**2. Theory**

This section describes the processes and theory that underlies the semi-analytical model, PerTy3 and perched water tables, more generally. More specifically, this section supports objectives 1(a–c); namely, the description of a conceptual model representing the main physical processes at the scale of the irrigation district and periods of seasons and years; provides continuity of modelling between conditions of perching and non-perching; and includes a limited number of additional parameters. Non-dimensionalisation is used to group related parameters and, therefore, simplify the model.

The motivation for this study is illustrated in Figure 1, which shows the hydrogeological model of the Loxton-Bookpurnong irrigation district in southern Australia. Irrigation development has led to a perched water table above a low-conductivity clay layer and a groundwater mound in the underlying regional groundwater system. The increased groundwater gradients lead to greater volumes of saline groundwater entering the River Murray.

Figure 2 shows the conceptual model for the fluxes in the vadose zone. There are three semi-infinite layers of homogeneous soils, in which the second layer is of lower permeability. The left boundary condition is a no flow boundary, which means this is a line of symmetry. The upper boundary is the base of the agronomic zone, the left side of which underlies irrigated agriculture, and to the right side underlies non-irrigated agriculture. The upper boundary condition is a downward water flux, irrigation accession, as determined by agricultural practices, including channel leakage and spillage. For the one-dimensional systems discussed below, the whole upper surface is irrigated. The lower boundary condition is the water table, which is assumed to be constant. The profile is assumed to be initially at steady-state, with the root zone drainage at the boundary condition being relevant to either native vegetation or non-irrigated agriculture. At time zero, irrigation is implemented leading to an increase in root zone drainage. As the vertical flux may also consist of leakage or spillage from channels, it will be referred to as irrigation accession. In the western Murray Basin context, the new irrigation accession rate is ~100–400 mm/year), while the pre-irrigation flux (~0.1 mm/year for native vegetation or 2–30 mm/year for dryland agriculture).

**Figure 1.** Conceptualizations of the Loxton-Bookpurnong Irrigation District, showing perched water table under the irrigation district, and groundwater flow to the River Murray. model used to simulate recharge under perched water tables.

**Figure 2.** Used to simulate recharge under perched water tables. The left-hand boundary is a no flow boundary, representing a line of symmetry. The variables are non-dimensionalised, with *x* = 1 being the outer limit of irrigation and *x* = *x*1 being the outer limit of perched water. Below layer 3 is the saturated zone of the aquifer.

This paper considers situations, where there is a reasonable probability of perched water tables under the irrigation area. This means that the irrigation accession should be sufficiently high or the saturated vertical conductivity sufficiently low for perched water tables to occur. Where it does occur, the ponded head builds up on the impeding layer and water moves laterally over the impeding layer, where it infiltrates into the impeding layer.

The layers of the unsaturated zone have been parameterised using the modified Mualem–Brooks–Corey model [21,22] for each layer. The water retention curve is given by:

$$
\partial \Theta = (\natural \psi \hbar\_{\flat})^{-\lambda}, \Downarrow \Psi \gg \hbar\_{\flat} \tag{1}
$$

$$
\Theta = 1, \psi \le h\_b \tag{2}
$$

where ψ is the soil suction, *hb* is the air-entry point, λ is a fitting parameter, and the relative saturation, Θ, is given by

$$
\Theta = \frac{\left(\theta - \theta\_r\right)}{\left(\theta\_s - \theta\_r\right)'}\tag{3}
$$

where θ is the volumetric water content, θ*r* is the residual volumetric water content and θ*s* is the saturation volumetric water content. The relative permeability, *Kr*, is given by:

$$K\_{\Gamma} = \Theta^{\text{m}},\tag{4}$$

where *m* is a fitting parameter related to the connectivity of soil pores.

The hydraulic conductivity, *K*, is obtained by multiplying*Kr* by the saturated hydraulic conductivity. These parameters will be different for the different layers and a subscript *i* will be subsequently used to distinguish layer *i*. The parameters are assumed to be the same for both vertical and horizontal properties, except for the saturated conductivity. A superscript *h* and *v* will be used with the saturated hydraulic conductivity to characterise anisotropy. The values given to the parameters is given in the Methods section.

