# One-Dimensional Consolidation of Multi-Layered Unsaturated Soil with Impeded Drainage Boundaries
## 4 Verifications and Examples
### Worked Examples  
In this part, two worked examples of three-layered unsaturated soil strata under the impeded drainage boundaries are investigated, and the influences stemming from the drainage parameter of air phase and that of the water phase are discussed. The specifications of the soil profile and material parameters listed in Table 1 for the second case are adopted in the calculations. Two typical types of external loadings including constant and exponential loads are incorporated in the calculation procedure. The excess pore-air and pore-water pressures are given as zeros at the initial time. Herein, at the bottom base, the drainage parameter of the air phase is taken as \(R_{\text{ba}}=10\) and that of the water phase is given as \(R_{\text{bw}}=5\), respectively; while at the top surface, they are changed one by one from 2 to 100. Based on the derived solutions, the excess pore-air and pore-water pressures as well as the ground settlement of these multi-layered unsaturated soil strata are obtained and the results are graphically demonstrated below. The first worked example takes a changing drainage parameter of air phase at the top surface while that of water phase is taken as \(R_{\text{tw}}=5\). For this situation, Figure 4a,b depict the evolutions of excess pore-air pressure induced by constant and exponential loads, respectively. As can be observed, the action of a constant load immediately results in a great pore-air pressure, which gradually decreases with time passing; while the pore  
Figure 3: Verification of the pore pressure evolutions: (**a**) \(u_{\text{a}}\); (**b**) \(u_{\text{w}}\). Figure 2: Verification of the pore pressure distributions: (**a**) \(u_{\text{a}}\); (**b**) \(u_{\text{w}}\). air pressure first gradually increases as the exponential load increases, and then deceases slowly after the increase in the exponential load becomes slow. In particular, the value of the drainage parameter with respect to the air phase has a pronounced effect on the evolution of pore-air pressure. A larger drainage parameter (i.e., greater drainage efficiency) of air phase delivers a faster dissipation of excess pore-air pressure, especially during the middle and later periods of the consolidation process. Similarly, the evolutions of pore-water pressure are shown in Figure 5a,b. As observed, the evolution of pore-water pressure has two distinct stages. The first one corresponds to the change of pore-air pressure and it is the result of the drainage of pore-air; while the later one is proceeded by the drainage of pore-water. The influence of the drainage parameter above mainly emerges on the first stage, and a larger drainage parameter of air phase leads to a faster dissipation of pore-water pressure. Once the value of this drainage parameter exceeds 100, the curves of pore pressures tend to change as the same way. In addition, for the situation of the constant load, Figure 6a,b demonstrate the isochrones of pore-air pressure and those of pore-water pressure at different times, respectively. As seen, being subjected to a constant load, both the pore-air and pore-water pressures induced in one soil layer are different from those caused in another soil layer. As shown in Table 1, the coefficient of soil volume change in the second layer is the largest one, indicating that the second layer is the softest one. Therefore, being subjected to an external load, this layer may provide a smaller resistance than the other two layers. Under this situation, the air and water phase in the second soil layer have to undertake more load, which means larger pore pressures. It is worthy to note that the drainage parameter (i.e., drainage efficiency) of the air phase shows a profound effect on the distribution of pore-air pressure and that of pore-water pressure. The larger drainage parameter of air phase can speed up their dissipation processes. This effect firstly appears at the position near the top surface, and later spreads far away gradually.