Modelling of landfill gas generation: a review

Abstract

Landfill gas generation models apply utilizes several different parameters to project Methane (CH₄) generation from a specific mass of disposed waste over a time period in a landfill site. These models are used for better estimating the size of landfill gas (LFG) collection systems, monitoring objectives, assessments, and prognostications. Compared to other options to estimate and control LFG production (such as the application of the test wells), models offer advantages due to the relatively prompt results and cost-effectiveness. Over the recent years, developing LFG generation models has become precedence for the LFG industry. The main trouble in designing and operating an LFG collection system is the uncertainty of LFG generation rates. The LFG generation rates are presently measured via models associated with the waste disposal history, moisture content, gas collection systems, and cover type, which have significant uncertainties. From literature studies, there is not sufficient data regarding the comparison between different models or calibrating them with CH₄-filled landfill data and the CH₄ generation and recovery models are not adequately progressive. The purpose of this study is to provide a comprehensive review of the commonly used models that predict LFG generated in landfill sites as well as discuss their specific characteristics. As a result, these numerical models are categorized in mathematical, numerical, and zero-, first-, and second-order decay models. Since first-order decay models are extensively used throughout the world, their consideration as multiphase models to make more appropriate projections is discussed in more detail.

1 Introduction

The landfill gas (LFG) is produced from a series of chemical and biological reactions that typically occur in the mass of a disposed waste through landfills. An LFG is generally composed of methane (CH₄) (50–60%), carbon dioxide (CO₂) (40–50%), nitrogen, water vapor, and other innumerable trace gases (Sabour et al. 2007). CH₄ and CO₂ are two of the foremost greenhouse gases (GHG), which makes the study of LFG generations very vital since their accumulation in the atmosphere may result in extreme climate change. CO₂, CH₄, ozone (O₃), vapor, and nitrous oxide (N₂O) are considered as the major compounds found in the atmosphere causing greenhouse effect (US EPA 2016). According to the US EPA (2016), the CH₄ accumulation in the atmosphere has doubled over the last two centuries and this trend keeps rising, although the rate of increase is decelerating. Over a 100-year time period, CH₄ has the global warming potential 21 times greater than CO₂ by mass (Cohen 2016). Sanitary landfilling is a common method of waste disposal. Figure 1 illustrates a landfill site including biogas collection systems, leachate collections and monitoring equipment. Landfill is considered a major anthropogenic CH₄ generation source, which contributes to the accumulation of CH₄ in the atmosphere (Kamalan 2009). Based on this alarming concern, several regulatory principles have been recently released to estimate and/or manage LFG including Transfer Registers (also called Kiev Protocol or PRTRs) and the Protocol on Pollutants Release (Scharff and Jacobs 2006). Additionally, CH₄ is widely recognized for its significant capacity as a source of energy. In order to use it as an energy source, especially in a large-scale generation, it is important to estimate its production in both quantity and quality (Sabour and Kamalan 2006). According to the literature, several studies have recently implemented modelling approaches to estimate the energy recovery potential of the generated LFG with the consideration of modelling uncertainties (Amini and Reinhart 2011).

Fig. 1

Different sections of a landfill site

All the aforementioned issues have paved the way for the development of various models to provide a better estimation of CH₄ generation in landfill sites. A majority of these models have been developed according to Monod equation and first order decay, such as Gassim, TNO, Afvalzorg, EPER, LandGEM, IPCC, and LFGEEN (Donovan et al. 2010; Ishii and Furuichi 2013; Krause et al. 2016; Penteado et al. 2012; Kumar et al. 2016; Kamalan 2016). Other models are defined based on Monod equation and zero order decay, such as EPER developed in France. A few models are based on the sequential biological growth to estimate CH₄ emissions such as Halvadakis model (Nastev 1998). Furthermore, some newly developed numerical models utilize neural network and weighted residual methods (Ozakaya et al. 2006; Shariatmadari et al. 2007). The main objective of this study is to offer a comprehensive review on all of these models coupled with their advantages and disadvantages to provide a guideline for researchers, scientists, and policy makers.

2 Landfill gas modelling

In order to recover LFG, the first and foremost stage is to come up with an appropriate estimation of gas generation within the landfill sites. To obtain this goal, numerous LFG generation models have implemented distinct approaches. Nevertheless, the ultimate aim of the LFG modelling is to offer the maximum accuracy for estimating LFG generation. Generally, the modelling of LFG generation could be accomplished using zero order, first order, second order, and/or multi-phase generation models. According to the data released in the previous studies, zero-order model results are not trustworthy due to relatively high levels of inaccuracy (Scharff and Jacobs 2006). When model outcomes are compared to the actual measurement data, higher order models indicate lower inaccuracies (Oonk et al. 1994) because replacing the first order with a second order or a multi-phase model makes the modelling process much more complicated; therefore, most users prefer using a first order model.

Although several mathematical and numerical models have been developed according to different approaches over the recent years, these models are not readily used since they are available only to the developers. The necessary parameters in these models are often unreliable and they adversely affect the accuracy of the model. Since mathematical and numerical models are complex, several simplified empirical models have also been developed. Although these models are based on the mathematical and numerical models, some processes have been modified to augment simplicity. In fact, the numerical models are not as accurate as the simplified models due to the uncertainty in the influential parameters that reduces the reliability of the numerical models. Some of these models are integrated with special computer software programs to make them more user-friendly. These include the IPCC, the E-PLUS, the French ADEME models, the Dutch Multiphase (AMPM), and the US EPA LandGEM model. Table 1 shows several widely used empirical models as well as their main parameters and their orders.

Table 1 Recently developed empirical LFG generation models (Faour 2003; Machado et al. 2009; Thompson et al. 2009)

For example, the US EPA LandGEM model estimates emission rates for CH₄, non-CH₄ organic compounds (NMOCs), CO₂, and other toxic air pollutants from landfill sites (Pierce et al. 2005). The Triangular model assumes that the waste decomposition occurs in two phases, while the E-PLUS model, also developed by the US EPA, estimates the benefits and costs of the LFG collection projects as well as predicts CH₄ flow, NMOCs emissions, and LFG flow (Pierce et al. 2005). The preliminary step of gas generation begins after the first year of deposition, followed by a linear increase from zero in the first year to a maximum amount in the sixth year. After 16 years of deposition, the rate linearly declines to zero (Mor et al. 2006). The ENCON model provides the users with the possibility of inputting specific waste environmental conditions associated with temperature, specific moisture input for individual refuse streams, the overall moisture conditions, and estimates the extraction efficacy (EMCON Associates 1980). According to Table 1, a majority of the presented empirical models are based on the first-order modeling. Landfill evaluators, designers, and operators have broadly used the Intergovernmental Panel on Climate Change (IPCC) and LFG generation models developed by the US EPA. Although the aforementioned models have small distinctions, the main parameters are defined similarly in all these models.

