A Multi-Field Coupling Model for Municipal Solid Waste Degradation in Landfills: Integrating Microbial, Chemical, Thermal, and Hydraulic Processes

Abstract

The degradation of municipal solid waste (MSW) in landfills involves complex physical, chemical, and biological interactions that span multiple spatial and temporal scales. To better understand these dynamics, this study develops a comprehensive model that couples microbial, chemical, thermal, and hydraulic fields. The model captures bidirectional feedback mechanisms, such as heat and acid production from microbial metabolism, which in turn influence microbial activity and reaction pathways. A simplified one-dimensional formulation was solved using the finite difference method and validated against historical temperature data from real landfills. Simulation results indicate that temperature peaks at approximately 45 °C around the fifth year, followed by a gradual decline. pH and substrate concentration decrease over time but exhibit minimal variation with depth. The degradation rate reaches its maximum within two years and subsequently declines. These trends highlight the critical roles of temperature in initiating rapid degradation and substrate concentration in determining the endpoint of the reaction. This model provides a theoretical foundation for interpreting energy and mass transformation processes in landfills and offers practical insights for optimizing landfill management, reducing pollution, facilitating resource recovery and providing a theoretical model and prediction tool for sustainable waste management.

1. Introduction

With the development of the economy and urbanization, the population and the scale of the cities are increasing, causing a rapid increase in municipal solid waste (MSW). According to the estimation of the World Bank, 2.24 billion tons of MSW was generated globally in 2020, and 3.88 billion tons of MSW will be produced in 2050 [1]. The fastest-growing regions for MSW are the developing countries and regions such as sub-Saharan Africa and South Asia. If it was not properly treated, MSW could occupy land resources, generate toxic gases and leachate, and eventually pollute the air, groundwater, and land [2,3]. In addition, a large amount of MSW is disposed into the ocean, which harms the marine animals and pollutes the marine environment [4]. The greenhouse gases produced by waste incineration, such as methane and carbon dioxide, also contribute to global climate change [5]. MSW must therefore be disposed of properly to reduce its impact on the human health, surrounding ecosystem, and global climate.

The main disposal methods of MSW include landfill, incineration and composting [3,6]. Among them, landfill is the simplest and cheapest method, and thus it is the most popular in developing countries. According to the World Bank, about 33% of MSW is disposed of in landfills [7]. However, against the backdrop of continuously increasing global MSW generation, a profound transformation is simultaneously occurring in the structure of waste treatment methods. Particularly in rapidly developing economies like China, the proportion of traditional landfill disposal has declined significantly, while waste-to-energy incineration and resource recovery rates have steadily increased [3,6]. This transition implies that the research focus must proceed along two parallel tracks, including optimizing emerging incineration and recycling technologies and addressing the long-term environmental challenges posed by a large number of existing landfill sites.

The degradation of MSW in landfills is a complex and lengthy process with simultaneous physical, chemical, and biological reactions [4]. This process may lead to changes in physical, chemical, and mechanical properties such as temperature, air pressure, pH value, stress, etc. [4,8,9,10]. However, it is difficult to predict the changes in its physical, chemical, and mechanical properties by accurate calculations, which may cause potential risks including explosion, contaminant leakage, and ground sedimentation [8,9,10]. Explaining the degradation process of MSW plays an important role in landfill design, pollutant reduction, risk control, gas and heat reuse, etc.

The degradation process of municipal solid waste (MSW) is predominantly driven by biological reactions involving diverse microorganisms [11]. Landfills operate as dynamic systems where biological, chemical, thermal, and flow fields are intricately coupled [12,13]. Temperature serves as a critical parameter affecting MSW degradation, with heat generation and accumulation observed in numerous cases [14]. The biochemical degradation process is highly sensitive to environmental conditions [15]; for instance, a moderate temperature increase can enhance enzymatic activity and reaction rates, whereas excessively high temperatures may inhibit microbial processes due to enzyme denaturation. However, many existing coupled models often simplify these interactions by focusing on unidirectional effects or partial field coupling, thereby neglecting essential bidirectional feedback mechanisms [13,14,15,16]. Specifically, current approaches may overlook how microbial activity drives heat production and chemical changes and how these changes in turn regulate microbial community dynamics and degradation pathways. This limited representation hinders a comprehensive understanding of landfill behavior.

To address these gaps, our work emphasizes the importance of a coupled framework that integrates microbial, chemical, thermal, and flow fields with bidirectional interactions. For example, microbial metabolism consumes substrates and produces heat and acids, leading to temperature rise and pH decline, which then feedback to influence microbial growth rates and reaction kinetics. Despite this need, the thermal effects of MSW degradation remain inadequately characterized, the systems of partial differential equations for multi-field coupling are challenging to solve, and the spatiotemporal distributions of these fields require further exploration. Therefore, this study pursues three primary objectives: (1) developing an integrated 3D theoretical framework representing the core biochemo-thermal processes; (2) employing a simplified 1D formulation for efficient numerical solution and validation of the key coupling mechanisms; and (3) benchmarking model predictions against available field data to evaluate its performance. This strategy ensures the model balances mechanistic comprehensiveness with computational tractability, maintaining scalability for future extension to complex scenarios, providing a numerical tool for the sustainable development of waste management.

