Numerical Simulation of Water and Mud Inrush Processes in Mountain Tunnels Using Coupled Lattice Boltzmann/Discrete Element Methods

Abstract

Investigating the mechanism of sudden water inrush and mudflow in mountain tunnels is crucial for implementing preventive measures. Tunnel excavation through a fault or fractured zone can easily trigger sudden water inrush or mudflow. In this paper, the coupled lattice Boltzmann method (LBM) and discrete element method (DEM) were employed to reproduce the process of water and mud inrush in mountain tunnels. The failure of tunnel mud burst and water inrush involves a fluid–solid coupling process. A two-dimensional Boltzmann method for fluids and DEM for particles were utilized, with the coupled LBM-DEM boundary adopting the immersed moving boundary method. For simulating the water inrush process, a numerical model was established to replicate the flow of water particles within karst pipelines, featuring dimensions of 7 cm length, 4 cm width, and consisting of 100 particles. Particles are transported through water flow to the outlet of karst pipelines under hydraulic gradient loading. When the hydraulic gradient exceeds 6, the Darcy velocity gradually tends to be constant. As for simulating the mud inrush process, a numerical model was developed with dimensions of 5 cm length and 4 cm height, incorporating 720 randomly generated particles. The results demonstrated the successful reproduction of the evolution process encompassing three consecutive stages of tunnel mud-burst failure: initiation, acceleration, and stabilization. The occurrence of mud inrush disasters is attributed to combined action involving disaster-causing geotechnical materials, groundwater pressure, and tunnel excavation.

1. Introduction

It is widely recognized that the occurrence of tunnel water inrush or mudslide has consistently posed significant challenges to tunnel excavation, and a comprehensive analysis of its generation mechanism remains imperative [1,2,3]. In comparison to tunnel water inrush, research on tunnel mud inrush commenced relatively late, necessitating targeted and systematic investigation into this issue. Tunnel water or mud inrush can be conceptualized as a complex fluid–particle coupling process, presenting considerable challenges for the development of an effective numerical modeling framework due to its inherent complexity.

The evolution of water and mud inrush disasters in mountain tunnels is a strongly coupled process of seepage erosion, essentially constituting fluid–particle coupling dynamics. Currently, numerical simulation is an important method for studying such phenomena. During the past half-century, a considerable amount of effort has been expended on developing diverse numerical techniques for simulating the interactions between fluids and particles, and there has also been remarkable progress in the efficient computational solvers of the Navier–Stokes equations [4,5,6,7,8]. In recent years, particle transport within fluid flows has found widespread applications in science and engineering domains [9]. Simulation methods involving fluid–particle coupling include lattice Boltzmann method (LBM)/discrete element method (DEM), discontinuous deformation analysis (DDA)/smoothed particle hydrodynamics (SPH), particle flow code (PFC)/computational fluid dynamics (CFD), DEM-CFD, and others [3,10,11,12]. Among these approaches, the DEM-CFD approach holds particular appeal owing to its computational convenience [13,14,15,16]. However, the weakness of this method lies in that it depends on the semi-empirical equations to handle the complex coupling between the fluid and the particles. Therefore, the coupled LBM-DEM has been demonstrated to be a potent method for resolving the problem of particle–fluid and particle–particle interactions [9,17,18].

The developed LBM-DEM serves as a robust tool for fundamental research, enabling the exploration of hydromechanical physics in flows through porous media [19]. The coupled LBM-DEM excels in providing a detailed depiction of the flow field surrounding each particle while maintaining a clear physical understanding of solid particle behaviors such as friction, collision, migration, and more. Furthermore, its implementation readily lends itself to parallel computing, significantly enhancing the speed and efficiency of simulations. The coupled LBM-DEM can effectively overcome the limitations of the CFD-DEM and is better suited for simulating complex fluid–particle systems at the pore scale. When solid particles in motion are partially or entirely submerged within the fluid, the particles will alter the velocity and pressure distribution in the surrounding flow field. On the contrary, fluids exert resistances on solid particles, like resistance, lift, and torque, thus influencing the mechanical behavior of solid particles. In the coupling process, to accurately capture the LBM-DEM coupling mechanism, pressure and velocity boundary conditions, no-slip conditions, and bounce-back boundaries are typically utilized for boundary treatment [20,21]. The mixing behaviors of particles in a sheared particle flow were investigated [22]. The particle flow method was used to study the particle transport mechanism in a basic soil–filter system [23]. A numerical study was presented on soil fluidization induced by local leakage of a buried pipe using the coupled LBM-DEM [5]. A collision model for phase-resolved direct numerical simulation of sediment transport, coupling fluid–particle via the immersed moving boundary method [24,25], was presented [26]. Four variants of boundary conditions were analyzed and compared in terms of their accuracy in the free-surface lattice Boltzmann method [27]. A numerical framework, rooted in a multiple relaxation time lattice Boltzmann model and innovative discrete techniques, has been proposed for the simulation of compressible flows, offering enhanced accuracy and versatility [28]. Macroscale lattice Boltzmann simulations were utilized to compare macroscopic-scale simulations with pore-scale analytical models, further validating coupling conditions [29]. More research on the coupled LBM-DEM can be found in Ref. [30]. Some open-source codes in the academic community, such as LIGGHTS, Yade, MatDEM, waLBerla, and Palabos, have promoted and accelerated research in the fields of LBM and DEM [31]. From existing research, the coupled LBM-DEM has been extensively studied in the field of fluid mechanics theory. However, further research is still required on the application of the coupled LBM-DEM in the study of water and mud inrush in mountain tunnels.

