Simultaneous estimation of effective and apparent diffusion coefficients in compacted bentonite

Effective diffusion coefficients (D(e)) are usually measured by means of "through-diffusion" experiments in which steady state is reached, and the "time-lag" methods are used to estimate the apparent diffusion coefficient (D(a)). For sorbing radionuclides (as caesium), the time needed to reach steady-state conditions is very large, and the precision in D(a) determinations is not satisfactory. In this paper, a method that allows determining simultaneously effective and apparent diffusion coefficients in compacted bentonite without reaching steady-state conditions is described. Basically, this method consists of an "in-diffusion" experiment in which the concentration profile in the bentonite sample is used to estimate D(a), and the temporal evolution of the solute concentration in the reservoir is used to estimate D(e). This method has several advantages over the typical "through-diffusion" experiments, in particular: (a) experiment duration is significantly shorter, (b) D(a) values are measured with greater precision and (c) it is not necessary to maintain a constant solute concentration in the reservoir. This new method has been used to estimate the effective and apparent diffusion coefficients for caesium in FEBEX bentonite and in order to validate it, the results have been compared with results previously obtained with standard methods. Experimental results have been satisfactorily modelled using a simple model of diffusion in porewater and the measured value of D(e)(Cs) is very similar to D(e)(HTO) in the same bentonite. There is no evidence of "surface diffusion" in FEBEX bentonite for caesium.     

Study on anisotropy of effective diffusion coefficient and activation energy for deuterated water in compacted sodium bentonite

To quantify the effects of temperature on the diffusivity of deuterated water (HDO) in compacted sodium bentonite, through-diffusion experiments were conducted at elevated temperatures ranging from 298 to 333 K. Kunipia F (Na-montmorillonite content>98 wt.%; Kunimine Industries) was compacted to a dry density of 0.9 or 1.35 Mg/m(3). As montmorillonite particles were oriented perpendicular to the direction of compaction, the anisotropy of diffusivity was investigated both parallel and normal to the preferred orientation of the montmorillonite. The effective diffusion coefficient D(e) of HDO was larger when the diffusional direction was parallel as opposed to normal to the preferred orientation for both dry densities. The magnitude of D(e) and the anisotropy for HDO were in good accordance with previously reported results for tritiated water at room temperature. Activation energies of D(e) were isotropic and increased with increasing dry density over the range of 19-25 kJ/mol. This relationship was considered to be due to both pore structure development and the high activation energy of water near the montmorillonite surface.     

Apparent diffusion coefficients and chemical species of neptunium (V) in compacted Na-montmorillonite

Diffusion of neptunium (V) in compacted Na-montmorillonite was studied through the non-steady state diffusion method. In this study, two experimental attempts were carried out to understand the diffusion mechanism of neptunium. One was to establish the diffusion activation energy, which was then used to determine the diffusion process in the montmorillonite. The other was the measurement of the distribution of neptunium in the montmorillonite by a sequential batch extraction. The apparent diffusion coefficients of neptunium in the montmorillonite at a dry density of 1.0 Mg m-3 were from 3.7 x 10(-12) m2 s-1 at 288 K to 9.2 x 10(-12) m2 s-1 at 323 K. At a dry density of 1.6 Mg m-3, the apparent diffusion coefficients ranged between 1.5 x 10(-13) m2 s-1 at 288 K and 8.7 x 10(-13) m2 s-1 at 323 K. The activation energy for the diffusion of neptunium at a dry density of 1.0 Mg m-3 was 17.5 +/- 1.9 kJ mol-1. This value is similar to those reported for diffusion of other ions in free water, e.g., 18.4 and 17.4 kJ mol-1 for Na+ and Cl-, respectively. At a dry density of 1.6 Mg.m-3, the activation energy was 39.8 +/- 1.9 kJ mol-1. The change in the activation energy suggests that the diffusion process changes depending on the dry density of the compacted montmorillonite. A characteristic distribution profile was obtained by the sequential extraction procedure for neptunium diffused in compacted montmorillonite. The estimated fraction of neptunium in the pore water was between 3% and 11% at a dry density of 1.6 Mg m-3 and at a temperature of 313 K. The major fraction of the neptunium in the montmorillonite was identified as neptunyl ions sorbed on the outer surface of the montmorillonite. These findings suggested that the activation energy for diffusion and the distribution profile of the involved nuclides could become powerful parameters in understanding the diffusion mechanism.     

