Convolution-based particle tracking method for transient flow

A convolution-based particle tracking (CBPT) method was recently developed for calculating solute concentrations (Robinson et al., Comput Geosci 14(4): 779–792, 2010). This method is highly efficient but limited to steady-state flow conditions. Here, we present an extension of this method to transient flow conditions. This extension requires a single-particle tracking process model run, with a pulse of particles introduced at a sequence of times for each source location. The number and interval of particle releases depends upon the transients in the flow. Numerical convolution of particle paths obtained at each release time and location with a time-varying source term is performed to yield the shape of the plume. Many factors controlling transport such as variation in source terms, radioactive decay, and in some cases linear processes such as sorption and diffusion into dead-end pores can be simulated in the convolution step for Monte Carlo-based analysis of transport uncertainty. We demonstrate the efficiency of the transient CBPT method, by showing that it requires fewer particles than traditional random walk particle tracking methods to achieve the same levels of accuracy, especially as the source term increases in duration or is uncertain. Since flow calculations under transient conditions are often very expensive, this is a computationally efficient yet accurate method.     

A hybrid approach to flow net generation

By following a simple set of rules, a flow net can be manually constructed to obtain a graphical solution to the Laplace equation for simple two‐dimensional (2‐D) flow problems. With the advent of numerical solutions such as the finite difference and finite element methods, it is more common to generate a flow net automatically using the nodal head and flow values output by the computer program. Two methods have been published for automatically generating flow nets from finite element solutions: the stream‐function method and the particle‐tracking method. The stream‐function method works well for many cases, but it does not work for problems with holes in the mesh or internal sources or sinks. The particle‐tracking method works for all cases, but previously published algorithms that utilize this method do not result in the properly sized flow channels. A new approach is presented in this paper that is a hybrid of the stream‐function and particle‐tracking approaches. This method works for all cases and generates the properly sized flow channels. Copyright © 2001 John Wiley & Sons, Ltd.     

Effects of Local Dispersion and Kinetic Sorption on Evolution of Concentration Variance in a Heterogeneous Aquifer

A Eulerian stochastic method is applied to develop a theory of concentration variance for solute transport in a heterogeneous medium. The study focuses on the effects of kinetic sorption and local dispersion on solute dissipation. Spatial distribution of the concentration variance is obtained by scaling the zero local dispersion form of σ_c ². The scaling function resulting from the local dispersion and kinetic sorption is derived in a closed integral form. It satisfies the measurement of total concentration variance resulting from the Eulerian mass balance using spatially integrated concentration moments. The spatially integrated moments bypass the need for classical closures applied to joint moments between concentration and velocity fields. The study results indicate that kinetic sorption reduces the total development of concentration variance in comparison with non-reactive solute transport. Kinetic sorption acts as a reduction mechanism, but not as a dissipating mechanism like the local dispersion. Kinetic sorption and local dispersion are not additive processes and their effects on the concentration variance depend on the stage of transport time.     

Analytic solutions to the advective contaminant transport equation with non-linear sorption

This paper considers advective transport of a soluble contaminant through saturated soil with non-linear sorption of the contaminant onto a stationary porous media. The non-linear sorption isotherms considered in the transport analysis are the Langmuir and Freundlich sorption isotherms. A special case of the Freundlich sorption isotherm is the linear sorption isotherm, and it is shown that in this case transport through a homogeneous soil results in the initial concentration profile simply being translated in the direction of the groundwater flow. However, when the sorption isotherm is non-linear the initial concentration profile distorts as it is translated with the groundwater flow, leading to the development of concentration shock fronts and rarefactions. Analytic solutions to the non-linear first-order hyperbolic equations are developed for a number of contaminant transport problems of practical significance. It is shown that in the case of the Langmuir sorption isotherms, shock fronts develop at the leading edge of the concentration profile while for the Freundlich sorption isotherm shock fronts may develop at either the leading or trailing edge of the concentration profile. Copyright © 1999 John Wiley & Sons Ltd.     

Transverse Dispersion of a Kinetically Sorbing Solute

A recursion formulation for the transverse spreading of a solute is developed, and under conditions of steady flow in a stratified aquifer, the transport of a linearly sorbing solute undergoing nonequilibrium sorption is studied. The effect of spatial variability in the velocity field and the sorption kinetics are modeled to see the combined effect of the two processes on the spreading of the solute injected at a point in the aquifer. The main result of this work is a transport model based on a discrete formulation that includes local dispersion and leads to nonasymptotic behavior in the spreading of the plume in a direction normal to the mean flow velocity.     

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.     

