IntroductionTop

Although many fire spread metrics can be analysed in the field of forest fire behaviour modelling, such as fuel time to ignition (Madrigal et al., 2011), flame residence time (Burrows, 2001), and flame geometry (Nelson & Adkins, 1988), spread rate (R) prediction is the focus of most studies. R estimates can be useful to assist fire management activities, such as prescribed burning (Fernandes et al., 2009) or wildfire suppression (Finney, 1998).

R models can be obtained via two distinct methods (Van Wagner, 1971): a physical approach, i.e., a mathematical description of the processes behind fire spread (Linn et al., 2002), or an empirical approach, i.e., the development of relationships between fuel and environmental parameters, derived from laboratory (Rossa et al., 2015a) or field fires (Fernandes et al., 2000). Nevertheless, because of key limitations associated with physical models (Cruz et al., 2017), such as complexity and high computation time, support to fire management operations is and will continue to be based on empirically-based predictions for the foreseeable future (Sullivan, 2009).

Typical empirical R formulations (Cruz et al., 2015) account for the fuel moisture content (M) effect through an M-damping function, hereafter called fuel content attenuation factor (fM). Most frequently, fM-functions are an exponential decay of the type exp(-b M) (Cheney et al., 1993; Fernandes, 2001), but a power law of the type a M-b is sometimes used (Cheney et al., 2012), where a and b are fitted coefficients. Both functional forms have advantages and shortcomings. Exponential decay fM vary between 0 (M = 8) and 1 (M = 0%) and allow obtaining a theoretical maximum R, i.e., when fuel is moisture-free. However, because exponentials do not fit well to wide M-variations (Rossa & Fernandes, 2017a), extrapolations far outside the development M-range can be inaccurate. On the other hand, power law fM provide a good fit to large M-intervals (Rossa, 2017), but do not offer reliable estimates for very low M-values because R tends rapidly to infinity when M approaches zero (Rossa & Fernandes, 2018a).

Although Rossa & Fernandes (2017a) show a very similar M-effect on R in no-wind and no-slope (R0), slope- (RS), and wind-driven (RU) laboratory fires, currently, no fM-function has been confirmed for the suitability to a general fire spread situation. In the present work, the hypothesis that a generic fM can be used in empirical R models was tested. fM was developed from the heat per unit mass of fuel requirements to ignite the fuel (Qi) and does not have the above-mentioned constraints of exponential decay and power law functions. fM was used to build R models from laboratory data and the ability to incorporate fM in existing field-based models was also verified.

MethodsTop

Fuel moisture content attenuation factor

Several factors beyond the heat needed to dry-out and ignite the fuel ahead of a flaming front have been attributed to the M-damping effect on R (Catchpole & Catchpole, 1991), such as the entrainment of moisture into the combustion zone and the attenuation of infra-red radiation by water vapour released from unburnt fuel. Still, not discarding those effects, in the present work Qi will be assumed as the main responsible for slowing down fire spread. Qi is given by (Rossa & Fernandes, 2018b):

where cf, cw, Ti, Tf, Tv, and Qw, are, respectively, fuel specific heat, water specific heat, fuel igniting temperature, fuel initial temperature, water boiling temperature, and water latent heat of evaporation. In physically-based formulations (Thomas, 1971; Rothermel, 1972), Qi is commonly used to account for the M-damping, as opposed to field-derived models. The relative M-effect on R, i.e., fM, results from dry-to-wet fuel ignition needs ratio:

Although exponential decay or power law fM-functions used in field-based R models are generally based solely on M, they implicitly account for the main variables determining the energy requirements to achieve ignition, i.e., Tf and M (Eq. [1]). But because Tf and M are correlated for dead fuels, and dead fuels are present in most real-world fuel beds, specific fM-factors work fine without explicitly accounting for Tf. This does not apply if fM is based on Qi. As a result, defining the numerator of Eq. [2] requires establishing Tf for which M will become 0%. Otherwise, predicted fM will be systematically above real fM values, causing an over-prediction bias. I assumed that fuel will attain moisture-free conditions at Tf = 100 ºC, which is water vaporization temperature and also roughly the temperature recommended to oven dry fuel samples (Matthews, 2010). If we consider the physical constants in Eq. [1] to be cf = 1.72 kJ kg-1 ºC-1 (Balbi et al., 2014), cw = 4.19 kJ kg-1 ºC-1, Ti = 320 ºC, Tv = 100 ºC, and Qw = 2260 kJ kg-1 (Catchpole & Catchpole, 1991), we obtain:

