A comparative analysis of capacity and non-capacity formulations for the simulation of unsteady flows over finite-depth erodible beds

Finite-depth sediment layers are common in natural water bodies. The presence of underlying bedrock strata covered by erodible bed layers is ubiquitous in rivers and estuaries. In the last years, the development of models based on the non-capacity sediment transport assumption, also called non-equilibrium assumption, has offered a new theoretical background to deal with complex non-erodible bed configurations and the associated numerical problems. Bedload non-capacity sediment transport models consider that the actual solid transport state can be different from the equilibrium state and depending on the temporal evolution of the flow. The treatment of finite-depth erodible bed layers, i.e. partially erodible beds, in bedload models based on the equilibrium approach has usually been made using numerical fixes, which correct the unphysical results obtained in some cases. Generally, the presence of a finite-depth erodible layer implies the introduction of a kind of non-equilibrium condition in the bedload transport state. Nevertheless, this common natural bed configuration has not been previously considered in the development of numerical models. In this work, a finite volume model (FVM) for bedload transport based on non-capacity approach and dealing with finite-depth erodible layers is proposed. New expressions for the actual bedload transport rate and the net exchange flux through the static-moving bed layers interface are used to develop a numerical scheme which solves the coupled shallow water and non-capacity bedload transport system of equations. The reconstruction of the intermediate states for the local Riemann problem at each intercell edge is designed to correctly model the presence of non-erodible strata, avoiding the appearance of unphysical results in the approximate solution without reducing the time step. The new coupled scheme is tested against laboratory benchmarking experiments in order to demonstrate its stability and accuracy, pointing out the properties of both equilibrium and non-equilibrium formulations.