**3. Methodology**

In this study, a simplified infiltration model widely used under ponding conditions was selected [15] This model is based on an infinite-series solution of the Richards flow equation [16] under the hypothesis of ponding conditions, which results in a two-term infiltration equation expressed as:

$$\mathbf{I} = \mathbf{S} \mathbf{t}^{1/2} + \mathbf{A} \mathbf{t} \tag{5}$$

where I is the cumulative infiltration, t is the time, S is the soil sorptivity, and A is a constant that is related to the saturated hydraulic conductivity (typically assumed as 0.4 Ks).

The sorptivity parameter can be defined as

$$\mathcal{S}(\mathbb{Y}\_0) = \lceil \gamma(\boldsymbol{\theta}(\mathbb{Y}\_0) - \boldsymbol{\theta}(\mathbb{Y}\_i)) \rceil \int\_{\mathbb{Y}\_i}^{\mathbb{Y}\_0} \mathcal{K}(\boldsymbol{\Psi}) \, d\boldsymbol{\Psi} \rceil^{\frac{1}{2}} \, \mathbb{Y}\_i < \mathbb{Y}\_0 < 0, \ \boldsymbol{\theta}(\mathbb{Y}\_i) = \boldsymbol{\theta}\_i < \boldsymbol{\theta}(\mathbb{Y}\_0) < \boldsymbol{\theta}\_i \tag{6}$$

where θ is the soil volumetric water content, the subscripts i and s refer to initial and saturated conditions, respectively, and γ =1.818 is a dimensionless empirical constant [17] related to the shape of the wetting front. Following Reynolds and Clarke Topp [18] and using Equation (2), the following equation holds:

$$\mathcal{S}(\mathbb{A}\_0) = \lceil \gamma(\theta(\mathbb{A}\_0) - \theta(\mathbb{A}\_i)) \frac{\mathbb{K}(\mathbb{A}\_0)}{\alpha(\mathbb{A}\_0)} \rceil \tag{7}$$

This equation highlights that sorptivity reduces with increasing antecedent water content, decreasing hydraulic conductivity and increasing sorptive number. For Ψ0 = 0 Equation (7) becomes:

$$\mathbf{S} = \lceil \gamma (\boldsymbol{\theta}\_{\text{s}} - \boldsymbol{\theta}\_{\text{i}}) \frac{\mathbf{K}\_{\text{s}}}{\alpha} \rceil^{\frac{1}{2}} \tag{8}$$

Based on Equation (8) values of sorptivity have been estimated for the three plots/treatments of this study (Table 3).

**Table 3.** Philip's model parameters estimated for the three experimental plots of this study. The di fference between saturated soil water content, θs, and initial soil water content, θi, is also given.


#### **4. Results and Discussion**

Based on the Philip model parameters estimated for the three experimental plots/treatments and shown in Table 3, it can be deduced that the sorptivity markedly decreases with increasing the period of irrigation using TWW. On the other hand, the A parameter is rather similar for all treatments. These results sugges<sup>t</sup> that A, which is related to the saturated hydraulic conductivity and connectivity of the largest pores, is not a ffected by the TWW movement. In contrast, the decrease of S could be due to the clogging of the small pores being this parameter mainly influenced by the sorptive number for invariant values of Ks (Equation (8)). In this context small di fferences of antecedent soil conditions in terms of θs − <sup>θ</sup>i—slightly decreasing from 0YR to 5YR treatments—were observed among the three plots (see Table 3).

Adopting the parameters of Table 3, the Philip model has been applied in the case of an irrigation process of duration 1.5 h and rate su fficiently high to determine quickly the saturation of soil surface in the three plots with di fferent earlier irrigation treatments. The results obtained are shown in Figure 3 in terms of cumulative infiltration. The simulated cumulative infiltrations (Figure 3) at the irrigation end for the 0YR, 2YR, and 5YR plots were 101, 61, and 54 mm, respectively. Statistical analysis, involving one-way ANOVA test at the probability level p < 0.001 and Tukey's Honestly Significant Di fference test at level p < 0.05, has shown that there are no significant di fferences between the two plots irrigated with TWW (2YR and 5YR treatments), while the cumulative infiltration in these two plots is significantly lower than that of the control plot (0YR). This significant decrease in cumulative infiltration in the 2YR and 5YR TWW irrigated soils could be explained by the high load of suspended solids present in the TWW. When soil is irrigated by TWW, these suspended materials settle in the smaller pores. With time, the micropores as well as the mesopores become smaller and disconnected producing a significant reduction in infiltration rate and cumulative infiltration. In agreemen<sup>t</sup> with this conclusion, Viviani and Iovino [19] showed a reduction in soil porosity which led to a decrease in soil infiltration rate as a result of pore-clogging due to the use of TWW. Similarly, Bardhan et al. [13] concluded that suspended solids loaded in the TWW reduced soil infiltrability due to pore-clogging.

**Figure 3.** Cumulative infiltration curves obtained by Philip's model for the three experimental plots subjected to different irrigation treatments (0YR, 2YR, and 5YR).

The aforementioned results can have relevant implications in arid and semi-arid regions where water is limited for the majority of irrigation farms. In this context, water use efficiency is an important issue to be considered because with its increase more crops can be irrigated. Based on our outcomes, for a fixed irrigation pattern able to produce approximately immediate soil surface ponding, the use of TWW reduces gradually in time the amount of water entering the soil and increases runoff. To better specify this element, Figure 4 shows the reduction in time of water amount that the soil can absorb in 2YR and 5YR plots with respect to the control plot (0YR) for irrigation duration up to 3 h. From this figure it can be observed that for an irrigation period of 1.5 h the water amount absorbed by the soil in 2YR and 5YR plots is reduced by 41 mm and 47 mm, respectively; for an irrigation stage of 3 h, the infiltrated water depth decreases by 58 and 68 mm, respectively. The water amount that the soil is not able to absorb becomes surface runoff. This implies that for an irrigation scheduled time a reduced water amount is requested in these two plots to avoid an increased runoff. The reduction of absorbable water is here interpreted as gained water and expressed in percentage, with respect to the water infiltrated in the control plot, is defined as irrigation efficiency. The trend of this quantity in the function of irrigation duration is represented in Figure 5, which highlights a decreasing advantage with increasing irrigation periods. For durations ranging from 30 min to 180 min, the irrigation efficiency tends to decrease from 50% to 44% and from 48% to 38% for 5YR and 2YR plots, respectively. In any case, the above values show how after just 2 years the efficiency is significantly increased and after a further 3 years, it becomes almost 50%.

**Figure 4.** Reduction of water amount absorbable for periods of irrigation up to 3 h in 2YR and 5YR plots if compared with the benchmark plot (0YR).

**Figure 5.** Irrigation efficiency obtained in 2YR and 5YR plots for periods of irrigation up to 3 h in terms of cumulative infiltration reduction if compared with the benchmark plot (0YR).
