9.1 Introduction
The development of least-cost water quality management policies requires an analysis of performance under a variety of design conditions and emission inputs. Thus, a model is needed to translate the control action to the ambient quality response (Chapter 5). This translation should take into account the processes which affect the instream concentrations of the constituents under consideration and route the input (emissions and the design natural conditions) to the desired output parameters (the ambient water quality).
The water quality constituents are subject to the following most important processes which may lead to a change in the instream concentrations (see also Somlyody and Varis, 1992):
Physical transport (advection and diffusion) Chemical and biological reactions
The transport process is caused (mainly) by the motion of water in which the substances are dissolved or suspended. It is therefore necessary to have a description of the flow field (in the case of river flow, it is called river hydraulics).
There is a variety of mathematical models available for the description of water quality phenomena in waterbodies under different conditions (Somlyody and Varis, 1992) which incorporate descriptions of the key processes. We will focus our attention on the models of rivers and river systems. Because cross-sectional mixing usually is intensive in rivers, one- dimensional (along the stream) mathematical models are most commonly used for description of river flow, advective-diffusive motion, and water quality processes in the river. The temporal changes may be taken into account (dynamic models), or the model can be aimed at the description of the steady-state situation. In both cases, the model is composed of the following equations or the equation sets (Somlyody and Varis, 1992):
Equation of water motion (hydraulics) Transport equation
Chemical and biological processes equation or reaction terms extending the transport equation
Selection of a model depends on the water quality problem addressed by the study. Focus of this study (Chapter 7) is organic pollution causing dissolved oxygen deficits, and the model should also simulate the processes affecting the oxygen household of the river. The key parameters related to this problem are oxidizable waste materials (biochemical oxygen demand, ammonia nitrogen, and organic nitrogen), dissolved oxygen, and to a lesser degree, phosphorus and nitrogen, which control algae proliferation (Thomann and Mueller, 1987).
The number of the water quality constituents considered by an oxygen model can range from 2-3 (simple BOD-DO oxygen models) to 20-30 (comprehensive ecosystem models). The reported work will be based on simple dissolved oxygen models with two or three components. The application of a more complex stream water quality model with about ten components (QUAL2) will also be discussed (for more detail , see Breithaupt and Somlyody,
1 994).
Modeling procedure involves the following common steps (Beck, 1983): selection of the appropriate model, parameter estimation (calibration), validation and the simulation under the desired conditions (scenario analysis). In this chapter, we will cover the model selection, calibration and validation issues. The set of data used for the model calibration was described in Chapter 8. First, the hydraulic model will be outlined (Section 9.2). Then, simple dissolved oxygen models and their calibration will be covered. The application of a more sophisticated model QUAL2E will be discussed later (Section 9.6). The "final" selected model (Section 9.5) will be used for control policy development (Chapters 10 and 13). Figure 9.1 illustrates use of water quality models in the framework of water quality management and planning.
7
and validation)
Water quality model
Water quality model for planning
Figure 9.1 Use of water quality models in the policy analysis framework.
9.2 Hydraulic Models and Their Calibration for the Nitra River
Water motion along a river or channel is often described by the set of one-dimensional equations of continuity and momentum, known also as Saint-Venant equations (Mahmood and Yevjevich, 1975):
where:
x is the coordinate along the river or channel,
Q is the stream flow rate,
z is the elevation of water surface, h is the water depth,
p is the water density, A is the cross sectional area,
q is the rate of the lateral inflow per unit of river length, g is the gravity constant (9.8 1 m/secZ),
T,, is the bottom shear stress, and
T, is the surface shear stress.
The surface shear (or wind) stress T, is rarely sigmficant for the river flow, which is being driven mainly by the gravitational forces. If the quadratic law of resistance is applied, one can write for the bottom shear T,, :
For the resistance coefficient C, a number of empirical formulations are available (Somlyody and Varis, 1992). The Manning's equation for the resistance coefficient was used in the model:
where n is the Manning's roughness coefficient.
For the water quality problems, the local fast movements of the fluid are generally not essential (Somlyody and Varis, 1992). In this case, the first two terms in the Eq. (2) could be disregarded, and a diffusive wave approximation equation can be derived from Eqs. (1) and (2) (Mahrnood and Yevjevich, 1975). However, the diffusive wave approximation requires a lot of input data as initial and boundary conditions, and simulation time is rather long. The diffusion wave approximation is justified when the temporal changes in the river hydraulic parameters (streamflow and water depth) are significant over the time frame of interest. When the main objective is water quality, the steady-state approximation often is acceptable (see Chapter 5 and Somlyody and Varis, 1992). The system of hydraulic equations for the steady state can be presented as follows:
and, for simplified evaluations, the following equation can be used instead of Eq. (6):
where
&
is the local slope of the river bottom. For simulating the hydraulics of the Nitra River, the last model was used .The morphometry data were available in the form of 299 cross-section profiles covering the major part of the Nitra River flow (from river km 155 to the mouth). The profiles are located at an average 0.5 km from each other, forming a comprehensive representation of the river morphometry. One of the typical cross-section profiles (at the river km 21.8) is shown in the Figure 9.2.
