Numerical Study of the Movement of Single Fine Particles in Porous Media Based on LBM-DEM

Abstract

The fine particle liquid–solid flow in porous media is involved in many industrial processes such as oil exploitation, geothermal reinjection and particle filtration. Understanding the migration characteristics of single fine particles in liquid–solid flow in porous media can provide micro-detailed explanations for the fine particle liquid–solid flow in porous media. In this paper, an existing lattice Boltzmann method–discrete element method (LBM-DEM) is improved by introducing a new boundary thickening direct forcing (BTDF) immersed boundary method (IBM) to replace the classical IBM. The new method is used to investigate the migrations of one, two or three fine particles in a flow field in porous media and the reactions of one, two or three fine particles on the flow field. It is found that the movement distance of a fine particle in porous media does not show a linear correlation with the fine particle’s density. A fine particle with a higher density may move a shorter distance and then stagnates. Although a fine particle with a smaller diameter has a better following performance in a flow field, it is also likely to be stranded in a low-infiltration area in porous media. Under the same increase ratio, the increase in the diameter of a fine particle causes an increased pressure drop of the liquid–solid flow. In some cases, the increase in the quantity of fine particles can intensify the disturbance of fine particles on the flow field, improving the movement of fine particles.

1. Introduction

Several materials in nature have porous media structure, including subsurface porous media, biological porous media and industrial porous media. Rock and soil are subsurface porous media. Microvascular networks and tissue spaces in the bodies of humans and other animals and in the roots, stems and leaves of plants are biological porous media. All types of filter meshes and granular bed filters are industrial porous media. Fine particles migrating in the flow field in these porous media form a typical liquid–solid flow in porous media. The migration, sedimentation and capture of fine particles in liquid–solid flow by porous media is a key problem. For example, artificial groundwater recharge systems will be blocked after a period of operation because there inevitably are some fine particle impurities in the system, resulting in a low efficiency of recharge even a premature discard of the system, so it is important to precisely predict the clogging process of fine particle impurities [1]. Slow sand filtration (SSF) is an effective water-treatment technology at a flow rate of 0.1–0.2 m³/h. It can efficiently remove turbidity, suspended solids and toxic metals in treated water and is efficient in removing microorganisms such as Giardia, Cryptosporidium, Salmonella, Escherichia coli, total and fecal coliforms, fecal streptococci, bacteriophages and MS2 virus from water/wastewater [2]. In the environmental protection industry, desulfuration wastewater containing fine particles must be cleaned by filtering through multilayer cloth or a granular bed filter before it is discharged into a river [3]. There are similar problems in the gas hydrate mining process [4] and the petroleum industry [5]. Therefore, it is important to investigate the fine particle liquid–solid flow in porous media. Further, in a few cases, the concentration of fine particles in liquid–solid flow in porous media is very dilute, and the explanations and analyses of some phenomena of fine particle group liquid–solid flow in porous media need to begin with the exploration of the phenomena of few fine particles liquid–solid flow in porous media.

Many experimental and theoretical studies have been reported. Gitis et al. [6] proposed a phenomenological deep-bed filtration model by cooperating an advection–dispersion equation with an equation of nonlinear multistage accumulation kinetics. This model can describe temporal and spatial changes of dispersion in media porosity. Khuzhayorov et al. [7] provided a mathematical model for the filtration of two-component suspensions in a dual-zone porous medium combining mass balance equations, the kinetic equations of each suspension component and Darcy’s law. It is shown that at the short distance between the medium and the input section causes the partial capacities in the passive zone of the concentration of deposited particles.

Alem et al. [8] experimentally studied a porous medium’s physical clogging by observing the migrations of suspended particles with diameters of 1.7 to 40 μm in a sand-filled column, and traced the dynamic behaviors of suspended particles at various flow velocities for a long time. It was shown that the flow velocity significantly affects the retention of the suspended particles. At low flow velocities, retention occurs in a limited length of porous medium, which insults in an evident changes of the porous medium’s hydraulic characteristics. Zhang et al. [9] examined the features of particle transport and deposition, and the penetration processes of a typical silica powder and fluorescein in saturated porous media under different pore structure and hydrodynamic forces using a series of column tests. They discovered that the effects of the hydrodynamics processes instead of pore structure on particle transport increase significantly with the increasing of seepage velocity. Du et al. [10] investigated the clogging behaviors of porous media with silt-sized suspended solids under different media size, size and concentration of suspended solids, and flow velocity on by classifying sand columns to controlled rates of flow and suspended-solid suspensions. The results showed that the size ratio of suspended solid particles to sand grains is the key influence factor for the position of physical clogging. As the pore velocity increases, the mobility of silt-sized suspended solids will improve, and the retention in the porous media will evidently decreases. The spatial retention profiles in porous media vary greatly at different flow velocities. Abbar et al. [11] studied the suspended particle transfer behaviors in the nonwoven flax fiber geotextiles with a sandy porous media in the saturated conditions. In the experiments they injected suspended particles in a column that is filled with sand containing or not containing flax fiber geotextiles at a constant flow rate. It was shown that the nonwoven flax fiber geotextiles can effectively enhance the flow homogeneity and the durability of the filtration system. The coarser particles mainly deposit at the inlet of the column and in the geotextiles. Bennacer et al. [12] studied the transport and deposition of suspended particles in saturated sand under the joint actions of ionic strength, particle size and flow velocity. They investigated three kinds of polydispersive suspended particle populations with a pulse injection technique. They found that the recovery and deposition of suspended particle strongly depends on the solution chemistry, the hydrodynamics, and the size of suspended particle. They proposed an empirical relationship about the influences of the ionic strength on the deposition kinetics, the sizes of particle and grain, the flow velocity.

In studies of liquid–solid flow, the analytical method often involves macroscopical law analysis and statistical models by observing experimental data and logical deduction, and numerical methods can provide more mesoscopic or microcosmic detailed information [13], which relies on the accuracy of mathematic model. Because it is difficult to observe the movement of a mesoscopic or microcosmic fine particle liquid–solid flow, a numerical method is a better choice.