Several researchers have previously investigated the accuracy of the LFG models to estimate gas generation rate from the landfill sites. For instance, Thompson et al. (2009) investigated 35 landfills in Canada equipped with LFG collection systems, considering only those with adequate waste data and removing outliers. To study the precision of the model outcomes compared to the actual data collected from LFG records, they applied five different LFG generation models to all 35 landfills. Having considered the inefficiency in collecting data, a 20% loss factor was also applied in their studies. The five generation models such as the TNO model, the Scholl Canyon model, the zero-order German EPER model, the LandGEM version 2.01 model, and the Belgium model. The TNO and the German EPER models overvalued the LFG production rate by two- to six-fold. The LandGEM version 2.01, the Belgium were used, and the Scholl Canyon models also showed a standard error of less than 100%. According to that study, only LandGEM version 2.01 model underestimated the LFG generation, while the other four models overestimated the gas generation.

In another study, Willumsen and Terraza (2007) studied LFG generated from six landfills in Europe and South America. The main objective of their research was to provide a comparison between the gas generation rates estimated via various models as compared to the collected data. Results demonstrated that the data values from the Dutch Multiphase First-order Model, the IPCC First-order Model, and the US EPA LandGEM Model were underestimated by 14, 44, and 15%, respectively. Other models such as the Scholl Canyon First-order Model, the Rettenberger First-order Model, and the E-PLUS Model provided outcomes that were considerably different (up to 100%) from the actual data.

The LFG generation models have paved the way for predicting gas generation and recovery according to the operational parameters and waste disposal history. The precision in modelling is necessary to use the outcomes to expand the existing systems, design new ones, and/or evaluate active gas recovery systems. However, the estimation of several parameters that influence the LFG generation is the main challenge in modelling. In the next sections, several major factors affecting the LFG generation are briefly discussed.

2.1 Amount of waste and its composition

The waste composition typically affects both LFG generation capacity and the lag time prior to the gas generation (Karanjekar et al. 2015). As a result, disposed waste could be classified based on inertness and degradability (high, moderate, and poor) of the material. For instance, food waste is a highly degradable and favorable material for generating more gas compared to refuses that contain non- or poorly-degradable substances such as wood, paper, and plastics.

The disposed waste amount can also directly affect the LFG generation. The high amount of refuse being disposed can result in the high availability of resources for gas generation.

2.2 Time

The time is an important factor due to two concerning aspects: the lag time prior to the beginning of gas generation and the overall period of LFG generation (Barlaz et al. 2009). These two factors can have dire consequences on the conceptual design of a landfill. Monitoring the lag time and gas generation period enables the operators to predict the precise time to start the gas collection systems and maintain stable and satisfactory collection efficiency.

2.3 Moisture content

Higher moisture content can increase the rate of gas generation up to a specific optimum point (Bilgili et al. 2007). Based on the climatic conditions and waste composition, the moisture content could be highly variable. Higher moisture contents could result in a faster gas generation in landfills (bioreactor landfills), making it possible to collect LFG in a shorter time interval. Over the recent years, the importance of this parameter has increased as many landfill sites have collected the leachate and recycled it into the landfills. Based on the previous studies, this will cause a considerable acceleration in gas generation (Reinhart and Al-Yousfi 1996; Bergin et al. 2005). The main reason for this acceleration is that the gas generation reaches its peak faster and generates the LFG over a shorter time interval. In terms of the biological aspects, this leachate recirculation provides the anaerobic bacteria with the appropriate moisture content required for degrading waste and consequently producing LFG. Corti et al. (2007) reported that employing the aforementioned action resulted in generation of 95% of the entire LFG in a landfill 10 years sooner as compared to a conventional landfill. Some studies have also reported the optimum values for the moisture content and leachate recirculation rate. Mehta et al. (2002) conducted a side-by-side comparison of two 8000 metric ton test cells to assess the effect of leachate recirculation on the waste decomposition. They investigated various moisture content values including 14.6, 19.2, 31.7, 34.8 and 38.8%. The optimum moisture content was reported as 38.8% since the maximum decay rate occurred at this level of moisture content. In another study, Sponza and Ağdağ (2004) investigated the influence of leachate recirculation and its rate on the anaerobic degradation of the MSW in three simulated landfill bioreactors. A single pass reactor was operated without leachate injection, while the other two were operated under leachate recirculation rates of 9 and 21 L/day. The leachate recirculation rate of 9 L/day was found to be optimum since it accelerated the MSW mixture stabilization in simulated anaerobic bioreactors.

2.4 Temperature

Generally, microbial activities are influenced by the changes in temperature. An approximation states that each 10 °C increase in the temperature doubles the microbial activity (Pierce et al. 2005). However, this trend is valid only in the optimal range between 30 and 40 °C. The increase in temperature henceforth results in a decline in the microbial activity (Gebert et al. 2003).

Overall, the empirical models combine all of the aforementioned factors into derivative parameters such as the gas generation rate constant (k) and the gas generation potential (L₀), which are widely used in first-order models.

3 LFG modelling parameters

The CH₄ generation potential (L₀) indicates the capacity of a waste stream to produce a specific volume of CH₄ per unit mass; consequently, it is considered as a function of the disposed waste composition. The gas generation rate constant (k) is a parameter that represents the CH₄ generation time span of a waste stream under certain site specific conditions including waste depth, waste moisture content, oxidation potential, temperature, pH, alkalinity, waste particle size, and density (Garg et al. 2006; Machado et al. 2009). For instance, in a landfill with deeper waste cells, the moisture is better retained at the bottom layers, which accelerates the LFG generation thereby resulting in a shorter time period (Garg et al. 2006). Another hypothesis is that in landfills with higher depth, the insulation improves and the temperature increases, which also accelerates the CH₄ generation rate (Huitric and Rosales 2005). Considering the uncertainties with each of the influential factors, it is imperative and the most challenging part of the modelling to select an appropriate value for k and L₀.

The gas generation potential may be too variable since the waste stream composition is continuously changing over the lifetime of a landfill site due to the community lifestyle changes or expanding recycling plans (Machado et al. 2009). Additionally, each component in the refuse stream also influences CH₄ generation potential. Waste disposed in landfills is typically composed of lignins, cellulose, hemicelluloses, and proteins that are recognized as the major organic components, which are converted to CH₄ through chemical, physical, and biological processes (Barlaz et al. 1989, 1997). Lignins and cellulose show a considerably variable trend in their rate of degradation under various landfill conditions. For instance, lignins are regarded as recalcitrant compounds under anaerobic conditions. Besides, pH and temperature could also have a significant impact on the microbial activities in the waste disposed in the landfills (McBean et al. 1995). The influence of all the aforementioned components has resulted in quantification of L₀ under different conditions by adjusting the waste quantity using a biodegradation factor. Table 2 represents the values of the biodegradation factor for different waste materials suggested by previous researchers. In addition, moisture content regulates CH₄ generation through microbial activities by providing better conditions for the microorganisms to propagate (Barlaz et al. 1990).

