# Research Progress in Methods for the Analysis of the Internal Stability of Landslide Dam Soils
## 3 Hydraulic Criteria
### Criteria Based on Critical Velocity  
In addition to using the critical hydraulic gradient to assess the internal stability of soil, the critical seepage velocity was also used in some criteria especially when dealing with coarse-grained soils ([PERSON], 2019; [PERSON], 2019). Some commonly used criteria are  
\[v_{cr}=\frac{\rho^{\prime}gdK}{18\mu\alpha\rho_{w}K+\rho_{w}gnd^{2}} \tag{2}\]  
\[v_{cr}=\frac{1}{2}n\frac{\frac{\sqrt{3}d^{2}}{9}(\rho_{s}-\rho_{w})g}{\frac{4} {n}n_{w}\left(\frac{d}{d_{0}}\right)^{2}+\frac{3\mu_{w}}{2}}\left|\sin\left( \theta+\frac{\pi}{6}\right)\right| \tag{3}\]  
In Equations (2) and (3), \(v_{cr}\) is the critical velocity, \(\rho_{w}\) is the density of water, \(g\) is the gravitational acceleration, \(d\) is the movable particle diameter, and \(n\) is the porosity of the soil.  
In Equation (2), \(\alpha\) is the correction factor of the particle shape, \(K\) is the hydraulic conductivity, \(\rho^{\prime}\) is the effective density of fine particles, and \(\mu\) is the dynamic viscosity of water. In Equation (3), \(\rho_{s}\) is the density of soil particles, \(\mu_{w}\) is the dynamic viscosity, \(d_{0}\) is the minimum diameter of the pore channel, and \(\theta\) is the inclination angle of the pore channel.  
The critical velocity formulas for internal erosion in non-cohesive soils were derived by [PERSON] (2019) and [PERSON] (2019), with Formula (2) derived by [PERSON] and Formula (3) derived by [PERSON]. Both derivations neglect the non-uniformity of pore channels, the water flow turbulence effects, and transient phenomena, but the handling details differ. [PERSON]'s approach uses a variable cross-sectional pipe model and spherical particle assumptions to simplify the physical model, mainly describing the forces and movement of particles under laminar flow conditions through force balance equations and the Poiseuille equation,making it suitable for practical engineering applications. In contrast, [PERSON]'s approach uses a probabilistic distribution model of pore sizes and detailed probabilistic analysis methods, combining force balance equations and momentum equations to detail the movement trajectories and conditions of particles in complex pore structures, making it suitable for complex seepage conditions. Although more complex, [PERSON]'s formula can more accurately reflect the actual conditions.  
The hydraulic gradient criterion is established through the force balance analysis of soil microelements, focusing on changes in the effective stress of soil particles under seepage action. When water flows through the soil, the effective stress of the particles is influenced by the seepage force, leading to the derivation of the critical hydraulic gradient. Conversely, the permeability flow velocity criterion is based on mechanical equilibrium, utilizing the torque balance or dynamic equation. At a certain critical flow velocity, the combined drag force and buoyancy acting on the particles exceed the particle gravity and friction, causing movement and thus deriving the critical seepage velocity. In both criteria, the seepage force plays a key role in assessing the critical movement conditions of particles. The hydraulic gradient criterion emphasizes changes in the internal stress distribution within the soil, while the permeability flow velocity criterion focuses on the direct transmission of kinetic energy and force to the particles.