Note: This is documentation for version 5.16 of Source. For a different version of Source, select the relevant space by using the Spaces menu in the toolbar above
Marker routing (Particle tracking) - SRG
Source requires functionality to model the movement of conservative constituents (eg. salt) along a river channel network, including exchange of constituent fluxes between floodplains, wetlands, irrigation areas and groundwater.
This functionality is required in Source to satisfy legislative requirements that exist in the Murray-Darling Basin. The MDBA’s Basin Salinity Management Strategy (BSMS) requires that the MDBA, state governments and the federal government contribute to reducing dryland and river salinity through land-use measures, salt interception schemes etc. To evaluate the impacts of these activities a reliable salinity modelling framework is required.
Model elements that require conservative constituent functionality include:
Link
Inflow node
Splitter node (Regulated and unregulated)
Confluence node
Minimum flow node (used to generate orders for dilution flows)
Storage node (These model reservoirs and can connect to wetlands)
Supply point node
Water user node and associated demand models
Loss Node
Gauge Node
Wetland Hydraulic Connector Node
Weirs (flows are routed through weirs using the same methodology as storage routing links)
This section describes the generic model for routing conservative constituents through links available in Source. It is based on a marker tracking method (Close, 1996).
"Markers" are established at the upstream end of the model and their movement downstream is modelled. The marker travels at the average speed of water in the river reach, thus is dependent on the routing of flow (inflow, outflow, storage, losses and gains for each division). The impact of inflows, evaporation and losses on the solute concentration of the marker is also described.
The method is appropriate for modelling the transport of most conservative constituents within stream channels. The method is inherently stable and is not affected by sudden changes in either concentration or flow.
The model will maintain a conservative constituent balance, even when a reach division ceases to flow and the dead storage evaporates, by allowing the modeller to maintain a small nominal storage volume in the division at all times.
Scale
Conservative constituent routing operates at the same spatial scale as link routing and operates at every model time-step when it is activated.
Principal developer
eWater CRC
Scientific provenance
The principles of the marker tracking method are described by Close (1996). Marker tracking has previously been implemented in BigMod.
Version
Source version 2.19
Dependencies
Each link where conservative constituent routing is activated requires a node at its upstream end to define constituent inputs, and a node at the downstream end to process constituent outputs as appropriate. It is also dependent on the routing of flow (inflow, outflow, storage, losses and gains for each division).
Assumptions and constraints
The parameters for particle tracking are summarised in to Table 1. Definitions of variables used in equations in following sections are listed in Table 2.
Table 1. Assumptions and Constraints
No | Assumption/Constraint |
|---|---|
1 | Rainfall, evaporation, diversions and inflows and outflows occurs uniformly over the model time-step. |
2 | Salinity concentration output at a node is the time weighted average of the marker concentrations that pass through that node over the model time-step. To obtain the load of salt passing through this point for the time-step, the time weighted average concentration is multiplied by the inflow to the node. |
3 | Constituent routing depends on flow properties in each division. Specifically, flow routing must be finalised for the division so that the inflow, outflow, change in storage and losses for the link over the time-step are known before constituent routing calculations start. It is assumed that the rates of inflow, outflow and losses for any link are constant over the time-step so that the storage in the link varies linearly over the time-step. |
4 | The marker routing method applies to weirs as well as river channel reaches. |
5 | Storages and wetlands are fully mixed. |
6 | Rainfall is assumed to contain no salt. |
7 | If a reach or a storage that is in dead storage evaporates fully, then it is possible for a mass of constituent to be left behind in the reach or storage. This mass of constituent is assumed to disappear from the system. |
Table 2. Equation variables - definitions
Variable | Definition | Units |
|---|---|---|
A | The cross-sectional area of the division at any given time t* during the model time-step dt. | m2 |
a | Rate of change in cross-sectional area of the division over the model time-step. | m2/s |