2. Model Development

2.1. Microbial Growth in Landfills

The MSW degradation is caused by physical, chemical, and biochemical reactions in landfills. Among them, the biochemical reactions play the most important role. The microbial growth in landfills is related to the temperature, pH value, substrate concentration, water content, and other factors [17]. All these factors are model variables that evolve over time and vary spatially (with depth), and their dynamic coupling constitutes the core of our following established model. The maximum environmental carrying capacity can be described as

$$K=k(T,pH,S,\cdots )K_{\mathrm{opt}}$$

(1)

Among all the factors, temperature, pH, and concentration of substrates have a more significant effect on the growth of the microorganisms. Only considering the effects of temperature, pH and substrate concentration, and other factors such as humidity are not considered constant and do not affect the growth of microorganisms. Assuming that the three factors are independent of each other, then

$$K=k_1(T)k_2(pH)k_3(S)K_{\mathrm{opt}}$$

(2)

It assumes that the microbial concentration is always at its instantaneous environmental carrying capacity. This means that microbial population growth is modeled as a quasi-steady-state process, where growth kinetics are considered instantaneous, thereby neglecting potential time-lag effects during population establishment. Since landfill degradation is a long-term process spanning decades, while microbial succession and population growth typically reach a relatively stable state within months, ignoring explicit growth kinetics is a reasonable simplification at the “year” scale. This approach significantly reduces model complexity and allows the computational focus to remain on the long-term coupled feedback mechanisms between microbial activity and environmental factors such as temperature, pH, and substrate concentration, then

$$N=K=k_1(T)k_2(pH)k_3(S)K_{\mathrm{opt}}$$

(3)

2.2. Consumption and Transformation of Substrates

The biodegradable MSW includes polysaccharides, proteins, fats, polymers, etc. The process of MSW degradation can be divided into aerobic and anaerobic reaction stages. In typical enclosed landfill environments, the aerobic phase is very brief (typically lasting from several days to a few weeks), whereas the long-term degradation process (persisting over decades) is predominantly governed by anaerobic reactions. Therefore, from the perspective of simulating long-term behavior, focusing the model on the anaerobic process represents a reasonable and commonly adopted simplification.

The anaerobic reaction stage can be divided into the hydrolysis process, hydrogen and acetic acid production process, and methane production process. The effect of the biochemical reaction is the conversion of MSW into the mass of the gases, microorganisms, acids, and the heat. In this section, we only discuss the changes in reactant mass and pH value, the heat and gas generation will be discussed in the subsequent sections.

The decomposition and consumption of substrates are mainly due to the biochemical reactions. If the decomposition rate of the ith kind of substrate is related only to the number of microorganisms, then we can achieve

$${\displaystyle \frac{\partial S_i}{\partial t}}=-v_{\mathrm{S},i}{K}_i$$

(4)

The ith substrate concentration can be calculated by subtracting the reaction consumption concentration from the initial concentration, which is

$$S_i=S_i^0-{\displaystyle {\int }_0^t{v}_{\mathrm{S},i}{K}_{i}dt}$$

(5)

By using the chemical reaction equation, we can achieve the ratio of the change in hydrogen ion concentration and the consumption of the ith reactant. Then we can calculate the pH value by

$$pH=-\lg \left((c_0^{{\mathrm{H}}^{+}}+{\displaystyle \sum _{i=1}^m{\displaystyle {\int }_0^t{v}_{{\mathrm{H}}^{+},i}{v}_{\mathrm{S},i}{K}_{i}dt)/1000}}\right)=-\lg \left(c_0^{{\mathrm{H}}^{+}}+{\displaystyle \sum _{i=1}^m{\displaystyle {\int }_0^t{v}_{{\mathrm{H}}^{+},i}{v}_{\mathrm{S},i}{K}_{i}dt}}\right)+3$$

(6)

It should be noted that the 1000 and 3 in Equation (6) are from the conversion of units, i.e., from mol/m³ to mol/L. Note that Equation (6) simplifies the complex chemical environment within landfills by failing to account for processes such as buffer systems, acid-base equilibria, and CO₂ dissolution. In the calculation, the pH calculation can be modified to incorporate buffer systems and acid-base equilibria to achieve more accurate results.

