Finite difference modeling of contaminant transport using analytic element flow solutions

The two-dimensional implementation of the analytic element method (AEM) is commonly used to simulate steady-state saturated groundwater flow phenomena at regional and local scales. However, unlike alternative groundwater flow simulation methods, AEM results are not ordinarily used as the basis for simulation of reactive solute transport. The use of AEM-simulated flow fields is impeded by the discrepancy between a continuous representation of flow and a typically discrete representation of transport, and requires translation of the flow solution to a discrete analog. This paper presents a variety of methods for analytically calculating conservative discrete water fluxes and integrated components of the dispersion tensor across cell interfaces. An Eulerian finite difference method based on these AEM-derived parameters is implemented for use in simulation of 2D (vertically averaged) solute transport. This implementation is first benchmarked against existing methods that use standard finite difference flow solutions, then used to investigate the effects of an inaccurate discrete water balance. It is shown that improper translation of AEM fluxes leads to significant water balance errors and inaccurate simulation of contaminant transport.     

Two-dimensional modeling of contaminant transport in porous media in the presence of colloids

It has long been known that colloids can facilitate the transport of contaminants in groundwater systems by reducing the effective retardation factor. A significant effort has been devoted to study colloid-facilitated contaminant transport during the past decade. Many of the previous studies were restricted to one-dimensional analyses and comparisons with finite-column experiments. In this work, a two-dimensional numerical model is developed and used to study the different interactions between colloids, contaminants, and porous media under homogeneous conditions. The numerical formulation of the model is based on discretizing mass balance equations and reaction equations using finite differences having a third-order, total variance-diminishing scheme for the advection terms. This scheme significantly reduces numerical dispersion and leads to greater accuracy compared to the standard central-differencing scheme. The model is tested against analytical solutions under simplified conditions as well as against experimental data, and the results are favorable. The model is used to investigate the impact of the various reaction rates and parameter values on the movement of contaminant plumes in two dimensions. The model is also used to investigate the hypothesis that colloids may increase the effective retardation factor of contaminant plumes. The analysis shows that assuming kinetic mass exchange between contaminant and colloids with constant reaction rate coefficients that are not related to the concentrations may lead to inaccurate results. These inaccurate results are exemplified in the finding that under the kinetic assumption the ratio of the initial concentration of colloids to the initial concentration of contaminant does not affect the amount of facilitation or retardation that occurs in the system. It is also found that colloids can increase the effective retardation factor for the contaminant under certain combinations of reaction rates and distribution coefficients. A quantitative empirical expression to identify whether colloids retard or facilitate the contaminant movement is presented.     

Well-balanced numerical modelling of non-uniform sediment transport in alluvial rivers

The last two decades have witnessed the development and application of well-balanced numerical models for shallow flows in natural rivers.However,until now there have been no such models for flows with non-uniform sediment transport.This paper presents a 1D well-balanced model to simulate flows and non-capacity transport of non-uniform sediment in alluvial rivers.The active layer formulation is adopted to resolve the change of bed sediment composition.In the framework of the finite volume Slope Llmiter Centred(SLIC) scheme,a surface gradient method is incorporated to attain well-balanced solutions to the governing equations.The proposed model is tested against typical cases with irregular topography,including the refilling of dredged trenches,aggradation due to sediment overloading and flood flow due to landslide dam failure.The agreement between the computed results and measured data is encouraging.Compared to a non-well-balanced model,the well-balanced model features improved performance in reproducing stage,velocity and bed deformation.It should find general applications for non-uniform sediment transport modelling in alluvial rivers,especially in mountain areas where the bed topography is mostly irregular.     

Reactive transport in stratified flow fields with idealized heterogeneity

A two-dimensional equation governing the steady state spatial concentration distribution of a reactive constituent within a heterogeneous advective–dispersive flow field is solved analytically. The solution which is developed for the case of a single point source can be generalized to represent analogous situations with any number of separate point sources. A limiting case of special interest has a line source of constant concentration spanning the domain’s upstream boundary. The work has relevance for improving understanding of reactive transport within various kinds of advection-dominated natural or engineered environments including rivers and streams, and bioreactors such as treatment wetlands. Simulations are used to examine quantitatively the impact that transverse dispersion (deviations from purely stochastic-convective flow) can have on mean concentration decline in the direction of flow. Results support the contention that transverse mixing serves to enhance the overall rate of reaction in such systems.     

