## 2 Previous Models
### The Effect of Moisture Content: A Multivalued Operator in Enthalpy  
The third improvement to the model was intended to reflect the effect of fuel moisture. The effect of the moisture content of the solid fuel was included through a multivalued maximal monotone operator relating enthalpy and temperature. The use of a multivalued operator was informed by the classical two-phase Stefan problem [27], and it was adapted to model the two well-defined phases in a wildfire combustion process [19]: the endothermic phase, which includes the dehydration of the solid fuel, and the exothermic phase, in which the flammable mixture from fuel pyrolysis begins to release energy. The Fuel Moisture Content (FMC) is one of the most-influential factors in fire spread, mainly through the process of heating and subsequently evaporating the water in the fuel, enabling it to attain combustion conditions. This process involves the consumption of the energy released by the adjacent combustion fuel and requires time, which reduces the fire Rate Of Spread (ROS)as moisture increases. The non-dimensional equations of this new model that appeared in [28] are, keeping the previous notation,  
\[\frac{\partial}{\partial t}e+\mathbf{w}\cdot\
abla e-\
abla(K(u)\
abla u)+\alpha u =f(u,c), \tag{11}\]  
\[e\in G(u), \tag{12}\]  
\[\frac{\partial}{\partial t}c=-\frac{\varepsilon}{q}f(u,c), \tag{13}\]  
where \(u\) and \(c\) are again the dimensionless temperature and mass fraction of solid fuel, respectively, and \(e\) is the dimensionless enthalpy.  
The change of the Frank-Kamenetskii variable is maintained since an Arrhenius-type expression continues to appear in the reactive term. The diffusive term function \(K(u)\), and the reactive term function \(f(u,c)\) are the same as in the previous model. The multivalued operator is given by  
\[G(u)=\left\{\begin{array}{lcl}u&\text{if}&u<u_{v},\\ \left[u_{v}\,\ u_{v}+\lambda_{v}\right]&\text{if}&u=u_{v},\\ u+\lambda_{v}&\text{if}&u_{v}<u<u_{p},\\ \left[u_{p}+\lambda_{v}\,\ u_{p}+\lambda_{v}+\lambda_{p}\right]&\text{if}&u=u_{p}, \\ u+\lambda_{v}+\lambda_{p}&\text{if}&u>u_{p},\end{array}\right. \tag{14}\]  
where \(u_{v}\) is the dimensionless water evaporation temperature and \(u_{p}\) is the dimensionless solid fuel pyrolysis temperature. The quantity \(\lambda_{v}\) is the dimensionless evaporation heat related to the latent heat of evaporation \(\Lambda_{v}\) and the fuel moisture content \(M_{v}\) (kg of water/kg of dry fuel); \(\lambda_{p}\) is the dimensionless pyrolysis heat related to the latent heat of pyrolysis \(\Lambda_{p}\).  
It is worth noting that, in the burnt area, the multivalued operator does not exactly represent the physical phenomena since water vapor is no longer in the porous medium. This inconvenience can be avoided by setting \(\lambda_{v}=0\) and \(\lambda_{p}=0\) in the burnt area.  
The maximal monotone property of this multivalued operator allows the implementation of a numerical algorithm with good convergence properties based on the use of duality methods [29]. Given an exact perturbation of the multivalued operator, its properties, and an appropriate choice of the parameters, we can define the resolvent and its Yosida approximation, whereby the new nonlinear univalued operator equivalent to the multivalued one can be solved by a fixed-point iteration. For further details of how to numerically treat this multivalued operator, see [30; 31], although Section 3.1 provides updated explanations about how the multivalued operator is numerically solved in the current model.