A self-adapting model for assessing hazardous environmental releases

Simulation results are presented by using an h-adaptive mass consistent finite element method (FEM) coupled with a Lagrangian particle transport technique (LPT) for dispersion associated with hazardous atmospheric releases. A three-dimensional wind field is first constructed from the adaptive FEM model. Lagrangian particles that define the contaminant dispersion are then produced with the LPT scheme, employing a random walk/stochastic approach. The application of FEM permits flow patterns with irregular geometries to be easily simulated, while the LPT permits contaminant particle dispersion patterns to be quickly depicted. The hybrid model is fast, runs on PCs, and appears well suited for emergency response dispersion predictions and assessment.     

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.     

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.     

Experimental study on the velocity-dependent dispersion of the solute transport in different porous media

The hydrodynamic dispersion is an important factor influencing the reactive solute transport in the porous media, and many previous studies assumed that it linearly varied with the average velocity of the groundwater flow. Actually, such linear relationship has been challenged by more and more experimental observations, even in homogeneous media. In this study, we aim to investigate the relationship between hydrodynamics dispersion and the flow velocity in different types of porous media through a laboratory-controlled experiment. The results indicate that(1) the dispersion coefficient should not be a linear function of the flow velocity when the relationship between the flow velocity and the hydraulic gradient can be described by Darcy's law satisfactorily;(2) Power function works well in describing the dispersion coefficient changing with the flow velocity for different types of porous media, and the power value is between 1.0-2.0 for different particle sizes.     

Experimental study on the velocity-dependent dispersion of the solute transport in different porous media

The hydrodynamic dispersion is an important factor influencing the reactive solute transport in the porous media, and many previous studies assumed that it linearly varied with the average velocity of the groundwater flow. Actually, such linear relationship has been challenged by more and more experimental observations, even in homogeneous media. In this study, we aim to investigate the relationship between hydrodynamics dispersion and the flow velocity in different types of porous media through a laboratory-controlled experiment. The results indicate that (1) the dispersion coefficient should not be a linear function of the flow velocity when the relationship between the flow velocity and the hydraulic gradient can be described by Darcy’s law satisfactorily; (2) Power function works well in describing the dispersion coefficient changing with the flow velocity for different types of porous media, and the power value is between 1.0-2.0 for different particle sizes.     

Numerical modeling of stress effects on solute transport in fractured rocks

The effects of stress/deformation on fluid flow and contaminant transport in fractured rocks is one of the major concerns for performance and safety assessments of many subsurface engineering problems, especially radioactive waste disposal and oil/gas reservoir fields. However, very little progress has been made to study this issue due to difficulties in both experiments and numerical modeling. The objective of this study is to systematically investigate the influence of stress on solute transport in fractured rocks for the first time, considering different stress and hydraulic pressure conditions. A hybrid approach combining discrete element method (DEM) for stress-flow simulations and a particle tracking algorithm is developed. The impact of matrix diffusion (diffusion of molecular size solutes in and out of the rock matrix, and sorption onto the surface of micropores in rock matrix) is also included. The numerical results show that stress not only significantly changes the solute residence time through the fracture networks, but also changes the solute travel paths. Matrix diffusion plays a dominant role in solute transport when the hydraulic gradient is small, which is often encountered in practice.     

Simulation of polychlorinated biphenyls transport in the vadose zone

Polychlorinated biphenyls (PCBs) are the main constituents of clophen (the liquid of the electric transformers and capacitors) and have been characterized as potential human carcinogens. PCBs can be a hazardous contaminant of soil and groundwater. We used the mathematical model variably saturated 2D flow and transport (VS2DT model) to simulate the transport of PCBs from the soil surface to groundwater for a time period of 30 years. We also used a mathematical model to simulate the colloid-facilitated PCB transport, under saturated flow conditions. The results showed that PCBs dissolved in water cannot be transported to large depths in unsaturated soils, because of their strong sorption onto soil and low solubility in water. For soils with very low or no organic matter content, PCB transport is much faster and the probability of groundwater contamination is much higher. PCBs can partition to colloids originating from dissolved organic matter in groundwater. Colloid-facilitated PCB transport is faster compared to PCB transport in aqueous solution with no colloids present.     

一维均质与非均质土柱中溶质迁移的分数微分对流-弥散模拟

分数微分对流-弥散方程(FADE)是模拟溶质迁移问题的新理论,但应用FADE来模拟溶质迁移时能否克服弥散的尺度效应尚待验证。利用长土柱实验资料结合FADE的解析解拟合推求FADE的弥散系数,并分析其与尺度之间的相关关系。研究结果表明,FADE的弥散系数具有随尺度增大而增大的现象,且均质土柱中FADE的弥散系数尺度效应小于非均质土柱中弥散系数尺度效应。在均质土柱中,弥散系数与尺度之间成指数相关关系,在非均质土柱中,弥散系数与尺度之间成幂相关关系。考虑了弥散系数分别与迁移时间和迁移距离呈线性递增两种相关关系,进而分别构建了3种考虑弥散尺度效应的FADE模型,并提出了求解的差分方法。利用上述3种考虑弥散尺度效应的FADE来模拟和预测不同空间位置处的溶质迁移过程。结果表明,对均质土柱中的溶质迁移可得到较好的模拟结果;对于非均质土柱,其模拟结果与实测结果仍然存在一定的差异。     