Wang et al. [14] studied the flow behavior of particles in porous media by means of Computational Fluid Dynamics (CFD) and Distinct Element Method (DEM). It was found that particles will be blocked and deposited in porous media at a low porosity. The increase in porosity prolongs the trajectory of particles but decreases the residence time, and prompts particles to perform reciprocating motion and flow out the microchannel. The lattice Boltzmann method (LBM) has fully demonstrated its high-efficiency in solving the mesoscopic-scale flow and heat transfer problems because of its simpler mathematic model and more convenient programming [15,16], so it has been popularized widely in the last 10 years. There are four kinds of liquid–solid coupling schemes for LBM, which are fluid drag model (seldomly used), standard bounce-back (SBB), the immersed moving boundary method and the immersed boundary method (IBM) [17,18]. IBM was created most recently, but it has been the fastest-developing and predominantly used method due to its high convenience [19,20]. An increasing number of researchers study the mesoscopic scale gas–solid/fluid–solid two-phase flow problems by coupling LBM, IBM and discrete element method (DEM). Jiang et al. [21] developed a numerical model for simulating the solid–liquid–gas three-phase flow in unconsolidated particle layers. They proposed a multiphase fluid–solid two-way coupling algorithm based on the DEM and the multiphase fluid model in the framework of the LBM. In this model, the phase-field method is used to handle the fluid–fluid interface, and the immersed boundary method is used for the multiphase fluid–particle interaction. Ma et al. [22] introduced the super-ellipsoid model to IBM-LBM-DEM for particles with arbitrary shapes to describe non-spherical particles, and adopted an appropriate method to distribute the discrete marker points on the surface of a super-ellipsoid. Farahani et al. [23] presented a pore-scale numerical model of fines liquid-solid in porous media by using LBM-DEM solver for the fluid flow with a rigid body physics engine describing the movement of particles. IBM was implemented in order to analyze the exerted force that the fluid acts on the particles and subsequent particulate movements. Li et al. [24] estimated the heat and mass transfer characteristic in Magneto-hydrodynamics (MHD) Williamson nanofluid flow on an exponentially porous stretching surface controlled by the heat generation/absorption and mass suction and discussed two different conditions of heat transfer. They pointed out the Sherwood number increases with the increase of the Prandtl number and Lewis number. Khan et al. [25] explored a two-dimensional unsteady radiative stagnation point flow of Casson fluid along a stretching/shrinking sheet dependent on the mixed convection, convective and slip condition. It had been found that the friction drags are bigger in the stable flow owing to the increase of Casson parameter, mixed convection, and slip velocity. Alshehri et al. [26] studied the two-dimensional unsteady radiative viscous nanofluid flow in an aligned magnetic field involving the suction, velocity slip, and heat source crossing a porous convective stretching/shrinking surface. They found that the rate of heat transfer continuously improves but the friction drag declines against the increase of stretching, respectively.

In this work, we concentrate on the migration characteristics of one, two or three fine particles in porous media using a modified LBM-DEM model because this study is a prerequisite for the exploration of migration characteristics of fine particle group in porous media. At the same time, our research practices show that the classical IBM [27] has poor stability when it is used to handle liquid–solid coupling with a big density difference between liquid and solid. We tried several kinds of modified IBM models, and determined that the boundary thickening direct forcing (BTDF) IBM proposed by Jiang et al. [28] is more precise and efficient. We introduce the BTDF IBM model into our existing LBM-DEM model [29] so as to combine LBM, DEM with BTDF IBM, thus propose a modified LBM-DEM platform with better precision and efficiency for fine particle liquid–solid flow. These two LBM-DEM platforms are both validated by the classical drafting–kissing–tumbling (DKT) effect of two circular particles (A and B) settling in a rectangular channel and the two-dimensional flow around a stationary cylinder. Moreover, the fine particle liquid–solid flow in porous media with one, two or three suspended fine particles at different working conditions is simulated, the mesoscopic-scale detailed information of the fine particle liquid–solid flow in porous media is traced, and the influences of the density and diameter of fine particle and the channel shape of porous media on the migrations of fine particles and the effects of the existing fine particles on the flow field in porous media are analyzed and discussed. The conclusions provide a solid foundation for the analyses of fine particle group liquid–solid flow phenomena.

2. Mathematic Model

2.1. Model of LBM

The standard Lattice-BGK (LBGK; B, G and K are the first letters of the names of Bhatnagar, Gross and Krook) model has compressible effect when used to solve incompressible flow, leading to a drastic change in the density of local fluid. Therefore, in this work, an incompressible D2G9 model [30] instead of the compressible standard LBGK D2Q9 model is employed to solve the flow field. Zou et al. [31] proposed the first LBM model, which was named Zou–Hou model. However, Lin et al. [32] found a contradiction in the Zou–Hou model, namely, that the density is variational when the pressure is calculated, and they presented a modified model, in which the density is set to be a constant 1. These two incompressible LBM models only can solve steady fluid flow problems. He and Luo [33] proposed a model that can solve unsteady fluid flow problems, and in this model, the higher-order Mach term in the equilibrium distribution function caused by density fluctuation is canceled. Guo et al. [30] pointed out that all of the above-mentioned LBM models have a common flaw, that is ${\displaystyle \sum f_{\alpha }^{\mathrm{eq}}}\ne \mathrm{const}$, so the continuous equations deduced by these LBM models do not meet incompressibility. Thus, they proposed an LBM model that can make ${\displaystyle \sum f_{\alpha }^{\mathrm{eq}}}=\mathrm{const}$, and in this model, pressure becomes a variable instead of density, whereas density is a constant. This LBM model was named the D2G9 model. As shown in Figure 1, the dimension and discrete velocities of the D2G9 model are the same as those of the LBGK D2Q9 model; the black nodes are the discrete nodes of the flow field, and 0–8 demonstrate 9 discrete velocity arrangements (described in Equation (10) below), respectively.