This study employs a two-dimensional coupled LBM-DEM to simulate the mechanism of water and mud inrush on the tunnel wall surface, considering inter-particle friction. The LBM and the DEM are utilized for simulating seepage and the interaction and motion of rock and soil particles in geotechnical materials, respectively. The immersed moving boundary method is employed to handle the interaction between moving solid particles and fluid. The numerical simulation replicates the process of water and mud inrush in mountain tunnels, providing valuable insights for geological disaster management in such environments.

The paper is organized as follows: Section 2 provides a brief introduction to the basic idea underlying the two-dimensional lattice Boltzmann method, while Section 3 presents the discrete element method. Section 4 and Section 5 introduce the coupled LBM-DEM and its verification, respectively. Section 6 discusses simulation illustrations and the final section concludes.

2. LBM

2.1. D2Q9 Formulations

The LBM, grounded in the principles of statistical mechanics, serves as a computational fluid dynamics approach that relies on mesoscopic simulation scales. It describes the evolution of particle distribution functions on a uniform grid [32,33]. In this study, the D2Q9 lattice model was employed to simplify the analysis and effectively illustrate the essence of the problem, as depicted in Figure 1a. Suppose a two-dimensional incompressible fluid flow in a planar rectangular domain is initially divided into squares of length h along the x and y directions, respectively. The fluid phase can then be represented by fluid particles at lattice nodes with discrete velocities at specific times, allowed to move only along predetermined directions to adjacent nodes during numerical calculation process. In the D2Q9 model, each regular node’s fluid particles are permitted to move to their eight immediate neighbors with eight different velocities ${\mathbf{e}}_i$, ($i=1,2,\dots ,8$), while the zero velocity particle ${\mathbf{e}}_0$ can remain on the node, as shown in Figure 1b. This will result in a total of nine distinct velocity vectors being defined as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathbf{e}}_0=c\left(0,0\right),\hfill \\ {\mathbf{e}}_1=c\left(1,0\right),{\mathbf{e}}_3=c\left(-1,0\right),\hfill \\ {\mathbf{e}}_2=c\left(0,1\right),{\mathbf{e}}_4=c\left(0,-1\right),\hfill \\ {\mathbf{e}}_5=c\left(1,1\right),{\mathbf{e}}_7=c\left(-1,-1\right),\hfill \\ {\mathbf{e}}_6=c\left(-1,1\right),{\mathbf{e}}_8=c\left(1,-1\right).\hfill \end{array}\right.\end{array}$$

(1)

where c is the lattice speed and given by

$$\begin{array}{c}\hfill c=h/\mathsf{\Delta }t\end{array}$$

(2)

in which $\mathsf{\Delta }t$ represents the discrete time step. Although the motion of a single fluid particle is irregular, the collective motion of multiple particles can impact the macroscopic parameters of fluid motion. The D2Q9 model can be depicted by nine fluid density distribution functions $f_i$($i=0,\dots ,8$). $f_i$ is associated with the possible quantity of fluid particles that move at velocity ${\mathbf{e}}_i$ along the ith direction at each node. During the collision process, fluid particles will move towards the nearest node in the given direction at each fluid time step. The evolution of the density distribution function ($f_i$) is given as follows [34]:

$$\begin{array}{c}\hfill f_i(\mathbf{x}+{\mathbf{e}}_i\mathsf{\Delta }t,t+\mathsf{\Delta }t)-f_i\left(\mathbf{x},t\right)=-\frac{1}{\tau }\left[f_i(\mathbf{x},t)-f_i^{\mathrm{eq}}(\mathbf{x},t)\right]\end{array}$$

(3)

where x represents the position of any fluid grid node and t represents time. $\mathbf{x}+{\mathbf{e}}_i\mathsf{\Delta }t$ represents its nearest neighbor in the direction i after a discrete time step. $\tau$ is the relaxation time, which is a dimensionless parameter. The equilibrium distribution function, $f_i^{\mathrm{eq}}(\mathbf{x},t)$, is described as follows:

$$\begin{array}{c}\hfill f_0^{\mathrm{eq}}=w_0\rho (1-\frac{3}{2c^2}\mathbf{u}·\mathbf{u})\end{array}$$

