One‐dimensional contaminant transport through a deforming porous medium: theory and a solution for a quasi‐steady‐state problem

Contaminant migration through soil is usually modelled mathematically using the dispersion–advection equation. This type of model finds application when planning the remediation of contaminated land, predicting the movement of polluted groundwater and designing engineered landfills. Usually the analysis assumes that the porous media through which the contaminant migrates is stationary. However, the construction of landfills on clay soils means that the soil beneath the landfill will undergo time‐dependent deformation as the soil consolidates. To date, there are no published data on the effect a deforming porous media may have on contaminant transport beneath a landfill; indeed, there appears to be no theory of contaminant migration through a deforming soil. In this paper, a one‐dimensional theory of contaminant migration through a saturated deforming porous media is developed based on a small and large strain analysis of a consolidating soil and conservation of contaminant mass. By selection of suitable parameters, the new transport equation reduces to the familiar one‐dimensional dispersion–advection equation for a saturated soil with linear, reversible, equilibrium controlled sorption of the contaminant onto the soil skeleton. Analytic solutions to a quasi‐steady‐state contaminant transport problem for a deforming media are presented, and a preliminary assessment made of the potential importance of soil deformation on the results of a contaminant migration analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

大变形黏土防渗层中的污染物迁移和转化规律研究

国内湖泊疏浚污染底泥堆场一般以较厚的黏土层作为主要防渗层,由于在上覆底泥作用下黏土层会发生较大的固结变形,因此,在研究黏土防渗层中的污染物运移和转化规律时,应该考虑土体变形的影响。基于Gibson一维大变形固结理论和饱和多孔介质中的污染物对流扩散方程,建立了二者耦合的可变形多孔介质中污染物的运移和转化模型,其中首次考虑了土体自重和生物降解作用的影响。利用所建立模型的数值解,研究了在可变形黏土防渗层中的污染物运移和转化规律,同时分析了模型中不同项和主要参数的作用和影响。研究结果表明,土体大变形对黏土防渗层中污染物的运移有着较复杂的影响,一方面土体变形会加速污染物的运移;另一方面土体固结带来的渗透性减小会增加污染物的穿透时间,二者的不同作用取决于众多的影响因素,如土层厚度和吸附作用等。研究结果对于评估天然黏土防渗层对污染物的阻隔作用有重要的指导意义。     

Analytical solutions of linear finite‐ and small‐strain one‐dimensional consolidation

Analytical solutions are presented for linear finite‐strain one‐dimensional consolidation of initially unconsolidated soil layers with surcharge loading for both one‐ and two‐way drainage. These solutions complement earlier solutions for initially unconsolidated soil layers without surcharge and initially normally consolidated soil layers with surcharge. Small‐strain solutions for the consolidation of initially unconsolidated soil layers with surcharge loading are also presented, and the relationship between the earlier solutions for initially unconsolidated soil without surcharge and the corresponding small‐strain solutions, which was not addressed in the earlier work, is clarified. The new solutions for initially unconsolidated soil with surcharge loading can be applied to the analysis of low stress consolidation tests and to the partial validation of numerical solutions of non‐linear finite‐strain consolidation. They also clarify a formerly perplexing aspect of finite‐strain solution charts first noted in numerical solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

连续时间随机游动理论模拟多孔介质中溶质运移的研究进展

近年来,基于连续时间随机游动(Continuous Time Random Walk, CTRW)理论所建立的模拟非均质多孔介质中溶质运移的方法已在大量的数值实验、室内实验、野外实验中得到了广泛的验证,为非均质多孔介质中的溶质运移行为提供了一种有效的模拟方法。简述了提出和发展CTRW的研究背景、基础理论以及与经典的对流-弥散方程等其他模拟方法的关系,综述了该理论在模拟溶质运移中的发展和应用,分析了实际应用中的关键问题,并展望了将其进一步发展应用于模拟反应性溶质运移的前景。     

Theoretical investigation of the effects of consolidation on contaminant transport through clay barriers

