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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