The evolution equation of the D2G9 model with the source term is:

$$g_{\alpha }\left(x+e_{\alpha }\Delta t,t+\Delta t\right)=g_{\alpha }\left(x,t\right)-{\displaystyle \frac{1}{\tau }}\left[g_{\alpha }\left(x,t\right)-g_{\alpha }^{\left(\mathrm{eq}\right)}\left(x,t\right)\right]+G_{\alpha }\left(x,t\right)\Delta t$$

(1)

in which $g_{\alpha }$ and $g_{\alpha }^{\left(\mathrm{eq}\right)}$ are the distribution function and the equilibrium distribution function in the $\alpha$-direction, respectively. $e_{\alpha }$ and $G_{\alpha }$ present the discrete velocity and the component of external force in the $\alpha$-direction, respectively. The expression of external force is introduced in Section 2.3, and the equilibrium distribution function is defined as:

$$g_{\alpha }^{\mathrm{eq}}=\left\{\begin{array}{l}{\rho }_0-4\sigma {\displaystyle \frac{p}{c^2}}+s_0\left(u\right),\hspace{1em}\alpha =0\\ \lambda {\displaystyle \frac{p}{c^2}}+s_{\alpha }\left(u\right),\hspace{1em}\hspace{1em}\hspace{1em}\alpha =1,2,3,4\\ \gamma {\displaystyle \frac{p}{c^2}}+s_{\alpha }\left(u\right),\hspace{1em}\hspace{1em}\hspace{1em}\alpha =5,6,7,8\end{array}\right.$$

(2)

where constant ${\rho }_0$ is the fluid density, $s_{\alpha }\left(u\right)$ is defined by Equation (3), and $\sigma$, $\lambda$ and $\gamma$ are some model parameters and satisfy the Equation (4):

$$s_{\alpha }\left(u\right)={\omega }_{\alpha }\left[{\displaystyle \frac{e_{\alpha }\cdot u}{c_s^2}}+{\displaystyle \frac{{\left(e_{\alpha }\cdot u\right)}^2}{2c_s^4}}+{\displaystyle \frac{u^2}{2c_s^2}}\right]$$

(3)

$$\left\{\begin{array}{c}\lambda +\gamma =\sigma \\ \lambda +2\gamma =1/2\end{array}\right.$$

(4)

where $c_s$ is the lattice sound speed, and ${\omega }_{\alpha }$ is the weight coefficient for the $\alpha$-direction. In D2G9, the discrete velocity model is given as:

$$c=\Delta x/\Delta t$$

(5)

$$c_s=c/\sqrt{3}$$

(6)

where $c$, $\Delta x$ and $\Delta t$ are the lattice velocity, lattice spacing and time step, respectively. The weighting factor ${\omega }_{\alpha }$ in the $\alpha$-direction is shown as:

$${\omega }_{\alpha }=\left\{\begin{array}{ll}{\displaystyle \frac{4}{9}},\hfill & \alpha =0\hfill \\ {\displaystyle \frac{1}{9}},\hfill & \alpha =1,2,3,4\hfill \\ {\displaystyle \frac{1}{36}}\hfill & \alpha =5,6,7,8\hfill \end{array}\right.$$

(7)

The macro pressure $p\left(x,t\right)$ and velocity $u\left(x,t\right)$ of the fluid can be calculated using:

$$p\left(x,t\right)={\displaystyle \frac{c^2}{4\sigma }}\left[{\displaystyle \sum _{i\ne 0}{g}_{\alpha }\left(x,t\right)+s_0\left(u\right)}\right]$$

(8)

$$u\left(x,t\right)={\displaystyle \sum _{\alpha }{e}_{\alpha }{g}_{\alpha }\left(x,t\right)}/{\rho }_0$$

(9)

The 9 discrete velocities are provided as:

$$e_{\alpha }=\left\{\begin{array}{l}\left(0,\hspace{0.17em}0\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha =0\\ c\left\{\cos \left[\left(\alpha -1\right){\displaystyle \frac{\pi }{2}}\right],\hspace{0.17em}\sin \left[\left(\alpha -1\right){\displaystyle \frac{\pi }{2}}\right]\right\},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha =1,2,3,4\\ \sqrt{2}c\left\{\cos \left[\left(2\alpha -1\right){\displaystyle \frac{\pi }{4}}\right],\hspace{0.17em}\sin \left[\left(2\alpha -1\right){\displaystyle \frac{\pi }{4}}\right]\right\},\hspace{1em}\hspace{1em}\alpha =5,6,7,8\end{array}\right.$$

(10)

2.2. Solid–Solid Coupled Model

Porous media are made up of many big particles, so the internal walls of porous media are just the surfaces of these big particles. The collision force between a fine particle and a large particle (an internal wall of porous media) is analyzed by the soft sphere model (DEM). In DEM, there will be a slight overlap between two discrete particles if they collide with each other, resulting in the deformation force, as shown in Figure 2.

When particles i and j collide with each other, there will be normal and tangential contact forces ${\mathit{F}}_{n,ij}$, and ${\mathit{F}}_{t,ij}$. ${\mathit{F}}_{n,ij}$ is calculated in accordance with the linear spring model:

$${\mathit{F}}_{n,ij}=\left(-k_n{\delta }_{n,ij}-{\eta }_n{\mathit{u}}_{ij}{\mathit{n}}_{ij}\right){\mathit{n}}_{ij}$$

(11)

where ${\delta }_{n,ij}$ is the normal overlap of particles i and j. $k_n$ is the displacement-dependent elastic normal stiffness, mainly depending on the contact radius of two particles, Young’s modulus and Poisson’s ratio. ${\mathit{n}}_{ij}$ is the unit vector from particle i to particle j. The normal damping coefficient ${\eta }_n$ is given as:

$${\eta }_n=2\alpha \sqrt{m_{ij}{k}_n}$$

(12)

$$\alpha =\beta /\sqrt{1+{\beta }^2}$$

(13)

$$\beta =-{\displaystyle \frac{1}{\pi }}\cdot ln{e}_n$$

(14)

where $e_n$ is the restitution coefficient.

