Discrete Element Method Simulation of Loess Tunnel Erosion

Abstract

The phenomenon of tunnel erosion is quite common in the Loess Plateau. Tunnel erosion can cause disasters such as landslides, mudslides, and ground collapses, resulting in significant economic losses and posing a threat to people’s safety. Therefore, understanding the evolution mechanism of tunnel erosion not only helps to analyze and predict the development law of erosion but also has a certain guiding role in engineering activities. Many scholars (including our team) have conducted field investigations and statistical analysis on the phenomenon of tunnel erosion in loess; however, these studies still have shortcomings in visual quantitative analysis. The combination of the Discrete Element Method (DEM) and Computational Fluid Dynamics (CFD) has significant advantages in studying soil seepage and erosion. Based on existing experimental research, this article combines the Discrete Element Method (DEM) with Computational Fluid Dynamics (CFD) to establish a CFD-DEM coupled model that can simulate tunnel erosion processes. In this model, by changing the working conditions (vertical cracks, horizontal cracks, and circular holes) and erosion water pressure conditions (200 Pa, 400 Pa, 600 Pa), the development process of tunnel erosion and changes in erosion rate are explored. The results indicate that during the process of fluid erosion, the original vertical crack, horizontal crack, and circular hole-shaped tunnels all become a circular cave. The increase in erosion water pressure accelerates the erosion rate of the model, and the attenuation rate of the particle contact force chain also increases, resulting in a decrease in the total erosion time. During the erosion process, the curve of the calculated erosion rate shows a pattern of slow growth at first, then rapid growth, before finally stabilizing. The variation law of the erosion rate curve combined with the process of tunnel erosion can roughly divide the process of tunnel erosion into three stages: the slow erosion stage, the rapid erosion stage, and the uniform erosion stage.

1. Introduction

The Loess Plateau, located in the upper and middle reaches of the Yellow River, is an extremely unique terrain unit area with a coverage area of up to 624,000 km². It is the region with the largest thickness and continuous distribution of wind-blown loess in the world. Due to year-round agricultural activities, soil erosion in this region is one of the most severe in the world, with an average annual soil loss of up to 15,000 t/km² [1,2,3,4]. Years of soil erosion have led to the interweaving of gullies on the Loess Plateau, and cavities formed by erosion can be seen everywhere. These erosion cavities not only lead to a reduction in farmland but also pose a threat to various engineering structures [5,6,7,8]. More seriously, soil erosion can trigger geological disasters such as landslides, mudslides, ground fissures, and ground collapses. Therefore, understanding the formation and evolution process of tunnel erosion is very important. It not only provides guidance for the widespread geological disasters on the Loess Plateau but also enables the development of effective measures for soil erosion prevention and control based on research results, which is beneficial for guiding engineering practice.

In tunnel erosion, also known as pipeline erosion [9,10,11], the erosive water flow preferentially erodes in soil fissures or weak areas, forming small fissures or cavities that gradually transform into large caves. Tunnel erosion is an important soil erosion process [12,13], and many scholars have conducted related research [14,15,16]. These studies often focus on the fields of agriculture and geomorphology, with a focus on changes in sediment profiles, sand migration, and the impact on the surrounding environment. The data used in research are usually based on observations of a single tunnel, with a slightly small sample size and a lack of research on engineering geology issues [17,18,19]. In nature, the process of tunnel erosion is extremely complex and slow, lasting for several decades. Complexity and unpredictability are the challenges of this research. The physical model experiment in the laboratory can be accelerated by artificially changing the sample size, hydraulic gradient, and other conditions to observe the process of tunnel erosion. Some scholars have studied the changes in soil infiltration and erosion rate under different hydraulic gradient conditions by enhancing the hydraulic gradient and summarizing the erosion laws of soil [20,21]. But as mentioned above, the process of tunnel erosion is complex and unpredictable, and the results obtained from these physical model experiments are different from the real situation.

