**2. Theory**

This section broadly provides the theory to meet the above objectives:


#### *2.1. Conceptual Model for Theory*

The conceptual model for this paper is similar to Ref. [18] and is shown in Figure 1. 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, i.e., this is a line of symmetry and conditions to the left reflect those on the right. 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 with 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.

**Figure 1.** Conceptual model for theory development. The darker brown layer (layer 2) represents the impeding layer. The blue layer represents the saturated zone. The base of layer 1 is the base of the agricultural zone, while the two lighter brown layers represent higher permeability zones. Saturated conditions build up on the base of layer 1 and top of layer 2 (i.e., perching). The irrigation field extends over the horizontal axis, *x*, from *x* = 0 to *x* = 1. The model dimensions extend beyond the irrigation field (i.e., *x* > 1) to investigate the lateral e ffects of perching that extend to *x*1.

The layers of the unsaturated zone have been parameterised using a modified Mualem–Brooks– Corey model of soil hydraulic properties for each layer [20,21]. The water retention curve is given by:

$$
\Theta = \left(\frac{\psi}{hb}\right)^{-\lambda}, \psi > h\_b \tag{1}
$$

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

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

$$O = \frac{\left(\theta - \theta\_r\right)}{\left(\theta\_\text{s} - \theta\_I\right)}.\tag{3}$$

The relative permeability, *Kr*, is given by:

$$K\_r = \Theta^m \tag{4}$$

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, *l*2, timescale *S*2 *l*2/*Ks*2*v*, horizontal length scale x0; where *li* is the thickness of the *i*th layer; x0 is the half-width of the irrigated area; *Ksi*v is the saturated vertical conductivity of the *i*th layer; and *S*2 is the specific yield for the second layer for the initial dry conditions. The purpose of the non-dimensionalisation is to simplify as much as possible using scaling and non-dimensional variables.

#### *2.2. Modelling Individual Actions*

In this section, the theory for recharge response to a reduction in irrigation accession is developed. The system is assumed to be initially in equilibrium and the change in irrigation accession occurs at time zero. The theory is adapted from the kinematic wave model [10] in which the initial and final states are unsaturated zone and processes are one-dimensional (vertical). This is described in Section 2.2.1. In Section 2.2.2, this theory is adapted to situations, in which the initial state is perched. For perched water tables, water can move laterally from the irrigation district and process can be two-dimensional. Under certain conditions, these two-dimensional processes can be reasonably represented by one-dimensional processes, for which the modelling is simpler.

#### 2.2.1. Kinematic Wave Theory for Unsaturated Zones

Where the original irrigation accession is sufficiently small, the soil zone is initially unsaturated for all layers. In the application of the kinematic wave approach [10] to the above conceptual model, the reduction in irrigation accession at the top of layer one causes the moisture content and soil potential there to reduce to the value for which the intrinsic vertical hydraulic conductivity, *Kr*1, equals the new non-dimensional irrigation accession flux, *A*. The kinematic value approach follows how any particular value of vertical hydraulic flux (and associated moisture content and soil potential) between the old and new irrigation accession fluxes travels through layer one. The higher value of flux (and associated moisture content and soil potential) travels more quickly, with the fastest being that for the old irrigation accession and the slowest being that for the new irrigation accession. When the fastest value reaches a particular depth, the flux at that depth begins to gradually reduce from the old irrigation accession until the slowest value reaches that depth and flux stabilizes at the new value. Any value of flux can be followed through the three layers to the water table. The theory implies that the speed of any value in the *i*th layer is given by:

$$\frac{dz}{dt} = d\mathbf{K}/d\boldsymbol{\theta} = \,\,\mathbf{K}\_{\dot{r}i}\left(\boldsymbol{\Theta}\right)/\left(\boldsymbol{\theta} - \boldsymbol{\theta}\_{\dot{r}i}\right)\,\mathbf{G}\_i\tag{5}$$

where *z* is the non-dimensional depth of the soil potential below the land surface, and θ is the moisture content, θ; in that layer that corresponds to the soil potential; and:

$$\mathbf{G}\_{i} = m\_{i} \, \mathbf{S}\_{i} \, \mathbf{K}\_{s i}^{\;\;\;\nu} / \mathbf{K}\_{s 2} \, \mathbf{C} \tag{6}$$

Recharge will remain at the old value until the fastest value reaches the water table and reduce until the slowest value reaches there and will then stabilize at the new value. The recharge between these values can be obtained by interpolating the times for other values to reach the water table.

