tailings-sand particles was minimal, whereas that in 0.25–0.5 mm particles was maximal. This indicates that, within a certain particle-size range, smaller tailings-sand particles have a greater impact on the permeability of the filter layer, although this influence diminishes beyond a certain threshold. This range can be assessed using the effective pore size of the filter layer. As this factor was analyzed in the previous section, it will not be elaborated here; however, the variation in the permeability coefficient observed in laboratory tests was significantly greater than that derived from numerical simulations. For instance, in Tests 1 and 2, the permeability coefficient of the 15–20 cm filter layer was greater than that of the 10–15 cm layer at the beginning but was lower at the end of the tests. The change in the permeability coefficient in Test 3 was also significantly greater than that of the numerical-simulation results. This discrepancy arises from the differences in the number of tailings-sand particles. In the numerical simulation, to maintain calculation speed, the number of tailings-sand particles must be limited; a larger number would significantly hinder computational efficiency. The model for the 0.075–0.25 mm group, which contained the greatest number of tailings-sand particles, is limited to only 50,000 particles. In the indoor tests, the number of tailings-sand particles was significantly greater, resulting in a different range of variation for the permeability coefficient.

5.4. The Influence of Particle Shape on the Permeability Coefficient of the Filter Layer

The conclusion of the previous section indicates that the permeability coefficient of the spherical-particle filter model is lower than that of the ellipsoidal-particle filter model, a conclusion consistent with findings in the literature [28]. This discrepancy is hypothesized to arise from the varying permeability coefficients of filter-layer models based on different particle-shape parameters. Consequently, this section simulates and validates the muddy-water seepage tests of ellipsoidal-filter-layer models under varying particle-shape parameters.

It has been confirmed that the particle-shape parameter S exhibits a linear relationship with the ratio of the long axis to the short axis, denoted by λ, within a reasonable range. For simplicity, this ratio will be referred to as λ in the following discussion.

Figure 24 shows the permeability coefficient of the model (without the inclusion of tailings-sand particles) under different particle-shape parameters, calculated using Darcy’s formula. With the increase in the ratio of the long axis to the short axis, λ, the permeability coefficient of the model initially decreases and then increases. When the λ value is between 1.3 and 1.4, the permeability coefficient of the model is minimized; at λ values of 1.7 to 1.8, the permeability coefficient reaches its maximum, and subsequently, the rate of change in permeability coefficient diminishes. The value of λ = 1.6 can be regarded as a boundary for the permeability coefficient of the model. Prior to this point, the permeability coefficient of each particle shape model is lower than that of spherical particles; beyond this threshold, it becomes greater.

The difference in the calculation results of the permeability coefficient for different particle-shape parameters is illustrated by Figure 25. When the particle-shape parameter is close to 1, as illustrated on the left side of Figure 25, the pore size of the filter layer decreases with decreasing particle-shape parameters, corresponding to the reduction in the permeability coefficient shown in Figure 24. After reaching a certain critical value, as the particle-shape parameter further decreases, as shown on the right side of Figure 25, the sizes of the pores of the filter layer begin to increase gradually, which corresponds to the subsequent increase in permeability coefficient observed in Figure 24.

5.5. Prediction of the Effect of Particle Shape on the Pore Structure of the Filter Layer

In Section 2.2.3, the pore structure of a filter layer with different particle shapes given a loose arrangement of single-size particles was examined. It was observed that under this arrangement, the porosity of the filter layer reached its minimum at λ = 1.2. Furthermore, for λ values less than 1.5, the porosity of the ellipsoidal-particle filter layer was lower than that of the spherical-particle filter layer. When λ exceeded 1.5, the porosity of the ellipsoidal-particle filter layer surpassed that of the spherical-particle filter layer, with an initial increase in growth rate that was followed by a decrease.

In Section 5.4, the seepage performance of the mixed-tailings model with different particle sizes and filter layers with varying particle shapes was investigated. It was observed that the permeability coefficient of the filter layer model was minimized when λ was between 1.3 and 1.4. For λ values less than 1.6, the permeability coefficient of the ellipsoidal-particle filter layer model was lower than that of the spherical-particle filter layer. Conversely, for λ values greater than 1.6, the permeability coefficient of filter-layer models for all particle shapes exceeded that of the spherical-particle filter layer. When λ was between 1.7 and 1.8, the permeability coefficient of the filter layer model reached its maximum; this was followed by a deceleration in the rate of change in the permeability coefficient.

The conclusions of these two sections are consistent; however, the λ values differ for two reasons. In Section 2.2.3, a filter layer with a single particle size was investigated, while in Section 5.4, a filter layer with a certain particle-size range was analyzed. Second, the ellipsoidal particles in Section 5.4 were approximated using spherical particles, while Section 2.2.3, by contrast, describes an approximation using ellipsoidal particles.

In these two sections, the shape of ellipsoidal particles was examined and described by λ; however, for various atypical particles (such as those in Figure 10), this representation is inadequate. Thus, these particles were characterized using the particle-shape parameter S introduced in Section 3.3.

A comprehensive analysis of the contents of these two sections reveals that the particle shape does have an impact on the pore structure. It is reasonable to speculate that irregular natural particles of various shapes may also influence the pore structure of the filter layer. With the decrease in the value of the particle-shape parameter S (corresponding to the increase in the value of the ratio of the long axis to the short axis, λ), the porosity of the filter layer initially decreased and subsequently increased, although the rate of increase gradually diminished. There exists a critical particle-shape parameter, S_a. Under the same particle arrangement, when S is less than S_a, the pore size of the ellipsoidal filter layer is smaller than that of the corresponding spherical filter layer. Conversely, when S exceeds S_a, the pore size of the ellipsoidal filter layer becomes larger than that of the corresponding spherical filter layer.

6. Conclusions

This paper primarily investigates the impact of particle shape on the pore structure and filtration characteristics of the filter layer. This paper primarily presents the following conclusions:

(1) The effective pore radius of the ellipsoidal particles in various spatial arrangements was calculated, and the relationship between the porosity of the filter layer and the ratio of the major to minor axes of the ellipsoidal particles in a loosely arranged configuration was established.

(2) A method for ellipsoidalizing arbitrarily shaped particles is presented, and the particle-shape parameter S was defined to characterize the shape of these particles.

(3) The relationship between the particle-shape parameter S of ellipsoidal particles and the ratio of the major axis to the minor axis λ is summarized. Within a certain range, the particle-shape parameter S is approximately linear with respect to the ratio of the major axis to the minor axis λ.

(4) The basis and construction method for creating ellipsoidal particles in particle-flow software are presented. Considering actual tailings particles from Jinduicheng Molybdenum Industry, the particle shape was analyzed. The foundational parameters for the construction of ellipsoidal particles are provided.

(5) In the particle-flow software, the differences between spherical particles and ellipsoidal particles in the muddy-water seepage simulation tests were compared and analyzed. The analysis revealed that the particle shape influenced the permeability coefficient of the filter-layer model. As the ratio of the major axis to the minor axis λ increased, the permeability coefficient of the filter layer model first decreased and then increased, as verified by laboratory tests.

(6) Through the analysis of the pore structure of spherical and ellipsoidal particles in this paper, the relationship between the pore structure and the particle-shape parameters of the irregular natural-particle filter layer with arbitrary shapes was predicted. It is proposed that there exists a boundary particle-shape parameter S_a. Under the same particle arrangement, when S is less than S_a, the pores of the filter layer are smaller than those of the corresponding spherical-particle filter layer. When S is greater than S_a, the pore size of the filter layer is larger than that of the corresponding spherical-particle filter layer.