(4)

$$\begin{array}{c}\hfill f_i^{\mathrm{eq}}=w_i\rho \left[1+3\frac{{\mathbf{e}}_i·\mathbf{u}}{c^2}+\frac{9}{2}\frac{{(\mathbf{e}·\mathbf{u})}^2}{c^4}-\frac{3}{2}\frac{{(\mathbf{u}·\mathbf{u})}^2}{c^2}\right](i=1,\dots ,8)\end{array}$$

(5)

where $\rho$ represents the fluid density, u denotes the fluid velocity, and $w_i$ represents the weighting factor in the direction i. The weighting factor should satisfy the following criterion:

$$\begin{array}{c}\hfill \sum _{i=0}^8{\omega }_i=1\end{array}$$

(6)

where $w_0=\frac{4}{9}$, $w_{1,2,3,4}=\frac{1}{9}$, and $w_{5,6,7,8}=\frac{1}{36}$.

The two most crucial steps in LBM calculation are collision and streaming. The right-hand side of Equation (3) signifies a collision occurring at position $\mathbf{x}$ and time t, specifically involving variables pertinent solely to the individual node $\mathbf{x}$, thereby rendering it a localized operation. Following a collision, new fluid particles occupy the positions of the original particles while the original particles migrate. The updated distribution functions at each node are explicitly propagated to their neighboring nodes located at $\mathbf{x}+{\mathbf{e}}_i\mathsf{\Delta }t$ during the streaming phase. In this process, no calculations are needed; only data exchange between neighboring nodes is required as represented on the left side of Equation (3). The explicit time-stepping nature of the LBM and its utilization of regular grids render it computationally efficient, straightforward to implement, and inherently parallelizable. It should be emphasized that although LBM belongs to the mesoscopic models, it still adheres to the principles of mass conservation, momentum conservation, and energy conservation.

In numerical calculations, the fluid kinematic viscosity ($\nu$), sound speed ($c_s=c/\sqrt{3}$), relaxation factor ($\tau$), and time step ($\mathsf{\Delta }t$) satisfy the following relation:

$$\begin{array}{c}\hfill \nu =c_s^2\left(\tau -\frac{1}{2}\right)\mathsf{\Delta }t\end{array}$$

(7)

The fluid density $\rho$ and velocity u are respectively determined by

$$\begin{array}{c}\hfill \rho =\sum f_i\end{array}$$

(8)

$$\begin{array}{c}\hfill \rho \mathbf{u}=\sum {\mathbf{e}}_i{f}_i\end{array}$$

(9)

The fluid pressure field p is determined by the following equation of state:

$$\begin{array}{c}\hfill p=\rho c_s^2\end{array}$$

(10)

2.2. Boundary Conditions

The accurate specification of boundary conditions is essential for simulating fluid flow processes. Commonly used surface boundary conditions for lattice Boltzmann include the slip, no-slip, imposed, and free boundary conditions. A detailed discussion on boundary conditions can be found in Ref. [20]. In this study, a simple case is considered, as shown in Figure 2, where the left boundary and right boundary are treated as the inlet and outlet for the flow, respectively. To enforce boundary conditions on all four sides of the channel, it is necessary to define eight sets of particle distribution functions: four at the corner nodes of the rectangle and an additional four at the midpoint of each side [35].

In Figure 2, it is assumed that the pressure boundary condition distribution is implemented at the left inlet and right outlet of the rectangular domain. These pressure boundary conditions are enforced by explicitly specifying the fluid densities at the flow boundaries, where ${\rho }_{\mathrm{in}}$ and ${\rho }_{\mathrm{out}}$ represent the fluid density at the inlet and outlet, respectively. To achieve equivalent LBM conditions, it is crucial to calculate $f_i$ ($i=0,\dots ,8$) at all boundary nodes. For example, for the left inlet boundary node, $f_1$, $f_5$, and $f_8$ can be determined based on the known density functions $f_i$ (i = 2, 3, 4, 6, 7) after streaming. The detailed derivation process is as follows:

$$\begin{array}{c}\hfill {\rho }_{\mathrm{in}}=f_1+f_5+f_8+f_0+f_2+f_3+f_4+f_6+f_7\end{array}$$

(11)

$$\begin{array}{c}\hfill {\rho }_{\mathrm{in}}{u}_x=f_1+f_5+f_8-\left(f_3+f_6+f_7\right)\end{array}$$

(12)

$$\begin{array}{c}\hfill f_5-f_8=-f_2+f_4-f_6+f_7\end{array}$$

(13)