2.3. Heat Generation, Accumulation, and Transport

The consumption of the reactants generates heat, which can be calculated by enthalpy change. The heat generated by the consumption of the ith substrate can be determined by

$$Q_i=\Delta H_i{\displaystyle \frac{\partial S_i}{\partial t}}$$

(7)

The heat transfer has three ways, which are heat conduction, heat convection, and heat diffusion. The thermal processes in landfills include heat generation, heat accumulation, heat conduction, heat convection, and heat loss. Among them, the heat generation, heat conduction and heat convection in the landfill determine the temperature distribution. Through the conservation of energy, we can obtain

$${\left(\rho C\right)}_{\mathrm{eff}}{\displaystyle \frac{\partial T}{\partial t}}+{\rho }_{\mathrm{f}}{C}_{\mathrm{f}}u\cdot \nabla T+\nabla \cdot \left(-k_{\mathrm{eff}}\nabla T\right)={\displaystyle \sum _{i=1}^m{Q}_i}$$

(8)

where ${\left(\rho C\right)}_{\mathrm{eff}}={\theta }_{\mathrm{S}}{\rho }_{\mathrm{S}}{C}_{\mathrm{S}}+{\theta }_{\mathrm{f}}{\rho }_{\mathrm{f}}{C}_{\mathrm{f}}$, and $k_{\mathrm{eff}}={\theta }_{\mathrm{s}}{k}_{\mathrm{s}}+{\theta }_{\mathrm{f}}{k}_{\mathrm{f}}$. The first term on the left side of Equation (8) is the heat accumulation, the second term is the fluid convection, and the third term is the heat conduction. The right side of Equation (8) represents the heat generation of the substrate degradation. The distribution of temperature in space and time can be obtained by solving Equation (8).

With the above equations, initial conditions, and boundary conditions, the temporal and spatial distributions of microorganisms, substrates, temperature, flow field, and pore volume can be solved under three-dimensional conditions.

2.4. Gas Generation and Transport

The state of a gas is related to various factors such as air pressure, temperature, molar mass, and volume. It can be measured by the ideal gas equation of state, which is

$$p={\displaystyle \frac{nRT}{V}}$$

(9)

The fluids in landfills are both the gases and liquids, and they all contribute to the convection. Since landfills are closed, the flow rate of the fluids inside is very slow. Therefore, only the effect of the gas convection was considered. The flow field of the gases can be described by Darcy’s law [18], as follows:

$$\mathbf{u}=-{\displaystyle \frac{\kappa }{\mu }}\nabla p=-{\displaystyle \frac{\kappa }{\mu }}\nabla {\displaystyle \frac{nRT}{V_{\mathrm{void}}}}$$

(10)

The mass of the generated gases can be calculated by the chemical reaction equations. The mass of the gases in the landfill is the sum of the mass of the gas generated and the mass of the gases transported due to the flow field, which is

$${\displaystyle \frac{\partial S_{\mathrm{gas}}}{\partial t}}={\displaystyle \sum _{i=1}^m\frac{v_{\mathrm{gas},i}\partial S_i}{\partial t}+{\rho }_{\mathrm{gas}}\mathbf{u}\mathbf{M}}$$

(11)

The molar mass of a gas is equal to the mass of the gas divided by the molecular weight of the gas, which is

$${\displaystyle \frac{\partial n}{\partial t}}={\displaystyle \sum _{i=1}^m\frac{\frac{v_{\mathrm{gas},i}\partial S_i(t)}{\partial t}+{\rho }_{\mathrm{gas}}\mathbf{u}\mathbf{M}}{{\lambda }_i}}$$

(12)

The decomposition of the substrate leads to an increase in pore volume, which can be expressed as

$${\displaystyle \frac{\partial V_{\mathrm{void}}}{\partial t}}={\displaystyle \sum _{i=1}^m\frac{v_{\mathrm{V},i}\partial S_i(t)}{\partial t}}$$

(13)

By using the above system of partial differential equations, initial conditions and boundary conditions, the distribution of microorganisms, substrates, temperature, flow field and pore volume under three-dimensional conditions can be solved.

3. Simplified Model for One-Dimension Condition

3.1. Model Simplification

Although a model has been obtained for the three-dimensional case, this model is difficult to solve, even by the finite element method. Considering the large area of the landfill, the values for each field in the central area of the landfill are relatively close in the horizontal plane. Therefore, we can explore the variation in the landfill properties with depth and time by studying the one-dimensional case.

To balance computational tractability with mechanistic insight, this study employs several simplifications: a 1D vertical geometry, a single-reaction representation for MSW degradation, and the neglect of horizontal fluid flow. These choices allow us to focus on validating the core microbial-chemical-thermal coupling mechanisms. While these simplifications limit direct field-scale predictive accuracy, they provide a tractable framework for initial validation and sensitivity analysis.

