Stepwise superposition approach for the analytical solutions of multi-dimensional contaminant transport in finite- and semi-infinite aquifers

Analytical solutions of contaminant transport in multi-dimensional media are significant for theoretical and practical purposes. However, due to the problems for which the solutions are sought which are complex in most of the cases, most available analytical solutions in multi-dimensional media are not given in their closed forms. Integrals are often included in the solution expressions, which may limit the practitioners to use the solutions. In addition, available multi-dimensional solutions for the third-type sources in bounded media are fairly limited. In this paper, a stepwise superposition approach for obtaining approximate multi-dimensional transport solutions is developed. The approach is based on the condition that the one-dimensional solution along the flow direction is known. The solutions are expressed in their closed forms without integrals. The transport media to the solutions are flexible and can be finite, semi-infinite, or infinite in the transverse directions. The solutions subject to the first- and third-type boundary conditions at the inlet with a distributed source over the domain are obtained. The integrals in some known solutions can also be evaluated by the approach if they can be derived to include known longitudinal integrals with respect to time. The accuracy and efficiency of the solutions proposed in this paper are verified through test problems and calculation examples.     

A numerical model for transient-hysteretic flow and solute transport in unsaturated porous media

A two-dimensional flow and transport model was developed for simulating transient water flow and nonreactive solute transport in heterogeneous, unsaturated porous media containing air and water. The model is composed of a unique combination of robust and accurate numerical algorithms for solving the Richards', Darcy flux, and advection-dispersion equations. The mixed form of Richards' equation is solved using a finite-element formulation and a modified Picard iteration scheme. Mass lumping is employed to improve solution convergence and stability behavior. The flow algorithm accounts for hysteresis in the pressure head-water content relationship. Darcy fluxes are approximated with a Galerkin and Petrov-Galerkin finite-element method developed for random heterogeneous porous media. The transport equation is solved using an Eulerian-Lagrangian method. A multi-step, fourth-order Runge-Kutta, reverse particle tracking technique and a quadratic-linear interpolation scheme are shown to be superior for determining the advective concentration. A Galerkin finite-element method is used for approximating the dispersive flux. The unsaturated flow and transport model was applied to a variety of rigorous problems and was found to produce accurate, mass-conserving solutions when compared to analytical solutions and published numerical results.     

Analytical solutions for multiple species reactive transport in multiple dimensions

Many numerical computer codes used to simulate multi-species reactive transport and biodegradation have been developed in recent years. Such numerical codes must be validated by comparison of the numerical solutions with an analytical solution. In this paper, a method for deriving analytical solutions of the partial differential equations describing multiple species multi-dimensional transport with first-order sequential reactions is presented. Although others have developed specific solutions of multi-species transport equations, here a more general analytical approach, capable of describing any number of reactive species in multiple dimensions is derived. A substitution method is used to transform the multi-species reactive transport problem to one that can be solved using previously published single-species solutions for various initial and boundary conditions. One- and three-dimensional examples are presented to illustrate the steps involved in extending single-species solutions to a four-species system with sequential first-order reactions.     

Semi-analytical solution of two-dimensional sharp interface LNAPL transport models

In this paper, we present semi-analytical solutions for two-dimensional equations governing transport of Light Non-Aqueous Phase Liquids (LNAPL) in unconfined aquifers. The proposed model is based on sharp interface displacement and steady groundwater flow assumptions, where both the water–LNAPL interface and the LNAPL–air interface are represented as sharp interfaces. In the case of steady groundwater flow, these equations can be reduced to a two-dimensional nonlinear solute transport equation, with the LNAPL thickness in the free product lens being the primary unknown variable. The linearized form of this solute transport equation falls into the category of two-dimensional transport equation with time-dependent dispersion coefficients. This equation can be solved analytically for an infinite domain region. In this paper, the general form of the analytical solution for the transport equation, as well as the solutions for some specific cases are presented. To demonstrate the utility of the proposed solution, numerical results obtained for two example problems are discussed and presented comparatively with a finite-element solution and other more restrictive solutions available in the literature. Although the solutions discussed in this paper have some simplifying assumptions, such as sharp-interfaces between fluid phases, steady groundwater flow and homogeneous aquifer properties, the semi-analytical solutions presented in this study may be used effectively as bench mark solutions in evaluating LNAPL migration in the subsurface. These solutions are simple and cost effective to implement and may be used in the calibration of other more complex numerical solutions that can be found in the literature.     

