A simple calculation method for predicting the leakage of landfill leachate
Patent Information
- Application Number
- CN202610890737.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-06-18
- Publication Date
- 2026-09-25
AI Technical Summary
目前，虽已有适用于指定工程场景的填埋场渗沥液外漏污染距离预测解析方法报道，但这些方法多依赖Laplace数值逆变换等复杂数学手段，存在不够简明的问题，无法直接给出污染物浓度随时间与二维空间位置变化的解析表达式，难以快速完成污染范围及距离预测
[0046]上述发明内容相关记载仅是本申请技术方案的概述，为了让本领域普通技术人员能够更清楚地了解本申请的技术方案，进而可以依据说明书的文字及附图记载的内容予以实施，并且为了让本申请的上述目的及其它目的、特征和优点能够更易于理解，以下结合本申请的具体实施方式及附图进行说明。
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Abstract
Description
Technical Field
[0001] This invention relates to the field of environmental geotechnical engineering technology, specifically to a simple calculation method for predicting leachate leakage pollution in landfills. Background Technology
[0002] Human production and daily life generate a large amount of solid waste, and sanitary landfill is a common method of solid waste disposal, especially suitable for municipal solid waste. For landfills formed by solid waste accumulation, the impermeable liner system is a core component. Normally, a pre-constructed impermeable liner system can effectively prevent the migration of pollutants from within the landfill, providing reliable protection for the surrounding ecological environment. However, after years of service, impermeable liner system failures still occur frequently due to the combined effects of high temperature and pressure inside the landfill, external wet / dry / freeze-thaw cycles, and non-uniform settlement caused by the solid waste stockpile. Impermeable liner system failure can lead to leachate leakage from the landfill, which in turn can cause pollutant migration and pose an ecological risk to the adjacent environment. In engineering practice, when leachate leakage occurs at a landfill, before taking measures such as constructing a vertical barrier on the outside to effectively prevent the leaked leachate from causing further pollution, it is necessary to predict the scope and distance of the pollution caused by the leachate leakage. This will provide guidance for the implementation and effectiveness evaluation of subsequent pollution prevention measures.
[0003] Analytical methods are commonly used to study the migration behavior of pollutants in soil layers, and are therefore frequently applied to predict the extent and distance of landfill leachate leakage pollution. Currently, although analytical methods for predicting the distance of landfill leachate leakage pollution applicable to specific engineering scenarios have been reported, these methods often rely on complex mathematical techniques such as Laplace numerical inverse transform, which are not concise enough and cannot directly provide analytical expressions for the changes in pollutant concentration over time and in two-dimensional spatial location, making it difficult to quickly predict the extent and distance of pollution. Therefore, there is an urgent need in this field to develop a concise calculation method that is highly accurate, efficient, and provides explicit analytical expressions to meet the practical engineering needs for rapid prediction of the extent and distance of landfill leachate leakage pollution. Summary of the Invention
[0004] In view of the above problems, this application provides a simple calculation method for predicting the pollution situation of landfill leachate leakage. It can directly provide an analytical expression of the change of pollutant concentration with time and two-dimensional spatial location, thus it can quickly predict the range and distance of landfill leachate leakage pollution. It has the advantages of high accuracy and simple operation, and can provide accurate preliminary data support for the formulation of pollution prevention and control measures after landfill leachate leakage.
[0005] To achieve the above objectives, this application provides a simple calculation method for predicting landfill leachate leakage pollution, including:
[0006] Obtain the property parameters of adjacent soil layers and the two-dimensional migration parameters of pollutants after leachate leakage from a landfill;
[0007] Based on the property parameters of adjacent soil layers and the two-dimensional migration parameters of pollutants, a two-dimensional migration mathematical model of pollutants in adjacent soil layers is constructed. The two-dimensional migration mathematical model includes the two-dimensional migration control equations and solution conditions.
[0008] The two-dimensional migration control equation and solution conditions are transformed and solved by using cosine series transformation and Laplace transform, generating a concise analytical solution expression for the change of pollutant concentration with time and two-dimensional spatial location;
[0009] Substitute the property parameters of the adjacent soil layer and the two-dimensional migration parameters of pollutants into the concise analytical solution expression to calculate the two-dimensional distribution data of pollutant concentration in the adjacent soil layer after a specified number of years.
[0010] Obtain pollutant concentration limits, and compare and analyze the two-dimensional distribution data of pollutant concentration with the pollutant concentration limits to determine the two-dimensional spatial range of contamination in adjacent soil layers;
[0011] The average contamination distance is calculated based on the two-dimensional spatial extent of the contaminated area and the vertical height of the adjacent soil layer, and the maximum contamination distance is determined based on the two-dimensional distribution data of pollutant concentration.
[0012] In some embodiments, a two-dimensional mathematical model of pollutant migration in adjacent soil layers is constructed, including:
[0013] A two-dimensional migration control equation for pollutants in adjacent soil layers after leachate leakage from a landfill is established, expressed by formula (1), which is as follows:
[0014]
[0015] In formula (1), The coordinates are horizontal, in meters. The coordinates are vertical, and the unit is meters. Time, in seconds; The concentration of pollutants in adjacent soil layers is expressed in mg / L. The horizontal hydrodynamic dispersion coefficient in the adjacent soil layer. The vertical hydrodynamic dispersion coefficient in adjacent soil layers, in meters. 2 / s; , The blocking factor is dimensionless. This is the distribution coefficient of the adjacent soil layer, expressed in L / mg. This refers to the dry density of the adjacent soil layer, expressed in g / cm³. 3 ; Porosity of adjacent soil layers, dimensionless; , The Darcy velocity in the outer soil mass is expressed in m / s. The convection velocity in the outer soil mass is expressed in m / s. , is the hydraulic permeability coefficient of the adjacent soil layer, in m / s; The hydraulic gradient in the adjacent soil layer is dimensionless. The first-order degradation constant of the pollutant is expressed in s. -1 ;
[0016] Based on the typical scenario of two-dimensional migration of pollutants in adjacent soil layers after leachate leakage from landfills, the solution conditions for the two-dimensional migration of pollutants are determined and expressed by formula (2), which is as follows:
[0017]
[0018] In formula (2), The concentration of contaminants in the externally leaked leachate is expressed in mg / L. This represents the vertical height of the adjacent soil layer, in meters (m).