Because it is not easy to measure or estimate Tf, air temperature (Ta) was used as a surrogate. fM can theoretically vary between 0 and 1, as in the case of an exponential decay. Throughout the remainder of the paper fM is Eq. [3], unless otherwise stated.

The M-effect on R will be restricted to fine fuels, which are responsible for ‘carrying the fire’ (Catchpole et al., 1993). M represents fuel bed overall water content and, hence, is obtained by weighing dead (Md) and live (Ml) fuel moisture contents based on mass fractions in fuel beds composed of dead and live fuels (Rossa & Fernandes, 2017b). Usually, fuel bed M < 20% is achieved when vegetation is composed only of dead fuels, which respond to Ta variations. As Md gets closer to zero, lowering its value requires an exponential Ta increase. On the other hand, fuel bed M > 20-30% is typically attained when vegetation also contains live fuels, whose Ml is insensitive to Ta. To obtain a continuous plot of fM as a function of M, I considered an exponential Tf decrease between 100 ºC for M = 0% and an arbitrary value of 15 ºC for M = 20%, and constant Tf = 15 ºC for M > 20%.

Laboratory data

A total of 282 laboratory fires were retrieved from several sources (Table 1). R0 tests (n = 181) compiled in Rossa & Fernandes (2018a) include experiments from Rossa (2009) and Oliveira (2010), and pertain to fire spread in litter, slash, and shrub-like fuel beds, i.e., vertically placed tree branches with or without a surface litter layer. Fuel beds were built using quasi-live, i.e., collected live with M decreasing as a function of storage time, and dead vegetation of several species (Pinus pinaster Ait., Eucalyptus globulus Labill., Eucalyptus obliqua L'Her., Acacia mangium Willd., Quercus robur L., Pinus resinosa Sol. ex Ait.).

Table 1. Data sources and summary of fuel bed, ambient, and fire spread metrics.

RS burns (n = 50) with slope angle (S) set to 20º were retrieved from Rossa et al. (2016). Fuel beds were made of vertically positioned quasi-live shrub and tree branches of four species: Acacia dealbata Link., Cytisus striatus (Hill) Rothm., P. pinaster, and E. globulus. In the A. dealbata tests, air-dried leaves had contracted folioles, because they fold inward when branches are cut from the plant and surface-to-volume ratio is greatly diminished, attaining a fire behaviour similar to the remaining fuel species.

The RU experiments (n = 51) from Rossa & Fernandes (2017a) were carried out under constant wind speed (U) of 8 km h-1 wind in shrub-like fuel beds, composed of vertically placed quasi-live tree branches over a dead litter layer. P. resinosa and P. pinaster needles were over-layered by P. pinaster branches, and E. globulus leaves were over-layered by E. globulus branches. In all laboratory trials (R0, RS, RU), only the foliar fuel component was considered for computing oven-dry fuel bed load (w) and density (?b) in vegetation containing woody elements.

Experimental field fires and wildfires data

The applicability of fM to real-world fire spread was tested based on 123 outdoors fires (experimental and wildfires). A comprehensive data set (n = 100), representative of global shrubland fire behaviour, was retrieved from Anderson et al. (2015), which compiled data from Catchpole (1987), Vega et al. (1998), Fernandes (2001), Vega et al. (2006), Anderson (2009), and Cruz et al. (2010).