Sensitivity analysis of radionuclide migration in compacted bentonite: a mechanistic model approach

Mechanistic model calculations for the migration of Cs, Ra, Am and Pb in compacted bentonite have been carried out to evaluate sensitivities with respect to different parameter variations. A surface chemical speciation/electric double layer model is used to calculate: (i) porewater composition and radionuclide speciation in solution and at the bentonite surface, yielding the distribution of mobile and sorbed species and (ii) interaction of diffusing species with negatively charged pore walls to obtain diffusion parameters. The basic scenario considers the interaction of compacted bentonite with a fresh-type groundwater; variations include the presence of bentonite impurities and saline groundwater. It is shown that these scenarios result in significant variations of porewater composition that affect migration via three mechanisms that can partly compensate each other: (1) effects on sorption through radionuclide complexation in solution, and competition of major cations for surface sites; (2) changes in radionuclide solution speciation leading to different diffusing species under different conditions; (3) effects on diffusion through changes in the electric double layer properties of the clay pores as a function of ionic strength.     

An axisymmetric diffusion experiment for the determination of diffusion and sorption coefficients of rock samples

Diffusion anisotropy is a critical property in predicting migration of substances in sedimentary formations with very low permeability. The diffusion anisotropy of sedimentary rocks has been evaluated mainly from laboratory diffusion experiments, in which the directional diffusivities are separately estimated by through-diffusion experiments using different rock samples, or concurrently by in-diffusion experiments in which only the tracer profile in a rock block is measured. To estimate the diffusion anisotropy from a single rock sample, this study proposes an axisymmetric diffusion test, in which tracer diffuses between a cylindrical rock sample and a surrounding solution reservoir. The tracer diffusion between the sample and reservoir can be monitored from the reservoir tracer concentrations, and the tracer profile could also be obtained after dismantling the sample. Semi-analytical solutions are derived for tracer concentrations in both the reservoir and sample, accounting for an anisotropic diffusion tensor of rank two as well as the dilution effects from sampling and replacement of reservoir solution. The transient and steady-state analyses were examined experimentally and numerically for different experimental configurations, but without the need for tracer profiling. These experimental configurations are tested for in- and out-diffusion experiments using Koetoi and Wakkanai mudstones and Shirahama sandstone, and are scrutinized by a numerical approach to identify favorable conditions for parameter estimation. The analysis reveals the difficulty in estimating diffusion anisotropy; test configurations are proposed for enhanced identifiability of diffusion anisotropy. Moreover, it is demonstrated that the axisymmetric diffusion test is efficient in obtaining the sorption parameter from both steady-state and transient data, and in determining the effective diffusion coefficient if isotropic diffusion is assumed. Moreover, measuring reservoir concentrations in an axisymmetric diffusion experiment coupled with tracer profiling may be a promising approach to estimate of diffusion anisotropy of sedimentary rocks.     

An alternate pseudospectral model for pollutant transport,diffusion and deposition in the atmosphere

A numerical model of atmospheric transport, diffusion and deposition has been developed for pollutants which are emitted into atmospheric boundary layer with upper boundary condition at the inversion layer and lower boundary condition at the surface. The numerical schemes used in the model are derived from the use of finite Fourier transform. However, instead of using FFT, alternate pseudospectral schemes for evaluations of space derivative terms in the model equation with the settling velocity are presented.The model involves the evaluation of a polynomial function for non-periodic boundary conditions in order to apply the alternate pseudospectral schemes to the periodic function. The numerical results were discussed for different profiles of vertical eddy diffusivity and for different absorptive deposition conditions at the ground.     