[0019] In some embodiments, the hydrodynamic dispersion coefficient in the horizontal direction and the hydrodynamic dispersion coefficient in the vertical direction of the adjacent soil layer are represented by formula (3), which is as follows:
[0020]
[0021] In formula (3), , The horizontal mechanical dispersion coefficient in adjacent soil layers, in meters. 2 / s; This represents the longitudinal dispersion of adjacent soil, in meters. This represents the molecular diffusion coefficient of pollutants in adjacent soil layers.
[0022] In some embodiments, the concentration of contaminants in the external leakage leachate is expressed by formula (4), which is as follows:
[0023]
[0024] In formula (4), This represents the maximum concentration of contaminants in the externally leaked leachate, expressed in mg / L. It is a dimensionless function that characterizes the non-uniform distribution of concentration in the vertical direction; A parameter characterizing the time-varying characteristics of concentration, with units of seconds. -1 .
[0025] In some embodiments, the two-dimensional migration control equation and solution conditions are transformed and solved using cosine series transformation and Laplace transformation to generate a concise analytical solution expression of the pollutant concentration changing with time and two-dimensional spatial location, which is expressed by formula (5), as follows:
[0026]
[0027] In formula (5), , It is an integer not less than zero; For functions Relevant parameters; To be related to convection rate First-order degradation constant and hydrodynamic dispersion coefficient A comprehensive parameter related to multiple coefficients;
[0028] And, the specified number of years since the landfill leachate leakage occurred shall be recorded as... Then, the concise analytical expression for the change of pollutant concentration in the adjacent soil layer with time and two-dimensional spatial location is expressed by formula (6), which is as follows:
[0029] .
[0030] In some embodiments, Equation (7) is expressed as follows:
[0031] .
[0032] In some embodiments, Equation (8) is expressed as follows:
[0033]
[0034] In formula (8), , For parameters Relevant intermediate parameters.
[0035] In some embodiments, pollutant concentration limits are obtained, and a comparative analysis is performed between the two-dimensional distribution data of pollutant concentrations and the pollutant concentration limits to determine the two-dimensional spatial extent of contamination in adjacent soil layers, including:
[0036] Based on the pollutant concentration limits for centralized drinking water sources and industrial and agricultural water use in the "Groundwater Quality Standard", the pollutant concentration limits for centralized drinking water sources and industrial and agricultural water use are denoted as follows: In adjacent soil layers, the concentration of pollutants is greater than or equal to the concentration limit. Defined as the two-dimensional spatial extent of contamination in adjacent soil layers.
[0037] In some embodiments, within a specified number of years The extent of contamination in the adjacent soil layer is determined by formula (9), which is as follows:
[0038] .
[0039] In some embodiments, within a specified number of years At that time, the average contamination distance in the adjacent soil layer is expressed by formula (10), which is as follows:
[0040]
[0041] In formula (10), For a specified number of years The average contamination distance of adjacent soil layers;
[0042] within the specified period At that time, the maximum pollution distance in the adjacent soil layer is determined based on the two-dimensional distribution data of pollutant concentration, and is expressed by formula (11), which is as follows:
[0043]
[0044] In formula (11), For a specified number of years The maximum contamination distance of the adjacent soil layer.
[0045] Unlike existing technologies, the above technical solution obtains the property parameters of adjacent soil layers and the two-dimensional migration parameters of pollutants after leachate leakage from a landfill. Based on these parameters, a two-dimensional migration mathematical model of pollutants in adjacent soil layers is constructed, including two-dimensional migration control equations and solution conditions. Cosine series transformation and Laplace transform are used to transform and solve the two-dimensional migration control equations and solution conditions, generating a concise analytical expression for the variation of pollutant concentration with time and two-dimensional spatial location. The property parameters of adjacent soil layers and the two-dimensional migration parameters are substituted into the concise analytical expression to calculate the two-dimensional distribution data of pollutant concentration in adjacent soil layers after a specified number of years. Pollutant concentration limits are obtained, and the two-dimensional spatial range of contaminated adjacent soil layers is determined by comparing and analyzing the two-dimensional distribution data with the limits. Based on the contaminated two-dimensional spatial range and the vertical height of adjacent soil layers, the average contamination distance is calculated, and the maximum contamination distance is determined based on the two-dimensional distribution data. To address the problems of existing technologies that rely on complex mathematical methods such as Laplace numerical inverse transform, cannot directly provide a clear analytical expression for the change of pollutant concentration with time and two-dimensional spatial location, and are difficult to rapidly predict the pollution range and distance, this invention transforms and solves the two-dimensional migration control equation and solution conditions using cosine series transform and Laplace transform. This directly generates a concise and rapidly calculable analytical expression for the change of pollutant concentration with time and two-dimensional spatial location, avoiding the cumbersome process of complex numerical inverse transform. This allows for the direct substitution of the property parameters of adjacent soil layers and the two-dimensional migration parameters of pollutants into this expression, quickly calculating the two-dimensional distribution data of pollutant concentration in adjacent soil layers after a specified number of years. Based on this, the two-dimensional spatial range of pollution, the average pollution distance, and the maximum pollution distance can be determined. Therefore, this invention has the advantages of high accuracy, simple operation, and rapid prediction, providing reliable preliminary data support for the scientific formulation of pollution prevention and control measures after landfill leachate leakage.
[0046] The above description of the invention is merely an overview of the technical solution of this application. In order to enable those skilled in the art to better understand the technical solution of this application and to implement it based on the description and drawings, and to make the above-mentioned objectives and other objectives, features and advantages of this application easier to understand, the following description is provided in conjunction with the specific embodiments and drawings of this application. Attached Figure Description
[0047] The accompanying drawings are only used to illustrate the principles, implementation methods, applications, features, and effects of specific embodiments of the present invention and other related contents, and should not be considered as limitations on this application.