Furthermore, the application of the bounce-back rule to the non-equilibrium component of the particle distribution, which is perpendicular to the left inlet of the rectangular domain, yields the following relationship:

$$\begin{array}{c}\hfill f_1-f_1^{\mathrm{eq}}=f_3-f_3^{\mathrm{eq}}\end{array}$$

(14)

The solution is obtained as follows:

$$\begin{array}{c}\hfill u_x=1-\frac{f_0+f_2+f_4+2\left(f_3+f_6+f_7\right)}{{\rho }_{\mathrm{in}}}\end{array}$$

(15)

$$\begin{array}{c}\hfill f_1=f_3+\frac{2}{3}{\rho }_{\mathrm{in}}{u}_x\end{array}$$

(16)

$$\begin{array}{c}\hfill f_5=f_7-\frac{1}{2}\left(f_2-f_4\right)+\frac{1}{6}{\rho }_{\mathrm{in}}{u}_x\end{array}$$

(17)

$$\begin{array}{c}\hfill f_8=f_6+\frac{1}{2}\left(f_2-f_4\right)+\frac{1}{6}{\rho }_{\mathrm{in}}{u}_x\end{array}$$

(18)

where $u_x$ represents the x-direction component of the fluid velocity vector u.

Similarly, the unknown density functions $f_3$, $f_6$, and $f_7$ can be obtained as follows:

$$\begin{array}{c}\hfill u_x=-1+\frac{f_0+f_2+f_4+2\left(f_1+f_5+f_8\right)}{{\rho }_{\mathrm{in}}}\end{array}$$

(19)

$$\begin{array}{c}\hfill f_3=f_1-\frac{2}{3}{\rho }_{\mathrm{out}}{u}_x\end{array}$$

(20)

$$\begin{array}{c}\hfill f_7=f_5+\frac{1}{2}\left(f_2-f_4\right)-\frac{1}{6}{\rho }_{\mathrm{out}}{u}_x\end{array}$$

(21)

$$\begin{array}{c}\hfill f_6=f_8-\frac{1}{2}\left(f_2-f_4\right)-\frac{1}{6}{\rho }_{\mathrm{out}}{u}_x\end{array}$$

(22)

The corner nodes within a rectangular domain can be determined using the following methodology. Taking the upper-left corner node in Figure 2 for instance, $f_2$, $f_3$, and $f_6$ are known, and $u_x$ = $u_y$ = 0 after streaming. Five unknown density functions, namely $f_1$, $f_4$, $f_5$, $f_7$, and $f_8$, require determination. By once again applying the bounce-back rule to the non-equilibrium part component perpendicular to the entrance distribution, the top boundary can be made use of.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{f}_1=f_3,\hspace{1em}{f}_4=f_2,\hspace{1em}{f}_8=f_6\\ f_5=f_7=\frac{1}{2}\left[\rho -f_0-2\left(f_1+f_2+f_6\right)\right]\end{array}\right.\end{array}$$

(23)

The other three corner nodes can also be treated similarly.

3. DEM

In contrast to conventional finite element methods, the DEM accommodates relative motion between elements, which may not necessarily adhere to the conditions of continuous displacement and deformation coordination. It boasts rapid computational speed and demands less storage space, rendering it particularly well suited for analyzing large deformations/displacements in rock and soil materials. Within DEM, each particle in a particle system adheres to Newton’s second law. While circular shapes are most prevalent among particles, elliptical and irregular shapes also exist. Contacting particles may exhibit slight overlap, with contact forces calculated based on specific force–displacement relationships and mutual interactions. The relationship between the normal force and contact displacement in the normal direction can be described as follows:

$$\begin{array}{c}\hfill F_n=k_n{d}_n\end{array}$$

(24)

where $F_n$ is the normal contact force, $k_n$ is the normal stiffness, and $d_n$ is the overlap between the pair of particles in contact.

Similarly, in the tangential direction, when the contact between particles is in an elastic state, the following equation can be used:

$$\begin{array}{c}\hfill \mathsf{\Delta }{F}_s=-k_s\mathsf{\Delta }{u}_s\end{array}$$

(25)

where $\mathsf{\Delta }{F}_s$ is the shear force increment, $k_s$ is the shear stiffness, and $\mathsf{\Delta }{u}_s$ is the increment of the tangential displacement of the particles. Once in an elastic–plastic state, the interaction between particles follows the Coulomb friction law and is expressed as follows:

$$\begin{array}{c}\hfill |F_s|>F_{n}tan\left(\mathsf{\Phi }\right)+C_s\end{array}$$

(26)

with $F_s$ denoting the tangential contact force, $\mathsf{\Phi }$ representing the contact friction angle, and $C_s$ indicating the shear cohesion. Shear cohesion, denoted as $C_s$, is set to zero when sliding occurs. More theoretical introductions about the DEM can be found in Ref. [35].

