# Multidisciplinary Study of Mud Emissions Following the 2016 Norcia Earthquake
## 5 Discussion
### Modeling Mudflow  
We used two equations to model the flow from the emission points: the [PERSON] and Lonergan model [56] to calculate the rise velocity of liquefied muds and the [PERSON] et al. model [57] to calculate the flow velocity on the substratum. In their study of sandstone intrusions due to seismicity-induced liquefaction, [PERSON] and [PERSON] [56] found that the forceful intrusion of remobilized clastic sediment, and its emplacement into the surrounding strata, forms sheets of sediment that are discordant to bedding (dykes) or concordant with bedding (sills). These intrusions can stay in the substrate as fissures, dykes, and sills and can reach the surface as sand volcanoes and flow on the substratum. The injection of high-pressure fluids (with entrained sand grains) into the surrounding sediments requires a sustained pressure differential between the fluid in the propagating fracture or intrusion and the fluid in the pores of the sediment or sedimentary rock, so that the fracture dilates and the sand-fluid mixture can flow through the fracture as liquefied in response to a sudden shock, e.g., an earthquake [56]. We used the minimum fluidization velocity (\(U_{m\!f}\)) proposed by [PERSON] and Lonergan ([56] and references therein):  
\[U_{m\!f}=\frac{0.00059\cdot\!d^{2}\left(\rho_{s}-\rho_{f}\right)\cdot\!g}{\eta} \tag{2}\]  
where \(d\) is the grain diameter, \(\rho_{f}\) is the density of the fluid, \(\rho_{s}\) is the density of the grains, \(g\) is the acceleration due to gravity, and \(\eta\) the dynamic viscosity of the fluid (water; 3\(\cdot\)10\({}^{-4}\) Pa-s). We considered grains with a minimum and maximum size of 2 and 8 \(\upmu\)m, respectively, based on the results of the granulometric analysis. \(\rho_{f}\) is the water density, and \(\rho_{s}\) was assumed equal to 1600 kg/m\({}^{3}\). The result shows a minimum fluidization velocity ranging between 0.05 and 0.74 m/s for grain sizes of 2 and 8 \(\upmu\)m, respectively.  
By selecting the parameters listed in Table 3, and assuming a Bingham fluid moving on an infinitely wide plane, we determined the velocity \(V\) of the selected suspensions by applying the following relation [57]:  
\[V=\frac{\rho\cdot\!g\cdot\!sen\theta}{2\cdot\eta}\cdot\!y_{0}^{2} \tag{3}\]  
where \(\rho\) is the density of the mixtures--we used 1400 kg/m\({}^{3}\), following Del Gaudio and Ventura [52], \(g\) is the gravity acceleration (9.8 m/s\({}^{2}\)), \(\theta\) is the slope angle (we considered \(2^{\circ}\), \(5^{\circ}\), and \(10^{\circ}\)), \(\eta\) the dynamic Bingham viscosity, and \(y_{0}\) the thickness of the shear layer calculated as:  
\[y_{0}\ =H-\frac{\tau_{B}}{\rho\cdot g\cdot sent\theta} \tag{4}\]  
where \(H\) is the flow thickness (here assumed as 0.7 m). This relation holds for \(H\geq y\geq y_{0}\) (y is measured from the surface in its perpendicular direction) and can be used to determine the velocity of non-Newtonian suspensions provided that the thickness \(H\), the density of the water, the suspended load, and the slope angle of the substratum are known. The results are summarized in Figure 4. The highest velocities are reached for sample AM7 (29.5 wt.% of water). If the AM7 flow moves on a substratum with a slope of \(2^{\circ}\), the maximum velocity is 0.12 mm/s; if \(5^{\circ}\), the maximum velocity is 0.37 mm/s; and if \(10^{\circ}\), the maximum velocity is 0.79 mm/s. In the range of 10-30 wt.% of water content, the flow velocity increases with increasing water content. By increasing the water content of the sample above 30 wt.%, the velocity does not increase. This could reflect the lack of particle aggregates within the water-solid mixture and the consequent movement of the flow as a particle-free water stream.