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Change in marker concentration within a division
The following section considers the change in concentration of a marker as it travels within a division. Changes in concentration of a marker as a result of additional inflow at the beginning of a division, such as from an inflow, are processed at the inflow node. This section deals with the interactions that occur within a division that affect marker concentration such as net evaporation, diversions, seepage/infiltration and groundwater interactions.
If a small fraction of the division, dx, is considered, then the load (mass) of solute in dx at any point in time can be defined as:
| Equation 37 |
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The load of solute in the division section dx after time dt* can then be expressed by taking into account the initial load, the additional solute inflow and the losses other than from net evaporation (Lo) that occur. This is expressed in Equation 38.
| Equation 38 |
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The volume of water in the division section of length x after time dt*is as shown in Equation 39.
| Equation 39 |
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The final concentration of division section dx is equal to c+dc; ie. the load in division section after time dx divided by the volume of water in division section dx after time dt* as shown below in Equation 40.
| Equation 40 |
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Equation 40 can be rearranged in terms of dc as follows:
Which can then be simplified to give Equation 41 and Equation 42:
| Equation 41 |
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| Equation 42 |
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A limit exists as dt*→0 of and A = At+a•t* as was established previously so Equation 42 then becomes:
| Equation 43 |
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Using the relationships established previously when defining the location of the marker:
then Equation 43 can then be rearranged as follows and integrated:
| Equation 44 |
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| Equation 45 |
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Integrating Equation 45 gives Equation 48 below through the following calculations, where Equation 48 defines c1.
| Equation 46 |
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| Equation 47 |
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| Equation 48 |
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By defining the following:
Equation 48 becomes:
And can be expressed as:
| Equation 49 |
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where:
Alternative Formulations
Equation 49 is the primary equation that defines the concentration of a marker, c1, at time t1*. However this equation is not valid for the following conditions:
- s = 0 and Ev = 0;
- s = 0 and Ev ≠ 0; and
- s ≠ 0 and Ev = 0.
The formulation for these conditions is contained in the sections below.
Formulation for when s = 0 and Ev = 0
When both s = 0 and Ev = 0, the concentration of the marker, c1, at time t1* may be calculated by adjusting Equation 44 as below:
Which leads to Equation 50, expressed in the same form as Equation 49.
| Equation 50 |
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Therefore:
Formulation for when s = 0 and Ev ≠ 0
When s = 0 and Ev ≠ 0, the concentration of the marker, c1, at time t1* may be calculated by adjusting Equation 44
| Equation 51 |
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| Equation 52 |
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Integrating Equation 51 leads to Equation 54 following the calculation steps shown next:
| Equation 53 |
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| Equation 54 |
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Therefore:
Formulation for when s ≠ 0 and Ev = 0
When s ≠ 0 and Ev = 0, the concentration of the marker, c1, at time t1* may be calculated by adjusting Equation 44 as follows:
| Equation 55 |
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| Equation 56 |
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| Equation 57 |
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Which leads to Equation 58 (shown next), expressed in the same form as Equation 49.
| Equation 58 |
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Therefore:
Summaries of equations
Using the standard equation (Equation 49, reproduced below) to define c1 as shown below:
We are able to calculate values of c1 by using the values of cev and cLoad, used to define marker concentration (c1) at time (t1*) defined for conditions of s and Ev in Table 5 and Table 6, respectively.
Table 5. Summary of equations to evaluate cev
cev Value | Conditions | Equation reference |
|---|---|---|
When s ≠ 0 and Ev ≠ 0 | Equation 49 | |
When s = 0 and Ev ≠ 0 | Equation 54 | |
When s = 0 and Ev = 0 | Equation 50 |
Table 86. Summary of equations to evaluate cLoad
cLoad Value | Conditions | Equation reference |
|---|---|---|
Whens ≠ 0 and Ev ≠ 0 | Equation 49 | |
When s ≠ 0 and Ev = 0 | Equation 58 | |
When s = 0 and Ev = 0 | Equation 50 |
Marker routing process with DODOC plug-in
Interaction approach
The DODOC is Source plugin for the blackwater and the details can be found here . DO represents Dissolved Oxygen and DOC is Dissolved Organic Carbon in DODOC model.
From version 5.1, the functions of interacting the Marker routing process with DODOC plugin was added. The approach of interacting the Marker Routing with DODOC plug-in are following
- Perform the constituent marker routing as per normal
- Determine the mass within the division level
- Provide the mass to plugin
- The plugin will return a modified mass.
- Adjust the 'faces' of the markers in that division proportionally to the change in mass.
The approach was used to develop the function for Marker routing process in stream (link), Storage and Weir
DO saturation concentration
Unlike other constituents, Dissolved Oxygen (DO) can be in the saturation state, and the saturation concentration can be a cap of the calculated concentration of DO. The equation 59 (Cox, 2003) is used for DO saturation concentration.
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Therefore:
Cs is the DO saturation concentration
T is the water temperature in degrees CelsiusInput data:
Details on data are provided in the Source User Guide.
Output data
Outputs can be viewed in the Recording Manager; details are provided in the User Guide.
Reference list
Close, A.F. (1996) A New Daily Model of Flow and Solute Transport in the River Murray, 23rd Hydrology and Water Resources Symposium, Institution of Engineers, Australia, Canberra, pp 173-178.
Cox, B.A. (2003) A review of currently available in-stream water-quality models and their applicability for simulating dissolved oxygen in lowland rivers. The Science of the Total Environment 314 –316 pp 335–377