Transport and retention of microparticles in packed sand columns at low and intermediate ionic strengths: experiments and mathematical modeling

Functional relationships correlating particle filtration coefficients and porewater ionic strength are herein proposed and validated, based on deposition experiments of micrometer-sized particles onto siliceous sand. Experiments were conducted using one-dimensional laboratory columns and stable monodisperse aqueous suspensions of negatively charged latex particles with a mean size of 1.90 μm. The role of ionic strength was systematically investigated and six different monovalent salt concentrations (1, 3, 10, 30, 100, 300 mM) were employed by addition of sodium chloride to the aqueous solution. A mathematical advection–dispersion-deposition transport model was adopted assuming that attachment and detachment of particles in the porous medium are concurrent mechanisms of particle filtration, and including a Langmuir-type blocking function to account for availability in deposition sites. The system of equations modeling colloid transport was solved numerically. Attachment rate and detachment rate coefficients were thereby determined for each employed ionic strength, as well as a blocking coefficient in the form of a maximum particle concentration in the solid phase. Therefore, functional relationships expressing the dependence of these coefficients on ionic strength were proposed, based on literature findings and present experimental observations. The existence of a critical salt deposition concentration (and release concentration) separating a favorable attachment (and detachment) regime from an unfavorable condition is assumed. In respect to the blocking coefficient, a power–law dependence on ionic strength is hypothesized. The proposed functional relationships proved adequate to reproduce the coefficient trends extrapolated from data fitting by the transport model. They may represent a powerful tool to describe and predict microparticle mobility in saturated porous media if embedded a priori in the related mathematical transport models.     

珊瑚砂三维孔隙微观特性研究

孔隙是多孔介质内渗流的发生场所,与介质渗透性存在必然的联系。珊瑚砂因其特殊的物质来源和形成过程,较陆源砂具有截然不同的孔隙特性。通过一系列微观研究手段,从本质上揭示了珊瑚砂特殊孔隙性质的原因。研究发现,从孔隙形状、孔喉尺寸和整体连通性3个角度描述孔隙性质较为合理。其中,孔隙形状用形状因子度量,孔喉尺寸包括孔隙半径和喉道半径,珊瑚砂多孔介质整体连通性利用配位数进行描述。而影响孔隙形状、孔喉尺寸和整体连通性的主导因素包括颗粒形状和颗粒表面粗糙度两方面。其中颗粒形状主要影响孔隙形状、喉道尺寸、孔喉尺寸离散性和介质内部连通性的均匀分布情况。颗粒表面粗糙度主要影响孔隙形状、孔隙形状离散性、孔隙尺寸和介质整体连通性。     

An expression for the permeability of anisotropic granular media

There are many expressions proposed for the permeability of isotropic media based on flow channel and pore size distribution concepts, but there are no such expressions for anisotropic media. In this paper an expression for the permeability of an anisotropic medium is proposed, which has been verified in the laboratory. The mechanism behind fluid flow through soil was investigated using microscopic computer simulations to propose an expression for macroscopic permeability. The soil was assumed to be a spatially periodic porous medium, and the Navier-Stokes equation was solved using the FEM with appropriate boundary conditions for several different arrangements of the porous medium. The basic variables influencing flow through soil at the microscopic level were identified as specific surface area, void ratio, particle shape, material heterogeneity and the arrangement of particles in a porous medium. A sensitivity analysis was carried out to obtain an expression for the permeability in terms of the above variables. The corresponding macroscopic variables for the above microscopic variables are average specific surface area, average void ratio, anisotropy, tortuosity due to material heterogeneity, and the arrangement of particles respectively. An expression for the directional permeability is proposed in terms of these variables for the most common occurrence of particles in a porous medium. For the verification of the proposed equation, the permeability values of a fine-grained sand were measured at different void ratios and were compared with those predicted by the proposed equation. The results show that the predicted permeability values from the proposed equation are very close to the measured values.     

