## 2 Materials and Methods
### Batch Experiments  
Batch leaching experiments were carried out doing parallel batch tests with different L/S ratios, as proposed by [PERSON] et al. (2020), to better understand the desorption dynamics of the soil-water system. Batch tests were performed using the SSE method described by [PERSON] and [PERSON] (1966), with the use of deionized water as a rainwater proxy (EC 15 \(\pm\) 2 \(\upmu\)S/cm, pH 6.84 \(\pm\) 0.01, ORP 154 \(\pm\) 11 mV). The contact time was set to 48 h. Each batch was replicated with four different L/S ratios (1, 2, 5, and 10) and for each configuration three replicas were made. Each experiment was performed in HDPE containers with rectangular base (3.9 \(\times\) 4.5 cm) and height of 8.5 cm, closed by a double stopper to isolate the content. Batches were stored under temperature-controlled condition (25 \(\pm\) 0.5 \({}^{\circ}\)C) in the dark to avoid algal proliferation. After preparation, the batches were shaken, mixed, and left stand unstirred, since a strong artificial shaking might cause the change of leaching behavior due to the abrasion of particles, the destruction of aggregates, and the mobilization of colloids ([PERSON], 2013).  
Batches were monitored with a HANNA Instruments multiparametric probe (HI98194) to record pH, electrical conductivity (EC), and oxidation-reduction potential (ORP) after 1, 12, 24, and 48 h. At the end of the experiment, water samples were collected and filtered with a 0.45 \(\upmu\)m polypropylene filter to analyze dissolved organic carbon (DOC) and NO3\({}^{-}\) via a Pharmacia Biotech Ultrospec 2000 UV/VIS spectrophotometer following the procedure of [PERSON] et al. (2014). Ca and Mg were analyzed with an ICP-OES (PerkinElmer, Waltham, USA) after acidification of the samples with ultrapure 1 M nitric acid and filtering on 0.45 \(\upmu\)m. Dissolved inorganic carbon (DIC) was determined by back-titration method using pH 3.5 as end point ([PERSON], 2013).  
One-way analysis of variance (ANOVA), computed with STATGRAPHICS v.18, was applied to calculate significant differences between treatments and between L/S ratios for the same treatment.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **Sand** & **Silt** & **Clay** & **Carbonates** & **pH** \\ \hline CS & 96.8 \(\pm\) 2.4 & 2.5 \(\pm\) 1.6 & 0.7 \(\pm\) 1.4 & 87.9 \(\pm\) 4.2 & 7.8 \(\pm\) 0.2 \\ SS & 96.9 \(\pm\) 0.2 & 2.7 \(\pm\) 0.1 & 0.4 \(\pm\) 0.2 & 5.3 \(\pm\) 0.6 & 8.1 \(\pm\) 0.2 \\ Compost & 90.6 \(\pm\) 2.0 & 8.8 \(\pm\) 1.0 & 0.5 \(\pm\) 0.1 & 0.2 \(\pm\) 0.03 & 6.7 \(\pm\) 0.1 \\ Biochar & 80.6 \(\pm\) 5.5 & 4.9 \(\pm\) 1.7 & 14.3 \(\pm\) 4.6 & 2.1 \(\pm\) 0.6 & 9.9 \(\pm\) 0.1 \\ Zeolites & 3.2 \(\pm\) 0.3 & 7.7 \(\pm\) 3.4 & 89.2 \(\pm\) 3.2 & 0.1 \(\pm\) 0.02 & 8.1 \(\pm\) 0.1 \\ Graphene & 0.0 \(\pm\) 0.0 & 0.0 \(\pm\) 0.0 & 100.0 \(\pm\) 0.0 & 0.0 \(\pm\) 0.0 & 7.1 \(\pm\) 0.1 \\ \hline \end{tabular}
\end{table}
Table 1: Grain size distribution, carbonates content expressed in percentage over dry weight, and pH for the CS and SS soils, and for the four improvers. Standard deviation from triplicate samples is also reported.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Compost** & **Zeolites** & **Biochar** & **Graphene** \\ \hline Type & Cow manure & \begin{tabular}{c} Clinoptiolite \\ vermicomost \\ \end{tabular} & \begin{tabular}{c} Wood \\ 67.5\% Mordenite \\ 32.5\% \\ \end{tabular} & \begin{tabular}{c} \\ bioschar \\ \end{tabular} & \begin{tabular}{c} GR006050 \\ \end{tabular} \\ Manufacturer & LTDG® & \
\begin{tabular}{c} SBM LIFE \\ SCIENCE® \\ \end{tabular} & BioDea® &
\begin{tabular}{c} Directa Plus® \\ \end{tabular} \\ Mix (\% d.w.) & 1.8 & 0.9 & 4.5 & 0.015 \\ \hline \end{tabular}
\end{table}
Table 2: Type of soil improvers used in this study and their percentage in the batch experiments (as dry weight).