### Basic Assumption
#### 2.2.1 Equations of Seepage Field  
(1) Basic definitions  
As shown in Figure 1, the micro-model for coarse-grained soil is composed of three parts, which includes solid medium ( \(s\), \(dm_{s}\) is mass of the solid and \(dV_{s}\) is volume of the solid), fluid medium (\(l\), \(dm_{l}\) is mass of the solid and \(dV_{l}\) is volume of the solid), and fine particles (\(p\), \(dm_{p}\) is mass of the solid and \(dV_{p}\) is volume of the solid). Porosity of porous media is defined as:  
\[\phi=\frac{dV_{V}}{dV}=\frac{dV_{l}+dV_{p}}{dV} \tag{1}\]  
where: \(\phi\) is the porosity of the micro-model; \(dV_{V}\) is the pore volume of the micro-model (m\({}^{3}\)); \(dV\) is the total volume of the micro-model (m\({}^{3}\)). The volume fraction of fine particles is:  
\[C=\frac{dV_{p}}{dV_{V}} \tag{2}\]  
where: \(C\) is the volume fraction of fine particles. (2) Equations of mass conservation  
According to the law of mass conservation of fluid, when considering the two-phase (fluid and fine particles) seepage in porous media, the mass conservation equations of fluid phase and fine particles phase can be expressed as:  
\[\frac{\partial}{\partial t}[\phi(1-C)\rho_{l}]+\
abla[(1-C)\rho_{l}v_{ls}]=q_ {l} \tag{3}\]  
\[\frac{\partial}{\partial t}(\phi C\rho_{p})+\
abla(C\rho_{p}v_{ps})=\dot{m} \tag{4}\]  
where: \(\rho_{l}\) is the inherent density of the fluid (kg/m\({}^{3}\)); \(\rho_{p}\) is the inherent density of fine particles (kg/m\({}^{3}\)); \(v_{ls}\) is the velocity vector of the fluid relative to the solid (m/s); \(q_{l}\) is the source (sink) item of the fluid (kg/m\({}^{3}\)); \(v_{ps}\) is the velocity vector of fine particles relative to solid (m/s); \(\dot{m}\) is the mass rate of fine particles migration (kg/m\({}^{3}\)s); \(\
abla=\frac{\partial}{\partial x}\mathbf{i}+\frac{\partial}{\partial y}\mathbf{ j}+\frac{\partial}{\partial z}\mathbf{k}\) is Hamiltonian operator. (3) Equations of motion  
The seepage of fluid conforms to [PERSON]'s law, as follows:  
\[v_{ls}=v_{l}-v_{s}=-\frac{k}{\mu_{l}}(\
abla p_{l}-\rho_{l}g) \tag{5}\]  
where: \(v_{l}\) is the velocity vector of the fluid (m/s); \(v_{s}\) is the velocity vector of solid (m/s); \(k\) is the permeability of porous media (m\({}^{2}\)); \(p_{l}\) is the pore pressure of the fluid (Pa); \(\mu_{l}\) is the dynamic viscosity coefficient of the fluid (Pa \(\cdot\) s); \(g\) is the gravitational acceleration vector (N/kg). The force of fine particles in liquid is complex. The fine particles is mainly affected by effective gravity, drag force, additional mass force, pressure gradient force, Basset force, Magnus force, Saffman force, and other force, among which additional mass force and pressure gradient force, Basset force, Magnus force, Saffman force have relatively small influence on the movement of fine particles and can be ignored. Therefore, considering the effect of effective gravity and drag force on fine particles, the schematic diagram of the force on a single fine particle is shown in Figure 2.  
We use \(v_{r}\) to represent the difference between the velocity vector of the fluid and the velocity vector of the fine particles (\(v_{p}\)). Assuming that the velocity of the fine particles differs little from that of fluid in the horizontal direction, and the velocity difference in the vertical direction can be expressed as:  
\[v_{rz}=v_{lz}-v_{pz} \tag{6}\]  
When a single particle is in force equilibrium in the fluid, there are:  
\[(\rho_{p}-\rho_{l})\frac{\pi d_{p}^{3}}{6}g=C_{D}\frac{\pi d_{p}^{2}}{4}\frac{ \rho_{l}v_{rz}{}^{2}}{2} \tag{7}\]  
then:  
\[v_{rz}=\sqrt{\frac{4(\rho_{p}-\rho_{l})d_{p}g}{3C_{D}\rho_{l}}} \tag{8}\]  
where, \(d_{p}\) is the diameter of fine particles (m);\(C_{D}\) is the coefficient of drag force, which is related to the Reynolds number of fine particles. \[\text{Re}=\frac{d_{p}\rho_{l}u_{l}}{\mu_{l}} \tag{9}\]  
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