## 4 Case Study--The Failure of the Gypsum Tailings Dam in East Texas in 1966 (FGT66)
### The Results from Three Different Rheological Models
The mud started to liquefy and collapse in a small region near the breach in the first 10 s. The spreading shape of the tailings was symmetric along the centerline of the breach during t = 0-20 s. Because the supply of the tailings from the impoundment was asymmetric, the spreading shape gradually became asymmetric when t > 20 s. The maximum velocity of the released tailings occurred at t = 30 s. Then, the flow velocity gradually decreased. After t = 90 s, the tailings slowed down and stopped moving at t = 110 s. The mean velocity of the tailings flow was estimated at around 2.8-3.4 m s\({}^{-1}\). Figure 6: The free-surface profiles of tailings (**a**) by Bingham model (BM) with \(\dot{\gamma}_{y}=0.0\) s\({}^{-1}\), (**b**) by conventional bi-viscosity model (CBM) with \(\dot{\gamma}_{y}=1\times 10^{-4}\) s\({}^{-1}\), (**c**) by conventional bi-viscosity model (CBM) with \(\dot{\gamma}_{y}=2\times 10^{-1}\) s\({}^{-1}\). Figure 7: The free-surface profile of tailings. (**a**) The simulated modified bi-viscous model (MBM) at the stoppage t = 110 s; (**b**) the aerial photo of the FGT66 [60]. Figure 8: Snapshots of free-surface velocity profiles simulated by BM. Figure 9: Snapshots of free-surface velocity profiles simulated by MBM. The observation and simulation results of inundation distance, freezing time, and mean velocity provided in this study, as well as some historical studies, are listed in Table 1 for comparisons. The simulation results of the MBM were not only accurate in the freezing time and the mean velocity, but also in the inundation distance. Figure 10 shows the strain rate and the un-yield/yield zones of MBM results in the center cross-section of the breach (y = 220 m). The vertical axis was ten times exaggerated. The yield strain rate \(\dot{\gamma}_{y}=0.2\) s\({}^{-1}\) was chosen to identify the interface between the un-yield/yield zones. The liquefied zone was located at the breach's front when t = 10 s, due to the large strain rate caused by gravitational force. The liquefied zone gradually shifted to the ground region after t = 10 s, resulting from the large shear stress that occurs near the bottom. However, the neighboring areas remained un-yield. The interface between the un-yield and yield zone was presented during the period of 10-40 s. It was caused by the viscosity's discontinuity of un-yield and yield zones. At t = 90-110 s, the liquefied zone shrunk gradually and disappeared at t = 110 s, due to the zero velocity in the entire flow field. This zero-velocity phenomenon was very close to the real landslide situation in which the velocity ceases to zero eventually. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \begin{tabular}{c} **Inundation** \\ **Distance (m)** \\ \end{tabular} & \
\begin{tabular}{c} **Freezing Time (s)** \\ \end{tabular} &
\begin{tabular}{c} **Mean Velocity (m s\({}^{-1}\))** \\ \end{tabular} \\ \hline Observed values [(60)] & 300 & 60–120 & 2.5–5.0 \\ Theoretical results from charts [(60)] & 550 & 132 & 4.2 \\ [PERSON] [(60)] & 670 & 116 & 6.0 \\ [PERSON] [(19)] & 330 & 120 & 2.75 \\ [PERSON] [(20)] & 360 & 120 & 3.0 \\ [PERSON] model (Figure 8) & 220 & 70–90 & 2.4–3.1 \\ Modified bi-viscosity model (Figure 9) & 310 & 90–110 & 2.8–3.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The summary key values in the case of FGT66. Figure 10: Snapshots on the centerline cross-section of the breach (y = 220 m) predicted by MBM. (**a**) The profiles of strain rate. The color bar is set from 0.0 to 1.0 s\({}^{-1}\) to emphasize the interface between the plug zone and liquefied zone at \(\dot{\gamma}_{y}=0.2\) s\({}^{-1}\). (**b**) The discontinuity of the plug zone and liquefied zone.