Present research concerns the physical background of a wild-fire propagation model
based on the split of the front motion into two parts - drifting and fluctuating. The drifting part is solved by the level set method and the fluctuating part describes turbulence
and fire-spotting. These phenomena have a random nature and can be modeled as a
stochastic process with the appropriate probability density function. Thus, wildland fire
propagation results to be described by a nonlinear partial differential equation (PDE) of
the reaction-diffusion type. A numerical study of the effects of the atmospheric stability
on wildfire propagation is performed through its effects on fire-spotting. Moreover, it
is shown that the solution of the PDE as an indicator function allows to construct the
energy balance equation in terms of the temperature.PhD Grant "La Caixa 2014

Egorova, V.

Pagnini, G.

Trucchia, A.

English

BCAM's Institutional Repository Data

Proceedings of the 17th International Conferenceon Computational and Mathematical Methodsin Science and Engineering, CMMSE 20174–8 July, 2017.Wildland fire propagation modeling:fire-spotting parametrisation and energy balanceVera N. Egorova1, Gianni Pagnini1, 2 and Andrea Trucchia1, 31 BCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo 14, E-48009Bilbao, Basque Country, Spain2 Ikerbasque Basque Foundation for Science, Calle de Maŕıa Dı́az de Haro 3, E-48013Bilbao, Basque Country, Spain3 University of the Basque Country UPV/EHU, Barrio Sarriena s/n, E-48940 Leioa,Basque Country, Spainemails: vegorova@bcamath.org, gpagnini@bcamath.org, atrucchia@bcamath.orgAbstractPresent research concerns the physical background of a wild-fire propagation modelbased on the split of the front motion into two parts - drifting and fluctuating. The drift-ing part is solved by the level set method and the fluctuating part describes turbulenceand fire-spotting. These phenomena have a random nature and can be modeled as astochastic process with the appropriate probability density function. Thus, wildland firepropagation results to be described by a nonlinear partial differential equation (PDE) ofthe reaction-diffusion type. A numerical study of the effects of the atmospheric stabilityon wildfire propagation is performed through its effects on fire-spotting. Moreover, itis shown that the solution of the PDE as an indicator function allows to construct theenergy balance equation in terms of the temperature.Key words: fire propagation, fire-spotting, level set method, energy balanceMSC 2000: 00A59, 35K57, 65C20, 70H201 IntroductionIn wildland fire propagation, fire-spotting phenomena cause isolated fire from the main fire.It is an important aspect because it affects the rate of spread of the fire and may causec�CMMSE ISBN: 978-84-617-8694-7Page 805 of  2288Wildland fire propagation modelingdangerous effects. The present study deals with fire propagation modelling and its physicalbackground.In the proposed model, the front motion is splitted into two parts - drifting and fluctu-ating. Each of them can be solved by using an appropriate method. In the present study,the Eulerian Level Set Method (LSM) is chosen for the drifting part while the fluctuatingpart is the result of a comprehensive statistical description of the physics of the systemand takes into account the randomness of the hot air turbulent transport and fire-spotting.Thus, in order to treat the fluctuating part, specific probability density function has to betaken into account [1].Fire-spotting is a complicated physical process due to many factors, such as wind, fireintensity, fuel characteristics, atmospheric conditions, etc. The statistical formulation offire-spotting has been proposed in [1] and completed by the physical parametrisation in [2].This formulation does not depend on the method used for fire propagation. In the presentstudy we extend this approach and include into account the change of wind direction andpossibility of the several secondary fires appearance.The main aim of the present study is to explain the physical background of the proposedmodel. The solution of the underlying PDE is connected to a temperature field. The transferof the temperature due to the turbulent flows is then described by the energy balanceequation. The rest of the paper is organized as follows. The wildland fire propagationmodel is proposed in the next section, including a brief description of the level-set methodand physical parametrisation of the fire-spotting. Section 3 deals with the energy balanceequation for the proposed model. Numerical examples for several test cases, such as differentwind conditions, and merging of the secondary fires, are provided in Section 4.2 Fire propagation model2.1 Level-set methodLSM is widely used as effective method for the front-tracking. For some computationaldomain S the fire front contour is represented by a closed curve Γ. The region bounded byΓ is denoted by Ω and represents the burnt area. Let us introduce an indicator functionφ(x, t):φ(x, t) =�1, x ∈ Ω(t),0, othervise.