The tangential contact force ${\mathit{F}}_{t,ij}$ is provided as:

$${\mathit{F}}_{t,ij}=\left\{\begin{array}{cc}-k_t{\delta }_t-{\eta }_t{\mathit{u}}_{t,ij}& \left|{\mathit{F}}_{t,ij}\right|\le \theta \left|{\mathit{F}}_{n,ij}\right|\\ -\mu \left|{\mathit{F}}_{n,ij}\right|{\mathit{t}}_{ij}& \left|{\mathit{F}}_{t,ij}\right|\ge \theta \left|{\mathit{F}}_{n,ij}\right|\end{array}\right.$$

(15)

where $k_t$ is the tangential stiffness coefficient. ${\delta }_t$, ${\eta }_t$ and $\theta$ represent the tangential displacement, damping coefficient and tangential friction coefficient of particles, respectively. ${\mathit{u}}_{t,ij}$ is the slip velocity of the contact point:

$${\mathit{u}}_{t,ij}=\left({\mathit{u}}_i-{\mathit{u}}_j\right)-\left({\mathit{u}}_{ij}\cdot {\mathit{n}}_{ij}\right){\mathit{n}}_{ij}+\left(R_i{\mathit{\omega }}_i+R_j{\mathit{\omega }}_j\right)\times {\mathit{n}}_{ij}$$

(16)

where R_i and R_j are the radii of particle i and j, and ${\mathit{\omega }}_i$ and ${\mathit{\omega }}_j$ are the angular velocities of particle i and j. ${\mathit{t}}_{ij}$ is the tangential unit vector and is expressed as:

$${\mathit{t}}_{ij}={\displaystyle \frac{{\mathit{u}}_{t,ij}}{\left|{\mathit{u}}_{t,ij}\right|}}$$

(17)

The contact force ${\mathit{F}}_c$ and torque ${\mathit{T}}_c$ of particle i is detailed as follows:

$$\left\{\begin{array}{l}{\mathit{F}}_c={\displaystyle \sum _{j=1}\left({\mathit{F}}_{n,ij}+{\mathit{F}}_{t,ij}\right)}\\ {\mathit{T}}_c={\displaystyle \sum _{j=1}{R}_i{\mathit{n}}_{ij}\times {\mathit{F}}_{t,ij}}\end{array}\right.$$

(18)

The parameters of the DEM are given in

Table 1. Parameters of the discrete element method (DEM).

Physical Property Parameter	Values
Normal elastic coefficient of particles $k_n$ (g/s2)	8 × 105
Coefficient of restitution of particles $e_n$	0.9
Friction coefficient of particles $\mu$	0.3
Time step (s)	0.0000012963

.

The masses and initial velocities of large particles are set to the infinite and 0 m/s so as to keep these particles static amid the simulation.

2.3. Liquid–Solid Coupled Model

Here the implementation steps of the modified boundary thickening direct forcing IBM is exhibited. The external force term G_α (x, t) in Equation (1) is defined as:

$$G_{\alpha }\left(x,t\right)=\left(1-{\displaystyle \frac{1}{2\tau }}\right){\omega }_{\alpha }\left[3{\displaystyle \frac{e_{\alpha }-u\left(x,t\right)}{c^2}}+9{\displaystyle \frac{e_{\alpha }\cdot u\left(x,t\right)}{c^4}}{e}_{\alpha }\right]\cdot \mathit{f}\left(x,t\right)$$

(19)

where $\mathit{f}\left(x,t\right)$ is the local force acting on fluid calculated by the modified IBM, which is given by Equation (22). $\tau$ is the dimensionless relaxation time.

Obtaining the IB-related force densities of the Lagrange points on the particle and then extending them to the adjacent Euler grid points are key steps. The Lagrange force density is expressed as:

$${\mathit{F}}_b={\displaystyle \frac{2\rho }{\Delta t}}\left({\mathit{U}}_b-D_I{\mathit{u}}^{✲}\right)$$

(20)

where ${\mathit{U}}_b$ is the velocity of boundary discrete point of the particle, $\rho$ is the fluid density, $D_I{\mathit{u}}^{✲}$ represents the velocity interpolation of the fluid points near the Lagrange point, and $D_I$ is a matrix with interpolation items, which is expressed as:

$$D_{I,ij}={\displaystyle \frac{1}{\mathrm{d}{h}^2}}{\varphi }_{ij}\left({\displaystyle \frac{x_i-X_j}{\mathrm{d}h}}\right){\varphi }_{ij}\left({\displaystyle \frac{y_i-Y_j}{\mathrm{d}h}}\right)\cdot \mathrm{d}{h}^2$$

(21)

in which $x_i$ and $y_i$ represent fluid Euler coordinates, $X_j$ and $Y_j$ represent solid Lagrange coordinates, and ${\varphi }_{ij}$ is the Dirac function; $\mathrm{d}h=\Delta x$. The additional force corresponding to the expansion of Lagrange force density to Euler point is given as:

$$\mathit{f}\left(x,t\right)=D_E{\mathit{F}}_b$$

(22)

where $D_E$ is a matrix with extensions, which is expressed as:

$$D_{E,ij}={\displaystyle \frac{1}{\mathrm{d}{h}^2}}{\varphi }_{ij}\left({\displaystyle \frac{x_i-X_j}{\mathrm{d}h}}\right){\varphi }_{ij}\left({\displaystyle \frac{y_i-Y_j}{\mathrm{d}h}}\right)\cdot \left(\mathrm{d}rs\cdot \mathrm{d}s\right)$$

(23)

in which drs is thickness of the solid forcing shell, and ds is the distance between two adjacent Lagrange points on the solid boundary. The velocity of the Euler point of the fluid is further corrected by the following equation:

$$\mathit{u}={\mathit{u}}^{✲}+{\displaystyle \frac{\Delta t}{2\rho }}\mathit{f}\left(x,t\right)$$

(24)

The fluid force and torque of the particle is given as:

$$\left\{\begin{array}{c}{\mathit{F}}_h=-{\displaystyle \sum _{i=1}^{N_L}{\mathit{F}}_{bi}\cdot \left(\mathrm{d}s\cdot \mathrm{d}rs\right)=-{\displaystyle \sum _{j=1}^{N_E}{\mathit{f}}_j\cdot \mathrm{d}{h}^2}}\\ {\mathit{T}}_h=-{\displaystyle \sum _{i=1}^{N_L}\left({\mathit{X}}_{bi}-{\mathit{X}}_C\right)\times {\mathit{F}}_{bi}\cdot \left(\mathrm{d}rs\cdot \mathrm{d}s\right)}\end{array}\right.$$

