## 3 Viscoelastic Parameter Prediction of the Multi-Layered CGS
### Predicting Viscoelastic Parameters of Multi-Layered CGS  
In the multi-layered CGS, the pebble and the sand-clay matrix are regarded as elastic and viscoelastic material, respectively. They are exploited from the local area, with the elastic modulus of pebble (\(Ep=50\) GPa) and density is 2.54 g/cm\({}^{3}\), and Poisson's ratio of pebble (\(v_{p}=0.25\)) and sand-clay matrix (\(v_{m}=0.25\)). Assuming the volume fraction of pore (\(p_{1}\)) is 20% [40], the volume fraction of pebble (\(p_{0}\)) and sand-clay matrix (\(p_{3}\)) can be calculated by the mass-density formula.  
Multi-layered CGS consists of several one-layered CGS and interface-layers (Figure 1). The volume fraction of one-layered CGS and the basic parameters of each component are in Table 3.  
Based on Table 3, viscoelastic parameters (\(\rho=1\)) of the multi-layered CGS can be calculated by the following steps:  
1. Calculate the elastic modulus (\(E^{T}\), superscript \(T\) means Laplace space) of the sand-clay matrix by the elastic-viscoelastic correspondence principle as Equation (19). \[E^{T}=\frac{E_{1}\eta_{1}(E_{2}+\eta_{2}s)s}{E_{1}E_{2}+(\eta_{1}E_{2}+E_{1}( \eta_{1}+\eta_{2}))s+\eta_{1}\eta_{2}s^{2}}\] (19) where \(s\) is Laplace transform parameter.
2. Calculate the equivalent elastic modulus (\(K_{s}\), \(E_{s}\)) of the solid phase (pebble, sand-clay matrix) by Equations (1) and (4).
3. Calculate the equivalent elastic modulus (\(K_{o}\), \(E_{o}\)) of the one-layered CGS by Equations (6) and (7).
4. Calculate the equivalent elastic modulus (\(K_{i}\), \(E_{i}\)) of the interface-layers by Equations (10)-(12).
5. Calculate the equivalent elastic modulus (\(E\)) of the multi-layered CGS by Equation (15).
6. Calculate the equivalent viscoelastic parameters (\(J\)) of the multi-layered CGS by Equation (16). and elastic-viscoelastic correspondence principle, as listed in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **Layer Combination** & **E\({}_{1}\)/MPa** & **E\({}_{2}\)/MPa** & **\(\eta_{1}\)/MPa-min** & **\(\eta_{2}\)/MPa-min** \\ \hline \multirow{2}{*}{One-layered CGS} & L1 & 58.82 & 166.67 & 19999.97 & 3703.70 \\  & L2 & 30.30 & 62.50 & 5000.05 & 236.71 \\  & L3 & 14.93 & 21.27 & 3999.91 & 167.53 \\ \multirow{2}{*}{Interface Layer (\(J_{i}\))} & L1/L2 & 46.24 & 149.21 & 12774.95 & 2176.75 \\  & L2/L3 & 18.89 & 19.14 & 4999.13 & 656.87 \\ \multirow{2}{*}{Two-layered CGS} & L1/L3 & 43.36 & 20.60 & 4368.95 & 87.73 \\  & L1 + L1/L2 + L2 & 39.30 & 104.07 & 7465.34 & 284.53 \\ \multirow{2}{*}{Two-layered CGS} & L2 + L2/L3 + L3 & 19.82 & 32.54 & 4401.50 & 218.38 \\  & L1 + L1/L3 + L3 & 23.67 & 38.83 & 6412.82 & 333.94 \\ \multirow{2}{*}{Three-layered CGS} & L1+ L1/L2 + L2 + L2/L3 + L3 & 26.41 & 48.72 & 5755.77 & 324.54 \\ \hline \end{tabular}
\end{table}
Table 4: Predicted viscoelastic parameters of multi-layered CGS.  
\begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Mass Ratio of Pebble to Sand-Clay Matrix** & **Volume Fraction of Pebble (\(p_{0}\))** & **Volume Fraction of Sand-Clay Matrix (\(p_{3}\))** \\ \hline L1 & 1:1.30 & 38.9\% & 41.1\% \\ L2 & 1:1.75 & 26.6\% & 53.4\% \\ L3 & 1:2.33 & 15.7\% & 64.3\% \\ \hline \end{tabular}
\end{table}
Table 3: Basic parameters of one-layered CGS.