Diffusive flux and pore anisotropy in sedimentary rocks

Diffusion of dissolved contaminants into or from bedrock matrices can have a substantial impact on both the extent and longevity of dissolved contaminant plumes. For layered rocks, bedding orientation can have a significant impact on diffusion. A series of laboratory experiments was performed on minimally disturbed bedrock cores to measure the diffusive flux both parallel and normal to mineral bedding of four different anisotropic sedimentary rocks. Measured effective diffusion coefficients ranged from 4.9×10(-8) to 6.5×10(-7)cm(2)/s. Effective diffusion coefficients differed by as great as 10-folds when comparing diffusion normal versus parallel to bedding. Differences in the effective diffusion coefficients corresponded to differences in the "apparent" porosity in the orientation of diffusion (determined by determining the fraction of pore cross-sectional area measured using scanning electron microscopy), with the difference in apparent porosity between normal and parallel bedding orientations differing by greater than 2-folds for two of the rocks studied. Existing empirical models failed to provide accurate predictions of the effective diffusion coefficient in either bedding orientation for all four rock types studied, indicating that substantial uncertainty exists when attempting to predict diffusive flux through sedimentary rocks containing mineral bedding. A modified model based on the apparent porosity of the rocks provided a reasonable prediction of the experimental diffusion data.     

Modeling the Reduction of Vapor Phase Emissions from Surface Soils Due to Soil Matrix Effects: Porosity/Tortuosity Concepts

ABSTRACT

A major route for transport of volatile organic compounds within porous media is vapor phase diffusion. The diffusion rate through a porous medium is less than that through free-air due to the decreased cross-sectional area available for gas movement and the increased path length due to pore tortuosity. Numerous empirical expressions have been published that relate the diffusion coefficient in porous media to the diffusion coefficient in free-air (unobstructed gas phase). Published measurements of relative diffusivity and air-filled porosity were combined into a database. Empirical expressions available in the literature, including the popular Millington-Quirk equation, were evaluated along with a fourth-degree polynomial expression developed by the authors to determine the best type of equation to predict relative diffusivity as a function of air-filled porosity over the domain of values for porosity ranging from 0.071 to 1 for different types of materials. Mean square deviations were used as the statistical test to compare equations. The polynomial expression developed in this project produced a significantly different effective diffusion coefficient (1.3 x 10^-6 m²/sec) compared to values of 9.2 x 10^-6 m²/sec and 3.1 x 10^-6 m²/ sec predicted by forms of the Millington-Quirk equation for a specific case.     

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.     

Volatilisation of o-xylene from sandy soil

The diffusive release of o-xylene from two soils with different contents of organic carbon (1.1 % and 0.11 % TOC) and with two different water contents (app. 5 % w/w and 15 % w/w was studied in the laboratory. The soils were spiked with o-xylene in the laboratory. The fluxes were measured over a period of 24 hours. The measured fluxes were compared to predictions by two models. Model I, which is an analytical model, assumed instant local equilibrium between soil air, water and solids. The distribution coefficients were measured for the two soils, and Henry's constant and the diffusion coefficient in air were taken from the literature. This model overestimated the flux for o-xylene for all the tested combinations. The ratios between estimated and observed fluxes at 1 h were between 1.7 and 7.3. Model II assumed that the mass transfer of o-xylene between the solids and the water phase was kinetically controlled and was solved numerically. However, the predictions by the more advanced model were not significantly better than the prediction by the simple analytical model. The results indicate that prediction of o-xylene volatilisation from unsaturated soil is associated with substantial uncertainty.     

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.     