At+1 | Cross-sectional area of division at end of model time-step | m2 |
At | Cross-sectional area of division at start of model time-step | m2 |
c | The concentration of a marker in a division, or the average | g/m3 |
c0 | The concentration of a marker in a division at a given time of t0* | g/m3 |
c1 | The concentration of a marker in a division at a given time of t1* | g/m3 |
cEv | A factor to apply to the marker concentration, c0 to account for the effects of evaporation over time dt* | - |
cLoad | A variable used to account for the change in concentration of a marker as a result of extra load being applied to a reach. | (m3/s)-1 |
dc | Change in concentration of a section of the division over a nominated time | g/m3/s |
dt | Model time-step | s |
dt* | Marker travel time-step within a division (t1* - t0*) | s |
dx | Change in marker position during dt* as a fraction of the division length | - |
Ev | The rate of evaporation loss in the division over the model time-step | m3/s |
fd | A nominal marker’s location relative to its downstream marker as a fraction of the time or distance between its upstream and downstream markers fd + fu = 1 | |
fu | A nominal marker’s location relative to its upstream marker as a fraction of the time or distance between its upstream and downstream marker fd + fu = 1 | |
fcd/s | An identifier to assign properties to the downstream face of a marker | |
fcu/s | An identifier to assign properties to the upstream face of a marker | |
Fradd | The additional inflow as a fraction of the total inflow at an inflow node | |
Fru/s | The upstream inflow as a fraction of the total inflow at an inflow node | |
gapd | A variable used to store the distance or time between a marker and its downstream marker | |
gapu | A variable used to store the distance or time between a marker and its upstream marker | |
gLD | A global load switch which is either turned on or off | |
I | Inflow to the division | m3/s |
Iadd | The additional inflow at an inflow node | m3/s |
Ic | The load of solute inflow distributed over the division in the model time-step. That is c(g/m3)•vol(m3) ÷ RL(m) ÷ dt(s).This value can be attributed to the concentration of groundwater inflow or a nominal mass of solute entered to get a satisfactory calibration. | g/m/s |
Iu/s | The upstream inflow at an inflow node | m3/s |
Intp | A flag to determine whether a marker is interpolated by time or position | |
Intpvol | The value used to interpolate between existing markers | |
Ls | Net loss from the division where loss is defined as water leaving the division other than outflow from the end of the division.-Includes net-evaporation, seepage/infiltration, highflow losses and any other outflows from the division eg. distributed diversions. | m3/s |
Lo | Net loss from the division other than net-evaporation ie. Lo = Ls - Ev | m3/s |
Load | The load of the solute in a division at any point within the model time-step. It is calculated based on the location of markers within the division and their respective concentrations. | g |
m | An identification for a nominal marker | |
md/s | An identification of the marker immediately downstream of marker m | |
mu/s | An identification of the marker immediately upstream of marker m | |
MAge | A value used to store a markers age ie how long since it was created | |
nc | Number of constituents being modelled | |
RL | Length of link | m |
s | Rate of change in storage volume over the model time-step | m3/s |
 | Division storage at time t1* | m3 |
| Division storage at time t0* | m3 |
St+1 | Division storage at end of model time-step | m3 |
St | Division storage at start of model time-step | m3 |
Sepmin | The minimum seperation of markers specified by the user as a fraction of either the time-step and distance between markers. eg. If Sepmin = 0.2 then markers that are located within 0.2 • dt or within distance fraction 0.2 are merged | |
t* | Elapsed time within a model time-step (s), t* = 0 at the beginning of a model time-step; and t* = dt at the end of a model time-step. | s |
t0* | The elapsed time (t*) within a model time-step when a marker enters a division (0 ≤ t0* ≤ dt), or the start of the model time-step t0* = 0. | s |
t1* | The elapsed time (t*) within a model time-step when a marker enters a division (0 ≤ t1* ≤ dt), or the start of the model time-step t1* = dt. | s |
x | Position of marker, expressed as a proportion of the total length of division, at time t*, (0 ≤ x ≤ 1) | |
x0 | Position of marker at t0* (0 ≤ x0 ≤ 1), as a proportion of total length of division. | |
x1 | Position of marker at t1* (0 ≤ x1 ≤ 1), as a proportion of total length of division. |
Definitions
The following definitions supplement those in the eWater Glossary:
Concentration
The mass of dissolved constituent per unit volume at a particular location at a particular point in time.