0 20 40 60 80 100
Width, rn
Figure 9.2 Cross-section profile at the river km 21.8 (Banov)
The calibration of the hydraulic model was based on the rating curves (elevation-streadow curves) available at seven locations. Both the Manning's roughness coefficient and the bottom elevation were calibrated to fit the elevation and stream flow data. The fitting was facilitated with the minimization procedure for the following function:
where N is the number of data points in the rating curve, zok is measured elevation,
z y s calculated elevation.
The elevation was calculated from the local bottom slope with Eq. (7). The function 0 was minimized with Powell's method, using z,, and n as the parameters to fit. Keeping in mind that the main purpose of the model is to describe low-flow conditions, only the parts of the rating curves corresponding to a river depth 0-2 m were used for calibration. The calibration results for the Banov location are shown in Figure 9.3 for illustration.
Stream flow rate, m3ls
x Rating curves data Calculated depth
Figure 9.3 Calibration of the hydraulic model for the B h o v location (river km 21.8)
To verifjl the calibration procedure, the steady-state hydraulic model (Eqs. (5) and (6)) was integrated keeping the streamflow rate constant along the river, and the results were compared with the rating curves' data points (Figs. 9.4-9.5). The results show certain irregularities in the water depth profile, caused by changes in the river morphometry. The rating curves' data points can be considered to adequately agree with the modeled depth profile, taking into account the mentioned irregularities.
River km
Simulation Rating curves data
Figure 9.4 Steady state hydraulic model of the Nitra River and the rating curves data.
Streamflow rate is 10 m3/s along the river
River km
Simulation Rating curves data
Figure 9.5 Steady state hydraulic model of the Nitra River and the rating curves data.
Streamflow rate 50 m3/s along the river.
9.3 Simple Dissolved Oxygen Models
The formulation of river water quality models of this class is rather conventional. The number of state variables representing the household of dissolved oxygen ranges between one and three, while the number of parameters varies between one and five. More specifically, the applied models include (see Thomann and Mueller, 1987):
The original DO-BOD Streeter-Phelps model (two parameters)
The same model with the incorporation of sedimentation of the particulate organic material (three parameters)
As above, but with sediment oxygen demand (four parameters) A three state variable model with nitrogenous BOD (five parameters)
The set of partial differential equations for the three state variablelfive parameter model can be written as follows (see assumptions in Sornlyody and Varis, 1992):
Where: L- carbonaceous biological oxygen demand (CBOD) in mgll N
-
nitrogenous biological oxygen demand (NBOD) in mgA C-
dissolved oxygen concentration in mgAx
-
coordinate along the river, m;t
-
travel time in days Q- streamflow in m3/d A-
cross-section area in m2 B-
stream width in m-
carbonaceous BOD removal rate in l/d k,-
oxygen exchange coefficient (see later) in m/d K,-
CBOD oxygenation rate in l/d& -
CBOD decay rate in l/d K,,-
NBOD oxygenation rate in l/d Ksm-
sediment oxygen demand in g/m2/dC,
-
saturation concentration of dissolved oxygen in mgAThe exchange coefficient across the water-atmosphere boundary, k, was calculated using the O ' C o ~ o r and Dobbins (1956) empirical relationship:
where: k,,
-
the reaeration coefficient in m s1/2/df(T)
-
dimensionless temperature correction factor U-
flow velocity expressed in m/sH
-
aeration depth in my defined as the A/B ratioThe reaeration rate
K,
(Ild) is defined as kJH and is dependent on the flow and stream morphometry at the current location.For every river stretch with relatively uniform characteristics, Equations (9)-(11) are solved analytically, fiom the most upstream location to the river's mouth. At confluence points. an assumption of immediate, complete mixing is utilized. The computation time is small enough to allow the model to be incorporated into relatively sophisticated parameter estimation and policy analysis fiameworks (Figure 9.1; see also Section 9.4 and Chapter 1 1, respectively).
9.4 Model Calibration Using Data from the Longitudinal Water Quality Profiles
The selection of proper parameter values for water quality models is a crucial step in developing a catchment-wide control policy. For the parameter estimation, the results of two experiments were used to produce longitudinal profiles and estimate mass balances (Chapter 8).
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