(25)

where $N_L$ is the number of Lagrange points on the particle boundary, $N_E$ is the number of local Euler meshes around the particle, ${\mathit{X}}_{bi}$ is the Lagrange point coordinate, and ${\mathit{X}}_C$ is the center coordinate of the particle mass. The Dirac function is expressed as:

$$\varphi \left(r\right)=\left\{\begin{array}{l}0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left|r\right|>1.5\\ {\displaystyle \frac{1}{6}}\left[5-3\left|r\right|-\sqrt{1-3{\left(1-\left|r\right|\right)}^2}\right],\hspace{1em}\hspace{1em}0.5<\left|r\right|<1.5\\ {\displaystyle \frac{1}{3}}\left(1+\sqrt{1-3{\left|r\right|}^2}\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{1em}\left|r\right|\le 0.5\end{array}\right.$$

(26)

2.4. Dynamics Model of Discrete Particle

The translation and rotation of particle i is calculated by Newton’s second law, which can be formulated as:

$$m_i{\displaystyle \frac{\mathrm{d}{\mathit{u}}_i}{\mathrm{d}t}}={\mathit{F}}_h+{\mathit{F}}_c+{\mathit{F}}_g$$

(27)

$${\mathit{I}}_i{\displaystyle \frac{\mathrm{d}{\mathit{w}}_i}{\mathrm{d}t}}={\mathit{T}}_h+{\mathit{T}}_c$$

(28)

where $m_i$ and ${\mathit{I}}_i$ represent the mass and rotational inertia of particle i, and ${\mathit{F}}_g$ is the gravity.

3. Physical Model and Numerical Simulation Conditions

The simulation area is 2 cm in length (X-axis) and 4 cm in width (Y-axis) The lattice resolution is N_x × N_y = 1600 × 3200. The density of fluid is 1.00 g/cm³, and the inlet fluid velocity (V_in) is −2 cm/s, which indicates that the fluid flows from the entrancement to the exit, as shown in Figure 3. The non-slip boundary condition is applied to the left and the right boundaries. The outlet boundary of the fluid is calculated by a fully developed scheme. The parameters of liquid–solid flow are given in

Table 2. Parameters of liquid–solid flow.

Parameter	Value
Density of fluid (g/cm3)	1.0
Inlet velocity of fluid Vin (cm/s)	−2.0
Length in X-axis L (cm)	2
Height in Y-axis W (cm)	4
Diameter of large particles (cm)	0.16

.

The diameter of large particles is 0.16 cm. Different quantities of large particles are stochastically placed in the appointed area, producing different porosities of porous media. The porosity of porous media is defined as the ratio of the volume of channels among large particles to the volume of the porous media zone. In a two-dimensional porous media zone, the porosity is given by the following equation:

$$\epsilon =1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^N{S}_i}}{A}}$$

(29)

where S_i is the area of large particle i, and A is the area of the porous media zone. In this work, a porous medium with a porosity of 0.5 composed of 149 large particles is chosen as the simulated porous medium.

The flow field with porous medium area is calculated first in order to generate an initial flow field, and then one, two or three fine particles are positioned in the area close to the entrance with an initial velocity of 0. The simulation of fine particle liquid–solid flow is then activated. The single fine particle in the cases of single fine particle liquid–solid flow and fine particle A in the cases of two or three fine particles liquid–solid flow are located at (1 cm, 3.1 cm) initially. Correspondingly, this moment is defined as the start-up time (i.e., t = 0 s), and the simulation ends with all fine particles stagnating somewhere in the porous media or passing through the porous media zone. The fine particles will be carried into the porous media zone by the fluid field and their own gravity after several time steps.

4. Model Validation

In order to validate the modified LBM-DEM framework, a simulation of 504 particles settling in the closed square cavity is implemented. Figure 4 shows the velocity distributions of the flow field and the position distributions of 504 particles settling with Rayleigh–Taylor instability over time, as presented in the literature [28] and in our simulation. Based on the literature, the Navier–Stokes equation was adopted as the incompressible fluid governing equations, and particle–particle coupling was handled by the repulsive force model presented by Glowinski et al. [34]. The simulated results of the LBM-DEM framework are very similar to those in the literature, so the LBM-DEM framework is considered to be sufficiently accurate and effective.

5. Numerical Simulation Results and Analysis

5.1. Migrations and Sedimentations of Fine Particles in Porous Media under Different Particle Densities

Once a fine particle enters a porous media zone, it will experience fluid drag, gravity and collision forces between itself and a large particle (i.e., an internal wall of porous media) simultaneously. The density of the fine particle is a very important factor influencing its movement in porous media. Here, the migrations and sedimentations of a fine particle in porous media under eight different particle densities are simulated. The densities of fine particles (ρ_p) are 1.0, 1.01, 1.50, 2.00, 2.10, 2.20, 2.50 and 3.00 g/cm³, respectively. The diameter of the fine particles (d_p) is 0.024 cm.

Precise information of the flow field in porous media is the foundation for analyzing the migration behaviors of single particles, so the velocity distribution of the flow field in porous media in and perpendicular to the Y-axis is given in Figure 5, which reveals that the flow field in porous media is disordered due to the disturbances of stochastically-distributed large particles. It can be seen that there are several areas in porous media in which the local fluid velocities on the Y-axis (V) approach 0, so a fine particle almost cannot pass through; however, in some other areas, the local fluid velocities on the Y-axis are very high, allowing a fine particle to pass through easily. These two kinds of areas are defined as low-infiltration and high-infiltration areas, respectively. The high-infiltration areas link with each other to form flow channels, and the flow channels basically match the streamlines of interstitial flow (red solid lines in Figure 5). Figure 6 shows the movement trajectories of a fine particle in porous media under eight different particle densities. The color of the track demonstrates the velocities of a fine particle on the Y-axis (v) at different positions. A fine particle under different densities generally migrates along the flow channel. However, the movement trajectory and route distance of fine particles with different densities vary, and the fine particles are ultimately deposited or stagnate in different positions.

Fine particles at a density of 1.00 or 1.01 g/cm³ can pass through some pore throats along the streamline of interstitial flow, but eventually, they will deviate from the streamline, then enter and stagnate in a low-infiltration area after passing through the pore throat at position b in Figure 5a. There is an evident velocity gradient of laminar flow in the pore throat, but the velocity distribution is not a paraboloid shape of Poiseuille interstitial flow, so the shearing field in the pore throat is uneven, resulting in the stochastic deviation of the moving fine particles. At the same time, Figure 6 shows that the velocity of a fine particle along the Y-axis attenuates seriously when it passes through the pore throat at position a; as a result, the fine particle lacks enough kinetic energy to escape from the low-infiltration area.