4. LBM-DEM

The application of coupled LBM-DEM requires the assumption that particles, whether in fluid lattices or solids, are rigid. The fluid phase is considered incompressible and Newtonian. Particle deformation is limited to the local surface area of the contact point. The interaction between particles primarily occurs through surface contact collisions while disregarding other weak forces. When considering issues pertaining to particle transport, solid particles are primarily propelled by the combined action of hydrodynamic forces exerted by the fluid, as well as body forces. In the context of LBM, fluid particles collide with the boundaries of solid particles exchanging momentum and altering their motion.

The first step in establishing a model of fluid and solid particle interaction involes representing particles using lattice nodes. Figure 3 categorizes solid and fluid nodes into three distinct groups: (1) Fluid boundary nodes, comprising fluid nodes that are directly adjacent to at least one solid node; (2) solid boundary nodes, representing solid nodes that are connected to at least one fluid node; and (3) internal solid nodes, which refer to solid nodes that are not linked to any fluid nodes. The connection between fluid boundary nodes and solid boundary nodes is referred to as a boundary link which constantly changes in an on–off manner when particles move, significantly impacting calculated interaction forces.

By incorporating collision operators into the immersed moving boundary (IMB) algorithm, boundary nodes undergo updates at every time step [24], as follows:

$$\begin{array}{c}\hfill f_i\left(\mathbf{x}+{\mathbf{c}}_i\mathsf{\Delta }t,t+\mathsf{\Delta }t\right)=f_i\left(\mathbf{x},t\right)-\frac{\mathsf{\Delta }t}{\tau }\left(1-B\right)\left(f_i\left(\mathbf{x},t\right)-f_i^{\mathrm{eq}}\left(\mathbf{x},t\right)\right)+\left(1-B\right)F_i\mathsf{\Delta }t+B{\mathsf{\Omega }}_{\mathit{i}}^{\mathbf{s}}\end{array}$$

(27)

where parameter B is defined as

$$\begin{array}{c}\hfill B=\frac{\epsilon \left(\tau -1/2\right)}{\left(1-\epsilon \right)+\left(\tau -1/2\right)}\end{array}$$

(28)

and where ${\mathsf{\Omega }}_{\mathit{i}}^{\mathbf{s}}$ is a novel collision operator for nodes that are partially or fully encompassed by a solid. $\epsilon$ represents the volume fraction of solids that overlaps with a lattice node. If $\epsilon$ = 1, the lattice node effectively assumes the properties of a solid node, causing fluid particles to rebound in their original incoming directions. If $\epsilon$ = 0, the collision equation reverts to the standard LBM collision equation.

The new collision operator ${\mathsf{\Omega }}_{\mathit{i}}^{\mathbf{s}}$ applies to nodes that have overlap with solids, as follows:

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_{\mathit{i}}^{\mathbf{s}}=f_{-i}\left(x_t,t\right)-f_i\left(x_t,t\right)+f_i^{\mathrm{eq}}\left(\rho ,u_{\mathrm{s}}\right)-f_{-i}^{\mathrm{eq}}\left(\rho ,u\right)\end{array}$$

(29)

where $u_{\mathrm{s}}$ denotes the velocity of the solid and the subscript $-i$ is the opposite direction to i.

The LBM fluid adds an additional hydrodynamic force ($F_{\mathrm{f}}$), which is given as follows:

$$\begin{array}{c}\hfill F_f=\frac{h^2}{\mathsf{\Delta }t}\sum B_n\sum _i{\mathsf{\Omega }}_{\mathit{i}}^{\mathbf{s}}{\mathbf{e}}_i\end{array}$$

(30)

and the torque is given by

$$\begin{array}{c}\hfill M_f=\frac{h^2}{\mathsf{\Delta }t}\sum _n\left(x_i-x_{\mathrm{s}}\right)\times \left(B_n\sum _i{\mathsf{\Omega }}_{\mathit{i}}^{\mathbf{s}}{\mathbf{e}}_i\right)\end{array}$$

(31)

where $x_{\mathrm{s}}$ represents the center location of the solid particle and $B_n$ denotes the new collision operator B at node n. The second summation in both equations is over the lattice directions.

The force and torque will be applied to coupled LBM-DEM simulations. In this calculation, both the interparticle contact force and the hydrodynamic force acting on the particle under analysis are taken into consideration, as follows:

$$\begin{array}{c}\hfill ma=F_f+mg\end{array}$$

(32)

$$\begin{array}{c}\hfill J\ddot{\theta }=M_f\end{array}$$

(33)

where m represents the mass, J represents the moment of inertia, $\ddot{\theta }$ represents the angular acceleration, and g represents the acceleration due to gravity. Taking into account all the forces acting on a solid particle, the dynamic equations can be formulated as follows:

$$\begin{array}{c}\hfill ma+cv=F_c+F_f+mg\end{array}$$

(34)

where $F_c$ represents the total contact forces from other particles and the walls. The term $cv$ accounts for a viscous force that considers the effect of all possible dissipation forces present in the system.

