2.1. 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:
where
is a generic magnitude,
represents the density,
is the velocity,
K is the diffusion coefficient, and
is the source term. Initially, we considered a three-dimensional problem, that is
.
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
. We considered the following boundary conditions: homogeneous Dirichlet on the lateral boundary of the three-dimensional domain, homogeneous Neumann on the lower boundary
, and non-homogeneous Neumann on the upper boundary
:
Denoting
and integrating Equation (
1) with respect to the variable
z, we can reduce the problem to two dimensions in space.
where, now,
and
.
We assumed that, away from the fire, the vertical variation of the corresponding magnitude is null near the fire, is not dependent on z, and the thickness of the three-dimensional domain is small enough to suppose that .
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.
2.2. First Model: Turbulent Diffusivity
The first simplified physical model for wildfire spread simulation proposed was a two-dimensional one-phase model considering turbulent flow, vertical heat lost, and a convective term representing the effect of wind [
22]. The original equations are
where
T is the average value of temperature and
M is the solid fuel load.
K is the turbulent diffusivity.
is the vertical convection heat transfer.
is the ambient temperature. The reaction rate is given by the Arrhenius law.
R is the universal gas constant.
represents the activation energy.
A is the pre-exponential or frequency factor of the reaction. We assumed that the activation energy for the gaseous phase is much lower than the one of the solid phase in order to propose a simplified one-phase chemical reaction.
Q is the global heat production in the reaction.
The PDEs to be solved are a non-dimensional version of Equations (
3) and (
4), based on the following change of variables: the dimensionless temperature is
, where
is a temperature up to which generalized combustion takes place; the dimensionless solid fuel is
, where
the initial solid fuel load. Dimensionless time and space variables were also introduced, although, for the sake of simplicity, we kept the same notation. The non-dimensional equations are
where
is the dimensionless wind velocity field,
is related to the turbulent diffusivity,
is the normalized vertical convection coefficient,
q depends on the heat released, and
is written in terms of the previous exponential expression.
The non-dimensional equations were completed with the initial and boundary conditions. We propose the Dirichlet boundary conditions assuming the domain is big enough to ensure that the fire does not arrive at the boundary. The initial conditions represent the initial fuel load and fire source.
The first numerical method proposed was an implicit and upwind finite difference scheme for the energy Equation (
5), assuming that the wind field velocity and diffusivity are constant along the domain. The matrix of the linear system obtained in each time step is block tridiagonal and strictly diagonally dominant, so the blockwise Gauss–Seidel method is convergent. The global Lipschitz condition that the source term satisfies ensures the stability of the scheme. An implicit Euler method was the scheme proposed for the fuel Equation (
6).
The second numerical method proposed for the partial differential problem given by Equations (
5) and (
6) was an Adaptive Finite-Element Method (AFEM). The idea of using an adaptive method aims to reduce the operational cost by adapting the mesh to the numerical solution where more precision is necessary, the fire front, as this takes up a small part of the whole domain. This method combines refinement and derefinement techniques to generate in each time step a sequence of nested meshes, allowing an easy application of multigrid acceleration techniques. The design of this first model and its numerical resolution already pursued the aim of simulating the fire spread in times much shorter than real-time, through a simplified model and efficient numerical methods.
2.3. Second Model: Rosseland Approximation for Local Radiation
The second model is again a one-phase 2D model, but it addresses radiation as the dominant thermal transfer mechanism, without forgetting convection, which represents the effect of wind and slope. In [
23], radiation was incorporated in the fire spread model by using a local radiation term using the Stefan–Boltzmann law and approximating the fourth power of the temperature by a Taylor expansion [
24]. This is known as the Rosseland radiation model; it is valid when the medium is optically thick. It is a diffusion approximation for the radiation, and it introduces a nonlinearity in the diffusive term of the energy equation. Another modification compared to the first model is the simplification of the vertical heat loss term or natural convection. In this second model, the one-phase simplification leads to introducing a logic expression depending on the phase change temperature, which we can assimilate with the pyrolysis temperature. The PDEs in this model are written as follows:
where
is the Stefan–Boltzmann constant,
is the optical path length for radiation, and
is the pyrolysis temperature.
