*6.2. Sediment Transport and Deposition Modeling*

Models for transport phenomena (dispersion, diffusion, and deposition) require the knowledge of the hydrodynamic field and the characteristics of the source term in order to produce reliable estimates of the spatial and temporal variability of suspended sediments (and of any associated contamination).

Numerical models for transport phenomena of suspended sediments are mainly distinguished in Eulerian and Lagrangian models on the basis of the selected approach to define the governing equation.

The Eulerian approach follows a formulation based on the description of the sediment concentration point by point, as given in Section 5 by Equation (4) in the special case of a two-dimensional approach. The resolution of the advection–diffusion equation allows for evaluating the space–time evolution of the SSC as a function of the hydrodynamic field and of the specific features of the source term. Similarly to hydrodynamic models, the governing equations can be simplified by averaging on small temporal or spatial scales (RANS and LES) by introducing parameters that represent the turbulence.

The Lagrangian method is based on a formulation that follows the spatial and temporal evolution of the position of individual particles, each representing a portion of the sediment plume. The main peculiarity of the mathematical formulation is that the effect of turbulence is represented by a stochastic formulation modeled by the random vector *d* - *h* (random walk models). As an example, the two-dimensional governing equation in the finite difference framework provided in [5,56] reads as follows: 

$$
\vec{r}\left(t+\delta t\right) = \vec{r}\left(t\right) + \delta t \left(\vec{v} + \sqrt{\frac{6k\_h}{\delta t}}d\_h^\*\right),
\tag{5}
$$

where*r* is the position vector of a specific individual particle, *δt* is the time step, *v* is the (horizontal) current field, *kh* is the horizontal eddy diffusivity, and *d* - *h* is a vector with dimensionless components uniformly distributed in the range [−1, <sup>+</sup><sup>1</sup>]. As an example, Figure 6 shows the particles dispersion due to the nearshore disposal during a beach nourishment intervention estimated by means of a random walk model.

**Figure 6.** Typical results of a random walk model. The plot (**right**) refers to the dispersion of fine sediments due to nourishment (**left**) projects at a coastal defense cell when submerged breakwaters are present. Contour lines refer to the bathymetric configuration, arrows to the 2DH nearshore circulation forced by a sea state propagating along the x-direction, white circles indicate the re-suspension sources, black dots refer to the instantaneous location of passive tracers.

The use of Equation (5) is based on the hypothesis that the sediment is a passive tracer, i.e., it does not alter hydrodynamics, but is simply advected by the current field and progressively dispersed in the water column. The higher the sediment concentration, the lower the validity of this hypothesis, since the rheological behavior of the sediment-water mixture varies. As a consequence, this approximation is likely to be more acceptable in the far field than close to the sediment release source. The use of this hypothesis makes it possible to describe the sediment diffusion and transport process, decoupled from the hydrodynamic model. Alternatively, this aspect can be taken into consideration by altering the local value of the fluid density (also dependent on temperature and salinity). Considering that density also affects the hydrodynamic equations, it is then necessary to solve in a coupled way the two systems of equations (hydrodynamics and transport/diffusion). Many numerical models for hydrodynamics simulation include specific modules (Eulerian and/or Lagrangian) for sediment transport (e.g., [44,57,58]).

In order to improve the accuracy of the solutions, it is possible to consider different granulometric classes. In this case it is necessary to solve the equations separately for each class. In particular, this approach is useful to separate and better reproduce the dynamics of the finest fraction of sediment that undergoes transport processes in larger areas.

As far as DEP is concerned, there are many formulations available in the literature (e.g., [59–64]). For the deposition of non-cohesive sediments, it is possible to refer to the formulation proposed by Stokes, based on the assumption that the flow is in a viscous regime. However, when dealing with cohesive sediment it tends to underestimate deposition (and consequently to overestimate SSC). Therefore, in some cases, it is necessary to resort to formulations that take into account the presence of cohesive sediment (e.g., [65]), which can generate floccules for the attraction between particles that causes aggregation (e.g., [66–68]). Flocculation influences not only the effective diameter of the settling particles, but also the density, since floccules have a lower density than sediment particles with the same diameter [69].

Re-suspension of the sediments is an intensively studied problem but is still of interest (e.g., [70]). It is important to underline the differences in the re-suspension process as a function of the sediment characteristics. In fact, the size and density of the particles are the main factors influencing the re-suspension of non-cohesive sediments. Whereas, cohesive sediments, depending on the composition and the cohesion levels, can be grouped in two types: those that tend to aggregate into floccules when re-suspending and those whose re-suspension occurs as a muddy mixture [68]. Furthermore, cohesive and non-cohesive sediments are generally characterized by different consolidation processes. Non-cohesive sediments tend to consolidate rapidly and form a layer characterized by constant erodibility at the same depths. Cohesive sediments, on the contrary, tend to consolidate slowly and form cohesive base layers characterized by variable erodibility over time and depth. Re-suspension of cohesive sediments is mostly studied in the case of unidirectional or slowly variable currents (e.g., tidal currents), although the action of surface waves sometimes plays a significant role. In particular, the fluctuation of pressure values induced by the waves propagation can weaken and fluidize the sediment at the bottom [71,72]. The erodibility can also vary in relation to other physical, chemical and biological factors, such as the mineralogical composition, the presence of interstitial water and the pH, the ionic composition, the quantity and the type of organic matter in the different types of sediment (e.g., [68]).

Biological activity can also cause a temporal and spatial variability of the sediment erodibility (e.g., [73]). Re-suspension formulations are generally based on the comparison between the tangential stress (on the bottom) due to the hydrodynamics and a critical value of the tangential tension beyond which the sediment is re-suspended. This critical value is related to the geotechnical characteristics of the sediment [74,75] and is often assessed on an empirical basis [69].