Should there be any further measures that reduces irrigation accession, then the above steps can be repeated. Since the group velocity for these new values are lower, there should be no interference between the steps.

#### 2.2.2. Soil Zone Initially with Perched Water

In this section, we consider the situation for which there is a perched water table at the base of the first layer but does not intercept the upper surface of layer 1 and this remains so throughout the transition from old accession rate to new accession rate. It will be shown in this section that most state variables of interest will move exponentially from the old equilibrium value to the new equilibrium value.

If layer one is su fficiently thick, equilibrium conditions can be reached in which the infiltration through the clay equals that of the irrigation accession [18]. The non-dimensional thickness of the perched water above the second layer, *heq*, under equilibrium conditions is given by:

$$h\_{eq} = (A - 1 - q\rho)(1 + \text{s}qrt(B))\tag{7}$$

where *A* is the non-dimensional irrigation accession; *B* and ϕ are dimensionless variables, defined by:

$$B = K\_{s1}{}^h l\_2 \, ^2/(K\_{s2}{}^v \, \propto\_0 2 \, )\tag{8}$$

$$\varphi = (A - 1) \int d\psi \langle (A/K\_{r2}{}^{\upsilon} \left(\psi\right) - 1 \rangle \, l\_2 \rangle \tag{9}$$

and the integral in Equation (10) is from *hb* to ∞ and ϕ is assumed to be insensitive to *A*.

Perching only occurs if

$$A > 1 + \varphi \tag{10}$$

and does not intercept the upper boundary condition if

$$l\_1/l\_2 > (A - 1 - \varphi)/(1 + \text{spt}(B))\tag{11}$$

Under conditions of quasi-steady state perched water, and a perched water of zero thickness under non-irrigated agriculture in the fringes of the irrigation areas, the transient condition is given by [18]:

$$
\beta \frac{\partial h}{\partial t} = A - 1 - q - h \left( 1 + \text{spt}(B) \right) \tag{12}
$$

where β is the ratio of *s*1 to *S2*; and *s*1 is the specific yield of layer 1 after wetting front has already passed through. This has the solution:

$$h = h\_{\text{eq}} + \exp(-t \left(1 + \text{sqrt}(B)/\beta\right) \left(h\_0 - h\_{\text{eq}}\right) \tag{13}$$

where *heq* and *h*0 are respectively the new equilibrium head and the old equilibrium head. This shows that perched head reduces exponentially to the new equilibrium conditions. The e ffect of two-dimensional flow is indicated by the dimensionless parameter *B*, where *B* = 0 corresponds to one-dimensional flow and large *B* corresponds to significant lateral movement. The e ffect of *B* is to not only reduce the steady-state ponded head, but to also quicken the rate at which the new equilibrium is attained. To understand this better, we consider the dimensioned time scale in the exponential function:

$$t\_5 = l\_2 \ S\_2 \ \beta/((1 + \text{s}qrt(B)) \ K\_{s2}\text{"})\tag{14}$$

As *B* becomes very large, *ts* becomes in dimensioned variables:

$$t\_s \sim \propto\_0 s\_1 / \text{sqrt}(K\_{s1}{}^h K\_{s2}{}^v) \tag{15}$$

Hence, the time scale involves a mixture of the horizontal conductivity of the sand layer and the vertical conductivity of the clay layer. This is not surprising given that these soil parameters influence both the magnitude of the ponded head and the ability of the water to move laterally. In parallel with the ponded head increasing exponentially to the equilibrium value, the vertical flux through the second layer under the irrigated agriculture would also approach the equilibrium value, *qeq*:

$$q(t) = q\_{\text{eq}} - (q\_{\text{eq}} - q\_0) \exp(-\left(t - t\_0\right) \left(1 + \text{sqrt}(B)\right) |\beta\rangle\tag{16}$$

where *q*0 is the vertical flux at initial equilibrium and is related to *h*0 by:

$$q\_0 = 1 + h\_0 + \varphi \tag{17}$$

There is a similar relationship between *qeq* and *heq* .The extent of the wetting outside of the irrigation field, *x*1, also decreases exponentially under the assumptions used from the original equilibrium value, *x*10, to the new equilibrium value, *<sup>x</sup>*1*eq*, in parallel with the ponded head.

