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.     

Consolidation of double‐layered ground with vertical drains

Vertical drains are usually installed in subsoil consisting of several layers. Due to the complex nature of the problem, over the past decades, the consolidation properties of multi‐layered ground with vertical drains have been analysed mainly by numerical methods. An analytical solution for consolidation of double‐layered ground with vertical drains under quasi‐equal strain condition is presented in this paper. The main steps for the computation procedure are listed. The convergence of the series solution is discussed. The comparisons between the results obtained by the present analytical method and the existing numerical solutions are described by figures. The orthogonal relation for the system of double‐layered ground with vertical drains is proven. Finally, some consolidation properties of double‐layered ground with vertical drains are analysed. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

An analytical solution for one-dimensional contaminant diffusion through multi-layered system and its applications

An analytical solution for one-dimensional contaminant diffusion through multi-layered media is derived regarding the change of the concentration of contaminants at the top boundary with time. The model accounts for the arbitrary initial conditions and the conditions of zero concentration and zero mass flux on the bottom boundary. The average degree of diffusion of the layered system is introduced on the basis of the solution. The results obtained by the presented analytical solutions agree well with those obtained by the numerical methods presented in the literature papers. The application of the analytical solution to the problem of landfill liner design is illustrated by considering a composite liner consisting of geomembrane and compacted clay liner. The results show that the 100-year mass flux of benzene at the bottom of the composite liner is 45 times higher than that of acetone for the same composite liner. The half-life of the contaminant has a great influence on the solute flux of benzene diffused into the underlying aquifer. Results also indicates that an additional 2.9–5.0 m of the conventional (untreated) compacted clay liner under the geomembrane is required to achieve the same level of protection as provided by 0.60 m of the Hexadecyltrimethylammonium (HDTMA)-treated compacted clay liners in conjunction with the geomembrane. Applications of the solution are also presented in the context of a contaminated two-layered media to demonstrate that different boundary and initial conditions can greatly affect the decontamination rate of the problem. The method is relatively simple to apply and can be used for performing equivalency analysis of landfill liners, preliminary design of groundwater remediation system, evaluating experimental results, and verifying more complex numerical models.     

Modelling of contaminant transport through landfill liners using EFGM

Modelling of contaminant transport through landfill liners and natural soil deposits is an important area of research activity in geoenvironmental engineering. Conventional mesh‐based numerical methods depend on mesh/grid size and element connectivity and possess some difficulties when dealing with advection‐dominant transport problems. In the present investigation, an attempt has been made to provide a simple but sufficiently accurate methodology for numerical simulation of the two‐dimensional contaminant transport through the saturated homogeneous porous media and landfill liners using element‐free Galerkin method (EFGM). In the EFGM, an approximate solution is constructed entirely in terms of a set of nodes and no characterization of the interrelationship of the nodes is needed. The EFGM employs moving least‐square approximants to approximate the function and uses the Lagrange multiplier method for imposing essential boundary conditions. The results of the EFGM are validated using experimental results. Analytical and finite element solutions are also used to compare the results of the EFGM. In order to test the practical applicability and performance of the EFGM, three case studies of contaminant transport through the landfill liners are presented. A good agreement is obtained between the results of the EFGM and the field investigation data. Copyright © 2009 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。     

地下水溶质迁移模拟研究进展

在地下水的相关研究中,农药和石油等地下水污染、土地盐碱化、海水入侵等诸多实际问题主要的研究方法都涉及地下水溶质迁移模拟. 相比地下水水流模拟的相对完善,对溶质迁移的模拟比较薄弱且迁移过程本身复杂性较高,目前地下水溶质迁移的研究工作还处在全面发展的阶段. 文中阐述了反映地下水溶质迁移机理和过程的数学模型,综述了溶质迁移模拟在地下水污染物防治、土地盐碱化、海水入侵、石油和放射性废物扩散等问题的诸多应用,归类了目前溶质迁移模拟所使用的对流迁移、对流-弥散模拟等主要数值方法,并对这些方法的优缺点和应用实例做了总结. 最后,分析了目前溶质迁移模拟中存在的不足,展望了未来在参数确定、裂隙介质运移机理和多相介质条件下运移模拟可能取得的突破.     