Table 2 Different biodegradable fraction values recommended by previous researchers (Machado et al. 2009)

The appropriate values for k and L₀ could be determined via collected LFG data, theoretical predictions, or laboratory experiments. According to Machado et al. (2009), the major challenge in predicting the values through laboratory experiments is to simulate the actual conditions of an LFG site at lab-scale. In addition, theoretical predictions come up with maximum values for L₀, which are never reached during an experiment. Machado et al. (2009) assumed the main reason for this issue is that not all parts of the organic refuse are biodegradable and thus, the application of biodegradability factor is not applicable for all components. In order to predict LFG generation, most simplified models follow Eq. (1) but with different k and L₀ values.

$${\text{Q}}_{\text{g}} = \beta {\text{kML}}_{0} {\text{e}}^{{ - {\text{k}}}}$$

(1)

where Q_g is the annual LFG generation (m³ year⁻¹), β is the inverse ratio of the fraction of CH₄ content, M is the average annual solid waste acceptance mass (Mg), t is the age of disposed waste (year), L₀ is the CH₄ generation potential (m³ Mg⁻¹), and k is the LFG generation rate constant (year⁻¹).

Several organizations including IPCC and US EPA have suggested default values for k and L₀. For instance, k values according to the literature typically range from 0.01 to 0.21 year⁻¹ with 0.04 year⁻¹ considered as the most commonly used value (Pierce et al. 2005; Garg et al. 2006). However, other values such as 0.3 and 0.5 year⁻¹ have also been mentioned under certain situations such as the rapidly degradable portions of waste and bioreactor landfills (Faour et al. 2007; Ogor and Guerbois 2005).

The US EPA has suggested the value of 0.3 year⁻¹ for wet landfills, 0.02 year⁻¹ for landfills receiving less than 63.5 cm (25 inches) of rainfall per year and 0.04 year⁻¹ for those receiving over 63.5 cm (25 inches) of rainfall per year (US EPA 2016). Although the US EPA recommended values that provided the best fit based on 40 different landfills, the estimated CH₄ generation still has a deviation from 30 to 400% from actual measurements (US EPA 2016). Machado et al. (2009) have reported that high k values of about 0.2 year⁻¹ are related to high moisture and higher portions of rapidly biodegradable refuse. They discussed that the high moisture and portions of rapidly biodegradable refuse are due to the presence of high food waste with their respective values of 0.26 kg water/dry-kg and 0.70. Table 3 represents various k values as suggested by IPCC.

Table 3 Different k values (year⁻¹) recommended by IPCC (Machado et al. 2009)

Similarly, the values suggested for L₀ also vary depending on the parameters. Different studies have revealed a range of L₀ values from 6 to 270 m³ Mg⁻¹ based on different case studies (US EPA 2016). The US EPA has recommended the L₀ value of 100 m³ Mg⁻¹ for “as received waste”. Machado et al. (2009) used an L₀ value of 70 m³ Mg⁻¹ for tropical landfilling situations for on-site and lab-scale measurements. Bentley et al. (2005) calculated both k and L₀ values for seven case studies involving landfills via the application of baro-pneumatic measurements, which are presented in Table 4. The average values for L₀ and k calculated in that study were 107 m³ Mg⁻¹ and 0.153 year⁻¹, respectively. Another study by Budka et al. (2007) also reported different values of L₀, which are represented in Table 5.

Table 4 Different L₀ and k values as well as other landfills conditions reported by Bentley et al. (2005)

Table 5 Different L₀ outcomes reported by Budka et al. (2007)

Bentley et al. (2005) have also proposed a new method of using barometric pressure data to calculate derivative parameters of LFG generation rate. This method is based on estimating the LFG generation via measuring pressure responses under the landfill cover in comparison with the atmospheric pressure changes. In this method, pressure changes are regulated under and above the landfill cover through the application of sensors installed at different depths and areas for several consecutive days. The data released in this study indicated that the barometric pressure in the deeper zones of the landfill cells was higher than the atmospheric pressure and this increase continued with the depth. Bentley et al. (2005) also reported that the inside landfill pressure ranged from approximately the atmospheric pressure (163 mm H₂O) to 8.6 atm (1400 mm H₂O). They mentioned two main benefits from their method. Firstly, the interpretation of the data was possible via a quantitative gas flow equation based on Darcy’s law and continuity equations. Secondly, the data obtained from the pressure response is imposed over a huge area which reduces the necessity to interpolate between spatial zones to provide accurate gas generation estimation.

In another study, Garg et al. (2006) developed a model to calculate the k value via the application of a fuzzy synthetic evaluation method. They provided a comparison between their calculated k value outcomes and the values calculated in 32 different studies, or estimated by LandGEM model using EPA’s 2E method for different case studies. The average k value was an overestimation of the given value by 79% with a regression coefficient of 0.79, the fuzzy-based model outcomes ranged from 43 (underestimated) to 287% (overestimated) of the values calculated in those studies. They also conducted a sensitivity analysis associated with k values and confirmed that these values were more sensitive to the depth of the landfill and biodegradable organic fraction of the waste than mean temperature and precipitation.

Tolaymat et al. (2010) have also evaluated the site-specific values for k and L₀ for the MSW landfills operated as bioreactors. To improve the LFG predictability, LFG flow rates and composition were monitored for a 4-year period using one conventional and two bioreactor landfill cells in a landfill located in Louisville, KY. The reported site-specific value for L₀ was approximately 48.4 m³ Mg⁻¹. The estimated k values for the conventional cell was 0.06 year⁻¹, while the average value for the two bioreactor cells was 0.11 year⁻¹ which was much higher.

4 Available models

There are generally three approaches to mathematically present LFG generation rate: (1) estimation as a summation of mathematical functions indicating the individual kinetics of the contemplated physico-chemical processes taking place during the waste anaerobic degradation, (2) estimation as a simple empirical function or as a mixture of simple functions of a general kinetic factor, and (3) numerical models that elucidate gas generation via digits.

Some models deal with a general kinetic parameter, which is the most prevalent kind of model available in the literature (Findikakis and Leckie 1979; Van Heut 1986; Gardner and Probert 1993; Hartz 1982). The overall kinetic parameter is empirically adjusted to the observed production rates for data comparison. It is simply assumed as a start-up period through which little or no CH₄ is generated, followed by active CH₄ generation period mostly demonstrated by exponential or linear relationships, or their combinations. The summation of the gas generation rates would result in obtaining the cumulative gas generation over the active generation period and can be associated with the assumed preliminary generation capacity of the landfill.

Derivation of most of these relatively simple models is defined via a theoretical approach, which is based on the general kinetic expression for the biodegradation processes commonly known as Monad^’s equation as follows:

$$\frac{\text{dC}}{\text{dt}} = \frac{\text{KCx}}{{{\text{Kc}} + {\text{C}}}}$$

(2)

where C is the concentration of remaining substrate at time t (kg of the waste degradable organic fraction per m³ of the waste), such as organic material, x is the microorganisms’ concentration (kg of the biomass per m³ of the waste), K_c is a waste concentration through which the rate is one-half of the maximum digestion of substrate, and K is the maximum substrate utilization per unit mass of microorganisms. There are two extreme functions through which the Monad^’s equation could be approximated depending on the concentration of the substrate: zero-order reaction and first- order reaction. For the zero-order reaction, if the concentration of the microorganisms (x) remains constant, the rate of substrate consumption (dC/dt) also remains constant. For the first-order reaction, if the concentration of the microorganisms (x) remains constant, the rate of substrate utilization could be considered as a linear function of the substrate’s concentration.