In recent years, computer technology has developed rapidly, and various numerical simulation software have emerged one after another. Numerical simulation methods have also been applied to study soil erosion problems. Among various numerical simulation methods, the Discrete Element Method (DEM) is an excellent and commonly used method. As a discontinuous analysis method, the advantage of DEM is that it can capture the motion process of particles. The coupling method of the Discrete Element Method (DEM) and Computational Fluid Dynamics (CFD) can accurately simulate the erosion process of soil. Many scholars use the CFD-DEM method to study soil erosion from different directions. Chen et al. [22] studied the erosion process of granular soils with different particle size distributions using a CFD-DEM coupled model and recorded changes in particle displacement, the force chain network, and stress anisotropy to analyze the microstructural changes of soil samples before and after erosion. Wang et al. [23] investigated the impact of particle size distribution changes on soil erosion and established a predictive model for the soil erosion rate, accurately predicting the effect of soil particle size on erosion rate. Zhang et al. [24] established a CFD-DEM coupled model to model the pipeline erosion of seabed sediment and calculated the critical erosion velocity of sediment erosion.

In summary, both physical model experiments and discrete element numerical simulations have achieved certain results in the study of tunnel erosion. However, existing research still has certain shortcomings. Firstly, there has been relatively little research on the erosion of loess tunnels, whether through physical model experiments or numerical simulations. Secondly, there is a lack of research combining examples and numerical simulations in the study of tunnel erosion issues. In order to fill this gap, the main purpose of this study is to establish a CFD-DEM coupled model to study the erosion process of loess tunnels and to compare and analyze it with our previous physical model experiments. This method of combining physical experiments with numerical simulation has more advantages in quantitative analysis. This article first introduces the methods of physical experiments and the design of the operating conditions. Then, while explaining the CFD-DEM method, a numerical model is established to characterize the erosion process of loess tunnels. The model is calibrated based on experimental cases, including the selection of contact parameters and comparison and verification with experimental results. Finally, quantitative analysis and discussions were conducted on the numerical simulation results, and the evolution process of the mechanical field during tunnel erosion was visualized. This analysis will help prevent tunnel erosion in the Loess Plateau of China and other loess regions around the world, as well as reduce damage to related geotechnical structures.

2. Materials and Methods

2.1. Materials

This study is a numerical simulation implementation of existing experimental research, which can be referenced in the literature [25]. During field research, we found that tunnel erosion in the Loess Plateau is quite serious, with vertical caves exposed on the surface and horizontal caves located on slopes visible everywhere (as shown in Figure 1). In response to the widespread problem of tunnel erosion in the Loess Plateau, based on field observations, indoor erosion experiments were designed for three working conditions: circular holes, longitudinal holes, and transverse holes (as shown in Figure 2). The soil sample used in the experiment was sourced from Malan loess Q₃ in Wanrong Township, Shanxi Province, and its physical properties are shown in

Table 1. Loess properties at the in situ experimental site.

Natural Density (g/cm3)	Natural Water Content (%)	Clay (mm), <0.005	Silt (mm), 0.075~0.0005	Sand (mm), 2~0.075
1.53	7.5	15.6	70.5	13.9

. The density of the soil sample is 1.53 g/cm³, and the natural moisture content is 7.5%. The particle size distribution of the soil sample is shown in Figure 3. In terms of particle size composition, Malan loess Q₃ is mainly composed of silt particles (0.005–0.075 mm), with a content of about 70%. The content of sand particles larger than 0.075 mm and clay particles smaller than 0.005 mm is relatively low, accounting for about 15%. The equipment used in the research institute is shown in Figure 4, consisting of water supply equipment, soil samples, and limit plates. In Wang’s study, the dry density and moisture content of loess samples were changed to investigate the influence of differences in loess physical properties on the development process and subsurface erosion rate of erosion tunnels, and relatively complete results were obtained. However, due to limitations in experimental conditions, Wang’s study [25] did not explore the impact of changes in hydraulic pressure on the development of erosion tunnels. In order to fill this gap, this article uses the Discrete Element Method to conduct further numerical simulation research.

