# Two-Dimensional Modelling Approach for Electrokinetic Water Transport in Unsaturated Kaolinite
## 2 Materials and Methods
### Description of the Simulation Exercises  
An extensive series of simulation exercises were carried out to analyse the capabilities of the presented numerical tool. All simulations were performed using a two-dimensional domain (50 cm wide by 70 cm long), a mid-plane of the mock-up installation used in previous works [35; 36; 39; 50]. Figure 1 shows a scheme of all the electrode configurations simulated in this work. First, the electrohydraulic behaviour was analysed as a function of the distribution of the electrodes in the soil to be treated. For this purpose, simulations were carried out using a simple electrode configuration with a single anode facing a cathode under two different approaches: both inside electrolyte reservoirs (Figure 1a) or both directly inserted into the soil (Figure 1b). In the former approach, it was assumed that the electrolyte has a high electrical conductivity, and therefore, potential losses between the electrode and the soil can be neglected. Table 2 shows the initial and boundary conditions for the simulation approach. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Description** & **Unit** & **Value** \\ \hline \(\alpha\) * & [PERSON] parameter & kPa\({}^{-1}\) & \(2.39\times 10^{-3}\) \\ \(n\) * & [PERSON] parameter & - & 1.8423 \\ \(m\) * & [PERSON] parameter & - & 0.4572 \\ \(a_{\text{A}}\) ** & [PERSON] parameter & \(\Omega\)m & 2.356 \\ \(m_{\text{A}}\) ** & Archie parameter & - & 5.639 \\ \(B\) ** & Saturation index & - & 2.744 \\ \(\rho_{\text{s}}\) & Density of mineral particles & kg m\({}^{-3}\) & 2650 \\ \(w_{\text{ini}}\) & Initial water content & - & 0.25 \\ \hline \end{tabular}
* From [31]; ** From [47].  
\end{table} 
Table 1: Characterisation parameters for the modelled kaolinite. \begin{table}
\begin{tabular}{c c c c} \hline  & & **Electrodes** & **Electrodes into the Soil** & **Electrolyte Reservoirs** \\ \hline  & & **Anode** & **Cathode** & **Analytic Reservoir** \\ \hline \multirow{2}{*}{Initial cond.} & Electrical (V) & \(\psi=0\) V & \(\psi=0\) V \\  & Hydraulic (kPa) & \(P_{\text{L}}=P_{\text{L, ini}}\) & \(P_{\text{L}}=P_{\text{L, ini}}\) \\ \hline Boundary & Electrical (V) & \(\psi=\psi_{\text{anode}}\) & \(\psi=0\) V & \(\psi=\psi_{\text{anode}}\) & \(\psi=0\) V \\ cond. & Hydraulic (kPa) & No flux & \(P_{\text{L}}=P_{\text{atm}}=100\) kPa \\ \hline \multirow{2}{*}{Electrode/reservoir radius} & \multirow{2}{*}{0.005 m} & \multirow{2}{*}{0.02 m} \\  & Electrode/reservoir height & & 0.1 m \\ \hline \end{tabular}
\end{table} 
Table 2: Initial and boundary conditions. The hydraulic initial condition is given by the initial value of the state variable of the water mass balance equation: the liquid pressure. The initial water content, the void ratio,  
Figure 1: Schemes of the simulated electrode configurations. (**a**) opposed electrodes in electrolyte reservoirs; (**b**) opposed electrodes inserted into the soil; (**c–n**) alternative distributions of electrodes inserted in electrolyte reservoirs. A in red is for anode(s) and C in blue is for cathode(s). and the density of solid particles are known, and therefore, with this information, the initial degree of saturation can be calculated as:  
\[Sr_{\text{ini}}=\frac{w_{\text{ini}}\rho_{\text{s}}}{e\rho_{\text{w}}} \tag{19}\]  
Based on the definition of \(Sr\), it is also possible to obtain the initial matric suction by rearranging the van Genuchten expression of the water retention curve, Equation (18), as  
\[s_{\text{ini}}=\frac{\left(Sr_{\text{ini}}^{-1/m}-1\right)^{1/n}}{\alpha} \tag{20}\]  
Assuming that \(P_{\text{G}}\) is constant, and equal to atmospheric pressure (\(P_{\text{atm}}\)), the initial liquid pressure, \(P_{\text{L, ini}}\) can be calculated by the expression:  
\[P_{\text{L, ini}}=P_{\text{atm}}-s_{\text{ini}} \tag{21}\]  
The initial electric potential gradient is zero in all simulations as the soil is initially electrically balanced.