$$\mathbf{x}\_1(t) = \mathbf{x}\_{1eq} - (\mathbf{x}\_{1eq} - \mathbf{x}\_{10}) \exp(-(t - t\_0) \left(1 + \text{sptt}(\mathbf{B})\right) |\beta\rangle \tag{18}$$

where *x*10 is related to *h*0 by:

$$\mathbf{x}\_{10} = h\_0 \operatorname{spt}(\mathbf{B}) + \mathbf{1} \tag{19}$$

However, the aggregated vertical flux through the clay external to the irrigation field is no longer proportional to the extent of wetting, as areas that had wetted previously will continue to drain. This additional term is ignored in the current modelling.

#### 2.2.3. Change from Perched Water Table to None

If the new irrigation accession is such that *A* < 1 + ϕ, there is no perched water in the new equilibrium situation. The solution to Equation (12) in that situation is still given by Equation (13), but with the term *heq* substituted by ( *A* − 1 − ϕ)/(1 + *sqr<sup>t</sup>*(*B*)). This allows the time for the perched head to go to zero to be estimated. After that time, layers 2 and 3 begin to drain; and Equations (5) and (6) apply.

#### 2.2.4. Initial State with Rejected Recharge

For the situation where the head of the perched water under the irrigated agriculture intersects the upper boundary (i.e., the first layer is saturated under irrigation), Equation (11) no longer applies and some of the irrigation accession is returned to the surface as there is no opportunity of any further infiltration. We refer to this situation as rejected recharge. If this is the starting situation, then a minor reduction in the irrigation accession may have no e ffect on recharge, but rather the volume of rejected recharge is reduced. It is only when the irrigation accession is reduced su fficiently for Equation (12) to hold that recharge will change in response to the reduced irrigation accession. The volume of rejected

recharge should reduce to zero almost immediately and the head of perched water will also begin to respond immediately. The Equations (12), (13), (16) and (18) should then apply.

#### *2.3. Modelling of Multiple Actions*

In Section 2.2.1, the theory for a soil zone with no perched water showed that the response of successive actions, which reduced irrigation accession, were independent of each other. This is unchanged for perched water tables. As can be seen in Equation (12), where it does not matter if the initial condition is an equilibrium situation. The value of the head at the time of the water use e fficiency change is used for *h*0. Taken together, the methodology across the range of situations described in Sections 2.2.1–2.2.4 does not change under multiple actions, which reduce irrigation accession.

For the first irrigation water use e fficiency improvement following a new development, the soil zone may not have come to a hydraulic equilibrium at the time this improvement occurs. Two situations are considered: (1) the wetting front has reached the water table, but perched water has not equilibrated and (2) wetting front is in layer three when new measure occurs.

In the first situation, the e ffect of reduced irrigation accession should almost be immediate. The perched head should move exponentially to the new equilibrium value or zero (if the new equilibrium state has no perched water). The flux through the second layer should change exponentially in response and the system behaves as in Section 2.2.2. The change in flux will then cause a change in recharge when the pressure e ffect reaches the water table. Where there is no perched water in the new equilibrium state, layers two and three begin to drain

In situation (2), the perched head will still respond exponentially, as will the flux through layer two. There is some chance that the pressure e ffect of this may reach the wetting front before it reaches the water table, causing a slight modification in the change in flux at the wetting front and the speed at which it moves. Once the wetting front has reached the water table, the system will respond as per Section 2.2.2.

#### *2.4. Transfer Function and Superposition*

Section 2.3 contributes to the second objectives; namely developing a way to implement transfer functions for the estimation of recharge from irrigation areas at district-wide scales. More specifically, it defines modified transfer functions for which superposition may apply. It also develops the theory for the linear reservoir model, in a way that is relevant to irrigation systems with perched water.

A transfer function is a mathematical model of a system that maps its input to its output (or response). A normalised transfer function *TF(t)* is normally defined as:

$$TF(t) = (R(t) - Ro) / (IA\_{\text{il}} - IA\_o) \tag{20}$$

An analogous transfer function can be defined for drainage volumes (*D(t)*). These transfer functions will mostly be used for displaying results in Section 4. Where there is no rejected recharge, *TF(t)* = 0 for *t* = 0; and approaches 1 for large times. Hence, it represents a cumulative probability distribution for time delays for pressure to travel through the vadose zone. Where there is rejected recharge, *TF(t)* is less than one for large times but the sum of transfer functions for recharge and drainage approaches one.