2.2. Overview of Discrete Element Method

The numerical simulation used in this study is the discrete element program PFC, which has the advantage of characterizing the laws of particle motion and their mutual contact. It has been widely used in the field of geotechnical engineering and can effectively simulate discontinuous phenomena such as material cracking and separation, with good application prospects. The software has multiple built-in contact models to express the mechanical strength parameters of different materials. Loess has a loose structure and weak cementation; therefore, the parallel bonding model (linear-pbond) is selected for model establishment. PFC comes with a fluid dynamics calculation module (CFD), which allows for the faster and simpler establishment of CFD-DEM coupled models for fluid particle interactions [26,27].

The basic principle of the CFD-DEM coupling model is the interaction between fluid and particles. The conceptual model diagram of the interaction between fluid and particles is shown in Figure 5. In Figure 5, the fluid flows through the particles and gradually dissolves the clay between the particles (as shown in Figure 5a). After the clay is completely dissolved, the particles will move; that is, the particles will be eroded (as shown in Figure 5b).

When running the CFD-DEM coupled model, the first step is to establish a DEM module, adjust the model size, and determine the contact parameters between particles. At the same time, the CFD module also needs to determine the size of the fluid grid and guide the equation. After the model preparation work is completed, the CFD module is started to calculate the changes in porosity and particle velocity during the particle erosion process. Afterward, the model will cycle through this process until the particles of the model are completely eroded, interrupting the cycle.

The erosion of soil particles in CFD modules follows the following formula (Equation (1)):

$$\xi =\left\{\begin{array}{cc}{k}_{\mathrm{er}}{(\left|\tau \right|-{\tau }_0)}^{\mathrm{b}}& if\left|\tau \right|>{\tau }_0,\\ 0\end{array}\right.$$

(1)

In the formula, ξ (kg·m⁻²·s⁻¹) represents the erosion rate; k_er (s/m) is the surface erosion coefficient, which is derived from experience; τ₀ is the critical shear stress; b is a constant, usually taking the value 1. τ is the fluid shear stress. When the fluid shear stress τ experienced by particles is greater than the critical shear stress τ₀, soil erosion occurs, and the value of erosion rate ξ is obtained. Conversely, when particles are not eroded, the erosion rate ξ is 0. The fluid shear stress τ is determined by the unidirectional shear stress τ_i, as shown in Equation (2) below.

$${\tau }_i={\displaystyle \frac{\mathsf{\Delta }P}{D}}\sqrt{{\displaystyle \frac{2K}{n}}}\left(i=1,2,3,\cdots \right)$$

(2)

In the formula, ΔP (Pa) represents the pressure drop. D (m) represents the size of the fluid mesh in the model, and ΔP/D (kN/m³) is the pressure gradient in the fluid cell. n represents the porosity of the fluid; K is the permeability coefficient, determined by the Kozeny Carman relationship, as shown in Equation (3) below:

$$K=\left\{\begin{array}{cc}{\displaystyle \frac{1}{180}}{\displaystyle \frac{n^3}{{\left(1-n\right)}^2}}{\left(d_p\right)}^2& n\le 0.7\\ K\left(0.7\right)& n>0.7\end{array}\right.$$

(3)

Among them, d_p is the effective diameter of the particles, which can be obtained from Equation (4) as follows:

$$d_p={\displaystyle \frac{6}{\eta }}$$

(4)

Among them, η represents the ratio of surface area to volume of particles in fluid cells, and the derivation process is shown in Equation (5) below:

$$\eta ={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^{n_p}4\pi r_i^2}}{{\displaystyle {\sum }_{i=1}^{n_p}4\pi r_i^3/3}}}$$

(5)

In the formula, r_i represents the particle radius, and n_p is the number of particles included in the fluid grid.

