# Research Progress in Methods for the Analysis of the Internal Stability of Landslide Dam Soils
## 3 Hydraulic Criteria
### Criteria Based on Critical Hydraulic Gradient  
The formula used to calculate the critical gradient proposed by [PERSON] [47], [PERSON] [48], [PERSON] [24], and [PERSON] [29; 49] are all based on force equilibrium, as shown in Table 3.  
In the formulas proposed by [PERSON] [47] and [PERSON] [48], the only difference between them is the shape coefficient \(\alpha\). [PERSON] [47] primarily referenced hydraulic principles during the derivation process, particularly the effects of dynamic water pressure on particles and the correction of the shape coefficient. [PERSON] [48], on the other hand, focused more on a comprehensive analysis of fluid mechanics and particle mechanics, performing detailed analyses of dynamic water pressure and friction. The introduction of the shape coefficient \(\alpha\) is intended to mitigate the impact of the particle shape on seepage resistance. The porosity \(n\) influences the size of the channels through which seepage flows through the soil. The equivalent particle size \(d_{\text{OK}}\) represents the diameter of hypothetical uniform particles with the same permeability characteristics, determined experimentally to accurately calculate the effects of seepage. The particle diameter \(d\) of eroded particles reflects the soil's resistance to erosion under seepage, and can be used to predict which \begin{table}
\begin{tabular}{c c c} \hline
**Researcher** & **Applicable Soils** & **Identification Method** \\ \hline [PERSON] [50] & \(i_{c}=(G_{\text{s}}-1)/(1+e)\) \\ [PERSON] [48] & \(i_{c}=(G_{\text{s}}-1)\frac{d}{d+e_{\text{OK}}}\) \\  & Non-cohesive soils & \(i_{c}=(G_{\text{s}}-1)/\frac{d}{d+e_{\text{OK}}}tan\varphi\) \\ [PERSON] [45] & \(i_{c}=(G_{\text{s}}-1)(1-n)\alpha d/d_{\text{OK}}\) \\ [PERSON] [49] & \(i_{c}=2.2(1-n)^{2}(G_{\text{s}}-1)\) \\ [PERSON] [24] & \(i_{c}=C(G_{\text{s}}-1)\frac{d}{d_{j}}\left[4p_{f}(1-n)\right]^{2}\) \\ \hline Skempton and Brogan [42] & Sand and gravel soil & \(i_{c}=\beta(G_{\text{s}}-1)(1+e)\) \\ \hline \end{tabular} Note: \(G_{\text{s}}\) is the specific gravity of the soil; \(e\) is the void ratio; \(d\) is the particle diameter of the eroded particles; \(d_{\text{OK}}\) is the equivalent particle diameter; \(\varphi\) is the internal friction angle of the soil; \(x\) is the particle shape coefficient; \(\beta\) is the reduction coefficient; \(C\) is the coefficient; and \(d_{j}\) is the maximum particle diameter of the filler.  
\end{table} 
Table 3: Methods for hydraulic identification. particles are most likely to be carried away during the seepage process. By integrating these parameters, the formulas can more accurately describe and predict the behavior of soil under seepage conditions. [PERSON] and [PERSON] (2018), addressing the issue of the significant loss of fine particles in unstable gravel long before reaching the critical hydraulic gradient, introduced a reduction coefficient \(\beta\) to modify Equation (1) based on experimental data and theoretical formulas. The reduction coefficient \(\beta=i_{c}/(r^{\prime}/r_{w})\), where \(i_{c}\) is the critical hydraulic gradient at which piping occurs in gravel, and \(r^{\prime}/r_{w}\) is the ratio of the buoyant unit weight of gravel to the unit weight of water. [PERSON] et al. (2018) considered the parameters of the diameter of eroded particles \(d_{f}\), the porosity \(n\), and the fine content filled in pores to comprehensively describe the movement behavior of movable particles under the action of water flow. This formula, based on classical mechanics principles, reasonably explains the particle behavior and enhances applicability by converting the critical flow velocity to a seepage gradient. The experimental calibration coefficient in the formula improves the prediction accuracy, making it suitable for various soil conditions. However, the empirical coefficient \(C\) in the formula depends on experimental calibration, adding complexity to its application. The formula assumes ideal particle shapes and ignores interactions between particles.