Experimental and modeling studies on sorption and diffusion of radium in bentonite

The sorption and desorption behavior of radium on bentonite and purified smectite was investigated as a function of pH, ionic strength and liquid to solid ratio by batch experiments. The distribution coefficients (Kd) were in the range of 10(2) to > 10(4) ml g-1 and depended on ionic strength and pH. Most of sorbed Ra was desorbed by 1 M KCl. The results for purified smectite indicated that Ra sorption is dominated by ion exchange at layer sites of smectite, and surface complexation at edge sites may increase Ra sorption at higher pH region. Reaction parameters between Ra and smectite were determined based on an interaction model between smectite and groundwater. The reaction parameters were then used to explain the results of bentonite by considering dissolution and precipitation of minerals and soluble impurities. The dependencies of experimental Kd values on pH, ionic strength and liquid to solid ratio were qualitatively explained by the model. The modeling result for bentonite indicated that sorption of Ra on bentonite is dominated by ion exchange with smectite. The observed pH dependency was caused by changes of Ca concentration arising from dissolution and precipitation of calcite. Diffusion behavior of Ra in bentonite was also investigated as a function of dry density and ionic strength. The apparent diffusion coefficients (Da) obtained in compacted bentonite were in the range of 1.1 x 10(-11) to 2.2 x 10(-12) m2 s-1 and decreased with increasing in dry density and ionic strength. The Kd values obtained by measured effective diffusion coefficient (De) and modeled De were consistent with those by the sorption model in a deviation within one order of magnitude.     

Efficient calculation of sensitivity coefficients for complex atmospheric models

Sensitivity coefficients describe how the solution to a set of equations depends on the parameters in the problem. We discuss here the direct method of calculating these coefficients for atmospheric models based on nonlinear, partial differential equations. A detailed analysis indicates for which models the method will be efficient and shows that calculating several sensitivity coefficients simultaneously improves the efficiency. A technique is given to derive an algorithm for implementing the direct method from the algorithm for solving the governing equations of the model. As a realistic test of the method, we have computed first-order sensitivity coefficients for a photochemical model describing the transport and chemical reactions of pollutants. In this example the direct method has a limiting speed six times faster than the indirect method.     

An evaluative model for lead distribution in roadside ecosystems

In roadside ecosystems lead concentrations in different environmental materials decrease with the distance from the road, following a typical pattern. A model that expresses this distribution pattern is proposed to evaluate lead contamination in these ecosystems.     

Measuring partition and diffusion coefficients for volatile organic compounds in vinyl flooring

Interactions between volatile organic compounds (VOCs) and vinyl flooring (VF), a relatively homogenous, diffusion-controlled building material, were characterized. The sorption/desorption behavior of VF was investigated using single-component and binary systems of seven common VOCs ranging in molecular weight from n-butanol to n-pentadecane. The simultaneous sorption of VOCs and water vapor by VF was also investigated. Rapid determination of the material/air partition coefficient (K) and the material-phase diffusion coefficient (D) for each VOC was achieved by placing thin VF slabs in a dynamic microbalance and subjecting them to controlled sorption/desorption cycles. K and D are shown to be independent of concentration for all of the VOCs and water vapor. For the four alkane VOCs studied, K correlates well with vapor pressure and D correlates well with molecular weight, providing a means to estimate these parameters for other alkane VOCs. While the simultaneous sorption of a binary mixture of VOCs is non-competitive, the presence of water vapor increases the uptake of VOCs by VF. This approach can be applied to other diffusion-controlled materials and should facilitate the prediction of their source/sink behavior using physically-based models.     

