## 2 Previous Models  
### Conservation Laws  
Combustion in a forest fire is a physical-chemical process that, as such, can be represented by conservation laws. These laws can be written in general as PDEs having the following form:  
\[\frac{\partial}{\partial t}(\rho\phi)+\
abla(\rho\phi\mathbf{v})-\
abla(K\
abla \phi)=S_{\phi} \tag{1}\]  
where \(\phi\) is a generic magnitude, \(\rho\) represents the density, \(\mathbf{v}=(v_{x},v_{y},v_{z})\) is the velocity, \(K\) is the diffusion coefficient, and \(S_{\phi}\) is the source term. Initially, we considered a three-dimensional problem, that is \(\
abla=(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial }{\partial z})\).  
A first paramount simplification to achieve an efficient simulation model from the computational point of view is to reduce the three-dimensional problem to a two-dimensional one. Equation (1) describes the evolution of the corresponding magnitude in a three-dimensional domain, which represents a bed of fuel of thickness \(\delta z\). We considered the following boundary conditions: homogeneous Dirichlet on the lateral boundary of the three-dimensional domain, homogeneous Neumann on the lower boundary \((z=0)\), and non-homogeneous Neumann on the upper boundary \((z=\delta z)\):  
\[K\frac{\partial\phi}{\partial\eta}\Big{|}_{\Gamma_{z=z}}=H_{\phi}.\]  
Denoting  
\[\overline{\phi}(x,y)=\int_{0}^{\delta z}\phi(x,y,z)\,\mathrm{d}z\]  
and integrating Equation (1) with respect to the variable \(z\), we can reduce the problem to two dimensions in space.  
\[\frac{\partial}{\partial t}(\rho\overline{\phi})+\
abla(\rho\overline{\phi} \mathbf{v})-\
abla(K_{\phi}\
abla\overline{\phi})=\overline{S}_{\phi}+H_{ \phi}, \tag{2}\]  
where, now, \(\
abla=(\frac{\partial}{\partial x},\frac{\partial}{\partial y})\) and \(\mathbf{v}=(v_{x},v_{y})\).  
We assumed that, \(v_{z}=0\) away from the fire, the vertical variation of the corresponding magnitude is null near the fire, \(K_{\phi}\) is not dependent on \(z\), and the thickness of the three-dimensional domain \(\delta z\) is small enough to suppose that \(\overline{S}_{\phi}\approx S_{\phi}\).  
Another aspect that adds complexity to an eventual fire spread simulation model is the fact that combustion in a forest fire is a positive feedback system, so we can distinguish two phases, an endothermic phase or solid phase, in which heat is used to release the volatile substances, and an exothermic phase or gas phase, in which the volatile substances mixed with oxygen react, producing more heat (see [19]). Bearing this in mind, a two-phase model can be proposed, with an energy conservation equation for each phase and mass conservation equations for solid fuel, gaseous fuel, and oxygen [20]. A two-phase model poses the drawback of correctly evaluating nonlinear terms. The correct evaluation of these terms would lead to introducing more elaborate models of turbulence, leading to much more complex systems, moving away from the goal of designing an efficient computational model.  
Each of the models that we reviewed depends on the simplifying hypotheses raised in each case. All of them attempt to represent the main heat transfer mechanisms in a wildfire, namely radiation and convection [21]. The predominant mode of heat transfer will depend on the wind conditions, terrain slope, fuel type, and location relative to the fire.