A more rational change of variables is also incorporated for the non-dimensionalization of the equations, the Frank-Kamenetskii change of variables [
25]. This change of variables involves setting reference values for temperature and fuel in which an equilibrium state can be assumed, as well as spatial and temporal variables, and permits elucidating the significant parameters. The reference values are the ambient temperature
and the initial solid fuel load
. Then, the dimensionless temperature is now
, and the dimensionless solid fuel load is
. We kept again the notation for the spatial and temporal variables. The non-dimensional PDEs of this second model are
where
is the dimensionless natural convection coefficient, and it depends on the natural convection coefficient, the density, and the specific heat. The nonlinear function
in the diffusive term represents the local radiation. The parameter
is the dimensionless inverse of the conductivity coefficient and depends on the Stefan–Boltzmann constant, the optical path length for radiation, the thermal conductivity, and the ambient temperature.
is the dimensionless inverse of the activation energy and depends on the activation energy, the universal constant of gases, and the ambient temperature. The function
is given by an Arrhenius type law, where
is the non-dimensional pyrolysis temperature, and the logic expression is equal to 1 if it is true and 0 if it is false.
q is the non-dimensional reaction heat and depends on the heat of combustion, the initial fuel, and the specific heat. For a deep understanding of the dimensionless process, see [
23].
Equations (
9) and (
10), together with appropriate initial and boundary conditions, provide challenging mathematical and numerical problems. Facing these types of mathematical questions is one of the challenges that should not be forgotten in the development of simulating models of processes as complex as wildfire spread. Some results about the existence and uniqueness of weak solutions of the non-convective version, that is the nonlinear reaction–diffusion problem, can be found in [
23]. From the numerical point of view, the proposed scheme for the full version of the model is based on the use of the Mixed Finite-Element Method (MFEM), which admits discontinuities in the temperature, preserving the continuity of the flux through the inter-element boundaries. This allows the representation of high gradients in the solution corresponding to the fire front with strong temperature gaps. The scheme proposed uses the lowest-order Raviart–Thomas elements [
26]. Finally, the convective term is solved by a splitting technique using Godunov’s method.
2.4. 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,
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
, and the reactive term function
are the same as in the previous model. The multivalued operator is given by
where
is the dimensionless water evaporation temperature and
is the dimensionless solid fuel pyrolysis temperature. The quantity
is the dimensionless evaporation heat related to the latent heat of evaporation
and the fuel moisture content
(kg of water/kg of dry fuel);
is the dimensionless pyrolysis heat related to the latent heat of pyrolysis
.
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 and 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.
2.5. Non-Local Radiation: Some 3D Effects
In fact, the model in
Section 2.4 was not implemented as-is, since another important improvement, the non-local radiation, was simultaneously incorporated. This is an essential improvement to reach the current version of the model, and it allows incorporating certain three-dimensional effects while maintaining the simplicity of a two-dimensional model.
In a first approximation, the idea of non-local radiation was introduced by means of a convolution operator, simultaneously maintaining the local radiation represented with the nonlinear term of the diffusive term of the energy equation; see [
30]. However, based on the results, it was decided to simplify the energy equation and choose only non-local radiation [
32]. This was a challenge from a computational point of view as it involved solving the radiation intensity equation in a 3D domain, the layer of air on the surface in which the fire develops.
The equations of the above models are defined over the surface
S where the fire takes place, defined by the mapping:
where
is a known function representing the topography of the surface
S.
In order to take into account some 3D effects, specifically non-local radiation, we shall consider the following 3D domain, representing the air layer of thickness
on the surface
S:
The non-dimensional simplified equations defined on the surface
S are
completed again with homogeneous Dirichlet boundary conditions, assuming that the surface
S is large enough so that the fire does not reach the boundary during the simulation interval
, the initial conditions representing the initial fuel load and fire source.
The unknowns , the dimensionless enthalpy, , the dimensionless temperature of the solid fuel, and , the mass fraction of solid fuel, are bidimensional variables defined in . Note that we abandoned the Frank-Kamenetskii change of variable since an Arrhenius-type expression no longer appears in the energy equation. The physical quantities E, T, and M are the enthalpy, the temperature of solid fuel, and the fuel load, respectively; C is the heat capacity of solid fuel; is a reference temperature, namely the ambient temperature; is the initial fuel load.
