## 2 Materials and Methods
### r.a workflow Numerical Code
\(p_{b_{s}}=\text{g}^{z}\) and \(p_{bw}=p_{b_{s}}/\left(1-\gamma\right)\) are effective basal pressures for the solid and fluid phases. \(\varepsilon\) is the typical height \(\left(H\right)\) to the typical extent of debris flow \(\left(L\right)\). \(X\) is the viscous shearing coefficient for fluid and \(\xi\) the vertical distribution of the particle concentration. \(N_{R}=\left(\sqrt{g\Gamma}\rho_{w}H\right)/\left(n_{w}\rho_{w}\right)\) and \(N_{R_{A}}=\left(\sqrt{g\Gamma}\rho_{w}H\right)/\left(A\rho_{w}\right)\) are the quasi-Reynolds numbers and the mobility Reynolds numbers, respectively. \(A\) is the mobility of fluid at the interface. In the special case in which \(A=n_{w}\), these quasi-Reynolds numbers coincide. \(C_{DG}\) is the generalized drag coefficient. Next, a set of additional functions and laws are needed to complete the conservation of mass and momentum equations presented in the current section. Here, we provide the following items: (1) the basal friction laws, (2) the interaction force laws, and (3) the erosion laws. Regarding the rheological models implemented in the r.avaflow code, the solid stress is computed based on Mohr-Coulomb plasticity, whereas the fluid stress is modeled as a solid volume-fraction-gradient-enhanced non-Newtonian viscous stress. These models rely on the empirical coefficients \(C_{AD}\) for ambient drag and \(C_{FF}\) for fluid friction. \(C_{AD}\) is a coefficient to be multiplied with the frontal surface and the velocity of the flow to derive air resistance. \(C_{FF}\) is Manning's coefficient used as the fluid friction coefficient, to consider the effects of the roughness of basal surface. Entrainment is computed through an empirical law in which a user-defined entrainment coefficient \(\left(E_{s}\right)\) is multiplied with the total momentum \(\left(M_{\text{s}}+M_{\text{w}}\right)\) at each raster cell. Then, the solid and fluid entrainment rates \(q_{\text{E,s}}\) and \(q_{\text{E,w}}\) can be obtained as follows:  
\[q_{\text{E,s}} =E_{\text{s}}|M_{\text{s}}+M_{\text{w}}|\tilde{n}_{\text{s,E}} \tag{5}\] \[q_{\text{E,w}} =E_{\text{s}}|M_{\text{s}}+M_{\text{w}}|\left(1-\tilde{n}_{\text{ s,E}}\right)\]  
where \(\tilde{n}_{\text{s,E}}\) is the solid fraction of the entrainable material. In the r.avaflow, the changes of the basal topography are considered by computing the solid and fluid erosion heights, \(H_{\text{E,s}}\) and \(H_{\text{E,w}}\), at each time step, as follows:\[ \begin{split} H_{\text{E,s,t}}&=\min\biggl{(}H_{\text{E,s} }(t-\Delta t)+\frac{q_{\text{E,s}}\Delta t}{\cos\beta},H_{\text{Emax,s}}\biggr{)} \\ H_{\text{E,w,t}}&=\min\biggl{(}H_{\text{E,w}}(t- \Delta t)+\frac{q_{\text{E,w}}\Delta t}{\cos\beta},H_{\text{Emax,w}}\biggr{)} \end{split}  \tag{6}\]  
where \(H_{\text{Emax,s}}\) and \(H_{\text{Emax,w}}\) are the maximum entraniable depths at the given cell, \(t\) is the time passed at the end of the time step, \(\Delta t\) is the time step length, and \(\beta\) is the local slope of the basal surface. The solid-fluid interaction is considered by using the two-phase generalized drag model proposed by [PERSON] (2012) [19], a modified Pitman and Le model [15]. It is capable of taking into account the effect of fluid viscosity on drag. The generalized drag coefficient (\(C_{DG}\)) is given by:  
\[C_{DG}=\frac{n_{s}n_{w}(\rho_{s}-\rho_{w})\text{g}}{\bigl{[}V_{T}\bigl{\{}PF \bigl{(}Re_{p}\bigr{)}+(1-P)G\bigl{(}Re_{p}\bigr{)}\bigr{\}}\bigr{]}^{m}} \tag{7}\]  
where \(V_{T}=\sqrt{\text{g}d/\gamma}\) is the terminal velocity. \(d\) the particle diameter. \(P\in(0,1)\) is the parameter for a combination of solid and fluid-like contributions to drag resistance. \(F=\bigl{(}\gamma(n_{w}/n_{s})^{3}\bigl{(}Re_{p}\bigr{)}\bigr{)}/180\). \(Re_{p}=(\rho_{w}du_{T})/\eta_{w}\) is the particle Reynolds number. \(\eta_{w}\) the fluid viscosity. \(G=\alpha_{f}^{M-1}\) where \(M=M\bigl{(}Re_{p}\bigr{)}\) varies from 4.65 to 2.4. \(m\) is the exponent for drag (1 for linear, 2 for quadratic).