Effects of internal pores on the mechanical properties of marine calcareous sand particles

Calcareous sand is a typical problematic marine sediment because of its angular and porous particles. The effects of internal pores on the mechanical properties of calcareous sand particles have rarely been investigated. In this paper, the apparent morphology and internal structure of calcareous sand particles are determined by scanning electron microscopy and computed tomography tests, finding that the superficial pores connect inside and outside of the particles, forming a well-developed network of cavities and an internal porosity of up to 40%. The effects of particle morphology and internal porosity on the mechanical responses of particle were investigated by conducting photo-related compression test and 3D numerical simulations. Two failure modes are observed for the porous calcareous sand, i.e., compressive failure indicates that the particle skeleton is continually compressed and fragmented into small detritus without obvious splitting, and tensile failure indicates that the particles are broken into several fragments when the axial force clearly peaks. Calcareous sand particles with a high internal porosity or with small and dense pores often exhibit compressive failure, and vice versa. The particle strength is considerably reduced by increasing the internal porosity, but affected by pore size in nonlinear correlation. The crushing stress–strain points can be well fitted by an exponential curve, which is supplied for discussion.

     

孔隙结构和水动力对饱和多孔介质中颗粒迁移和沉积特性的耦合影响

为了研究孔隙结构和水动力对悬浮颗粒在饱和多孔介质中沉积和迁移特性的影响,对天然硅粉（悬浮颗粒）和荧光素钠（示踪剂）在饱和多孔介质中的渗流迁移特性进行土柱试验,分别得到了5种不同渗流速度（0.033、0.066、0.132、0.199、0.265 cm/s）、两种不同多孔介质（石英砂和玻璃球）的悬浮颗粒和示踪剂全组合下的20条穿透曲线。根据试验结果,研究孔隙结构、渗流速度对饱和多孔介质中颗粒迁移和沉积过程中水动力作用机制、弥散效应、加速效应的影响。研究表明,悬浮颗粒的穿透曲线可以用一阶沉积动力学对流弥散方程的解析解来描述。随着渗流速度的增大,水动力学作用对颗粒出流浓度的影响越来越大,而孔隙结构的影响则相对减弱。同时,存在一个临界渗流速度值。当渗流速度超出该值时,悬浮颗粒迁移要快于示踪剂,而且临界渗流速度对于玻璃球和石英砂两种多孔介质是不同的;其次,在两种介质中,随渗流速度增大,弥散度增加,回收率和回收悬浮颗粒粒径增大,沉积系数先增大后减小。此外,在孔隙比相近的情况下,悬浮颗粒在玻璃球介质中的回收率要大于其在石英砂中的。可见,孔隙结构和渗流速度是影响饱和多孔介质中颗粒输运的重要因素,渗流速度越大,孔隙结构的作用越明显。     

An experimental and theoretical study of the seepage migration of suspended particles with different sizes

This study experimentally investigates the effect of particle size, particle concentration and flow velocity on the migration of suspended particles of size 1.02–47 μm in porous media. The results show that at the same flow velocity, the peak values of the breakthrough curves decrease and corresponding pore volumes increase slightly with increasing particles size. The migration velocity of smaller suspended particles is even greater than water flow velocity, which is attributed to the size exclusion effect. With increase of the injected particle concentration, the deposition coefficients of small single particles increase at first and then tend to a steady state or even decrease slightly, explained by the maximum retention concentration. The dispersivity of small particles decreases with increasing velocity. However, at a high flow velocity, the hydrodynamic dispersivity becomes increasingly dominant with the increase of particle size. The deposition coefficients for large-sized particles are higher than those for small-sized particles, which is attributed to considerable mass removal due to straining. An analytical solution, considering the release effect of sorbed particles, is developed to account for the one-dimensional flow and dispersive effect using a source function method, and then three transport parameters—dispersivity, deposition coefficient and release coefficient—are fitted using the experimental results. Finally, suspended-particle migration is predicted by the proposed model for short-time constant-concentration injection and repeated three-pulse injection. Overall, particle size has a significant effect on the seepage migration parameters of suspended particles in porous media such as the particle velocity, dispersivity and deposition coefficient.     