An analytical diffusion model for long distance transport of air pollutants

A steady-state two-dimensional diffusion model suitable for predicting ambient air pollutant concentrations averaged over a long time period (e.g., month, season or year) and resulting from the transport of pollutants for distances greater than about 100 km from the source is described. Analytical solutions are derived for the primary pollutant emitted from a point source and for secondary pollutant formed from it. Depletion effects, whether due to wet or dry deposition or chemical conversion to another species, are accounted for in these models as first order processes. Thus, solutions for multiple point sources may be superimposed.In this model the time-averaging of the random trajectories of pollutant-contaminated air parcels is represented by horizontal diffusion in a steady, two-dimensional flow field of the time-averaged wind. The resulting concentration isopleths for a point source show significant dispersion both upwind and cross wind of the source with respect to the mean wind field.The analytical theory for the dispersion of a primary pollutant is compared with the numerical predictions of a plume trajectory model for the case of steady emission from a point source. Good overall agreement between the two models is achieved whether or not depletion by wet and dry deposition is included.The theory for the dispersion of a secondary pollutant is compared with measurements of the annual average sulfate concentration in the U.S. Calculations are carried out using SO₂ emissions from electric power plants in the United States as a source inventory. Using optimum values of the dispersion parameters, the correlation coefficient of observed and calculated ambient concentrations is 0.87 for the eastern United States and 0.69 for the western region. The optimum dispersion parameters used are comparable to values quoted in the literature.The horizontal length scale characterizing the sulphate concentration distribution from a single source is about 500 km, being noticeably larger than that characterizing the primary (sulfur dioxide) distribution. Using optimum dispersion parameters, a point source of 33 kg s⁻¹ of sulfur dioxide would give rise to a maximum annual average sulfate concentration of 1 μ m⁻³.A calculation of annual average SO₂ concentrations in the United States is carried out using previously derived optimal values of the parameters from the sulfate calculation. The resulting isopleths are similar to measured values in the eastern U.S.     

Measurement of effective air diffusion coefficients for trichloroethene in undisturbed soil cores

In this study, we measure effective diffusion coefficients for trichloroethene in undisturbed soil samples taken from Picatinny Arsenal, New Jersey. The measured effective diffusion coefficients ranged from 0.0053 to 0.0609 cm2/s over a range of air-filled porosity of 0.23-0.49. The experimental data were compared to several previously published relations that predict diffusion coefficients as a function of air-filled porosity and porosity. A multiple linear regression analysis was developed to determine if a modification of the exponents in Millington's [Science 130 (1959) 100] relation would better fit the experimental data. The literature relations appeared to generally underpredict the effective diffusion coefficient for the soil cores studied in this work. Inclusion of a particle-size distribution parameter, d10, did not significantly improve the fit of the linear regression equation. The effective diffusion coefficient and porosity data were used to recalculate estimates of diffusive flux through the subsurface made in a previous study performed at the field site. It was determined that the method of calculation used in the previous study resulted in an underprediction of diffusive flux from the subsurface. We conclude that although Millington's [Science 130 (1959) 100] relation works well to predict effective diffusion coefficients in homogeneous soils with relatively uniform particle-size distributions, it may be inaccurate for many natural soils with heterogeneous structure and/or non-uniform particle-size distributions.     

Consistent interpretation of the results of through-, out-diffusion and tracer profile analysis for trace anion diffusion in compacted montmorillonite

Literature data for anion diffusion in compacted swelling clays contain systematic inconsistencies when the results of through-diffusion tests are compared with those of out-diffusion or tracer profile analysis. In the present work we investigated whether these inconsistencies can be explained by taking into account heterogeneities in the compacted samples; in particular increased porosities at the clay boundaries. Based on the combined results of out-diffusion, tracer profile analysis and the spatial distribution of the electrolyte anion in the clay, we conclude that the inconsistencies can indeed be resolved by taking into account a heterogeneous distribution of the total and the anion-accessible porosity. This, by definition, leads to a position dependence of the effective diffusion coefficient. Neglecting these effects results in a rather subordinate systematic error in the determination of effective diffusion coefficients of anions from through-diffusion tests with clay thicknesses in the centimetre range. However, stronger errors in terms of absolute values and conceptual interpretation may be introduced in out-diffusion tests and profile analyses of the diffused tracer. We recommend that anion diffusion tests should be accompanied by measurements of the total and anion-accessible porosity as a function of position in the direction of diffusion.     

