3. Water quality model

Water quality, which includes its physical, chemical, biological, and other characteristics, is predominantly controlled by the hydrodynamic processes and the various mechanisms governing the fate and transport of contaminants. Once a contaminant enters a water system, its concentration is determined by its chemical and biological reactions and hydrodynamic transport processes, including advection, dispersion, and vertical mixing. Water quality models are widely used to reflect these processes. Some examples of water quality model applications include the simulation and prediction of water temperature, dissolved oxygen (DO), biochemical oxygen demand (BOD), the nitrogen cycle (including levels of organic nitrogen, ammonia, nitrite and nitrate), the phosphorous cycle (including levels of organic phosphorous and phosphates), algae growth and decay, cohesive/noncohesive suspended sediment, salinity, heavy metals, and pathogens, among others [27]. Depending on the results for advection, dispersion, and turbulent mixing obtained from the hydrodynamic model, water quality models may also incorporate the sources/sinks, chemical and biological reactions of contaminants.

#### 3.1. Governing equations for contaminant fate and transport

The governing equations for contaminant fate and transport are as follows [16, 21, 23]:

3.1.1. 1D equation

2.3.3. The side surface

2.3.4. Other boundary conditions

2.3.5. Initial conditions

ΓWY) and frictional stress (ΓBX and ΓBY).

94 Applications in Water Systems Management and Modeling

simulation for a period from this initial state.

3. Water quality model

2.4. Parameters and data for the hydrodynamic model

The side surface refers to the shoreline of the water body or a defined boundary of the model such as the entrance or exit of a river reach. The shoreline conditions are zero normal velocity, that is, no leakage across the boundary surface. Information on the water surface elevation, velocities, or flow rate need to be specified for the defined boundaries, but combinations of

In addition to the required boundary conditions mentioned earlier, some frictional stresses Γ<sup>i</sup> in the governing equations may act on the surfaces of water body so the corresponding conditions should be added to the model. Common stresses include wind stress (ΓWX and

A dynamic model requires a set of initial values to be input in order to begin to solve the governing equations. The model needs a "best guess" set of conditions for all the nodes in the mesh at the beginning of the iterative process; a bad initial guess that is far from the real conditions will adversely affect the convergence, slowing down the process immensely. A commonly used strategy is to have the whole water body at rest with a constant water surface elevation (WSE) when t = 0. The real initial scenario generally emerges after running the

Many different types of data are needed as input or to determine the controlling parameters when running a hydrodynamic model. The data used as direct input include the geographic coordinates of the shoreline, bathymetry, flow rate, water surface elevation, meteorological data such as wind speed and direction, radiation intensity, air pressure, precipitation and the evaporation rate, among others. In addition to the input data, data on the flow rate, velocities, water depth, and water surface elevation at other locations than the boundaries are also needed to calibrate the model to determine hard to measure parameters such as the roughness of the bottom and the horizontal momentum diffusion coefficient [16], and to confirm that the model accurately reflects the real scenario. The modeling results are compared to the data measured either in the laboratory or in the field for the model calibration and verification.

Water quality, which includes its physical, chemical, biological, and other characteristics, is predominantly controlled by the hydrodynamic processes and the various mechanisms governing the fate and transport of contaminants. Once a contaminant enters a water system, its concentration is determined by its chemical and biological reactions and hydrodynamic transport processes, including advection, dispersion, and vertical mixing. Water quality models

different conditions should be avoided on the same boundary.

$$\frac{\partial(A\_s \mathbb{C})}{\partial t} + \frac{\partial(A u \mathbb{C})}{\partial x} - \frac{\partial}{\partial x} \left( D\_x A \frac{\partial \mathbb{C}}{\partial x} \right) - k A\_s \mathbb{C} \pm G = 0 \tag{18}$$

#### 3.1.2. 2D depth-averaged equation

$$\begin{aligned} \frac{\partial(h\mathbf{C})}{\partial t} + u \frac{\partial(h\mathbf{C})}{\partial x} + v \frac{\partial(h\mathbf{C})}{\partial y} - \frac{\partial}{\partial x} \left( D\_x h \frac{\partial \mathbf{C}}{\partial x} + D\_{xy} h \frac{\partial \mathbf{C}}{\partial y} \right) \\ - \frac{\partial}{\partial y} \left( D\_{xy} h \frac{\partial \mathbf{C}}{\partial y} + D\_y h \frac{\partial \mathbf{C}}{\partial y} \right) - kh \mathbf{C} \pm G = 0 \end{aligned} \tag{19}$$

#### 3.1.3. 2D laterally averaged equation

$$\begin{cases} \frac{\partial(BC)}{\partial t} + u \frac{\partial(uBC)}{\partial x} + v \frac{\partial(wBC)}{\partial z} - \frac{\partial}{\partial x} \left( BD\_x \frac{\partial C}{\partial x} \right) - \frac{\partial}{\partial x} \left( BD\_z \frac{\partial C}{\partial z} \right) \\ -khC \pm G = 0 \end{cases} \tag{20}$$

