A depth-integrated SPH framework for slow landslides

Slow and very slow landslides can cause severe economic damage to structures. Due to their velocity of propagation, it is possible to take action such as programmed maintenance or evacuation of affected zones. Modeling is an important tool that allows scientists, engineers, and geologists to better understand their causes and predict their propagation. There are many available models of different complexities which can be used for this purpose, ranging from very simple infinite landslide models which can be implemented in spreadsheets to fully coupled 3D models. This approach is expensive because of the time span in which the problems are studied (sometimes years), simpler methods such as depth-integrated models could provide a good compromise between accuracy and cost. However, there, the time step limitation due to CFL condition (which states that the time step has to be slower than the ratio between the node spacing $\Delta x$ and the physical velocity of the waves results in time increments which are of the order of one-10th of a second on many occasions. This paper extends a technique that has been used in the past to glacier evolution problems using finite differences or elements to SPH depth-integrated models for landslide propagation. The approach is based on assuming that (i) the flow is shallow, (ii) the rheological behavior determining the velocity of propagation is viscoplastic, and (iii) accelerations can be neglected. In this case, the model changes from hyperbolic to parabolic, with a time increment much larger than that of classic hyperbolic formulations.