Fire Spread modelling
modelling Level 4

Fire Spread modelling

How fast will this fire spread and in which direction? Fire spread models predict rate and direction of propagation using fuel properties, topography, and weather. This model derives the Rothermel equation, implements elliptical fire growth, demonstrates cellular automaton simulation, and maps fire perimeters over time.

Prerequisites: rothermel equation, fire spread rate, elliptical fire growth, cellular automata

Updated 12 min read

1. The Question

Where will this fire be in 6 hours, and what communities are at risk?

Fire spread modelling:

Predicts fire perimeter expansion over time.

Inputs:

  • Fuel type and loading
  • Fuel moisture
  • Wind speed and direction
  • Topography (slope, aspect)
  • Weather forecast

Outputs:

  • Rate of spread (m/min)
  • Fire perimeter at time $t$
  • Fireline intensity
  • Flame length
  • Arrival time at locations

Models:

Empirical: Rothermel (1972)
Physical: FIRETEC, WFDS (computational fluid dynamics)
Statistical: Machine learning on historical fires
Hybrid: Couple multiple approaches

Operational systems:

  • FARSITE (USA Forest Service)
  • Phoenix RapidFire (Australia)
  • Prometheus (Canada)
  • FlamMap (landscape fire potential)

2. The Conceptual Model

Rothermel Fire Spread

Basic principle:

Fire spreads when radiant/convective heat preheats adjacent fuel to ignition.

Rate of spread:

\[ROS = \frac{I_R \xi (1 + \Phi_w + \Phi_s)}{\rho_b \epsilon Q_{ig}}\]

Where:

  • $ROS$ = rate of spread (m/min)
  • $I_R$ = reaction intensity (kW/m²)
  • $\xi$ = propagating flux ratio
  • $\Phi_w$ = wind coefficient
  • $\Phi_s$ = slope coefficient
  • $\rho_b$ = bulk density (kg/m³)
  • $\epsilon$ = effective heating number
  • $Q_{ig}$ = heat of preignition (kJ/kg)

Reaction intensity:

Energy release rate per unit area.

\[I_R = \Gamma' w_n h\]

Where:

  • $\Gamma’$ = optimum reaction velocity (min⁻¹)
  • $w_n$ = net fuel loading (kg/m²)
  • $h$ = heat content (kJ/kg, ~18,600 for wood)

Elliptical Fire Growth

Wind-driven fires: Elongated downwind.

Head fire: Fastest spread (downwind)
Flank fire: Perpendicular to wind
Back fire: Upwind (slow)

Ellipse parameters:

\[a = \frac{ROS_h + ROS_b}{2}\] \[b = \frac{ROS_h + ROS_b}{2 \times LB}\]

Where:

  • $a$ = semi-major axis
  • $b$ = semi-minor axis
  • $ROS_h$ = head fire rate
  • $ROS_b$ = backing fire rate
  • $LB$ = length-to-breadth ratio (~3-4 typical)

Area at time $t$:

\[A(t) = \pi a b t^2\]

Fireline Intensity

Energy release per unit length:

\[I = h w ROS\]

Where:

  • $I$ = fireline intensity (kW/m)
  • $h$ = heat content (kJ/kg)
  • $w$ = fuel consumed (kg/m²)
  • $ROS$ = rate of spread (m/s)

Byram’s flame length:

\[L = 0.0775 I^{0.46}\]

Where $L$ = flame length (m)

Suppression difficulty:

  • $I < 500$ kW/m: Direct attack possible
  • $I = 500-2000$ kW/m: Indirect attack
  • $I = 2000-4000$ kW/m: Very difficult
  • $I > 4000$ kW/m: Suppression ineffective

3. Building the Mathematical Model

Rothermel Coefficients

Wind coefficient:

\[\Phi_w = C \left(\frac{\beta}{\beta_{op}}\right)^{-E} \left(\frac{U_m}{60}\right)^B\]

Where:

  • $C, E, B$ = fuel-specific constants
  • $\beta$ = packing ratio (fuel density / particle density)
  • $\beta_{op}$ = optimum packing ratio
  • $U_m$ = midflame wind speed (ft/min)

Typical: $B \approx 1.5$ (spread ∝ wind^1.5)

Slope coefficient:

\[\Phi_s = 5.275 \beta^{-0.3} \tan^2\theta\]

Where $\theta$ = slope angle

Example: 20° slope

\[\Phi_s = 5.275 \times (0.01)^{-0.3} \times \tan^2(20°)\] \[= 5.275 \times 2.15 \times 0.133 = 1.51\]

Slope multiplies spread by factor of 2.5 (1 + 1.51)!

Cellular Automaton Fire Spread

Discretize landscape:

Grid cells (30-100 m resolution).

