Modeling shallow geological flows on steep terrains using a specific differential transformation

Shallow geological flows such as landslides, debris flows, and floods are frequently modeled mathematically in the depth-averaged framework. As the bed topography becomes steep and irregular, the classical shallow-water model loses accuracy. How to handle bed topography and gravity effects in depth-averaged models becomes important, and the validity of the model needs verifying. In this study, a differential transformation called the G2L transformation linking the global height to the local depth of the surface material is proposed. This transformation helps to reconstruct the landslide/avalanche surface for one-dimensional and two-dimensional terrains that are obviously curved as well as real landslide examples such as the Huangtian and Rongsong landslides with shallowness ratios as high as 1/4. With this G2L transformation and closures of the lateral and bed stress, two versions of a model governing the geological flow dynamics are developed to avoid difficulties in terrain treatment when using differential geometry. Version 1 of the model is a coupled hyperbolic–differential system; version 2 is a modified hyperbolic system. The two proposed models are verified by a simple one-dimensional granular flow problem on a steep slope. Both mitigate problems associated with computational differential geometry of geological flows and the singularity problem of shallow-water models. In particular, the reduced gravity of several current models in landslide dynamics is a special instance of version 2. The two proposed models are well able to mathematically model the gravity geological hazards and may provide the tools required for prediction and mitigation of natural geological hazards.