An analytical solution for two-dimensional contaminant transport during groundwater extraction

We obtain an analytical solution for two-dimensional steady-state transport of conservative contaminant between injecting and pumping wells. Flow and transport are considered in the vertical cross-section. The Dupuit approximation and conformal mapping onto the complex potential domain are employed to determine the velocity and concentration distributions, respectively. We use this solution to derive a priori conditions under which widely used 1-D analytical solutions with constant velocity and dispersion coefficients provide accurate approximations. These conditions are formulated in terms of aquifer parameters, such as hydraulic conductivity, porosity and dispersivities, and remediation strategy, e.g., well spacing and pumping regimes.     

A risk assessment tool for contaminated sites in low-permeability fractured media

A risk assessment tool for contaminated sites in low-permeability fractured media is developed, based on simple transient and steady-state analytical solutions. The discrete fracture (DF) tool, which explicitly accounts for the transport along fractures, covers different source geometries and history (including secondary sources) and can be applied to a wide range of compounds. The tool successfully simulates published data from short duration column and field experiments. The use for risk assessment is illustrated by three typical risk assessment case studies, involving pesticides, chlorinated solvents, benzene and MTBE. The model is compared with field data and with results from a simpler approach based on an Equivalent Porous Media (EPM). Risk assessment conclusions of the DF and EPM approaches are very different due to the early breakthrough, long term tailing, and lower attenuation due to degradation associated with fractured media. While the DF tool simulates the field data, it is difficult to conclude that the DF model is superior to an EPM model because of a lack of long term monitoring data. However, better agreement with existing field data by the DF model using observed physical fracture parameters favors the use of this model over the EPM model for risk assessments.     

Approximate evaluation of contaminant transport through vertical barriers

Containment of groundwater contamination using physical barriers can be an important element of a subsurface remediation program. This work presents simple analytical tools for predicting the performance of barriers in terms of the steady-state contaminant flux across the barrier, the duration of the transient period following barrier installation, and the time-dependent contaminant concentration distribution within the barrier. The analytical expressions are developed from approximate boundary layer (BL) solutions to the advective–dispersive equation subject to conservative fixed concentration boundary conditions. Critical ranges of important dimensionless quantities are identified for use in barrier performance assessment, for both steady-state and transient conditions. Comparative calculations made with the BL equations and more exact semi-analytical solutions are used to characterize the accuracy and applicability of the BL approach.     

Scaling of spontaneous imbibition data with wettability included

Wettability is a dominant parameter governing spontaneous imbibition. However less attention has been paid to the effect of wettability on the scaling of spontaneous imbibition data. Actually few models can include wettability in scaling of spontaneous imbibition data. To this end, a scaling model has been developed for NAPL (oil)-saturated porous media with different wettability based on the fluid flow mechanisms in porous media. Relative permeability, capillary pressure, initial water saturation, and wettability are considered in the scaling model. Theoretically this scaling model is suitable for both cocurrent and countercurrent spontaneous imbibition. The experimental data of countercurrent spontaneous water imbibition at different wettability cannot be scaled using the frequently used scaling model but can be scaled satisfactorily using the scaling model developed in this study. An analytical solution to the relationship between recovery and imbibition time for linear spontaneous imbibition has also been derived in the case in which gravity is ignored. The analytical solution predicts a linear correlation between the recovery by spontaneous water imbibition and the square root of imbibition time, which has been verified against experimental data.     

Analytical model of solute transport by unsteady unsaturated gravitational infiltration

Penetration of reactive solute into a soil during a cycle of water infiltration and redistribution is investigated by deriving analytical closed form solutions for fluid flux, moisture content and contaminant concentration. The solution is developed for gravitational flow and advective transport and is applied to two scenarios of solute applications encountered in the applications: a finite pulse of solute dissolved in irrigation water and an instantaneous pulse broadcasted onto the soil surface. Through comparison to simulations of Richards' flow, capillary suction is shown to have contrasting effects on the upper and lower boundaries of the fluid pulse, speeding penetration of the wetting front and reducing the rate of drying. This leads to agreement between the analytical and numerical solutions for typical field and experimental conditions. The analytical solution is further incorporated into a stochastic column model of flow and transport to compute mean solute concentration in a heterogeneous field. An unusual phenomenon of plume contraction is observed at long times of solute propagation during the drying stage. The mean concentration profiles match those of the Monte-Carlo simulations for capillary length scales typical of sandy soils.     