Modeling tracer diffusion in fractured and unfractured, unsaturated, porous media

A proposed tracer diffusion test for the Exploratory Shaft Facility at Yucca Mountain, NV, is modeled. For the proposed test, a solution containing conservative tracers will be introduced into a borehole in the geologic medium of interest. The tracers will diffuse and advect from the saturated source region into the unsaturated matrix in the surrounding tuff. After some time, the borehole is to be overcored, and tracer concentrations in the fluid will be measured in the core as a function of distance from emplacement. The data will be used to evaluate diffusive behavior and to derive effective diffusion coefficients for the tracers in the specific tuff. Numerical simulations are used to study the effects of effective diffusion coefficient, porosity, saturation, and fracturing on tracer transport. Results are reported for numerical simulations of tests in the Topopah Spring Member and the Tuff of Calico Hills, which have significantly different porosities and saturations. The simulations make the following predictions: The spread of tracer during the test will be sensitive to the effective diffusion coefficient of the tracer. Tracer will diffuse farther in the Topopah Spring Member than in the Tuff of Calico Hills because of the former's lower porosity and saturation. Tracer transport by advection into the Topopah Spring Member will be greater than that into the Tuff of Calico Hills because of capillary effects. While advection will be a significant mechanism for tracer penetration into the Topopah Spring tuff, it will be less significant for tracer penetration into the Calico Hills tuff. The proximity of a single vertical fracture to the source region determines its effects on tracer transport, especially if the fracture diverts fluid flowing from the source region into the matrix.     

Determination of spatially-resolved porosity, tracer distributions and diffusion coefficients in porous media using MRI measurements and numerical simulations

This work is focused on measuring the concentration distribution of a conservative tracer in a homogeneous synthetic porous material and in heterogeneous natural sandstone using MRI techniques, and on the use of spatially resolved porosity data to define spatially variable diffusion coefficients in heterogeneous media. The measurements are made by employing SPRITE, a fast MRI method that yields quantitative, spatially-resolved tracer concentrations in porous media. Diffusion experiments involving the migration of H(2)O into D(2)O-saturated porous media are conducted. One-dimensional spatial distributions of H(2)O-tracer concentrations acquired from experiments with the homogeneous synthetic calcium silicate are fitted with the one-dimensional analytical solution of Fick's second law to confirm that the experimental method provides results that are consistent with expectations for Fickian diffusion in porous media. The MRI-measured concentration profiles match well with the solution for Fick's second law and provide a pore-water diffusion coefficient of 1.75×10(-9)m(2)s(-1). The experimental approach was then extended to evaluate diffusion in a heterogeneous natural sandstone in three dimensions. The relatively high hydraulic conductivity of the sandstone, and the contrast in fluid density between the H(2)O tracer and the D(2)O pore fluid, lead to solute transport by a combination of diffusion and density-driven advection. The MRI measurements of spatially distributed tracer concentration, combined with numerical simulations allow for the identification of the respective influences of advection and diffusion. The experimental data are interpreted with the aid of MIN3P-D - a multicomponent reactive transport code that includes the coupled processes of diffusion and density-driven advection. The model defines local diffusion coefficients as a function of spatially resolved porosity measurements. The D(e) values calculated for the heterogeneous sandstone and used to simulate diffusive and advective transport range from 5.4×10(-12) to 1.0×10(-10)m(2)s(-1). These methods have broad applicability to studies of contaminant migration in geological materials.     

Diffusive transport through compacted Na- and Ca-bentonite

The effect of exchangeable cation — Na⁺ and Ca ²⁺ — on the diffusive transport of I⁻, Sr ²⁺ and ³H (as HTO) in compacted bentonite was examined using a through-diffusion method. Total intrinsic diffusion coefficients, D_i, were determined from the steady-state flux of the diffusants through the clays, and apparent diffusion coefficients, D_a, were obtained from the time lag technique. The clays were compacted to a dry bulk density of 1.3 Mg/m³, and Na-bentonite was saturated with a solution of 100 mol NaCl/m3 and Ca-bentonite with one of 50 mol CaCl₂/m³. The D_i values for all diffusants are 2 to 6 times higher in the Ca- than Na-clay. We attribute this to the larger quasicrystal, or particle, size of Ca- compared to Na-bentonite. Hence, Ca-bentonite has a greater proportion of relatively large pores; this was confirmed by Hg intrusion porosimetry. This means the diffusion pathways in Ca-bentonite are less tortuous than those in Na-bentonite. Moreover, in some cases the effective porosity, or the porosity available for diffusive transport, may be greater in Ca-bentonite. The D_a values are inversely proportional to the distribution coefficients of the diffusants with the clays.     

