## 4 Case Study--The Failure of the Gypsum Tailings Dam in East Texas in 1966 (FGT66)
### Difference between the BM and MBM  
Figure 11 shows the velocity magnitude profiles obtained from BM and MBM in the centerline cross-section (y = 220 m). In the BM results, the tailings moved faster than that in MBM. The maximum velocity of the tailings' front at t = 10 s was approximately 6.0-8.0 m s\({}^{-1}\) and decreased sharply during t = 10-40 s. It made the inundation distance (around 220 m) at t = 110 s shorter than that in MBM. In the MBM results, the maximum moving velocity was about 5.0-7.0 m s\({}^{-1}\), which was slightly smaller than that in the results of BM. However, the MBM tailings flow took a longer time to reach the zero-velocity stage and the resulting inundation distance was longer than that in the BM results. The inundation distance (310 m) predicted by MBM was very close to the field observed (300 m). The yield strain rate played an important role in MBM. As long as the strain rate was smaller than the yield strain rate, a large viscosity was applied to slowing down the deformation. Due to the absence of measuring techniques, the yield strain rate in this study was obtained from the sensitivity analysis by taking the flood distance from the field observation as the criterion.  
Figure 12 illustrates the strain rate profile of the initiation process of the tailings flow. The strain rate profiles in BM results showed a smooth and continuous feature. A large amount of tailing material deformed and slid down (Figure 12a). On the other hand, in MBM results, the yield strain rate \(\dot{\gamma}_{y}=0.2\) s\({}^{-1}\) was introduced as the indicator to identify the plug and sheared zone. Because the un-yield viscosity \(\mu_{A}=10^{10}\) Pa s was much greater than \(\tau_{y}/\dot{\gamma}_{y}\), a discontinuous pattern of the strain rate could be observed in Figure 12b. The yield strain rate \(\dot{\gamma}_{y}=0.2\) s\({}^{-1}\) kept the plug zone rigid. The initiation process of the mudslide in MBM results was different from that in BM results. A high strain rate appeared not only near the toe of the breach but also in the gate area, which caused the sliding process and formed a slip surface. The slip surface was the interface between the un-yield and  
Figure 11: Snapshots of velocity magnitude on the centerline cross-section of the breach (y = 220 m) (**a**) Bingham model (BM); (**b**) modified bi-viscosity model (MBM).  
yield regions. In the bank of homogeneous mud, the slip surface of failure could be determined by the empirical method, which follows the arc of a circle that usually intersects the toe of the bank [52; 53]. However, the slip surface was developed automatically by MBM. It is worth a more profound study in the future.  
Figure 13 shows the strain rate profiles of BM and MBM. The slip surface (Figure 13b) at t = 10 s) and the interface between the plug/sheared zones (Figure 13b) at t = 40 s can be identified in the results of MBM. The slip surface was relatively sharp in the MBM results compared to the ones in MB (Figure 13a).  
Because of adopting the larger un-yield viscosity \(\mu_{A}\), larger yield strain rate \(\dot{\gamma}_{y}\), same yield stress \(\tau_{y}\), and same yield viscosity \(\mu_{B}\) to simulate the FGT66, the un-yield zone of tailings in MBM was sturdier than that in the BM. However, once tailings yielded, the rheology returned to the conventional Bingham properties.  
Figure 12: Snapshots of strain rate profile at the early stage on the centerline cross-section of the breach (y = 220 m). (**a**) Bingham model (BM); (**b**) modified bi-viscosity model (MBM).