In many cases, obtaining the concentration of microorganisms and the rate of decomposition per unit concentration is difficult since the complexity and heterogeneity of microbial growth. The decomposition rate of the ith substrate under the optimal conditions $r_{\mathrm{opt},i}$ can be calculated when $k_1(T)$, $k_2(pH)$ and $k_3(S)$ are known from Equations (3) and (4). The decomposition rate of the ith substrate under a certain condition can be obtained by

$${\displaystyle \frac{\partial S_i}{\partial t}}=-k_1(T)k_2(pH)k_3(S)r_{\mathrm{opt},i}$$

(14)

For the one-dimensional case, the flow of gas in the horizontal plane is caused by the gas wall. The natural convection of the gas transfer to the external air space can be described by Newton’s law of cooling. Substituting Newton’s law of cooling for gas convection in Equation (8), we can obtain

$${\left(\rho C\right)}_{\mathrm{eff}}{\displaystyle \frac{\partial T}{\partial t}}+{\displaystyle \frac{\partial v_{\mathrm{ext}}h(T-T_{\mathrm{ext}})}{\partial z}}+{\displaystyle \frac{\partial }{\partial z}}\left(-k_{\mathrm{eff}}{\displaystyle \frac{\partial T}{\partial z}}\right)={\displaystyle \sum _{i=1}^m{Q}_i}$$

(15)

3.2. Model Parameterization

Peleg [17,19] proposed that the effects of temperature and pH and reactant concentration on microbial growth can be described by

$$k(X)={({\displaystyle \frac{\alpha }{\alpha +\beta }})}^{-\alpha }{(1-{\displaystyle \frac{\alpha }{\alpha +\beta }})}^{-\beta }{X}^{\alpha }{(1-X)}^{\beta }$$

(16)

$$X_{\mathrm{opt}}={\displaystyle \frac{x_{\mathrm{opt}}-x_{\min }}{x_{\max }-x_{\min }}}={\displaystyle \frac{\alpha }{\alpha +\beta }}$$

(17)

$$X={\displaystyle \frac{x-x_{\min }}{x_{\max }-x_{\min }}}$$

(18)

$$\beta ={\displaystyle \frac{\alpha }{X_{\mathrm{opt}}}}-\alpha$$

(19)

The model in Equations (16)–(19) originally stems from the field of food science, where it is used to describe microbial growth responses under non-ideal conditions. It must be pointed out that directly applying it to the highly heterogeneous and complex system of MSW degradation carries a risk. Therefore, in this work, it is primarily regarded as a simplified model with a parametric advantage that it uses a small number of shape parameters to flexibly characterize nonlinear inhibition effects, providing a feasible framework for the preliminary exploration of multi-field coupling mechanisms. The ultimate applicability of the model in this specific context requires subsequent validation and parameter calibration through experimental data targeted at MSW degradation.

Only 4 parameters including $x_{\mathrm{opt}}$, $x_{\max }$, $x_{\min }$ and $\alpha$ need to be determined since $\beta$ is decided by $\alpha$ and $x_{\mathrm{opt}}$. Then we can the influence of temperature and pH value from Equations (20) and (21), which are shown below:

$$k_1(T)={(X_{\mathrm{opt}}^T)}^{-{\alpha }_T}{(1-X_{\mathrm{opt}}^T)}^{-{\beta }_T}{(X^T)}^{{\alpha }_T}{(1-X^T)}^{{\beta }_T}$$

(20)

$$k_2(pH)={(X_{\mathrm{opt}}^{pH})}^{-{\alpha }_{pH}}{(1-X_{\mathrm{opt}}^{pH})}^{-{\beta }_{pH}}{(X^{pH})}^{{\alpha }_{pH}}{(1-X^{pH})}^{{\beta }_{pH}}$$

(21)

where $X^T={\displaystyle \frac{T-T_{\min }}{T_{\max }-T_{\min }}}$, $X_{\mathrm{opt}}^T={\displaystyle \frac{T_{\mathrm{opt}}-T_{\min }}{T_{\max }-T_{\min }}}$, $X^{pH}={\displaystyle \frac{pH-p{H}_{\min }}{p{H}_{\max }-p{H}_{\min }}}$, and $X_{\mathrm{opt}}^{pH}={\displaystyle \frac{p{H}_{\mathrm{opt}}-p{H}_{\min }}{p{H}_{\max }-p{H}_{\min }}}$.

The effect of reactant concentration on microbial growth can be expressed by

$$k_3(S)={\displaystyle \frac{e}{e-1}}(1-\exp [{(-{\displaystyle \frac{S}{S_{\mathrm{opt}}}})}^l])$$

(22)

The results of Equations (20)–(22) are shown in Figure 1. When in the optimal condition, $k_1(T)$, $k_2(pH)$ and $k_3(S)$ are equal to 1. ${\alpha }_T$, ${\alpha }_{pH}$ and $l$ determine the shape of the curves, while they do not change the start, end and highest points of the curves.

Although the results in Figure 1 show the impact of changing a single influencing factor on microbial growth while keeping other factors constant, all the influencing factors exhibit a strong indirect correlation mediated by microbial activity in the established model and real landfill sites. The consumption of substrate concentration promotes the production of gases and heat, driving the entire cycle, while the accumulation of product concentration is the direct cause of the decrease in pH and the shift from alkaline to acidic conditions. The coupling effects between multiple factors will be further discussed in the following sections.

After obtaining the above relationship in Equations (20)–(22), all the equations are determined. By selecting the parameters according to the literature, the numerical model can be finalized. The parameters are selected as shown in

Table 1. The parameter selection required for the model calculation.

Parameters	Expressions or Values	Data Source and Notes
$k_1(T)$	${\alpha }_T=2$, ${\beta }_T=2.8$, $T_{\max }=70 °\mathrm{C}$, $T_{\min }=10 °\mathrm{C}$, $T_{\mathrm{opt}}=35 °\mathrm{C}$	[17,19]
$k_2(pH)$	${\alpha }_{pH}=2$, ${\beta }_{pH}=3$, $p{H}_{\max }=10$, $p{H}_{\min }=5$, $p{H}_{\mathrm{opt}}=8$	[17,19]
$k_3(S)$	$S_{\mathrm{opt}}=100 {\mathrm{kg}/\mathrm{m}}^3$, $l=2$	[19]
$S_i^0$	300 kg/m3	The initial reactant concentration is 300 kg/m3.
$c_0^{{\mathrm{H}}^{+}}$	1 × 10−4 mol/m3	The initial pH value is equal to 7.
$r_{\mathrm{opt},i}$	8/(3600 × 24 × 365) kg/(m3∙s)	The decomposition rate of the ith substrate under the optimal conditions is 8 kg/(m3∙year)
$v_{{\mathrm{H}}^{+},i}$	1 × 10−6 mol/kg	Considering that the product acetic acid is a weak acid, it cannot be completely hydrolyzed.
$m$	1	Consider the whole reaction process as 1 reaction.
$\Delta H_i$	2 × 106 J/kg	[21]
${\left(\rho C\right)}_{\mathrm{eff}}$	1.4 × 106 J/(m3∙°C)	Calculation by ${\theta }_{\mathrm{f}}$, ${\rho }_{\mathrm{f}}$, $C_{\mathrm{f}}$, ${\theta }_{\mathrm{S}}$, ${\rho }_{\mathrm{S}}$ and $C_{\mathrm{S}}$.
${\rho }_{\mathrm{f}}$	0.7 kg/m3	[22] From methane.
$C_{\mathrm{f}}$	2.23 × 103 J/(kg∙°C)	[22] From methane.
${\theta }_{\mathrm{S}}$	0.7	Assume that the porosity of the MSW is 0.3.
${\rho }_{\mathrm{S}}$	1000 kg/m3	[23]
$C_{\mathrm{S}}$	2000 J/(kg∙°C)	[23]
${\theta }_{\mathrm{f}}$	0.3	Assume that the porosity of the MSW is 0.3.
$k_{\mathrm{eff}}$	0.29 W/(m∙°C)	Calculation by ${\theta }_{\mathrm{S}}$, $k_{\mathrm{s}}$, ${\theta }_{\mathrm{f}}$ and $k_{\mathrm{f}}$.
$k_{\mathrm{s}}$	0.40 W/(m∙°C)	[24]
$k_{\mathrm{f}}$	0.03 W/(m∙°C)	[23] From methane.
$v_{\mathrm{gas},i}$	0.300	[25]
${\rho }_{\mathrm{gas}}$	0.7 kg/m3	[22] From methane.
${\lambda }_i$	16 kg/mol	[22] From methane.
$v_{\mathrm{V},i}$	1.39 m6/kg	[25] Calculation through the mass and density of reactants and products of the chemical reaction equations.
$v_{\mathrm{ext}}$	0.006	Calculation by the density of the gas wall.
$h$	4 W/((m2∙°C)	[26]
$T_{\mathrm{ext}}$	25 °C	Calculation by the temperature in the air.
$M$	0.45 m2	Calculation by the porosity of the MSW.
$\mathit{\kappa }$	10−14 m2	Calculation by the porosity of the MSW.
$\mu$	1.1 × 10−5 Pa∙s	Calculation by the porosity of the MSW.

, and some of them are obtained from Tansel [20].

3.3. Equation Solving by Finite Difference Method

To minimize the influence of the upper boundary conditions on the results, the height of the landfill is set to 50 m and the operation time of the landfill is set to 30 years. By comparing the variation in results under different mesh densities, we confirmed that the 20 × 200 grid scheme ensures solution stability. This scheme was ultimately selected as it maintains a good balance between computational accuracy and efficiency.

In the process of establishing the equations, the interactions between the internal parts of the system are considered, as well as the external actions on the internal part of the system. A defined physical process, however, is also subject to the influence of historical circumstances and the constraints of the surrounding environment on the motion through the boundaries. The choice of initial and boundary conditions directly affects the final calculation results. The boundary conditions were set with the same initial material concentration, temperature, and pH values in all parts of the landfill. Their specific conditions were taken as shown in

Table 2. The initial and boundary conditions for the model calculation.

Parameters	Expressions or Values	Notes
$T(t,z)$	$T(0,z)=20 ℃$ $T(t,0)=20 ℃$ ${\displaystyle \frac{\partial T(t,z_{\max })}{\partial z}}=0$	Assume that the temperature at the top is always equal to the temperature of the air and heat exchange between the landfill base and the underlying soil layer is negligible.
$pH(t,z)$	$pH(0,z)=7$ ${\displaystyle \frac{\partial pH(t,0)}{\partial z}}=0$ ${\displaystyle \frac{\partial pH(t,z_{\max })}{\partial z}}=0$	Assume that the exchange of pH at the top and bottom boundaries are negligible.
$S(t,z)$	$S(0,z)=300 {\mathrm{kg}/\mathrm{m}}^3$ ${\displaystyle \frac{\partial pH(t,0)}{\partial z}}=0$ ${\displaystyle \frac{\partial pH(t,z_{\max })}{\partial z}}=0$	Assume that the exchange of substrates at the top and bottom boundaries are negligible.

.

3.4. Model Validation and Result Analysis

Model Validation

Figure 2 illustrates a comparative analysis between the simulation results generated by our model and the empirical temperature data published by Yeşiller et al. [27]. The close alignment observed between the calculated values and the measured data strongly supports the rationality and accuracy of the proposed coupled model. Specifically, the temperature profile demonstrates a characteristic trend of increasing to a peak value of approximately 45 °C around the fifth year, followed by a gradual decline. In the subsequent years, the temperature continues to decrease steadily until it asymptotically approaches the ambient temperature level. The calculated results are not completely consistent with the monitoring results, caused by calculation errors, which will be analyzed in the section of Discussion.

Due to the potential challenges including data availability and the complexity of field monitoring, this paper lacks direct comparison with pH and gas generation data. One of the key future works is seeking independent datasets that include pH and gas flux data to conduct a more comprehensive validation of the model. Despite these limitations, the model’s success in predicting the overall trend and magnitude of temperature changes, affirms its value as a reasonable exploratory tool.

Figure 3a illustrates the spatial and temporal distribution of temperature, and the peak value (approximately 44.3 °C) was observed at the fifth year. The trend of k₁ was not the same as the temperature, especially in the fifth year in Figure 3b. When the temperature exceeds the optimum, k₁ decreases with temperature, leading to a valley in the fifth year. In the initial stage, k₁ increases rapidly due to the optimal temperature, which results in a rapid rate of heat production in the landfill. The results in Figure 3c follow the same time distribution pattern at different depths. The temperature at the top is lower in Figure 3d, which is caused by the boundary condition that the upper MSW is closer to the ground. When the depth exceeds 10 m, the temperature tends to be constant. In a real case from Yeşiller et al. [27], the temperature at the bottom will be lower than in the middle due to the lower temperature of the liner and soil at the bottom. The effect of the liner and soil at the bottom is not considered in this calculation, so the temperature at the bottom is higher.

The pH value exhibits a clear temporal decline while maintaining remarkable consistency across different depths, as illustrated in Figure 4a,d. This spatial homogeneity suggests that acid production and distribution processes occur relatively uniformly throughout the landfill profile under the modeled conditions. As the pH deviates from the optimal range for microbial activity, the k₂ demonstrates a corresponding monotonic decrease until reaching equilibrium, as shown in Figure 4b.

The observed stabilization of pH at approximately 6.3 represents a dynamic equilibrium state in Figure 4a. This plateau occurs due to the balance between continuous acid generation from ongoing degradation processes and the inherent buffering capacity provided by the weak acid characteristics of degradation products like acetic acid. The appropriate concept is the incomplete ionization of weak acids, which results in a buffering effect that moderates pH changes.

This buffering mechanism, combined with the feedback inhibition of the k₂, creates a self-regulating system where excessive acid production ultimately suppresses the microbial activity responsible for acid generation. The consistent pH patterns observed across different depths in Figure 4c further confirm that the balance operates throughout the landfill volume.

The spatial and temporal distribution pattern of the substrate concentration in Figure 5a is similar to that of temperature. The substrate concentration decreases with time in Figure 5c, while it is almost the same at different depths in Figure 5d. The decreasing rate of the substrate concentration is greatest in 5 years, representing the maximum rate of decomposition of MSW at this time. Since k₃ increases with decreasing concentration of the substrate, it decreases with time in Figure 5b. When the concentration of the substrate approaches 0, k₃ also approaches 0, indicating that the biochemical reaction of MSW has stopped at this time. This suggests that the endpoint of the decomposition reaction in landfills is controlled by the substrate concentration.