4.1 Zero-order model

In this type of model, the biogas generation rate produced in landfill sites remains steady with the change in time. As a result, the type and age of the waste have no effects on the gas generation. The following subsections elucidate some zero-order models that are readily used by researchers.

4.1.1 IPCC

This model provides a zero-order kinetic reaction based on the degradable material within the waste, population, and CH₄ correction factor as indicated in Eq. (3) (IPCC 1996):

$${\text{E}}_{{{\text{CH}}_{ 4} }} = \left( {{\text{MSWT}} \times {\text{MSWF}} \times {\text{MCF}} \times {\text{DOC}} \times {\text{DOCF}} \times 16/12 - {\text{R}}} \right) \times \left( {1 - {\text{OX}}} \right)$$

(3)

where E_CH4 is CH₄ emissions from landfills, MSWT is the total municipal solid waste (MSW), MSWF is a fraction of MSW in a landfill, MCF is the CH₄ correction factor (fraction) (IPCC default value is 0.6), DOC is the degradable organic carbon which is recommended to be 15% w/w as the default value by the IPCC, DOCF is the fraction of actually decomposed DOC in the waste which is recommended to be 77% w/w by the IPCC, F is the fraction of the CH₄ in the LFG (the IPCC default is 0.5 v/v), 16/12 is the conversion factor from C to CH₄, R is the recovered CH₄, and OX is the oxidation factor (fraction, IPCC default is 0). Table 6 represents different default values for MCF in IPCC model based on the site type and Table 7 represents values for DOC parameter in the IPCC model for different types of waste.

Table 6 Different MCF default values for the IPCC model (IPCC 2006)

Table 7 Different DOC default values for the IPCC model (IPCC 2006)

The way to calculate the MSWT is to multiply the population by the annual MSW generation rate per capita, which is available for different countries and regions. These distinctions could play a key role in the estimation of the resulting emissions, as every waste stream is likely to have a different DOC content and subsequently CH₄ production capacity. Generally, countries should include three types of waste streams for the estimation of MSWT generated: commercial or market waste, garden and yard waste, and household waste.

4.1.2 SWANA zero order

A zero-order model has been developed by Solid Waste Association of North America (SWANA) which considers different kinds of waste. This model formula is represented by Eq. (4) (SWANA 1998):

$${\text{Q}} = \frac{{{\text{ML}}_{0} }}{{({\text{t}}_{ 0} - {\text{t}}_{\text{f}} )}}$$

(4)

where Q is the rate of CH₄ generation, M is the mass of waste disposed in the landfill, L₀ is the potential of CH₄ generation per mass of the waste, t₀ and t_f are the lag time and time until CH₄ is generated, respectively.

4.1.3 EPER

The EPER model is another zero-order model, which was developed in Germany, and it is represented by Eq. (5):

$${\text{M}}_{\text{e}} = {\text{M}} \times {\text{BDC}} \times {\text{BDC}}_{\text{f}} \times {\text{F}} \times {\text{D}} \times {\text{C}}$$

(5)

where M_e is the amount of CH₄ emission diffused, M is the amount of annual waste in a landfill, BDC is the proportion of biodegradable carbon (the default value is 0.15), BDC_f is the proportion of biodegradable carbon converted to CH₄ (default value is 0.5), F is a calculation factor of carbon converted into CH₄ (the default value is 1.33), D is the efficiency of collection with default values for active LFG recovery and cover, active degassing, and no recovery as 0.1, 0.4, and 0.9, respectively. C is the CH₄ concentration with the default value of 0.5. The outcomes of this model would be the same as a first-order degradation model if the constant amount of waste with the same composition were landfilled.

4.2 First-order model

Most of the worldwide and available models commonly used to estimate the rate of LFG generated in landfills have been developed based on the first-order decay reaction. Generally, the landfill conditions (precipitation, temperature, and climate), waste quantity, and waste quality (waste degradation over time, carbon content, moisture content, and age of waste) have been implicitly considered in these models. In other words, the first-order kinetic model considers the effect of carbon depletion in the waste over time (Ozakaya et al. 2006). The first-order models assume that there is an exponential relationship between the waste carbon and the decay rate with time (Gardner and Probert 1993).

Although the current models focusing on LFG generation are mostly based on first order kinetics (exponential degradation), the long-term emissions of landfills have been typically underestimated in the literature. Tintner et al. (2012) aimed to mitigate this problematic issue by focusing on the quantification and curve fitting of the LFG production throughout the final phase of degradation under optimal anaerobic conditions. To achieve this objective, the mechanically and biologically treated waste materials were examined under long-term LFG generation period from 240 to 1830 days. The observed gas sum curve resulted in greater values compared to the widely used exponential decay models. Here are some of the most commonly used first order kinetic models.

4.2.1 TNO

This model estimates LFG generation according to the waste organic carbon degradation as represented in Eq. (6):

$$\alpha_{\text{t}} = \zeta 1.87{\text{AC}}_{ 0} {\text{k}}_{ 1} {\text{e}}^{{ - {\text{k}}_{1} {\text{t}}}}$$

(6)

where α_t is the LFG generation at a given time, ζ is the dissimilation (or catabolism) factor of the organic waste (the default value is 0.58), A is the amount of waste placed in the landfill, C₀ is the waste organic carbon content, and k₁ is the degradation constant with the default value of 0.094. There are seven categories defined for the term C₀ in this model including shredder waste, contaminated soil, compost and sewage sludge, street cleaning waste, construction and demolition waste, coarse household waste, commercial waste with the default values of 130, 11, 90, 90, 11, 130, and 11, respectively.

4.2.2 SWANA first order

In this model, the effect of waste age on the gas generation is incorporated to facilitate the exponential decrease in LFG generation rates for the unit waste amount. The formula for this model is represented in Eq. (7) (SWANA 1998):

$${\text{Q}} = {\text{ML}}_{0} {\text{e}}^{{ - {\text{kt}}}}$$

(7)

where Q is the CH₄ generation rate, k is a constant for the first-order degradation and L₀ is the CH₄ generation potential of the waste unit of mass.

