1 Introduction

A standardized approach is essential for forecasting the spread and intensity of fires in forests, brush fields, and grasslands [1,2,3,4,5,6], as well as for forest managers and researchers studying these environments [7, 8]. The fire spread model presented in this study provides, for the first time, a technique for calculating quantitative assessments of fire intensity and spread rate in fuels that adhere to the model's fundamental assumptions [9, 10]. This model includes gas and field-measurable meteorological characteristics as inputs. It is acknowledged that this steady-state fire situation model is just the beginning in simulating wild land fires, but its initial applications to the National Fire-Danger Rating System and its support in the evaluation process affirm its broad suitability [11].

2 Methodology

To maximize the model's applicability, it was developed on a robust theoretical basis. This foundation employed the principle of energy conservation to determine the quantity of fuel required to propel a unit volume of fire through a uniformly fueled bed. His findings revealed the following:

$$R = \frac{{I_P + \smallint_{ - \infty }^0 \left( {\frac{\partial I_z }{{\partial z}}} \right)_{z_c } dx}}{{\rho_{bl } Q_{gn} }}$$
(1)

where:

R: represents the quasi-steady spread rate as a random percolation, measured in feet per minute.

\(I_P\) is the amount of heat absorbed per unit volume of fuel during horizontal heat flux at ignition, expressed in British Thermal Units (BTU) per square foot per minute.

\(\rho_{bl }\) denotes the optimal bulk density, which is the amount of fuel per unit volume of the fuel bed that is brought to ignition ahead of the advancing fire, measured in pounds per cubic foot.

\(Q_{gn}\) is the pre-ignition heat, indicating the heat required to bring a unit weight of fuel to the point of ignition, in BTU per pound.

\(\left( {\frac{\partial I_z }{{\partial z}}} \right)_{z_c }\) is the vertical intensity gradient measured at a plane at a specific depth \(z_c\) within the fuel bed, in BTU per cubic foot per minute.

In the mathematical framework, the horizontal and vertical coordinates are denoted as x and z, respectively. Frandsen's analysis focuses on a unit volume moving towards the interface at x = 0 from x =  − ∞, with the fuel-reaction zone contact maintained at a constant depth, \(z_c\). At the interface, ignition of the unit volume occurs. According to Eq. (1), the spread rate during the quasi-steady state can be understood as a ratio where the numerator represents the heat flux received from the source, and the denominator signifies the heat required for ignition by the potential fuel. In Eq. (1), there are heat flux terms for which the heat transmission mechanisms remain unidentified.

As a result, an analytical solution was currently unattainable. It became crucial to evaluate each term and establish experimental and analytical methods for their assessment in order to address Eq. (1). This required the introduction of new concepts, which eventually provided a preliminary solution to Eq. (1).

2.1 Beta bulk density random variable

The objective of computational science is to develop models that accurately predict natural phenomena. However, these models often rely on parameters that are uncertain. In this study, we consider the effect of randomness, assuming it follows a beta bulk density random variable. We aim to calculate and simulate the mean and variance of this random variable. A real random variable X is defined on the probability space (Ω, F, P) and has the probability density function:

$$f\left( x \right) = \frac{{\left( {x - a} \right)^{\alpha - 1} \left( {b - x} \right)^{\beta - 1} }}{{B\left( {\alpha ,\beta } \right)\left( {b - a} \right)^{\alpha + \beta - 1} }},\,a \le x \le b,\alpha ,\beta > 0$$

where a and b are the lower and upper bounds of the distribution, and α and β are the shape parameters. The term B(α,β) represents the beta function and \(B\left( {a,b} \right) = \int_0^1 {x^{\alpha - 1} \left( {1 - x} \right)^{\beta - 1} dx}\). If a = 0 and b = 1, it is referred to as the standard beta distribution. The expectation value of X, denoted by E[X], is given by \(\frac{\alpha }{\alpha + \beta }\).

2.2 Random an effective heating number

The pre-ignition heat (\(Q_{gn}\)) is a function of the fuel moisture to ovendry weight ratio (\(M_f\)) and the ignition temperature (\(T_{ig}\)).

This relationship indicates that the energy required to initiate combustion (pre-ignition heat) is dependent on both the moisture content of the fuel and its ignition temperature. Higher moisture content generally requires more heat to evaporate the water before the fuel can ignite, and different materials have different ignition temperatures. The function f (\(M_f\), \(T_{ig}\)) mathematically encapsulates this interplay, which is a critical factor in understanding and predicting fire behavior.

$${\text{Q}}_{{\text{gn}}} = f\left( {M_f ,T_{ig} } \right),B.t.u./lb$$
(2)

where:

\(M_f\) represents the ratio of fuel moisture to its oven-dry weight.

\(T_{ig}\) denotes the ignition temperature.

