We present a numerical model for the simulation of 3D mono-dispersed sediment dynamics in a Newtonian flow with free surfaces. The physical model is a macroscopic model for the transport of sediment based on a sediment concentration with a single momentum balance equation for the mixture (fluid and sediments). The model proposed here couples the Navier-Stokes equations, with a volume-of-fluid (VOF) approach for the tracking of the free surfaces between the liquid and the air, plus a nonlinear advection equation for the sediments (for the transport, deposition, and resuspension of sediments). The numerical algorithm relies on a splitting approach to decouple diffusion and advection phenomena such that we are left with a Stokes operator, an advection operator, and deposition/resuspension operators. For the space discretization, a two-grid method couples a finite element discretization for the resolution of the Stokes problem, and a finer structured grid of small cells for the discretization of the advection operator and the sediment deposition/resuspension operator. SLIC, redistribution, and decompression algorithms are used for post-processing to limit numerical diffusion and correct the numerical compression of the volume fraction of liquid. The numerical model is validated through numerical experiments. We validate and benchmark the model with deposition effects only for some specific experiments, in particular erosion experiments. Then, we validate and benchmark the model in which we introduce resuspension effects. After that, we discuss the limitations of the underlying physical models. Finally, we consider a one-dimensional diffusion-convection equation and study an error indicator for the design of adaptive algorithms. First, we consider a finite element backward scheme, and then, a splitting scheme that separates the diffusion and the convection parts of the equation.