[0048] In the accompanying drawings of the instruction manual:
[0049] Figure 1 This is a flowchart illustrating steps S101 to S106 of the simplified calculation method described in the specific implementation.
[0050] Figure 2 This is a schematic diagram of a two-dimensional migration model of pollutants in adjacent soil layers after leachate leakage from a landfill, as described in a specific implementation method.
[0051] Figure 3 This is a schematic diagram comparing the pollutant concentration variation with the horizontal coordinate described in the specific implementation method with the results of COMSOL Multiphysics numerical simulation.
[0052] Figure 4 This is a schematic diagram comparing the results of pollutant concentration variation with vertical coordinates in the specific implementation method with the results calculated by the semi-analytical method;
[0053] Figure 5 This is a schematic diagram illustrating the two-dimensional distribution of pollutants in adjacent soil layers, the two-dimensional spatial range of contamination, the average contamination distance of adjacent soil layers, and the maximum contamination distance when landfill leachate leakage is predicted to occur 2 years later, as described in the specific implementation method. Detailed Implementation
[0054] To illustrate the possible application scenarios, technical principles, implementable specific solutions, and achievable objectives and effects of this application in detail, the following description, in conjunction with the listed specific embodiments and accompanying drawings, provides a detailed explanation. The embodiments described herein are merely illustrative of the technical solutions of this application and are therefore intended to limit the scope of protection of this application.
[0055] In this document, the term "embodiment" means that a specific feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of this application. The term "embodiment" appearing in various places throughout the specification does not necessarily refer to the same embodiment, nor does it specifically limit its independence or connection with other embodiments. In principle, in this application, as long as there are no technical contradictions or conflicts, the technical features mentioned in each embodiment can be combined in any way to form corresponding implementable technical solutions.
[0056] Unless otherwise defined, the technical terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains; the use of related terms herein is merely for the purpose of describing particular embodiments and is not intended to limit this application.
[0057] In the description of this application, the term "and / or" is used to describe the logical relationship between objects, indicating that three relationships can exist. For example, A and / or B means: A exists, B exists, and A and B exist simultaneously. Additionally, the character " / " in this document generally indicates that the preceding and following objects have an "or" logical relationship.
[0058] In this application, terms such as “first” and “second” are used only to distinguish one entity or operation from another, and do not necessarily require or imply any actual quantity, hierarchy or order relationship between these entities or operations.
[0059] Without further limitations, the use of terms such as “comprising,” “including,” “having,” or other similar open-ended expressions in this application is intended to cover non-exclusive inclusion, which does not exclude the presence of additional elements in a process, method, or product that includes the stated elements, such that a process, method, or product that includes a list of elements may include not only those defined elements but also other elements not expressly listed, or elements inherent to such a process, method, or product.
[0060] As understood in the Examination Guidelines, in this application, expressions such as "greater than," "less than," and "exceeding" are understood to exclude the stated number; expressions such as "above," "below," and "within" are understood to include the stated number. Furthermore, in the description of the embodiments in this application, "multiple" means two or more (including two), and similar expressions related to "multiple" are also understood in this way, such as "multiple groups" and "multiple times," unless otherwise explicitly specified.
[0061] The processor described in the embodiments of this application can be implemented by hardware, firmware, software, or a combination thereof. It can be a circuit, one or more of an application-specific integrated circuit (ASIC), a digital signal processor (DSP), a digital signal processing device (DSPD), a programmable logic device (PLD), a field-programmable gate array (FPGA), a central processing unit (CPU), a controller, a microcontroller, or a microprocessor. It also includes other physical, biological, or chemical structures that can implement the same or equivalent functions as the processors listed above, such as biological neurons, quantum computing units, DNA computing units, etc., so that the processor can execute some or all of the steps in the computer program or method involved in the various embodiments of this application, or any combination of the steps mentioned therein.
[0062] The computer program involved in the embodiments can be stored in a computer device readable storage medium, which includes, but is not limited to, disks, magnetic tapes, magnetic cards, floppy disks, flash memory, optical disks, optical cards, read-only memory (ROM), random access memory (RAM), erasable programmable ROM (EPROM), and electrically erasable programmable ROM (EEPROM), etc., and also includes other biological, physical, or chemical structures that can achieve the same or equivalent functions as the storage media listed above, such as DNA, RNA, proteins, and other units with information storage capabilities. In specific embodiments, the storage medium involved can be one of the above-mentioned media types, or a combination of the above-mentioned media types. In different embodiments, the computer program involved in the embodiments can be centrally stored in a single medium, or distributed and stored in multiple media. The memory containing the computer device readable storage medium can be non-volatile memory or random access memory. These computer device readable storage media can be built into the device, or can be connected to the device involved in the embodiments as an external device or part of an external device. In some embodiments, the memory having a computer device readable storage medium is deployed locally; in other embodiments, the memory may be deployed remotely from the processor, for example, as a network-attached memory accessed via RF circuitry or an external port and a communication network, wherein the communication network may be the Internet, one or more intranets, a local area network (LAN), a wide area network (WLAN), a storage area network (SAN), or a suitable combination thereof, as long as computer device access to the memory is enabled. Furthermore, the computer program involved in the embodiments may be stored in plaintext / ciphertext form, or it may be designed as training data, integrated and recombined through model training and implicitly stored in the parameter states of a deep neural network or other machine learning model.
[0063] Please see Figures 1 to 5 This embodiment provides a simple calculation method for predicting landfill leachate leakage pollution, including:
[0064] S101. Obtain the property parameters of adjacent soil layers and the two-dimensional migration parameters of pollutants after leachate leakage from the landfill.
[0065] S102. Based on the property parameters of adjacent soil layers and the two-dimensional migration parameters of pollutants, a two-dimensional migration mathematical model of pollutants in adjacent soil layers is constructed. The two-dimensional migration mathematical model includes the two-dimensional migration control equations and solution conditions.