Application of continuous time random walk theory to nonequilibrium transport in soil

Continuous time random walk (CTRW) formulations have been demonstrated to provide a general and effective approach that quantifies the behavior of solute transport in heterogeneous media in field, laboratory, and numerical experiments. In this paper we first apply the CTRW approach to describe the sorbing solute transport in soils under chemical (or) and physical nonequilibrium conditions by curve-fitting. Results show that the theoretical solutions are in a good agreement with the experimental measurements. In case that CTRW parameters cannot be determined directly or easily, an alternative method is then proposed for estimating such parameters independently of the breakthrough curve data to be simulated. We conduct numerical experiments with artificial data sets generated by the HYDRUS-1D model for a wide range of pore water velocities (υ) and retardation factors (R) to investigate the relationship between CTRW parameters for a sorbing solute and these two quantities (υ, R) that can be directly measured in independent experiments. A series of best-fitting regression equations are then developed from the artificial data sets, which can be easily used as an estimation or prediction model to assess the transport of sorbing solutes under steady flow conditions through soil. Several literature data sets of pesticides are used to validate these relationships. The results show reasonable performance in most cases, thus indicating that our method could provide an alternative way to effectively predict sorbing solute transport in soils. While the regression relationships presented are obtained under certain flow and sorption conditions, the methodology of our study is general and may be extended to predict solute transport in soils under different flow and sorption conditions.     

An interpretation of potential scale dependence of the effective matrix diffusion coefficient

Matrix diffusion is an important process for solute transport in fractured rock, and the matrix diffusion coefficient is a key parameter for describing this process. Previous studies have indicated that the effective matrix diffusion coefficient values, obtained from a large number of field tracer tests, are enhanced in comparison with local values and may increase with test scale. In this study, we have performed numerical experiments to investigate potential mechanisms behind possible scale-dependent behavior. The focus of the experiments is on solute transport in flow paths having geometries consistent with percolation theories and characterized by multiple local flow loops formed mainly by small-scale fractures. The water velocity distribution through a flow path was determined using discrete fracture network flow simulations, and solute transport was calculated using a previously derived impulse-response function and a particle-tracking scheme. Values for effective (or up-scaled) transport parameters were obtained by matching breakthrough curves from numerical experiments with an analytical solution for solute transport along a single fracture. Results indicate that a combination of local flow loops and the associated matrix diffusion process, together with scaling properties in flow path geometry, seems to be an important mechanism causing the observed scale dependence of the effective matrix diffusion coefficient (at a range of scales).     

Transport of particulate material and dissolved tracer in a highly permeable porous medium: comparison of the transfer parameters

We are experimentally studying, by means of short-pulse injection, the transport and deposition kinetics of suspended particles (silts of the order of 10 microm) in a highly permeable medium consisting of a column of gravel. In our experiments, the breakthrough curves (BTCs) are well described by analytical solutions of a convection/dispersion model with first-order deposition kinetics. All the transport parameters calculated by the model for both particles and dissolved tracer depend on the flow rate. We demonstrate the existence of a critical flow rate, determined experimentally, beyond which the transfer time for the particles is longer than that for the tracer. This phenomenon is unusual in comparison with the results available in the literature. The increase in transfer time of particles in comparison to tracer leads us to assume a purely mechanical phenomenon, that is, collision between particles and grains of the medium with instantaneous reset in motion when the flow rate is sufficient to avoid settling. Thanks to the polydispersivity of the injected suspension and the control of grain size at the outlet, it can also be determined that the coarser particles are recovered before the finer particles, as expected when one considers the size-exclusion effect.     

