Modeling fast flows with variable water content: A depth-integrated SPH approach

Water content is one important factor on which velocities, heights, and runout of debris flows and similar phenomena depend. To this purpose, we need two ingredients (i) a mathematical model describing the incorporation of water into the moving soil, which results in a change of water content, and (ii) a rheological model with properties depending on the water content. Modeling of such problems can be done either by using either a: (i) two phases approach, where velocities of solid and water may be different, using two sets of nodes, or (ii) two phases approach where velocities of both are assumed to be the same, using a single set of nodes. In both cases, the models have to implement a mechanism for the water inflow -or outflow-. We will modify both types of two phases models (one or two sets of nodes for solid and water) to include the change of water content due to water inflow. Implementation in the SPH requires extending the algorithm and updating the smoothing length because it is based on the mass of particles and their relative position. Updating the smoothing length when only changes mass is to be avoided. Regarding the rheological model, we will introduce a new model for frictional debris flows implementing a Voellmy coefficient which depends on water content. Alternatively, we will propose a more consistent model based on Bagnold's idea of introducing a 1D concentration parameter (λ). Finally, we will illustrate the proposed model capability with two examples, a dam break problem, and a real case in El Salvador where the water content played an important role in the propagation properties of a debris flow.