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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 |
Note about the algorithm change during the period from v5.0 to v5.30:
- Marker merge functionality for distance did not have brackets in the right place before, and this reduced the number of markers being merged. Now this is revised, and the revised equation is one for the parameter of usGap.
- The confluence condition in the CanMerge function (in Marker.cs) is now shifted, so it allows the merging of markers arriving with the start time of beginning (0) to occur; it was not hitting this condition and setting it to false.
- The merge functionality at the confluence constituent provider is updated such as Constituent Interpolation Type is changed from Distance to Time.
- In the algorithm for the storage mass in a division (divisionOutput.StoredMass), the minimum value for the division storage now is 0.00001 instead of the previous zero.
Marker routing interaction with instream processing
From Source version 5.1, the functionality was updated to allow interaction between the Marker routing process and the instream water quality process models. Core models such as decay are supported as well as external plug-in models.
The approach of interacting the Marker Routing with the instream processing models is as follows:
- Perform the constituent marker routing as per normal in Source
- Determine the mass within the division level of Source stream (/link)
- Provide the mass from Source to the instream processing model
- The plug-in model will return a modified mass to Source
- Adjust the 'faces' of the markers in that division proportionally to the change in mass.
Marker routing in Weirs
To ensure the mass balance of constituents within a weir, the conveyance flux and constituent mass is considered as:
1) when inflowing into a weir, it is added as an inflow into the upstream end of the weir's most downstream division
2) when outflowing from a weir it is treated as an outflow from the downstream end of the weir's most downstream divisionThis facilitates calculating the inflowing mass and outflowing mass since the model is adjusting markers at a point, not across the division within the timestepof the weir's most downstream division
This facilitates calculating the inflowing mass and outflowing mass since the model is adjusting markers at a point, not across the division within the timestep.
Note about the algorithm change in Weirs during the period from v5.0 to v5.30 :
- Weir used division.conveyanceflux in calculations before. Now this has been changed to only include conveyance inflows, and the adjusted calculation equations include those for the flow and fraction add, conveyance inflow concentration and conveyance mass.
- Weir was creating inflow markers under certain conditions, but it was not wiping the inflow marker flag after the processing was happening. Now all inflow marker flags are resetting as false after the processing.
Marker Age in Storages
The calculation of the marker age in a storage considers three sources of the constituents: (a) in the storage, (b) in upstream inflow and (c) in the conveyance flux. The fully mixed status is assumed in the storage. When multiple constituents are existed, only the marker age for one constituent is calculated and the marker ages of all remaining constituents are same as that of the calculated constituent.
The calculation can be expressed by the follow equation:
Where:
| MAgeStorage,t: | The marker age of the storage at time step t. |
| MAgeStorageVol,t: | The marker age of the constituent only from the storage volume at time step t and it is equal to 1 plus the marker age of the storage at time step t-1. |
| StorageVolt-1: | The storage volume at the beginning of time step t. |
| MAgeUSFlowVol,t | The marker age of the constituent only from the upstream inflow to the storage at time step t. |
| USFlowVolt: | The volume of the upstream inflow to the storage at time step t. |
| MAgeConveyVol,t: | The marker age of the constituent only from the conveyance flux to the storage at time step t. |
| ConveyVolt: | The volume of the conveyance flux to the storage at time step t. |
Note that if the sum volume (i.e., StorageVolt-1 + USFlowVol USFlowVolt + ConveyVol ConveyVolt) in the equation is less than the required minimum storage volume at step t, the sum value will use the required minimum storage volume.
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