Division
Part of a link. Each division in a link has the same length.
Face
Side of a marker, ie. upstream or downstream.
Load
The mass of constituent entering the river reach laterally.
Marker
Point in the river network at which conservative constituent concentration, marker age and load factors are calculated. Markers move during the model simulation at the average flow velocity of the river.
Marker travel time-step
Proportion of the model time-step that the marker resides within the current division.
Overview
The movement of conservative constituents in stream channels (links in Source) is modelled by tracking the movement of markers of water within the river.
Conservative constituents such as salt move through the river system in slugs. To model such discontinuities in concentration, each marker has a concentration recorded for two faces (or sides) - upstream and downstream.
In Source, links are divided into divisions for routing purposes. Initially, the model starts with a marker at the end of each division in every link. Every time-step, a new marker is created for each division, to mark the concentration at the end of that division at the end of the time-step. Markers travel through the river network until they are either merged with adjoining markers, or leave the river network.
The concentrations of conservative constituents such as salt at each marker changes as the marker moves within a link as a result of evaporation, groundwater inflows/losses and rainfall. To allow for this, each marker’s constituent concentrations as well as its position are tracked within the link over each model time-step.
The concentrations of markers are also changed at nodes in the river model to account for:
Hence, in Source, marker concentration is set or recalculated at nodes used to model the items in the list above. In addition, there is the option for the modeller to set concentrations at gauge nodes.
Tracking the movement of a marker within a link
The marker methodology is executed for each link routing division immediately following the flow routing phase and before processing the flow phase of the next downstream division or node. The distance a marker moves is driven by the velocity in the division over the current time-step. While the flow rate is assumed constant over the time-step, the velocity within the division will change as a result of change in reach storage and cross sectional area.
A key point is that the residence time of a marker within a division is typically different to the model time-step. Therefore, the concept of a marker travel time-step is introduced. The marker travel time-step represents the proportion of the model time-step that the marker resides within the current division. Each marker is likely to have a different marker travel time in each division and for subsequent model time-steps. There are four potential trajectories a marker can follow within a model time-step.
A marker starts the model time-step within the division and its final position at the end of the model time-step is within the same division;
A marker is in the upstream division (n-1) at the start of the model time-step, moves into the division (n) part way through the model time-step, and finishes within division (n) at the end of the model time-step;
A marker is in division n at the start of the model time-step and moves to division (n+1) part way through the model time-step; and
A marker is in division (n-1) at the start of the model time-step, moves into division (n) part way through the model time-step, and moves into division (n+1) prior to the end of the model time-step.
These scenarios are illustrated in Figure 1 (t is time at start of model time-step, dt =model time-step, RL = reach length, t* = elapsed time since start of model time-step, x = distance from start of division), where x is used to define the marker position within the division. t* is the elapsed time since the start of the model time-step and is used to track the time that markers enter and leave the division.
Figure 1. Movement of water markers relative to sub reach (division) n
A marker moves at average water velocity. Inflow to the division upstream of where the marker is located, losses upstream of the marker or change in reach storage will cause a change in marker position.
The distance a marker of water moves within a model division over dt* can be calculated from the water balance in the portion of the division upstream of the marker. The volume upstream of the marker in the division after dt* is equal to the volume upstream of the marker at the start of the marker travel time-step, plus inflows to the division over dt*, less net losses (Ls) from the division upstream of the marker (see Equation 1).
Equation 1 | |
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Expanding Equation 1 yields Equation 2:
Equation 2 | |
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Cancelling RL•A•x terms on each side of Equation 2 and assuming RL•A•dx•dt* ≈ 0 leads to Equation 3:
Equation 3 | |
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Equation 3 may then be rearranged to give Equation 4: