## 2 Theoretical Background
### Hydrus 1D and 2D Simulation Codes  
Accurate and continuous estimations of SWC dynamics can be obtained with deterministic soil-water flux models, such as those solving the Richards equation [31; 32]. In this context, the Hydrus model [33] simulates the movement of water in variably saturated soil using the Richards equation, that in the mono-dimensional dimension can be written as:  
\[\frac{\partial SWC}{\partial t}=\frac{\partial}{\partial x}\bigg{[}K\bigg{(} \frac{\partial h}{\partial x}+\cos\alpha\bigg{)}\bigg{]}-S \tag{7}\]  
where \(h\) [m] is the water pressure head, \(SWC\) [m\({}^{3}\)m\({}^{-3}\)] is the volumetric soil water content, \(t\) [s] is time, \(x\)[m] is the spatial coordinate (positive upward), \(\alpha\) is the angle between flow direction and the vertical axis (\(\alpha\) = 0\({}^{\circ}\) for vertical flow, 90\({}^{\circ}\) for horizontal flow and 0\({}^{\circ}\) < \(\alpha\) < 90\({}^{\circ}\) for the inclined flow), \(S\) [m\({}^{3}\)m\({}^{-3}\) s\({}^{-1}\)] is the sink term and \(K\) [ms\({}^{-1}\)] is the unsaturated hydraulic conductivity function.  
In order to simulate soil water movement and solute transport in two-dimensional directions, Hydrus-2D allows solving the governing equations of water movement by considering a bi-dimensional vertical plane or three dimensional regions exhibiting radial symmetry around the vertical axis. Under these conditions, Richards equation can be rewritten in its two-dimensional form as:  
\[\frac{\partial SWC}{\partial t}=\frac{\partial}{\partial x_{i}}\bigg{[}K\bigg{(} K_{i,j}^{A}\frac{\partial h}{\partial x}+K_{i,j}^{A}\bigg{)}\bigg{]}-S \tag{8}\]  
where, \(x_{i}\) and \(x_{j}\) [m] are the spatial coordinates, and \(K_{i,j}^{A}\) are the components of a dimensionless anisotropy tensor \(K^{A}\).  
The model requires the knowledge of the soil water retention curve, \(SWC(h)\) and the hydraulic conductivity function, \(K(h)\), that can be expressed based on the van Genuchten-Mualem model [34; 35]:  
\[SWC=SWC_{r}+(SWC_{s}+SWC_{r})\frac{1}{\big{[}1+\left|\alpha h\right|^{n}\big{]} ^{m}} \tag{9}\]  
\[K(SWC)=K_{s}\bigg{[}\frac{(SWC-SWC_{r})}{(SWC_{s}-SWC_{r})}\bigg{]}^{\lambda} \bigg{[}1-\bigg{(}1-\bigg{(}\frac{(SWC-SWC_{r})}{(SWC_{s}-SWC_{r})}\bigg{)}^{ \frac{1}{m}}\bigg{)}^{m}\bigg{]}^{2} \tag{10}\]  
where, \(SWC_{r}\) and \(SWC_{s}\) [m\({}^{3}\) m\({}^{-3}\)] are the residual and saturated soil water content, \(K_{s}\) [m s\({}^{-1}\)] is the saturated hydraulic conductivity, \(\alpha\) [m\({}^{-1}\)], \(n\) [-], \(m\) [-] and \(\lambda\) [-] are fitting parameters.  
The sink term (S), representative of actual root water uptake (volume of soil water removed per unit of time), can be described as a fraction \(\alpha_{s}\) (\(h\)) (0 \(\leq\alpha\) (\(h\)) \(\leq\)1) of the potential water uptake rate, \(S_{p}\) [m\({}^{3}\) m\({}^{-3}\) s\({}^{-1}\)] [36]. Actual root water uptake is assumed to be zero, \(\alpha_{s}\) (\(h\)) = 0, when the soil-water content is equal to the wilting point, whereas it is maximum, \(\alpha_{s}\) (\(h\)) = 1, in the absence of crop water-stress, so that the soil water status is not limiting the plant growth [37; 38].  
The Hydrus model was applied to assess SWC during both a laboratory experiment and a field application (see Section 3).