[0066] S103. The two-dimensional migration control equation and solution conditions are transformed and solved by using cosine series transformation and Laplace transformation to generate a concise analytical solution expression of the pollutant concentration changing with time and two-dimensional spatial location.
[0067] S104. Substitute the property parameters of the adjacent soil layer and the two-dimensional migration parameters of pollutants into the simplified analytical solution expression to calculate the two-dimensional distribution data of pollutant concentration in the adjacent soil layer after a specified number of years.
[0068] S105. Obtain pollutant concentration limits, and compare and analyze the two-dimensional distribution data of pollutant concentration with the pollutant concentration limits to determine the two-dimensional spatial range of contamination in adjacent soil layers.
[0069] S106. Based on the two-dimensional spatial range of the contaminated area and the vertical height of the adjacent soil layer, the average contamination distance is calculated, and the maximum contamination distance is determined based on the two-dimensional distribution data of pollutant concentration.
[0070] In step S101, the adjacent soil layer property parameters refer to the basic data used to describe the physical and hydraulic properties of the natural soil layers outside the landfill. Specifically, these include the vertical height, hydraulic permeability coefficient, longitudinal dispersion, hydraulic gradient, distribution coefficient, dry density, and porosity of the adjacent soil layers. These parameters can be obtained through on-site drilling sampling and in-situ testing combined with laboratory geotechnical tests. The two-dimensional pollutant migration parameters refer to the key data used to describe the migration behavior of pollutants in the soil layers. Specifically, these include the molecular diffusion coefficient, first-order degradation constant, maximum pollutant concentration in the leaked leachate, functions characterizing the non-uniform distribution of concentration in the vertical direction, and parameters characterizing the time-varying characteristics of concentration. These parameters can be determined by consulting pollutant migration handbooks or through batch tests and soil column tests. By obtaining these two types of parameters, a complete input data foundation is provided for the subsequent construction of a mathematical model of pollutant migration, ensuring that the model can accurately reflect the actual situation of the target landfill engineering scenario.
[0071] In step S102, the two-dimensional migration control equation refers to a partial differential equation established based on convection-dispersion theory and the law of conservation of mass, used to describe the evolution of pollutant concentration over time in both horizontal and vertical dimensions. The solution conditions refer to the initial state and boundary constraints set to obtain a unique solution to the control equation, including initial concentration distribution conditions and concentration or concentration gradient conditions at the landfill leakage boundary, infinite boundary, surface boundary, and bottom impermeable boundary. Substituting the obtained parameters into the general two-dimensional migration control equation framework and combining them with the actual boundary characteristics of the leakage scenario to determine the solution conditions, a mathematical problem capable of accurately characterizing the pollutant migration behavior under the target engineering scenario is constructed.
[0072] In step S103, the cosine series transformation refers to expanding the concentration function into an infinite series of cosine functions along the vertical direction, thereby transforming the two-dimensional partial differential equation into a one-dimensional partial differential equation. Then, the corresponding concise analytical solution expression can be directly obtained using the Laplace transform. The concise analytical solution expression is the concentration function explicitly expressed in series form, directly obtained through the above transformation; its form is simple and does not involve complex calculation steps such as numerical inverse transformation. Utilizing the orthogonality of the cosine series transformation in the vertical direction, the complex two-dimensional partial differential equation is solved by reducing its dimension, ultimately obtaining an analytical expression that can be directly substituted with parameters for numerical calculation, avoiding the cumbersome process of relying on the Laplace numerical inverse transform in traditional methods.
[0073] In step S104, the specified timeframe refers to the target time point for pollution prediction after a landfill leachate leak occurs, such as 2, 5, or 10 years after the leak. The two-dimensional distribution data refers to the set of pollutant concentration values at different horizontal and vertical positions in adjacent soil layers at this specified timeframe. By substituting the specific parameter values obtained in step S101 into the analytical solution expression obtained in step S103, the concentration distribution result within the entire soil layer area can be quickly obtained through a finite series summation calculation. This process only involves algebraic operations and error function calculation, without the need for iterative solutions or numerical inversion.
[0074] In step S105, the pollutant concentration limit refers to the pollutant concentration threshold corresponding to centralized drinking water sources and industrial and agricultural water use in the "Groundwater Quality Standard." This value can be obtained by consulting the limit table for the corresponding pollutants in the standard. The contaminated two-dimensional spatial range refers to the set of spatial locations in adjacent soil layers where the pollutant concentration is greater than or equal to the concentration limit. This range can be defined by comparing the concentration value of each spatial point calculated in step S104 with the limit one by one. Using the concentration limit in the national environmental protection standard as the basis for judgment, the theoretically calculated continuous concentration distribution is transformed into a contaminated area boundary with clear engineering significance, thereby quantitatively defining the spatial range affected by pollution.
[0075] In step S106, the average pollution distance is the ratio obtained by dividing the area of the contaminated two-dimensional space by the vertical height of the adjacent soil layer. This index is used to characterize the average horizontal spread of pollution. The maximum pollution distance is the furthest horizontal distance from the pollution source within the contaminated two-dimensional space. This value can be obtained by searching for the maximum horizontal coordinate on the boundary of the contaminated area determined in step S105. By compressing the two-dimensional contaminated area into two one-dimensional distance indices with clear physical meaning, an intuitive and quantitative basis for evaluating the degree of pollution is provided for engineering decisions.
[0076] This embodiment obtains the property parameters and pollutant migration parameters of adjacent soil layers to provide data support for subsequent modeling. Based on these parameters, a two-dimensional migration mathematical model including governing equations and solution conditions is constructed. The model is solved using cosine series transformation and Laplace transform to directly obtain a concise analytical solution expression. Substituting the parameters into this expression allows for the rapid calculation of the two-dimensional concentration distribution after a specified number of years. Based on the concentration limits of national standards, the two-dimensional spatial range of pollution, the average pollution distance, and the maximum pollution distance are extracted from the calculated concentration distribution. By using cosine series transformation and Laplace transform, complex inverse numerical transformations are avoided, enabling rapid prediction of the entire process from parameter input to pollution index output. This approach offers advantages such as high accuracy, ease of operation, and high computational efficiency, providing reliable preliminary data support for the scientific formulation of pollution prevention and control measures after landfill leachate leakage.