Direct simulation of heterogeneous diffusion and inversion procedure applied to an out-diffusion experiment. Test case of Palmottu granite

An out-diffusion laboratory experiment using a non-reactive tracer was fitted using the Time Domain Diffusion (TDD) method. This rapid particle tracking method allows simulation of the heterogeneous diffusion based on pore-scale images and local values of diffusivities. The superimposed porosity and mineral 2D maps act as computation grids to condition diffusion pathways. We focused on a Palmottu granite sample, in which the connected pore space has a composite microstructure with cracks linking microporous minerals and is above the percolation threshold. Three main results were achieved: (i) When compared to the fitting obtained with one coefficient (best mean square residual R = 1.6 x 10(-2)), diffusion is shown to be suitably characterised with two coefficients related to cracks and microporous minerals (best R = 6.5 x 10(-4)), (ii) rather than imposing a local apparent diffusion coefficient D(a) independent of the local porosity Phi, a best fit is obtained by applying Archie's relationship D(a) = D(0) x G with G = Phi(m) to each pixel of the calculation grids (G is the geometry factor, D(0) is the diffusion coefficient in free fluid, and m is Archie's exponent), and (iii) the order of magnitude of the fitted diffusion coefficient or Archie's exponents (m=0 for microcracks and m=1.82 for microporous minerals) is physically realistic.     

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.     

Matrix diffusion coefficients in volcanic rocks at the Nevada test site: influence of matrix porosity, matrix permeability, and fracture coating minerals

Diffusion cell experiments were conducted to measure nonsorbing solute matrix diffusion coefficients in forty-seven different volcanic rock matrix samples from eight different locations (with multiple depth intervals represented at several locations) at the Nevada Test Site. The solutes used in the experiments included bromide, iodide, pentafluorobenzoate (PFBA), and tritiated water ((3)HHO). The porosity and saturated permeability of most of the diffusion cell samples were measured to evaluate the correlation of these two variables with tracer matrix diffusion coefficients divided by the free-water diffusion coefficient (D(m)/D*). To investigate the influence of fracture coating minerals on matrix diffusion, ten of the diffusion cells represented paired samples from the same depth interval in which one sample contained a fracture surface with mineral coatings and the other sample consisted of only pure matrix. The log of (D(m)/D*) was found to be positively correlated with both the matrix porosity and the log of matrix permeability. A multiple linear regression analysis indicated that both parameters contributed significantly to the regression at the 95% confidence level. However, the log of the matrix diffusion coefficient was more highly-correlated with the log of matrix permeability than with matrix porosity, which suggests that matrix diffusion coefficients, like matrix permeabilities, have a greater dependence on the interconnectedness of matrix porosity than on the matrix porosity itself. The regression equation for the volcanic rocks was found to provide satisfactory predictions of log(D(m)/D*) for other types of rocks with similar ranges of matrix porosity and permeability as the volcanic rocks, but it did a poorer job predicting log(D(m)/D*) for rocks with lower porosities and/or permeabilities. The presence of mineral coatings on fracture walls did not appear to have a significant effect on matrix diffusion in the ten paired diffusion cell experiments.     