By combining Equations (2)–(5), the fluid shear stress τ can be obtained, as shown in Equation (6) below. In the equation, V_i is the volume of the overlapping part between the particle and the fluid grid; V is the volume of the fluid mesh.

$$\tau ={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{V}_i{\tau }_i}}{V}}$$

(6)

Finally, combining Equations (1) and (6) yields the complete formula for particle erosion.

2.3. Model Establishment

Based on the above guidance formula, a preliminary CFD-DEM coupling model can be established (as shown in Figure 6). The size of the model is consistent with the size of the physical experiment (0.06 m × 0.12 m × 0.06 m). The boundary part of the model is a rigid wall, which serves to constrain the left and right of the particles (as shown in Figure 6a). The model is filled with particles, and the particle size distribution of the particles is adjusted according to Table 1. In loess particles, the proportion of silt particles is relatively large, while the proportion of clay particles and sand particles is relatively small. To simplify the model and reduce its computational complexity, the selection of model particles is powder particles, which have been enlarged to a size of 0.001~0.005 m. Figure 6b shows the water pressure model, where the erosion fluid enters from the back end of the model and flows out from the front end. The water pressure can be freely adjusted through the pressure module built into the model. A total of 1024 fluid grids (8 × 16 × 8) are set in the model, with each grid having the same size (0.0075 m × 0.0075 m × 0.0075 m). Figure 6c shows a fluid model with a built-in monitoring module that can monitor the erosion rate changes of particles in real time during the erosion process.

After the initial establishment of the CFD-DEM coupling model, adjustments need to be made according to the experimental conditions shown in Figure 2. The adjusted erosion model of the loess tunnel is shown in Figure 7. Figure 7a shows working condition a, with a particle count of 15899. There is a longitudinal hole (longitudinal crack) in the middle of the model, with a size of 0.015 m × 0.12 m × 0.03 m. The erosion fluid flows out from the hole. Figure 7b shows working condition b, with a particle count of 15876. There is a transverse hole (transverse crack) in the middle of the model, with a size of 0.03 m × 0.12 m × 0.015 m. The erosion fluid flows out of the hole, and the volume of the hole is the same for condition a and condition b. Figure 7c shows working condition c, with a particle count of 14545. There is a circular hole (circular cavity) in the middle of the model, with a diameter of 0.03 m and a length of 0.12 m. The hole in condition c is larger than the holes in conditions a and b. The water pressure models for the three working conditions need to adjust the water pressure field of the model so that water flows in from the front end of the hole in the middle of the model and flows out from the back end. The different colors of streamlines in fluid models can represent the mutual influence between particles and flow fields. The models for three working conditions were simulated under three different water pressure conditions: 200 Pa (low water pressure), 400 Pa (medium water pressure), and 600 Pa (high water pressure).

2.4. Calibration of the Model

2.4.1. Calibration of Contact Parameters

After establishing the tunnel erosion model for loess, it is necessary to adjust the parameter calibration and adjust the contact parameters of particles to make their properties similar to loess. The parameters required to calibrate the CFD-DEM coupled model are divided into solid module parameters (DEM) and fluid module parameters (CFD), which need to be calibrated separately. Among them, the calibration of solid module parameters is based on strength experiments, and, here, the shear test of loess is used as the standard. The calibration process is shown in Figure 8. In the discrete element program PFC^3D, a three-dimensional sample model with a diameter of 61.8 mm and a height of 20 mm was generated, and the model size was consistent with the direct shear experiment. To reduce computational complexity and increase the parameter calibration rate, the generated particles are rigid spheres with a certain proportion of enlarged particle size ranging from 0.6 mm to 0.9 mm. The number of particles generated is 15,200, and the particle density is 1500 kg/m³. The normal stress values are changed by controlling the vertical compression of the upper and lower loading plates on the model. The normal stress values are selected as 100 kPa, 200 kPa, 300 kPa, and 400 kPa, respectively. The model before failure is shown in Figure 8a. The shear failure of the model is achieved by applying shear stress on the side walls of the model, and the model after failure is shown in Figure 8b. Conduct multiple simulation experiments and continuously adjust the model parameters to obtain the shear stress–strain simulation curve. Compare the simulation curve with the physical experimental curve. When the two approximately coincide, the model parameters obtained are the particle contact parameters used for subsequent simulations, namely, the shear stress–strain curve (as shown in Figure 8c). This method can obtain parameters such as bond modulus (kPa), effective modulus (Pa), stiffness ratio (k_n/k_s), porosity, friction coefficient μ, etc. This type of parameter obtained, based on a trial and error method, has high accuracy and stronger persuasiveness of simulation results. The calibrated particle contact parameter values are shown in