4.2.3 LandGEM

A software named LandGEM has been developed by the US EPA, which calculates the LFG generation based on the same formula as MSW. This model estimates the annual gas generation based on the first-order degradation kinetic over a specified period of time (US EPA 2016). This model is represented in Eq. (8).

$${\text{Q}}_{{{\text{CH}}_{4} }} = \sum {{\text{kL}}_{ 0} {\text{M}}_{\text{i}} \left( {{\text{e}}^{{ - {\text{kt}}}} } \right)}$$

(8)

where Q is the total gas generation rate, n is the number of years for the placement of the waste, and M_i is the mass of solid waste placed in year i. According to the US EPA protocols, the waste composition through this model reflects the composition of the MSW in the US, inert compounds, and other non-hazardous materials. In some cases, when landfills contain inert materials (i.e. non-biodegradable waste), such as ash resulting from waste combustion, this portion is likely to be subtracted from the waste acceptance rates. This model suggests subtracting inert materials when the documentation is provided and permitted by the regulatory authority. This case does not include those landfills containing MSW that may or may not be decomposable. The LandGEM model provides the values for the gas generation rate constant and gas generation potential for both AP42 and CAA standards (Clean Air Act 1963). For standard landfills, it is highly recommended to use AP42 default values. The CAA default value of 180 m³ CH₄ tone⁻¹ waste recommends a significant amount of gas generation potential (L₀) (Scharff and Jacobs 2006). LandGEM as a first-order kinetic model contains inputs including the years of waste acceptance, the annual waste acceptance rate or the amount of waste in place, the landfill design capacity, placement of the hazardous waste in the landfill, the potential values for k and L₀, and the total and individual concentration of the non- CH₄ organic compounds (NMOCs). Default values for k and L₀ may be implemented into the model or site-specific values could be developed through field measurements.

According to Thorneloe et al. (1999), LandGEM outputs are as follows: the individual pollutant emissions, two sets of default values incorporated in the model to calculate emissions (the first set is for specifying the Federal regulatory requirements known as “Clean Air Act defaults” and the other set is for development of emission inventories known as “AP-42 defaults”), estimation of the landfill closure according to the waste acceptance rate and landfill capacity, estimation of the CH₄, NMOCs, and selected air pollutants’ emission rates on an annual basis over the active lifespan of a landfill and for a certain number of years after the landfill is closed, and emission graphs for individual pollutant.

LandGEM has gained high attention in the literature as a reliable model to predict LFG generation, especially over the recent years. In one study, Amini et al. (2012) evaluated four methods to quantify first-order model parameters (L₀ and k) using LandGEM model. The results obtained from five case study landfills in Florida showed that L₀ ranged between 56 and 77 m³ Mg⁻¹ and k ranged from 0.04 to 0.13 year⁻¹ for conventional landfills while the value of 0.10 year⁻¹ was obtained for wet cells. They also reported that the model outcomes for LFG generation rate were on average less than the actual values collected.

4.2.4 Afvalzorg

This type of first-order kinetic multiphase model has the capability of estimating LFG generation based on eight waste categories and three fractions. Each fraction is separately taken into account to estimate LFG generation. For some categories of waste, no carbon content or organic matter data were provided (Scharff and Jacobs 2006). The amount of organic matter content for eight waste categories used in the Afvalzorg model are shown in Table 8. Equation (9) represents the mathematical format of this model.

$$\alpha_{\text{t}} = \zeta \sum\nolimits_{(i = 1)}^{3} {1.87{\text{AC}}_{{0,{\text{i}}}} {\text{k}}_{{1,{\text{i}}}} {\text{e}}^{{ - {\text{k}}_{{1,{\text{i}}}} {\text{t}}}} }$$

(9)

where α_t is the LFG generation at time t, ζ is the dissimilation factor, i is the waste fraction corresponded with k_1,i degradation rate constant, which ranges between 0.03 and 0.231, and A is the amount of waste, and C_0,i is the primary concentration of the waste fraction i.

Table 8 Organic matter content for waste categories used in the Afvalzorg model

4.2.5 GasSim

The GasSim model is another first-order kinetic multiphase model, which was developed by Scheepers and van Zantan (1994). According to the manual Version 2.0 (Golder Associates 2006), there is no complete set of equations applicable to the GasSim model. Thus, in the study done by Scheepers and van Zantan (1994), it was impossible to reproduce the equations for this model, since the calculation modules in this software are secure. This model’s requirements are specific to the year of disposal and waste input is in Mg. Different categories of waste and their carbon content typically used in GasSim model are represented in Table 9. The degradability of the three waste factions and different k values used in GasSim are represented in Table 10 (Scharff and Jacobs 2006).

Table 9 Categories of waste as well as their carbon content used in the GasSim model

Table 10 Degradability of the three waste factions and different k values used in the GasSim model

4.2.6 EPER France

The EPER France model provides the estimation of CH₄ emissions for landfill sites whether it is joined or not joined to an LFG collection system via a multiphase model called “ADEME” (version 15/12/2002). The formula for this multiphase model is mathematically represented in Eq. (10):

$${\text{FE}}_{{{\text{CH}}_{ 4} }} = \sum {{\text{FE}}_{0} } \times \left( {\sum {{\text{A}}_{\text{i}} \times {\text{p}}_{\text{i}} \times {\text{k}}_{\text{i}} \times {\text{e}}^{{ - {\text{k}}_{\text{i}} {\text{t}}}} } } \right)$$

(10)

where, ${\text{FE}}_{{{\text{CH}}_{4} }}$ is the annual CH₄ generation rate, FE₀ is the CH₄ generation potential, p is the fraction of waste with degradation constant of k_i, and A_i is the normalization factor.

The left and right sides of the Eq. (10) do not match dimensionally and it seems that the waste amount in Mg year⁻¹ is absent on the right side of the equation. In the spreadsheet of this model, the amount of waste on an annual basis is utilized in the calculations. The normalization factor is not included in the spreadsheet, whereas it is defined in the equation for the model. This model defines three waste categories and each category has a particular CH₄ generation potential per Mg of waste. Table 11 represents the waste categories as well as their respective CH₄ generation potential. Table 12 shows the fractions and k values for different waste categories (Scharff and Jacobs 2006).

Table 11 Categories of waste and CH₄ production potential in the ADEME model

Table 12 Different fractions as well as k values used in ADEME model

4.2.7 Mexico

The basic assumption in this model is that there is a 1-year lag between the waste disposed in the cells and LFG production. The model also assumes that an exponential decline occurs in the CH₄ generation rate for each unit of waste after 1 year due to the consumption of the waste organic fraction. The model estimates the LFG generation rate for a landfill in a given year with known (or estimated) year-to-year rate of solid waste disposal according to the equation shown in Eq. (11):

$${\text{Q}}_{\text{M}} = \mathop \sum \limits_{{{\text{i}} = 1}}^{\text{n}} 2{\text{kL}}_{0} {\text{M}}_{\text{i}} \left( {{\text{e}}^{{ - {\text{kt}}}} } \right)$$

(11)

This formula has been published in Title 40 of the US Code of Federal Regulations (CFR) Part 60 (Stege and Murray 2003). In order to estimate the gas generation rate, this model needs site-specific data for all the information required, except for the L₀ and k values. Actually, this model uses default values for L₀ and k that were obtained based on three aspects: the relationship between L₀ and k values; the average precipitation rate measured at US landfills annually, and the specific data collected from representative landfill sites in Mexico (Stege and Murray 2003). These values are represented in Table 13 with an assumption that the LFG composition generated from landfills is about 50% CH₄ and the rest is comprised of other gases such as CO₂ and trace amounts of other components.