Analytical solutions of a finite two‐dimensional fluid‐saturated poroelastic medium with compressible constituents

An analytical solution is proposed for transient flow and deformation coupling of a fluid‐saturated poroelastic medium within a finite two‐dimensional (2‐D) rectangular domain. In this study, the porous medium is assumed to be isotropic, homogeneous, and compressible. In addition, the point sink can be located at an arbitrary position in the porous medium. The fluid–solid interaction in porous media is governed by the general Biot's consolidation theory. The method of integral transforms is applied in the analytical formulation of closed‐form solutions. The proposed analytical solution is then verified against both exact and numerical results. The analytical solution is first simplified and validated by comparison with an existing exact solution for the uncoupled problem. Then, a case study for pumping from a confined aquifer is performed. The consistency between the numerical solution and the analytical solution confirms the accuracy and reliability of the analytical solution presented in this paper. The proposed analytical solution can help us to obtain in‐depth insights into time‐dependent mechanical behavior due to fluid withdrawal within finite 2‐D porous media. Moreover, it can also be of great significance to calibrate numerical solutions in plane strain poroelasticity and to formulate relevant industry norms and standards. Copyright © 2014 John Wiley & Sons, Ltd.     

Parametric sensitivity analysis of coupled mechanical consolidation and contaminant transport through clay barriers

In this paper, an extensive parametric sensitivity analysis of coupled consolidation and solute transport in composite landfill liner systems has been undertaken. The analysis incorporates results of more than 3000 simulations for various combinations of barrier thickness, waste loading rate, initial void ratio, compression index, hydraulic conductivity and dispersion coefficient. However, it is noted that to limit the extent of the study a constant coefficient of consolidation is assumed in the analysis presented here, though this assumption is easily relaxed. Results of the parametric sensitivity analysis are succinctly presented using dimensionless plots, which allow the comparison of results for a large number of parameter values, and so the clear identification of the most important determinants on contaminant transport through the liner system. The dimensionless plots demonstrate a pessimum (for which the ‘breakthrough time’ is minimised). Numerical results reveal that in cases of extreme liner compressibility an order of magnitude reduction in contaminant transit time may arise due to coupling between solute transport and consolidation, while for barriers of low compressibility and porosity (such as well-engineered composite compacted clay landfill liners), it is found that the contaminant transit time may still be reduced by more than 30%. The numerical results suggest that the use of coupled consolidation–contaminant transport models are sometimes required for informed and conservative landfill liner design.     

Contaminant transport with non‐equilibrium processes in unsaturated soils and implicit characteristic Galerkin scheme

A non‐equilibrium sorption—advection—diffusion model to simulate miscible pollutant transport in saturated–unsaturated soils is presented. The governing phenomena modelled in the present simulation are: convection, molecular diffusion, mechanical dispersion, sorption, immobile water effect and degradation, including both physical and chemical non‐equilibrium processes. A finite element procedure, based on the characteristic Galerkin method with an implicit algorithm is developed to numerically solve the model equations. The implicit algorithm is formulated by means of a combination of both the precise and the traditional numerical integration procedures. The stability analysis of the algorithm shows that the unconditional stability of the present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results illustrate good performance of the present algorithm in stability and accuracy, and in simulating the effects of all the mentioned phenomena governing the contaminant transport and the concentration distribution. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

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.     

Comparison Between Analytical and Numerical Methods in Evaluating the Pollution Transport in Porous Media

Contaminant transport modelling in environmental engineering is generally conducted to evaluate the potential impact of contaminant migration on the subsurface environment or for interpreting tracer tests or groundwater quality data. In the past few decades a number of mathematical models have been established for evaluating the migration of pollution as indicated in the literature. This paper presents a comparison between a number of analytical and numerical models in evaluating pollution transport in soils. Three analytical models and a finite element model developed in this research are used for comparing four numerical examples under different conditions. Four cases of advection dominated problem with line source boundary, advection dominated problem with semi-line source boundary, advection–dispersion–sorption problem with line source boundary and advection–dispersion–sorption problem with semi-line source are considered. Based on the results the best analytical model that has a higher accuracy is recommended for practical applications.     