It should be noted that there exist two time steps in the coupled LBM-DEM calculation program. These two times need to be filled with the following relationships:

$$\begin{array}{c}\hfill \mathsf{\Delta }t=\left(\tau -\frac{1}{2}\right)\frac{h^2}{3\nu }\end{array}$$

(35)

$$\begin{array}{c}\hfill \mathsf{\Delta }{t}_s=\frac{\mathsf{\Delta }t}{n_s}(n_s=[\mathsf{\Delta }t/\mathsf{\Delta }{t}_D]+1)\end{array}$$

(36)

where $\mathsf{\Delta }t$ is the time interval for the fluid flow and $\mathsf{\Delta }{t}_D$ is the time interval for the particles. Since $\mathsf{\Delta }{t}_D$ may generally be less than $\mathsf{\Delta }t$, it needs to be reduced to a new value $\mathsf{\Delta }{t}_s$ such that the ratio between $\mathsf{\Delta }t$ and $\mathsf{\Delta }{t}_s$ is an integer $n_s$ [4].

For the sake of simplicity in calculations, all units in the LBM are typically dimensionless, meaning that lattice units are employed.

5. Verification

The classical incompressible fluid flow around a cylinder is used to verify the correctness of the program. The flow around a cylinder refers to the flow pattern of low-speed, steady flow around a two-dimensional cylinder. This flow pattern is only determined by the Reynolds number ($Re=\rho v\mathrm{L}/\nu$), where v represents the typical macroscopic velocity of the flow and $\mathrm{L}$ is a characteristic length of the fluid domain. If $Re$ exceeds 100, it will result in turbulent flow in the channel.

Assume a two-dimensional region with dimensions of 12 cm in length and 4 cm in width is established on the $xoy$ plane. A cylinder with a radius of 0.5 cm and a central coordinate located at the point (x = 4 cm, y = 2 cm) is positioned within this plane, as illustrated in Figure 4. The left and right boundaries of the region are set as periodic boundaries, while the upper and lower boundaries are designated as no-slip boundaries. The viscosity of water $\nu$ is 1 × ${10}^{-6}$ m²/s, and the maximum flow rate is 5 cm/s. The resolution of the cylinder diameter is 30 cells. The calculation results for different Reynolds numbers depicting the flow around a cylinder are shown in Figure 5. At low Reynolds numbers, laminar flow occurs, as shown in Figure 5a,b. As the Reynolds number increases, there is no separation behind the cylinder. Instead, a pair of symmetric vortices appear, and then the vortices begin to detach due to instability, as shown in Figure 5c,d. Furthermore, the finite element method was implemented to compare the fluid dynamics around a cylinder, as shown in Figure 6. There are altogether 14,652 finite element meshes. The numerical results demonstrate that for Reynolds numbers lower than 100, the fluid flow presents a relatively stable pattern. When the Reynolds number surpasses 100, a significant amount of turbulence emerges, similar to the coupled simulation results witnessed in LBM-DEM analysis. The turbulence results calculated using the LBM method are essentially consistent with existing conclusions [36], confirming the accuracy of the previous formula derivation.

6. Simulation Illustrations

6.1. Simulation of the Water Flow Process and the Movement of Particles in a Karst Pipeline Flow

In this subsection, the coupled LBM-DEM is utilized to model the particle flow in karst pipelines within mountain tunnels. The primary challenge arises from the need for more analytical solutions for estimating the forces exerted on particles by fluid flow containing a large number of particles. In a fluid-filled system comprising numerous particles, each submerged particle is subject to not only the hydrodynamic forces and torques exerted by the surrounding fluid field but also engages in intricate interactions with its neighboring particles. This system holds significant importance in civil engineering. The selection of interactions between a smaller number of individual particles and fluids within simple geometric shapes represents an initial step in the progressive exploration of such systems.

For instance, the Xiaozhai Tunnel is a Class I high-risk tunnel on the China Yunnan–Guangxi Railway (from Yunnan Province to Guangxi Province). It has a total length of 6700 m and a maximum burial depth of 270 m. The survey area mainly consists of four lithologies: Quaternary silty clay, Triassic sandstone mixed with shale and mudstone, Devonian limestone mixed with dolomite, and Cambrian dolomitic limestone. The regional geological structure is relatively complex, with multiple faults developed. The rock mass is strongly influenced by its structure, with highly developed joints and fissures. The groundwater types in the tunnel area are mainly divided into clastic rock fissure water and carbonate rock karst water. Advanced geological prediction during tunnel excavation is essential for high-risk railway tunnels. It primarily involves the elastic wave reflection method and electromagnetic wave reflection method [37]. During the construction of the Xiaozhai Tunnel, Tunnel Seismic Prediction (TSP) was utilized for advanced geological prediction for the 100 m ahead of the section DK412+026∼DK411+926 of the Xiaozhai Tunnel. It was found that from the chainage DK411+950 on the palm face, the longitudinal and transverse wave velocities of the rock mass ahead decreased, while the Poisson’s ratio increased; the dynamic elastic modulus decreased with decreasing chainage, as shown in Figure 7. Subsequent excavation revealed a karst pipeline in the middle and lower part of the face, which experienced water with mud gushing, as shown in Figure 8a. In response to this phenomenon, a 2D simulation model of karst piping flow using coupled LBM-DEM was presented to verify if it accurately describes the linear relationship between erosion rate and fluid shear stress. A 2D simulation model of a karst pipeline flow, measuring 7 cm in length and 4 cm in width, is established in Figure 8b.

