## 2 Stochastic Simulation of 1D Consolidation  
### Random Field Generation: Hydraulic Conductivity  
Coefficient of consolidation \(\mathrm{C}_{\mathrm{v}}=\mathrm{k}/(\gamma_{\mathrm{w}}\mathrm{m}_{\mathrm{v}})\), that incorporates hydraulic conductivity, \(\mathrm{k}\), into coefficient of volume compressibility, mv, defines the dissipation of excess pore water pressure. However, the statistical analysis of consolidation parameters observed that coefficient of variation (COV) value of \(\mathrm{m}_{\mathrm{v}}\) (\(\approx\)30%) is smaller than that of \(\mathrm{k}\) (\(\approx\)300%), and thus, \(\mathrm{k}\) has a more pronounced effect on the soil consolidation process [15; 16]. Thus, the hydraulic conductivity, \(\mathrm{k}\), is selected as a random variable, and mv is assumed constant. The hydraulic conductivity \(\mathrm{k}\) can be obtained from different probability distribution functions that satisfy with non-negative values and large variability. The inherent variability of \(\mathrm{k}\) is generated using lognormal distribution [11; 17]. The hydraulic conductivity, \(\mathrm{k}\), following lognormal distribution, is identified by mean \(\mathrm{\mu}_{\mathrm{k}}\) and standard deviation \(\mathrm{\sigma}_{\mathrm{k}}\). The \(\mathrm{COV}_{\mathrm{k}}\) can be expressed as  
\[\mathrm{COV}_{\mathrm{k}}=\frac{\mathrm{\sigma}_{\mathrm{k}}}{\mathrm{\mu}_{ \mathrm{k}}} \tag{1}\]  
The mean \(\mathrm{\mu}_{\mathrm{ln}\;\mathrm{k}}\) and standard deviation \(\mathrm{\sigma}_{\mathrm{ln}\;\mathrm{k}}\) following normal distribution have the relation with lognormal parameters as follows:  
\[\mathrm{\sigma}_{\mathrm{ln}\;\mathrm{k}}=\sqrt{\mathrm{ln}\Big{(}1+\mathrm{ COV}_{\mathrm{k}}^{2}\Big{)}} \tag{2}\]  
\[\mathrm{\mu}_{\mathrm{ln}\;\mathrm{k}}=\mathrm{ln}(\mathrm{\mu}_{\mathrm{k}})- \frac{1}{2}\mathrm{\sigma}_{\mathrm{ln}\;\mathrm{k}}^{2} \tag{3}\]  
The multiple realizations of the one-dimensional correlated random variability in the hydraulic conductivity, \(\mathrm{k}\), can be generated with the preselected values (\(\mathrm{\mu}_{\mathrm{k}}\) and \(\mathrm{\sigma}_{\mathrm{k}}\)) through the matrix decomposition technique. The procedure is as follows [18]:  
1. Assign index properties \(\mathrm{i}\) and \(\mathrm{j}\) to every location at the given geometry.
2. Build distance matrix, \(\mathrm{d}\), where \(\mathrm{d}_{\mathrm{ij}}\) is the distance between point \(\mathrm{i}\) and \(\mathrm{j}\).
3. Compute the covariance matrix, \(\mathrm{A}\), with correlation length \(\mathrm{\theta}_{\mathrm{ln}\;\mathrm{k}}\): \[\mathrm{A}_{\mathrm{ij}}=\mathrm{\sigma}_{\mathrm{ln}\;\mathrm{k}}^{2} \mathrm{e}^{-\frac{\mathrm{\sigma}_{\mathrm{ij}}}{\mathrm{\theta}_{\mathrm{ln} \;\mathrm{k}}}}\] (4)
4. Decompose the matrix \(\mathrm{A}\) into matrix \(\mathrm{C}\): \[\mathrm{A}=\mathrm{CC}^{\mathrm{T}}\] (5)
5. Calculate the correlated Gaussian random field \(\mathrm{G}\): \[\mathrm{G}=\mathrm{C}\varepsilon+\mathrm{T}\] (6) where \(\varepsilon\) is uncorrelated Gaussian random field of zero mean and unit variance for local averages over the domain, and \(\mathrm{T}\) is the trend following \(\mathrm{\mu}_{\mathrm{ln}\;\mathrm{k}}\).
6. Transform the generated Gaussian random filed to the lognormal distribution \[\mathrm{k}=\mathrm{exp}(\mathrm{G})\] (7)The spatial correlation length \(\theta_{\ln\mathrm{k}}\) is the distance where the spatial autocorrelation decays by 1/e, and provides the spatial scale of material heterogeneity [5; 19]. The formulated one-dimensional correlated random heterogeneity reflects the spatial variation of the consolidation coefficient with depth. The relative correlation length \(\Theta=\theta_{\ln\mathrm{k}}/\mathrm{H}\) is expressed by combining internal fluctuation scale \(\theta_{\ln\mathrm{k}}\) and specimen height H.