## 3 Coupled Fluid-Solid Simulation Model
### Seepage-Induced Erosion of Clay  
Due to the extremely small diameter of clay particles, it is impossible to model clay particles in the actual size to reflect their mechanical properties in the DEM model. However, the presence of clay greatly affects the physical and mechanical properties of the soil, especially because the process of seepage erosion is also accompanied by the loss of clay. To solve the problem, clay degradation is considered within the coupled flow-solid model. In the model test, to obtain soil samples with different properties, quartz sand was mixed with different proportions of clay. When seepage erosion occurs in the mixed soil, the clay that wraps the quartz sandy particles is lost, accompanied by the loss of fine sand particles. The erosion and degradation process of the mixed soil is shown in Figure 10. In the simulation, to reflect the process of clay erosion and degradation, the gradual loss of clay by the seepage forces is realized by decreasing the contact strength between discrete element particles [29; 30]. Figure 10: Schematic diagram of clay erosion degradation [29; 30]: (**a**) erosion degradation of clay; (**b**) realization with DEM. Figure 9: Fluid–solid coupling calculation process. Loss of clay occurs when the hydraulic shear stress exceeds the critical shear stress of the clay; otherwise, no loss of clay mass occurs. The lost clay mass during seepage erosion can be calculated by Equation (4) [29; 30]. \[m= \begin{cases}k_{\sigma}\left(\left|\tau\right|-\tau_{c}\right)^{b},if\left| \tau\right|>\tau_{c}\\ 0,\ otherwise\end{cases}  \tag{4}\]  
where \(m\) is the mass of lost soil per unit area and time (\(\text{kg}/\left(\text{m}^{2}\text{-s}\right)\)), \(b\) is an exponent. [PERSON], et al. [31] compiled many tests conducted on the erosion of cohesive soils, such as open channel tests and streambed tests. Based on these tests, the coefficient \(b\) was suggested to be 1. \(k_{\sigma}\) is the erosion rate, \(\tau\) is the shear stress at the interface, and \(\tau_{c}\) is the critical shear stress. In DEM modeling, when simulating soils with clay addition, the contact model is usually chosen to be the linear contact bond model, while the contact model adopted in simulating sands is commonly the linear model [26]. The main difference between the two contact models is that tensile and shear bond strengths are considered in the linear contact bond model. Therefore, it is assumed that the shear and tensile bond strengths in the linear contact bond model come from the contribution of the added clay. When the clay is completely lost, the shear and tensile bond strengths are completely reduced to zero and the linear contact bond model degrades to the linear contact model. Based on this assumption, the ratio of the reduction in the shear and tensile bond strengths to the original strength is equal to the ratio of the mass of clay lost to the mass of the original clay. Taking the tensile strength \(\sigma\) as an example, Equation (5) is satisfied:  
\[\frac{\Delta\sigma}{\sigma}=\frac{\Delta m_{j,clay}}{m_{j,clay}} \tag{5}\]  
where \(\Delta\) denotes the amount of change in the corresponding physical quantity, \(\sigma\) is the tensile strength of the contact between the particles, and \(m_{j,clay}\) is the mass of clay attached to the surface of the particle \(j\). Based on this idea, the corresponding decay ratio of the contact bond strength is further determined. Assuming that the clay is uniformly distributed on the surface of all sand particles according to the surface area of the sand particles, all particles share a clay content equal to the proportion of the surface area of the particles themselves. Assuming that there are a total number of \(N\) particles and the total mass of clay is \(m_{clay}\), then the mass of clay shared by the particle \(j\left(m_{j,clay}\right)\) can be calculated according to Equation (6). \[m_{j,clay}=4\pi R_{j}^{2}\frac{m_{clay}}{\sum_{i=1}^{N}4\pi R_{i}^{2}} \tag{6}\]  
where \(R\) is the radius of the particle.