This system serves as a simplified model that disregards gravity effects and is spatially discretized utilizing a square lattice to facilitate the implementation of the LBM algorithm, with voids being filled with water. One hundred particles, randomly generated in a two-dimensional box, have an average radius of 0.155 cm, maximum radius of 0.206 cm, and minimum radius of 0.105 cm, as shown in Figure 9a. It should be noted that the flow in karst pipelines follows Darcy’s law, while the physical and mechanical parameters of fluid and solid particles are detailed in

Table 1. Physical and mechanical parameters of fluid and solid particles using coupled LBM-DEM.

Method	Property (Units)	Value
LBM	Water density (kg ${\mathrm{m}}^{-3}$)	${\rho }_0$ = 1000
	Fluid kinematic viscosity (m2/s)	$\nu$ = 1 $\times {10}^{-6}$
	Relaxation parameter	$\tau$ = 1.1
	Lattice size (m)	h = 2.77241 $\times {10}^{-5}$
	Interval time (s)	$\mathsf{\Delta }t$ = 1.53726 $\times {10}^{-4}$
DEM	Solid grain density (kg ${\mathrm{m}}^{-3}$)	${\rho }_n$ = 2650
	Normal stiffness (MPa)	$k_n$ = 3
	Tangential stiffness (MPa)	$k_s$ = 1.6
	Interval time (s)	$\mathsf{\Delta }{t}_D$ = 5.12419 $\times {10}^{-5}$
	Internal friction angle (°)	21

. The x and y directions include 255 and 147 lattice grids, respectively. A pressure gradient is imposed between its inlet and the outlet as follows:

$$\begin{array}{c}\hfill \mathsf{\Delta }P=p_{\mathrm{in}}-p_{\mathrm{out}}=c_s^2{\rho }_0({\rho }_{\mathrm{in}}-{\rho }_{\mathrm{out}})\end{array}$$

(37)

where ${\rho }_{\mathrm{out}}=1$ and ${\rho }_0$ is the physical density of the fluid. The y-direction component, $v_y$, is equal to 0 at the inlet and outlet, respectively. The x-direction component, $v_x$, can be obtained using Equation (15). The velocity at both the upper and lower boundaries is zero. The ${\rho }_{\mathrm{in}}$ value is fixed according to Equation (15). Here, corner nodes belong to fixed walls, and they are initialized in the same way as wall nodes ($v_x=0$ and $v_y$ = 0).

A series of snapshots is presented in Figure 9, illustrating the detachment of particles under hydraulic loading and their subsequent transport towards the outlet of the karst pipeline by the flow. Upon removal of a particle, no specific treatment is required for the resulting fluid nodes, as they were already treated as such when part of the solid particle. With an increase in hydraulic gradient, there is a more pronounced initial oscillation in Darcy velocity of particles, with a maximum value of 0.022 cm/s and a minimum value of −0.015 cm/s, where ‘-’ indicates the opposite direction of flow from the inlet to the outlet. When the hydraulic gradient exceeds 6, Darcy velocity gradually tends to become constant, as shown in Figure 10a. As the hydraulic gradient increases further, the particles at the upper and lower boundaries move away from the inlet of the karst pipeline and towards the outlet on the right-hand side. At a hydraulic gradient greater than 6, upper boundary particles exhibit sharp back-and-forth movement. In contrast, lower boundary particles remain at a constant position of 7 cm from the inlet, as depicted in Figure 10b. The return movement of upper boundary particles results from setting this boundary as bounce-back during numerical simulation calculations.