Analytical solutions for transport of decaying solutes in rivers with transient storage

Analytical solutions are presented for solute transport in rivers including the effects of transient storage and first order decay. The solute transport model considers an advection–dispersion equation for transport in the main channel linked to a first order mass exchange between the main channel and the transient storage zones. In case of a conservative tracer, it is shown that different analytical solutions presented in the literature are mathematically identical. For non-conservative solutes, first order decay reactions are considered with different reaction rate coefficients in the main river channel and in the dead zones. New analytical solutions are presented for different boundary conditions, i.e. instantaneous injection in an infinite river reach, and variable concentration time series input in a semi-infinite river reach. The correctness and accuracy of the analytical solutions is verified by comparison with the OTIS numerical model. The results of analytical and numerical approaches compare favourably and small differences can be attributed to the influence of boundary conditions. It is concluded that the presented analytical solutions for solute transport in rivers with transient storage and solute decay are accurate and correct, and can be usefully applied for analyses of tracer experiments and transport characteristics in rivers with mass exchange in dead zones.     

A parallel mesh-free contaminant transport model based on the Analytic Element and Streamline Methods

This paper introduces a new method for simulating large-scale subsurface contaminant transport that combines an Analytic Element Method (AEM) groundwater flow solution with a split-operator Streamline Method for modeling reactive transport. The key feature of the method is the manner in which the vertically integrated AEM flow solution is used to construct three-dimensional particle tracks that define the geometry of the Streamline Method. The inherently parallel nature of the algorithm supports the development of reactive transport models for spatial domains much larger than current grid-based methods. The applicability of the new approach is verified for cases with negligible transverse dispersion through comparisons to analytic solutions and existing numerical solutions, and parallel performance is demonstrated through a realistic test problem based on the regional-scale transport of agricultural contaminants from spatially distributed sources.     

An adaptive hybrid EnKF-OI scheme for efficient state-parameter estimation of reactive contaminant transport models

Reactive contaminant transport models are used by hydrologists to simulate and study the migration and fate of industrial waste in subsurface aquifers. Accurate transport modeling of such waste requires clear understanding of the system’s parameters, such as sorption and biodegradation. In this study, we present an efficient sequential data assimilation scheme that computes accurate estimates of aquifer contamination and spatially variable sorption coefficients. This assimilation scheme is based on a hybrid formulation of the ensemble Kalman filter (EnKF) and optimal interpolation (OI) in which solute concentration measurements are assimilated via a recursive dual estimation of sorption coefficients and contaminant state variables. This hybrid EnKF-OI scheme is used to mitigate background covariance limitations due to ensemble under-sampling and neglected model errors. Numerical experiments are conducted with a two-dimensional synthetic aquifer in which cobalt-60, a radioactive contaminant, is leached in a saturated heterogeneous clayey sandstone zone. Assimilation experiments are investigated under different settings and sources of model and observational errors. Simulation results demonstrate that the proposed hybrid EnKF-OI scheme successfully recovers both the contaminant and the sorption rate and reduces their uncertainties. Sensitivity analyses also suggest that the adaptive hybrid scheme remains effective with small ensembles, allowing to reduce the ensemble size by up to 80% with respect to the standard EnKF scheme.     

Analytical models of contaminant transport in coastal aquifers

The Henry formulation, which couples subsurface flow and salt transport via a variable-density flow formulation, can be used to evaluate the extent of sea water intrusion into coastal aquifers. The coupling gives rise to nontrivial flow patterns that are very different from those observed in inland aquifers. We investigate the influence of these flow patterns on the transport of conservative contaminants in a coastal aquifer. The flow is characterized by two dimensionless parameters: the Péclet number, which compares the relative effects of advective and dispersive transport mechanisms, and a coupling parameter, which describes the importance of the salt water boundary on the flow. We focus our attention on two regimes – low and intermediate Péclet number flows. Two transport scenarios are solved analytically by means of a perturbation analysis. The first, a natural attenuation scenario, describes the flushing of a contaminant from a coastal aquifer by clean fresh water, while the second, a contaminant spill scenario, considers an isolated point source.     

