## 3 Result and Analysis
### Consolidation of Spatially Correlated Heterogeneous Soils  
Figure 4 shows three selected realizations of the spatially varying consolidation coefficient profiles. For the realizations, the mean value \(\upmu\)[k] is 1.8 \(\times\) 10\({}^{-9}\) m/s, COV[k] is 0.6, and the correlation length \(\theta_{\text{lm\,k}}\) is 1 m. The total thickness of the compressible layers D is 10 m, the maximum radial drainage length r is 2 m, and the radius of the vertical drain is 0.1 m. The consolidation behavior of spatially varying random soils with different drainage conditions is compared in the following Figure 5. The results show that a different configuration with the same even statistic promotes a different consolidation process, and such differences due to the different heterogeneity configurations are reduced by the radial drainage, as discussed in the section on binary mixtures. Case C is the fastest, with a 90% consolidation by a single upward vertical drainage (for 90% consolidation, case A: 2800 days; case B: 3300 days; case C: 2600 days); however, Case C is the slowest for the case of coupled drainage (for 90% consolidation, case A: 520 days; case B: 580 days; case C: 590 days), which indicates that the contributions of spatial variation to the global consolidation process differ with respect to the drainage conditions. The 100 realizations for each case are tested to quantitatively investigate the effect of random spatial variability in the hydraulic conductivity on the consolidation process. Figure 6 presents the consolidation process with the preselected variability and correlation length. Clearly, the spatial variability in k and drainage conditions have a pronounced effect on the global consolidation process. For further analysis of numerical results, 50% consolidation time, t\({}_{50}\), and 90% consolidation time, t\({}_{90}\), are computed, and data are gathered for statistical analyses. The computed t\({}_{50}\) and t\({}_{90}\) for heterogeneous soils are normalized by those of the homogeneous soil, which are called the normalized time. Results are presented with the ensemble mean and the standard deviation of the normalized consolidation time. The effects of COV in the spatially varying consolidation coefficient COV[k] on the normalized consolidation times, t\({}_{50}\) and t\({}_{90}\), are studied, while keeping the correlation length (\(\Theta\) = 0.1) and the radial drainage length ratio (r/D = 0.2) constant. Results show that a greater value of  
Figure 4: Realizations of spatially varying correlated consolidation coefficient profiles. All the cases are with coefficient of variation COV[k] = 0.6 and dimensionless correlation length \(\Theta\) = 0.1. COV[k] corresponds to a more delayed consolidation time, and greater standard deviation of t\({}_{50}\) and t\({}_{90}\). However, such variations are significantly reduced when the radial drainage is allowed, as shown in Figure 7b. The mean values of the normalized t\({}_{50}\) and t\({}_{90}\) for the realizations with COV[k] = 1.0, are about 2.0 for the case without radial drainage; those values go down to about 1.3 if coupled drainage is allowed. Therefore, spatial variability should be carefully considered when analyzing the consolidation process of soils with high variability, especially for the cases without vertical drains. Figure 7 also shows that the mean values for both normalized t\({}_{50}\) and t\({}_{90}\) are very similar, but the variation in normalized t\({}_{90}\) is smaller than that in normalized t\({}_{50}\), which shows that the mean consolidation time does not differ much with degree of consolidation, but the relative variation in consolidation time becomes smaller at a higher degree of consolidation.