Modelling the non-uniform sediment transport by means of the statistical moment approach

One-dimensional mathematical modelling of non-uniform sediment transport in freesurface, mobile-bed flows, is commonly faced adding to the De St. Venant equations, the mass-balance equations for each grain-size class in which the statistical distribution of the bed material is divided on the basis of a given choice. This method is called Bed Material Fraction (BMF) approach. A possible alternative is the so-called Statistical Moment (SM) approach, in which the non-uniformity of the sediments is described in terms of the statistical moments of the distribution. In this work, we show how the two methods can be obtained as different approximations of the same differential equation for the bed material distribution in the space of the diameters. Moreover, because of the strong similitude of the relevant differential equation in the x-t space, the two methods can be treated in a unified way from a numerical point of view. Finally, an idealized test case allows comparing the performances of the two approaches.