# Two-Dimensional Modelling Approach for Electrokinetic Water Transport in Unsaturated Kaolinite
## 2 Materials and Methods  
### Mathematical Model  
#### 2.1.1 Water Transport  
The temporal and spatial distribution of the mass of water contained in a partially saturated soil was obtained by solving a standard mass conservation equation, as shown below:  
\[\frac{\partial m_{\rm w}}{\partial t}+\
abla\cdot\mathbf{l}_{\rm w}=0 \tag{1}\]  
where \(m_{\rm w}\) is the mass of water per unit of total volume, \(\mathbf{l}_{\rm w}\) is the mass flux of water, and \(\
abla\cdot\) is the divergence operator.  
The mass of water per unit of total volume is defined as:  
\[m_{\rm w}=\phi Sr\rho_{\rm w} \tag{2}\]  
where \(\phi\) is the porosity of the soil, \(Sr\) is the degree of soil saturation, and \(\rho_{\rm w}\) is the water density. In a general form, the mass flux of water in this system, \(\rm I_{w}\), is the sum of the hydraulic flux, \(\rm I_{w}^{h}\), and the electroosmotic flux, \(\rm I_{w}^{eo}\), defined by the following expressions:  
\[\rm I_{w}=\rm I_{w}^{h}+\rm I_{w}^{eo} \tag{3}\]  
\[\rm I_{w}^{h}=\rho_{\rm w}\rm q_{w}^{h} \tag{4}\]  
\[\rm I_{w}^{eo}=\rho_{\rm w}\rm q_{w}^{eo} \tag{5}\]  
where \(\rm q_{w}^{h}\) and \(\rm q_{w}^{eo}\) are the hydraulic and electroosmotic volumetric fluxes, respectively. The hydraulic volumetric flux is defined using [PERSON]'s law (without including the effect of gravity since the two-dimensional domain is a horizontal plane), which considers the liquid pressure gradient as the driving force for the transport of water by this phenomenon:  
\[\rm q_{w}^{h}=-\frac{K^{h}}{\rho_{\rm w}g}\
abla P_{L} \tag{6}\]  
where \(\
abla\) is the gradient differential operator, \(g\) is the gravitational constant, \(P_{L}\) is the liquid pore pressure, and \(K^{h}\) is the effective hydraulic permeability of the porous media. The electroosmotic volumetric flux is calculated as:  
\[\rm q_{w}^{eo}=-K^{eo}\
abla\psi \tag{7}\]  
where \(\psi\) is the electric potential and \(K^{eo}\) is the effective electroosmotic permeability of the porous media. Both effective permeabilities are calculated as the product of a saturated conductivity (\(K^{h}_{\rm sat}\) for the hydraulic conductivity and \(K^{eo}_{\rm sat}\) for the electroosmotic conductivity) and a relative permeability function, \(k_{\rm rel}\), expressed as a Brooks and Corey [45] type power function with an exponent of 3. In this case, the driving force for water transport by electroosmosis is the electric potential gradient.  
To simulate the temporal evolution of the volume from electrolyte reservoirs, additional water balance equations in these compartments were formulated using the following expression:  
\[\rm\frac{\partial V_{R}}{\partial t}=\overline{q}_{R}H_{R} \tag{8}\]  
where \(H_{R}\) is the height of the electrolyte reservoir and \(\overline{q}_{R}\) is calculated by means of the closed line integral:  
\[\rm\overline{q}_{R}=\oint\rm\mathbf{q}_{w}\cdot\mathbf{n}\;\textit{ds} \tag{9}\]  
where \(\mathbf{n}\) is the outwards vector normal to the reservoir boundary (reservoir wall) and \(\rm\mathbf{q}_{w}\) can be defined as:  
\[\rm\mathbf{q}_{w}=\rm\mathbf{q}_{w}^{h}+\rm\mathbf{q}_{w}^{eo} \tag{10}\]