State: Unburned / Burning / Burned

Transition rules:

\[P_{\text{ignite}} = f(\text{fuel, weather, distance, time})\]

Simplified:

\[P = P_0 \times \exp\left(-\frac{d}{ROS \times \Delta t}\right)\]

Where:

  • $P$ = ignition probability
  • $P_0$ = base probability
  • $d$ = distance to burning cell
  • $ROS$ = rate of spread
  • $\Delta t$ = time step

Algorithm:

For each time step:
  For each burning cell:
    For each neighbor:
      Calculate P_ignite
      If random() < P_ignite:
        Ignite neighbor
  Burning cells → Burned

Advantages:

  • Computationally fast
  • Handles complex landscapes
  • Stochastic variation

Disadvantages:

  • Requires calibration
  • Less physical basis than Rothermel

Fire Arrival Time

Minimum time path:

\[T(x,y) = \min_{\text{path}} \int_{\text{path}} \frac{ds}{ROS(s)}\]

Where:

  • $T$ = arrival time
  • $ds$ = path element
  • $ROS(s)$ = rate of spread along path

Solved by:

Level-set methods or Dijkstra’s algorithm on grid.

Isochrones:

Contours of equal arrival time.

4. Worked Example by Hand

Problem: Calculate fire spread and perimeter.

Conditions:

  • Fuel: Grass (short, dry)
  • Fuel load: 0.5 kg/m²
  • Fuel moisture: 6%
  • Wind speed: 30 km/h (8.3 m/s)
  • Slope: 15° downwind
  • Initial ignition: Point source

Constants (grass fuel model):

  • $\xi = 0.4$ (propagating flux ratio)
  • $\epsilon = 0.7$ (effective heating)
  • $Q_{ig} = 300$ kJ/kg (heat of preignition)
  • $h = 18,600$ kJ/kg (heat content)

Calculate ROS, fire perimeter after 1 hour.

Solution

Step 1: Wind coefficient

Simplified: $\Phi_w \approx 0.15 U^{1.5}$ where $U$ in m/s

\[\Phi_w = 0.15 \times 8.3^{1.5} = 0.15 \times 23.9 = 3.6\]

Step 2: Slope coefficient

\[\Phi_s = 5.275 \times (0.01)^{-0.3} \times \tan^2(15°)\] \[= 5.275 \times 2.15 \times 0.072 = 0.82\]

Step 3: Reaction intensity

Assume $\Gamma’ = 15$ min⁻¹ (typical grass):

\[I_R = 15 \times 0.5 \times 18600 = 139,500 \text{ kW/m}^2\]

Step 4: Rate of spread

\[ROS = \frac{139500 \times 0.4 \times (1 + 3.6 + 0.82)}{500 \times 0.7 \times 300}\] \[= \frac{139500 \times 0.4 \times 5.42}{105000}\] \[= \frac{302,472}{105000} = 2.88 \text{ m/min}\]

Head fire ROS = 2.88 m/min = 173 m/h

Step 5: Backing fire

Assume $ROS_b = 0.1 \times ROS_h = 0.29$ m/min

Step 6: Ellipse parameters

\[a = \frac{2.88 + 0.29}{2} = 1.59 \text{ m/min}\]

Length-to-breadth ratio $LB = 3$:

\[b = \frac{1.59}{3} = 0.53 \text{ m/min}\]

Step 7: Perimeter after 1 hour

Semi-axes at $t = 60$ min:

\[a_{60} = 1.59 \times 60 = 95.4 \text{ m}\] \[b_{60} = 0.53 \times 60 = 31.8 \text{ m}\]

Area:

\[A = \pi \times 95.4 \times 31.8 = 9,540 \text{ m}^2 \approx 1 \text{ hectare}\]

Fire spread 95 m downwind, 32 m on flanks in 1 hour.

Step 8: Fireline intensity (head)

\[I = 18600 \times 0.5 \times \frac{2.88}{60} = 446 \text{ kW/m}\]

Flame length:

\[L = 0.0775 \times 446^{0.46} = 0.0775 \times 12.3 = 0.95 \text{ m}\]

Moderate intensity (direct attack still possible if resources available quickly)

5. Computational Implementation

Below is an interactive fire spread simulator.

Head ROS: -- m/min

Fire area: -- ha

Fireline intensity: -- kW/m

Suppression: --

Observations:

  • Fire spreads fastest downwind (head fire)
  • Elliptical shape elongated in wind direction
  • Higher wind dramatically increases spread rate
  • Slope amplifies spread when aligned with wind
  • Low moisture enables rapid spread
  • Fireline intensity determines suppression feasibility

Key insights:

  • Exponential area growth over time
  • Wind and slope effects multiplicative
  • Even moderate winds create elongated fires
  • Suppression window closes quickly with high intensity

6. Interpretation

Operational Fire Prediction

FARSITE workflow:

  1. Initialize: Ignition point, time, weather
  2. Propagate: Calculate ROS at fire perimeter
  3. Advance: Move perimeter based on ROS, timestep
  4. Repeat: Until forecast end or fire containment

Outputs:

  • Fire perimeter evolution (hourly)
  • Arrival time maps
  • Intensity maps
  • Flame length maps

Used for:

  • Evacuation planning
  • Resource pre-positioning
  • Backfire/burnout strategy
  • Structure triage

Example - 2018 Camp Fire (California):

FARSITE predicted Paradise threatened within 6 hours.