At a density of 1.50 or 2.00 g/cm³, the fine particle deviates from the streamline of interstitial flow at position b because of its greater inertia. The fine particle continuously collides with the internal walls of porous media, losing some kinetic energy and thus being deposited on an internal wall.

At a density of 2.10 or 2.20 g/cm³, as the fine particle enters the porous media zone, although it temporarily stagnates at position c on the internal wall of the porous media, it will slide along the internal wall due to gravity and fluid drag and eventually leaves the internal wall, thereby entering a flow channel again. After that, the fine particle deviates from the streamline of interstitial flow at position a in Figure 5a (i.e., position d in Figure 6) again because an obvious contortion of the streamline of interstitial flow takes place here. However, the fine particle’s gravity is stronger, so the fine particle enters a low-infiltration area and is deposited there after colliding with the internal wall several times. At a density of 2.10 g/cm³, a fine particle shows a trend of moving along the main stream direction on account of gravity, but an eddy that occurs in the local flow field at position e provides a force along the Y-axis. This force causes the fine particle to be in an equilibrium state, as shown in Figure 7, so that it is always suspended in a low-infiltration area.

The migration trajectory and velocity on the Y-axis of the fine particle at a density of 2.50 g/cm³ differ from those of fine particles at densities of 2.10 or 2.20 g/cm³. As shown at position b in Figure 5 (position f in Figure 6), the channel turns right and prompts fine particles to turn right, but the phenomenon weakens with the increase in the fine particle density, because a higher density means higher inertia and gravity. Therefore, at a density of 2.50 g/cm³, the fine particle deviates from the streamline of interstitial flow when arriving at position f in Figure 6 and then moves to position g under gravity instead of being accelerated by the local flow field in the channel.

Due to stronger gravity, when the fine particle at a density of 3.00 g/cm³ reaches position h, it slides a long distance on the internal wall after colliding with the wall several times, and finally reaches position I, where it is deposited under gravity, as shown in Figure 5. A fine particle with a different density migrates along its own movement trajectory when moving from position h to i; fine particles with a density of 1.00 or 1.01 g/cm³ smoothly migrate along the streamline of interstitial flow without colliding with the internal wall due to the smaller mass (that is, a smaller Stokes number); fine particles with a density of 1.50 or 2.00 g/cm³ diverge from the streamline and are deposited at position i; fine particles with densities of 2.10, 2.20 or 2.50 g/cm³ do not deviate from the streamline but move longer distances towards the right with an increase in the fine particles’ densities. In order to further analyze the phenomena, the velocities perpendicular to the Y-axis (u) and along the Y-axis (v) of fine particles with densities of 1.50, 2.00, 2.10, 2.20, 2.50 or 3.00 g/cm³ in the process of migrations in porous media are extracted and shown in Figure 8. Figure 8a shows that when a fine particle’s density increases, the velocity perpendicular to the Y-axis of the fine particle becomes smaller at position h but dampens less after the fine particle leaves position h; the reason for this is that the higher mass results in greater inertia. Figure 8b demonstrates that the velocity along the Y-axis (v) of a fine particle with different density is similar except for that of the fine particle with a density of 3.00 g/cm³ during its moving from position h to i. Analysis suggests that the movement of a fine particle is affected by two factors: (1) the sliding distance after leaving position h and (2) the fine particle’s density. A fine particle with a higher density will slide longer distances and be accelerated over a shorter distance in the channel after leaving position h, but at the same time its acceleration of gravity is bigger, leading to the similar velocity on the Y-axis. While moving from position h to i, a fine particle with a higher density, except a fine particle with a density of 3.00 g/cm³, horizontally migrates farther because the attenuation of the velocity perpendicular to the Y-axis (u) is slower due to higher inertia.

5.2. Migration and Sedimentation of a Fine Particle in Porous Media under Different Diameters

It is beyond doubt that the size (diameter) of fine particles is a very important factor influencing the migration of a single fine particle in porous media. For example, a fine particle cannot pass through a pore throat and then block the pore throat if its size is bigger than that of the pore throat, and the pore throat and the local fluid in the pore throat are clogged concurrently. The migrations and deposition behaviors of a fine particle in porous media under different particle diameters are investigated by simulating the movement of a fine particle under different particle sizes. The diameters of sample fine particles are 0.016, 0.024, 0.032, 0.040 and 0.048 cm, respectively.

Figure 9 shows the movement trajectories of fine particles of different diameters in porous media. It can be seen that fine particles with different diameters have different movement trajectories and migration velocities; a fine particle with a smaller diameter has less inertia and can pass through a pore throat more easily. For example, at position a, the fine particle with a diameter of 0.016 cm is accelerated in the pore throat; in contrast, a fine particle with a diameter of 0.024 cm will in the end tarry in a low-infiltration area owing to the shortage of enough kinetic energy after deviating from the streamline of the interstitial flow. A fine particle with a diameter of 0.032 cm moves along a very flexural route during its migration in porous media. It will deviate from the streamline of interstitial flow because of inertia and will be deposited at position c after passing through the pore throat at position b; moreover, it migrates very slowly at positions d and e, and it diverges the streamline of interstitial flow and is then deposited again at position f. The increase in a fine particle’s diameter weakens its penetration performance and thus seriously hampers the migration of the fine particle in porous media.

The velocity of a fine particle with a diameter of 0.040 cm is clearly lower than that of a fine particle with a diameter of 0.032 cm when it passes through the pore throat at position g; in addition, the fine particle with a diameter of 0.040 cm is blocked in the pore throat at position h due to its large size. In the same way, the fine particle with a diameter of 0.048 cm is blocked when passing the pore throat at position i. The larger size of this particle brings about its earlier blockage, which in turn reshapes the flow field in the porous medium. The blockage is mutual, as the local fluid channel in the porous medium is blocked while a fine particle is lodged there. A fine particle with a smaller diameter is also likely to be stranded in a low-infiltration area while it passes through a pore throat more easily.

