# Bulk and Rayleigh Waves Propagation in Three-Phase Soil with Flow-Independent Viscosity
## 4 Numerical Examples and Parametric Analysis
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(n^{a}\) & 0.4 & \(K_{s}\) & 36 GPa & \(\lambda_{e}\) & 120 MPa \\ \(S^{\prime}\) & 0.6 & \(K_{l}\) & 2.2 GPa & \(\mu_{e}\) & 120 MPa \\ \(S_{ns}\) & 0.05 & \(K_{a}\) & 0.1 MPa & \(\mu^{l}\) & 0.001 Pa-s \\ \(\rho^{s}\) & 2650 kg\(\cdot\)m\({}^{-3}\) & \(X\) & 0.0001 & \(\mu^{d}\) & \(1.8\times 10^{-5}\) Pa-s \\ \(\rho^{l}\) & 1000 kg\(\cdot\)m\({}^{-3}\) & \(m\) & 0.5 & \(\tau_{l}\) & 1.0 \\ \(\rho^{a}\) & 1.3 kg\(\cdot\)m\({}^{-3}\) & \(d\) & 2.0 & \(\tau_{a}\) & 1.0 \\ \(k\) & \(1.0\times 10^{-11}\) m\({}^{2}\) & \(f\) & 100 Hz & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of unsaturated soil. Figure 4: P\({}_{1}\) wave propagation for different saturations and relaxation times. (**a**) Wave speed. (**b**) Attenuation coefficient. Figure 5: S wave propagation for different saturations and relaxation times. (**a**) Wave speed. (**b**) Attenuation coefficient. Figure 3: Comparisons of the wave speed between this study and the works done by [PERSON] [2] and [PERSON] [37]. (**a**) Comparison with [PERSON] (2005). (**b**) Comparison with [PERSON] (1984). The dependency of \(v_{p1}\), \(v_{s}\), \(v_{R}\), \(\delta_{p1}\), \(\delta_{s}\), and \(\delta_{R}\) on \(t_{s}\) and \(f\) are depicted in Figures 8-10, therein \(f\) ranges from 0.01 Hz to 150 Hz, \(t_{s}\) is taken to be 0 s, \(0.5\times 10^{-3}\) s, and 1.0 \(\times 10^{-3}\) s, respectively, and the value of other physical parameters refers to Table 1. It can be captured from Figures 8-10 that each wave presents positive relativity between both the wave speeds and attenuation coefficients and the frequency. In contrast, this positive relativity will be hidden when the soil skeleton viscosity is not considered. This is a vital issue worthy of attention in practical engineering applications. Meanwhile, Figures 8-10 show that the increase of \(v_{p1}\), \(v_{s}\), \(v_{R}\), \(\delta_{p1}\), \(\delta_{s}\), and \(\delta_{R}\) will accompany the enlargement of \(t_{s}\). As mentioned in Equations (25)-(27), the flow-dependent viscosities from pore-liquid and pore-air are characterized by the parameters \(b_{w}\) and \(b_{g}\) in Equations (26) and (27) and finally represented by the intrinsic permeability \(k\). For comparison, Figures 11-13 demonstrate the effects of \(k\) and \(t_{s}\) on \(v_{p1}\), \(v_{s}\), \(v_{R}\), \(\delta_{p1}\), \(\delta_{s}\), and \(\delta_{R}\). In this example, the value range of \(k\) is \(10^{-12}\)-\(10^{-7}\) m\({}^{2}\), while \(t_{s}\) ranges from 0 \(\times\)\(10^{-3}\) s to 2.0 \(\times\)\(10^{-3}\) s. It can be seen from Figures 11a, 12a and 13a that \(v_{p1}\), \(v_{s}\), and \(v_{R}\) remain unchanged when \(k\) is in the range of \(10^{-12}\)-\(10^{-9}\) m\({}^{2}\) but increase gradually with the enlargement of \(k\) in the high permeability zone (i.e., \(k=10^{-9}\) m\({}^{2}\)-\(10^{-7}\) m\({}^{2}\)). For \(\delta_{p1}\), \(\delta_{s}\), and \(\delta_{R}\), the variation trends in Figures 11b, 12b and 13b display an approximately normal distribution in the entire permeability range. Furthermore, as above-mentioned discussion, the increase of \(t_{s}\) will make the dependences of \(v_{p1}\), \(v_{s}\), \(v_{R}\), \(\delta_{p1}\), \(\delta_{s}\), and \(\delta_{R}\) on \(k\) move upward along the positive direction of the vertical axis. Apparently, through comparison, it can be seen that compared to the fluid viscosity, the soil skeleton viscosity has a more prominent contribution to the motion behaviors (speed and attenuation) of elastic waves and is more significant for practical engineering applications. Figure 6: Rayleigh wave propagation for different saturations and relaxation times. (**a**) Wave speed. (**b**) Attenuation coefficient. Figure 7: Soil–water characteristic curve (SWCC). Figure 8: P\({}_{1}\) wave propagation for different frequencies and relaxation times. (**a**) Wave speed. (**b**) Attenuation coefficient.