## 2 Materials and Methods  
### Mathematical Model Development and Implementation  
A mathematical model was developed comprising the description of transport and transformation of linear saturated petroleum hydrocarbons in a flooded soil column. The model assumed mass conservation for the chemical species, with mass partition between solid and liquid phases. Hydrocarbons in the liquid phase may be dissolved in the soil moisture or surrounded by other hydrocarbon molecules, forming a non-aqueous phase liquid (NAPL) ([PERSON] and [PERSON], 2016). In this model, these two possibilities were considered together, i.e., hydrocarbon concentration in the liquid phase refers to the mass of both dissolved and NAPL forms per unit of liquid volume. Both forms are considered to be available for microorganisms' growth. In fact, other authors have already showed that microorganismscan directly access hydrocarbons in the NAPL phase and the hydrocarbon-water interphase and use it for growth [14]. The model was implemented in AQUASIM [24], a software for the analysis and simulation of aquatic systems, which allows to define the spatial configuration of the system as a set of compartments. Another advantage of AQUASIM is the fact that it allows the definition of new variables and processes at any time [25]. In the present case, we used the saturated soil column (with sorption and pore volume exchange) compartment [24]. Advective dispersive transport of substances present in the liquid phase was considered (first and second terms of Equation (1), respectively), as well as transformation (third term of Equation (1)) by adsorption/desorption and biodegradation processes [26; 27]. Biodegradation was assumed to occur only in the liquid phase. \[\frac{\partial C_{mobj}}{\partial t}=-\frac{1}{A\theta}\frac{\partial}{ \partial x}(QC_{mobj})+\frac{1}{A\theta}\frac{\partial}{\partial x}\bigg{(} A\theta E\frac{\partial C_{mobj}}{\partial x}\bigg{)}+rC_{mobj} \tag{1}\]  
\(C_{mobj}\)_-concentration of component \(i\) in the liquid phase; \(t\)--time; \(A\)--soil column cross-sectional area; \(\theta\)--porosity; Q--water flow through the column, \(E\)--longitudinal dispersion coefficient; \(r\)--reaction term. The model is able to calculate phase concentrations and fluxes as a function of space (soil column depth) and time for the different components. In the present model, hydrocarbons biodegradation processes were based on the availability of different electron acceptors (Figure 1). Briefly, the presented mathematical model considered aerobic and anoxic biodegradation of total petroleum hydrocarbons (TPH), the latter occurring in the presence of iron (III), nitrate or sulfate as electron acceptors. Anaerobic biodegradation of hydrocarbons to methane, which occurs in the absence of any electron acceptor other than bicarbonate/CO\({}_{2}\), was also considered. Hydrocarbon biodegradation to methane is generally performed by complex microbial communities, where close syntrophic relationships between bacteria and methanogenic archaea have been reported as essential [28; 29]. Complete hydrocarbon degradation to methane is thermodynamically limited by the activity of the methanogens [29]. Moreover, within anaerobic microbial communities, methanogens are generally the most sensitive and slow-growing microorganisms [30]. Therefore, in this  
Figure 1: Schematic representation of hydrocarbon biodegradation processes considered in the mathematical model. work, petroleum hydrocarbon biodegradation to methane was modelled in only one step, according with the reaction presented in Table S1, admitting that methanogenesis is the limiting step. The model also considered the adsorption/desorption of hydrocarbons, nitrate and sulfate to soil particles as transformation processes. The state variables and the processes rate equations of the developed model are given in Tables 1 and 2, respectively. In Annex B of Supporting Information, the full AQUASIM system implementation of the present mathematical model can be consulted.