Table 2. Parameters for tunnel erosion model.

Computation Model	Parameter	Numerical Value
Solid State Systems (DEM)	Particle density (kg/m3)	1500
	Bond modulus (kPa)	4.0 × 106
	Effective modulus, Ec (Pa)	1.0 × 107
	porosity	0.35
	Stiffness ratio (kn/ks)	1.5
	Coefficient of friction	0.3
Fluid Dynamics (CFD)	Density (kg/m3)	1.0 × 103
	Fluid viscosity coefficient (Pa·s)	1.0 × 10−3

. The parameters that need to be determined for the Fluid Dynamics Module (CFD) are fluid density (kg/m³) and the fluid viscosity coefficient (Pa · s), which are determined by fluid properties. The specific values are shown in Table 2.

2.4.2. Adjustment of the Number of Fluid Grids

In conventional CFD-DEM coupled models, the number of fluid grids is also an important factor affecting the accuracy of the model. Therefore, grid adjustments are required before conducting model calculations. Taking working condition c as an example, Figure 9 shows three numerical models with different grid sizes for working condition c. The sizes of the three models are the same; however, the number of fluid grids is different. Among them, the number of fluid grids in Figure 9a is 4 × 8 × 4, the number of fluid grids in Figure 9b is 8 × 16 × 8, and the number of fluid grids in Figure 9c is 16 × 32 × 16. The change in the number of fluid grids results in a significant variation in the density of streamlines.

For the convenience of calculation and comparison of differences, the calculation formula for particle erosion rate is introduced here, as shown in Equation (7) below:

$$\eta ={\displaystyle \frac{m_d}{m_w+m_d}}$$

(7)

where η—erosion rate, m_d—mass of particles, and m_w—mass of water.

The erosion rate time curve can be plotted with particle erosion rate (η) as the vertical axis and erosion time (t) as the horizontal axis. The resulting curves are shown in Figure 10, where the erosion water pressures in Figure 10a–c are 200 Pa, 400 Pa, and 600 Pa, respectively. From the graph, it can be seen that the change in the number of fluid grids affects the growth law of the erosion rate. From the total erosion time, models with more fluid grids take less time to be completely eroded. When the erosion time is the same, the model with more fluid grids has a higher erosion rate.

Although the variation in the number of fluid grids in the model has an impact on its erosion rate, overall, this difference is not very significant. At the same time, considering that the increase in the number of fluid grids makes it more difficult for the model to run and calculate, and the model running time increases, this study chooses a model with 8 × 16 × 8 fluid grids (as shown in Figure 9b) as the final solution.

2.4.3. Comparison and Verification with Physical Experiments

In previous studies, the model was calibrated by adjusting the contact parameters and the number of fluid grids in the model. However, compared to the above two calibration methods, comparing and verifying with the results of physical experiments is more accurate.

Figure 11 shows the comparison between numerical simulation results and experimental data. The numerical simulation in the figure shows the variation in the particle erosion rate under a water pressure of 200 Pa. The obtained data were analyzed and compared from the testing system with the numerical simulation results. Overall, the numerical results are in good agreement with the indoor experimental results, and the described phenomenon is consistent: from the beginning to the end of the experiment, the erosion rate gradually increases in the early stage of erosion but shows a non-linear increase and finally tends to stabilize. There are significant differences in erosion curves under different working conditions: Figure 11a,b have been completely eroded for a longer period of time, and the amount of eroded soil in the experiment is also relatively small. The time for complete erosion in Figure 11c is relatively short, and the amount of eroded soil in the experiment is also relatively large.

