## 2 Materials and Methods
### Numerical Investigations--Set Up  
The numerical model for the supporting simulations under variable conditions is conceptualized based on the column experiment with a similar spatial configuration (Figure 1). A one-dimensional (1D) model has been realized using HYDRUS 1D (V 4.17.0140, Co. PC-Progress, Prague, CZE, [21]), applying the Richards equation [30] for unsaturated flow with Mualem-van-Genuchten retention and relative hydraulic conductivity model [31,32]. The discretization of the space domain led to 146 mesh elements with a finer discretization at the topsoil. The TWWE and/or precipitation infiltration was implemented as a variable, time-dependent flux boundary condition (BC) on top of the column. The infiltration rates had been adopted from the experiments. The lower boundary condition was assumed to be a free drainage BC. Two materials with specific hydraulic properties were implemented in the model. The first one represented individual soil types in each column, and the other one used glass beads at the bottom of all the columns. Soil retention functions and hydraulic conductivities of these two materials were defined using the initial parameters extracted from laboratory tests (Table 1). An inverse analysis based on observed water contents during joint infiltration of TWWE and precipitation events was performed by HYDRUS internally to optimize the initial parameters (Table 1) and to determine a final parameter set for each soil. Inverse methods are typically based upon the minimization of a suitable objective function, which expresses the discrepancy between the observed values and the predicted system responses. The system response was represented here by the numerical solution of the Richards equation, augmented with single porosity hydraulic model (Mualem-van-Genuchten) and mentioned boundary conditions. Initial estimates were then iteratively improved during the minimization process until the desired degree of precision was obtained. Here, water content observed by FDR probes with every 10 min interval during experimental runtime considered as an observed value. For the optimization of the objective function, a weighted least-squares approach based on the Levenberg-Marquardt nonlinear method is used in HYDRUS 1D [33]. The final parameter set is presented in Table 5; noticeable differences in residual and saturated water content can be identified as compared to the laboratory data. The pore connectivity parameter in the conductivity function was defined as l = 0.5 for all soils, following [31] respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Soil 1** & **Soil 2** & **Soil 3** & **Soil 4** \\ \hline Hydraulic Conductivity under full saturation \(K\) (cm day\({}^{-1}\)) & 360 & 504 & 3939 & 2651 \\ Bulk density \(\rho_{b}\) (g cm\({}^{-3}\)) & 1.68 & 1.70 & 1.67 & 1.65 \\ Van Genuchten shape parameter \(\alpha\) (cm\({}^{-1}\)) & 0.03 & 0.035 & 0.049 & 0.045 \\ Van Genuchten shape parameter \(n\) (-) & 2.58 & 1.96 & 2.75 & 2.05 \\ Residual water content \(\theta_{\tau}\) (cm\({}^{3}\) cm\({}^{-3}\)) & 0.07 & 0.005 & 0.021 & 0.005 \\ Saturated water content \(\theta_{8}\) (cm\({}^{3}\) cm\({}^{-3}\)) & 0.27 & 0.25 & 0.32 & 0.25 \\ Pore connectivity parameter in the conductivity function \(l\) (-) & 0.5 & 0.5 & 0.5 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Final parameter set for the used soil materials based on inverse solution, the value for the pore connectivity parameter in the conductivity function; here, 0.5 was selected after [31]. The total simulation time for the scenarios with joint infiltration of TWWE and additional precipitation was 10,080 min (7 days). The precipitation events in the scenarios started for event 1 after 5230 min (Soil 1, 2 and 3)/5080 min (soil 4), for event 2 after 5040 min (soil 1, 2 and 3)/5140 min (soil 4) and for event 3 after 3900 min (soil 1, 2 and 3)/3700 min (soil 4). In general, they all started between morning and evening TWWE infiltration and lasted according to the event duration mentioned in Table 4.