Figure 6a illustrates the variation in k (degradation rate relative to its maximum potential value under optimal conditions) with time and space. The decomposition rate of MSW increases rapidly over 2 years and then decreases rapidly later in Figure 6b. The variation in k with depth is not significant in Figure 6c. Combining the results in Figure 3b, Figure 4b and Figure 5b, the peak is mainly due to the increase in temperature. The main factor affecting the pre-reaction rate is the temperature, and the factor affecting the final reaction rate is the concentration of the substrate. pH value has a small range of variation and has a limited effect on the reaction rate.

Figure 7a and Figure 7b, respectively, illustrate the spatiotemporal evolution patterns of cumulative methane production and pressure within the landfill body. As shown in Figure 7a, the cumulative methane production gradually increases with time and depth, while its spatial distribution exhibits a clear inverse relationship with the consumption pattern of substrate concentration (Figure 5c). Methane, as the end product of anaerobic degradation of substrates, has a yield directly dependent on substrate consumption. Consequently, in regions with high substrate concentration (such as the central part of the landfill), degradation activity is intense, leading to higher methane production. As the substrate becomes depleted (corresponding to the decrease in concentration shown in Figure 5c), the rate of methane generation slows, and the cumulative curve gradually stabilizes. This confirms that gas production is a direct result of substrate consumption.

The pressure distribution in Figure 7b closely resembles the pattern of degradation rate (Figure 6a), rather than aligning directly with the temperature distribution (Figure 3a). This indicates that within the model’s temperature variation range (20 °C to 45 °C), the contribution of thermal expansion to gas pressure is relatively minor. Instead, pressure dynamics are primarily controlled by the gas generation rate. During the peak degradation period (around year 2, corresponding to the peak in Figure 6a), the large volume of gas produced per unit time cannot dissipate instantaneously, causing a rapid increase in pressure within the landfill body. As degradation activity weakens, the gas production rate declines, and with the slow migration of gas, the internal pressure gradually decreases. This phenomenon highlights that biological gas production is the main factor driving changes in the pressure change process.

4. Discussion

The model developed in this study represents a significant step toward understanding the complex interactions within landfills by integrating microbial, chemical, thermal, and flow fields. While the results demonstrate the model’s capability to capture key trends, such as the temperature peak and the evolution of pH and substrate concentration, a critical discussion of its limitations is essential for contextualizing its findings and guiding future research. The primary sources of uncertainty stem from intentional simplifications, parameter selection, and numerical implementation.

4.1. Impact of Model Simplifications on Predictive Capability

The model’s framework involved several necessary simplifications to ensure computational tractability, which consequently influence its predictive accuracy, including 1D model simplification, single-reaction representation and neglect of horizontal flow.

The adoption of a one-dimensional geometry, while valid for analyzing dominant vertical processes in deep landfills, inherently neglects lateral heterogeneities. In reality, factors such as non-uniform waste placement, localized leachate recirculation, and the presence of gas extraction wells create significant horizontal variations in moisture content, temperature, and degradation rates. Our 1D approach likely smooths over these variations, potentially leading to an overestimation of heat accumulation in the central core and an underestimation of edge effects.

Furthermore, representing the highly heterogeneous MSW with a single, aggregated chemical reaction is a major simplification. This approach cannot capture the distinct degradation kinetics of different waste components, such as rapidly decomposing food waste versus slowly degrading lignocellulosic materials. A multi-reaction strategy would likely yield a more complex timeline for gas and heat production, characterized by multiple peaks corresponding to different waste fractions, rather than the single broad peak predicted by the current model (Figure 6a). This simplification is most applicable to landfills with highly uniform waste streams.

The decision to neglect the porosity change and vertical fluid flow also causes some errors. As shown in Figure 3d, Figure 4c and Figure 5c, the variations in parameters along the depth are insignificant except for temperature, indicating a weak vertical pressure gradient within the landfill body. The calculation results in Figure 7b further support this observation, demonstrating that the error introduced by neglecting vertical gas flow is limited. Furthermore, the current model does not account for variations in permeability, thus failing to reflect the increase in permeability resulting from microstructural changes due to substrate consumption. This simplification may lead to an overestimation of the simulated temperature. Note that for landfills with horizontal discharge or replenishment of substrates and gases or significant non-uniformity in the horizontal direction, the accuracy of calculations can be improved by adding a convection term to simulate the movement of substrates and gases in the horizontal plane.