[0077] Please see Figure 2 In some embodiments, a two-dimensional mathematical model of pollutant migration in adjacent soil layers is constructed, including:
[0078] A two-dimensional migration control equation for pollutants in adjacent soil layers after leachate leakage from a landfill is established, expressed by formula (1), which is as follows:
[0079]
[0080] In formula (1), The coordinates are horizontal, in meters. The coordinates are vertical, and the unit is meters. Time, in seconds; The concentration of pollutants in adjacent soil layers is expressed in mg / L. The horizontal hydrodynamic dispersion coefficient in the adjacent soil layer. The vertical hydrodynamic dispersion coefficient in adjacent soil layers, in meters. 2 / s; , The blocking factor is dimensionless. This is the distribution coefficient of the adjacent soil layer, expressed in L / mg. This refers to the dry density of the adjacent soil layer, expressed in g / cm³. 3 ; Porosity of adjacent soil layers, dimensionless; , The Darcy velocity in the outer soil mass is expressed in m / s. The convection velocity in the outer soil mass is expressed in m / s. , is the hydraulic permeability coefficient of the adjacent soil layer, in m / s; The hydraulic gradient in the adjacent soil layer is dimensionless. The first-order degradation constant of the pollutant is expressed in s. -1 ;
[0081] Based on the typical scenario of two-dimensional migration of pollutants in adjacent soil layers after leachate leakage from landfills, the solution conditions for the two-dimensional migration of pollutants are determined and expressed by formula (2), which is as follows:
[0082]
[0083] In formula (2), The concentration of contaminants in the externally leaked leachate is expressed in mg / L. This represents the vertical height of the adjacent soil layer, in meters (m).
[0084] In some embodiments, the hydrodynamic dispersion coefficient in the horizontal direction and the hydrodynamic dispersion coefficient in the vertical direction of the adjacent soil layer are represented by formula (3), which is as follows:
[0085]
[0086] In formula (3) , The horizontal mechanical dispersion coefficient in adjacent soil layers, in meters. 2 / s; This represents the longitudinal dispersion of adjacent soil, in meters. This represents the molecular diffusion coefficient of pollutants in adjacent soil layers.
[0087] In this embodiment, the mathematical model is constructed using a two-dimensional rectangular coordinate system. The horizontal and vertical coordinates in formula (1) are used to define the spatial location, time is used to define the calculation time, and the pollutant concentration in the adjacent soil layer is the dependent variable to be solved. The horizontal and vertical hydrodynamic dispersion coefficients respectively characterize the comprehensive diffusion and dispersion capabilities of pollutants in the horizontal and vertical directions. The horizontal hydrodynamic dispersion coefficient is contributed by both mechanical dispersion and molecular diffusion, while the vertical hydrodynamic dispersion coefficient is contributed only by molecular diffusion. The retardation factor is used to characterize the delayed effect of pollutants due to adsorption during migration. It is calculated by the distribution coefficient, dry density, and porosity. The distribution coefficient reflects the distribution ratio of pollutants between soil particles and water phase, while the dry density and porosity describe the compactness and pore space characteristics of the soil. The Darcy velocity in the outer soil refers to the seepage flow rate through a unit cross-sectional area of soil per unit time. It is calculated by the hydraulic permeability coefficient and hydraulic gradient. The hydraulic permeability coefficient characterizes the permeability of the soil, and the hydraulic gradient characterizes the pressure difference driven by seepage. The convection rate in the outer soil mass refers to the actual speed at which pollutants move with the pore water flow, obtained by the ratio of Darcy velocity to porosity. The first-order degradation constant of pollutants is used to characterize the rate at which pollutants decay due to chemical reactions or biodegradation. The physicochemical processes of convection, dispersion, adsorption, and degradation of pollutants in the soil layer are unified into a mathematical description, thus laying a theoretical foundation for subsequent solutions.
[0088] The solution conditions refer to the initial state and boundary constraints set to obtain a unique solution to the governing equations. The initial conditions assume that the contaminant concentration in the adjacent soil layer is zero before leakage occurs, i.e., assuming the initial state is uncontaminated. The left boundary condition sets the concentration at the leakage boundary to be equal to the contaminant concentration in the leaked leachate; this concentration varies with vertical position and time, characterizing the input characteristics of the pollution source. The right boundary condition sets the concentration gradient to zero at infinity, assuming the contaminant concentration tends to stabilize far from the pollution source. Both the upper and lower boundary conditions set the concentration gradient to zero, assuming no contaminant flux crosses the surface and bottom impermeable layer. The lower boundary condition involves the vertical height of the adjacent soil layer, which is used to define the calculation range of the soil layer in the vertical direction. By reasonably setting the initial and boundary conditions, the governing equations have a definite and unique solution in the specified engineering scenario, ensuring the mathematical rigor and physical rationality of subsequent transformation solutions.
[0089] In some embodiments, the concentration of contaminants in the external leakage leachate is expressed by formula (4), which is as follows:
[0090]
[0091] In formula (4), This represents the maximum concentration of contaminants in the externally leaked leachate, expressed in mg / L. It is a dimensionless function that characterizes the non-uniform distribution of concentration in the vertical direction; A parameter characterizing the time-varying characteristics of concentration, with units of seconds. -1 .
[0092] In this embodiment, the pollutant concentration in the leaked leachate refers to the concentration of pollutants contained in the leachate leaking from the damaged part of the landfill's anti-seepage liner system into the adjacent soil layer. This concentration varies with vertical position and time. In formula (4), the maximum concentration of pollutants in the leaked leachate refers to the highest value that the pollutant concentration in the leachate may reach during the entire leakage process. This parameter can be determined by on-site sampling and testing or by consulting historical data of similar landfills, and is used to characterize the magnitude of the pollution source. The function characterizing the non-uniform distribution of concentration in the vertical direction is a mathematical function describing the variation law of pollutant concentration along the vertical direction. Its dimensionless form is used to reflect the vertical non-uniformity of concentration caused by factors such as leachate density differences or stratified distribution within the landfill. The specific form of this function can be determined by fitting actual detection data or based on an empirical distribution model.