Table 13 Different L₀ and k values used in the Mexico model

4.2.8 LFGGEN

The University of Central Florida developed this model and used it for the first time (Keely 1994). Basically, this model estimates LFG generation based on three assumptions: first, methanogenesis process starts after a short period of lag time; second, the first phase of methanogenesis process follows a linearly increasing generation rate; and third, the second phase of methanogenesis process follows a first-order kinetic represented by an exponentially decreasing generation rate. Other features of this model are discussed as follows.

There are 11 categories for the biodegradable solid waste in this model. The methods of analysis include Biochemical CH₄ Potential (BMP), EPA Tire3, biodegradability factors, and theoretical stoichiometric production of CH₄ and CO₂. The solid waste is divided into three categories in terms of moisture content: wet, moderate, and dry. The waste biodegradation is divided into three categories, which include rapid, moderate, and slow (biodegradability rate is also a function of moisture).

This model considers the lag time to establish anaerobic conditions. After this delay, the gas generation rate follows a linear increase to a specific peak rate (Q_s), which occurs at the end of the year (t_p). After the peak is observed, the generation rate follows an exponential decline from the peak to a near-zero value at the end of the biodegradation time (t₉₉). During this time (t₉₉), the gas generation rate falls to approximately one percent of the peak rate. The assumption in this model is that the characteristic times (t₀, t_p, and t₉₉) are dependent on the moisture content and waste type. The specific peak rate (Q_s) is also a function of the CH₄ generation potential according to the Eq. (12) (Reinhart and Faur 2004):

$${\text{Q}}_{\text{s}} = {\text{L}}_{ 0} \left[ {{{2{\text{k}}} \mathord{\left/ {\vphantom {{2{\text{k}}} {\left( {{\text{k}}\left( {{\text{t}}_{\text{p}} - {\text{t}}_{ 0} } \right) + 2} \right)}}} \right. \kern-0pt} {\left( {{\text{k}}\left( {{\text{t}}_{\text{p}} - {\text{t}}_{ 0} } \right) + 2} \right)}}} \right]$$

(12)

The biodegradation constant k for the second phase of methanogenesis is related to the assumed times as represented in Eq. (13):

$${\text{k}} = {{\left( { - { \ln }\,0.01} \right)} \mathord{\left/ {\vphantom {{\left( { - { \ln }\,0.01} \right)} {\left( {{\text{t}}_{99} - {\text{t}}_{\text{p}} } \right)}}} \right. \kern-0pt} {\left( {{\text{t}}_{99} - {\text{t}}_{\text{p}} } \right)}}$$

(13)

4.3 Second-order models

These models explain the complicated reactions that occur during the waste degradation period via several first-order reactions with different kinetic rates. The LFG generation rate could be modelled via the second-order kinetics using a complex set of different reactions (Reinhart and Faur 2004).

4.4 Complex mathematical models

These models explicitly consider some or most of the well-known physical, chemical and biochemical variables affecting the gas generation rate in complex mathematical functions. Some of these important variables include temperature, composition, moisture content, adsorption, absorption, volatilization, evaporation, complications, filtration, dilution, neutralization, oxidation, reduction, and precipitation. Without a detailed characterization of the actual conditions in a landfill, it is not possible to evaluate the gas generation rate with any certainty. This model needs a very detailed analysis and a considerable amount of measurements (Nastev 1998).

For instance, the Halvadakis model developed by El-Fadel et al. (1989) is based on the sequential biological growth and is considered as a representative complex mathematical model used for estimation of LFG generation. A system of first-order equations is used to describe a complicated microbial landfill ecosystem. They were described in terms of pathways and sinks, carbon sources, digestion of the aqueous carbon for the acidogenic and methanogenic biomass growth, which explain the hydrolysis of the biogasifiable and hydrolysable components of waste, acetate utilization, and subsequent generation of CH₄ and CO₂. This is equivalent to a simplified microbial food chain. Nevertheless, this model ignores the amount of CH₄ generated by the CO₂ reduction with hydrogen, which constitutes roughly 25–30% of the total CH₄ generated. This model also considers carbon in seven possible forms based on the mass balance equations as follows:Aqueous carbon is calculated according to Eq. (14):

$${{{\text{dC}}_{{ ( {\text{aq)}}}} } \mathord{\left/ {\vphantom {{{\text{dC}}_{{ ( {\text{aq)}}}} } {\text{dt}}}} \right. \kern-0pt} {\text{dt}}} = \sum {{\text{k}}_{\text{ha}} {\text{C}}_{{\left( {\text{s}} \right){\text{i}}}} {-}\left( {1/{\text{Y}}_{\text{A}} } \right)} \, \left( {{{\mu_{\text{A}} {\text{KC}}_{{ ( {\text{aq)}}}} } \mathord{\left/ {\vphantom {{\mu_{\text{A}} {\text{KC}}_{{ ( {\text{aq)}}}} } {\left( {{\text{K}}_{\text{sa}} - {\text{C}}_{{ ( {\text{aq)}}}} } \right)}}} \right. \kern-0pt} {\left( {{\text{K}}_{\text{sa}} - {\text{C}}_{{ ( {\text{aq)}}}} } \right)}}} \right){\text{C}}_{\text{xa}}$$

(14)

where C(aq) is the aqueous carbon concentration, k_ha is the hydrolysis rate of refuse component a, Y_A is the mass of acidogenic biomass carbon formed per mass of substrate dissolved organic carbon metabolized (a dimensionless fraction), µ_A is the acidogenic biomass maximum specific growth rate constant, and K_sa is the half saturation constant for acidogens (carbon basis).

Solid carbon is calculated according to Eq. (15):

$${{{\text{dC}}_{{ ( {\text{s)}}}} } \mathord{\left/ {\vphantom {{{\text{dC}}_{{ ( {\text{s)}}}} } {\text{dt}}}} \right. \kern-0pt} {\text{dt}}} = - {\text{k}}_{\text{ha}} {\text{C}}_{\text{s}}$$

(15)

Carbon in acidogenic biomass is calculated according to Eq. (16):

$${{{\text{dC}}_{{ ( {\text{xa)}}}} } \mathord{\left/ {\vphantom {{{\text{dC}}_{{ ( {\text{xa)}}}} } {\text{dt}}}} \right. \kern-0pt} {\text{dt}}} = \left[ {\left( {{{\mu_{\text{A}} {\text{C}}_{{ ( {\text{aq)}}}} } \mathord{\left/ {\vphantom {{\mu_{\text{A}} {\text{C}}_{{ ( {\text{aq)}}}} } {\left( {{\text{k}}_{\text{ha}} - {\text{C}}_{{ ( {\text{aq)}}}} } \right)}}} \right. \kern-0pt} {\left( {{\text{k}}_{\text{ha}} - {\text{C}}_{{ ( {\text{aq)}}}} } \right)}}} \right) - {\text{k}}_{\text{da}} } \right]{\text{C}}_{\text{xa}}$$