The underlying equations are non-dimensionalised using the vertical length scale, *l2*, timescale *S2l2*/*Ks2v*, horizontal length scale *x*0, and vertical flux *Ks2*v; where *li* is the thickness of the *i*th layer; *x*0 is the half-width of the irrigated area; *Ksi*v is the saturated vertical conductivity of the *i*th layer; and *S2* is the specific yield for the 2nd layer for the initial dry conditions. The purpose of the non-dimensionalisation is to simplify the range of situations as much as possible using scaling and non-dimensional variables.

#### *2.1. Unsaturated Zone Conditions*

The PerTy3 model, as it relates to new developments, is adapted from the wetting front model [5]. In that model, the movement of water through the vadose zone occurs via gravity, causing a pressure (or wetting) front. Behind the wetting front, the flux of water is equal to the new flux, while below it, the flux equals the old flux. The wetting front moves with the speed (non-dimensional):

$$dz\_{w\theta}dt = (A\_n - A\_o) \langle (\theta\_n - \theta\_o) \rangle \tag{5}$$

where *zwf* is the depth of the wetting front below the land surface; *An* and *Ao* are the non-dimensional irrigation accessions for the irrigated and pre-existing agriculture respectively; and θ*n* and θ*o* are the volumetric water contents above and below the wetting front. Their values are such that the relative vertical hydraulic conductivity equals *An* and *Ao*, respectively. When the pressure front reaches the water table, the recharge (dimensioned) increases from *IAo* to *IAn*. Equation (5) can be used to estimate the time delay between the change in land use and the change in groundwater recharge. The above theory, or variants of it, has been used to estimate time delays for changes in non-irrigated agriculture to affect the underlying groundwater. In this paper, we look to adapt this model to the situation, where the soil conductivity of parts of the vadose zone is sufficiently low to not allow the new water flux to move vertically by gravity alone. The simplicity of the model is appropriate for our knowledge of soil properties and input fluxes over representative scales. The parameters in Equations (1)–(4) will change for each layer. A subscript '*i*' will be used to denote these parameters for layer *i*.

#### *2.2. Situations Where Perched Water Tables Occur*

The conceptual model in PerTy3 considers five stages for the pressure front to move through to the water table and for the new recharge rate to be attained. These are:

Stage 1: the pressure front moves through first layer according to Equation (5).

Stage 2: the interface between layers 1 and 2 needs to become saturated for the perched situation to occur. A wetting front continues to move through layer 2 while the moisture content about the interface is increasing. However, the dimensionless vertical flux across a broad zone from above the interface to below the pressure front is reducing from *An* to *Ao.*

Stage 3: saturated conditions develop at the interfaces of the first and second layers. This causes a saturation front to move downward behind the wetting front into layer 2, while the perched layer begins to build up in layer 1. The zone between wetting and saturation fronts is near-saturated and can be broad. The perched layer causes the hydraulic gradient in the saturated zone to be greater than that of gravity alone. As the ponded head rises, the hydraulic gradient continues to increase and the flux behind the wetting front increases. Water begins to move laterally above the interface between layers one and two and begins to infiltrate into the impeding layer.

Stage 4: the wetting front has reached the base of layer 2 and begins to move through layer three. The surface of the perched water table continues to rise towards an equilibrium, as does the flux behind the wetting front. As the flux increases, the saturation zone moves slowly towards the base of layer 2.

Stage 5: the wetting front has reached the water table. The recharge rises from the old irrigation accession rate. The recharge continues to rise until the perched water table reaches a new equilibrium. This occurs when the increased gradient through the second layer and the increased area of infiltration means that the recharge is equal to the irrigation accession. Where layer one is su fficiently thin, water from the perched water table is intercepted either by evapotranspiration or sub-surface drainage. This prevents the recharge from reaching the irrigation accession flux.

In adapting the wetting front model to perched situations, the following changes are incurred:


The sections below describe these stages in more mathematical detail. Table 1 lists all the parameters used, their symbols and units and the equation number, where first used.



## *2.3. Stage 1*

During stage 1, the rate of movement of the wetting front is given by Equation (5). The dimensionless time for the wetting front to reach the top of the clay layer is given by:

$$t\_1 = l\_1(\theta\_{n1} - \theta\_{o1}) / ((A\_n - A\_o)l\_2) \tag{6}$$

The variable *t*1 is sensitive to θ*n*1*,* this needs to be calculated for the new flux.