A new numerical method of considering local longitudinal dispersion in single fractures

The solutions of advection–dispersion equation in single fractures were carefully reviewed, and their relationships were addressed. The classic solution, which represents the resident or flux concentration within the semi‐infinite fractures under constant concentration or flux boundary conditions, respectively, describes the effluent concentration for a finite fracture. In addition, it also predicts the cumulative distribution of solute particle residence time passing through a single fracture under pulse injection condition, based on which a particle tracking approach was developed to simulate the local advection–dispersion in single fractures. We applied the proposed method to investigate the influence of local dispersion in single fractures on the macrodispersion in different fracture systems with relatively high fracture density. The results show that the effects of local dispersion on macrodispersion are dependent on the heterogeneity of fracture system, but generally the local dispersion plays limited roles on marodispersion at least in dense fracture network. This trend was in agreement with the macrodispersion in heterogeneous porous media. Copyright © 2013 John Wiley & Sons, Ltd.     

New method and equations for determining attachment tenacity and particle size limit in flotation

A new method and simple, yet accurate, equations for determining the tenacity of particle attachment and the particle size limit in flotation were developed by applying the force analysis of the gravity–capillarity coupling phenomena controlling the bubble–particle stability and detachment. Approximate solutions to the Young–Laplace equation were used to develop simple equations for the tenacity of attachment of particles with diameter up to 20 mm. Simple equations for the maximum size of floatable particles were derived as explicit functions of the particle contact angle, the surface tension, the particle density and the mean centrifugal acceleration of turbulent eddies. For the typical particle size and contact angle encountered in flotation, the analysis showed that the bubble size has little effect on the tenacity of particle attachment. The prediction for the largest size of floatable particles is compared with the experimental data and signifies influence of turbulence on the particle detachment.     

Single-particle crushing behaviour of carbonate sands studied by X-ray microtomography and a combined finite–discrete element method

This paper investigates the particle breakage behaviour of a carbonate sand based on single-particle compression experiments with in situ X-ray microtomography scanning (μCT) and a combined finite–discrete element method (FDEM). Specifically, X-ray μCT is applied to extract the information on grain morphology and intra-particle pores of carbonate sand particles to establish an FDEM model. The model is first calibrated by comparing the simulation results of two carbonate sand grains with the corresponding single-particle compression experiment results and then applied to model the stress evolution, cracking propagation and failure of other carbonate sand particles under single-particle compression. To study the influence of intra-particle pores, FDEM modelling of carbonate sands with completely filled intra-particle pores is also performed. The particle strength of carbonate sands both with and without pore filling is found to follow a Weibull distribution, with that of the sand with pore filling being considerably higher. This behaviour is associated with lower stress concentration, resulting in later crack development in the pore-filled sand than in the sand without pore filling. The cracks are found to usually pass through the intra-particle pores. Consequently, a larger proportion of particles fail in the fragmentation mode in the sand without pore filling.

     

Finite element evaluation of diffusion and dispersion tensors in periodic porous media with advection

This paper presents three-dimensional finite element simulations to evaluate diffusion and dispersion tensors in periodic porous media in the presence of an advective velocity field. These tensors are evaluated in the framework of the double-scale expansion technique. Two problems, a Newtonian flow and a vector-valued advection–diffusion equation, have to be sequentially solved at the pore scale. Finite element techniques to approximate these problems are proposed and analyzed. Numerical results in three-dimensional networks of spheres are presented to quantitatively assess the impact of the pore morphology and of the advection velocity on the diffusion and dispersion tensors.     

Implication of surface modified NZVI particle retention in the porous media: Assessment with the help of 1-D transport model

Retention of surface-modified nanoscale zero-valent iron (NZVI) particles in the porous media near the point of injection has been reported in the recent studies. Retention of excess particles in porous media can alter the media properties. The main objectives of this study are, therefore, to evaluate the effect of particle retention on the porous media properties and its implication on further NZVI particle transport under different flow conditions. To achieve the objectives, a one-dimensional transport model is developed by considering particle deposition, detachment, and straining mechanisms along with the effect of changes in porosity resulting from retention of NZVI particles. Two different flow conditions are considered for simulations. The first is a constant Darcy’s flow rate condition, which assumes a change in porosity, causes a change in pore water velocity and the second, is a constant head condition, which assumes the change in porosity, influence the permeability and hydraulic conductivity (thus Darcy’s flow rate). Overall a rapid decrease in porosity was observed as a result of high particle retention near the injection points resulting in a spatial distribution of deposition rate coefficient. In the case of constant head condition, the spatial distribution of Darcy’s velocities is predicted due to variation in porosity and hydraulic conductivity. The simulation results are compared with the data reported from the field studies; which suggests straining is likely to happen in the real field condition.     

