**2.4 Mathematical modelling**

Numerical modelling is a multifaceted tool that enables a better understanding of physical, chemical and biological processes in the water bodies, based on a "simplified version of the real" described by a set of equations, which are usually solved by numerical methods.

The models to be used for the implementation of the WFD management strategies should ideally have the highest possible degree of integration to comply with the integrated river basin approach, coupling hydrological, hydrodynamic, water quality and ecological modules as a function of the specific environmental issues to analyse.

The *Mondego Estuary (MONDEST)* model was conceptualized (Fig. 13) as an integrated hydroinformatic tool, linking hydrodynamics, water quality and residence time (*TempResid*) modules (Duarte, 2005).

Fig. 13. The *MONDEST* model conceptualization

The formulation of an accurate model requires the best possible definition of the geometry and bathymetry of the water body and the interactions with the boundary conditions, as stated in previous items.

This model is based on generalized computer programmes RMA2 and RMA4 (WES-HL, 1996; 2000), which were applied and adapted to this specific estuarine ecosystem. The CEWES version of RMA4 is a revised version of RMA4 as developed by King & Rachiele (1989).

The RMA2 programme solves depth-integrated equations of fluid mass and momentum conservation in two horizontal directions by the finite element method (FEM) using the Galerkin Method of weighted residuals. The shape (or basis) functions are quadratic for velocity and linear for depth. Integration in space is performed by Gaussian integration. Derivatives in time are replaced by a nonlinear finite difference approximation.

The RMA4 programme solves depth-integrated equations of the transport and mixing process using the Galerkin Method of weighted residuals. The form of the depth averaged transport equation is given by equation (1)

$$h\left(\frac{\partial c}{\partial t} + \mu \frac{\partial c}{\partial x} + v \frac{\partial c}{\partial y} - \frac{\partial}{\partial x} D\_x \frac{\partial c}{\partial x} - \frac{\partial}{\partial y} D\_y \frac{\partial c}{\partial y} - \sigma + kc + \frac{R(c)}{h}\right) = 0 \tag{1}$$

Where

14 Hydrodynamics – Natural Water Bodies

Ever increasing computational capacities provide the development of powerful and userfriendly mathematical models for the simulation and forecast of quality changes in receiving

The results of several research works have showed that the linkage of tracer experimental approach with mathematical modelling can constitute a power and useful operational tool to establish better warning systems and to improve management practices for the efficiently

Numerical modelling is a multifaceted tool that enables a better understanding of physical, chemical and biological processes in the water bodies, based on a "simplified version of the real" described by a set of equations, which are usually solved by numerical methods. The models to be used for the implementation of the WFD management strategies should ideally have the highest possible degree of integration to comply with the integrated river basin approach, coupling hydrological, hydrodynamic, water quality and ecological

The *Mondego Estuary (MONDEST)* model was conceptualized (Fig. 13) as an integrated hydroinformatic tool, linking hydrodynamics, water quality and residence time (*TempResid*)

**RESULTS RESULTS HYDRODYNAMIC**

The formulation of an accurate model requires the best possible definition of the geometry and bathymetry of the water body and the interactions with the boundary conditions, as

This model is based on generalized computer programmes RMA2 and RMA4 (WES-HL, 1996; 2000), which were applied and adapted to this specific estuarine ecosystem. The CEWES version of RMA4 is a revised version of RMA4 as developed by King & Rachiele

The RMA2 programme solves depth-integrated equations of fluid mass and momentum conservation in two horizontal directions by the finite element method (FEM) using the Galerkin Method of weighted residuals. The shape (or basis) functions are quadratic for velocity and linear for depth. Integration in space is performed by Gaussian integration.

Derivatives in time are replaced by a nonlinear finite difference approximation.

**SCENARIO SCENARIO SCENARIO**

**Dispersive characteristics** *Salinity distribution* **Saltwater intrusion** 

**TRANSPORT MODULE**

*TempResid MODULE*

• **spatial distribution** *Residence time:*

• **discharge type effect**

waters after land runoff, mining and wastewater discharges.

**2.4 Mathematical modelling** 

modules (Duarte, 2005).

**Bathymetry**

**Boundary**

**Mesh generation**

stated in previous items.

(1989).

protection of water supply sources and, consequently, public health.

modules as a function of the specific environmental issues to analyse.

**MODULE**

**Tidal prism and flows**

*Currents velocity*

**Wetlands**

**Nutrients balance**

Fig. 13. The *MONDEST* model conceptualization

h =water depth;

c = concentration of pollutant for a given constituent;

t = time;

u, v = velocity in x direction and y direction;

Dx, Dy, = turbulent mixing (dispersion) coefficient;

k = first order decay of pollutant;

σ = source/sink of constituent;

R(c) = rainfall/evaporation rate.