4.2. Parameter Sensitivity and Uncertainty

Although all parameters in principle can be determined experimentally, the values used in this study were primarily sourced from literature. This introduces uncertainty, as these values may not be fully representative of the specific waste composition and environmental conditions in all landfills. A formal sensitivity analysis would be highly beneficial, and the results were shown in Figure 8a. It compares the relative sensitivity coefficients of 7 parameters (temperature boundary condition at the top (T (t,0)), initial pH value (pH (0,z)), initial substrate concentration (S (0,z)), the optimal temperature (T_opt), the optimal pH value (pH_opt), the decomposition rate of the substrate under the optimal conditions (r_opt) and the convective heat transfer coefficient (h)) to 5 different output (final porosity, final pH value, final substrate degradation rate, peak time and peak temperature). From Figure 8a, the optimal temperature (T_opt) and the maximum substrate degradation rate (r_opt) are the two parameters that have the greatest impact on almost all output variables, especially peak temperature and final degradation rate. This indicates that the microbial processes are the main driving force in the system. On the contrary, some boundary condition parameters, such as initial pH value (pH (0,z)), have very little effect on most outputs (sensitivity coefficient close to 0). This is a positive signal, meaning that within the reasonable range of these parameters, the model’s prediction results are relatively robust, reducing reliance on these difficult to measure parameters accurately.

The relationships between the two most sensitive parameters (the optimal temperature (T_opt) and the maximum substrate degradation rate (r_opt)) and the two most affected output variables (the peak temperature and substrate degradation rate) were further analyzed, shown in Figure 8b and Figure 8c, respectively. The peak temperature is mainly controlled by the heat generation potential, and its value is strongly positively correlated with both r_opt (substrate degradation and heat generation rate) and T (t,0) (environmental energy input). This indicates that the more heat generated or the higher the ambient temperature causes greater substrate assumption and higher the maximum temperature that the system can reach. Future work should prioritize determining the most influential parameters and quantifying their impact on key outputs like peak temperature, time to peak, and final degradation extent through experiments.

4.3. Model Validation and Future Directions

The model validation relied on a single historical temperature dataset. While the agreement with the data from Yesiller et al. [27] is encouraging (Figure 2), the model’s generalizability needs to be tested against a wider range of conditions, particularly datasets including pH and gas generation measurements from landfills with varying waste compositions and climates. The good fit for temperature trends confirms the model’s strength as a mechanistic tool for exploring fundamental couplings rather than a precise predictive tool for specific sites.

Noted that due to the significant simplification of the 1D model, it is only applicable to large deep landfills with relatively uniform waste compositions where vertical gradients dominate. Its applicability to MSW characteristic of developing countries, which features high organic content and moisture and requires further assessment [28,29,30]. For instance, a higher proportion of readily degradable components may lead to a sharper and earlier predicted peak in heat generation, potentially triggering more severe acidification inhibition phenomena. Furthermore, high moisture content significantly alters the pore structure and permeability of the landfill mass, considerations which are currently insufficiently coupled in the present model.

Future research should focus on relaxing the key simplifications discussed herein. The logical next steps include: (1) developing a multi-reaction framework to account for the distinct degradation pathways of major waste components; (2) incorporating dynamic porosity and coupled hydro-mechanical processes to better simulate long-term structural changes and fluid flow; and (3) extending the model to two or three dimensions to account for spatial heterogeneities. Implementing these enhancements will transform the model from a valuable conceptual tool into a more robust predictive platform for landfill management and risk assessment.

5. Conclusions

This study has established an integrated multi-field coupling model to simulate the degradation process of municipal solid waste in landfills, incorporating microbial, chemical, thermal, and hydraulic mechanisms. Through simplification into a one-dimensional framework and solution via the finite difference method, the model was validated against empirical data, demonstrating its ability to capture key dynamic trends. The following conclusions are drawn:

(1)

The landfill temperature peaks at approximately 45 °C in the fifth year, after which it gradually decreases. The temperature influence factor k₁ (T) effectively reflects thermal impacts on degradation rates, showing a suppressed effect when temperatures exceed the optimal range.

(2)

Both pH and substrate concentration decrease over time while remaining largely uniform along the depth dimension. The limited variation in pH leads to a relatively minor influence on the degradation rate. The substrate-related factor k₃ (S) decreases as the substrate is consumed and approaches zero upon depletion, indicating that substrate availability ultimately determines the endpoint of the degradation process.

(3)

The rate of waste decomposition peaks within two years and subsequently declines. Temperature is the dominant factor controlling the initial rapid degradation, while substrate concentration becomes the limiting factor in later stages.

The model offers value as a tool for predicting temperature trends, guiding leachate treatment, simulating gas generation scenarios, and assessing landfill stability, which provides a theoretical model and numerical tool for waste sustainability. However, the current simplifications—including the 1D approach, single-reaction representation, and neglect of porosity changes and two-phase flow—limit direct field applicability. Future work should introduce multi-reaction kinetics, dynamic permeability, and 3D transport processes to improve predictive accuracy and generalize the model to a wider range of landfill configurations and waste types.