## 3 Result and Analysis  
### Random Field Generation: Hydraulic Conductivity  
The excess pore water dissipation is significantly affected by the distance from a free-drainage boundary to permeable layer and relative boundary conditions formed by relative hydraulic conductivity k difference among layers. Figure 1 presents the effects of binary mixture sequence and thickness on 1D consolidation process. At the given binary mixture thickness ratio, the excess pore water pressure dissipates faster for the soils in which the layers with higher k are located closer to the drainage boundary (case: \([\mathrm{k}]_{\mathrm{A}}/[\mathrm{k}]_{\mathrm{B}}=5.0\)), even if the volumetric composition of each material is equal [20]. On the contrary, lower k at a top drainage delays the consolidation process (case: \([\mathrm{k}]_{\mathrm{A}}/[\mathrm{k}]_{\mathrm{B}}=0.2\)). Furthermore, introducing internal material heterogeneity (layer thickness) tends to delay or accelerate the dissipation, depending on the k magnitude of the top layer. Figure 2 shows that the relative k magnitude differences between hydraulic conductivity of the layers cause a more delayed dissipation of excess pore water pressure. Smaller layer thickness ratios and lower magnitude differences reduce those specific patterns, yet initial heterogeneity configurations are quite different from the equivalent homogeneous one during dissipation of excess pore pressure. The vertical drainage condition has a pronounced effect on the consolidation process of the binary mixtures. The internal heterogeneity promotes the delayed dissipation of the excessive pore water pressure, yet such differences in consolidation time delay are significantly reduced when radial drainage is allowed (Figure 3a). If the vertical drains are more narrowly spaced, the global consolidation process is controlled more by the radial drainage, and thus the differences in consolidation time are reduced more with a shorter maximum radial drainage path (Figure 3b).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Drainage Condition** & **Topic** & **COV** & **Relative Correlation Length** \\ \hline \multirow{2}{*}{No vertical drainage} & Variability & 0.1, 0.2, 0.4, 0.6, 0.8, 1.0 & 0.1 \\  & Correlation length & 1.0 & 0.01, 0.03, 0.05, 0.1, 0.2 \\ \hline \multirow{2}{*}{Vertical drainage} & Variability & 0.1, 0.2, 0.4, 0.6, 0.8, 1.0 & 0.1 \\  & Correlation length & 1.0 & 0.01, 0.03, 0.05, 0.1, 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parametric studies. Random variable is hydraulic conductivity, k, following lognormal distribution in all cases. The global mean value of k case is kept the same (\(\mathrm{\mu[k]}=1.8\times 10^{-9}\) m/s).  
Figure 1: Effects of heterogeneity configurations on one-dimensional consolidation process of soils with binary layers. All the cases have the same arithmetic mean of consolidation coefficient.  
Figure 3: Effect of a vertical drainage conditions on the consolidation of the binary mixtures; (**a**) vertical drain (r/D = 0.2); (**b**) vertical drainage distance. The vertical drain is installed along the centerline and radial drainage is allowed. [k]\({}_{\rm A}\)/[k]\({}_{\rm B}\) = 0.2 in all cases. The radial consolidation coefficient is assumed to be identical to the vertical one.  
Figure 2: Effects of k variability on one-dimensional consolidation process of soils with binary layers. All the cases have the same mean of consolidation coefficient and the same thickness ratio t/H = 0.05.