# Experimental and Empirical Studies to Evaluate Liquefaction Resistance of Partially Saturated Sands
## 2 Prediction Models for Partially Saturated Soils  
### Previous Literature  
Liquefaction is directly associated with the build-up of excess pore pressures under cyclic loading, and therefore its prediction is of engineering significance. For this purpose, different stress-based [42; 43; 44], and energy-based predictive models [45] have been proposed in the literature. These models are intended for nearly or fully saturated sands with or without fines, and their robustness relies heavily on the soil and test-dependent empirical coefficients.  
The gas bubbles in pore spaces reduce the bulk modulus of the pore fluid (gas-water mixture) and therefore affect the generation of excess pore pressures in partially saturated sands. Previous studies [46; 47] suggested that excess pore pressures during one loading cycle (\(\Delta u\)) can be correlated with the volumetric strain increment (\(\Delta e_{\text{pd}}\)), the rebound modulus of the soil skeleton (\(E_{r}\)), the porosity of the soil deposit (\(n\)), and the bulk modulus of the pore fluid (\(K_{f}\)):  
\[\Delta u=\frac{\Delta e_{\text{pd}}}{\frac{1}{E_{r}}+\frac{n}{K_{f}}} \tag{1}\]  
This model requires the determination of several soil parameters and experimental constants that define the magnitude of volume change and elastic rebound of soils under drained conditions.  
For partially saturated sands with occluded gas bubbles (\(S_{r}\geq 80\%\)), the surface tension between the gas and water can be ignored, and the gas bubble pressure is considered to be almost equal to pore water pressure. The bulk modulus of the pore fluid (\(K_{f}\)) for this particular case can be estimated using the relationship given in Equation (2) proposed by [PERSON] [48]:  
\[K_{f}=\frac{1}{\frac{S_{r}}{K_{w}}+\frac{1-S_{r}}{K_{g}}} \tag{2}\]  
where \(K_{w}\) and \(K_{g}\) are the bulk modulus of water and gas, respectively.  
The occluded gas bubbles in pore spaces of partially saturated sand initially exist in equilibrium with the surrounding pore water. During cyclic loading in undrained conditions, the change in pore water (\(\Delta u_{w}\)) and gas pressure (\(\Delta u_{g}\)) will be the same as the excess pore pressure (\(\Delta u\)). In this condition, the volumetric strains (\(\varepsilon_{v}\)) suffered by partially saturated sands can be estimated using the theoretical formulation in Equation (3):  
\[\varepsilon_{v}=\frac{\Delta u}{P_{0}+\Delta u}\cdot(1-S_{r})\cdot\frac{e}{e+ 1}\leq\frac{\sigma_{c}^{\prime}}{P_{0}+\sigma_{c}^{\prime}}\cdot(1-S_{r}) \cdot\frac{e}{e+1}=\varepsilon_{v}^{*} \tag{3}\]  
where \(e\) is the void ratio of the soil and \(P_{0}\) is the absolute pore water pressure. The sand suffers the largest (potential or maximum) volumetric strain (\(\varepsilon_{v}^{*}\)) when the excess pore pressure reaches the initial effective confining stress (\(\sigma_{c}^{\prime}\)).  
[PERSON] and [PERSON] [17] showed that a unique relationship exists between the maximum volumetric strain (\(\varepsilon_{v}^{*}\)) and liquefaction resistance ratio (\(LRR\)) that is defined as the liquefaction resistance of partially saturated soils normalized with that of fully saturated soils. Few strain-based or stress-based empirical models were proposed for excess pore pressure generation [26; 49] and for settlement in partially saturated soils [50; 51].