## 3 Permeability of the Rare Earth Pile under Lateral Liquid Injection  
### Variation of the Permeability Coefficient with Hydraulic Head  
According to [PERSON]'s law, the permeability coefficient of the rare earth pile in the model tests was determined by:  
\[K=QL/\,A\Delta h \tag{1}\]  
where \(K\) denotes the permeability coefficient, \(Q\) denotes the seepage flow, \(L\) is the distance between the adjacent observation points, \(A\) denotes the cross-sectional area, and \(\Delta h\) denotes the difference of the hydraulic head between the adjacent observation points. The permeability coefficients between the adjacent observation points of the rare earth pile are expressed as \(K_{\rm AB}\), \(K_{\rm BC}\), and \(K_{\rm CD}\), respectively. The variations of these permeability coefficients with the increasing hydraulic head for the four soil samples are shown in Figure 6. With an increase in the hydraulic head of leaching liquid, the permeability coefficients show a trend of increase on the whole, and especially, the permeability coefficient \(K_{\rm AB}\) near the liquid injection end varies most significantly. The variation of \(K_{\rm AB}\) with the hydraulic head is clearly divided into four stages. In the first stage, namely, the curve e-f in Figure 6, the permeability coefficient is small and remains almost constant. This indicates that the seepage of the leaching liquid within the rare earth pile is in a stable state when the hydraulic head is small. In the second stage (see the curve f-g), the permeability coefficient increases slowly with the hydraulic head. This is because as the hydraulic head increases, the fine particles start to move in the pores of the soil samples due to the increasing seepage force. This enhances the connectivity between the pores, and accordingly, the permeability is increased. This stage is also the startup stage of the fine particle migration. In the third stage (see the curve g-h), the permeability coefficient tends to stabilize again. In the process of increasing the hydraulic head, more and more particles with various sizes participate in the migration. The co-migration of the particles of different sizes can easily cause the blockage of the pores, thereby preventing the connectivity of the pores and the increase of the permeability in the rare earth pile. In the fourth stage (see the curve h-i), the permeability coefficient increases substantially. This is because when the hydraulic head of leaching liquid is increased to a certain level, the particles that block the pores are driven to migrate again by the greater seepage force, and the connectivity of the pores is further increased. This stage is the development stage of preferential flow paths. Figure 5: Arrangements of the observation points within the rare earth pile (unit: cm). For the coarse-fine particle soil sample S4, the particle size varies widely between the coarse particles and the fine particles. The fine particles are not large enough to block the pores between the coarse particles during the migration. Once the migration of the fine particles is initiated, the permeability coefficient increases continuously. Therefore, the variations of the permeability coefficients for the soil sample S4 have no secondary stable stage, i.e., the stage of the curve g-h. Furthermore, the hydraulic head corresponding to the fine particle migration initiation for the soil sample S4 is the smallest among the four soil samples. This indicates that under the lateral liquid injection conditions, the absence of medium particles makes it easier for the fine particles to migrate and they cannot block the pore channels. In this case, it is easy to form preferential flow paths for the leaching liquid. In our previous publication [7], we have experimentally and numerically studied the formation of the preferential flow paths and its influences on the leaching efficiency of the rare earth under lateral liquid injection conditions. It is found that the preferential flow paths mainly develop in the lower part of the rare earth pile.