A simple analytical solution to one‐dimensional consolidation for unsaturated soils

This paper presents a simple analytical solution to Fredlund and Hasan's one‐dimensional (1‐D) consolidation theory for unsaturated soils. The coefficients of permeability and volume change for unsaturated soils are assumed to remain constant throughout the consolidation process. The mathematical expression of the present solution is much simpler compared with the previous available solutions in the literature. Two new variables are introduced to transform the two coupled governing equations of pore‐water and pore‐air pressures into an equivalent set of partial differential equations, which are easily solved with standard mathematical formulas. It is shown that the present analytical solution can be degenerated into that of Terzaghi consolidation for fully saturated condition. The analytical solutions to 1‐D consolidation of an unsaturated soil subjected to instantaneous loading, ramp loading, and exponential loading, for different drainage conditions and initial pore pressure conditions, are summarized in tables for ease of use by practical engineers. In the case studies, the analytical results show good agreement with the available analytical solution in the literature. The consolidation behaviors of unsaturated soils are investigated. The average degree of consolidation at different loading patterns and drainage conditions is presented. The pore‐water pressure isochrones for two different drainage conditions and three initial pore pressure distributions are presented and discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Theoretical and numerical analyses of chemical‐dissolution front instability in fluid‐saturated porous rocks

The chemical‐dissolution front propagation problem exists ubiquitously in many scientific and engineering fields. To solve this problem, it is necessary to deal with a coupled system between porosity, pore‐fluid pressure and reactive chemical‐species transport in fluid‐saturated porous media. Because there was confusion between the average linear velocity and the Darcy velocity in the previous study, the governing equations and related solutions of the problem are re‐derived to correct this confusion in this paper. Owing to the morphological instability of a chemical‐dissolution front, a numerical procedure, which is a combination of the finite element and finite difference methods, is also proposed to solve this problem. In order to verify the proposed numerical procedure, a set of analytical solutions has been derived for a benchmark problem under a special condition where the ratio of the equilibrium concentration to the solid molar density of the concerned chemical species is very small. Not only can the derived analytical solutions be used to verify any numerical method before it is used to solve this kind of chemical‐dissolution front propagation problem but they can also be used to understand the fundamental mechanisms behind the morphological instability of a chemical‐dissolution front during its propagation within fluid‐saturated porous media. The related numerical examples have demonstrated the usefulness and applicability of the proposed numerical procedure for dealing with the chemical‐dissolution front instability problem within a fluid‐saturated porous medium. Copyright © 2007 John Wiley & Sons, Ltd.     

Two-dimensional meshfree modelling of contaminant transport through saturated porous media using RPIM

This paper presents a new numerical tool to model the two-dimensional contaminant transport through saturated porous media using a meshfree method, called radial point interpolation method (RPIM) with polynomial reproduction. In RPIM, an approximate solution is constructed entirely in terms of a set of nodes and no characterisation of the interrelationship of the nodes is needed. The advection–dispersion equation with sorption is considered to illustrate the applicability of the RPIM. The Galerkin weak form of the governing equation is formulated using 2D meshfree shape functions constructed using thin plate spline radial basis functions. MATLAB code is developed to obtain the numerical solution. Three numerical examples are presented and the results are compared with those obtained from the finite element method and analytical solutions. In order to test the practical applicability and performance of the RPIM, two case studies of contaminant transport through landfill liners are presented. A good agreement is obtained between the results of the RPIM and the field investigation data.     

