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Definitions

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.

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; and
  • 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.

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	Image Modified

Expanding Equation 1 yields Equation 2:

Equation 2	Image Modified

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

Equation 3	Image Modified

Equation 3 may then be rearranged to give Equation 4:

Equation 4	Image Modified

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

Equation 5	Image Modified

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

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

Equation 6	Image Modified

Equation 7	Image Modified

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	Image Modified

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

Equation 9	Image Modified

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	Image Modified

Equation 11	Image Modified

Equation 12	Image Modified

Equation 13	Image Modified

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

Image Modified

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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.

Image Modified

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When t₁^* = dt:

Image Modified

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	Image Modified

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	Image Modified

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.

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.

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.

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

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

Equation 18	Image Modified

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	Image Modified

Equation 20	Image Modified

Recalling that

Image Modified

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And that when t₁^* = dt then:

Image Modified

Equation 21	Image Modified

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	Image Modified

Equation 23	Image Modified

Equation 24	Image Modified

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	Image Modified

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	Image Modified

Equation 27	Image Modified

Equation 28	Image Modified

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	Image Modified

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Equation 30	Image Modified

Equation 31	Image Modified

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	Image Modified

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	Image Modified

Equation 34	Image Modified

Equation 35	Image Modified

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	Image Modified

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	Image Added

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	Image Added

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

39.

Equation 39	Image Added

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	Image Added

Equation

40 can be rearranged in terms of dc as follows:

Image Modified

Which can then be simplified to give Equation

41 and Equation

42:

Equation 41	Image Added

Equation 42	Image Added

A limit exists as dt^*→0 of _{Image Modified} and A =

At+a•t^* as was established previously so Equation

42 then becomes:

Equation 43	Image Added

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

s = RL•a

Image Modified

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S_t = RL • A_t

S_t+1 = RL • A_t+1

then Equation 43 can then be rearranged as follows and integrated:

Equation 44	Image Added

Equation 45	Image Added

Integrating Equation

45 gives Equation 48 below through the following calculations, where Equation

48 defines c₁.

Equation 46	Image Added

Equation 47	Image Added

Equation 48	Image Added

By defining the following:

Image Modified

Equation

48 becomes:

Image Modified

And can be expressed as:

Where:

Image Modified

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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:

Image Modified

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

Therefore:

Image Modified

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:

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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:

Image Modified

Using the standard equation (Equation 425, reproduced below) to define c1 as shown below:

Image Modified

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.