The multivalued maximal monotone operator
G is slightly different:
where, again,
,
, and
are the same as above. We simplified the model assuming that the maximum value that the dimensionless temperature of the solid fuel
u reaches is the dimensionless solid fuel pyrolysis temperature
.
We must specify that, in this model, only the solid phase of the combustion process is considered: the mass fraction of solid fuel
c is a dimensionless variable between 0 and 1, and as just mentioned, the maximum value of the dimensionless solid fuel temperature
u is the dimensionless pyrolysis temperature
. Note that this is not the fire temperature, since the gaseous phase in this model is parameterized through the flame temperature
and the flame height
F in the radiation term, which at the moment are considered input data dependent on the type of fuel. In
Section 2.6, we explain how to improve this parameterization.
We also assumed that the dimensionless solid fuel begins to be lost when it reaches this temperature,
. This means a simplification of the right-hand side of Equation (
18), which represents the loss of solid fuel due to combustion. Now,
when
, and
is constant when
, where this constant is inversely proportional to the half-life time of the combustion of each type of fuel.
It remains to explain how the non-local radiation is computed, that is the
r term on the right-hand side of Equation (
16). Since radiation essentially comes from flames, we considered a simplified physical model in which the gases produced by pyrolysis burn above the fuel layer, producing a flame above that layer. The flame may eventually be tilted due to the wind or the slope of the ground and emits radiation, which reaches points ahead of the flame, heating the surrounding unburned fuel and allowing the fire to spread.
The term
r describes the thermal radiation, which reaches the surface
S from the flame above the layer
where
is a time scale that appears during the dimensionless process.
R represents the incident energy at a point
of the surface
S, due to radiation from the flame over the surface per unit time and per unit area, obtained by summing up the contribution of all directions
, i.e.,
where
is the solid angle and
is the unit normal vector to the surface
S. We considered only the hemisphere above the fuel layer, and each contribution depends, among others, on the flame height
F and the flame temperature
.
I is the total radiation intensity, described by the following differential equation:
where
is the black body total radiation intensity, and it is governed by the Stefan–Boltzmann law:
where
is the Stefan–Boltzmann constant and
a is the radiation absorption coefficient inside the flame. The path
s is within the 3D domain
D reaching any point on the surface
S and eventually passing through a flame. We assume that the temperature
T reaches the flame temperature
.
Details on how to approximate Equation (
22) and calculate
for different types of flame, vertical or tilted, can be found in [
32]. We include here some aspects of the calculation of the radiation term for the case of a vertical flame in order to clarify some concepts that were introduced to reduce the computational time and to enhance the efficiency of the simulation process.
In our model, the radiation term R for a vertical rectangular flame has the form:
where
is the characteristic function of the flames, i.e.,
if
and
otherwise, and
represents the pyrolysis temperature.
We used the P1 finite-element method for the spatial discretization, so using the finite-element basis to represent the
f function, the radiation term can be written for each node
of the mesh as follows:
where
is the
Radiation Matrix, representing the nodes of the finite element mesh, which are reached by the radiation emitted by each node, and it is computed only once (outside the time loop).
In practice, the radiation term is only calculated in a neighborhood of the fire front, which we call
Active Nodes. At the beginning of the numerical process, we define a uniform and fine spatial mesh for spatial discretization, and for each time step
of the time discretization, we define a set of
Active Nodes formed by those nodes located inside a sufficiently large area close to the fire front (burning area); we solved the corresponding equations only in this set of nodes, as well as the calculation of the radiation term. Specifically, a node is
Active if the dimensionless temperature in the node is positive, and the dimensionless solid fuel is
, or if the node belongs to the
Radiation Molecule associated with a node fulfilling the previous conditions. The
Radiation Molecule is a round set of nodes formed by the node itself and neighboring nodes that define the area affected by the radiation emitted by the node concerned. The
Radiation Molecule was initially a set of 89 nodes formed by the node itself and 88 nodes surrounding it forming a circle of nodes of radius 5 nodes (see Figure 6 in [
32]).