6.2. Simulation of the Mud Inrush Procedure in a Mountain Tunnel

During the excavation of mountain tunnels, mud inrushes may occur, as shown in Figure 11a. The mechanism of mud inrush in mountain tunnels remains unclear due to challenges in accurately assessing geological conditions. A two-dimensional fluid–solid coupling numerical model with dimensions of 5 cm in length and 4 cm in height is used to replicate the process of tunnel mud inrush, as shown in Figure 11b. Particle aggregates within this range were generated to mimic the formation process through self-weight deposition, equilibrium, and bonding, with particle sizes ranging from 0.06 to 0.1 cm. A total of 720 particles were randomly generated within the box. The left boundary features a water flow inlet and pressure boundary while employing a specific density value and non-equilibrium rebound method to simulate water pressure within the formation. The upper boundary is open, and the lower boundary is a no-slip boundary. The tunnel face located on the right boundary serves as a mudslide outlet. It should be noted that, in order to enhance the computational efficiency of coupled LBM-DEM simulation, the particle size in the numerical model is larger than that of typical mud bodies in practical engineering. When the particles and flow field are stable, the calculation stops. The detailed physical and mechanical parameters of fluid and solid particles are the same as in Table 1, except for the relaxation parameter $\tau$ = 0.504.

Throughout the simulation process, data such as particle distribution, inter-particle bonding distribution, flow velocity, water pressure, and mud-burst volume are recorded every 4000 time steps. The damage caused by tunnel mud burs is a result of the combined effects of various factors, including the soil medium properties, water pressure dynamics, and excavation activities. It is only under specific conditions that these three factors combine to cause mud-burst disasters in tunnels. Numerical calculations are solely used for simulating the phenomenon of mud inrush, which differs somewhat from actual engineering scenarios involving mud inrush.

Figure 12 illustrates the process of tunnel mud inrush, which can be categorized into three stages: initiation, acceleration, and stability. During the initiation stage, as particles near the mud burst gradually enter, the upper particles collapse downward due to the loss of effective support, leading to the formation of a free surface that inclines towards the mud burst and creates a wider water passage. Under the erosion of water flow, a large number of particles migrate along the tunnel’s palm face towards the direction of mud burst, resulting in a continuous and rapid increase in the amount of mud burst. In the accelerating stage, there is an increase in bonding strength between particles, causing rapid collapse of the tunnel face and scouring of many particles with a maximum distance between them exceeding 12 cm. The time for the acceleration stage becomes shorter with a slightly smaller rate of increase. At the stable stage, the free surface has essentially reached the lower edge of the mud mouth, and most of the water flows near the right boundary and exits through the tunnel palm face. The water flow on the free surface weakens, causing a slowdown in the growth of the tunnel palm face, which ultimately remains unchanged. At this point, both the particles and the flow field have stabilized, entering a steady state. The numerical simulation results are compared with Ref. [38]. The comparison results indicate that the coupled LBM-DEM can be used for analyzing tunnel mud bursts. The sudden mud process was divided into four stages, including starting stage, acceleration stage, slow stage, and stable stage, as shown in Figure 13 [38]. In the present study, there is no slow stage. This difference may be related to the selection of simulation parameters and the setting of boundary conditions.

Figure 14 shows the position distribution curves of three solid particles (P1, P2, P3) as shown in Figure 11b, respectively. On the $xoy$ plane, the initial positions of points P1, P2, and P3 were located at (3.81, 36.14), (44.87, 38.20), and (31.81, 20.88), respectively. After the mud burst occurred in the mountain tunnel, the following points were located: (8.89, 23.96), (126.69, 2.78), and (67.03, 10.23), respectively. The particles in the upper part of the tunnel face, such as P2, have the farthest distance of movement. P1 has the shortest movement distance. It is necessary to promptly avoid the farthest point of particle movement when mud bursts occur in the tunnel.

7. Conclusions

The paper presents numerical procedures for modeling particle transport in tunnel water and mud inrush using the coupled LBM-DEM. The following main conclusions are drawn:

(1) Numerical tests have confirmed the feasibility of the coupled LBM-DEM for solving fluid–particle interaction problems dominated by the presence of many particles and the occurrence of channels. The TSP was utilized to identify karst water inrush ahead of the Xiaozhai Tunnel, with subsequent excavation confirming the forecast’s accuracy. Additionally, the coupled LBM-DEM was employed to simulate water flow processes in karst pipelines with a small amount of sediment. Particles are transported through water flow to the outlet of karst pipelines under hydraulic gradient loading. When the hydraulic gradient exceeds 6, the Darcy velocity gradually tends to be constant.

(2) The mud outburst on the tunnel face can be summarized into three stages: initiation stage, acceleration stage, and the stable flow stage. Especially during the acceleration stage, rapid collapse of the tunnel working face occurs as bonding strength between particles increases, leading to a large number of particles being washed away. This highlights the importance of reinforcing advanced support and providing timely support during tunnel construction.

(3) Tunnel mud-burst damage is attributed to combined effects from various factors including geological conditions, water pressure, and excavation activities. Only when these three components reach certain conditions can a tunnel experience mud-burst disasters.

The methodology is described for 2D problems, but extending it to 3D cases has proven straightforward. The overall computational cost can be quite significant as a sufficiently fine lattice is frequently demanded and millions of time steps also have to be carried out.