## 2 Materials and Methods
### Numerical Methods
#### 2.4.2 SPH-Smoothed Particle Hydrodynamics  
In this section, we perform the GeoFlow-SPH simulation model which has been developed in Madrid by an expert research team for almost a decade. It has previously been applied to theoretical, experimental, and real case histories. First, we briefly describe a Lagrangian meshless numerical method of Smoothed Particle Hydrodynamics (SPH) that is used to transform the problems that are basically in the form of partial differential equations (PDEs) to a form suitable for particle-based simulation. The SPH method was first invented by [30; 31] to model astrophysical problems. This technique has been applied in many areas due to its capability to model complicated cases that involve large-displacement deformations, such as modeling fast landslides in Solid Mechanics [32; 33; 34]. Good reviews can be found in the texts of [35; 36]. In this model, these 3D problems are transformed into a 2D form by applying a depth-integrated model. As a result, the depth-integrated model provides an excellent combination of accuracy and computational time. This technique has been successfully applied to landslides by [15; 16; 37; 38; 39; 40; 41]. The two-phase depth-integrated SPH model we follow here is proposed based on the mathematical model proposed by [PERSON] and [PERSON] [42] and is similar to those of [42; 43; 44]. To discretize the propagating mass in the SPH method, the first step is to present them as a set of nodes, as depicted in Figure 8, exhibiting individual material properties. Then, an interpolation process calculates the relevant properties on each node over neighboring nodes through a kernel function \(\big{(}W_{ij}=W\big{(}x_{i}-x_{j},h\big{)}\big{)}\), without having to define any element. The formulation can be expressed as  
\[f(x_{i})=\int f\big{(}x_{j}\big{)}W\big{(}x_{i}-x_{j},h\big{)}dx_{j} \tag{6}\]  
where the function \(f(x)\) is approximated at a position vector \(x\) in space. By applying the particle approximation technique and considering that the infinitesimal volume \(\left(dx_{j}\right)\) is related to the mass and density of the particle \(j\), the function at particle \(i\) can be written as  
\[f(x_{i})\overset{\simeq}{=}\sum_{j=1}^{N}f(x_{j})\left(x_{i}-x_{j},h\right) \frac{m_{j}}{\rho_{j}} \tag{7}\]  
Then, in the numerical SPH method, the ordinary differential equations (ODEs) of the balance of mass and momentum can be produced in discretized form with respect to, respectively,  
\[\frac{\overline{d}h_{i}}{dt}+h_{i}{\sum_{j=1}^{Nh}}\frac{m_{j}}{\rho_{j}}v_{j} \text{grad}W_{ij}=\overline{n}_{i}e_{R} \tag{8}\]  
\[ \begin{split}\frac{\overline{d}^{(\text{a})}\overline{v}_{ai}}{ dt}&=-{\sum_{j=1}^{Nh}}m_{aj}\left(\frac{\overline{P}_{ai}}{h_{ai}^{2}}+ \frac{\overline{P}_{aj}}{h_{aj}^{2}}\right)\text{grad}W_{ij}-\left(\frac{1}{2} \frac{\rho_{w}}{\rho_{\alpha}}b_{3}h_{ai}^{2}-\frac{\Delta\overline{p}_{w}h_ {ai}}{\rho_{\alpha}}\right){\sum_{j=1}^{Nh}}m_{aj}\\ &\qquad\qquad\left(\frac{\overline{n}_{ai}}{h_{ai}^{2}}+\frac{ \overline{n}_{aj}}{h_{aj}^{2}}\right)\text{grad}W_{ij}+\frac{1}{\rho_{\alpha} h_{ai}}\tau_{B}^{(\text{a})}+b_{i}+\frac{1}{\rho_{\alpha}}\overline{R}_{ \alpha}-\frac{1}{h_{ai}}\overline{n}_{ai}e_{R}\end{split}  \tag{9}\]  
where \(\alpha\) denotes the phase (\(s\) or \(w\)), \(\overline{n}_{\alpha}\) are the volume fractions of solid and fluid phases (\(\overline{n}_{s}=1-\overline{n}\) and \(\overline{n}_{w}=\overline{n}\)), and \(h_{\alpha}=h\overline{n}_{\alpha}\). \(\overline{P}_{\alpha}\) is the averaged pressure (\(\overline{P}_{\alpha}\)) acting on solid or fluid phases and can be defined as  
\[\overline{P}_{s}=\frac{1}{2}b_{3}hh_{s}+\frac{\Delta\overline{p}_{w}h \overline{n}}{\rho_{s}}\text{ For Solid} \tag{10}\]  
\[\overline{P}_{w}=\frac{1}{2}b_{3}hh_{w}-\frac{\Delta\overline{p}_{w}h \overline{n}}{\rho_{w}}\text{ For Fluid} \tag{11}\]  
where \(\Delta\overline{p}_{w}\) is the excess pore-water pressure. The interested reader will find in the article by [40] a detailed description of this two-phase SPH model.