Wildfires in fully cured grasslands (n = 23), com­piled by Cheney et al. (1998), were used to test fM for fire spread in very low M conditions, seldom attained in experimental fires. Data provenance was Cheney et al. (1998) own observations, McArthur (1966), Finocchiaro et al. (1970), Douglas (1970), McArthur et al. (1982), Rawson et al. (1983), Keeves & Douglas (1983), Maynes & Garvey (1985), and Noble (1991). Fuel beds were undisturbed, cut or grazed, and eaten-out pastures. Because Cheney et al. (1998) did not report M, the Noble et al. (1980) equation describing the McArthur (1977) model: Md = (97.7 + 4.06 RH) / (Ta + 6.0) – 0.00854 RH, where RH is relative humidity, was used to obtain M estimates.

Data analysis and modelling

fM was used to develop R0, RS, and RU models from the laboratory fire spread data. In the Rossa & Fernan­des (2018a) R0 formulation based on fuel bed height (h) and M, the h-exponent is close to unity. So, for the sake of simplicity, a linear h-effect was assumed. The present R0 model was obtained by linear fitting R0 to h fM. In the case of RS and RU data, structural fuel bed metrics of most trials were close to the experimental mean, despite some variation between observed minimum and maximum h and w values. Also, both S and U were kept constant. As a result, M was the parameter with most influence on RS and RU, and both models were obtained by establishing a linear relationship between R and fM.

Both studies where field fires were compiled (Che­ney et al., 1998; Anderson et al., 2015) pro­vide R models accounting for the M-effect through an exponential decay fM, which, like Eq. [3] fM, varies in the 0–1 range. Thus, the concept of using a generic fM-function was tested by using the original R models, substituting their original (specific) fM by the proposed generic fM. In mixed live and dead fuel complexes, this exchange can only be done if the specific fM-function accounts for both Md and Ml, as in Anderson et al. (2015). Specific fM were plotted against generic fM-values and predictions using both fM-functions were evaluated for comparison.

Goodness of fit of linear regressions was assessed based on the coefficient of determination (R2). All predictions (laboratory and field fires) were evaluated using deviation measures: root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and mean bias error (MBE) (Willmott, 1982).

ResultsTop

Both in the laboratory (6.0–179.3%) and outdoors (2.6–101.9%) fires the M-range was very wide (Table 1). Wildfires allowed testing fM for extreme fire spread conditions with U10 (measured at a 10-m height) up to 55 km h-1 and an impressive R of 383.4 m min-1 (23 km h-1). As expected, fM evolution with M (Fig. 1) resembles the M-damping plots obtained using power law fM-functions (Rossa, 2017), which are able to describe the M-effect well over wide ranges.

All laboratory R relationships yielded a good fit to the data (Fig. 2) with R2 between 0.651 and 0.9. Model evaluation (Table 2) confirms these figures, with MAE and MAPE, respectively, in the range 0.06–0.19 m min-1 and 16.2–28.9%. fM testing with field fires showed highly significant correlations (p<0.0001) between specific and generic fM-derived values (Fig. 3), respectively of 0.457 for shrubland and 0.995 for grassland fires. The lower correlation for shrubland suggests a diminished sensitivity of the generic fM to M. Nevertheless, generic fM produced accurate predictions of all field data (Fig. 4) and, in fact, allowed for an overall improvement in model performance, for example with a decrease in MAPE from 70.6 to 63.4% in shrubland fires and 26.7 to 24.8% in grassland wildfires. Of course, the quality of predictions is mostly dictated by the original R formulation and these results only demonstrate that the proposed generic fM is a reasonable surrogate for the specific fM.

Table 2. Model evaluation metrics (see Table 1 for details on fire spread data).

DiscussionTop

fM performance and applicability

Laboratory-based R models built with the generic fM showed good agreement with data. They yielded R2 slightly below those obtained using the original power law fM-based models (0.667–0.947), but significantly above the 0.566 and 0.665 values obtained for the RS and RU models using exponentials (Rossa et al., 2016; Rossa & Fernandes, 2017a, 2018a). Despite a small decrease in accuracy, when compared with the use of power laws, the generic fM provides important benefits, such as not becoming extremely sensible at very low M-values and allowing extrapolation to moisture-free conditions. The generic fM allowed improved prediction ability in relation to the specific fM-functions used in existing field-based models for shrubland experimental fires and grassland wildfires.