Hybrid analytical and finite element numerical modeling of mass and heat transport in fractured rocks with matrix diffusion

Quantification of mass and heat transport in fractured porous rocks is important to areas such as contaminant transport, storage and release in fractured rock aquifers, the migration and sorption of radioactive nuclides from waste depositories, and the characterization of engineered heat exchangers in the context of enhanced geothermal systems. The large difference between flow and transport characteristics in fractures and in the surrounding matrix rock means models of such systems are forced to make a number of simplifications. Analytical approaches assume a homogeneous system, numerical approaches address the scale at which a process is operating, but may lose individual important processes due to averaging considerations. Numerical stability criteria limit the contrasts possible in defining material properties. Here, a hybrid analytical–numerical method for transport modeling in fractured media is presented. This method combines a numerical model for flow and transport in a heterogeneous fracture and an analytical solution for matrix diffusion. By linking the two types of model, the advantages of both methods can be combined. The methodology as well as the mathematical background are developed, verified for simple geometries, and applied to fractures representing experimental field conditions in the Grimsel rock laboratory.     

The analysis of pollutant migration through soil with linear hereditary time-dependent sorption

This paper presents a non-equilibrium sorption dispersion–advection transport model for the analysis of pollutant migration through soil. The formulation involves a convolution integral of the product of the rate of change of concentration and a time-dependent sorption coefficient, suggesting an integral transformation of the governing equations. This facilitates the primary purpose of this paper, to incorporate a time-dependent solute sorption process into a computationally efficient and accurate semi-analytic Laplace transform method. An application of the non-equilibrium sorption model for backfiguring dispersion–advection equation parameters from experimental data is presented, and the implications of non-equilibrium sorption on the design of landfill liners is explored by means of an illustrative example.     

Sensitivity of incipient particle motion to fluid flow penetration depth within a packed bed

A discrete element method is applied to a three‐dimensional analysis related to sediment entrainment on a micro‐scale. Sediment entrainment is the process by which a fluid medium accelerates particles from rest and advects them upward until they are either transported as bedload or suspended by the flow. Modelling of the entrainment process is a critically important aspect for studies of erosion, pollutant resuspension and transport, and formation of bedforms in environmental flows. Previous discrete element method studies of sediment entrainment have assumed the flow within the particle bed to be negligible and have only allowed for the motion of the topmost particles. At the same time, micro‐scale experimental studies indicate that there is a small slip of the fluid flow at the top of the bed, indicating the presence of non‐vanishing fluid velocity within the topmost bed layers. The current study demonstrates that the onset of particle incipient motion, which immediately precedes particle entrainment, is highly sensitive to this small fluid flow within the topmost bed layers. Using an exponential decay profile for the inner‐bed fluid flow, the discrete element method calculations are repeated with different fluid penetration depths within the bed for several small particle Reynolds numbers. For cases with slip velocity corresponding to that observed in previous experiments with natural sediment, the predicted particle velocity is found to be a few percent of the fluid velocity at the top of the viscous wall layer, which is a reasonable range of velocities for observation of incipient particle motion. This method for prescribing the fluid flow within the particle bed allows for the current discrete element method to be extended in future studies to the analysis of sediment entrainment under the influence of events such as turbulent bursting. Additionally, predictions for the slip velocities and fluid flow profile within the bed suggest the need for further experimental studies to provide the data necessary for additional improvement of the discrete element method models.     

Study on numerical simulation of organic pollutant transport in groundwater northwest of Laixi

The construction and hydrogeological area in the low hill, which is northwest of Laixi City and in the east of Shandong Province more precisely, is one of Dagu River’s groundwater source. Regarding COD as the typical pollutant according to general situation of groundwater pollution in aforementioned area, a three-dimensional advection-dispersion model was established to simulate the transport of organic pollutants under two accident conditions. In addition, the effect of corresponding dispersity was analyzed. The results show that COD transport is mainly in accordance with the direction of groundwater flow, pollutant concentration and its gradient as well as the rule of narrower pollution range in deeper stratum. Moreover, COD is mainly transported and diffused in groundwater of sandy soil in the first and second layers. However, under accident condition II, the pollutant concentration begins to decay gradually after its transport and diffusion tends to be stable. Besides, in terms of dominance, dispersion is to transverse transport of pollutants what advection is to longitudinal transport. If considering random dispersion, the final results see higher peak concentration of COD and longer transverse distance from pollution center compared to transport route. What’s more, the pollution plume changes and concentration isocline becomes slightly irregular.