# Quantitative Assessment of the Environmental Impacts of Dredging and Dumping Activities at Sea
## 2 Materials and Methods
### Description of Study Site
2D solves the advection-dispersion equation as follows:  
\[\frac{\partial\overline{c}}{\partial t}+u\frac{\partial\overline{c}}{ \partial x}+\
u\frac{\partial\overline{c}}{\partial y}=\frac{1}{h}\frac{ \partial}{\partial x}\bigg{(}hD_{x}\frac{\partial\overline{c}}{\partial x} \bigg{)}+\frac{1}{h}\frac{\partial}{\partial y}\bigg{(}hD_{y}\frac{\partial \overline{c}}{\partial y}\bigg{)}+Q_{L}C_{L}\frac{1}{h}-S \tag{5}\]  
where \(\overline{c}\) is depth averaged mass concentration (kg\(\cdot\)m\({}^{-3}\)) \(u\); \(\
u\) is depth averaged flow velocities (m\(\cdot\)s\({}^{-1}\)); \(D_{x}\) and \(D_{y}\) are dispersion coefficients; \(h\) is water depth; \(S\) is the deposition/erosion term (km\(\cdot\)m\({}^{-3}\)\(\cdot\)s\({}^{-1}\)); \(Q_{L}\) is the source discharge per unit horizontal area (m\({}^{3}\)\(\cdot\)s\({}^{-1}\)\(\cdot\)m\({}^{-2}\)); and \(C_{L}\) is the concentration of the source discharge (kg\(\cdot\)m\({}^{-3}\)). The bottom shear stress \(\tau_{b}\) (N\(\cdot\)m\({}^{-2}\)) is calculated with respect to current and waves using following equation:  
\[\tau_{b}=\frac{1}{2}\rho f_{w}\Big{(}{U_{b}}^{2}+{U_{b}}^{2}+2U_{b}U_{b}\cos \beta\Big{)} \tag{6}\]  
where \(\rho\) is the density of water (kg\(\cdot\)m\({}^{-3}\)); \(f_{w}\) is the wave friction factor; \(U_{b}\) is the horizontal mean wave orbital velocity at the bed (m\(\cdot\)s\({}^{-1}\)); \(U_{b}\) is the current velocity at top of water boundary layer (m\(\cdot\)s\({}^{-1}\)); and \(\beta\) is the angle between the average wave direction and the observed wave direction (degree). Formula for sediment deposition was originally proposed by [PERSON] (1962) [45] as  
\[S_{D}=w_{s}C_{b}p_{d} \tag{7}\]  
where \(S_{D}\) is deposition rate (kg\(\cdot\)m\({}^{-3}\)\(\cdot\)s\({}^{-1}\)); \(w_{s}\) is settling velocity (m\(\cdot\)s\({}^{-1}\)); \(C_{b}\) is near bed concentration (kg\(\cdot\)m\({}^{-3}\)); \(p_{d}\) is probability of deposition which is calculated by  
\[p_{d}=1-\frac{\tau_{b}}{\tau_{cd}},\tau_{b}\leq\tau_{cd} \tag{8}\]  
where \(\tau_{b}\) is bed shear stress (N\(\cdot\)m\({}^{-2}\)) and \(\tau_{cd}\) is critical bed shear stress for deposition (N\(\cdot\)m\({}^{-2}\)). Settling velocity, \(w_{s}\), described flocculation process. Flocculation is when the concentration of sediment is high enough for the sediment flocs to influence each other settling velocity. The modification of settling velocity also computed due to salinity variation. \[w_{s,s}=w_{s}\Big{(}1-C_{1}e^{\varepsilon\cdot C_{2}}\Big{)} \tag{9}\]where \(w_{s,s}\) is settling velocity due to salinity variation; \(C_{I}\) and \(C_{2}\) are calibration parameters; and S is salinity. Erosion formula was described as soft partly consolidated sediment [46]. \[S_{E}=E\textit{exp}\left[\alpha(\tau_{b}-\tau_{ce})^{1/2}\right],\tau_{b}>\tau_{ce} \tag{10}\]  
where \(S_{E}\) is the erosion rate (g\(\cdot\)m\({}^{-2}\)); \(E\) is the erosion coefficient (g\(\cdot\)m\({}^{-2}\)\(\cdot\)s\({}^{-1}\)); \(\tau_{ce}\) is critical bed shear stress for erosion (N\(\cdot\)m\({}^{-2}\)); and \(\alpha\) is coefficient (m\(\cdot\)N\({}^{-0.5}\)). The overall transport of particles during a time interval, \(\Delta t\), results from an advective component and a dispersive component, which accounts for the non-resolved flow processes. The particle transport equation at the \(i\)'th time step can be expressed as:  
\[\overline{x}_{i+1}=\overline{x}_{i}+\overline{v}\cdot\Delta t+\underline{D} \cdot\overline{V}+\overline{\gamma} \tag{11}\]  
where  
\[\overline{V}=\left( \begin{array}{c}u_{x}\\ u_{y}\\ -V_{sett}\end{array} \right);\left|\overline{U}\right|=\sqrt{u_{x}^{2}+u_{y}^{2 }};\underline{D}=\frac{1}{\left|\overline{U}\right|}\left( \begin{array}{ ccc}\Delta D_{L}&-\Delta D_{T}&0\\ \Delta D_{L}&\Delta D_{T}&0\\ 0&0&0\end{array} \right);\overline{\gamma}=\left( \begin{array}{c}\Delta D_{0 }\\ \Delta D_{0}\\ \Delta D_{0W}\end{array}