Assessing impacts of partial mass depletion in DNAPL source zones: II. Coupling source strength functions to plume evolution

Analytical solutions, describing the time-dependent DNAPL source-zone mass and contaminant discharge rate, derived previously in Part I [Falta, R.W., Rao, P.S., Basu, N., this issue. Assessing the impacts of partial mass depletion in DNAPL source zones: I. Analytical modeling of source strength functions and plume response. J. Contam. Hydrol.] are used as a flux-boundary condition in a semi-analytical contaminant transport model. These analytical solutions assume a power relationship between the flow-averaged source concentration, and the source DNAPL mass; the empirical exponent (gamma) is a function of the flow field heterogeneity, DNAPL architecture, and the correlation between them. The DNAPL source strength terms can account for partial source remediation, either at time zero, or at some later time after the DNAPL release. The transport model considers advection, retardation, three-dimensional dispersion, and sequential first-order decay/production of several species. A separate solution is used to compute the time-dependent mass of each contaminant in the plume. A series of examples using different values of gamma shows how the benefits of partial DNAPL source remediation can vary with site conditions. In general, when gamma>1, relatively large short-term reductions in the plume concentrations and mass occur, but the source longevity is not strongly affected. Conversely, when gamma<1, the short-term reductions in the plume concentrations and mass are smaller, but the source longevity can be greatly reduced. In either case, the source remediation effort is much more effective if it is undertaken at an early time, before much contaminant mass has entered the plume. If the remediation effort is significantly delayed, the leading parts of the plume are not affected by the source remediation, and additional control or remediation of the plume itself is required.     

A comparison of hydraulic and transport parameters measured in low-permeability fractured media

Data from 90 tracer experiments performed in low-permeability fractured media have been studied to explore correlations among parameters controlling flow and transport. The original data had been interpreted by different authors using different models, which prevents direct comparison of their estimated parameters. In order to produce comparable parameters, the data have been reexamined using simple models (homogeneous domain, steady-state flow regime, single porosity). Specifically, hydraulic conductivity has been derived as the ratio of water flux to head gradient and apparent porosity as the ratio of water velocity to water flux; the former estimated from both first and peak arrival times. Hydraulic conductivity and porosity correlate along a straight line of slope 1:3 in log scale. While the regression is too noisy to be of predictive use, it lends some support to the use of a generalized cubic law. The fact that correlation for first arrival time porosity (0.77) is larger than for peak arrival porosity (0.62) suggests that first arrival is controlled by the same flow paths as hydraulic conductivity. Apparent porosity derived from peak arrival time is found to grow with travel time along a line of 0.55 slope (again log scale). The correlation coefficient ranges between 0.73 and 0.80 (depending on the data set) for hard rocks. The fact that this correlation is maintained when varying the flow rate at a given site leads us to suggest that it is caused by diffusion mechanisms. This conclusion is further supported by the increase of apparent porosity with the matrix porosity of the rock on which the experiments were performed.     

Effects of sorption on transverse mixing in transient flows

Transverse mixing has been identified as a potentially limiting factor for natural attenuation of plumes originating from continuously emitting sources. Under steady-state flow conditions, dispersion is the only process leading to lateral mixing. This process is very slow and cannot explain the lateral spread of plumes observed in the field. When the flow direction fluctuates with time, transverse dispersion is slightly enhanced, but not very pronounced. Under these flow conditions, however, sorption can contribute to mixing into the mean transverse direction. If the reacting compounds differ in their strength of sorption, chromatographic mixing and separation alternate in time-periodic flows. For instantaneous sorption, the plumes may overlap within a stripe of fixed width. In contrast to sorption in local equilibrium, kinetic sorption contributes to mixing also for compounds with identical sorption strength. I derive an analytical expression for the equivalent transverse dispersion coefficient of a kinetically sorbing compound in a spatially uniform flow field undergoing sinusoidal fluctuations in time. This expression may be used for reactive transport calculations in an equivalent domain with constant flow. The effects are the strongest for compounds with a dimensionless partitioning coefficient of about unity, slow sorption kinetics, and slowly fluctuating velocities. For realistic parameters, kinetic sorption contributes to transverse mixing in the same range as heterogeneity.     

Pore-scale modeling of dispersion in disordered porous media

We employ a direct pore-level model of incompressible flow that uses the modified moving particle semi-implicit (MMPS) method. The model is capable of simulating both unsteady- and steady-state flow directly in microtomography images of naturally-occurring porous media. We further develop this model to simulate solute transport in disordered porous media. The governing equations of flow and transport at the pore level, i.e., Navier-Stokes and convection-diffusion, are solved directly in the pore space mapped by microtomography techniques. Three naturally-occurring sandstones are studied in this work. We verify the accuracy of the model by comparing the computed longitudinal dispersion coefficients against the experimental data for a wide range of Peclet numbers, i.e., 5×10(-2)相似文献   

Modelling of diffusion-limited retardation of contaminants in hydraulically and lithologically nonuniform media