The equation \(\varepsilon = \frac{{\rho_{bl} }}{\rho_b },\) represents the ratio of the effective bulk density \(\rho_{bl}\) to the actual bulk density \(\rho_b\) of the fuel. This ratio, denoted as ε, is a crucial parameter in understanding fire dynamics. The ratio ε is significant because it provides insight into the efficiency and effectiveness of the fuel bed for sustaining a fire. A higher value of ε indicates that a larger proportion of the available fuel contributes effectively to the fire, which can influence the intensity and spread rate of the fire. Understanding this ratio is essential for predicting fire behavior and develo** accurate fire spread models. In a physical context, especially when considering the effective heating number as a random variable, it reflects the variability in the fuel's capacity to absorb and retain heat, which is crucial for ignition in fire spread models. This ratio, representing the effective bulk density to the actual bulk density, essentially captures the efficiency of the fuel bed in terms of its heat absorption and retention characteristics.

In more practical terms, it quantifies the difference between the theoretical or 'ideal' fuel density that would be required for efficient combustion, and the actual observed fuel density in a natural setting. As a random variable, it acknowledges that this efficiency can vary due to numerous factors such as moisture content, fuel type, and environmental conditions. Thus, it serves as a critical factor in predicting fire behavior under varying and uncertain conditions, making it a valuable tool for analysis and interpretation in fire management and research. A dimensionless quantity known as the effective heating number will be close to one for fine fuels and fall toward zero as fuel size rises. Therefore, we can assume that, the effective heating number has a beta random variable.

3 Results and Discussion

3.1 Flux Propagation under the randomness of ε

Flux propagation refers to the transmission or movement of flux (e.g., heat flux or mass flux) through a medium. In the context of fire dynamics or physical systems, it particularly pertains to how heat or mass moves through different materials or over space.

In fire science, understanding flux propagation is crucial for modeling and predicting how fires spread. This involves analyzing how heat transfers from the combustion zone to unburned fuel, potentially igniting it and contributing to the fire's progression. The mechanisms of this propagation can involve conduction (heat transfer through direct contact), convection (heat transfer through fluid motion, like air or gases), and radiation (heat transfer through electromagnetic waves).

Flux propagation is a key component in the behavior of fires, influencing the rate and direction of fire spread, the intensity of the heat generated, and the overall impact on the environment. Accurately modeling this phenomenon is essential for develo** effective fire management strategies, predicting the behavior of fires, and ensuring safety measures in fire-prone areas.

The effect of randomness in ε (the ratio of the effective bulk density to the actual bulk density of the fuel) can significantly influence flux propagation, particularly in the context of fire spread and heat transfer.

Randomness in ε introduces variability in how efficiently the fuel bed can propagate the heat flux. A higher ε suggests that a larger proportion of the fuel is effectively contributing to the fire, potentially leading to more intense and faster-spreading fires. On the other hand, a lower ε indicates that less of the available fuel contributes to the fire, possibly resulting in slower spread rates and less intense heat flux.

Fire spread models often assume certain conditions about the fuel and its environment. Randomness in ε introduces uncertainty into these models, affecting the accuracy of predictions about fire behavior, spread rates, and intensity.

The propagation of heat flux involves conduction, convection, and radiation. Variability in ε can alter the relative importance or effectiveness of these mechanisms. For instance, a denser, more effective fuel bed (higher ε) may conduct and radiate heat more efficiently, affecting the spread rate and the way heat is distributed in the surrounding area.

Understanding and predicting the behavior of fires is crucial for effective fire management and mitigation strategies. Randomness in ε makes it challenging to predict how quickly a fire will spread and how intense it will become, complicating efforts in fire suppression, evacuation planning, and resource allocation.

In scenarios where human safety is at risk, understanding the potential variability in fire behavior due to randomness in ε is crucial for develo** effective evacuation protocols and safety measures.

Therefore, randomness in ε introduces a layer of complexity and uncertainty in understanding and predicting flux propagation in fires. It affects the efficiency of fuel beds in propagating heat, influences the mechanisms of heat transfer, and poses challenges to fire modeling, management, and safety protocols. Addressing this randomness and its implications is essential for accurate fire behavior predictions and effective fire management strategies.

The propagating flow has units of heat per unit area and per unit time and is the numerator of the RHS of Eq. (1). \(I_P\) represents the propagating flux.

$$I_{FP} = I_P + \mathop \smallint \limits_{ - \infty }^0 \left( {\frac{\partial I_z }{{\partial z}}} \right)_{z_c } dx,B.t.u/ft^2 - \min .$$
(3)

The propagating flux \(I_{FP}\), quantifies the rate at which heat is transferred across a given area over time. In fire science, this is critical for understanding how quickly heat is moving to unburned fuel, potentially leading to ignition and contributing to the spread of the fire.

For the no-wind case then, from Eq. (3) into Eq. (1) and let \(I_{FP} = (I_{FP} )_0\) and \(R = R_0\), we have:

$$(I_{FP} )_0 = R_0 \rho_P \varepsilon Q_{gn} ,B.t.u./ft^2 - \min$$
(4)

3.2 Effect of slope and wind

The effect of slope and wind, under the influence of randomness in ε (the ratio of the effective bulk density to the actual bulk density of the fuel), introduces additional layers of complexity in understanding and predicting fire behavior. ε represents inherent uncertainties in fuel properties and their interaction with heat, and when you factor in the variable nature of slope and wind, the behavior of fire becomes even more complex and unpredictable. Here’s how slope, wind, and randomness in ε might interact:

Slope can significantly influence the spread of fire. Fires tend to spread more rapidly uphill due to the pre-heating of fuel above the fire by the rising hot air. The randomness in ε can affect how this pre-heating interacts with the fuel's capacity to absorb and transmit heat, leading to variability in the rate of uphill fire spread.