水平非均匀基流中行星波的传播

行星波传播理论虽然已有很多研究,但是大多以纬向对称基流为主,无法解释东西风带之间相互作用的事实.鉴于此,本文从理论上系统讨论了纬向对称和水平非均匀基流中定常和非定常波动的传播特征.首先,对纬向对称基流中波动传播的周期特征进行分析后发现,西风中位相东传超长波周期大于30 d,而东风中位相西传超长波的周期则小于30 d.之后,从传播的空间以及周期特征等方面系统研究了水平非均匀基流中球面波动传播理论,得到以下结论:经向基流使得定常波可以穿越东风带,在南北两半球间传播,为东西风带之间的相互作用提供了理论解释;强的经向流使得波动传播具有单向性;亚澳季风区低层纬向1波呈低频特征.     

Probabilistic sensitivity and modeling of two-dimensional transport in porous media

A reliability approach is used to develop a probabilistic model of two-dimensional non-reactive and reactive contaminant transport in porous media. The reliability approach provides two important quantitative results: an estimate of the probability that contaminant concentration is exceeded at some location and time, and measures of the sensitivity of the probabilistic outcome to likely changes in the uncertain variables. The method requires that each uncertain variable be assigned at least a mean and variance; in this work we also incorporate and investigate the influence of marginal probability distributions. Uncertain variables includex andy components of average groundwater flow velocity,x andy components of dispersivity, diffusion coefficient, distribution coefficient, porosity and bulk density. The objective is to examine the relative importance of each uncertain variable, the marginal distribution assigned to each variable, and possible correlation between the variables. Results utilizing a two-dimensional analytical solution indicate that the probabilistic outcome is generally very sensitive to likely changes in the uncertain flow velocity. Uncertainty associated with dispersivity and diffusion coefficient is often not a significant issue with respect to the probabilistic analysis; therefore, dispersivity and diffusion coefficient can often be treated for practical analysis as deterministic constants. The probabilistic outcome is sensitive to the uncertainty of the reaction terms for early times in the flow event. At later times, when source contaminants are released at constant rate throughout the study period, the probabilistic outcome may not be sensitive to changes in the reaction terms. These results, although limited at present by assumptions and conceptual restrictions inherent to the closed-form analytical solution, provide insight into the critical issues to consider in a probabilistic analysis of contaminant transport. Such information concerning the most important uncertain parameters can be used to guide field and laboratory investigations.     

Probabilistic sensitivity and modeling of two-dimensional transport in porous media

A reliability approach is used to develop a probabilistic model of two-dimensional non-reactive and reactive contaminant transport in porous media. The reliability approach provides two important quantitative results: an estimate of the probability that contaminant concentration is exceeded at some location and time, and measures of the sensitivity of the probabilistic outcome to likely changes in the uncertain variables. The method requires that each uncertain variable be assigned at least a mean and variance; in this work we also incorporate and investigate the influence of marginal probability distributions. Uncertain variables includex andy components of average groundwater flow velocity,x andy components of dispersivity, diffusion coefficient, distribution coefficient, porosity and bulk density. The objective is to examine the relative importance of each uncertain variable, the marginal distribution assigned to each variable, and possible correlation between the variables. Results utilizing a two-dimensional analytical solution indicate that the probabilistic outcome is generally very sensitive to likely changes in the uncertain flow velocity. Uncertainty associated with dispersivity and diffusion coefficient is often not a significant issue with respect to the probabilistic analysis; therefore, dispersivity and diffusion coefficient can often be treated for practical analysis as deterministic constants. The probabilistic outcome is sensitive to the uncertainty of the reaction terms for early times in the flow event. At later times, when source contaminants are released at constant rate throughout the study period, the probabilistic outcome may not be sensitive to changes in the reaction terms. These results, although limited at present by assumptions and conceptual restrictions inherent to the closed-form analytical solution, provide insight into the critical issues to consider in a probabilistic analysis of contaminant transport. Such information concerning the most important uncertain parameters can be used to guide field and laboratory investigations.     

