Comparative analysis of <i>μ</i> ( <i>I</i> ) and Voellmy-type grain flow rheologies in geophysical mass flows: insights from theoretical and real case studies

Abstract. The experimentally based μ(I) rheology is now prevalent in describing the movement of gravitational mass flows. We reinterpret the μ(I) rheology as a Voellmy-type relationship to highlight its connection to grain flow theory and demonstrate its practical applications. Using one-dimensional block modeling and two real-world case studies – the 2017 Piz Cengalo rock–ice avalanche and an experimental snow avalanche at the Swiss Vallée de la Sionne test site – we demonstrate the relationship between the dimensionless number I and the granular temperature R, establishing the equivalence between μ(I) and widely used Voellmy-type grain flow rheologies μ(R). Results indicate that μ(I) rheology utilizes the dimensionless inertial number I to mimic contributions of granular temperature/fluctuation energy to flow behavior. In terms of Voellmy, the μ(I) rheology contains a velocity-dependent turbulent friction coefficient that models shear-thinning behavior. This turbulent friction assumes the production and decay of fluctuation energy are in balance, exhibiting no difference during accelerative and depositional phases of avalanche flow. The constant Coulomb friction coefficient prevents μ(I) rheology from accurately modeling the dispositional characteristics of actual mass flows. The modeled evolution of the snow avalanche using the μ(I) rheology is too slow, lagging 5 s behind the measured values. More importantly, the calculated runout extends approximately 200 m beyond the observed limits, with significant deposit anomalies in the valley. By incorporating non-steady production and decay of fluctuation energy in the μ(R) framework, it becomes possible to achieve a good match with both the measured velocities and the observed runout. Our results highlight the strengths and limitations of both μ(I) and Voellmy μ(R) rheologies, bolstering the theoretical foundation of mass flow modeling while revealing practical engineering challenges.