Consolidation of clayey contaminant barriers such as landfill liners has been postulated as a cause of early breakthrough of contaminants. In this paper we theoretically investigate this proposition. For this purpose a sophisticated one‐dimensional, large‐deformation model of coupled mechanical consolidation and solute transport is employed. This new model is a generalization of existing coupled consolidation and solute transport models described in the literature. It takes into account both non‐linearities in geometry as well as constitutive relations. The latter relate the compressibility, hydraulic conductivity and coefficient of effective diffusivity to the deformation of the soil. The model is applied to a case study of a clay liner and geomembrane system. Results obtained from numerical solution of the model equations are compared with those from various simplified models, including a ‘diffusion only’ (i.e. a rigid soil) model traditionally used in contaminant barrier design. For barriers incorporating low compressibility soils (as for many well compacted clays), there is little difference between contaminant transit (i.e. breakthrough) times predicted by the two models. However, for contaminant barriers incorporating more compressible soils, consolidation is shown to significantly accelerate transport. These results indicate the potential importance of accounting for the effects of soil consolidation and highlight the limitations of existing models when modelling solute transport through composite barriers utilizing soft soils. Based on these limited results, we suggest a possible way of taking into account soil consolidation using simplified models. Copyright © 2008 John Wiley & Sons, Ltd.     

Advective transport from a penny-shaped crack in a porous medium and an associated uniqueness theorem

This paper examines the problem of the advective transport of a contaminant from sources in the shape of either a penny-shaped crack or an elongated needle-shaped cavity located in a porous medium of infinite extent. The advective transport is induced by Darcy flow in the porous medium, where the internal boundary is maintained at a constant potential. The paper presents an approximate analytical solution to this problem, which is deduced from a formulation that models a cavity in the shape of either an oblate or a prolate spheroid. The results also represent one of the few spatially three-dimensional exact analytical solutions for the, albeit linear, hyperbolic problem governing the contaminant transport problem. The paper also presents a canonical proof of uniqueness for advective contaminant transport problems associated with media of infinite extent. Copyright © 2004 John Wiley & Sons, Ltd.     

非饱和土中溶质迁移参数反演的HISR方法

以非饱和土中溶质迁移参数反演问题为背景, 依据正则化方法的思路, 以Itakura Saito距离作为同伦函数中的平凡问题, 将同伦方法引入非线性参数反演问题的求解, 进而提出一种求解非线性参数反演问题的大范围收敛(HomotoyItakura SaitoRegularization, HISR) 方法.为保证迭代稳定性, 并同时削弱观测噪声的影响, 同伦参数的修正采用了连续化修正方法.本文将HISR方法应用于求解带有平衡及非平衡吸附效应的一维非饱和土中溶质迁移参数反演问题, 计算结果表明HISR方法具有大范围收敛性及计算稳健性, 同时有较强的抵抗观测噪声的能力.      

Modelling Transport of Linearly Sorbing Solutes in a Single Fracture: Asymptotic Behavior of Solute Velocity and Dispersivity

The asymptotic behavior of the solute velocity and dispersivity for a system of parallel fractures with matrix diffusion is made using numerical modeling and theoretical analyses. The study is limited to linearly sorbing solutes with a constant continuous source boundary condition. Expressions are provided for solute velocity and effective dispersivity in terms of fracture porosity during asymptotic stage using spatial moment analyses. The importance of matrix porosity and fracture porosity on solute velocity as well as the relationship governing effective dispersivity and fracture porosity is discussed for both non-reactive and linearly sorbing solutes. By using a dimensionless effective dispersivity parameter it is shown that the relationship between the fracture porosity and dimensionless effective dispersivity is linear for non-reactive solutes. It is also shown that this holds true for the linearly sorbing solutes with the same proportionality constant.     

Numerical study of boundary conditions for solute transport through a porous medium

A transition region may be defined as a region of rapid change in medium properties about the interface between two porous media or at the interface between a porous medium and a reservoir. Modelling the transition region between different porous media can assist in the selection of the most appropriate boundary conditions for the standard advection–dispersion equation (ADE). An advantage of modelling the transition region is that it removes the need for explicitly defining boundary conditions, though boundary conditions may be recovered as limiting cases. As the width of a transition region is reduced, the solution of the transition region model (TR model) becomes equivalent to the solution of the standard ADE model with correct boundary conditions. In this paper numerical simulations using the TR model are employed to select the most appropriate boundary conditions for the standard ADE under a variety of configurations and conditions. It is shown that at the inlet boundary between a reservoir and porous medium, continuity of solute mass flux should be used as the boundary condition. At the boundary interface between two porous media both continuity of solute concentration and solute mass flux should be used. Finally, in a finite porous medium where the solute is allowed to advect freely from the exit point, both continuity of solute concentration and solute mass flux should be used as the outlet boundary condition. The findings made here are discussed with reference to a detailed review of previous relevant theoretical and experimental observations. Copyright © 2001 John Wiley & Sons, Ltd.     

粘性土壤溶质运移新模型的应用

将Moldrup等提出的模拟水流和溶质运移的运动平均斜率模型和运动浓度斜率模型联合,模拟了重粘土在入渗和蒸发条件下的水盐动态并与特征-有限差分数值模型的计算结果进行了比较。在模拟过程中新模型显示出更加高效的优点,这对粘性土壤地区盐渍化趋势的长期预报具有重要的实际意义。     

孔隙尺度多孔介质流体流动与溶质运移高性能模拟

深入探究孔隙尺度下的流体流动特性和溶质运移规律对石油开采、农田养分管理、地下水污染修复有着重要意义。以人工构建的多孔介质结构和同步辐射X射线显微CT扫描的土壤团聚体(分辨率3.7μm)为研究对象,在空间节点数多达64 000 000的情况下,基于格子Boltzmann模型和GPU并行技术计算得到多孔介质流体运动和溶质运移过程的关键参数,并据此探究多孔介质空间异质性对水力学特性的影响。通过对3组不同结构的多孔介质比较发现,结构复杂程度最高的土壤样品和不规则堆叠的圆球结构的渗透率在100 mD(即10^-13m^2)量级,远低于规则堆叠的圆球结构(>20 000 mD);土壤的迂曲度为1.40~1.60,明显高于规则堆叠的圆球结构。研究结果表明,渗透率大的样品具有较小的迂曲度,这与结构的空间异质性有较强的关系;土壤的渗透率和迂曲度呈现各向异性;在水力梯度一定的前提下,渗透率较大的样品,纵向弥散系数也较大;同时,结构的异质性也会影响溶质的穿透曲线。本研究提出的模拟方法可在土壤结构中进行高效的水流运动和溶质运移模拟,可用于土壤多孔介质在孔隙尺度下的水力学特性研究。     

基于图像法的多孔介质双分子反应溶质运移模拟

采用图像分析法确定溶质浓度,以硫酸铜和EDTA二钠作为双分子反应物,在多孔介质模型中开展了不同粒径(1.52.0,2.53.0,3.54.0 mm)和流量(1.0,1.5,2.0 mL/s)下反应性溶质运移实验,探讨了应用不完全混合的对流弥散模型(IM-ADRE)对双分子反应溶质运移的模拟和预测,并进行了参数敏感性分析。结果表明:图像分析法可准确获取多孔介质中显色反应性溶质的浓度,灰度值与浓度的决定系数R²大于0.96;用IM-ADRE模型能够准确预测双分子反应性溶质硫酸铜和EDTA二钠在3种不同多孔介质中的运移过程,误差低于3.71%;实验条件的改变对IM-ADRE模型参数D、m和β₀的影响显著,说明模型参数依赖于环境条件,其变化规律需要根据实际环境条件进一步率定,便于IM-ADRE模型的进一步推广应用。     

孔隙尺度多孔介质流体流动与溶质运移高性能模拟

深入探究孔隙尺度下的流体流动特性和溶质运移规律对石油开采、农田养分管理、地下水污染修复有着重要意义。以人工构建的多孔介质结构和同步辐射X射线显微CT扫描的土壤团聚体（分辨率3.7 μm）为研究对象，在空间节点数多达64 000 000的情况下，基于格子Boltzmann模型和GPU并行技术计算得到多孔介质流体运动和溶质运移过程的关键参数，并据此探究多孔介质空间异质性对水力学特性的影响。通过对3组不同结构的多孔介质比较发现，结构复杂程度最高的土壤样品和不规则堆叠的圆球结构的渗透率在100 mD（即10^-13m²）量级，远低于规则堆叠的圆球结构（>20 000 mD）；土壤的迂曲度为1.40~1.60，明显高于规则堆叠的圆球结构。研究结果表明，渗透率大的样品具有较小的迂曲度，这与结构的空间异质性有较强的关系；土壤的渗透率和迂曲度呈现各向异性；在水力梯度一定的前提下，渗透率较大的样品，纵向弥散系数也较大；同时，结构的异质性也会影响溶质的穿透曲线。本研究提出的模拟方法可在土壤结构中进行高效的水流运动和溶质运移模拟，可用于土壤多孔介质在孔隙尺度下的水力学特性研究。     

饱和水流溶质运移问题数值解法综述

本文总结了饱和水流中溶质运移方程求解的各种数值方法，分析各种方法的本质特征以及各自的优缺点，并指出了求解对流—弥散方程的各种数值方法的研究进展和值得重视的问题。研究结果表明，自适应欧拉—拉格朗日法(EM)是溶质运移问题中，求解对流—弥散方程是比较有发展潜力的方法之一。以MMOC法为基础在陡峰值高价插值和其它区域低价插值相结合的ELM法，将是未来发展的趋势。而寻求非规则网格上高精度的空间单元插值模式，已开始成为求解对流问题数值方法研究的重点和关键问题。     

Chemically induced deformation of a porous layer coupled with advective–dispersive transport. Analytical solutions

In this paper a chemically induced deformation of porous material taking place during advective–dispersive transport of a chemical is considered. Linearized governing equations are derived and analytical solutions of 2 one‐dimensional problems for a homogeneous layer with drained boundaries are developed. Numerical results for a particular clayey material and a chemical migrating through the layer showing distributions of concentration of chemical, changes in porosity of the material and pore fluid pressure, and evolution of settlement of the layer as functions of time are discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

The use of laboratory experiments for the study of conservative solute transport in heterogeneous porous media

 Laboratory experiments on heterogeneous porous media (otherwise known as intermediate scale experiments, or ISEs) have been increasingly relied upon by hydrogeologists for the study of saturated and unsaturated groundwater systems. Among the many ongoing applications of ISEs is the study of fluid flow and the transport of conservative solutes in correlated permeability fields. Recent advances in ISE design have provided the capability of creating correlated permeability fields in the laboratory. This capability is important in the application of ISEs for the assessment of recent stochastic theories. In addition, pressure-transducer technology and visualization methods have provided the potential for ISEs to be used in characterizing the spatial distributions of both hydraulic head and local water velocity within correlated permeability fields. Finally, various methods are available for characterizing temporal variations in the spatial distribution (and, thereby, the spatial moments) of solute concentrations within ISEs. It is concluded, therefore, that recent developments in experimental techniques have provided an opportunity to use ISEs as important tools in the continuing study of fluid flow and the transport of conservative solutes in heterogeneous, saturated porous media. Received, December 1996 · Revised, July 1997 · Accepted, August 1997     

Solute transfer during consolidation based on a solid‐fluid‐solute coupling model

The migration of contaminant through soil is usually modeled using the advection‐dispersion equation and assumes that the porous media is stationary without introducing a constitutive equation to represent soil structure. Consequently, time‐dependent deformation induced by soil consolidation or physical remediation is not considered, despite the need to consider these variables during planning for the remediation of contaminated ground, the prediction of contaminated groundwater movement, and the design of engineered landfills. This study focuses on the numerical modeling of solute transfer during consolidation as a first step to resolve some of these issues. We combine a coupling theory‐based mass conservation law for soil‐fluid‐solute phases with finite element modeling to simulate solute transfer during deformation and groundwater convection. We also assessed the sensitivity of solute transfer to the initial boundary conditions. The modeling shows the migration of solute toward the ground surface as a result of ground settlement and the dissipation of excess pore water pressure. The form of solute transport is dependent on the ground conditions, including factors such as the loading schedule, contamination depth, and water content. The results indicate that an understanding of the interaction between coupling phases is essential in predicting solute transfer in ground deformation and could provide an appropriate approach to ground management for soil remediation.     