(16)

where kDa is the acidogenic biomass death rate constant.Carbon in methanogenic biomass is calculated according to Eq. (17):

$${{{\text{dC}}_{{ ( {\text{xm)}}}} } \mathord{\left/ {\vphantom {{{\text{dC}}_{{ ( {\text{xm)}}}} } {\text{dt}}}} \right. \kern-0pt} {\text{dt}}} = \left[ {\left( {{{\mu_{\text{m}} {\text{C}}_{{ ( {\text{Ac)}}}} } \mathord{\left/ {\vphantom {{\mu_{\text{m}} {\text{C}}_{{ ( {\text{Ac)}}}} } {\left( {{\text{K}}_{\text{sm}} + {\text{C}}_{{ ( {\text{Ac)}}}} } \right)}}} \right. \kern-0pt} {\left( {{\text{K}}_{\text{sm}} + {\text{C}}_{{ ( {\text{Ac)}}}} } \right)}}} \right) - {\text{k}}_{\text{dm}} } \right]{\text{C}}_{\text{xm}}$$

(17)

where µ_m is the methanogenic biomass maximum growth rate, K_sm is the half saturation constant for methanogens (carbon basis).

Carbon in acetate is calculated according to Eq. (18):

$$\begin{aligned} {{{\text{dC}}_{{ ( {\text{Ac)}}}} } \mathord{\left/ {\vphantom {{{\text{dC}}_{{ ( {\text{Ac)}}}} } {\text{dt}}}} \right. \kern-0pt} {\text{dt}}} \,=\, & {\text{Y}}_{\text{Ac}} \left[ {\left( {{{\left( {1 - {\text{Y}}_{\text{Ac}} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - {\text{Y}}_{\text{Ac}} } \right)} {{\text{Y}}_{\text{Ac}} }}} \right. \kern-0pt} {{\text{Y}}_{\text{Ac}} }}} \right)\mu_{\text{A}} \left( {{{{\text{C}}_{{ ( {\text{aq)}}}} } \mathord{\left/ {\vphantom {{{\text{C}}_{{ ( {\text{aq)}}}} } {\left( {{\text{K}}_{\text{sa}} + {\text{C}}_{{ ( {\text{aq)}}}} } \right)}}} \right. \kern-0pt} {\left( {{\text{K}}_{\text{sa}} + {\text{C}}_{{ ( {\text{aq)}}}} } \right)}}} \right) + {\text{k}}_{\text{da}} } \right] \\ - \,\left[ {\left( {1/{\text{Y}}_{\text{m}} } \right) \, \mu_{\text{m}} \left( {{{{\text{C}}_{{ ( {\text{Ac)}}}} } \mathord{\left/ {\vphantom {{{\text{C}}_{{ ( {\text{Ac)}}}} } {\left( {{\text{K}}_{\text{sm}} - {\text{C}}_{{ ( {\text{Ac)}}}} } \right)}}} \right. \kern-0pt} {\left( {{\text{K}}_{\text{sm}} - {\text{C}}_{{ ( {\text{Ac)}}}} } \right)}}} \right)} \right]{\text{C}}_{\text{xm}} \\ \end{aligned}$$

(18)

where Y_Ac is the acetate carbon fractional formation yield coefficient, and Y_m is the mass of methanogenic biomass carbon formed per mass of acetate carbon utilized (dimensionless).

Carbon in CO₂ is calculated according to Eq. (19):

$$\begin{aligned} {{{\text{dC}}_{{ ( {\text{CO}}_{ 2} )}} } \mathord{\left/ {\vphantom {{{\text{dC}}_{{ ( {\text{CO}}_{ 2} )}} } {\text{dt}}}} \right. \kern-0pt} {\text{dt}}} \,=\, & \left( {1 - {\text{Y}}_{\text{Ac}} } \right)\left[ {\left( {{{\left( {1 - {\text{Y}}_{\text{Ac}} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - {\text{Y}}_{\text{Ac}} } \right)} {{\text{Y}}_{\text{Ac}} }}} \right. \kern-0pt} {{\text{Y}}_{\text{Ac}} }}} \right)\mu_{\text{A}} \left( {{{{\text{C}}_{{ ( {\text{aq)}}}} } \mathord{\left/ {\vphantom {{{\text{C}}_{{ ( {\text{aq)}}}} } {\left( {{\text{K}}_{\text{sa}} + {\text{C}}_{{ ( {\text{aq)}}}} } \right)}}} \right. \kern-0pt} {\left( {{\text{K}}_{\text{sa}} + {\text{C}}_{{ ( {\text{aq)}}}} } \right)}}} \right) + {\text{k}}_{\text{da}} } \right]{\text{C}}_{\text{xa}} \\ + \,\left( {1 - {\text{Y}}_{{{\text{CH}}_{ 4} }} } \right)\left[ {\left( {{{\left( {1 - {\text{Y}}_{\text{m}} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - {\text{Y}}_{\text{m}} } \right)} {{\text{Y}}_{\text{m}} }}} \right. \kern-0pt} {{\text{Y}}_{\text{m}} }}} \right)\mu_{\text{m}} \left( {{{{\text{C}}_{{ ( {\text{Ac)}}}} } \mathord{\left/ {\vphantom {{{\text{C}}_{{ ( {\text{Ac)}}}} } {\left( {{\text{K}}_{\text{sm}} + {\text{C}}_{{ ( {\text{Ac)}}}} } \right)}}} \right. \kern-0pt} {\left( {{\text{K}}_{\text{sm}} + {\text{C}}_{{ ( {\text{Ac)}}}} } \right)}}} \right) + {\text{k}}_{\text{dm}} } \right] \\ {\text{C}}_{\text{xm}} \left( {1 - {\text{Y}}_{{{\text{CH}}_{ 4} }} } \right)\left[ {\left( {{{\left( {1 - {\text{Y}}_{\text{m}} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - {\text{Y}}_{\text{m}} } \right)} {{\text{Y}}_{\text{m}} }}} \right. \kern-0pt} {{\text{Y}}_{\text{m}} }}} \right)\mu_{\text{m}} \left( {{{{\text{C}}_{{ ( {\text{Ac)}}}} } \mathord{\left/ {\vphantom {{{\text{C}}_{{ ( {\text{Ac)}}}} } {\left( {{\text{K}}_{\text{sm}} + {\text{C}}_{{ ( {\text{Ac)}}}} } \right)}}} \right. \kern-0pt} {\left( {{\text{K}}_{\text{sm}} + {\text{C}}_{{ ( {\text{Ac)}}}} } \right)}}} \right) + {\text{k}}_{\text{dm}} } \right]{\text{C}}_{\text{xm}} \\ \end{aligned}$$

(19)

where ${\text{Y}}_{{{\text{CH}}_{ 4} }}$ is the CH₄ carbon fractional formation yield coefficient (dimensionless).

