**3. Description of the AquaCrop Model**

The model describes soil, water, crop, and atmosphere interactions through four sub-model components: (i) the soil with its water balance; (ii) the crop (development, growth, and yield); (iii) the atmosphere (temperature, evapotranspiration, and rainfall), and carbon dioxide (CO2) concentration; and (iv) the management, such as irrigation and crop fertilization soil fertility.

The AquaCrop model is based on the relationship between the relative yield and the relative evapotranspiration [22] as follows

$$\frac{\mathbf{Y\_{\bar{X}} - \mathbf{Y\_{\bar{a}}}}}{\mathbf{Y\_{\bar{X}}}} = \mathbf{K\_{\mathcal{Y}}} \left( \frac{\mathbf{ET\_{\bar{X}} - \mathbf{ET\_{\bar{a}}}}{\mathbf{ET\_{\bar{X}}}} \right) \tag{1}$$

where Yx is the maximum yield, Ya is the actual yield, ETx is the maximum evapotranspiration, ETa is the actual evapotranspiration, and Ky is the yield response factor between the decrease in the relative yield and the relative reduction in evapotranspiration.

The AquaCrop model does not take into account the non-productive use of water for separating evapotranspiration (ET) into crop transpiration (T) and soil evaporation (E)

$$\text{ET} = \text{E} + \text{Tr} \tag{2}$$

where ET = actual evapotranspiration, E = soil evaporation and Tr = the sweating of crop.

At a daily time step, the model successively simulates the following processes: (i) groundwater balance; (ii) development of green canopy (CC); (iii) crop transpiration; (iv) biomass (B); and (v) conversion of biomass (B) to crop yield (Y). Therefore, through the daily potential evapotranspiration (ETo) and productivity of water (WP\*), the daily transpiration (Tr) is converted into vegetal biomass as follows

$$\mathbf{B}\_{\mathbf{i}} = \text{WP}^{\star}\_{\left(\frac{\mathbf{T}\_{\text{ri}}}{\text{ETo}\_{\mathbf{i}}}\right)} \tag{3}$$

where WP\* is the normalized water productivity [32,33] relative to Tr. After the normalization of water productivity for different climatic conditions, its value can be converted into a fixed parameter [34]. The estimation and prediction of performance are based on the final biomass (B) and harvest index (HI). This allows a clear distinction between impact of stress on B and HI, in response to the environmental conditions

$$
\mathbf{Y} = \mathsf{HI} \* \left(\mathbf{B}\right) \tag{4}
$$

where: Y = final yield; B = biomass; HI = harvest index.

During the calibration and testing of the model, we calculated water productivity (WP) as presented by Araya et al. [5]

$$\text{W}\text{P} = \begin{bmatrix} \text{Y} \\ \frac{\sum \text{Tr}}{\sum \text{Tr}} \end{bmatrix} \tag{5}$$

where Y is the yield expressed in kg ha−<sup>1</sup> and Tr is the daily transpiration simulated by the model.