Distribution of suspended sediment concentration in wide sediment‐laden streams: A novel power‐law theory

The diffusion equation of suspended sediment concentration in a wide sediment‐laden stream flow is dependent on the vertical gradient of streamwise velocity and the sediment diffusivity. This study aims at investigating the influence of the streamwise velocity laws on the suspended sediment concentration distributions, resulting from the solution of the diffusion equation. Firstly, the sediment concentration distributions are obtained numerically from the solution of the diffusion equation using different velocity laws and compared with the experimental data. It is found that the power‐law approximation produces good computational results for the concentration distributions. The accuracy of using a power‐law velocity model is comparable with the results obtained from other classical velocity laws, namely log‐law, log wake‐law and stratified log‐law. Secondly, a novel analytical solution is proposed for the determination of sediment concentration distribution, where a power‐law, wall‐concentration profile is coupled with a concentration wake function. The power‐law model (for velocity and concentration) is calibrated using the experimental data, and then a generalized wake function is obtained by choosing a suitable law. The developed power‐law model involving the wake function adjusted by an exponent predicts the sediment concentration distributions quite satisfactorily. Finally, a new explicit formula for the suspended‐load transport rate is derived from the proposed theory, where numerical computation of integrals, as needed in the Einstein theory, is avoided.     

Steady‐state analytical solutions of flow and deformation coupling due to a point sink within a finite fluid‐saturated poroelastic layer

An exact steady‐state closed‐form solution is presented for coupled flow and deformation of an axisymmetric isotropic homogeneous fluid‐saturated poroelastic layer with a finite radius due to a point sink. The hydromechanical behavior of the poroelastic layer is governed by Biot's consolidation theory. Boundary conditions on the lateral surface are specifically chosen to match the appropriate finite Hankel transforms and simplify the transforms of the governing equations. Ordinary differential equations in the transformed domain are solved, and then the analytical solutions in the physical space for the pore pressure and the displacements are finally obtained by using finite Hankel inversions. The analytical solutions at some special locations such as the top and bottom surfaces, lateral surface, and the symmetrical axis are given and analyzed. And a case study for the consolidation of a water‐saturated soft clay layer due to pumping is conducted. The analytical solution is verified against the finite element solution. Meanwhile, an analysis of coupled hydromechanical behavior is carried out herein. The presented analytical solution is an exact solution to the practical poroelastic problem within an axisymmetric finite layer. It can provide us a better understanding of the poroelastic behavior of the finite layer due to fluid extraction. Besides, it can be applied to calibrate numerical schemes of axisymmetric poroelasticity within finite domains. Copyright © 2017 John Wiley & Sons, Ltd.     

The transport of heat and matter by fluids during metamorphism

Non-dimensional solutions to the equations for the combined advective and diffusive one-dimensional transport of heat and solute in a layer are derived for fixed temperature/concentration on the boundaries and initial conditions of a linear gradient across the layer or a step function at the lower boundary. The solutions allow distinction of regimes in which advective or diffusive transport of either heat or solute predominate as a function of fluid flux, time and a length scale. The much lower diffusive coefficients for solute than heat results in a significant range of length scales and fluid flux rates characterised by advection of matter and diffusion of heat. The advective velocity of a component is a function of its fluid:rock partition coefficient. The most rapidly transported tracers which partition largely into the fluid phase, such as He, will travel orders of magnitude faster than heat or compatible solutes such as oxygen. Geochemical profiles in boundary layer regions where both advective and diffusive transport are significant are shown to be particularly informative as to properties of the rocks related to fluid flow such as porosity, permeability, time scales and fluid flux rates. The importance of advection can be directly estimated from the asymmetry of the geochemical profiles across individual layers.     

Assessment of advective–dispersive contaminant transport in heterogeneous aquifers using a meshless method

Groundwater solute transport phenomena typically occur in water-bearing zones with heterogeneous solute dispersive characteristics and/or media hydraulic properties. A radial basis function collocation method (RBFCM)-based numerical method was developed in order to investigate the ability of RBFCM to accurately portray solute transport phenomena under heterogeneous conditions. Simulations were performed for 1-D and 2-D transport scenarios in which scale-dependent dispersivity fields were taken into consideration and compared with available analytical solutions. Different radial basis functions (RBFs) were employed for assessing the sensitivity of the present method on the selected RBFs. The simulation results were also compared with the results of MT3DMS which is a modular three-dimensional transport model with alternative solution schemes including the method of characteristics, the implicit central finite difference and the third order total variation diminishing finite volume. The proposed model was also used to simulate a real case condition where solute transport through a two-layer soil medium had been investigated experimentally. The results showed that RBFCM represented a powerful tool for predicting the solute transport occurrence under heterogeneous conditions with high accuracy.