[0093] The parameter characterizing the time-varying characteristics of concentration is a coefficient used to describe the rate of decay or change of pollutant concentration over time. This parameter can be obtained through exponential fitting of long-term monitoring data and is used to reflect the concentration change over time caused by factors such as degradation, dilution, or source strength attenuation. The pollutant concentration in the leaked leachate is decomposed into the product of three independent factors: maximum concentration, vertical distribution function, and time decay function. This allows for a concise mathematical expression that simultaneously characterizes the spatial and temporal changes of the pollution source, providing boundary input conditions that conform to actual engineering scenarios for the subsequent two-dimensional migration model.
[0094] In some embodiments, the two-dimensional migration control equation and solution conditions are transformed and solved using cosine series transformation and Laplace transformation to generate a concise analytical solution expression of the pollutant concentration changing with time and two-dimensional spatial location, which is expressed by formula (5), as follows:
[0095]
[0096] In formula (5), , It is an integer not less than zero; For functions Relevant parameters; To be related to convection rate First-order degradation constant and hydrodynamic dispersion coefficient A comprehensive parameter related to multiple coefficients;
[0097] And, the specified number of years since the landfill leachate leakage occurred shall be recorded as... Then, the concise analytical expression for the change of pollutant concentration in the adjacent soil layer with time and two-dimensional spatial location is expressed by formula (6), which is as follows:
[0098] .
[0099] In this embodiment, the cosine series transformation refers to expanding the concentration function vertically into an infinite series with cosine functions as the basis functions, thereby transforming the original partial differential equation into a series of ordinary differential equations about the horizontal coordinates and time for solution. The core advantage of this transformation is that it can automatically satisfy the zero flux condition at the upper and lower boundaries, thus simplifying the solution process.
[0100] In formula (5), A non-zero integer, representing the index of the terms in the series expansion, as... As the value increases, the contribution of the series term to the concentration distribution gradually decreases. In actual calculations, the required accuracy can be achieved by truncating a finite number of terms. For functions The relevant parameters map the non-uniform distribution characteristics of pollutant concentration in the external leachate along the vertical direction to the coefficients of the series expansion, thereby incorporating the information of the boundary conditions into the analytical solution. To be related to convection rate First-order degradation constant and hydrodynamic dispersion coefficient The comprehensive parameters related to multiple coefficients collectively reflect the combined effects of convection, degradation, and vertical dispersion on pollutant migration, and their specific form is further determined by subsequent formulas. The residual error function included in the expression is the core mathematical tool for describing the characteristics of concentration front propagation during the convection-dispersion process. Its two components correspond to the concentration propagation contributions along the horizontal positive and negative directions, respectively, and together constitute the complete convection-dispersion solution form.
[0101] This embodiment solves the two-dimensional partial differential equation by reducing its dimension through cosine series transformation and Laplace transform, and explicitly gives a complete mathematical expression in series form of the pollutant concentration as a function of horizontal coordinates, vertical coordinates and time. This expression is concise and does not involve complex calculation steps such as numerical inverse transformation, and can be directly substituted into parameters for numerical calculation.
[0102] Furthermore, formula (6) is completely identical to formula (5) in form, the difference being that the time variable is changed. Replace with the specified year value By assigning the time variable in formula (5) to the target prediction time, the two-dimensional spatial distribution expression of pollutant concentration in the adjacent soil layer at that specific time can be directly obtained, providing a direct calculation basis for subsequent prediction of pollution range and distance.
[0103] For example, using toluene as a representative pollutant, the relevant parameters of adjacent soil layer properties and the relevant parameters of two-dimensional pollutant migration were measured and input as follows: vertical height of adjacent soil layer. 10m, hydraulic permeability coefficient 1.0×10 -6 m / s, longitudinal dispersion 1.52m, hydraulic gradient The distribution coefficient is 0.1. 0.0 mL / g, dry density It is 1.19 g / cm³ 3 Porosity The molecular diffusion coefficient is 0.55. 4.25×10 -10 m 2 / s, first-order degradation constant 2.2×10 -10 s -1 Maximum concentration of pollutants in externally leaked leachate The function representing the non-uniform distribution of concentration in the vertical direction is 20 mg / L. , where the parameters Take 5m, Take 2 meters as a parameter to characterize the time-varying characteristics of concentration. Set to 0.0s -1 .
[0104] Furthermore, taking 2 years as the number of years since the landfill leachate leakage occurred, and substituting this into formula (6), the two-dimensional distribution of pollutant concentration in the adjacent soil layer 2 years after the landfill leachate leakage occurs can be seen. Figure 5 .
[0105] In some embodiments, Equation (7) is expressed as follows:
[0106] .
[0107] In some embodiments, Equation (8) is expressed as follows:
[0108]
[0109] In formula (8), , For parameters Related intermediate parameters.
[0110] In this embodiment, with function Relevant parameters As expressed by formula (7), this parameter is the key bridge for mapping the non-uniform distribution characteristics of pollutant concentration in external leachate along the vertical direction to the coefficients of the cosine series expansion. In formula (7), when hour, equal The average value within the soil layer height range corresponds to the coefficient of the zeroth-order term in the series expansion, reflecting the overall average level of the concentration distribution. When hour, equal Multiplying the integral average of the product with the corresponding order cosine function over the soil layer height by 2, these values correspond to the coefficients of higher-order terms, reflecting the details of the vertical fluctuations in concentration distribution. The continuous vertical distribution function is then transformed through integral transformation. Discretize into a series of series coefficients This allows the exposed boundary conditions to be incorporated into the analytical solution expression in a concise coefficient form, thereby achieving the parameterized expression of boundary conditions while ensuring mathematical accuracy.
