Particle tracking

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, wetlands and re-regulating weirs. Flows are routed through weirs using the same methodology as links)
• Supply point node
• Water user node and associated demand models
• Loss Node
• Gauge Node
• Wetland Hydraulic Connector Node

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.

Conservative constituent routing operates at the same spatial scale as link routing and operates at every model time-step when it is activated.

eWater CRC.

The principles of the marker tracking method are described by Close (1996). Marker tracking has previously been implemented in BigMod.

Source version 2.19.

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).

The parameters for particle tracking are summarised in to Table 81. Definitions of variables used in equations in following sections are listed in Table 82

The following definitions supplement those in the eWater Glossary:

Concentration

Constituent 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.

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)

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.

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 103 (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 377).

Expanding Equation 377 yields Equation 378:

Cancelling RL•A•x terms on each side of Equation 378 and assuming RL•A•dx•dt* ≈ 0 leads to Equation 379:

Equation 379 may then be rearranged to give Equation 380:

Rearranging Equation 380 in terms of dx results in Equation 381:

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

and substituting these into Equation 381, allows dx to be expressed in terms of Equation 382 and Equation 383:

The storage volume of the link at any given time can be given as RL•A = RL•(At+a•t*). This allows Equation 383 to be rewritten as Equation 384:

Then Equation 384 may be rearranged so that the integral of dx and dt* is calculated as shown in Equation 385:

Integrating Equation 385 will lead to a value of x1 shown in Equation 390 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 390:

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 389 gives Equation 390 which defines the location of the marker, x1, within the division at the end of the model time-step:

If the calculated value of x1 in Equation 390 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 388 in the form below.

Time t1* can be determined from Equation 391, where t1* is the time at which the marker will arrive at the end of the division.

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 392 becomes Equation 393.

Equation 390 is the primary equation that defines the location of a marker, x1, in a division when t1* = dt, while Equation 393 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 390 and using the established relationship of A = At + a • t*, the following relationship is obtained.

RL+(At + a • t*)•dx = (I - Ls + s) • x) • dt*

However as Ls = -s this leads to Equation 394:

Recalling that

And that when t1* = dt then:

If the value of x1 calculated in Equation 397 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 396 and can be expressed in terms of t1* in Equation 400 through the process outlined below:

Formulation for when s = 0 and Ls ≠ 0.

For the case where s = 0 and Ls ≠ 0 the formulation starts from Equation 385 which then becomes:

In this case, g0 = Ls and g1 = I/Ls and therefore x1 may be expressed as shown in Equation 404 following the calculation process below:

If the value of x1 calculated in Equation 404 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 402 and can be expressed in terms of t1* in Equation 407 through the process outlined below:

Formulation for when s = 0 and Ls = 0

For the case where s = 0 and Ls = 0 the formulation starts from Equation 380 which then becomes:

Equation 408 may then be arranged in terms of dx and dt* as shown in Equation 409 so that the integrals may be taken to arrive at a value of x1 in Equation 411 through the steps outlined below:

If the value of x1 calculated in Equation 411 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 411 and can be expressed in terms of t1* in Equation 412 below:

Table 83 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 84 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.

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:

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 below in Equation 414.

The volume of water in the division section of length x after time dt* is as shown in Equation 415.

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 416.

Equation 416 can be rearranged in terms of dc as follows:

Which can then be simplified to give Equation 417 and Equation 418:

A limit exists as dt*→0 of and A = At+a•t* as was established previously so Equation 418 then becomes:

Using the relationships established previously when defining the location of the marker:

then Equation 419 can then be rearranged as follows and integrated:

Integrating Equation 421 gives Equation 424 below through the following calculations, where Equation 424 defines c1.

By defining the following:

Equation 424 becomes:

And can be expressed as:

Where:

Equation 425 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
• 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 420 as below:

Which leads to Equation 426 below, expressed in the same form as Equation 425.

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 420 as below:

Integrating Equation 427 leads to Equation 430 following the calculation steps below:

Which leads to equation Equation 430 below, expressed in the same form as Equation 425.

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 420 as follows:

Which leads to Equation 434 below, expressed in the same form as Equation 425.

Therefore:

Using the standard equation (Equation 425, 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 85 and Table 86, respectively.

Details on data are provided in the Source User Guide.

Outputs can be viewed in the Recording Manager; details are provided in the User Guide.

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.