A new reactive transport modelling approach and examples of its application are presented, dealing with the impact of sorption/desorption kinetics on the spreading of solutes, e.g. organic contaminants, in groundwater. Slow sorption/desorption is known from the literature to be strongly responsible for the retardation of organic contaminants. The modelling concept applied in this paper quantifies sorption/desorption kinetics by an intra-particle diffusion approach. According to this idea, solute uptake by or release from the aquifer material is modelled at small scale by a "slow" diffusion process where the diffusion coefficient is reduced as compared to the aqueous diffusion coefficient due to (i) the size and shape of intra-particle pores and (ii) retarded transport of solutes within intra-particle pores governed by a nonlinear sorption isotherm. This process-based concept has the advantage of requiring only measurable model parameters, thus avoiding fitting parameters like first-order rate coefficients.In addition, the approach presented here allows for modelling of slow sorption/desorption in lithologically nonuniform media. Therefore, it accounts for well-known experimental findings indicating that sorptive properties depend on (i) the grain size distribution of the aquifer material and (ii) the lithological composition (e.g. percentage of quartz, sandstone, limestone, etc.) of each grain size fraction. The small-scale physico-chemical model describing sorption/desorption is coupled to a large-scale model of groundwater flow and solute transport. Consequently, hydraulic heterogeneities may also be considered by the overall model. This coupling is regarded as an essential prerequisite for simulating field-scale scenarios which will be addressed by a forthcoming publication.This paper focuses on mathematical model formulation, implementation of the numerical code and lab-scale model applications highlighting the sorption and desorption behavior of an organic contaminant (Phenanthrene) with regard to three lithocomponents exhibiting different sorptive properties. In particular, it is shown that breakthrough curves (BTCs) for lithologically nonuniform media cannot be obtained via simple arithmetic averaging of breakthrough curves for lithologically uniform media. In addition, as no analytical solutions are available for model validation purposes, simulation results are compared to measurements from lab-scale column experiments. The model results indicate that the new code can be regarded as a valuable tool for predicting long-term contaminant uptake or release, which may last for several hundreds of years for some lithocomponents. In particular, breakthrough curves simulated by pure forward modelling reproduce experimental data much better than a calibrated standard first-order kinetics reactive transport model, thus indicating that the new approach is of high quality and may be advantageously used for supporting the design of remediation strategies at contaminated sites where some lithocomponents and/or grain size classes may provide a long-term pollutant source.     

Successive identification of biodegradation rates for multiple sequentially reactive contaminants in groundwater.

At the field scale, the biodegradation rate is usually estimated from analytical solutions to single species transport with first-order reactions, using measured data as input. Because many contaminants, e.g., chlorinated solvents, are degraded in a sequential pattern, with degradation products further reacting to produce new species, it is of great interest to quantify the transformation rate of every reaction. The conventional inverse solutions for identifying the transformation rates are limited to single species problems. In the present study, we propose a successive optimization approach to identify the biodegradation rate for each species by using a previously developed analytical solution to multi-species first-order reactive transport using data obtained at the field scale. By specifying a link between analytical solutions to sequentially reactive transport problems and optimization methods and assuming constant transport parameters (velocity, dispersivities, and retardation factors), the first-order transformation rates are optimized successively from parent species to its daughter species.     

Flow and transport in fractured tuff at Yucca Mountain: numerical experiments on fast preferential flow mechanisms

Recent discovery of bomb-related ³⁶Cl at depth in fractured tuff in the unsaturated zone at the Yucca Mountain candidate high-level waste (HLW) repository site has called into question the usual modeling assumptions based on the equivalent continuum model (ECM). A dual continuum model (DCM) for simulating transient flow and transport at Yucca Mountain is developed. In order to ensure properly converged flow solutions, which are used in the transport simulation, a new flow solution convergence criteria is derived. An extensive series of simulation studies is presented which indicates that rapid movement of solute through the fractures will not occur unless there are intense episodic infiltration events. Movement of solute in the environs of the repository is enhanced if the properties of the tuff layer at the repository horizon are modified from current best-estimate values. Due to a large advective–dispersive coupling between the matrix and fractures, the matrix acts as a major buffer which inhibits rapid transport along the fractures. Consequently, fast movement of solutes through the fractures to the repository depth can only be explained if the matrix–fracture coupling term is significantly reduced from a value that would be calculated on the basis of data currently available.