Variability in Fuel Characteristics: On slopes, fuel distribution and characteristics can vary greatly, which can further influence ε. This variability can impact how consistently the fuel contributes to the fire, making the behavior of the fire on slopes even more unpredictable.

Wind can dramatically alter a fire's behavior, affecting its direction and rate of spread. The wind can carry heat to unburned fuel, pre-heating it and making conditions more conducive to fire spread.

The randomness in ε means that the fuel's response to wind-driven heat transfer can be unpredictable. The efficiency of heat absorption and transmission by the fuel can vary, affecting how the fire responds to changing wind conditions.

The combined effects of slope, wind, and randomness in ε create highly complex and dynamic fire behavior. The unpredictability of fuel properties, along with varying topographical and meteorological conditions, makes it challenging to predict fire spread and intensity accurately.

Incorporating the variability of ε, along with the dynamic nature of slope and wind conditions into fire spread models requires sophisticated modeling techniques. It necessitates a comprehensive understanding of the interplay between these factors and advanced computational methods to simulate realistic fire behavior.

Effective fire management in the presence of variable slope, wind, and ε requires advanced prediction tools that can account for these complexities. It's crucial for develo** effective evacuation plans, fire containment strategies, and resource allocation during fire incidents.

Risk Assessment and Mitigation: Understanding the combined effects of these factors is vital for assessing fire risk accurately and implementing appropriate mitigation strategies, especially in areas prone to wildfires.

Let \(\varphi_W\) and \(\varphi_S\) indicate the extra propagating flux that the slope and wind cause. They are dimensionless coefficients that depend on the fuel parameters, the slope, and the wind. They must be assessed using the results of the experiments. The phrase, which is how the total propagating flux is represented,

$$I_{FP} = (I_{FP} )_0 \left( {1 + \varphi_W + \varphi_S } \right)$$
(5)

Inserting the approximate relationships, Eq. (1) becomes:

$$R = \frac{{(I_{FP} )_0 + \left( {1 + \varphi_W + \varphi_S } \right)}}{{\rho_{b } \varepsilon Q_{gn} }}$$
(6)

We can discuss the relation between {\(R, \varphi_W ,\varphi_S\)} as in Fig. 1 under the consideration of randomness of \(\varepsilon\) for 100, 1000 sample sizes and, we can also find some statistical measures such as mean, variance, and the probability density function of the rate spreading variable as in Fig. 2.

Fig. 1
figure 1

The relationship between the wind factor (\(\varphi_W\)), the slope factor (\(\varphi_S\)), and the fire spread rate (R) for 10, 100, 1000 sample size,\(\varepsilon\) has Beta distribution

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Fig. 2
figure 2

The statistical measures of t fire spread rate (R) as \(\varepsilon\) has Beta distribution

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The colors in Fig. 1 represent the density of points, providing insights into the distribution of probable values for R based on specific sets of \(\varphi_W\) and \(\varphi_S\) values. It shows how both the wind factor and the slope factor can influence the fire spread rate. The colors represent the density of points in a certain area of the space. Warmer colors (like yellow and red) indicate a higher concentration of points, suggesting a higher probability for those values of R, while cooler colors (like blue) indicate a lower concentration. By analyzing how the values of R change with \(\varphi_W\) and \(\varphi_s\), one can understand how these two factors interact with each other and how they influence the intensity and speed of fire spread. This information provides valuable insights for individuals responsible for fire management and planning preventive measures. It can be used to estimate the most critical scenarios and refine strategies for fire response.

Figure 2 is a meticulously crafted histogram that represents the probability density of fire spread rates (R). This visualization is marked by a spectrum of blue shades, reflecting the distribution of R values across the dataset. At the core of the plot, a prominent red dashed line denotes the mean of R, serving as a central reference point that signifies the average fire spread rate under the considered conditions. Flanking this mean, two green dashed lines demarcate the bounds of one standard deviation from the mean, encapsulating the spread or variance of the data. These lines vividly illustrate the central tendency and dispersion of the fire spread rates, offering a clear, visual understanding of the distribution's shape and spread. The plot not only provides a quantitative analysis of the fire spread rates but also a visual narrative of how these rates vary under the influence of inherent stochastic factors in the model.

4 Conclusions

This study highlights the crucial role of stochastic variability, particularly in the parameter ε, in understanding and predicting fire spread rates. By integrating various probability distributions to model the randomness inherent in fuel properties and environmental factors, we have provided a nuanced perspective on the interactions between wind, slope, and fuel density. The resulting 3D visualizations and density plots reveal a complex, multidimensional landscape of fire behavior, underscoring the importance of incorporating stochastic elements into fire behavior models. This approach not only enriches our comprehension of fire dynamics but also offers a solid foundation for refining fire management and mitigation strategies, thereby contributing to the safety and resilience of fire-prone ecosystems.