River/Stream Decay Modeling*


Understanding and predicting the impact of various sources are essential components of a typical course on Analysis of Stream and Estuary Pollution in an engineering curriculum. Dispersal and decay of contaminants in lakes, streams, estuaries, and oceans. Effects of pollutants on chemical quality and ecology of receiving waters.

Point Source (Nonconservative Substances)

Many substances exhibit decay or nonconservative behavior including oxidizable organic matter, nutrients, volatile chemicals, and bacteria. A very useful assumption is that the substance decays according to a first-order reaction, that is, the rate of loss of the substance is proportional to the concentration at any time. The mass balance equation, at steady state, is a first-order linear differential equation:

  • The boundary condition: S = So at x = 0
  • Assuming no change of flow
  • A uniform cross-sectional area
If the derivative is expanded and assumptions are made, then:
This equation shows that for a nonconservative substance decaying at a rate K, the downstream distribution of the substance will drop exponentially and approach zero. Figure 1(a) shows the distribution and (b) shows a plot of lnS vs. x or t.

    Figure 1. Decay of nonconservative substance
Distributed Sources

Streams are often subjectd to sources or sinks of a substance which are distributed along the length of the stream. Given a constant magnitude of the distributed source which originates at x = 0, and given a stream with constant parameters (flow, area, depth) over a given length, the mass balance equation at steady state is:

Solution of the linear first-order differential equation results in:
Figure 2 below shows the distributed sources.
Figure 2. Distributed Sources - Type and Water Quality Response

Multiple Sources

Differential equations which govern the concentration S along a stream x are said to be linear. For such linear systems, the concentration in a river or stream due to multiple point and/or distributed sources is the linear summation of the responses due to the individual sources plus the responses due to any upstream boundary conditions. For a stream with constant flow and velocity subjected to a point source W at x = 0 as well as a distributed source originating at x = 0, the water quality response would be:

Figure 3 below shows examples of water quality responses to multiple sources.
Figure 3. Examples of Water Quality Responses to Multiple Ssources

Time Variable Analysis - Nondispersive Streams

For some problem contexts, it is important to be able to describe the time variable behavior of water quality in a river downstream of an outfall or turbidity input. The basic principle of the time variable response in a river or stream can be quickly seen by making the initial assumption that there is no mixing in the longitudinal direction. If there is no mixing of the water parcels, then each parcel does not interact with with the parcel in front of it or behind it. This type of condition is called plug flow. Figure 4(a) shows the schematic to plug flow at time t = 0 and (b) shows time t = t1.

Figure 4. Schematic of Water Quality Responses to Plug Flow

Time Variable Analysis - Effect of Dispersion

In any real river, however, there is some mixing that occurs along the length of the river due to the horizontal and vertical gradients of velocity. In addition, river channel changes and twists add further to this mixing. The phenomenon is called longitudinal dispersion. The mass balance equation for constant cross-section area, river flow, dispersion, and no ther inputs of S expect at the outfall is:

where Ex = Longitudinal Dispersion Coefficient
    -- Instantaneous Input
    If an instantaneous spill of waste occurs, the water quality response would be:

    Note that the spread increases as time increases and that the overall peak decays. Figure 5(a) shows the stream quality respone to pulse input as a function of distance and (b) shows the response over time..

Figure 5. Schematic of the stream quality respone to pulse input

    --  Continuous Input
    With a steady input, the water quality response would be:


    Figure 6 shows the dispersive stream quality respone to a square wave input load and time response at x = x1 and x2.

Figure 6. Dispersive stream quality respone

Estimate of River Dispersion Coefficient

A variety of theoretical and empirical relationships has been proposed, the following dispersion coefficient in real streams had been proposed by Fischer et at. (1979):

McQuivey and Keefer proposed the following for conditions where the Froude number ( ) is less than 0.5:
Model Examples * Contents and Pictures are from "Principles of Surface Water Quality Modeling and  Control" by Robert V. Thomann and John A. Mueller, Harper-Collins, 1988

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