(1)Let γ : S × [0,∞) → R be a level-set function, such that for some fixed γ∗ at themoment t the fire front can be described by Γ(t) = {x ∈ S|γ(x, t) = γ∗}. If γ(x, t) > γ∗,then the ignition is observed at the point x. The level-set function γ(x, t) evolves accordingto the following ordinary level-set equationc�CMMSE ISBN: 978-84-617-8694-7Page 806 of  2288V.N. Egorova, G. Pagnini, A. Trucchia∂γ∂t= V (x, t)||∇γ||, (2)where V (x, t) is a rate of spread (ROS) of the fire front. The ROS value depends on manyelements, such as the intensity and direction of the wind, fuel conditions, etc.2.2 Turbulence and fire-spotting modellingTogether with other factors, turbulence and fire-spotting cause the random motion of thefire-front. Thus, the random front contour can be defined by the effective indicator function[1]:φe(x, t) =�Ω(t)f(x; t|x̄)dx̄, (3)where f(x; t|x̄) is the probability density function (PDF) that accounts for turbulence andfire-spotting effects. Note that the point is labelled as burnt, if φe(x, t) exceeds somethreshold value φthe .In accordance with the Reynold transport theorem, the evolution of the effective indi-cator function φe(x, t) takes the form∂φe∂t=�Ω(t)∂f∂tdx̄+�Ω(t)∇x̄ [V (x̄, t)f(x; t|x̄)] dx̄. (4)Evolution equation for the PDF f(x; t|x̄) takes the form∂f∂t= �f, (5)where � = �(x) is a generic evolution operator. Hence, (4) can be rewritten in the followingform∂φe∂t= �φe +�Ω(t)∇x̄ [V (x̄, t)f(x; t|x̄)] dx̄. (6)In (6) the front-line velocity is controlled by the ROS, while random process, such asturbulence and fire-spotting, are modelled by modifying PDF. Thus, if f(x; t|x̄) = δ(x− x̄),equation (6) reduces to the deterministic case described by (2).Assuming that the downwind phenomenon of fire-spotting is independent of turbulence,the random process handled by f(x; t|x̄) in (6) can be defined as followsf(x; t|x̄) =��∞0 G(x− x̄− ln̂; t)q(l)dl, if downwind,G(x− x̄; t), otherwise.(7)c�CMMSE ISBN: 978-84-617-8694-7Page 807 of  2288Wildland fire propagation modelingThe shape of the PDF is defined by the isotropic bi-variate Gaussian function (consid-ering turbulence effects) G(x − x̄; t) and the firebrand landing distribution q(l) is definedby a lognormal distribution as followsq(l) =1√2πσlexp−(ln l/µ)22σ2, (8)where µ is the ratio between the square of the mean of landing distance l and its standarddeviation, σ is the standard deviation of ln l/µ.In [3] authors complete the study of the fire-spotting proposed in [1] by describing thisphenomenon in terms of the fire intensity, wind velocity and fuel characteristics. It includesonly the vital ingredients, each firebrand is assumed to be spherical of the constant size.Then lognormal parameters µ and σ in (8) take the following formµ = H�3ρaCd2ρfrg�1/2, σ =12zpln�U2rg�, (9)where U is the wind velocity and H is the maximum loftable height, that is according to [4],H = αHABL + β�IdPf0�γexp�−δN2FTN20�, (10)where all the parameters are defined in Table 1.3 Energy balance equationWildfire model can be formulated based on balance equations for energy and fuel, as it isproposed in [5,6]. In the present study we follow the level-set formulation with the stochasticprocess. However, there is a connection between these two formulations. In order to derivethe energy balance equation, the physical laws are used, mainly, conservation of energy andfuel reaction. In present study we focus on the solution in the form of indicator function.Thus, it is important to show, that the found solution is connected to the temperature andthe energy balance equation can be derived by using the indicator function.An important part of study deals with the temperature field. Temperature is transferreddue to the turbulent flows, and it can be modelled by the diffusion process. The heating-before-burning mechanism, that is accumulation in time of potential fire, can be associatedwith an amount of heat:ψ(x, t) =� t0φe(x, η)dητ, (11)where τ is the ignition delay, that can be understood as a resistance to the hot-air heatingand firebrand landing in parallel.c�CMMSE ISBN: 978-84-617-8694-7Page 808 of  2288V.N. Egorova, G. Pagnini, A. TrucchiaNotation Descriptionα Part of ABL passed freely, α < 1 0.24 [4]β [m] Contribution of the fire intensity, β > 0 170 [4]γ Power-law dependence on FRP, γ < 0.5 0.35 [4]δ Dependence on stability of the FT, δ ≥ 0 0 [4]Habl [m] Height of the atmospheric boundary layer (ABL) 1200d [m ] Unit depth of the combustion zone 1Pf0 [MWm−2] Ratio of reference fire power 1 [4]N2FT [s−2] Brunt-Väisälä frequency in the FT 2.789 · 1e− 4N20 [s−2] Brunt-Väisälä frequency 2.5 · 1e− 4H [m] The maximum loftable heightρa [kg/m3] Density of the ambient air 1.1ρf [kg/m3] Density of the wild-land fuels 542Cd Drag coefficient 0.45zp p-th percentile 0.45r [m] Brand radius 0.015g [ms−2] Acceleration due to gravity 9.81Table 1: Physical parameters of the atmospheric boundary layer and fire-spotting.The amount of heat is proportional to the increasing of the temperature T (x, t). Forthe sake of simplicity, we can assume thatψ(x, t) =T (x, t)− Ta(x)Tign − Ta(x), T < Tign, (12)where ambient temperature is denoted