3. Results and Discussion

3.1. Erosion Process

The process of model erosion is shown in Figure 12 (water pressure 200 Pa, low pressure). In the simulation process, the change in water pressure mainly affects the erosion rate and has a relatively small impact on the erosion process. Therefore, the image here takes a low water pressure of 200 Pa as an example. It can be clearly observed from the figure that the particles around the hole are first eroded. In the early stage of erosion, erosion fluid flows out from the model hole, carrying away particles around the hole and causing the model hole to become larger (as shown in Figure 12(a1–c1)). As the fluid continues to erode, the erosion hole of the three models becomes larger, and the shapes of the longitudinal holes in condition a and the transverse hole in condition b gradually become circular (as shown in Figure 12(a2,b2)). In the middle stage of erosion, compared to before the experiment, the hole in the three working conditions significantly increased, and the shapes of the longitudinal hole in working condition a and the transverse hole in working condition b completely changed to circular shapes (as shown in Figure 12(a3,b3)). In the later stage of erosion, a large number of particles around the model hole are eroded, and the erosion range of the fluid also increases accordingly (as shown in Figure 12(a4–c4)).

Figure 13 shows the erosion process of the particle contact force chain during tunnel erosion (taking condition c as an example), with changes in erosion water pressure ranging from 200 Pa to 600 Pa. The green bars in the figure represent the contact force chain of particles, indicating that multiple particles are agglomerated together. When the number of green bars decreases, it indicates that the bonding effect of particles is weakened, and the separation of agglomerated particles is eroded by the fluid.

From the erosion process of the contact force chain, it can be seen that the model is not uniformly eroded. The particles at the back of the model are first eroded and flow away from the middle hole. When the erosion time is the same, the attenuation rate of the contact force chain of the model with higher erosion water pressure is faster. It can be observed that, at 80 s, the number of contact force chains in the model with an erosion water pressure of 600 Pa is extremely small, indicating that the particles have been completely eroded.

Figure 14 shows the erosion process captured in Wang’s experiment. Comparing Figure 12, it is found that the erosion process in the physical experiment is different from the simulation results. This is related to the water pressure and gravity of the erosive fluid. During the experiment, the erosion water flows in from the back end of the hole and then flows out from the front end of the hole. The erosion water flows downwards under the action of gravity, causing the erosion channel to appear below the hole. At the same time, due to the low water pressure of the erosive fluid, the water flow from the front end of the hole corrodes backward, forming an irregular tunnel with a narrow upper part and a wide lower part.

During the simulation process, due to the freedom to adjust the erosion water pressure, the erosion water quickly flows out of the hole, reducing the impact of gravity on the water flow. Therefore, particles around the hole are all eroded, and the hole changes from a rectangle to a more regular circle until the model particles are completely eroded.

Although the process of numerical simulation differs from the experimental process, in reality, the numerical simulations of erosion processes are relatively accurate. During our field investigation, we found a large number of erosion tunnels, which are actually wider at the tunnel entrances and narrower at the depths of the caves. The shape of erosion tunnels is mostly circular (as shown in Figure 15). Erosion fluids often slowly infiltrate through the dominant channels of loess (tension cracks or plant roots), widening the existing cracks and gradually forming circular channels. In addition to the circularization of holes, the rectangular voids in the experimental process (Figure 12a,b) are also developed from cracks. The volume of the voids is small, and they are in the early stage of loess erosion. After a longer period of erosion, the rectangular voids will become circular channels.

