## 2 Materials and Methods
### GeoFlow-SPH Numerical Code
It is based on an algorithm in which the total volume of debris increases in accordance with a specified rate and is given by:  
\[e_{R}=E_{s}h\bar{\sigma} \tag{12}\]  
which indicates a direct proportionality between the entrainment rate and the product of depth average velocity \((\bar{\sigma})\) and mobilized soil depth \((h)\). \(E_{s}\) can be obtained directly from the initial and final volumes of the material and the distance traveled as \(E_{s}\approx\ln((V_{\text{final}}/V_{0})/(\text{distance}))\). The interaction between solid and fluid phases is given by the depth-integrated value of \(\bar{R}\). It is provided by the Anderson and Jackson law [27], which is capable of considering large relative velocities. It is given by:  
\[\bar{R}=\frac{\bar{n}(1-\bar{n})}{V_{T}\bar{n}^{m}}(\rho_{s}-\rho_{w})g(\bar{ \sigma}_{w}-\bar{\sigma}_{s}) \tag{13}\]  
where \(V_{T}\) is the terminal velocity, g is the acceleration of gravity, and \(m\) is a constant. In depth-integrated models, the vertical structure of the magnitudes is lost as the only available information is their depth-integrated values. To overcome this limitation, an additional equation has been implemented in GeoFlow-SPH's model to describe how pore pressure evolves over time and depth [28]. Here, we recall the consolidation equation describing the evolution of pore pressure along the vertical axis, as follows:  
\[\frac{d^{(\ s)}\Delta p_{w}}{dt}=\ -\rho_{d}^{{}^{\prime}}b_{3}\frac{dh}{dt}+c_{ v}\frac{\partial^{2}\Delta p_{w}}{\partial x_{3}{}^{2}}-E_{m}\frac{1}{(1-\bar{n})} \frac{d^{(\ s)}\bar{n}}{dt} \tag{14}\]where \(\rho_{d}^{{}^{\prime}}(=(1-n)(\rho_{s}-\rho_{w}))\) is the effective density, \(c_{v}\) the consolidation coefficient, and \(E_{m}\) the odometric modulus. Based on the above equation, the pore-water pressures are also influenced by the propagation height and its velocity, and the porosity variations. The consolidation parameter of \(c_{v}=k_{v}k_{w}\) plays an important role, as a high value of the coefficient allows the rapid dissipation of excess pore-water pressure through the consolidation process. In the article by [PERSON] (2021) [26], the interested reader will find a detailed explanation of the consolidation equation. Next, we briefly describe a Lagrangian meshless numerical method that is used to discretize the presented depth-integrated equations. Smoothed particle hydrodynamics (SPH) uses it to transform the problems that are basically in the form of partial differential equations (PDEs) to forms suitable for particle-based simulation. SPH was first invented by [PERSON] (1977) [29] and [PERSON] (1977) [30] to model astrophysical problems. This technique has been applied in many areas due to its ability to model complicated cases which involve large-displacement deformations, such as the modeling of fast landslides in solid mechanics [4; 31; 32]. Good reviews can be found in the texts of [PERSON] (2004) [33] and [PERSON] (2003) [34]. In this model, these 3D problems are transformed into 2D forms by applying a depth-integrated model. As a result, it provides an excellent combination of accuracy and computational time. This technique has been successfully applied to debris flows by [PERSON] et al. (2014) [10] and [PERSON] et al. (2014, 2016) [35; 36], and debris avalanches by [PERSON] et al. (2014, 2016) [37; 38]. To discretize the propagating mass in the SPH method, the first step is to present them as a set of nodes, as depicted in the figure, having individual material properties. In this paper, the mathematical equations are discretized while using the two-phase SPH technique to deal with two phases of flow involving solid and water particles. Therefore, two sets of nodes are introduced, one to represent the solid particles' movement and another to represent the movement of the fluid particles, as depicted in Figure 2.  
Then, an interpolation process calculates the relevant properties on each node over neighboring nodes through a kernel function \(\left(W_{IJ}=W\big{(}x_{I}-x_{J},h\big{)}\right)\) without defining any element.