Carbon in CH₄ is calculated according to Eq. (20):

$${{{\text{dC}}_{{({\text{CH}}_{4} )}} } \mathord{\left/ {\vphantom {{{\text{dC}}_{{({\text{CH}}_{4} )}} } {\text{dt}}}} \right. \kern-0pt} {\text{dt}}} \,=\, {\text{Y}}_{{{\text{CH}}_{ 4} }} \left[ {\left( {{{\left( {1 - {\text{Y}}_{\text{m}} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - {\text{Y}}_{\text{m}} } \right)} {{\text{Y}}_{\text{m}} }}} \right. \kern-0pt} {{\text{Y}}_{\text{m}} }}} \right)\mu_{\text{m}} \left( {{{{\text{C}}_{{ ( {\text{Ac)}}}} } \mathord{\left/ {\vphantom {{{\text{C}}_{{ ( {\text{Ac)}}}} } {\left( {{\text{K}}_{\text{sm}} + {\text{C}}_{{ ( {\text{Ac)}}}} } \right)}}} \right. \kern-0pt} {\left( {{\text{K}}_{\text{sm}} + {\text{C}}_{{ ( {\text{Ac)}}}} } \right)}}} \right) + {\text{k}}_{\text{dm}} } \right]{\text{C}}_{\text{xm}}$$

(20)

4.5 Numerical models

Numerical models are strong implements to simulate and model the whole phenomena that occur during the waste degradation and gas generation (Afshar 2002). Therefore, it could be incorporated into a simpler method. In this case, a technique called “Weighted Residual Method (WRM)” has been successfully implemented to promote the simplicity. The application of this method based on the data collected from a real landfill site has confirmed the reliability of numerical models. The formula represented in Eq. (21) could be utilized for every landfill site with a few CH₄ measurements as follows:

$$\iint\limits_{\text{Domain}} {{\text{Wlp}}({\text{G}} - \psi )\,{\text{dtdW}}}=\iint\limits_{\text{Domain}} {\text{Amn}}\left( {{\text{Wlp}}\sum {\text{Nmn}} } \right) \leftrightarrow {\text{f}}_{1} = {\text{k}}_{1} \sum {\text{Amn}}$$

(21)

where w_lp is a weighting function, G is the approximated CH₄ generation, Ψ is a function to comply the boundary condition, N_mn is considered as a trail function that is assumed to be zero on the boundaries, k₁ is the coefficient matrices, f₁ is the right hand side matrix which is known and A_mn is a coefficient that should be specified and is an unknown matrix. The ultimate goal of WRM is to select Amn in a way that the residue (R) declines over a determined domain. R is the difference between the left and right side of the Eq. (21).

4.6 Comparison of the studied models

In order to provide a comparison between different types of models, Table 14 represents a brief review of all the models discussed in this study as well as their basic assumptions, advantages and disadvantages. According to Table 14, most of the empirical models predict the LFG generation based on the degradable organic carbon, portion of the degradable carbon, or carbon content of the waste. Some of these models including Afvalzorg and GasSim have classified the waste into different categories. In terms of the ease of operation, some models are more user-friendly compared to others. For instance, Land GEM and Mexico have a user-friendly operation in the datasheet environment, while EPER France model contains a normalization factor that is not considered in the spreadsheet.

Table 14 An overview of the discussed models, their assumptions, advantages, and disadvantages

In order to estimate the LFG generation, several studies have recently investigated different models in terms of their application for various objectives. For instance, Cakir et al. (2016) have used three different methodologies and one approach to investigate the LFG potential as an energy source to produce heat and electricity recovered from the MSW. The methods they used include Afvalzorg multiphase model, LandGEM, and the IPCC (2006). The LFG generated using the multiphase and the IPCC (2006) models were 291,897,215 and 491,752,247 m³, respectively. The LFG generated via the LandGEM model with three k values of 0.35, 0.1 and 0.35 year⁻¹ were 792,073,359, 769,734,749 and 681,685,027 m³, respectively.

Mahar et al. (2016) provided a new approach to simulate the cumulative LFG generation and its rate for a pretreated MSW landfill using four models. These models include first order exponential model, modified Gompertz model, single component combined growth and decay model, and Gaussian function. They developed a new multi-component model based on the single component combined growth and decay model using an anaerobic landfill reactor acting as a simulator to treat the pretreated MSW. Their results confirmed that in terms of cumulative LFG generation simulation, modified Gompertz model offered better fit as compared to the first- order exponential model. They also reported that newly developed multi-component model (which has been developed based on biochemical processes) was more precise as compared to the single component combined growth and decay model and Gaussian function.

In another study, Chakma and Mathur (2016) developed a new method to predict the optimum generation of long-term spatial–temporal landfill gas components on post closure landfill. The model used in this study was a first order kinetic model that incorporated the chemical and biological processes degrading the MSW. The model has also considered the spatial moisture content and spatial variation of the density in different landfill depth values. The optimum pH and temperature that improve the biodegradation have also been incorporated into the model. They reported that this model could be applied to any sites that provide appropriate proportion of heterogeneity of the MSW. They introduced heterogeneity as the percentage of rapidly, moderately, slowly, and non-biodegradable fraction of the MSW.

In brief, different models have different applications as well as different assumptions, advantages, and disadvantages. Therefore, different physical, chemical, and biological aspects of the waste degradation and LFG generation should be taken into account while selecting an appropriate model to obtain respectively precise outcomes.

5 Conclusions and recommendations

Several investigations have been conducted on estimating the LFG generation from sanitary landfills in the literature. These estimations are mainly based on Monod first-order kinetic, which are well known as first-order decay models. In these models, there is a linear relationship of the maximum gas generation potential per waste weight unit as well as an exponential relationship of the waste degradation rate and time.

Zero-order models assume the gas generation rate to be constant versus time. This assumption is the main reason for the significant inaccurate outcomes of these models. Complex mathematical models are based on the carbon mass balance in CH₄ generation chain consisting of solid carbon to aqueous carbon, carbon in acidogenic biomass, carbon in methanogenic biomass, carbon in acetate, carbon in CO₂, and carbon in CH₄. Numerical models are also found to be powerful implements to estimate CH₄ generation from landfills and WRM is typically applied in these models for simplicity.

Although the research focus on the estimation of the LFG via modelling has increased over the last decade, there are still several research gaps in the literature that should be taken into account. In most studies, the field measurement data and consequently the modelling are based on the LFG collected from the gas collection systems, and is not based on the actual gas generation rate in the landfills. Therefore, more in-depth research is required to evidently distinguish between the actual data and the estimations from modelling. Besides, in the landfill sites that are installed with biocovers, biofilters, or biowindows, a proportion of CH₄ is oxidized through the landfill cover soils due to the microbial activity. Since none of the models developed so far has considered CH₄ oxidation, further research should be carried out to incorporate its effect in the modelling basic concepts. In addition, most of the studies focused on using the empirical models to estimate LFG generation rate, and less attention has been paid to the application of the numerical models. However, the recent studies have shown that the numerical models are much closer to the actual collected data than the empirical models. Therefore, it is recommended that more investigations should be carried out on developing new numerical models.