#### 3.1.4. 3D stratified equation

h ∂C <sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>h</sup> ∂ð Þ uC ∂x þ h ∂ð Þ vC ∂y þ ð Þ b � a ∂ð Þ wC <sup>∂</sup><sup>z</sup> � ð Þ <sup>b</sup> � <sup>a</sup> Tx ∂ð Þ uC ∂z �ð Þ b � a Ty ∂ð Þ vC <sup>∂</sup><sup>z</sup> � <sup>h</sup> <sup>∂</sup> ∂x Dx ∂C ∂x þ Dxy ∂C ∂y <sup>þ</sup> <sup>h</sup> <sup>∂</sup> ∂x ð Þ b � a <sup>h</sup> DxTx <sup>þ</sup> DxyTy <sup>∂</sup><sup>C</sup> ∂z þð Þ b � a Tx ∂ ∂z Dx ∂C ∂x þ Dxy ∂C ∂y � ð Þ b � a Tx ∂ ∂z ð Þ b � a <sup>h</sup> DxTx <sup>þ</sup> DxyTy <sup>∂</sup><sup>C</sup> ∂z �<sup>h</sup> <sup>∂</sup> ∂y Dxy ∂C ∂x þ Dy ∂C ∂y <sup>þ</sup> <sup>h</sup> <sup>∂</sup> ∂y ð Þ b � a <sup>h</sup> DxyTx <sup>þ</sup> DyTy <sup>∂</sup><sup>C</sup> ∂z þð Þ b � a Ty ∂ ∂z ð Þ b � a <sup>h</sup> DxyTx <sup>þ</sup> DyTy <sup>∂</sup><sup>C</sup> ∂z �ð Þ b � a ∂ ∂z Dz ð Þ b � a h ∂C ∂z � khC � G � ð Þ b � a ∂ð Þ vsC <sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>0</sup> (21)

where C is the concentration of the contaminant or temperature, A is the cross-sectional area, As is the storage cross-sectional area, Di is the turbulent diffusion coefficient in the i direction, G is a general term representing sources or sinks, and k is the first order decay or reproductive rate coefficient [23]. These equations represent the first order case; other reaction rates (such as zero order or second order) are described using different expressions for the source term. The terminology used for the other variables is same as that used in Section 2.

risk to human health, triggering many beach closures and advisories every year with a consequent significant loss to local economies along the shoreline. Predictive modeling has been suggested as an effective approach to enhancing measurements of water quality both temporally and spatially, thus reducing the damage caused by improving the beach management. This case study was therefore conducted to develop a nearshore transport model for fecal pollution in Lake Michigan. The resulting model can be used to inform beach goers promptly in order to protect them from any potential exposure to waterborne pathogens and to help develop a better understanding of the key processes and factors influencing the fate and transport of fecal pollution in the nearshore reaches of the lake. As the indicators of fecal bacteria, Escherichia coli (EC) and enterococci (ENT) were used to evaluate the water quality at a recreational beach. In such environments, modeling nearshore, wind-driven circulation and the transport of EC and ENT are particularly challenging due to the interactions with complex lake-wide circulation. Originally, a 2D depth-averaged model was developed to simulate the entire lake with finer meshes close to the shoreline to emphasize the nearshore region [21]. The modeling results show that the current in nearshore region flew predominately along the shoreline direction and the cross-shoreline flow was fairly weak. Therefore, within an acceptable error tolerance range, the entire lake can be simplified as a narrow channel along the shoreline to significantly save computation efforts for nearshore process investigations. With this simplification, current flows only in clockwise or anticlockwise direction along the shoreline, and the exchanging process of mass and momentum in crossshoreline direction only occur within the channel [26]. Later investigation indicates that the current in vertical direction sometimes cannot be neglected in such lake environment, and thus, a stratified 3D model may be necessary to explain the hydrodynamic and transport processes along the water depth. In this case study, a new 3D model for the domain model was developed based on the aforementioned channelized 2D model. Some 3D modeling

Application of a Hydrodynamic and Water Quality Model for Inland Surface Water Systems

http://dx.doi.org/10.5772/intechopen.74914

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This case study focused on the nearshore regions along approximately 100 km of the shoreline of southern Lake Michigan. Based on the channelization simplification [26], the computational domain included a 5-km wide channel throughout the entire Lake Michigan shoreline (Figure 1). To fairly delineate the complex shoreline, a finite element model with a 3D nonuniform mesh was used. The mesh was gradually refined from a resolution of approximately 1–2 km at the locations far from the research domain to 100 m in the research area. Four streams are the primary tributaries discharging into southern Lake Michigan: Trail Creek (TC) at the Michigan City Harbor (USGS 04095380), Kintzele Ditch (KD) nearby Michigan City, Burns Ditch at Portage, IN (BD, USGS 04095090), and Indiana Harbor Canal (IHC) at East Chicago. As KD and TC both discharge combined sewer flows (CSOs), these are the most significant sources of EC and ENT so Mt. Baldy beach, which is located between the two, was chosen as the nearshore beach

Major factors considered in the model included the bathymetry, the shape of the lake shoreline, wind stress, hydrological flows from the tributaries, and water temperature. Bathymetric data

results are reported in the following sections.

4.1.2. Investigation domain and model mesh

for this investigation (Figure 1).

4.1.3. Data collection
