An overview of debris-flow mathematical modelling

Debris flows are among the most catastrophic natural phenomena, attracting the interest of researchers, engineers and government agencies. The complexity of the physical process has led to the development of numerous mathematical models to simulate and deeper understand the propagation of debris flows. Nevertheless, the intrinsic characteristics of models are not all suitable for every process observed in nature. Hence, this review identifies the main features of the different modelling approaches for debris flows and proposes recommendations for a more rigorous selection of models. The classification of the models is based on the following features: (1) phases of the flow; (2) entrainment of materials into the flow; (3) constitutive relationships; (4) spatial dimensionality; and (5) the solution methods. We discuss the advantages and limitations of the most remarkable contributions by comparing the available mathematical models for debris flows based on each feature. A selection strategy is proposed to be supported by the main assumptions of the models, the classification features and some dimensionless numbers evoked in debris flow research. As a result of this review, the principal considerations can be summarised as follows. First, the complexity of the models has been increasing with the enhanced understanding of debris flows and advances in technology. Thus, the multiphase approach, more sophisticated constitutive relationships, three-dimensional representations and numerical methods requiring higher computational cost are becoming noteworthy. Secondly, three-dimensional representations are better suited for the study of specific and small-scale debris-flow processes, while the two-dimensional depth-averaged approach is still the most appropriate for field applications. Finally, the most recognised and recent models are highlighted and concisely compiled in a table to provide an overview of the options.