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 include those where thresholds are involved, such as the initiation of perching or rejected recharge. The addition of two actions, that do not meet individually exceed the threshold, but in aggregation do so, is not linear. The transfer functions, as defined in Equation (20), is clearly not appropriate for superposition, because of the threshold where rejected recharge occurs. For the purpose of superposition, a modified transfer function for the recharge is defined:

$$TF'(t) = (R(t) - Ro) / (IA\_n{\*}{\*} - IA\_o{\*}{\*}) \tag{21}$$

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

$$TF''(t) = (D(t) - D\_o) / (IA\_n """-IA\_o """) \tag{22}$$

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

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

$$TF'(t) = (\sum\_{j} \left(IA^\*\_{j+1} - IA^\*\_{j}\right) TF'\_{j+1}) \langle (IA^\*\_{j+1} - IA^\*\_{0}) \tag{23}$$

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*, but with (a) time delays; (b) "smearing" and (c) capping at the maximum irrigation accession for which there is no rejected recharge. Where superposition applies, this would simplify the numerical process by generating the transfer function for the combination of actions from the transfer function of individual actions.

Within the parameter range used, superposition provided a reasonable approximation to new developments because of the lack of hydraulic interaction between developments [18] i.e., spatial superposition. In this paper, we will test the principle of superposition for a set of actions acting on the one irrigation development at different times, i.e., temporal superposition. For the modelling in this paper, a new development is assumed to be followed by a succession of water use efficiency improvements.

#### *2.5. Theory for the Linear Reservoir Model*

The transfer functions for application in Equation (23) can be generated from PerTy3, using the methodology described in Section 2.1. However, one of the objectives of the paper is to explore the application of conceptual models for the transfer function and, in particular, the linear reservoir model. The separation between physically based and conceptual models is not black and white. 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 [6,10].

The physical processes above are consistent with a linear reservoir, for which the inputs, *IA*, and outputs, *R*, are related through the storage, *M*:

$$\text{l.d}\!M/\text{dt} = \text{I}A^\* - q\tag{24}$$

where *M* is the total amount of water in the unsaturated zone, including the perched water table and *IA*\* is the irrigation accession, adjusted for rejected recharge, if it occurs. If *R* is linear with respect to *M* i.e.,

$$q = cM + d\tag{25}$$

This leads to M being exponential with respect to time with coefficient <sup>−</sup>*c*. There is a time lag between *q* and *R* due to the time for pressure fronts to move through the unsaturated zone. This leads to:

$$
\Delta R = \Delta I A \left(1 - \exp(-\mathcal{c}\left(t - t\_5\right)\right)\right), t > t\_6 \tag{26}
$$

where Δ represents the change in the parameter, and *c*, *t*5 and *t*6 become fitted parameters. While the theory in previous sections give physical meaning to parameters *c* and *tl* or fitted to model outputs, they can be also fitted to groundwater responses.

The exponential nature of the linear reservoir function is similar to that of many of the parameters in PerTy3. This is perhaps not surprising, given that the mass balance Equation (24) bears striking similarity to Equation (12).

#### *2.6. Criteria for Rejected Recharge*

Rejected recharge occurs when Equation (11) is not satisfied. The criterion for rejected recharge can be written in dimensioned form:

$$IA > K\_{s2} \text{ }^\upsilon \left(1 + q\nu + l\_1/l\_2 + \text{sqrt}(K\_{s1}{}^h/K\_{s2}{}^\upsilon) \text{ } l\_1/\text{x}\_0\right) \tag{27}$$

Equation (22) separates soil-related parameters (right-hand side) from irrigation accession (left hand side). This aids the parameterization of the transfer function, as over time, irrigation accession will change and gradually decrease. For many areas, rejected recharge will require drainage of some form for agriculture to be sustained. The presence or absence of drainage and the drainage volume for di fferent irrigation accessions and di fferent soils to calibrate the parameters in the transfer function. Where Equation (23) applies, the drainage volume, *D*, can be written as:

$$D = IA - K\_{s2}{}^{v} \left(1 + q + l\_1/l\_2 + \text{sqrt}(K\_{s1}{}^h/K\_{s2}{}^v) \; l\_1/\mathbf{x}\_0\right) \tag{28}$$

where data for drainage volumes exist, they can also be used for calibration.