Enabled evacuation (though still 85 deaths, rapid spread).

Prescribed Burn Planning

Objective: Reduce fuel loads safely.

Model used to:

Determine burn window (weather conditions for control).

Required:

  • Low intensity (< 1000 kW/m)
  • Slow spread (< 1 m/min)
  • Favorable wind direction (away from structures)

Example conditions:

  • Fuel MC: 12-15%
  • Wind: 10-15 km/h
  • Atmospheric stability: Stable
  • RH: 40-60%

Model confirms burn will remain controllable.

Post-Fire Analysis

Reconstruct fire behavior:

Given final perimeter, weather records.

Calibrate models:

Adjust parameters (fuel models, moisture, wind reduction factors).

Improve predictions:

Better local fuel characterization.

Identify extreme fire behavior:

Rates of spread » model predictions indicate:

  • Spotting (long-range ember transport)
  • Crown fire (not surface fire)
  • Fire whirls (tornadic circulation)

7. What Could Go Wrong?

Crown Fire Transition

Surface fire models (Rothermel) don’t predict crown fire.

Crown fire: Spreads through tree canopy, 10-100× faster.

Transition: Occurs when:

\[I > I_{\text{critical}}\]

Critical intensity:

\[I_{\text{crit}} = \frac{(460 + 25.9 \times MC)^{3/2}}{60} \times 0.01 \times \rho_{\text{canopy}}\]

Once in crown:

ROS can exceed 100 m/min (vs 2-10 m/min surface).

Solution: Van Wagner crown fire models, FIRETEC 3D physics.

Spotting Ignored

Embers travel ahead of fire front.

Distance: Function of ember size, height lofted, wind.

Model (Albini):

\[d_{\text{spot}} = f(\text{flame length, wind, terrain})\]

Typically: 0.5-2 km, extreme cases 10-30 km.

Creates secondary ignitions ahead of modelled perimeter.

Catastrophic when embers land in receptive fuel.

Solution: Spotting submodels, but highly stochastic.

Fuel Heterogeneity

Models assume uniform fuel in each type.

Reality:

Variable loading, moisture, species within same classification.

Result: Under/overprediction locally.

Solution:

  • Fine-scale fuel mapping (LiDAR-derived)
  • Stochastic variation in parameters

Weather Forecast Uncertainty

Fire spread depends on forecast wind, temperature, humidity.

Forecast errors propagate:

10% wind error → 15% ROS error → 25% perimeter error.

Solution:

  • Ensemble forecasts (multiple weather scenarios)
  • Probabilistic fire prediction

8. Extension: Machine Learning Fire Spread

Data-driven approaches:

Train on historical fires (perimeter progression, weather, fuels).

Neural networks:

Input: Fuel, weather, terrain
Output: ROS or probability of burning

Advantages:

  • Capture complex nonlinear relationships
  • No need to specify physics
  • Fast inference

Disadvantages:

  • Requires large training datasets
  • Difficult to extrapolate beyond training conditions
  • Less interpretable than physics-based

Hybrid: Use physics model as baseline, ML for corrections.

9. Math Refresher: Ellipse Geometry

Standard Form

\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]

Where:

  • $a$ = semi-major axis (larger)
  • $b$ = semi-minor axis (smaller)

Area:

\[A = \pi a b\]

Perimeter (approximate):

\[P \approx \pi (3(a+b) - \sqrt{(3a+b)(a+3b)})\]

Eccentricity

\[e = \sqrt{1 - \frac{b^2}{a^2}}\]

Circle: $e = 0$ ($a = b$)
Elongated: $e \to 1$ ($b \ll a$)

Fire ellipses:

Typical $e = 0.8-0.95$ (highly elongated in strong wind).

Summary

  • Fire spread rate determined by fuel, weather, and topography via Rothermel equation
  • Wind and slope effects multiplicative increasing spread exponentially
  • Elliptical fire growth results from directional spread differences
  • Fireline intensity determines suppression difficulty with threshold at 500 kW/m
  • Cellular automaton models provide computationally efficient landscape-scale simulation
  • Operational systems like FARSITE predict fire perimeter evolution for planning
  • Applications span wildfire prediction, prescribed burn planning, post-fire analysis
  • Challenges include crown fire transition, spotting, fuel heterogeneity, weather uncertainty
  • Machine learning approaches complement physics-based models
  • Critical tool for emergency management and firefighter safety

References