# An Historical Review of the Simplified Physical Fire Spread Model PhyFire: Model and Numerical Methods

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## Abstract

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## 1. Introduction

## 2. Previous Models

#### 2.1. Conservation Laws

#### 2.2. First Model: Turbulent Diffusivity

#### 2.3. Second Model: Rosseland Approximation for Local Radiation

#### 2.4. The Effect of Moisture Content: A Multivalued Operator in Enthalpy

#### 2.5. Non-Local Radiation: Some 3D Effects

#### 2.6. Flame Submodel

#### 2.6.1. Flame Height Submodel

#### 2.6.2. Flame Temperature Submodel

#### 2.7. Fire-Spotting

## 3. Current Model and Numerical Algorithm

#### 3.1. Model Description

#### 3.2. Numerical Method

- Build the set of Active Nodes.
- Compute the Radiation Heat.
- Prediction step: semi-implicit Euler method.
- Update the set of Active Nodes.
- Update the Radiation Heat.
- Correction step: modified Crank–Nicolson method.

#### 3.2.1. Convection Step

#### 3.2.2. Predictor Step

#### 3.2.3. Corrector Step

## 4. GIS Integration

## 5. Real Case

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

EFFIS | European Forest Fire Information System |

GFAS | CAMS Global Fire Assimilation System |

PDE | Partial Differential Equation |

FEM | Finite-Element Method |

AFEM | Adaptive Finite-Element Method |

MFEM | Mixed Finite-Element Method |

GIS | Geographical Information System |

FMC | Fuel Moisture Content |

ROS | Rate Of Spread |

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**Figure 1.**Burned hectares in the most-affected European countries (more than 10,000 ha) during 2022. Data from EFFIS on 28 September 2022.

**Figure 2.**Simulation area (black rectangle), contour lines, fire ignition point, IFN4 fuel type distribution, actual final perimeter (red line), and firefighters’ firebreaks (blue lines).

**Figure 3.**Simulation area (black rectangle), fire ignition point, actual final perimeter (red line), firefighters’ firebreaks (blue lines), simulated burned areas each hour, and fire front at 20.00, considering the effect of fuelbreaks by modifying the initial fuel load.

**Figure 4.**Simulation area (black rectangle), fire ignition point, actual final perimeter (red line), simulated burned areas each hour, and fire front at 20.00. No firefighting actions were considered.

Physical Variable | Symbol | Units | Dimensionless Variable |
---|---|---|---|

Enthalpy | E | J m${}^{-2}$ | $\phantom{\rule{2.em}{0ex}}e=E/MC{T}_{\infty}$ |

Solid fuel temperature | T | K | $\phantom{\rule{2.em}{0ex}}u=(T-{T}_{\infty})/{T}_{\infty}$ |

Solid fuel load | M | kg m${}^{-2}$ | $\phantom{\rule{2.em}{0ex}}c=M/{M}_{0}$ |

Input Variable | Symbol | Units |
---|---|---|

Heat capacity | C | J K${}^{-1}$ kg${}^{-1}$ |

Maximum initial fuel load | ${M}_{0}$ | kg m${}^{-2}$ |

Fuel moisture content | ${M}_{v}$ | kg water/kg fuel |

Maximum flame temperature | ${T}_{f,max}$ | K |

Pyrolysis temperature | ${T}_{p}$ | K |

Combustion half-life | ${t}_{1/2}$ | s |

Flame length independent factor | ${F}_{H}$ | m |

Flame length wind correction factor | ${F}_{v}$ | m${}^{1/2}$s${}^{1/2}$ |

Flame length slope correction factor | ${F}_{s}$ | − |

Parameter | Symbol | Units |
---|---|---|

Natural convection coefficient | H | J s${}^{-1}$m${}^{-2}$K${}^{-1}$ |

Convective term factor | $\beta $ | − |

Mean absorption coefficient | a | m${}^{-1}$ |

Fuel Type (NFFL/[34]) | ${\mathbf{M}}_{0}$ | ${\mathbf{M}}_{\mathbf{v}}$ | ${\mathbf{T}}_{\mathbf{f}}$ | ${\mathbf{T}}_{\mathbf{p}}$ | ${\mathbf{t}}_{1/2}$ | C | ${\mathbf{F}}_{\mathbf{H}}$ | ${\mathbf{F}}_{\mathbf{v}}$ | ${\mathbf{F}}_{\mathbf{s}}$ |
---|---|---|---|---|---|---|---|---|---|

Timber grass (2/Pa-06) | $1.0$ | $10\%$ | 1300 | 500 | 100 | 2000 | $1.1100$ | $0.4712$ | $0.6759$ |

Brush (5/Eu-06) | $2.3$ | $10\%$ | 1300 | 500 | 200 | 2300 | $3.7780$ | $0.5075$ | $2.8280$ |

Dormant brush (6/Cl-02) | $2.2$ | $10\%$ | 1300 | 500 | 200 | 2300 | $3.3240$ | $0.4888$ | $2.6880$ |

Inflammable brush (7/Ea-08) | $2.4$ | $15\%$ | 1300 | 500 | 300 | 2300 | $3.9320$ | $0.6752$ | $3.0150$ |

Local Time | Temperature | Humidity | Wind Speed (m/s) | Wind Direction |
---|---|---|---|---|

3.45–5.00 p.m. average | 32.02 | 27.17 | 2 | 260 |

5.00–6.00 p.m. average | 32.01 | 27.43 | 3.3 | 300 |

6.00–7.00 p.m. average | 31.50 | 27.43 | 4.75 | 300 |

7.00–8.00 p.m. average | 31.42 | 27.50 | 4.75 | 360 |

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**MDPI and ACS Style**

Asensio, M.I.; Cascón, J.M.; Prieto-Herráez, D.; Ferragut, L.
An Historical Review of the Simplified Physical Fire Spread Model PhyFire: Model and Numerical Methods. *Appl. Sci.* **2023**, *13*, 2035.
https://doi.org/10.3390/app13042035

**AMA Style**

Asensio MI, Cascón JM, Prieto-Herráez D, Ferragut L.
An Historical Review of the Simplified Physical Fire Spread Model PhyFire: Model and Numerical Methods. *Applied Sciences*. 2023; 13(4):2035.
https://doi.org/10.3390/app13042035

**Chicago/Turabian Style**

Asensio, María Isabel, José Manuel Cascón, Diego Prieto-Herráez, and Luis Ferragut.
2023. "An Historical Review of the Simplified Physical Fire Spread Model PhyFire: Model and Numerical Methods" *Applied Sciences* 13, no. 4: 2035.
https://doi.org/10.3390/app13042035