*2.6. Stage 4*

After the wetting front reaches the base of layer 2, Equations (14) and (26) can be combined to estimate the vertical flux:

$$q = h + 1 + q \tag{30}$$

Incorporating Equation (30) into (20), the mass balance for the perched layer becomes:

$$
\beta \frac{\partial \hbar}{\partial t} = A - 1 - \varphi - \hbar \left( 1 + \text{sptt}(B) \right) \tag{31}
$$

where ϕ is calculated using Equation (27). This has the solution:

$$h = h\_{\text{eq}} + \exp(-(t - t\_0)(1 + \text{spt}(B)) / \beta)(h\_0 - h\_{\text{eq}}) \tag{32}$$

where *h*0 and *t*0 are, respectively, the head and time at which the wetting front breaks through the clay layer Equation (29). The equilibrium head, *heq,* is given by:

$$h\_{\text{eq}} = (A - 1 - q)/(1 + \text{spt}(B))\tag{33}$$

The effect of *B* is to not only reduce the steady-state ponded head, but it also quickens the rate at which it is attained. To understand this better, we consider the dimensioned time scale:

$$t\_{\mathbb{S}} = l\_2 \mathbb{S}\_2 \beta/((1 + \text{s}qrt(B))\mathbb{K}\_{\mathbb{S}\mathbb{Z}}\text{"})\tag{34}$$

As *B* becomes very large, *ts* becomes:

$$t\_s \sim \propto \propto\_0 s\_1 / \text{sqrt}(K\_{s1} \,^h K\_{s2} \,^v) \tag{35}$$

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 both soil parameters influence both the magnitude of the ponded head and the ability of the water to move laterally.

Equation (30) implies that the vertical flux through the clay layer would also approach exponentially the equilibrium value *qeq* from *q*0*.* Equation (30) and Equation (32). The extent of the wetting outside the irrigation field also increases exponentially to the equilibrium value *x1eq* from *x10* in parallel with the ponded head (Equation (18) and Equation (32)). Equation (16) implies that the aggregated vertical flux through the clay external to the irrigation field is proportional to the extent of wetting.

As the flux at the base of layer 2 increases, the velocity of the wetting front can increase. The speed of the wetting front is given by Equation (5), but with *A* replaced by *q.* Unlike the situation where there is no perching, the water content at the wetting front, <sup>θ</sup>*wf,* is likely to change as the flux at the bottom of layer 2 changes gradually. The change in flux (and associated water content) will move at a speed determined by the group velocity, *dK*/*d*θ. This will continue until the change reaches either the wetting front or the capillary fringe (in the case, where the wetting front has already reached the capillary fringe). In the former situation, the time at which the change reaches the wetting front is given by:

$$\mathbf{t} \cdot \mathbf{t} - \mathbf{t}\_4 = z\_{w\mathfrak{f}} \langle d\mathbf{K} \rangle d\varOmega(\Theta(\mathbf{t}\_0)) \rangle = z\_{w\mathfrak{f}} \langle \mathbf{v}\_{\mathfrak{g}'} \tag{36}$$

where *t*4 is the time at which the change at the change occurs at the bottom of layer 2 and *dK*/*d*θ(θ*(t*4*))* = *vg* is the group velocity as determined there.

**Figure 3.** Figures showing two aspects of the modelling: (**a**) near-saturated zone between wetting front and saturation front, as described in Equation (26); and (**b**) changing speed of wetting front, as flux behind increases.

Figure 3b shows the process of calculating the rate of movement of the wetting front. Following the logic of that diagram allows *zwf* to be calculated by integrating the following equation:

$$
\Delta z\_{wf} = \upsilon\_{wf} \Delta t\_4 / (1 + \upsilon\_{wf} \psi\_{\mathcal{K}}) \tag{37}
$$

where *vwf* is the velocity of the wetting front and *vg* is the group velocity. If *dK*/*d*θ >> Δ*K*/Δθ, as is the case for our parameterisation of layer 3, this simplifies to:

$$z\_{wf} = \int v\_{wf} dt\_4\tag{38}$$

Equation (38) implies that changes in flux and water content at the base of layer 2 are transmitted quickly to the wetting front. This causes the wetting front to move more quickly. Once the wetting front reaches the water table, the changes in flux should transmit quickly to the water table. Under the assumption that *dK*/*d*θ >> Δ*K*/Δθ, the gap in time between the flux at the base of layer 2 disappears. The effect of the higher fluxes at the base of layer 2 during the propagation of the wetting front through layer 3 is to quicken the pace of the wetting front but also to increase the flux at the time the wetting front reaches the water table. The speed at which the wetting front moves through layer 3 is less than that for the unperched situation as the flux at the base of layer 2 is less than for the unperched situation.

Computationally, the inclusion of the assumption simplifies and quickens the algorithms but not including the assumption is still computationally viable. The simpler assumption is made in PerTy3.