5.3. Effect of Fine Particles on the Flow Field in Porous Media

There are distinct reactions in the flow field if one or more fine particles exist in the flow field. When a fine particle moves within the flow field in porous medium, especially when it blocks a local fluid channel, it will disturb the entire flow field. The effects of fine particles on the flow fields in porous media are examined by detecting changes in the pressure drop of the flow field between the exit and entrance of porous media.

Figure 10 shows the changes in pressure drop of the flow field between the exit and entrance of the porous media over time under different working conditions. When the density varies, the diameter is 0.024 cm; when the diameter varies, the density is 1.00 g/cm³. As shown in Figure 10, there are several peaks, which indicate that several fluctuations in pressure drop of the flow field will occur during the migration of a fine particle. Figure 11 shows the movement trajectories of a fine particle under different working conditions. Figure 10a–d correspond one-to-one with Figure 11a–d; point a in Figure 10a corresponds to point a in Figure 11a, and so on. Figure 10a displays the change in pressure drop of the flow field over time when the fine particle’s density is 1.00 g/cm³. Comparing Figure 10a and Figure 11a, it can be observed that the peak at position a occurs when the fine particle enters the porous medium. When a fine particle stays somewhere in the porous medium, the local flow channel will be clogged to some extent, resulting in the decrease in sectional area of the flow channel and therefore an increase in the local fluid velocity; at the same time, the resistance coefficient of the fluid also increases because the fluid must bypass the fine particle. The pressure drop of the flow field is in direct proportion to the resistance coefficient and the square of fluid velocity. There will be an evident momentum transfer between the local flow field and the fine particle moving locally by drag forces and buoyancy, and the higher velocity difference between the local flow field and the fine particle brings about a greater momentum transfer. When the fine particle passes through a larger pore throat, the pressure drop of the flow field will also fluctuate, but the fluctuation intensity is smaller, as shown at positions c and d in Figure 10a and Figure 11a. However, when the fine particle passes through the pore throat at position e, the pressure drop of the flow field changes little. It may be that the velocity difference between the local flow field and the fine particle is not large enough, so that the fine particle does not exert any resistance force on the local flow field, as shown in Figure 11. Obviously, the pressure drop of the flow field is the highest at position f. The pore throat at position f (position b in Figure 5) is very small, so the local fluid velocity is very large; at the same time, it can be seen from Figure 10a and Figure 11a that the migration velocity of the fine particle in the pore throat at position f is very small and the residence time is long, which is responsible for the highest pressure drop in the flow field.

Figure 10b shows the change in the pressure drop of the flow field over time when a fine particle with a density of 2.20 g/cm³ migrates in porous media. Compared to a fine particle with a density of 1.00 g/cm³, when the 2.20 g/cm³ particle migrates, the peak values of pressure drop occur at same positions except for position e but the occurrence time is slightly lagged; there is one more peak value of pressure drop at position e. By carefully tracking the movement trajectory, when the fine particle with a density of 2.20 g/cm³ reaches the pore throat at position e, it collides with a large particle (an internal wall of porous media), losing some momentum, so it requires more hydrodynamic force to pass through the pore throat, resulting in the occurrence of the peak value of pressure drop at position e. It can also be seen from Figure 12 that at position e, the velocity of the fine particle with a density of 2.20 g/cm³ is slightly less than that of the fine particle with a density of 1.00 g/cm³. It can be concluded that the more resistance force the fine particle exerts on the flow field, the more the pressure drop fluctuates.

Figure 10c shows the change in the pressure drop of the flow field over time when the diameter of the fine particle is 0.032 cm. The fluctuation in the pressure drop of the flow field before the fine particle arrives at position g is more drastic than that when the fine particle’s diameter is 0.024 cm (as shown in Figure 10a). When a larger fine particle passes through a pore throat, it will exert a greater resistance force on the local flow field in the pore throat. When the fine particle arrives at position g and the pore throat is clogged, the pressure drop of the flow field increases greatly. The pressure drop of the flow field is greater when the fine particle blocks the pore throat at position h than when it blocks the pore throat at position g because the local fluid velocity in the pore throat at position h is higher, so the flow field suffers more. When the fine particle passes through the pore throats at positions i and j, a pressure drop in the flow field obviously appears but recovers soon because of the lack of clogging.

Figure 10d shows the change in the pressure drop of the flow field over time when the diameter of the fine particle is 0.040 cm. The fine particle blocks the pore throat at position k for a long time and eventually is lodged in the pore throat at position i owing to its large particle size; at the same time, the pressure drop in the flow field remains very high.

From Figure 10 we can see that, when the diameter of the fine particle changes from 0.032 cm to 0.040 cm, a 1.25× increase, the peak value of the pressure drop changes from about 255 Pa to about 262 Pa; meantime, when the density of the fine particle changes from 1.00 g/cm³ to 2.20 g/cm³, a 2.20× increase, the peak value of the pressure drop changes from about 257 Pa to about 258 Pa. The increase in the diameter of the fine particle causes a greater increase in pressure drop of the flow field compared with the increase in the density of the fine particle.

5.4. Migration and Sedimentation of Two Fine Particles in Porous Media

The migration of a fine particle in porous media is not only affected by its density and diameter, the size of pore throats and the local fluid velocity, but also on other existing fine particles. The mutual influencing mechanisms between two or three non-contact fine particles in porous media are examined by tracking the migrations of two non-contact fine particles in this section and the migrations of three non-contact fine particles in the next section at a fine particle density of 1.00 g/cm³.

We calculated four initial conditions, which are formed by four initial positions of two fine particles. The initial positions of particle B are (0.5 cm, 3.1 cm), (0.75 cm, 3.1 cm), (0.875 cm, 3.1 cm) and (1.25 cm, 3.1 cm).

Figure 13 displays the migration trajectories of two fine particles under four calculated initial conditions, along with the migration trajectory of a single fine particle, A. We can clearly see that the participation of fine particle B causes an alteration in the migration trajectory of fine particle A, and different initial positions result in different influences.