Laboratory data included a great number of tests in several fire spread conditions over a wide M-range, and fuel beds were very diverse in terms of species and structure. R0 laboratory tests are representative of field R0 and a reasonable surrogate for backing fires R (Rossa, 2017; Rossa & Fernandes, 2018a). That is not the case of slope and wind-driven laboratory trials, in which R is limited by the fire front width (Fernandes et al., 2009). Shrubland and grassland outdoors fires enabled the positive testing of fM in RU conditions free of scaling issues. There is no apparent reason for fM not to hold for slope-driven field fires as well. Not excluding the need of further assessing fM with additional field data, its overall performance in all tested fire spread situations lends strong support to its ability of successfully incorporating empirically-based R models in generic fire spread conditions.

If Eq. [3] were developed without assuming that moisture-free conditions will be attained at Tf = 100 ºC, i.e., with the numerator becoming 1.72 (320 – Tf) instead of 378.4, using the generic fM in the field-derived R models would yield MBE of 3.92 and 41.7 m min-1, respectively for shrubland and grassland fires. The arising of this substantial over-prediction bias lends support to the supposition that fM-functions based only on M implicitly account for Tf. In other words, this means that in a hypothetical situation of fire spread through a dry fuel bed at, for example, Tf = 20 ºC, predicted R using typical empirical field-based models would be higher than observed because the M-functions were fitted in conditions where the decrease in M is concurrent with increasing Tf. As a result, estimated fM attains its maximum, i.e., fire spread attenuation is minimum, although fuel conditions will delay fuel ignition more than expected in an extrapolation to M = 0%, where Tf was supposed to grow concomitantly with diminishing M. It is important to notice that this rationale was derived from results using a limited field data set, hence further testing with additional data would benefit its confirmation.

Advantages and limitations

Md of field fuels is easy to sample. Overall M determination requires measuring both Md and Ml (Rossa et al., 2015b), as well as assessing dead and live fuel mass fractions, which may be problematic in very heterogeneous fuel complexes. This is a limitation of using the generic fM, when compared to fM-functions accounting for only the Md-effect. Most empirical fuel-dependent models rely on the sole use of Md (Cruz et al., 2015) to provide a satisfactory R explanation, which restricted the data available to test the specific fM-function proposed in the present work. Field-based models based only on Md work well because, usually, Ml is either constant or correlated with Md for a given fuel complex (Rossa & Fernandes, 2017b).

Nevertheless, especially for experimental programs composed of a limited number of tests, possible diffi­culties in assessing overall M might pay-off in terms of the advantages of using a generic fM. The use of experimental outdoors fires as a source of development data is appealing because of the strong resemblance to real-world fire-spread. However, this option is often challenged by heterogeneity in fuel bed properties and correlated fuel descriptors, which elude the correct quantification of specific effects (Rossa & Fernandes, 2017b). Establishing a priori the M-effect through the use of fM significantly simplifies the proper assessment of the remaining influent variables.

ConclusionsTop

A generic fM-function for empirical R models was developed based on the assumption that the main M-damping effect is a function of Qi. fM was successfully used to derive R models from laboratory fire spread in no-wind and no-slope, slope-, and wind-aided conditions. The ability to incorporate fM in existing field-based models was also positively assessed. Possible difficulties in assessing overall M due to fuel complex heterogeneities, might pay-off in terms of the advantages of using a tested generic fM. For example, establishing a priori the M-effect benefits the proper quantification of the remaining variables influence. Not excluding the need of further assessing fM with additional field data, its overall performance in all tested fire spread situations lends strong support to its ability of successfully incorporating empirically-based R models in generic fire spread conditions.

AcknowledgmentsTop

The author acknowledges the anonymous reviewers and the section editor for the thoughtful comments and suggestions that contributed to improve an early version of the manuscript.