Influence d un Paysage Fractal sur la Propagation d un Feu de Forêt

Transcription

1 1 Influence d un Paysage Fractal sur la Propagation d un Feu de Forêt F. Aguayo*, A. Fuentes, J.P. Clerc et B. Porterie Polytech Marseille Université de Provence IUSTI UMR 6995 du CNRS andres.fuentes@polytech.univ-mrs.fr * Financée en partie par ALFA-SCAT projet 9 ème Rencontre du GDR Incendie

2 2 Contents 1. Fractal dimension (D f ) 2. Percolation 3. Fractal surface (S f ) 3.1. Generation of a S f 3.2. Determination of D f 4. Percolation 3D an example 5. Propagation model 6. How S f affects the propagation of a forest fire?

3 3 Overview Goal: Introduce a real landscape (fractal) and to study the influence in the forest fire propagation Know-how: The small world local network To improve: Modeling the fractal landscape A representative dynamic of fire propagation behavior

4 4 Fractal landscape generation Weierstrass Model: A k and B k : are independent, uniformly distributed in [0;2π). C k : are independent, normally distributed with mean 0 and variance 1. λ : a parameter >1. α : the index of the surface.

5 5 Fractal landscape generation Weierstrass Model: Some examples D f =2.1 D f =2.2

6 6 Fractal dimension (D f ) determination Definition of D f : The Koch curve => Divide each line segment into three (b=3) segments and replace the center segment with two segments the same length as the segment being replaced, and then to repeat (Gefen 1987). n = b D D f = ln(n)/ln(b) D f = ln(4)/ln(3) D f = 1.26

7 7 Fractal dimension (D f ) determination Definition of D f : More information is available in: Les Chroniques de Rose Polymath Par Ian Stewart

8 8 Fractal dimension (D f ) determination Box counting method in 3D: In Euclidean space the volume V of an object scales as: where L stands for the side of the object and d is its usual Euclidean dimension. For this we enclose the object in a cube with dimension d E Then we divide the side of the cube in segments of length ε The volume V will be directly proportional to the number of little cubes N ε of volume1/ε de needed to cover the object

9 9 Fractal dimension (D f ) determination Box counting method:

10 10 Fractal dimension (D f ) determination Box counting method in 3D: Only for given values of λ, the mean fractal dimension is: D f = (3-α)

11 11 Additional dimension role in percolation Transition line: The bond percolation in a 2D square mesh. The occupancy probability p inside that mesh will be called p. Several of those meshes one above the other and will link the corresponding sites with a probability p forming a common 3D square mesh

12 12 Additional dimension role in percolation Transition line:

13 13 Dynamic propagation model C i : the amount of fuel in the site i on time t, relative to the initial amount of fuel. C i =1 and C i =0 are stationary states of the site if no external agent is considered. C i th : the threshold linked to the ignition of the tree (time necessary to obtain a critical pyrolysis mass flux). U i : parameter associated to mass loss when the tree is burning. W ij : the interaction between trees (here proportional to radiative heat transfer).

14 14 Dynamics propagation model An example with only 4 trees: t 0 C i th t 1 t 2

15 15 Some results Propagation of a forest fire in a fractal landscape: D f = 2.2 Doping = 0.7 C th i = 0.85

16 16 Some results Evolution the D f of the fire front Evolution the D f of the burned mass

17 17 Some results Mean time of fire <T s > for different surfaces D s The P c in function of D s

18 18 Conclusion and perspectives The fractal landscape and simplified propagation model was introduced to study the influence in fire propagation front, active sites, burning area, etc The effects of different type of vegetation can be (geometries, fuel load, ) The effect of D f in the propagation of a forest fire is being analyzed The universality of the phenomena is investigated now trough the critical exponent. The presented framework could have a very interesting application in practical situations

19 19 Merci de votre attention! En espérant que la présentation ne vous a pas paru trop plane sur un sujet aussi fractal

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21 21 Dynamic propagation model (annexes) The term Ui

22 22 Dynamic propagation model (annexes) The term Wij

23 23 Dynamic propagation model (annexes) The term Wij

24 24 Dynamic propagation model (annexes) The view factor f

25 25 Dynamic propagation model (annexes) The view factor f

26 26 Dynamic propagation model (annexes) The view factor f

27 27 Dynamic propagation model (annexes) Relation between energy and degradation

28 28 Dynamic propagation model (annexes) Relation between energy and degradation