一种适用于孔隙体积应变的有效应力方程

土力学奠基石Terzaghi有效应力原理被广泛应用于油藏孔隙和渗透率应力敏感研究中,然而其对于岩石孔隙体积应变的适用性存在争议。对颗粒不可压缩和颗粒可压缩的多孔介质分别进行了受力分析,推导了总体积、颗粒骨架、孔隙体积的有效应力表达式,与Biot、Skepmton有效应力方程对比,建立了适用于孔隙体积应变的新型有效应力方程,并进行了试验论证和应用举例。研究表明:在颗粒不可压缩多孔介质中,有效应力为超出平衡孔隙流压之外的颗粒间宏观等效应力;在颗粒可压缩变形多孔介质中,有效应力为其相同应变下的等效应力,有3种有效应力分别适用于总体积应变、颗粒体积应变、孔隙体积应变;新提出的孔隙体积有效应力方程与孔隙度、岩石总体积压缩系数、颗粒压缩系数、总应力和流压相关,4个理论计算式计算结果在3种多孔介质试验测试结果中的偏差均在5%以内;孔隙体积有效应力系数解决了如何定量增总应力来等效模拟储层降流压生产过程这一关键问题,3个压缩系数关系式理论计算准确方便。     

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.     

基于孔隙网络模型的非水溶相液体运移实验研究进展

进行多孔介质中非水溶相液体（Non Aqueous Phase Liquids，NAPLs）运移的微观机理研究，微观孔隙网络模型实验是目前应用比较广泛且行之有效的方法。通过网络模型实验，获得对NAPLs在多孔介质中运移更深入的认识。从多孔介质孔隙结构测量、孔隙网络模型制作、NAPLs运移网络模型实验和数值模拟4个方面评述了该方向的研究进展，结果显示测量孔隙结构方法、图像刻蚀技术、可视化测量实验数据方法等有力地促进了本实验研究的发展。分析了孔隙网络模型实验存在的问题以及未来的发展趋势，对开展孔隙网络模型实验研究有一定的启发作用。     

Transport and deposition of suspended particles in saturated porous media: hydrodynamic effect

A laboratory study was undertaken to determine the transport and deposition rate of suspended particles in columns of saturated porous media (gravel and glass beads), where the porous media were subjected to steady-state flow. Silt particles with a mode of 14 μm diameter (used as the suspended particles) and fluorescein (as the conservative tracer) were injected into the columns in short pulses. The breakthrough curves were competently described with the analytical solution of a convection–dispersion equation with a first-order deposition rate. The experiments were performed using different flow rates. The suspended particle size distribution, the porous media, and the flow rates themselves were the main factors retained in this study to investigate the mechanisms governing the transport and deposition kinetics in detail. The results showed the existence of a flow rate, beyond which suspended particles travel faster than the conservative tracer. A decrease of the deposition rate of suspended particles beyond a critical flow velocity was also observed. Such behaviour led to consideration of the couple hydrodynamic-gravity forces at high flow rates. As the hydrodynamic force increases, particle deposition rates are reduced due to the effect of hydrodynamic forces inhibiting the deposition.     

Three-Dimensional Modeling of Mass Transfer in Porous Media Using the Mixed Hybrid Finite Elements and the Random-Walk Methods

A three-dimensional (3D) mass transport numerical model is presented. The code is based on a particle tracking technique: the random-walk method, which is based on the analogy between the advection–dispersion equation and the Fokker–Planck equation. The velocity field is calculated by the mixed hybrid finite element formulation of the flow equation. A new efficient method is developed to handle the dissimilarity between Fokker–Planck equation and advection–dispersion equation to avoid accumulation of particles in low dispersive regions. A comparison made on a layered aquifer example between this method and other algorithms commonly used, shows the efficiency of the new method. The code is validated by a simulation of a 3D tracer transport experiment performed on a laboratory model. It represents a heterogeneous aquifer of about 6-m length, 1-m width, and 1-m depth. The porous medium is made of three different sorts of sand. Sodium chloride is used as a tracer. Comparisons between simulated and measured values, with and without the presented method, also proves the accuracy of the new algorithm.     