Estimation of mechanical dispersion and dispersivity in a soil-gas system by column experiments and the dusty gas model

In a previous study, column experiments were carried out with Toyoura sand (permeability 2.05×10(-11)m(2)) and Toyoura sand mixed with bentonite (permeability 9.96×10(-13)m(2)) to obtain the molecular diffusion coefficient, the Knudsen diffusion coefficient, the tortuosity for the molecular diffusion coefficient, and the mechanical dispersion coefficient of soil-gas systems. In this study, we conducted column experiments with field soil (permeability 2.0×10(-13)m(2)) and showed that the above parameters can be obtained for both less-permeable and more-permeable soils by using the proposed method for obtaining the parameters and performing column experiments. We then estimated dispersivity from the mechanical dispersion coefficients obtained by the column experiments. We found that the dispersivity depended on the mole fraction of the tracer gas and could be represented by a quadratic equation.     

Sandstones of unexpectedly high diffusibility

Measurements have been made of diffusion coefficients (D(i)=-mass flux/concentration gradient) using a double reservoir, steady-state method with two tracers, CaBr(2) and amino-G-acid, on intact samples of Triassic red-bed sandstone from northwest England. Diffusibility (D'=D(i)/diffusion coefficient in water) averages 0.124, ranging between 0.075 and 0.215 (porosity 0.1 to 0.24), very similar for the two tracers. Implied tortuosities (actual path length/straight line length) average 1.21 (range 1.06 to 1.47), with constrictivities close to 1. In comparison with limited red-bed sandstone data from elsewhere, these D' values are up to 4 times greater, and tortuosity correspondingly lower. Re-interpretation of formation factor data from previous studies on shallow sandstone samples also from northwest England confirms that diffusibility is significantly higher in these sandstones than others from similar palaeoenvironment/stratigraphic units. The lower tortuosities appear to result from the relatively high permeability, open fabric of the rock, properties likely to be present in shallow sandstone systems used for water supply. It is concluded that diffusion rates may, in some shallow freshwater-containing continental sandstone systems, be significantly greater than is implied by estimates of sandstone diffusibility current in the literature.     

Influence of temperature on formaldehyde emission parameters of dry building materials

The diffusion coefficient, D, partition coefficient, K, and the initial volatile organic compounds (VOCs) in dry building materials, are the three key parameters used to predict the VOC emissions. D and K may be strongly affected by temperature. We have developed a new and simple method, the C-history method, to measure the diffusion coefficient, D and the partition coefficient, K of formaldehyde in dry building materials at temperatures of 18, 30, 40 and 50 °C. The measured variations of the diffusion coefficients and the partition coefficients with temperature for particle board, vinyl floor, medium- and high-density board are presented. A formula relating the partition coefficient and related factors is obtained through analysis. This formula can predict the partition coefficient in principle and provide an insight for fitting experimental data, and it agrees well with the experimental results.     

Contributions of advective and diffusive oxygen transport through multilayer composite caps over mine waste

The relative contributions of four mechanisms of oxygen transport in multilayer composite (MLC) caps placed over oxygen-consuming mine waste were evaluated using numerical and analytical methods. MLC caps are defined here as caps consisting of earthen and geosynthetic (polymeric) components where a composite barrier layer consisting of a geomembrane (1-2 mm thick polymeric sheet) overlying a clay layer is the primary barrier to transport. The transport mechanisms that were considered are gas-phase advective transport, gas-phase diffusive transport, liquid-phase advective transport via infiltrating precipitation and liquid-phase diffusive transport. A numerical model was developed to simulate gas-phase advective-diffusive transport of oxygen through a multilayer cap containing seven layers. This model was also used to simulate oxygen diffusion in the liquid phase. An approximate analytical method was used to compute the advective flux of oxygen in the liquid phase. The numerical model was verified for limiting cases using an analytical solution. Comparisons were also made between model predictions and field data for earthen caps reported by others. Results of the analysis show that the dominant mechanism for oxygen transport through MLC caps is gas-phase diffusion. For the cases that were considered, the gas-phase diffusive flux typically comprises at least 99% of the total oxygen flux. Thus, designers of MLC caps should focus on design elements and features that will limit diffusion of gas-phase oxygen.     

Volatilization of 1,2-dibromo-3-chloropropane (DBCP) from soils.

The volatilization of DBCP from soils, as affected by the soil characteristics and application techniques, was studied in a laboratory experiment. The volatilization rate of DBCP applied in water was higher from sandy and silty loam soils than from clay soil. Water added after DBCP application acted as a soil cover, decreasing the volatilization rate. The results obtained with DBCP application in hexane to air-dry soils, indicate that adsorption could be an important factor in reducing the volatilization losses. Diffusion coefficients were calculated from the volatilization parameters, by using a simplified relationship between volatilization losses and diffusion through soil.     

Determination and analysis of distribution coefficients of 137Cs in soils from Biscay (Spain)

The distribution coefficient of (137)Cs has been determined in 58 soils from 12 sampling points from Biscay by treating 10 g with 25 ml of an aqueous solution with an activity of 1765 Bq in the radionuclide, by shaking during 64 h and measuring the residual activity with a suitable detector. Soils were characterised by sampling depth, particle size analysis and the usual chemical parameters. Soils were thereafter treated to fix the chemical forms of (137)Cs speciation by successive extractions in order to determine fractions due to exchangeable, associated with carbonates, iron oxide and organic matter fractions, obtaining by difference the amount taken by the rest of the soil constituents. For this research, 16 soils from four points were selected from the previous samples. The greatest mean percentages of (137)Cs sorption were with the rest (69.93), exchangeable (13.17) and organic matter (12.54%) fractions. This paper includes also the calculation of partial distribution coefficients for chemical species as well as relations of distribution coefficients both among them and with soil parameters.     

Rate-limited mass transfer of octane, decane, and dodecane into nonionic surfactants solutions under laminar flow conditions

A key component to predicting the success of utilizing surfactants to enhance the removal of organic liquids from soil system is quantifying micellar solubilization kinetics. In this study, a flow reactor was employed to investigate the influence of surfactant ethoxylate chain length on the rates of solubilization of octane, decane, and dodecane in micellar solutions of a homologous series of purified dodecyl alcohol ethoxylates. Effluent concentration data were fit using a finite element model utilizing a linear-driving-force model to represent mass transfer at the interface. For flow rates between 0.1 and 2 ml min(-1), mass transfer coefficients ranged from 5 x 10(-8) to 7 x 10(-7)m s(-1) and did not vary in a systematic way with either solute structure or surfactant ethoxylate chain length and were lower than those found in pure water. Correlations developed for the Sherwood number based on diffusion coefficients of surfactant micelles containing organic material (organic-laden micelle) exhibit a velocity dependence similar to that found for systems based on aqueous diffusion. These results suggest that under gentle flowing conditions, the mass transfer is limited by diffusion of the organic-laden micelle. Although these trends are specific for this experimental system, the results demonstrate the importance of selecting the proper diffusion coefficient when modeling surfactant solubilization processes.     

RHIZOtest: A plant-based biotest to account for rhizosphere processes when assessing copper bioavailability

The ability of the free ion activity model (FIAM), the terrestrial biotic ligand model (TBLM), the diffusive gradients in thin films (DGT) technique and a plant-based biotest, the RHIZOtest, to predict root copper (Cu) concentration in field-grown durum wheat (Triticum turgidum durum L.) was assessed on 44 soils varying in pH (3.9-7.8) and total Cu (32-184 mg kg⁻¹). None of the methods adequately predicted root Cu concentration, which was mainly correlated with total soil Cu. Results from DGT measurements and even more so FIAM prediction were negatively correlated with soil pH and over-estimated root Cu concentration in acidic soils. TBLM implementation improved numerically FIAM prediction but still failed to predict adequately root Cu concentration as the TBLM formalism did not considered the rhizosphere alkalisation as observed in situ. In contrast, RHIZOtest measurements accounted for rhizosphere alkalisation and were mainly correlated with total soil Cu.