by Ta, and Tign stands for the ignition temperature.From (12) one can see, that ψ(x, t) = 1 entails that T (x, t) = Tign and the spacial point xat the moment t belongs to the burning area.Temperature T (x, t) can be found from (12) as followsT (x, t) = Ta(x) + ψ(x, t) (Tign − Ta(x)) , (13)that by the differentiation leads to∂T∂t=(Tign − Ta(x))τφ(x, t). (14)According to the heat-before-burning mechanism, the temperature field is described bythe following reaction-diffusion type equation [1]∂T∂t= �T +Tign − Taτ(IΩ0(x) +W (x, t)) , (15)c�CMMSE ISBN: 978-84-617-8694-7Page 809 of  2288Wildland fire propagation modelingwhere Tign is the ignition temperature, Ta(x) is the ambient temperature, IΩ0(x) = φe(x, 0)andW (x, t) =� t0��Ω(θ)∇x̄ · [V(x̄, θ)f(x; θ|x̄)] dx̄�dθ. (16)Equation (15) can be understood as the energy balance equation associated to themodel. In (15) � = �(x) is a generic evolution operator, associated to the evolution off(x; t|x̄), and it models the turbulent heat transfer due to radiation. Moreover, the secondterm on the RHS corresponds to the convective heat lost to the atmosphere and the third tothe rate of fuel consumed by the fire with Rate of Spread V(x̄, t) in the outward direction.4 Numerical resultsIn this section some numerical examples are considered in order to study effects of the windand the choice of underlying PDF on the rate of spread. It is natural that if there is nowind, for the homogeneous moisture the fire front would grow slower and in all directions,as it is shown in Figure 1. If there is wind (we consider U = 6.7ms−1, the fire intensityI = 20MW ), then the fire propagates in the downwind direction (see Figure 2).� ���� �����������������������������������Figure 1: Fire propagation with zero wind.c�CMMSE ISBN: 978-84-617-8694-7Page 810 of  2288V.N. Egorova, G. Pagnini, A. Trucchia� ���� ���� ���� ���� ���� �������������������������������������� ��������������Figure 2: Fire propagation with mean wind U = 6.7ms−1.The model treats the fire-spotting. In order to model this phenomenon the lognormalPDF (8), as it is shown in Figure 3. The point of ignition of the secondary fire also dependson the wind direction. If at some moment t = t∗ the wind changes the direction, then newsecondary fires appear in new direction, as it is shown in Figure 4. For the sake of simplicity,the initial conditions for new ignitions are chosen the same as for the main fire zone, thatcauses the same form of the secondary fires.However, not for any set of the parameters the sire-spotting can be observable. In somecases the fire intensity is not high enough to let the firebrands jump from the main columnand produce new independent fire. Moreover, it can occur the situation when the firebrandjumps not that far, so it merges to the main fire changing the curvature of the fire front(see Figure 5).AcknowledgementsThis research is supported by the Basque Government through the BERC 2014-2017 pro-gram and by Spanish Ministry of Economy and Competitiveness MINECO through BCAMSevero Ochoa excellence accreditation SEV-2013-0323 and through project MTM2016-76016-R MIP and by the PhD grant ”La Caixa 2014”.References[1] G. Pagnini and A. Mentrelli, “Modelling wildland fire propagation by trackingc�CMMSE ISBN: 978-84-617-8694-7Page 811 of  2288Wildland fire propagation modelingFigure 3: Fire-spotting effect (initial moment t = 0 and t = 74 min).random fronts,” Natural Hazards and Earth System Sciences, vol. 14, no. 8, pp. 2249–2263, 2014.[2] I. Kaur, A. Mentrelli, F. Bosseur, J.-B. Filippi, and G. Pagnini, “Turbulenceand fire-spotting effects into wild-land fire simulators,” Communications in NonlinearScience and Numerical Simulation, vol. 39, pp. 300 – 320, 2016.[3] I. Kaur and G. Pagnini, “Fire-spotting modelling and parametrisation for wild-landfires,” International Congress on Environmental Modelling and Software, Paper 55,2016.[4] M. Sofiev, T. Ermakova, and R. Vankevich, “Evaluation of the smoke-injectionheight from wild-land fires using remote-sensing data,” Atmospheric Chemistry andPhysics, vol. 12, no. 4, pp. 1995–2006, 2012.[5] M. I. Asensio and L. Ferragut, “On a wildland fire model with radiation,” Inter-nationalL Journal for Nuerical Methods in Engineering, vol. 54, pp. 137-157,2002.[6] J. Mandel, L. S. Bennethuma, J. D. Beezley, J. L. Coenb, C. C. Douglas, M.Kima, and A. Vodacek, “A wildland fire model with data assimilation,”Mathematicsand Computers in Simulation, vol. 79, pp. 584-606,2008.c�CMMSE ISBN: 978-84-617-8694-7Page 812 of  2288V.N. Egorova, G. Pagnini, A. TrucchiaFigure 4: Fire-spotting effect with wind changing (t = 90 and t = 110 min).� ���� ���� ���� ���� ������ ����������������������������������������������������������Figure 5: Secondary fire merges to the main fire.c�CMMSE ISBN: 978-84-617-8694-7Page 813 of  2288

Wildland fire propagation modeling: fire-spotting parametrisation and energy balance

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Wildland fire propagation modeling: fire-spotting parametrisation and energy balance

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