纬向非均匀基流对大气长波调整的作用

大气长波的发展和演变影响着大气的可预报性,并对提高天气预报和气候预测水平有重要的意义.在影响大气长波演变的因子中,除波与波非线性相互作用外,基流的作用也非常重要.本文利用非均匀基本场下Rossby波运动方程,通过数值求解,分析了基本场结构和初始场对Rossby波演变的影响,揭示了纬向非均匀基本场对长波调整的作用.研究结果表明:基流纬向非均匀时,线性Rossby波也会出现长波调整现象,基流随纬向变化是长波发生调整的又一个重要机制;大气长波调整对波动的初始振幅不敏感,但基本场振幅影响着长波调整能否出现和出现的时间;基本场纬向平均西风基流的大小除影响波动传播的速度和方向外,还影响长波调整出现的时间和规律;长波调整的出现还与基本场和初始场的结构有关,不同基本场时,波动是否发生调整、向高波数还是向低波数调整都决定于基本场结构,相同基本场时,不同初始结构的波动也有着不同的演变过程.     

One-dimensional numerical simulation of non-uniform sediment transport under unsteady flows

One-dimensional numerical models are popularly used in sediment transport research because they can be easily programmed and cost less time compared with two- and three-dimensional numerical models. In particular, they possess greater capacity to be applied in large river basins with many tributaries. This paper presents a one-dimensional numerical model capable of calculating total-load sediment transport. The cross-section-averaged sediment transport capacity and recovery coefficient are addressed in the suspended load model. This one-dimensional model, therefore, can be applied to fine suspended loads and to hyperconcentrated flows in the Yellow River. Moreover, a new discretization scheme for the equation of unsteady non-uniform suspended sediment transport is proposed. The model is calibrated using data measured from the Yantan Reservoir on the Hongshui River and the Sanmenxia Reservoir on the Yellow River. A comparison of the calculated water level and river bed deformation with field measurements Shows that the improved numerical model is capable of predicting flow, sediment transport, bed changes, and bed-material sorting in various situations, with reasonable accuracy and reliability.     

Variable-density groundwater flow and solute transport in porous media containing nonuniform discrete fractures

Variations in fluid density can greatly affect fluid flow and solute transport in the subsurface. Heterogeneities such as fractures play a major role for the migration of variable-density fluids. Earlier modeling studies of density effects in fractured media were restricted to orthogonal fracture networks, consisting of only vertical and horizontal fractures. The present study addresses the phenomenon of 3D variable-density flow and transport in fractured porous media, where fractures of an arbitrary incline can occur. A general formulation of the body force vector is derived, which accounts for variable-density flow and transport in fractures of any orientation. Simulation results are presented that show the verification of the new model formulation, for the porous matrix and for inclined fractures. Simulations of variable-density flow and solute transport are then conducted for a single fracture, embedded in a porous matrix. The simulations show that density-driven flow in the fracture causes convective flow within the porous matrix and that the high-permeability fracture acts as a barrier for convection. Other simulations were run to investigate the influence of fracture incline on plume migration. Finally, tabular data of the tracer breakthrough curve in the inclined fracture is given to facilitate the verification of other codes.     

Series solutions for flow in stratified aquifers with natural geometry

An analytical series solution method is presented for modeling regional steady-state groundwater flow in a two-dimensional stratified aquifer cross-section where the water table is well-characterized. The aquifer system may have any number of contiguous or non-contiguous layers and the geometry of each layer is restricted only by the requirement that the elevation of the stratigraphic unconformities between layers is a function of the x-coordinate alone. Various techniques may be used to handle pinching layers, faults, and other discontinuities. The solutions are obtained by minimizing head and flow continuity errors between layers and errors in the Dirichlet surface at a set of control points along these unconformities; the governing equation is met exactly. The solutions are derived and demonstrated on multiple test cases. The errors for some specific, geometrically challenging cases are assessed and discussed.     

