Hydrogeological instability is among the effects of climate change with the highest impact on the safety of people and of the built environment. In particular, landslides are responsible for significant human and economic losses worldwide. The capability to predict landslides and to assess the risk connected with extreme events is of paramount importance to the safety of people and infrastructures. In this sense, Computer simulations in geomechanics have nowadays become essentials, as they are increasingly replacing expensive and time-consuming real experiments. Despite Grid-based numerical methods, like Finite Element Method, are widely used in the context of landslides simulations, these approaches have to deal with the extreme changes of geometry associated to problems like large deformations, crack propagation and multi-phase interactions. From this perspective, particle-based methods, such as Material Point Method (MPM) are proven to be suitable to manage this kind of problems, while preserving physical accuracy and details of the material front during the simulations. In this work we have adapted the classical conservative shallow water equations in the context of the MPM framework, developing a semi-conservative variant of the Depth Averaged Material Point Method (DAMP) in order to take into account the hydrostatic pressure gradient. Both bed friction and the rheology are considered in this framework, following the Voellmy model and the depth integrated Bingham visco-plastic stress model respectively. The tracking of the material front is carried out at every time step, by coupling the use of alpha-shapes and the polygonal line defined by the material boundary particles. After verifying the performances of the numerical method through different benchmarks and idealized settings, it has been tested on a realistic scenario.