Quoted from: Prieto Herráez, D., María Isabel Asensio Sevilla, Luis Ferragut Canals, José Manuel Cascón Barbero, and A. Morillo Rodríguez. "A GIS-based fire spread simulator integrating a simplified physical wildland fire model and a wind field model." *International Journal of Geographical Information Science* 31, no. 11 (2017): 2142-2163. https://doi.org/10.1080/13658816.2017.1334889

The PhFFS is the current version in a series of physical fire propagation models. It has its origin in a simple 2D one-phase physical model, based on the principles of energy and mass conservation, and considered convection and diffusion. In due course, heat transfer by radiation was incorporated into the model with a local radiation term (Asensio and Ferragut 2002). The influences of fuel moisture content and heat absorption by pyrolysis were included by Ferragut *et al*. (2007b) with an operator representing enthalpy. At the same time, the non-local radiation from the flames above the vegetal layer was added to the model (Ferragut *et al*. 2007a), enabling it to deal with the effect that wind and slope had over flame tilt and thus increasing heat transfer. Fresh efforts have been made to improve the suitability of the PhFFS model for the simulation of real fires in Ferragut *et al*. (2014) and experimental fires in Prieto *et al*. (2015), with the introduction of data assimilation techniques in Ferragut *et al*. (2015).

The partial differential equations describing the PhFFS model are based on the energy and mass conservation equation on the surface where the fire takes place, and the radiation equation on the air layer over this surface. The model equations are not described here because of their mathematical complexity. A detailed explanation of the current PhFFS model equations can be found in Prieto *et al*. (2015), although here we briefly describe the physical meaning of each of the equations terms. We also outline the equations unknowns, its input variables and its three parameters, that are summarized in Section 2.4, in order to understand how the PhFFS model is coupled with GIS.

The surface where the fire develops is given by a function *h* representing the topography; that is, the function *h* gives the land surface height.

The physical quantities involved as the unknowns in the model equations are enthalpy *E* , solid fuel temperature *T* and fuel load *M* . The following input variables are also required: the heat capacity of the solid fuel , the maximum solid fuel load , both depending on fuel type, and the reference temperature, which is the ambient temperature .

The influence of the vegetation’s moisture content is modelled through an operator depending on enthalpy, the latent heat of evaporation , and fuel moisture content (kg of water/kg of dry fuel).

The model also takes into account the energy lost by natural free convection through a term in the energy conservation equation. This term is related to the natural convection coefficient *H* , the first of the three model parameters.

Our model considers wind effect in two different ways: through the convective term itself and through the flame tilt caused by wind that affects the radiation term. The energy conservation equation incorporates a term representing convective heat. This term depends on the surface wind velocity, **V** , rescaled by a correction factor , which is the second model parameter. To deeply understand the meaning of this parameter , see Prieto *et al*. (2015), where it is explained how this one solid phase model is simplified from a two phase solid–gas model, and how the assumptions of this simplification allow to estimate this parameter. Surface wind velocity **V** is provided by meteorological sources. The HDWM (or any other wind model) is used to compute a 3D wind velocity field from these meteorological data. Then, the wind velocity at flame average weight is supplied as input data to PhFFS. Alternatively, surface wind velocity **V** can be considered as constant wind along the simulation area.

The thermal radiation reaching the surface from the flame is included in the energy conservation equation, taking into account the influence of wind and slope over flame tilt. The radiation equation contains the third and final model parameter, the radiation absorption coefficient *a* , and two model variables, flame temperature denoted by and flame length *F* , with both depending on fuel type. For further details about radiation computation, see Ferragut *et al*. (2015).

An important simplification of the PhFFS model is that only the solid phase of the combustion process is considered: the solid fuel mass *M* varies between 0 and its maximum value , and the maximum value of solid fuel temperature *T* is the pyrolysis temperature . The gaseous phase is parameterized in the radiation term through flame temperature , and flame length *F*.

The loss rate of solid fuel due to combustion is represented in the mass conservation equation. It is null when the pyrolysis temperature has not been reached, and constant once it has been exceeded. This constant value is inversely proportional to the solid fuel half-life of the combustion, (s), of each type of fuel, measured from the moment of ignition.

The numerical solution of the model is obtained by solving the corresponding non-dimensional partial differential equations depending on the non-dimensional solid fuel temperature , and the non-dimensional solid fuel mass fraction . The numerical methods used are the finite element method combined with various finite difference schemes in time, the characteristic method for the convective term and an exact integration method assuming a rectangular flame shape section (under windy conditions we assume tilted flames) for the non-local radiation equation. With a view to reducing the computational cost of the 3D radiation equation, active nodes are defined for solving this equation only where necessary in the vicinity of the flames. These efficient numerical methods, together with different parallel computing techniques, ensure that the computational cost of running the PhFFS model has been significantly reduced, whereby it can compete with other simpler models.

The PhFFS model is implemented in C++, using API OpenMP (Chapman *et al*. 2008) in order to exploit today's multiprocessor platforms for reducing computational time. This implementation is entered in the Spanish Registry of Intellectual Property on 16 July 2015 under record entry 00/2015/4720.