Summary
This paper presents a model which can be used for fast landslides where coupling between solid and pore fluid plays a fundamental role. The proposed model is able to describe debris flows where the difference of velocities between solid grains and fluid is important. The approach is based on the mathematical model proposed by Zienkiewicz and Shiomi, which is similar to those of Pitman and Le and Pudasaini. The novelty of the present work is the numerical technique used, the smoothed particle hydrodynamics (SPH). We propose to use a double set of nodes for soil and water phases, the interaction between them being described by a suitable drag law. The paper presents both mathematical and numerical models, describing the main assumptions and their limitations. Then, the model is applied to (1) a simple case where shocks and expansion waves appear, (2) a dam break problem on a horizontal plane with a frictional soil phase, and (3) a debris flow which happened in Hong Kong. The main conclusions that can be drawn from the applications are: 

Debris flows having 2 phases with important relative mobility present a rich structure of shocks and rarefaction waves, which has to be properly modeled. Otherwise, the model will have numerical damping or dispersion.
Dambreak exercises provide interesting information in simple and controlled situations. We can see how both phases move relative to each other.
Real debris flows can be simulated with the proposed model, obtaining reasonable results.

A two-phase SPH model for debris flow propagation

SPH-FDM propagation and pore water pressure modelling for debris flows in flume tests

Two-phase SPH numerical study of pore-water pressure effect on debris flows mobility: Yu Tung debris flow

Debris flows are a type of fast landslides where a mixture of soil and water propagates along narrow channels.
 The main characteristics are (1) important relative displacements between the solid and fluid phases, and (2) development of pore-water pressures in excess to hydrostatic. The ratios between vertical and horizontal displacements of the flow, from the triggering point to the deposition, indicate that friction angles are much smaller than those measured in laboratories. Debris flows are modeled as two phases flow, but implementing pore-water pressure is an important issue. The purpose of this paper is to improve the existing two phases debris flow models by implementing pore-water pressures in excess to hydrostatic. It is found that pore pressure evolution depends on consolidation, changes in the flow depth, and changes and gradients of porosity. The proposed depth integrated mathematical model is discretized using two sets of SPH nodes (solid and fluid), with a set of finite difference meshes associated with each solid material SPH point. The paper presents two examples from where it is possible to gain insight into the differences between the models (with and without excess pore water pressure).

https://link.springer.com/content/pdf/10.1007/s11440-020-01114-4.pdf

A depth integrated, coupled, two-phase model for debris flow propagation

Source areas, connectivity, and delivery rate of sediments in mountainous-forested hillslopes: A probabilistic approach

This paper deals with modelling of landslide propagation. Its purpose is to present a methodology of analysis based on mathematical, constitutive and numerical modelling, which includes both well-established theories together with some improvements which are proposed herein. Concerning the mathematical model, it is based on Biot–Zienkiewicz equations, from where a depth-integrated model is developed. The main contribution here is to combine a depth-integrated description of the soil–pore fluid mixture together with a set of 1D models dealing with pore pressure evolution within the soil mass. In this way, pore pressure changes caused by vertical consolidation, changes of total stresses resulting from height variations and changes of basal surface permeability can be taken into account with more precision. Most of rheological models used in depth-integrated models are derived either heuristically (the case of Voellmy model, for instance), or from general 3D rheological models. Here, we will propose an alternative way, based on Perzyna’s viscoplasticity. The approach followed for numerical modelling is the SPH method, which we have enriched adding a 1D finite difference grid to each SPH node, in order to improve the description of pore water profiles in the avalanching soil. This paper intends to be a homage to Professor Felix Darve, who has very much contributed to the field of modern geomechanics.

A viscoplastic approach to the behaviour of fluidized geomaterials with application to fast landslides

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A viscoplastic approach to the behaviour of fluidized geomaterials with application to fast landslides