When the initial position distance between fine particles A and B is 0.500 cm, as shown in Figure 13a, the migration trajectory changes at position a in comparison with the case of single fine particle A. Fine particle A collides with large particles (internal walls of porous media) and loses some kinetic energy; therefore, it is not able to continue moving along the streamline of interstitial flow. It deviates from the streamline and then collides with large particles and loses kinetic energy again. Ultimately, it is lodged at position c. After fine particle B is lodged at position d, the fluid flowing in the pore throat at position d must enter other channels, increasing the local fluid velocities in channels I and II, as shown in Figure 14. When fine particle A passes through position e, it obtains a lot of kinetic energy in the direction perpendicular to the mainstream because channel II at position e is curved, as shown in Figure 15, so fine particle A deviates from the streamline and then collides with larger particles before it ultimately becomes stuck.

When particle A is 0.250 cm from particle B on the left, as shown in Figure 13b, the movement of particle A is not impeded and its migration trajectory is the same as in the absence of particle B. The reason for this may be that particle B migrates smoothly in porous media and does not damage the channels through which particle A travels. At the same time, particle A is suspended at position g before particle B is lodged at position f. The clogging of particle B does not change the local flow field around particle A, and therefore does not change the movement situation of particle A.

When particle A is initially 0.125 cm away from particle B, as shown in Figure 13c, the velocity on the Y-axis of particle A is higher than that when only particle A exists. This is because when particle A passes through position h, particle B moves very slowly in the pore throat at position i and hinders the flow of fluid here; as a result, the local fluid velocity in the channel where particle A resides increases, transferring more kinetic energy to particle A. As particle A passes through position j, the velocity of particle A on the Y-axis is higher, because before particle A arrives at position j, particle B has been blocked at position k. This blockage increases the local fluid velocity around particle A, thus promoting particle A to obtain more kinetic energy. With higher kinetic energy, particle A passes through the pore throat position j easier and moves along the streamline, changing the migration trajectory of particle A.

By observing the migration trajectories of particle B under initial distances of 0.250 and 0.125 cm, we can see that when a fine particle moves in a porous medium, if the velocity of the particle is higher and the channel is curved, the particle is very likely to deviate from the streamline, as shown in positions l and m, which is unfavorable for the movement of the particle.

If fine particle B is 0.250 cm away from fine particle A on the right, as shown in Figure 13d, before particle A reaches position n, its movement trajectory is similar to that when only particle A exists. However, when particle B arrives at position o, particle A ends its stagnation and resumes its migration. Perhaps the blockage by particle B changes the local flow field around particle A, prompting particle A to enter the fluid channel again.

In brief, when two fine particles simultaneously exist in a porous medium, even when they are not in direct contact, they will impede each other indirectly by the fluid–solid coupling. When one fine particle is lodged, the flow field in the porous medium will adjust, altering the migration of the other particle. In some cases, an increase in the quantity of fine particles can intensify the disturbance of fine particles in the flow field, improving the movement of fine particles.

5.5. Migration and Sedimentation of Three Fine Particles in Porous Media

In this section, the migrations of three non-contact fine particles (A, B and C) in porous media are investigated. The initial positions of particles B and C are (0.5 cm, 3.1 cm) and (1.5 cm, 3.1 cm).

The movement routes of the three particles are shown in Figure 16. In comparison with Figure 13a, particle C brings few effects to bear on particles A and B, and the movement trajectories of particles A and B change little as particle C joins in the migration. The changes in pressure drop of the liquid–solid flow containing one, two or three fine particles with time are demonstrated in Figure 17. Obviously, the changes in pressure drop over time of the liquid–solid flow containing three fine particles are much more complex than those of the liquid-solid flow containing one or two fine particles. In 0–0.1 s, the changes in pressure drop of these three liquid–solid flows are similar, and the occurrence of the peak value of the pressure drop of the liquid–solid flow at position a is due to particle B blocking the pore throat at position a. After particle B has been lodged at position a, and particle A begins to enter the pore throat at position b, the pressures drop sharply and the peak values are much higher when two or three fine particles coexist than when only one fine particle exists, as shown in Figure 17. Compared with the evolution of the pressure drop of the liquid–solid flow containing two fine particles, there is one more fluctuation in that of the liquid–solid flow containing three fine particles around position c, because particle C reaches the pore throat at position c and blocks the local fluid flow. When one, two or three fine particles stagnate somewhere, the pressure drop of the liquid–solid flow containing two or three fine particles is slightly greater than that of the liquid–solid flow containing one fine particle. One reason is that particles A and B both block the pore throats when two or three fine particles coexist, whereas particle A does not block the pore throat when only one fine particle exists; another reason is that the presence of particle C enhances the energy loss of the flow field.

6. Summary and Conclusions

In this work, a new boundary thickening direct forcing (BTDF) immersed boundary method (IBM) was introduced into the existing LBM-DEM framework, the migration behaviors of one, two or three fine particles in the flow field in porous media were investigated, and the effects of the density and diameter of fine particles and the internal shape of porous media on the movements of one, two or three fine particles and the changes in pressure drop between the exit and entrance of porous media under different conditions were analyzed. The specific results are listed in the following.

(1)

A fine particle with a density of 2.1–2.5 g/cm³ can travel longer distance under fluid drag, gravity and collision force, although it is more likely to deviate from the channel and collide with the internal wall of porous media; however, a fine particle with a density of 3.0 g/cm³ will be deposited on an internal wall after a shorter route because of its greater gravity.

(2)

The migration distance of a fine particle is not linearly correlated with its density. A fine particle with a density less than 2.0 g/cm³ is liable to stagnate in a low-infiltration area in porous media.

(3)

A fine particle with a smaller diameter is also likely to be stranded in a low-infiltration area even though its Stokes number is small.

(4)

A fine particle blocking the pore throat in porous media results in an evident pressure fluctuation of the flow field, intensifying the momentum transfer between the flow field and fine particles. An increase in diameter of a fine particle causes a greater increase in the pressure drop of the liquid–solid flow compared to an increase in the fine particle’s density.

(5)

If the horizontal distance between two non-contact fine particles is less than 0.25 cm, the presence of one fine particle may improve the migration of the other, and the other will travel a longer distance in the flow field in porous media.

(6)

When the pressure drop of the liquid–solid flow approaches a stable state, the values of the pressure drop of the liquid–solid flow containing two or three fine particles in porous media are slightly greater than that of the liquid–solid flow containing only one fine particle.