Coupled consolidation of a porous medium with a cylindrical or a spherical cavity

This paper presents a theoretical approach to analyse coupled, linear thermoporoelastic fields in a saturated porous medium under radial and spherical symmetry. The governing equations account for compressibility and thermal expansion of constituents, heat sink due to thermal dilatation of water and thermal expansion of the medium, and thermodynamically coupled heat–water flow. It has been reported in the literature that thermodynamically coupled heat–water flows known as thermo-osmosis and thermal filtration have the potential to significantly alter the flow fields in clay-rich barriers in the near field of a underground waste containment scheme. This study presents a mathematical model and examines the effects of thermo-osmosis and thermal-filtration on coupled consolidation fields in a porous medium with a cavity. Analytical solutions of the governing equations are presented in the Laplace transform space. A numerical inversion scheme is used to obtain the time-domain solutions for a cylindrical cavity in a homogeneous or a non-homogeneous medium. A closed form time-domain solution is presented for a spherical cavity in a homogeneous medium. Selected numerical solutions for homogeneous and non-homogeneous media show a significant increase in pore pressure and displacements due to the presence of thermodynamically coupled flows and a negligible influence on temperature. © 1998 John Wiley & Sons, Ltd.     

非均匀土壤中溶质运移的两区模型及其解析解

给出了非均匀土壤中考虑水动力弥散尺度效应的一维溶质运移两区模型。然后在初始浓度为零,半无限一维空间内常数通量边界条件下,推导出了可动区和不动区溶质浓度分布的解析表达式。并用Mathematica软件包作了数值计算。计算结果表明,该模型的模拟结果符合溶质运移的一般规律,因而该模型在土壤溶质运移的研究和应用中有一定的价值。     

Solute transport in a single fracture with negligible matrix permeability: 1. fundamental mechanisms

This report describes the fracture-scale mechanisms acting on solute transport in fractured aquifers under natural-flow conditions. It focuses on low-permeability rocks where advection in the matrix is negligible compared with that in fractures. The relevant transport mechanisms detailed have been identified by experimental and theoretical studies over the past 30 years: advection and hydrodynamic dispersion, channeling effects, matrix diffusion, and sorption reactions. This review is intended to emphasize the fundamental concepts and to draw up a reader's guide through an extensive bibliography by linking key problems to key papers. These concepts might be integrated into transport models, but their influence at the large scale, however, remains an open question that is not dealt with in this review.

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Resumen Este informe describe los mecanismos que actúan a escala de fractura en el transporte de solutos en acuíferos fracturados, bajo condiciones de flujo natural. Se centra en rocas de baja permeabilidad en las que la advección en la matriz es despreciable en relación con la advección en las fracturas. Los mecanismos de transporte relevantes han sido identificados mediante estudios experimentales y teóricos realizados en los últimos 30 años. Consisten en advección y dispersión hidrodinámica, influencia de los canales, difusión en la matriz y reacciones de sorción. Esta revisión pretende enfatizar en los conceptos fundamentales y generar una guía para el lector por medio de una bibliografía abundante que relaciona los problemas clave con artículos clave. Estos conceptos podrían ser integrados en modelos de transporte, pero su influencia a gran escala es todavía un tema no resuelto, el cual no se aborda en este trabajo.

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Résumé Ce rapport décrit les mécanismes agissant sur le transport de soluté dans les aquifères fissurés en régime d'écoulement naturel. L'accent est mis sur les roches peu perméables, où le transport par convection dans la matrice est négligeable devant la convection dans les fractures. Les mécanismes de transport sont analysés à l'échelle de la fracture: convection et dispersion hydrodynamique, chenalisation, diffusion dans la matrice et réactions de sorption. L'objectif de ce travail est double: d'une part, mettre en avant les concepts fondamentaux pouvant être intégrés dans les modèles de transport, et d'autre part, établir un "guide" permettant au lecteur de s'orienter à travers une bibliographie abondante, en reliant chaque sujet abordé à quelques articles clés. Notons que l'influence de ces mécanismes sur le transport à grande échelle reste un problème clé qui ne sera pas abordé ici.

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