## 2 Materials and Methods
### r.a workflow Numerical Code  
r.aavaflow, developed by [PERSON] et al. (2017, 2018) ([PERSON] et al. (2017, 2018), 17), is an open-source computational tool designed for simulating the dynamics of rapid mass flows, including avalanches and debris flows. It represents a GIS-based computational framework, and the sourcing mass and topography can be defined through raster maps. The tool employs the NOC-TVD numerical scheme ([PERSON] et al., 2018) along with the Pudasaini two-phase model ([PERSON], 2019), or a simple single-phase, Voellmy-type model. Besides, it considers the interactions between the phases and the entrainment of material along a path. The interested reader will find in the article by [PERSON] et al. (2017) ([PERSON] et al., 2017), full details about the r.aavaflow numerical code. In this tool, the flow's evolution is computed through depth-averaged, two-phase balance equations developed by [PERSON] (2012) ([PERSON], 2019). The mass balance equations for the solid and fluid phases, respectively, are given by:  
\[ \begin{split}&\frac{\partial}{\partial t}(n_{s}h)+\frac{\partial}{ \partial x}\Big{(}n_{s}hu_{s})+\frac{\partial}{\partial y}(n_{s}hv_{s})=0\\ &\frac{\partial}{\partial t}(n_{w}h)+\frac{\partial}{\partial x} (n_{w}hu_{w})+\frac{\partial}{\partial y}(n_{w}hu_{w})=0\end{split}  \tag{2}\]  
The momentum equations for the solid and the fluid phases are written in conservative form, respectively, as:  
\[ \begin{split}&\frac{\partial}{\partial t}[n_{s}h(u_{s}-u_{s}^{vm})]+ \frac{\partial}{\partial x}\bigg{[}n_{s}h\bigg{(}{u_{s}}^{2}-uu_{s}^{vm}+\wp_ {s}^{s}\frac{h}{2}\bigg{)}\bigg{]}\\ &\hskip 14.226378 pt+\frac{\partial}{\partial y}[n_{s}h(u_{s}v_{s}- uv_{s}^{vm})]=hS_{s}\\ &\frac{\partial}{\partial t}[n_{w}h(u_{w}-u_{w}^{vm})]+\frac{ \partial}{\partial x}\bigg{[}n_{w}h\bigg{(}{u_{w}}^{2}-uu_{w}^{vm}+\wp_{w}^{s }\frac{h}{2}\bigg{)}\bigg{]}\\ &\hskip 14.226378 pt+\frac{\partial}{\partial y}[n_{w}h(u_{w}v_{w}- uv_{w}^{vm})]=hS_{w}\end{split}  \tag{3}\]  
where \(u_{s}^{vm}=\gamma C\Big{(}u_{f}-u_{s}\Big{)}\) and \(u_{f}^{vm}=(\alpha_{s}/\alpha_{s})C\Big{(}u_{f}-u_{s}\Big{)}\) are the virtual masses for the solid and fluid phases, respectively. \(\gamma\) is the fluid to solid density ratio. \(C=(N_{vm}-1)/\left(n_{s}/n_{w}+\gamma\right)\)  
Figure 1: Scheme for obtaining partial heights of both phases. is the virtual mass force coefficient. \(N_{vm}=\left(m_{w}/h_{w}-m_{s}/h_{s}\right)/\left(u_{w}-u_{s}\right)\) is the virtual mass number. \(\beta_{s}=K_{s}\text{g}^{z}(1-\gamma)\) and \(\beta_{w}=\text{g}^{z}\) are the hydraulic pressure coefficients for the solid and fluid phases, respectively. \(K\) is the earth pressure coefficient and \(\text{g}^{z}\) the gravity. The source terms of Equation (3) are as follows:  
\[S_{s} =\alpha_{s}\left[\text{g}-\frac{u_{s}}{|\mathbf{u}_{s}|}\tan \delta p_{b_{s}}-\varepsilon p_{b_{s}}\frac{\partial b}{\partial x}\right] \tag{4}\] \[\qquad-\varepsilon\alpha_{s}\gamma p_{b_{f}}\left[\frac{\partial h }{\partial x}+\frac{\partial b}{\partial x}\right]+C_{DG}\left(u_{f}-u_{s} \right)\left|\mathbf{u}_{f}-\mathbf{u}_{s}\right|^{J-1}\] \[S_{w} =\alpha_{w}\left[\text{g}-\varepsilon\left[\frac{1}{h}\frac{ \partial}{\partial x}\left(\frac{h^{2}}{2}p_{bw}\right)+p_{bw}\frac{\partial b }{\partial x}\right.\right.\] \[\qquad-\left.\frac{1}{\alpha_{w}N_{R}}\left\{2\frac{\partial^{2 }u_{w}}{\partial x^{2}}+\frac{\partial^{2}v_{w}}{\partial y\partial x}+ \frac{\partial^{2}u_{w}}{\partial y^{2}}-\frac{Xu_{w}}{\varepsilon^{2}h^{2}}\right\}\] \[\qquad+\left.\frac{\partial}{\partial y}\left(\frac{\partial n_{ s}}{\partial x}(v_{w}-v_{s})+\frac{\partial n_{s}}{\partial y}(u_{w}-u_{s}) \right)\right\}\] \[\qquad-\frac{\xi n_{s}(u_{w}-u_{s})}{\varepsilon^{2}n_{w}N_{R}h^{ 2}}\right]\right]+\frac{1}{\gamma}C_{DG}\left(u_{f}-u_{s}\right)|\mathbf{u}_{w }-\mathbf{u}_{s}|^{J-1}\]  
where \(\text{g}\) is the non-dimensional gravitational acceleration. \(\delta\) the internal friction angle.