## 2 Materials and Methods
### Numerical Models
#### 2.2.1 Equations  
The mathematical model was based on the shallow-water equations of the two-dimensional numerical solution method and the Navier-Stokes equations of the three-dimensional incompressible Reynolds value uniformity. This approach is subject to the Boussinesq hypothesis and the assumption of hydrostatic pressure. The finite volume and computational grid unstructured methods were used in the mathematical model along with a flexible boundary surface and dynamic complex terrain. The following equations were used:  
(1) Hydrodynamic control equations:  
Continuous equation:  
\[\frac{\partial\xi}{\partial t}+\frac{\partial P}{\partial x}+\frac{\partial q }{\partial y}=\frac{\partial d}{\partial t} \tag{1}\]  
Momentum equations:  
\[\begin{array}{l}\frac{\partial p}{\partial t}+\frac{\partial}{\partial x}( \frac{p^{2}}{T})+\frac{\partial}{\partial y}(\frac{pq}{h})+gh\frac{\partial x} {\partial x}+\frac{gp\sqrt{p^{2}+q^{2}}}{C^{2}\cdot h^{2}}-\frac{1}{\rho_{w}} \Big{[}\frac{\partial}{\partial x}(h\tau_{xx})+\frac{\partial}{\partial y} \big{(}h\tau_{xy}\big{)}\Big{]}\\ -\Omega_{q}-fVV_{x}+\frac{h}{\rho_{w}}\frac{\partial}{\partial x}(p_{a})=0\\ \\ \frac{\partial p}{\partial t}+\frac{\partial}{\partial y}(\frac{p^{2}}{T})+ \frac{\partial}{\partial x}(\frac{pq}{h})+gh\frac{\partial x}{\partial y}+ \frac{gp\sqrt{p^{2}+q^{2}}}{C^{2}\cdot h^{2}}-\frac{1}{\rho_{w}}\Big{[}\frac {\partial}{\partial y}\big{(}h\tau_{yy}\big{)}+\frac{\partial}{\partial x} \big{(}h\tau_{xy}\big{)}\Big{]}\\ -\Omega_{q}-fVV_{y}+\frac{h}{\rho_{w}}\frac{\partial}{\partial y}(p_{a})=0\\ \end{array} \tag{2}\]  
(2) Water quality equation:  
\[\frac{\partial(hC)}{\partial t}+\frac{\partial(uhC)}{\partial x}+\frac{ \partial}{\partial y}=\frac{\partial}{\partial x}\bigg{(}hE_{x}\frac{\partial C }{\partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(}hE_{y}\frac{\partial C }{\partial y}\bigg{)}+F(C)+S \tag{4}\]  
where \(h(x,y,t)\) is the water depth (m); \(d(x,y,t)\) is the water depth variation over time (m); \(\xi(x,y,t)\) is the free surface elevation; \(p,q(x,y,t)\) are the discharge per unit width in the \(x\) and \(y\) directions (m\({}^{3}\)/s\(\cdot\)m), respectively; \(g\) is the gravitational acceleration (m/s\({}^{2}\)); \(C(x,y)\) is the Chezy resistance (m\({}^{1/2}\)/s); \(f=2\Omega\sin\phi\) is the Coriolis force coefficient (\(\Omega\) is the angular velocity of earth rotation, and \(\phi\) is the latitude of Earth); \(V\) is the flow velocity (m/s); \(u,v\) are the mean flow velocity along the water depth in the \(x\) and \(y\) directions (m/s), respectively; \(V_{x},V_{y}\) are the flow velocity components in the \(x\) and \(y\) directions (m/s), respectively; \(\Omega(x,y)\) is the coefficient of the Coriolis force; \(\rho_{w}\) is the water density (kg/m\({}^{3}\)); \(P_{a}(x,y,t)\) is the atmospheric pressure (kg/m\(\cdot\)s); \(x\) and \(y\) make up the cartesian coordinate system; \(\tau_{xx},\tau_{xy},\tau_{yy}\), are the tangential stresses; \(C\) is the pollutant concentration (mg/L); \(E_{x},E_{y}\) are the turbulent diffusion coefficient and dispersion coefficient in the \(x\) and \(y\) directions (m/s\({}^{2}\)), respectively; \(S\)--are the source or sink terms for flow; and \(F(C)\) is the biochemical reaction.