3.2. Erosion Rate

The erosion rate curves under different water pressures are shown in Figure 16. Figure 16a, Figure 16b, and Figure 16c, respectively, show the erosion rate changes of a longitudinal hole (condition a), transverse hole (condition b), and circular hole (condition c). The difference in the growth of erosion rate curves for different water pressures under three working conditions is very significant. When the water pressure is 600 Pa, the time required for the complete erosion of working conditions a and b is close, about 100 s. The hole volume of working condition c is relatively large, and particles are completely eroded at about 70 s. When the water pressure is 400 Pa, conditions a and b are completely eroded at around 200 s, and condition c is completely eroded at 120 s. When the water pressure is 200 Pa, conditions a and b are completely eroded at around 400 s, and condition c is completely eroded at 250 s. This indicates that changes in water pressure have a significant impact on particle erosion, with higher water pressure leading to faster erosion rates. The size of the tunnel hole also has a model influence on the time required for erosion, with larger hole volumes requiring less time for erosion.

Figure 17 shows the variation curve of the particle erosion rate under the same water pressure conditions and different working conditions. Figure 17a–c shows the erosion rate changes under low-pressure conditions (200 Pa), medium-pressure conditions (400 Pa), and high-pressure conditions (600 Pa), respectively. When the water pressure is 200 Pa, the growth curves of erosion rates in working conditions a and b are close (with a total erosion time of 400 s), while the growth rate of erosion rate in working condition c is much higher than that in working conditions a and b (with a total erosion time of 200 s). When the water pressure is 400 Pa, the growth curves of erosion rates in working conditions a and b are still close (total erosion time is 150 s), and the growth rate of erosion rate in working condition c is still higher than that in working conditions a and b; however, this difference has narrowed (total erosion time is 100 s). When the water pressure is 600 Pa, the growth curves of erosion rates for working conditions a, b, and c are basically similar (with a total erosion time of 80 s).

When the water pressure is low, the size of the tunnel hole has a greater impact on the erosion rate. As the erosion water pressure increases, the size of the tunnel hole has a smaller impact on the erosion rate. When the erosion water pressure reaches a large value, the influence of the volume of the tunnel hole on the erosion rate basically disappears. The increase in erosion water pressure causes particles to be completely eroded in a very short period of time, and rapid erosion reduces the impact of the pore size on the erosion rate.

From Figure 17, it can also be observed that in the initial stage of erosion, the growth rate of erosion is relatively slow. In the mid-term of erosion, the growth of the erosion rate will accelerate. In the later stage of erosion, the erosion rate no longer increases and tends to stabilize. This pattern is particularly evident at high water pressures (600 Pa) (as shown in Figure 17c). Therefore, the erosion of particles can be divided into the following three stages: the slow erosion stage, the rapid erosion stage, and the stable stage. Slow erosion stage: This stage is the initial stage of erosion and takes up a relatively short amount of time. It is not obvious at low water pressure, but at high water pressure, the time required for this stage increases significantly (as shown in Figure 17c). Rapid erosion stage: This stage is the middle stage of erosion and lasts for a long time. The curve of erosion rate in this stage shows a straight line and is the main stage of particle erosion. Stable stage: This stage is the later stage of erosion, where particles are basically completely eroded and the erosion rate curve shows almost no growth. This stage takes longer in low water pressure conditions.

3.3. Discussion

This study reproduced the erosion process of loess tunnels using the Discrete Element Method and compared the numerical simulation results with the physical experimental results, obtaining satisfactory results. However, there is still room for improvement in this study.

We have conducted small-scale physical experiments [25] and used the CFD-DEM coupling method to study the process of erosion in loess tunnels [28,29]. This study combined physical experiments and numerical simulations and compared and demonstrated the two methods, filling the gap in previous conclusions from physical experiments and numerical simulations and obtaining satisfactory results. But, there is still room for improvement in this study. The following are the problems solved and the problems that need to be solved in this study:

Problems solved:

1. From the comparison between simulation results and physical experimental results, it can be seen that the erosion model developed in this study for loess tunnels has better performance. In terms of model calibration, detailed explanations have been provided for the calibration of contact parameters, determination of fluid mesh number, and comparison with physical experimental data. This calibration method can be appropriately adjusted for simulations similar to cohesive soils. In terms of comparing the simulation results with the experimental results, although there is a slight difference between the erosion process obtained from the experiment and the simulated erosion process, the simulated particle erosion process and particle erosion rate curve are close to the model experiment, indicating the high accuracy of the simulation.

