A two-layer shallow water model for bedload sediment transport: convergence to Saint-Venant-Exner model

TL;DR

This paper introduces a two-layer shallow water model for bedload sediment transport that converges to a Saint-Venant-Exner system with gravitational effects, offering computational efficiency and applicability to various transport regimes.

Contribution

It proposes a novel two-layer model with generalized friction laws that converges to a SVE system including gravity, reducing computational costs for sediment transport simulations.

Findings

Model converges to SVE with gravity in small ratio regimes

Numerical tests show good performance in low transport rate scenarios

Inclusion of gravitational effects without high computational cost

Abstract

A two-layer shallow water type model is proposed to describe bedload sediment transport. The upper layer is filled by water and the lower one by sediment. The key point falls on the definition of the friction laws between the two layers, which are a generalization of those introduced in Fern\'andez-Nieto et al. (ESAIM: M2AN, 51:115-145, 2017). This definition allows to apply properly the two-layer shallow water model for the case of intense and slow bedload sediment transport. Moreover, we prove that the two-layer model converges to a Saint-Venant-Exner system (SVE) including gravitational effects when the ratio between the hydrodynamic and morphodynamic time scales is small. The SVE with gravitational effects is a degenerated nonlinear parabolic system. This means that its numerical approximation is very expensive from a computational point of view, see for example T. Morales de Luna et al. (J. Sci. Comp., 48(1): 258-273, 2011). In this work, gravitational effects are introduced into the two-layer system without such extra computational cost. Finally, we also consider a generalization of the model that includes a non-hydrostatic pressure correction for the fluid layer and the boundary condition at the sediment surface. Numerical tests show that the model provides promising results and behave well in low transport rate regimes as well as in many other situations.