Travel-time distribution from a finite line contamination source to an extraction well with regional flow

Advective transport from a finite line contamination source to an extraction well with regional flow depends on interplay of radial and regional flows, a scheme commonly encountered in capture zone delineation. We have investigated travel-time distribution from a finite line contamination source and the associated breakthrough curves (BTCs) observed at an extraction well. The resulting travel-time distribution and BTCs depend on dimensionless source length, dimensionless pumping rate, and inclined angle of the source with respect to the regional flow, where the dimensionless terms are lumped parameters involving source length, pumping rate, distance between the source and the extraction well, aquifer thickness, and regional flow discharge. The observed concentration at the extraction well increases with time in a sub-linear manner. When the source orientation is perpendicular to the regional flow, the dimensionless first arrival time only depends on the dimensionless pumping rate whereas the dimensionless steady-state arrival time depends on both the dimensionless pumping rate and the dimensionless source length. The steady-state concentration at the extraction well is sensitive to the dimensionless source length and the inclined angle of the source with respect to the regional flow, but not sensitive to the dimensionless pumping rate. Two special cases where the extraction well is very close to the source and the regional flow can be negligible have also been discussed.     

Evidence of distinct contaminant transport patterns in rivers using tracer tests and a multiple domain retention model

Solute transport in rivers is controlled by surface hydrodynamics and by mass exchanges with distinct retention zones. Surface and hyporheic retention processes can be accounted for separately in solute transport models with multiple storage compartments. In the simplest two component model, short term storage can be associated to in-channel transient retention, e.g. produced by riparian vegetation or surface dead zones, and the long-term storage can be associated to hyporheic exchange. The STIR (Solute Transport In Rivers) multiple domain transport model is applied here to tracer test data from three very different Mediterranean streams with distinctive characteristics in terms of flow discharge, vegetation and substrate material. The model is used with an exponential residence time distribution (RTD) to represent surface storage processes and two distinct modeling closures are tested to simulate hyporheic retention: a second exponential RTD and a power-law distribution approximating a known solution for bedform-induced hyporheic exchange. Each stream shows distinct retention patterns characterized by different timescales of the storage time distribution. Both modeling closures lead to very good approximations of the observed breakthrough curves in the two rivers with permeable bed exposed to the flow, where hyporheic flows are expected to occur. In the one case where the occurrence of hyporheic flows is inhibited by bottom vegetation, only the two exponential RTD model is acceptable and the time scales of the two components are of the same magnitude. The significant finding of this work is the recognition of a strong signature of the river properties on tracer data and the evidence of the ability of multiple-component models to describe individual stream responses. This evidence may open a new perspective in river contamination studies, where rivers could possibly be classified based on their ability to trap and release pollutants.     

Field studies of sediment transport by overland flow

Field studies on sandy soils of the Cottenham Series in mid-Bedfordshire show that the mean annual rate of sediment transport by overland flow on an 11° mid-slope is 98 g cm¹. The feasibility of using sediment transport equations to predict erosion by overland flow on a storm basis is examined by comparing the observed values of sediment yield with values predicted by four sediment transport equations and a regression equation which relates soil loss to runoff energy and rainfall energy. An expression combining Engelund's sediment transport capacity equation and the Manning equation for flow velocity, as modified by Savat for disturbed flow, best reflects field conditions. Although there is a significant correlation (r = 0.69; n 30) between the observed and predicted values using this expression, the coefficient of determination is too low for predictive purposes. Reasons for this are presented.     

Relation of sediment transport capacity to stone cover and size in rain‐impacted interrill flow

This study examines how the sediment transport capacity of interrill overland flow varies with stone cover and stone size at two flow intensities. Six series of flume experiments were conducted on two slopes (2° and 10°) with stones of three sizes (28·0, 45·5 and 91·3 mm) serving as roughness elements. Bed sediment size, water discharge and simulated rainfall intensity were the same in all experiments. It was found (1) that transport capacity is positively related to stone size, with the relation becoming stronger as stone cover increases and flow intensity decreases; and (2) that transport capacity is negatively related to stone cover at the high flow intensity and curvilinearly related to stone cover at the low flow intensity. The curvilinear relations are concave‐upward with the lowest transport capacities occurring at stone covers between 0·40 and 0·60. The highest transport capacities are found at stone covers of 0 and 1, with the transport capacity being greater at the former stone cover than at the latter. Copyright © 2000 John Wiley & Sons, Ltd.