2. Numerical simulation solves a problem that is difficult to achieve in experiments. In terms of experimental variables, due to limitations in experimental equipment and site factors, previous physical experiments have only studied the effects of changes in dry density and moisture content on soil erosion. However, the methods of numerical simulation are different, as they can freely adjust experimental variables. In this study, multiple comparative studies were conducted by controlling erosion water pressure. In terms of analyzing the simulation results, the changes in the particle contact force chain during tunnel erosion were analyzed, which provides a good micro-level analysis of the entire process of tunnel erosion and reveals more useful information.

Problems to be solved:

1. The issue of particle size. In similar soil CFD-DEM coupling studies [22,23,24], the influence of particle size differences on the CFD-DEM coupling model was considered. But, the research objects of these studies consist mostly of gravel soil, with significant differences in particle size, while the particle size of loess is extremely small. Therefore, neither previous experimental studies nor numerical simulations have considered the impact of particle size differences on erosion in loess tunnels. In addition, due to the small size of loess particles, the particle size selected for this numerical simulation study has been enlarged, which will also affect the accuracy of the numerical simulation to some extent.

2. Lack of comparison in on-site experiments. In the introduction of the article, it is mentioned that the erosion problem of tunnels in the Loess Plateau is a slow and complex process that lasts for more than ten years. In our initial research design, this study described the erosion process of loess tunnels in detail from three aspects: small-scale physical experiments, discrete element numerical simulations, and on-site, in situ experiments. This not only ensures the accuracy of the results but also allows for further research based on actual engineering cases. However, due to various reasons, the in situ experiments on tunnel erosion did not achieve ideal results, which resulted in an insufficient comparison between small-scale physical experiments and numerical simulations developed based on small-scale physical experiments and actual cases, leading to insufficient research results.

4. Conclusions

Based on previous experimental research on the erosion of loess tunnels, this article combines the Discrete Element Method (DEM) with Computational Fluid Dynamics (CFD) to establish a CFD-DEM coupled model that can simulate the erosion process of loess tunnels. The formation process and erosion rate changes of tunnel erosion widely present in the Loess Plateau were studied using numerical simulation methods, and the following conclusions were drawn:

1. During the process of tunnel erosion, fluid flows through the model hole, and the particles around the hole are first eroded. With the increase in erosion time, the original longitudinal hole (longitudinal crack), transverse hole (transverse crack), and circular hole (circular cave) gradually become circular (the “circularization” of tunnel), and, finally, all become a circular hole. From the change pattern of the particle contact force chain during the erosion process, it can be seen that erosion water pressure has a significant impact on the attenuation of the particle contact force chain. At the same time, the greater the erosion water pressure, the faster the attenuation rate of the particle contact force chain.

2. The increase in erosion model holes and erosion water pressure significantly affect the total erosion time. The larger the holes in the erosion model, the faster the increase in erosion rate, and the shorter the time it takes for the model to completely erode. Similarly, the higher the erosion water pressure, the faster the increase in erosion rate, and the shorter the time it takes for the model to completely erode.

3. The growth rate of the erosion rate curve of tunnel erosion increases from slow to fast, then gradually slows down and tends to stabilize, which is particularly evident at high water pressure. According to the variation law of the erosion rate curve, the erosion process of the loess tunnel can be divided into three stages: the slow erosion stage, the rapid erosion stage, and the stable stage.

4. The erosion process and erosion rate of the numerical simulation are basically consistent with the comparison between the erosion process and erosion rate in previous experiments. The results of this numerical simulation are quite successful, and the model can be adjusted appropriately for simulating other characteristic soils.