[0111] In this embodiment, comprehensive parameters This is expressed by formula (8). This parameter is the core comprehensive parameter in the analytical solution expression, reflecting the comprehensive influence of multiple physical factors such as convection rate, first-order degradation constant, vertical hydrodynamic dispersion coefficient, retardation factor, soil height, and time-varying decay parameter on the pollutant migration process. In formula (8), The convection rate in the outer soil mass. As a blocking factor, This represents the hydrodynamic dispersion coefficient in the horizontal direction. (Intermediate parameter) The physical significance lies in unifying the vertical dispersion effect, degradation effect, and source strength time-varying effect into a single series term with the same order. Related comprehensive indicators. By performing square root operations, the convection term is coupled with a comprehensive term that includes dispersion, degradation, and time-varying effects to generate a comprehensive parameter with velocity dimensions. This parameter directly determines the propagation characteristics of the exponential term and the residual error function term in the analytical solution, thereby precisely controlling the horizontal migration distance and concentration distribution of pollutants.
[0112] In some embodiments, pollutant concentration limits are obtained, and a comparative analysis is performed between the two-dimensional distribution data of pollutant concentrations and the pollutant concentration limits to determine the two-dimensional spatial extent of contamination in adjacent soil layers, including:
[0113] Based on the pollutant concentration limits for centralized drinking water sources and industrial and agricultural water use in the "Groundwater Quality Standard", the pollutant concentration limits for centralized drinking water sources and industrial and agricultural water use are denoted as follows: In adjacent soil layers, the concentration of pollutants is greater than or equal to the concentration limit. Defined as the two-dimensional spatial extent of contamination in adjacent soil layers.
[0114] In this embodiment, the concentration limits for toluene, a pollutant, in centralized drinking water sources and industrial and agricultural water use are determined according to the "Groundwater Quality Standard" (GB / T 14848-2017). It is 0.7 mg / L.
[0115] In some embodiments, within a specified number of years The extent of contamination in the adjacent soil layer is determined by formula (9), which is as follows:
[0116] .
[0117] In some embodiments, within a specified number of years At that time, the average contamination distance in the adjacent soil layer is expressed by formula (10), which is as follows:
[0118] ;
[0119] In formula (10), For a specified number of years The average contamination distance of adjacent soil layers;
[0120] within the specified years At that time, the maximum pollution distance in the adjacent soil layer is determined based on the two-dimensional distribution data of pollutant concentration, and is expressed by formula (11), which is as follows:
[0121] ;
[0122] In formula (11), For a specified number of years The maximum contamination distance of the adjacent soil layer.
[0123] Furthermore, in this embodiment, the predicted two-dimensional spatial extent of contaminated adjacent soil layers two years after the landfill leachate leakage occurs is shown in the following figures. Figure 5 Two years after a landfill leachate leak occurs, the average contamination distance to adjacent soil layers is... The predicted maximum pollution distance is 39.1m. It is 46.0m, and its size is as follows Figure 5 As shown.
[0124] The above technical solution obtains the property parameters of adjacent soil layers and the two-dimensional migration parameters of pollutants after leachate leakage from a landfill; based on the property parameters of adjacent soil layers and the two-dimensional migration parameters of pollutants, a two-dimensional migration mathematical model of pollutants in adjacent soil layers is constructed, including two-dimensional migration control equations and solution conditions; the two-dimensional migration control equations and solution conditions are transformed and solved using cosine series transformation and Laplace transform to generate a concise analytical solution expression of pollutant concentration changes with time and two-dimensional spatial location; the property parameters of adjacent soil layers and the two-dimensional migration parameters of pollutants are substituted into the concise analytical solution expression to calculate the two-dimensional distribution data of pollutant concentration in adjacent soil layers after a specified number of years; pollutant concentration limits are obtained, and the two-dimensional spatial range of contaminated adjacent soil layers is determined by comparing and analyzing the two-dimensional distribution data of pollutant concentration with the pollutant concentration limits; based on the two-dimensional spatial range of contaminated adjacent soil layers and the vertical height of adjacent soil layers, the average contamination distance is calculated, and the maximum contamination distance is determined based on the two-dimensional distribution data of pollutant concentration. To address the problems of existing technologies that rely on complex mathematical methods such as Laplace numerical inverse transform, cannot directly provide a clear analytical expression for the change of pollutant concentration with time and two-dimensional spatial location, and are difficult to rapidly predict the pollution range and distance, this invention transforms and solves the two-dimensional migration control equation and solution conditions using cosine series transform and Laplace transform. This directly generates a concise and rapidly calculable analytical expression for the change of pollutant concentration with time and two-dimensional spatial location, avoiding the cumbersome process of complex numerical inverse transform. This allows for the direct substitution of the property parameters of adjacent soil layers and the two-dimensional migration parameters of pollutants into this expression, quickly calculating the two-dimensional distribution data of pollutant concentration in adjacent soil layers after a specified number of years. Based on this, the two-dimensional spatial range of pollution, the average pollution distance, and the maximum pollution distance can be determined. Therefore, this invention has the advantages of high accuracy, simple operation, and rapid prediction, providing reliable preliminary data support for the scientific formulation of pollution prevention and control measures after landfill leachate leakage.
[0125] Finally, it should be noted that although the above embodiments have been described in the text and drawings of this application, this should not limit the scope of patent protection of this application. Any technical solutions that are based on the essential concept of this application and utilize the content described in the text and drawings of this application, resulting in equivalent structural or procedural substitutions or modifications, as well as the direct or indirect application of the technical solutions of the above embodiments to other related technical fields, are all included within the scope of patent protection of this application.