Determination of time-dependent partition coefficients for several pesticides using diffusion theory

Diffusion-retarded partitioning of pesticides with aggregated soils results in a time-dependent partition coefficient (Kd') which is different at equilibrium from the partition coefficient derived from conventional 24-h batch studies (Kd) measured on dispersed soil. An experiment was undertaken to determine the importance of Kd' for the prediction of pesticide concentrations in solutions bathing artificial soil aggregates and to determine whether diffusion theory could accurately predict the concentrations. Two clay soils were mixed with polyacrylamide to create artificial aggregates of 0.8, 1.4 and 1.7 cm diameter when dry. After saturation, the aggregates were immersed in solutions containing isoproturon or a mixture of isoproturon, chlorotoluron and triasulfuron. The decline with time of the pesticide concentrations in the bathing solution was monitored and the results were compared with predictions from a diffusion-based model. The effective diffusion coefficients of the compounds were obtained by either fitting the non-linear diffusion model to the data (D(ef)) or by independent calculations based on the properties of the compounds and of the aggregates (D(ec)). The diffusion model was able to predict the temporal variation in pesticide concentrations in the bathing solution reasonably well whether D(ef) or D(ec) values were used. However, equilibrium concentrations in solution were sometimes overestimated due to increased sorption with time at the particle scale. Overall, the ratio between D(ef) and D(ec) ranged from 0.23 to 0.95 which was a reasonable variation when compared to the range of aggregate sizes used in the experiments and of the Kd values of the compounds.     

An adsorption diffusion model for removal of para-chlorophenol by activated carbon derived from bituminous coal

Batch adsorption experiments were carried out to study the adsorptive removal and diffusion mechanism of para-chlorophenol (p-CP) onto Calgon Filtrasorb 400 (F400) activated carbon. The external mass transfer resistance is negligible in the adsorption process carried out under different conditions in batch operation. Intraparticle diffusion model plots were used to correlate the batch p-CP adsorption data; three distinct linear sections were obtained for every batch operation. The textural properties of F400 activated carbon showed that it has a large portion of supermicropores, which is comparable to the size of the p-CP molecules. Due to the stronger interactions between p-CP molecules and F400 micropores, p-CP molecules predominantly diffused and occupied active sites in micropore region by hopping mechanism, and eventually followed by a slow filling of mesopores and micropores. This hypothesis is proven by the excellent agreement of the intraparticle diffusion model plots and the textural properties of F400 activated carbon.     

A turbulent energy model for diffusion in the convective boundary layer

A turbulent energy model developed by the authors to describe atmospheric flows is used to study diffusion in the convective boundary layer. The model is based on the turbulent energy transport equation coupled with eddy diffusivity expressions for momentum and heat transfer. The diffusion model assumes equality of the eddy diffusivity for heat and mass and Gaussian diffusion in the cross-stream direction. The model is shown to reproduce satisfactorily the main features of diffusion in convective flows, and its predictions compare well with the measurements of the laboratory experiments of Willis and Deardorff, as well as with field data.     

Tracer diffusion coefficients in sedimentary rocks: correlation to porosity and hydraulic conductivity

Matrix diffusion is an important transport process in geologic materials of low hydraulic conductivity. For predicting the fate and transport of contaminants, a detailed understanding of the diffusion processes in natural porous media is essential. In this study, diffusive tracer transport (iodide) was investigated in a variety of geologically different limestone and sandstone rocks. Porosity, structural and mineralogical composition, hydraulic conductivity, and other rock properties were determined. The effective diffusion coefficients were measured using the time-lag method. The results of the diffusion experiments indicate that there is a close relationship between total porosity and the effective diffusion coefficient of a rock (analogous to Archie's Law). Consequently, the tortousity factor can be expressed as a function of total porosity. The relationship fits best for thicker samples (> 1.0 cm) with high porosities (> 20%), because of the reduced influence of heterogeneity in larger samples. In general, these correlations appear to be a simple way to determine tortuosity and the effective diffusion coefficient from easy to determine rock porosity values.     

A simple model for diffusion of SO2 in Bergen

A model for computing daily mean SO₂ concentrations in Bergen, Norway, is developed and tested using SO₂ measurements from seven winter seasons during the 1970s. The meteorological predictors used are daily mean temperature, wind speed and temperature at two levels. Source strength is estimated from daily mean temperature. Correlations between observed and estimated SO₂ concentrations ranged between 0.8 and 0.9. Observed SO₂ levels declined by about 40 % from beginning to end of the observation period due to decreased SO₂ emission.