*3.1. Model Implementation*

To achieve these objectives, a series of implementations of the PerTy3 and the numerical model, FEFLOW (finite-element subsurface flow simulation system), are used. The default parameters used for both the one- and two-dimensional (2D) modelling are shown in Table 2. One-dimensional (1D) situations represents those, where lateral movement of water is minimal, and hence where *B* approximates zero. The default parameters have been derived using published estimates of soil hydraulic parameters for the Mallee region [23] and defining equivalent parameters for the Brooks–Corey–Mualem model. The irrigation flux pre-development, *IAo*, has been assumed to be 10 mm/year for all experiments. For the two-dimensional experiments, the half-width of the irrigation area is assumed to be 500 m. Pertinent information on the irrigation water balance is from [24].

The numerical modelling is undertaken using FEFLOWTM [25]. FEFLOW solves the governing flow equations in porous media for variable saturation. Richards' equation is solved for a single dominant fluid phase (in this case water) with an assumed stagnant air phase that is at atmospheric pressure everywhere. FEFLOW implements a number of empirical and spline models for variable saturation. Fine mesh refinement is used to achieve stable numerical solutions for the adopted choices of the Mualem–Brooks–Corey model parameters. The vertical mesh size for both the 1D and 2D modelling is 10 cm (250 (1D) or 150 (2D) elements), while for the 2D modelling, the horizontal mesh size is 10 m (200 elements).


**Table 2.** Default soil parameters used in the modelling.

Table 3 details the various modelling experiments and the associated parameters. The modelling has been designed to achieve the various objectives:




\* experiments where no PerTy3 modelling was performed, but FEFLOW (finite-element subsurface flow simulation system) modelling and superposition experiments were performed.

#### *3.2. Transfer Functions and Superposition*

In this section, we describe the concept of a transfer function for application to superposition, to support objective 2(a). For this purpose, a modified transfer function for the recharge is defined:

$$TF'(t) = (R(t) - Ro) \langle IA\_n ^\*-IA\_o ^\* \rangle \tag{42}$$

where the superscript \* indicates the minimum of *IA* and the maximum irrigation accession without rejected recharge. We can define a similar transfer function for drainage, *TF*", where:

$$TF''(t) = (D(t) - D\_o) \langle IA\_n \text{ \*\*} - IA\_o \text{\*\*} \text{\*\*} \rangle \tag{43}$$

where *Do* is the original drainage rate, *IA*\*\* is the maximum of *IA* and the maximum irrigation accession that occurs without rejected recharge.

The general aim is to generate outputs, with appropriate accuracy, for a range of inputs. In line with information theory, we would look to see whether simplifications are possible, by using, for example, processes, such as superposition. As the system is not necessarily linear, there is no reason to believe that superposition would necessarily apply to a range of situations. Some situations are not expected to follow superposition including those where thresholds are involved, such as the initiation of perching or rejected recharge. The addition of two actions, that do not individually exceed the threshold but in aggregation do so, is not linear.

*Water* **2020**, *12*, 944

If superposition does apply, the aggregate transfer function for a sequence of actions that a ffect irrigation accession: 

$$TF'(t) = \left(\sum\_{j} (IA^\*\_{\ j+1} - IA^\*\_{\ j}) \left. TF'\_{\ j+1} \right\rangle \langle IA^\*\_{\ j+1} - IA^\*\_{\ 0} \rangle \right. \tag{44}$$

where *IAj* is a sequence of modified irrigation accessions that occur from *j* = 0 to *j* = *p* + 1 and *TF'j*+<sup>1</sup> is the modified transfer function that applies for a change of irrigation accession from *IA\*j* to *IA\*j*+1.

The pattern of the aggregate *TF*' with time would broadly follow that of *IA*, perhaps with time delays and some 'smearing' and capping at the maximum irrigation accession for which there is no rejected recharge. Where superposition applies, this could simplify the numerical process. The transfer function could be generated for individual actions alone and this allows the overall aggregated transfer function to be generated. While the theory above describes physical processes, the analytical model is simplified, with assumptions such as spatial homogeneity, flat surface elevations etc. It is possible to consider transfer functions as conceptual models, using enough complexity to broadly replicate the recharge response. Such models have previously been used for recharge [16,18].

For this paper, we will compare the recharge under a brownfield development and the original greenfield development with the superposition of the two individual recharge outputs. Brownfield sites were thought previously to have shorter time delays, as they have already been wetted. Should the superposition be a good approximation, this may simplify the estimation of recharge under a complex irrigation district by allowing each development to be considered individually and then aggregated.