Claims
1. A simplified calculation method for predicting leachate leakage pollution in landfills, characterized in that, include: Obtain the property parameters of adjacent soil layers and the two-dimensional migration parameters of pollutants after leachate leakage from a landfill; Based on the property parameters of the adjacent soil layer and the two-dimensional migration parameters of pollutants, a two-dimensional migration mathematical model of pollutants in the adjacent soil layer is constructed. The two-dimensional migration mathematical model includes two-dimensional migration control equations and solution conditions. The two-dimensional migration control equation and solution conditions are transformed and solved by using cosine series transformation and Laplace transform to generate a concise analytical solution expression of the pollutant concentration changing with time and two-dimensional spatial location. Substitute the adjacent soil layer property parameters and the two-dimensional migration parameters of pollutants into the simplified analytical solution expression to calculate the two-dimensional distribution data of pollutant concentration in the adjacent soil layer after a specified number of years. Obtain pollutant concentration limits, and compare and analyze the two-dimensional distribution data of pollutant concentration with the pollutant concentration limits to determine the two-dimensional spatial range of contamination in adjacent soil layers; Based on the contaminated two-dimensional spatial range and the vertical height of the adjacent soil layer, the average contamination distance is calculated, and the maximum contamination distance is determined based on the two-dimensional distribution data of the pollutant concentration.
2. The simplified calculation method for predicting landfill leachate leakage pollution according to claim 1, characterized in that, A two-dimensional mathematical model of pollutant migration in adjacent soil layers is constructed, including: A two-dimensional migration control equation for pollutants in adjacent soil layers after leachate leakage from a landfill is established, which is expressed by formula (1), as follows: In formula (1), The coordinates are horizontal, in meters. The coordinates are vertical, and the unit is meters. Time, in seconds; The concentration of pollutants in adjacent soil layers is expressed in mg / L. The horizontal hydrodynamic dispersion coefficient in the adjacent soil layer. The vertical hydrodynamic dispersion coefficient in adjacent soil layers, in meters. 2 / s; , The blocking factor is dimensionless. This is the distribution coefficient of the adjacent soil layer, expressed in L / mg. This refers to the dry density of the adjacent soil layer, expressed in g / cm³. 3 ; Porosity of adjacent soil layers, dimensionless; , The Darcy velocity in the outer soil mass is expressed in m / s. The convection velocity in the outer soil mass is expressed in m / s. , is the hydraulic permeability coefficient of the adjacent soil layer, in m / s; The hydraulic gradient in the adjacent soil layer is dimensionless. The first-order degradation constant of the pollutant is expressed in s. -1 ; Based on the typical scenario of two-dimensional migration of pollutants in adjacent soil layers after leachate leakage from landfills, the solution conditions for the two-dimensional migration of pollutants are determined and expressed by formula (2), which is as follows: In formula (2), The concentration of contaminants in the externally leaked leachate is expressed in mg / L. This represents the vertical height of the adjacent soil layer, in meters (m).
3. The simplified calculation method for predicting landfill leachate leakage pollution according to claim 2, characterized in that, The hydrodynamic dispersion coefficient in the horizontal direction and the hydrodynamic dispersion coefficient in the vertical direction of the adjacent soil layer are expressed by formula (3), which is as follows: In formula (3), , The horizontal mechanical dispersion coefficient in adjacent soil layers, in meters. 2 / s; This represents the longitudinal dispersion of adjacent soil, in meters. This represents the molecular diffusion coefficient of pollutants in adjacent soil layers.
4. The simplified calculation method for predicting landfill leachate leakage pollution according to claim 2, characterized in that, The concentration of contaminants in the external leachate is expressed by formula (4), which is as follows: In formula (4), This represents the maximum concentration of contaminants in the externally leaked leachate, expressed in mg / L. It is a dimensionless function that characterizes the non-uniform distribution of concentration in the vertical direction; A parameter characterizing the time-varying characteristics of concentration, with units of seconds. -1 .
5. The simplified calculation method for predicting landfill leachate leakage pollution according to claim 1, characterized in that, The two-dimensional migration control equation and solution conditions are transformed and solved by using cosine series transformation and Laplace transformation to generate a concise analytical solution expression of the pollutant concentration changing with time and two-dimensional spatial location, which is expressed by formula (5), as follows: In formula (5), , It is an integer not less than zero; For functions Relevant parameters; To be related to convection rate First-order degradation constant and hydrodynamic dispersion coefficient A comprehensive parameter related to multiple coefficients; And, the specified number of years since the landfill leachate leakage occurred shall be recorded as... The concise analytical expression for the change of pollutant concentration in adjacent soil layers with time and two-dimensional spatial location is expressed by formula (6), which is as follows: 。 6. The simplified calculation method for predicting landfill leachate leakage pollution according to claim 5, characterized in that, Equation (7) is expressed by the following formula: 。 7. The simplified calculation method for predicting landfill leachate leakage pollution according to claim 5, characterized in that, Equation (8) is expressed by the following formula: In formula (8), , For parameters Relevant intermediate parameters.
8. The simplified calculation method for predicting landfill leachate leakage pollution according to claim 5, characterized in that, Obtain pollutant concentration limits, and based on the two-dimensional distribution data of pollutant concentrations and the pollutant concentration limits, determine the two-dimensional spatial range of contamination in adjacent soil layers, including: Based on the pollutant concentration limits for centralized drinking water sources and industrial and agricultural water use in the "Groundwater Quality Standard", the pollutant concentration limits for centralized drinking water sources and industrial and agricultural water use are denoted as follows: In adjacent soil layers, the concentration of pollutants is greater than or equal to the concentration limit. Defined as the two-dimensional spatial extent of contamination in adjacent soil layers.
9. The simplified calculation method for predicting landfill leachate leakage pollution according to claim 8, characterized in that, within the specified years The extent of contamination in the adjacent soil layer is determined by formula (9), which is as follows: 。 10. The simplified calculation method for predicting landfill leachate leakage pollution according to claim 9, characterized in that, within the specified period The average contamination distance in adjacent soil layers is expressed by formula (10), which is as follows: In formula (10), For a specified number of years The average contamination distance of adjacent soil layers; within the specified period At that time, the maximum pollution distance in the adjacent soil layer is determined based on the two-dimensional distribution data of the pollutant concentration, and is expressed by formula (11), which is as follows: In formula (11), For a specified number of years The maximum contamination distance of the adjacent soil layer.