As with the hydrodynamic model RMA2, the transport model RMA4 handles onedimensional segments or two-dimensional quadrilaterals, triangles or curved element edges. Spatial integration of the equations is performed by Gaussian techniques and the temporal variations are handled by nonlinear finite differences consistent with the method described for RMA2.

The numerical computation was carried out for all Mondego estuary spatial domains. Several sections were carefully selected and used for calibrating and analysis of the simulation results (Duarte, 2005). The legend includes the designation, section code and their distance to the mouth of the estuary (Fig. 14).

Fig. 14. The *MONDEST* model finite elements mesh and outline of the control sections

A Hydroinformatic Tool for Sustainable Estuarine Management 17

This graph presents the concentration decrease of a conservative constituent, in three control points (N0 - estuary mouth; S1 - Gala bridge/Lota; and S3- Pranto river mouth), due to estuarine flushing currents, considering the well known re-entrance phenomena at the

For hydrodynamic modelling purpose, a wide range (sixteen) of management scenarios were judiciously selected covering a representative set of hydraulic conditions (Table 2), resulting from the combination of typical tidal amplitudes (0.60, 1.15, and 1.60 m) and

Mondego Pranto Medium Spring Neap

75 0 H 6 H 7 H 8

500 30 - H 14 - 800 30 - H 15 H 16

For the *Mondest* transport model calibration and validation, the salinity was adopted as a natural tracer. Several management scenarios (nine) were also carefully selected (Table 3) considering the most representative hydrodynamic conditions in order to estimate salt wedge propagation into the estuary and to identify the areas (in both arms) where favourable salinity values for macroalgae growth can potentiate the estuarine

Mondego Pranto Medium Spring Neap

75 0 SL 4 SL 7 - 340 15 SL 5 SL 8 -

For the RT values calculation using the *TempResid* module, the simulated management scenarios (fourteen) were defined considering not only the most critical hydrodynamic conditions, but also by carefully selecting distinct pollutant load characteristics (e,g. location, duration and type of the discharge event, instant of tidal cycle when the release occurs) and

0 H 1 H 2 H3 15 H 4 - - 30 H 5 - -

0 H 9 H 10 H 11 15 H 12 - - 30 H 13 - -

0 SL 1 SL 6 SL 9 15 SL 2 - - 30 SL 3 - -

**Freshwater flow** (m3.s-1) **TIDE** 

Table 2. Simulated management scenarios for the hydrodynamic modelling

**Freshwater flow** (m3.s-1) **TIDE** 

Table 3. Simulated management scenarios for the hydrodynamic modelling

estuary mouth.

**2.5 Simulated management scenarios** 

15

340

eutrophication vulnerability.

15

freshwater flow inputs (from Mondego and Pranto).

The size of the elements to consider in the spatial discrimination of the simulated domain of numerical models must be established as a function of larger or smaller spatial gradients than those displayed by the variables (water level and velocity) in that domain. In the case of the Mondego estuary, since the south arm was the preferred object for studying, the network of finite elements was refined in that sub-domain, thereby reducing the maximum area of its (triangular) elements to 500 m2 (Duarte, 2005).

In the *MONDEST model*, the hydrodynamic module provides flow velocities and water levels for the water quality module, whose results acts as input on the *TempResid* module, feeding the constituents concentration over the aquatic system. The post-processing and mapping of model results was performed using SMS package (Boss SMS, 1996).

The *TempResid* module was integrally developed in this research work aiming to compute RT values of each water constituent (conservative or not) and allowing to map its spatial distribution over all the estuarine system, considering different simulated management scenarios.

RT value of a substance was calculated for each location and instant, as an interval of time that is necessary for that corresponding initial mass to reduce to a pre-defined percentage of that value. In this work, a value of 10% was adopted for the residual concentration of the substance, attending to the fact that the effect of the re-entry of the mass in the estuary during tidal flooding is considered (a significant effect for dry-weather river flow rates).

The determination of the RT in several stations along the estuary, where the eutrophication gradient occurred, was carried out by applying the *TempResid* programme to the results of the simulations that were performed with the transport module of the MONDEST model. Figure 15 shows an example of the MONDEST model transport module results for the management scenario considered as the most favourable to macroalgae blooms occurrence (Duarte, 2005), due to low freshwater inputs and consequent reduction of estuarine waters renovation (scenario RT1).

Fig. 15. Residence time computation using *TempResid* module

This graph presents the concentration decrease of a conservative constituent, in three control points (N0 - estuary mouth; S1 - Gala bridge/Lota; and S3- Pranto river mouth), due to estuarine flushing currents, considering the well known re-entrance phenomena at the estuary mouth.