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.     

A 2D mixed fracture–pore seepage model and hydromechanical coupling for fractured porous media

A novel two-dimensional mixed fracture–pore seepage model for fluid flow in fractured porous media is presented based on the computational framework of finite-discrete element method (FDEM). The model consists of a porous seepage model in triangular elements bonded by unbroken joint elements, as well as a fracture seepage model in broken joint elements. The principle for determining the fluid exchange coefficient of the unbroken joint element is provided to ensure numerical accuracy and efficiency. The mixed fracture–pore seepage model provides a simple but effective tool for solving fluid flow in fractured porous media. In this paper, examples of 1D and 2D seepage flow in porous media and porous media with a single fracture or multiple fractures are studied. The simulation results of the model match well with theoretical solutions or results obtained by commercial software, which verifies the correctness of the mixed fracture–pore seepage model. Furthermore, combining FDEM mechanical calculation and the mixed fracture–pore seepage model, a coupled hydromechanical model is built to simulate fluid-driven dynamic propagation of cracks in the porous media, as well as its influence on pore seepage and fracture seepage.

     

Physical model study on the clay–sand interface without and with geotextile separator

In most marine reclamation projects, sand fill is placed directly on soft marine seabed soils. The sand particles can easily penetrate into the soft marine soils, and the soft soil can also move into the pore spaces inside the sand at the initial contact interface between the sand and the soft marine soil. In this case, the permeability and the volume of the sand above the initial surface are reduced. To avoid this problem, a geotextile separator is often placed on the surface of the soft marine soils before placing the sand. In this study, a two-dimensional physical model is utilized to study the geotextile separator effects. The initial conditions of a clayey soil, sand fill, and surcharge loading were kept the same in the physical model test with the only difference being that a geotextile separator was either placed on the clay surface or omitted. The settlements of the initial interface were recorded and compared for the two cases without or with the geotextile separator. The particle size distribution of the soils taken across the interface zone for different time durations was then measured, analyzed, and compared. Based on an analysis of the results, the sand percolation depth was 40 mm and fine particle suffusion was apparent when the sand was placed directly on the marine slurry surface without a geotextile separator. However, when a geotextile separator was used sand percolation was avoided, and the fine particle suffusion was effectively diminished. A relative fine particle fraction is defined to illustrate the migration of fine particles from the clay to the sand soils. The fine particle percentages of the Hong Kong Marine Deposits–sand mixtures were calculated for the cases with and without a geotextile separator using an empirical formula and micromechanical modeling to obtain a better understanding of the effects of geotextile separators in practice.

     

Numerical simulation of fully saturated porous materials

Fully coupled, porous solid–fluid formulation, implementation and related modeling and simulation issues are presented in this work. To this end, coupled dynamic field equations with u?p?U formulation are used to simulate pore fluid and soil skeleton (elastic–plastic porous solid) responses. Present formulation allows, among other features, for water accelerations to be taken into account. This proves to be useful in modeling dynamic interaction of media of different stiffnesses (as in soil–foundation–structure interaction). Fluid compressibility is also explicitly taken into account, thus allowing excursions into modeling of limited cases of non‐saturated porous media. In addition to these features, present formulation and implementation models in a realistic way the physical damping, which dissipates energy. In particular, the velocity proportional damping is appropriately modeled and simulated by taking into account the interaction of pore fluid and solid skeleton. Similarly, the displacement proportional damping is physically modeled through elastic–plastic processes in soil skeleton. An advanced material model for sand is used in present work and is discussed at some length. Also explored in this paper are the verification and validation issues related to fully coupled modeling and simulations of porous media. Illustrative examples describing the dynamical behavior of porous media (saturated soils) are presented. The verified and validated methods and material models are used to predict the behavior of level and sloping grounds subjected to seismic shaking. Copyright © 2008 John Wiley & Sons, Ltd.