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.

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:

• • Inflows;
  • Confluences;
  • Storages and wetlands (Storage node), and
  • Weirs

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.

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

Expanding Equation 1 yields Equation 2:

Equation 2

Cancelling RL•A•x terms on each side of Equation 2 and assuming RL•A•dx•dt^* ≈ 0 leads to Equation 3:

Equation 3

Equation 3 may then be rearranged to give Equation 4:

Equation 4

Rearranging Equation 4 in terms of dx results in Equation 5:

Equation 5

Creating the two following values, g₀ and g₁, which are constant over the model time-step:

and substituting these into Equation 5, allows dx to be expressed in terms of Equation 6 and Equation 7:

Equation 6

Equation 7

The storage volume of the link at any given time can be given as RL•A = RL•(A_t+a•t^*). This allows Equation 7 to be rewritten as Equation 8:

Equation 8

Then Equation 8 may be rearranged so that the integral of dx and dt^* is calculated as shown in Equation 9:

Equation 9

Integrating Equation 9 will lead to a value of x₁ shown in Equation 14 that gives the location of the marker in the division at time t₁^* = dt; ie. the location of the marker at the end of the model time-step. The following equations show the integration process to arrive at Equation 14:

Equation 10

Equation 11

Equation 12

Equation 13

The storage volumes in the division at two specific times, t₀^* and t₁^*,are represented by the following relationships:

The term a is calculated by assuming that the rate of change in cross sectional area is constant through-out the model time-step. As reach length is constant, the change in cross sectional area can be expressed in terms of the reach storage at the start and end of the model time-step and is equivalent to the term s, the rate of change of the storage over the model time-step.

When t₁^* = dt:

Substituting the above into Equation 13 gives Equation 14 which defines the location of the marker, x₁, within the division at the end of the model time-step:

Equation 14

If the calculated value of x₁ in Equation 14 is greater than 1 then the marker has exited the division within the model time-step. Therefore, the time at which the marker exits the division needs to be defined. This time can be calculated by rearranging Equation 12 in the form shown.

Equation 15

Time t₁^* can be determined from Equation 15, where t₁^* is the time at which the marker will arrive at the end of the division.

Equation 16

As the marker has arrived at the end of the division, therefore x₁ = 1, and by substituting in the relationships between area and storage, Equation 16 becomes Equation 17.

Equation 17

Alternative formulations

Equation 14 is the primary equation that defines the location of a marker, x₁, in a division when t₁^* = dt, while Equation 17 defines the time a marker exits a division, t₁^*, when x₁ > 1. However, these equations are not defined for the following conditions:

• s ≠ 0 and Ls = -s; therefore g₀ = 0 and g₁ is undefined;
• s = 0 and Ls ≠ 0; and
• s = 0 and Ls = 0; therefore g₀ = 0 and g₁ is undefined

The next sections detail how x₁ and t₁^* are calculated for the the above conditions.

Formulation for when s ≠ 0 and Ls = -s

From Equation 14 and using the established relationship of A = A_t + a • t^*, the following relationship is obtained.

However as Ls = -s this leads to Equation 18:

Equation 18

Equation 18 may be then arranged in terms of dx and dt^* as shown in Equation 19 so that the integrals may be taken to arrive at a value of x₁ in Equation 21 through the steps outlined next:

Equation 19

Equation 20

Recalling that

And that when t₁^* = dt then:

Equation 21

If the value of x₁ calculated in Equation 21 is greater than 1, then the marker has exited the division during the model time-step. The time elapsed when a marker exits the division (x₁ = 1) can be calculated using Equation 20 and can be expressed in terms of t₁^* in Equation 24 through the process outlined next:

Equation 22

Equation 23

Equation 24

Formulation for when s = 0 and Ls ≠ 0.

For the case where s = 0 and Ls ≠ 0 the formulation starts from Equation 9 which then becomes:

Equation 25

In this case, g₀ = Ls and g₁ = I/Ls and therefore x₁ may be expressed as shown in Equation 28 following the calculation process described next:

Equation 26

Equation 27

Equation 28

If the value of x₁ calculated in Equation 28 is greater than 1, then the marker has exited the division during the model time-step. The time elapsed when a marker exits the division (x₁ = 1) can be calculated using Equation 26 and can be expressed in terms of t1* in Equation 31 through the process outlined next:

Equation 29

Equation 30

Equation 31

Formulation for when s = 0 and Ls = 0

For the case where s = 0 and Ls = 0 the formulation starts from Equation 4 which then becomes:

Equation 32

Equation 32 may then be arranged in terms of dx and dt^* as shown in Equation 33 so that the integrals may be taken to arrive at a value of x₁ in Equation 35 through the steps outlined next:

Equation 33

Equation 34

Equation 35

If the value of x₁ calculated in Equation 35 is greater than 1, then the marker has exited the division during the model time-step. The time elapsed when a marker exits the division (x₁ = 1) can be calculated using Equation 35 and can be expressed in terms of t₁^* in Equation 36, shown next:

Equation 36

Table 3 summarises the key equations that define the location of a marker of water in the division at the end of the model time-step. The conditions for the use of each formula are also contained in the table. It defines marker location (x₁) at the end of the model time-step (t₁^* = dt) for conditions of s and Ls.

Table 3. Summary of equations (location)

Table 4 summarises the key equations that define the time that a marker of water leaves the division. The conditions for the use of each formula are also contained in the table. It defines the time (t₁^*) when a marker exits a division within the model time-step (x₁ > 1 and t₁^*< dt) for conditions of s and Ls.

Table 4. Summary of equations (time)

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

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

The volume of water in the division section of length x after time dt^*is as shown in Equation 39.

Equation 39

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

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

Equation 42

A limit exists as dt^*→0 of and A = At+a•t^* as was established previously so Equation 42 then becomes:

Equation 43

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

Equation 45

Integrating Equation 45 gives Equation 48 below through the following calculations, where Equation 48 defines c₁.

Equation 46

Equation 47

Equation 48

By defining the following:

Equation 48 becomes:

And can be expressed as:

Equation 49

where:

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

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

Equation 52

Integrating Equation 51 leads to Equation 54 following the calculation steps shown next:

Equation 53

Equation 54

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

Equation 56

Equation 57

Which leads to Equation 58 (shown next), expressed in the same form as Equation 49.

Equation 58

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 c_ev and c_Load, 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 c_ev

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 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.

Input data: