Diffusion of nonreactive and reactive solutes through fine-grained barrier materials

Diffusion of nonreactive and reactive solutes through fine-grained barrier materials

R. W. GILLHAM, M. J. L. ROBIN, AND D. J. DYTYNYSHYN Department of Earth Sciences, University of Waterloo, Waterloo, Ont., Canada N2L 3GI

AND

H. M. JOHNSTON Soils Section Division, Civil Research Department, Ontario Hydro Research Department, 800 Kipling Avenue,

Toronto, Ont., Canada M8Z 5S4 Received January 3, 1984 Accepted April 23, 1984

In many cases where fine-grained geologic materials are used as barriers to the migration of contaminated fluids, the principal mechanism of contaminant transport is molecular diffusion. Thus the effective molecular diffusion coefficient is the parameter of greatest importance when predicting migration rates and contaminant fluxes. Diffusion coefficients were measured for two non- reactive solutes ( 3 6 ~ 1 and 3 ~ ) and one reactive solute ( 8 5 ~ r ) in seven mixtures of bentonite and silica sand ranging from 0 to 100% bentonite by weight. Tortuosity factors were calculated from the results of the nonreactive diffusion experiments, and retardation factors for the reactive solute from measured distribution coefficient (Kd), bulk density, and porosity values. The results showed the diffusive transport of both the reactive and nonreactive solutes to be consistent with a Fickian diffusion equation. For practical purposes, and at the low values of bulk density used in the experiments, the effective diffusion coefficient of the reactive solute could be calculated with a reasonable degree of certainty from the measured retardation factor and an estimated value of tortuosity. The results showed that because of the interaction between the distribution coefficient, bulk density, and porosity, an increase in clay content beyond about 5-10% did not result in a further reduction of the diffusion coefficient of the reactive solute.

Key words: diffusion, adsorption, retardation, tortuosity, clay liners.

Dans de nombreux cas ou des matCriaux gCologiques B grains fins sont utilists comrne barrikre contre la migration de fluides contaminis, le mecanisme principal de transport des polluants est la diffusion molCculaire. Ainsi, le coefficient effectif de diffusion molCculaire est le paramktre le plus important dans la prediction des vitesses de migration et des flux de polluant. Des coefficients de difusion ont CtC mesurCs pour deux solutCs non-rCactifs ( 3 6 ~ 1 et 3 ~ ) et un solutC riactif (85Sr) dans sept mClanges de bentonite et sable de silice, la proportion de bentonite variant de 0 B 100% en poids. Les facteurs de tortuositC ont CtC calculCs B partir des rksultats des essais de diffusion avec les solutCs non-rkactifs, et les facteurs de retardement pour le solutC rCactif, B partir des mesures de coefficient de distribution (Kd), de densit6 et de porositk. Les rCsultats ont montrk que le transport par diffusion, aussi bien pour les solutCs non-rCactifs que pour le solutC rCactif, correspondait B 1'Cquation de diffu- sion de Fick. A des fins pratiques, et aux faibles valeurs de densit6 utiliskes dans les essais, le coefficient effectif de diffusion du solutC rCactif pouvait &tre calculC avec une fiabilitC acceptable B partir du facteur de retardement mesurC et d'une valeur estimCe de la tortuositC. Les rCsultats ont monk6 qu'en raison de l'interaction entre le coefficient de distribution, la densit6 et la porositC, une augmentation de la teneur en argile au deli2 d'environ 5 B 10% ne produisait pas de rkduction additionnelle du coefficient de diffusion du solute rCactif.

Mots elks: diffusion, adsorption, retardement, tortuositk, membranes en argile. [Traduit par la revu]

Can. Geotech. J. 21,541-550 (1984)

Introduction Fine-grained geological materials are finding in-

creased applications as barrier materials for preventing the migration of contaminants from waste management sites into local hydrogeologic regimes. Possible applica- tions include liners for waste lagoons and landfills, grout curtains, and barrier materials surrounding buried waste containers. Geologic barriers may be constructed from local clay materials; in the event that suitable materials are not available, they are frequently constructed by mixing bentonitic clay with local materials. Significant design characteristics include the hydraulic conductivity of the barrier material and the rate at which the contaminants of concern will be transmitted through the

barrier. If bentonitic clay is used in the construction of the barrier, the proportion of clay that must be mixed with the local materials to give the desired transport characteristics can be a very significant economic factor.

The licensing of waste management facilities generally requires a prediction of the effect of the facility on the environment. Where barriers are used as a significant component of the waste containment design, knowledge of the rate at which contaminants will migrate through the barrier is required. This of course requires an appreciation of the mechanisms of contaminant trans- port through fine-grained materials and an ability to predict contaminant fluxes through the barrier.

The recognized mechanisms that affect the transport

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542 CAN. GEOTECH. J. VOL. 21. 1984

of solutes through saturated geologic materials include transport as a result of the bulk motion of the fluid phase (advection); dispersive transport caused by velocity variations about the mean velocity, and by molecular diffusion; and geochemical retardation processes. Of these, there is a growing body of evidence to indicate that in fine-grained geologic materials the principal mechanism of solute transport is molecular diffusion (Desaulniers et al. 1981). In particular, Gillham and Cherry (1982a, b) show that if the average groundwater velocity is less than about 1.6 X lO-'Orn/s, then molecular diffusion would be the dominant transport mechanism. Assuming a hydraulic gradient of and a porosity of 0.4, this would correspond to a hydraulic conductivity in the order of 5 X lo-' m/s. Since natural clay materials have conductivity values in the range of about 5 X loW9 to 5 X 10-'Om/s, and the hydraulic conductivity of bentonite is generally less than about

m/s, it is reasonable to expect that the principal mechanism of contaminant transport through many liner materials will be molecular diffusion.

This paper describes a series of laboratory experi- ments that examined the diffusive transport of both nonreactive and reactive solutes through sand-bentonite mixtures. The specific objectives of the study were to determine the applicability of the existing diffusion theory for predicting the migration characteristics, and to examine the effects of changes in the bentonite content on the migration characteristics.

Theory The equation describing diffusive transport of a

nonreactive solute in a saturated porous medium is analogous to Fick's second law and, following Bear (1972), can be written in one-dimensional form as

where c is the concentration of solute in the solution phase (M/L~) , t is time (T), D is the effective diffusion coefficient of the solute in the porous medium ( L ~ / T ) , and x is the space coordinate (L). Although there are several empirical forms (Manheim 1970; Lerman 1978), the effective diffusion coefficient (D) is commonly related to the free-solution diffusion coefficient (Do) by

where T is the tortuosity of the medium. Bear (1972) suggests 0.67 as a reasonable value for T in granular materials, and, from a survey of measured values, Per- kins and Johnston (1963) suggest that T generally falls within a rather narrow range, from 0.5 to 0.8.

In the present study, diffusion coefficients were measured by bringing into contact two half-cells packed

with the soil material, one containing tracer and one tracer-free. After sufficient time had elapsed for a diffuse concentration profile to develop, the cells were separated and sectioned, and the concentration of the tracer in each section was measured. A solution to [ I ] was fitted "by eye" to the resulting data in order to determine the diffusion coefficient.

Provided the diffusion profile at the time of sectioning does not extend to the ends of the cells, the medium can be treated as infinite, in which case the initial and boundary conditions are

[3a] c(x,O) = co for x < 0, and c(x,O) = 0 for x > 0

and

[3b] c(-m,t)=co, and c(m,t)=O

where co is the initial concentration of solute in the "tagged" part of the porous medium. Solving [ l ] for these conditions gives

[4] c/co = 112 erfc (x/2<t)

If the diffusion profile extends to the end of the cell, another solution must be used. In this case, for a porous medium of length L, the applicable initial and boundary conditions are

[5a] c(x,O) = co for 0 < x < xo, and c = 0 forxo < x < L

and

dc [5b] -=O, at x = O and x = L

dx

The solution to [I.] is then given by (Carslaw and Jaeger 1959)

r l rx rlnxo x cos (T) sin (T) Note that [4] gives the plane through which diffusion occurs at x = 0, whereas the equivalent plane is situated at x = xo for [6].

The migration characteristics of solutes that undergo geochemical reactions in the porous medium can differ substantially from those of nonreactive solutes and thus for these solutes, [ l ] is not an appropriate model. Although there are several types of geochemical pro- cesses that can influence contaminant migration in geo- logic materials (Cheny et al. 1983), this study was restricted to those reactions that can be represented by a constant partitioning parameter, the distribution coefficient.

The distribution coefficient ( K d ) is defined as the concentration of solute in the solid phase divided by the concentration in the solution phase, and is applicable for

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{AM ET AL. 543

reactions that are instantaneous and reversible and in which there is a functional relationship between the solution- and solid-phase concentrations. If the functional relationship is linear through the origin, then

[7] i;/c = constant = Kd

where i; is the concentration of solute in the solid phase (MIM). Under these conditions, a retardation factor (R) can be defined by

where p is the bulk density of the porous medium (M/L~) and q is the porosity (L3/L3). It can further be shown that [ l ] becomes

where D, is the effective diffusion coefficient of the reactive solute in the porous medium ( L 2 / ~ ) and is given by

It should be noted that [9] and [I ] are identical in form and thus solutions of the equation for the diffusion of nonreactive solutes are equally applicable for the diffusion of reactive solutes.

For situations in which the Kd is very large, direct measurement of D, from diffusion cells requires very long equilibration times. Alternatively, the parameters of [lo] can all be estimated or measured with a reasonable degree of certainty. It is then apparent that [lo], when applicable, provides a convenient means of evaluating the effective diffusion coefficient for a retarded species.

The applicability of the Kd parameter is of course limited by the underlying assumptions and, because of its empirical nature, has several other difficulties associated with it. Many of these are discussed in Cherry et al. (1983) and in Reardon (1981). Nevertheless, it is generally viewed as a reasonable model for representing the sorption processes of trace constituents and has been applied to a wide range of solutes, including cationic radionuclides, trace organics, heavy metals, ammonium, calcium, magnesium, and phosphorous (Cherry et al. 1983).

The specific objectives of this study were to examine the applicability of [ l ] and [9] in materials of high clay content, to determine if [lo] provides reasonable estimates of the effective diffusion coefficients for reactive solutes in these types of materials, and to examine the effect of clay content on the effective diffusion coefficient. Chlorine-36 and tritium were used as nonreactive solutes and 85Sr as the reactive solute. The porous media included mixtures of silica sand and bentonitic clay. In general, the procedure involved the

measurement of diffusion coefficients for the three tracers in the sand-bentonite mixtures. These were compared with values calculated from [lo].

Methods of investigation Porous media

The porous media (soils) used in this study were seven mixtures of silica sand and bentonite clay. The silica sand was sieved and particles larger than 63 pm were retained and washed with deionized water. The bentonite clay was air-dried and passed through a 62.5 pm mesh. The sand-bentonite mixtures contained 0 ,5 , 10, 15,25,50, and 100% bentonite by oven-dried weight. The clay used was a sodium bentonite from the Avonlea mine in Saskatchewan, Canada. A 2:l solution-to-soil water extract of the clay contained 363, 20.3, 1602, and 2 1 pg /mL of Ca2+, ~ g ~ + , Na2+, and K+ , respectively.

Kd measurements Distribution coefficients for 8 5 ~ r were determined in

duplicate for all soils. The soil-to-solution ratios used were close to those of the diffusion experiments. The 85Sr solution used was a 1.15 X mol/L solution of strontium chloride. The appropriate amount of this solution was added to duplicate 10 g samples of soil and the mixture was allowed to equilibrate, with frequent stirring, for a period of 5 days in plastic beakers with tight-fitting lids. Removal of a portion of the solution phase from the clay-paste materials was problematic in that normal vacuum filtering and squeezing methods proved to be inadequate. Consequently, soil solution was removed by the filtering-centrifuge device shown schematically in Fig. 1. For the high clay content materials, centrifuging at 1 1 000 r/min (20 000 G) for 25 min generally gave about a 2-3 mL sample. One millilitre of the extracted soil solution was diluted with 4.0mL of deionized water. Three millilitres of the resulting solution was placed in a gamma radiation counter. The 8 5 ~ r activity obtained was expressed in counts per minute (cpm) per millilitre of soil solution. Kd values were then calculated from

where ci is the number of cpm/mL of stock solution added to the soil initially, c is the number of cpm/mL of equilibrium soil solution, V is the volume of " ~ r solution added, and M is the mass of oven-dried soil used.

DifSusion experiments Each diffusion cell consisted of two Plastipak 50 mL

plastic syringes with an inner diameter of 2.65 cm. The syringes were modified by removing the needle fittings and the ridge around the inner diameter to provide for easy sample extraction. When assembled, the two

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CAN. GEOTECH.

MODIFIED 5 0 m L DISPOSABLE SYRINGE

# 5 4 0 FILTER PAPER

2 5 4 c m x 2 . 2 2 c m RUBBER O-RING

0 . 4 5 m MEMBRANE FILTL'R PERFORATED NYLON PLUG

GLASS SCINTILLATION V I A L

PLEXIGLAS HOLDER

FIG. 1. Centrifugation device used for extracting water from soil for soluble cation and Kd determinations.

syringes were held together by machine screws passing through holes drilled in the finger grips. A Plexiglas plug, 2.0cm long, was placed at each end of the diffusion cell, and held in place by stainless steel pins passing through the syringe barrel. A schematic of an assembled diffusion cell is given in Fig. 2.

Tracer-free soils were prepared by adding a known volume of deionized water sufficient to saturate the soil. Tagged soils were saturated with solutions of either 3H, 3 6 ~ 1 , or 8 5 ~ r . The specific activity of the 3~ solution used was approximately 3.7 x lo5 Bq/L (10 pCi/L), and that of the 36Cl solution was approximately 9.25 X

lo5 Bq/L (25 pCi/L). The 85Sr solution used was the same as that used in the batch tests.

The tagged and tracer-free soil-solution mixtures were allowed to equilibrate for a minimum of 2 weeks in parafilm-covered plastic beakers. The soils were mixed with a spatula periodically to ensure a uniform distribution of the tracer.

Following equilibration, the moisture content of each soil was determined by drying small duplicate samples to a constant weight at 105°C. Half-cells were weighed and the soil was then packed to achieve as high a bulk density as possible. care was taken to ensure that the bulk density was the same for tagged and tracer-free half-cells of the same soil, and to avoid air bubbles within the packed soil. Once packed, each half-cell was reweighed, and the mass of the soil determined. Problems were encountered when packing the silica sand in that some solution would seep out between the plug and syringe wall. Attempts were made to correct this by using paraffin as a sealant.

The tracer-free and tagged half-cells were then brought together so that the soil surfaces made complete contact, and were held in place by bolts passing through the plastic grips. To reduce moisture loss during the

J. VOL. 21, 1984

MODIFIED 5 0 rnL DISPOSABLE SYRINGE PLEXIGLAS PLUG

SOIL f

STAINLESS STEEL RETAINING PIN

FIG. 2. Schematic of an assembled diffusion cell.

diffusion period, Dow Coming high-vacuum grease was added to the face of the moulded plastic grip of each syringe before assembly and the assembled cells were dipped in molten paraffin wax. After the wax hardened, the cells were placed horizontally on racks in a controlled-environment chamber at 22 f 1°C where a high humidity was maintained by trays of water in the chamber.

Estimates of the sectioning time (i.e., termination time of the diffusion experiment) were obtained from [ I ] using D = 9.1 X l ~ ~ c m ~ / s for both nonreactive solutes. The D, values for 85Sr were estimated from [lo] andsetting7 = 0.67, Do = 1.3 X 10-5cm2/s, q = 0.33, and using the bulk density values calculated for each soil and the Kd values obtained from the batch experiments. The time ( t ) required to obtain c/co = 0.9 at a distance (x) from the interface equal to 213 of the length of the half-cell was calculated from [8]. These values gave diffusion times that were too large, allowing the diffusion profiles to reach the ends of the cells. It was therefore necessary to use [6] for the data analysis.

At the end of the diffusion experiment (i.e., at the termination time) the half-cells were separated. Each half-cell was then placed in a specially designed jig and sectioned by removing the pins from the Plexiglas plug and pushing the plug outward with the aid of a long threaded bolt. The bolt, turned by hand, extruded 0.254 cm of soil per revolution. About 24 samples were collected from each half-cell, with the length of each section being either 0.254 or 0.508 cm.

The soil was sectioned using a stainless steel spatula run along the surface of the plastic grip. Samples were put into preweighed 20 mL glass scintillation vials, each containing 10 mL of water. For soils containing 85Sr, the vials contained 10 mL of 6 mol/L HC1 instead of water. Acid extraction was used to release s5Sr from the exchange sites, and, therefore, concentrations obtained for 85Sr in the extract were in terms of total 8 5 ~ r in the sample. Expressing the concentration of 85Sr in this manner is equivalent to expressing it in terms of 85Sr in solution only because Kd is assumed to be constant.

Two samples from each half-cell were taken to determine the moisture content. One s a m ~ l e was taken a few sections from the contact plane and the other was taken at the opposite end of the half-cell, adjacent to the

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GILLHAM ET AL. 545

plug. Following equilibration, the supernatant solution in each glass vial was decanted into a Sorvall poly- ethylene centrifuge tube and centrifuged at 15 000 r/min for 20min. Samples containing more than 25% bentonite usually required an additional decanting and centrifugation before an appropriate volume of clear solution could be recovered. After centrifuging, 5 mL of the supernatant solutions of the two nonreactive tracers were transferred, along with lOmL of scintillation cocktail, into glass scintillation vials. The vials were allowed to cool for a few hours in a Mark I11 Automatic Liquid Scintillation Counter (Model 68802) and then counted to 10 000 counts.

The "Sr was analysed by transferring 3.0mL of supernatant solutions to counting tubes and counting to 10000 counts, using a Nuclear-Chicago automatic gamma counter (Model 1085). The data for all solutes were then transformed to counts per minute per gram of soil, and plotted as a function of distance from the interface.

Data reduction In order to transform the data into normalized concen-

tration profiles, values of co, the initial concentration in the tagged half-cell, are required. There are several ways of obtaining co from the "absolute" concentration profiles. The usual and most straightforward way is either directly from the absolute concentrations at the smallest values of x (if conditions [3a] and [3b] are applicable) or to obtain and double the concentration at the interface. For a combination of reasons, neither method could be used consistently for all experiments. Most concentration profiles showed enough scatter at low values of x to make direct reading difficult and, in many experiments, diffusion extended to the ends of the diffusion cells, in which case the first method is not applicable. In addition to the diffusion profile reaching the ends of the cell, the concentration profiles in some diffusion cells were asymmetrical so that neither method was applicable. Consequently, co was obtained using a mass balance on the concentration profiles, using the equation

where n is the total number of sample sections analysed, no is the number of sections in the tagged half-cells, ci is the concentration in the ith section (in cpm/g of soil), and gi is the number of grams of soil in the ith section. The effective diffusion coefficients were then deter- mined by fitting [6] to the measured profiles.

Results and discussion Retardation parameters

The porosity and bulk density were determined for six diffusion cells of each soil mixture; duplicates for each of the three tracers. There was very little variability in

POROSITY 171

BULK DENSITY l p l

I I 0 M 40 60 80 100

% BENTONITE

FIG. 3. Porosity and bulk density of sand-bentonite mixtures versus bentonite content.

the data for a particular soil and thus the values were averaged and are plotted as a function of bentonite content in Fig. 3. The porosity decreased from about 0.34 at 0% bentonite to 0.30 at 5% and then increased to 0.84 at 100% bentonite. The initial decrease can be attributed to clay-size particles filling the pore space between the sand grains. As the proportion of clay increased, the porosity reached a minimum when all the large pores between the sand grains were occupied by the clay particles. The porosity increased again as the clay occupied more space than that provided by the large pores between the sand grains.

The bulk density values followed the opposite trend, in- creasing from about 1.64 g/cm3 at 0% bentonite to about 1.73 g/cm3 at 5%, and then decreasing to 0.39 g/cm3 at 100% bentonite.

The results of the 8 5 ~ r batch Kd determinations are given in Fig. 4 as a function of bentonite content. The values increased almost linearly with clay content, with values ranging from 0.29 mL/g at 0% bentonite to 20.85 mL/g at 100% bentonite.

Included in Fig. 4 are values of plq for the diffusion cells containing 85Sr and the calculated retardation factors (R). Equation [8] shows that the retardation factor is proportional to the distribution coefficient (Kd) and to the bulk density (p), but inversely proportional to the porosity (q). Although Kd increased by about 70 times as the bentonite content increased from 0 to loo%, the bulk density decreased and the porosity increased. The resulting bulk density to porosity ratios increased from about 5.0 at 0% bentonite to about 5.9 at 5% bentonite, then decreased to 0.45 at 100%. The resulting R values increased sixfold, from about 2.45 at 0% bentonite to 14.35 at 5% bentonite, and then decreased to a value of 10.46 at 100% bentonite.

It is interesting to note that in the 5-100% bentonite range, the bulk density to porosity ratio affected the R values to about the same degree as Kd. In fact, an

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546 CAN. GEOTECH. J . VOL. 21, 1984

increase of almost an order of magnitude in Kd was counterbalanced by the decrease in the bulk density to porosity ratio. For coarse-grained geological materials, it is frequently assumed that the bulk density to porosity ratio varies within relatively narrow limits (two to six). The above results show that this may not be the case for repacked fine-textured materials, and variations in the ratio may have a very significant effect on the retardation factor.

-

2 0 -

DISTRIBUTION COEFFICIENT L K ~ I

RETARDATON FACTOR I R l

a

BULK DENSITY/ WROSITYLp/?)

o 20 40 6 0 80 100

Diffusion of nonreactive tracers Tortuosity factors were calculated from the results of

the nonreactive solute diffusion experiments. The data for some of the diffusion cells were discarded when air entrapment at the interface was obvious. An example of a nonreactive diffusion profile is shown in Fig. 5 for tritium in the 50% bentonite mixture.

The experimental data showed a fair amount of scatter, particularly in the tagged half-cell, where the concentrations were higher. This trend was present in all diffusion cells, and it is difficult to assess whether it is an indication of the actual variation in concentration or is the result of the extraction and (or) analysis procedures. Generally, the data showed a high degree of symmetry about x = 0, c/co = 0.5, for all nonreactive diffusion cells.

The diffusion coefficient for each nonreactive diffusion cell was determined by superimposing several "calculated diffusion profiles" onto the experimental data. Calculated profiles were obtained from [I ] by varying the diffusion coefficient. The coefficient giving the best "eye-fit" of the experimental data was taken as

the "experimental" diffusion coefficient. More weight was placed on the data at positive values of x, where the scatter was usually less, than on the tagged half-cell side of the interface. This procedure is illustrated in Fig. 5 for tritium in the 50% bentonite mixture where a value of D = 1.1 X lop5 cm2/s gives the best fit.

Tortuosity factors were calculated for each nonreactive diffusion cell as the ratio of the experimental diffusion coefficient of the solute in the soil (D) to the molecular diffusion coefficient of the solute in free solution (Do) (see [2]). The values of Do used were 1.19 X cm2/s for 3 6 ~ 1 , which is the molecular diffusion coefficient of 0.1 mol/L CaC12 at 25°C (Handbook of Chemistry and Physics 1975) and 2.44 X 1 0 - ~ c m ~ / s for tritium (Klitzsche et al. 1976).

The values of the experimental diffusion coefficient and tortuosity factor obtained for each diffusion cell of the nonreactive solutes are presented in Table 1. In spite of the scatter in the data, the duplicate diffusion

the experimental diffusion coefficient, and thus of the tortuosity factors, carry an uncertainty that could be as high as k 10%.

The tortuosity factors for 3 6 ~ 1 first decreased from 0.84 to 0.59 as the clay content increased from 0 to 10% bentonite and then increased to 0.84 at 100%. The trend was similar for tritium. This behaviour is expected, if one views the tortuosity factor as the ratio of the net distance travelled in the porous medium to the length of the tortuous pathway followed by the molecules in the medium. At 0% bentonite, the molecules have to diffuse around the sand grains through fairly large pores. When a small amount of clay is added, the clay particles occupy the space between the sand grains and the length of the molecular pathway increases. As the clay content increases further, the effect of the sand grains decreases, and the path length decreases.

The similarity in the shapes of the measured and calculated diffusion profiles indicates that the diffusion of nonreactive solutes in clay materials of the type used in this study is consistent with the Fickian diffusion model. The observation that the trends were similar for the two nonreactive solutes as a function of clay content indicates that they behaved in a similar manner in the soil. Therefore, for these experimental conditions, phenomena such as anion exclusion, anion adsorption, or the exchange of 3H with bound water in the clay did not appear to have major effects on the diffusion of the nonreactive solutes. The fact that the tortuosity factors were usually lower for 3~ than for 3 6 ~ 1 may be due to the uncertainty in the free-solution diffusion coefficient (Do) for tritium.

% B E N T O N I T E coefficients seldom differed by more than 10%. Thus FIG. 4. Strontium-85 distribution coefficients, bulk density Table 1 includes only the mean value of the duplicates.

to porosity ratios, and retardation factors versus bentonite Because of the moderate insensitivity of the calculated content. diffusion profiles to small variations in D, the values of

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GILLHAM ET AL

1.12 r

-5 2 D = 1.5 x10 cm /s

* EXPERIMENTAL RESULTS

CALCULATED PROFILES

- 8 - 4 0 4 8 12

DISTANCE (cm)

FIG. 5. Diffusion data for 3~ in the 50% bentonite mixture.

TABLE 1. Experimental diffusion coefficients and tortuosity factors for the nonreacting solutes 3 6 ~ 1 and 3~

Chlorine-36 Tritium

Experimental Experimental diffusion diffusion

coefficient Tortuosity coefficient Tortuosity Soil cm2/s) factor (low5 cmvs) factor

0%* 1 .O 0.84 1.7 0.70 5 % 0.8 0.67 0.8 0.33

10% 0.7 0.59 0.9 0.36 15% 0.8 0.67 1 .O 0.41 25 % 0.8 0.67 1 .O 0.41 50% 0.9 0.76 1.1 0.45

100% 1 .O 0.84 1.2 0.49

*Percent bentonite in the bentonite - silica sand mixture.

Calculated and experimental strontium-85 diflusion co- eficients

The tortuosity factors obtained from the 3 6 ~ 1 and the 3~ diffusion experiments were used in [lo] to calculate values of the 8 5 ~ r diffusion coefficient (D,) for each soil. The other parameters used in [lo] were the measured porosity and bulk density in the 85Sr diffusion cells, the average Kd values obtained from the batch tests for each soil, and a value of 1.3 x cm2/s for the molecular diffusion coefficient of 8 5 ~ r in free solution. The tortuosity factors for 36Cl and for tritium, the retardation factors, and the resulting 8 5 ~ r diffusion coefficients are

plotted in Fig. 6 as a function of bentonite content. The calculated diffusion coefficient (D,) was influenced very strongly by the retardation factor (R), but somewhat less by the tortuosity factor. This was expected, since R was usually higher than 10, making 1/R less than 0.1, whereas the tortuosity factor was usually higher than 0.3 (see [lo]). It should be noted that the calculated values of D, carry, through the tortuosity factor, the uncertainties of the experimental diffusion coefficients of the non- reactive solutes.

An example of an experimental diffusion profile for "Sr is given in Fig. 7 for the 25% bentonite mixture.

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548 CAN. GEOTECH. J. VOL. 21, 1984

RETARDATION FACTOR I R I

TORTUOSITY (71

CALCULATED D,

- FROM 3 6 ~ 1

--- FROM ) H - 40 60

% BENTONITE

FIG. 6. Retardation factors, tortuosity factors, and calculated D, values for 8 5 ~ r versus bentonite content.

The same general observations may be made for these profiles as were made for those of the nonreactive solutes; that is, the scatter was usually more pronounced at the higher concentration values (tagged half-cells), and the data were usually fairly symmetrical about the point x = 0 , c/co = 0.5 for symmetrical cells. This symmetry is expected since the reaction term is assumed to be a function of neither concentration nor time.

Experimental values of the diffusion coefficient of

85Sr for each diffusion cell were obtained as for the nonreactive solutes by "eye-fitting" calculated curves to the experimental data. More weight was placed on the data at positive values of x. The experimental diffusion coefficient values thus carry an uncertainty of about 5-10% for the same reasons as explained for the nonreactive solutes. The calculated concentration profile judged to give the best fit to the data is given by the solid line in Fig. 7. Also presented in this figure are the concentration profiles obtained using each of the two calculated diffusion coefficients (dashed lines).

Figure 8 shows that the experimental and calculated diffusion coefficients behaved in a similar manner with increasing clay content. In all cases, the diffusion coefficients decreased from high values (greater than 20 x 10-7cm2/s) at 0% bentonite to a minimum of less than 5 X cm2/s between 5 and 10% bentonite. The diffusion coefficients then increased and leveled off between 50 and 100% bentonite. At 100% bentonite, the experimental D, value and the D, value calculated from the 3 6 ~ 1 tortuosity factors were about 7.0 and 8.0 X

cm2/s, respectively, and about 4.5 X cm2/s for the Ds calculated from the 3~ tortuosity factors.

Between 5 and 100% bentonite one would expect the diffusion coefficient to decrease as the clay content increased because the cation exchange capacity, and thus the Kd value of the porous material, would increase. However, because of the strong influences of bulk density and porosity on the retardation factor, the

-5 2 - "EXPERIMENTAL" DIFFUSION COEFFICIENT : 6.5 x 10 cm /s

--- DIFFUSION COEFFlClEM FROM 3 6 ~ ~ TORTUOSITY : 9.1 x 1c5 cm2/s

DIFFUSION COEFFICIENT FROM 3~ TORTUOSITY : 5.3 x 1 6 ~ cm2/s

E X P E R I M E N T A L RESULTS " r

-10 -6 -2 2 6 10 14 18 22

DISTANCE (cm)

FIG. 7. Diffusion data for "sr in the 25% bentonite mixture.

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GILLHAM ET AL. 549

CALCULATED 0. C3'c9 TORTUOSITYI

, I 0 20 40 60 80 100

% BENTONITE

FIG. 8. Strontium-85 diffusion coefficients: experimental and calculated using 36Cl and 3H tortuosities versus bentonite content.

opposite trend was observed. It would seem that, if a maximum s 5 ~ r retardation (i.e., a minimal diffusion coefficient) is desired, a mixture containing 5-10% bentonite would be preferable to either pure sand or a mixture containing more than 10% bentonite. Improved retardation at higher bentonite contents could only be realized in situations where higher degrees of compac- tion could be achieved.

Conclusions The results indicate that diffusive transport of both

nonreactive and reactive solutes in the clays of this study is consistent with the Fickian diffusion equation. The results further indicate that, for practical purposes, distribution coefficients ( K d ) provide a convenient and simple means of estimating the reaction tern of the diffusion equation for 8 5 ~ r in sand-bentonite mixtures. One would expect a similar behaviour for other solutes and soil materials in which the reaction with the porous medium can be described by a distribution coefficient.

The results also showed that a small amount of clay mixed with sand (5-10% bentonite) was more effective in retarding 85Sr than either pure silica sand or mixtures of higher clay content. Within the experimental constraints of this study, as the clay content increased from 5 to loo%, even though the soils were packed as tightly as possible by hand, the effect of increasing porosity and decreasing bulk density on the retardation

factor countered the effect of a ninefold increase in the distribution coefficient. This could have significant implications in the optimum design of contaminant barriers. There appears to be little to be gained by increasing the clay content above about lo%, in that minimum values of hydraulic conductivity would also be reached when the pore spaces between the sand grains are filled with clay. This may not be the case in situations where the barrier material is highly compacted.

The insensitivity of the calculated concentration profiles to variations in the diffusion coefficient was such that, in many cases, variations of as much as twofold still produced profiles giving reasonable fits to the experimental data. The tortuosity factors obtained from the nonreactive solute diffusion experiments, though variable, were generally close to 0.67, the commonly assumed value. It therefore appears that, for practical purposes, using the value 0.67 as a tortuosity factor, together with the free-solution diffusion coefficient and the retardation factor, could yield very satisfactory predictions of the effective diffusion coefficient. It remains to be shown that this would be the case in highly compacted soil materials.

Acknowledgements The authors wish to acknowledge the support of

Atomic Energy of Canada Limited for this study. The authors are grateful to Dr. W. E. Inniss and Dr. J. C. Carlson of the Department of Biology, University of Waterloo, for their assistance in several of the analytical aspects. Finally, the authors wish to thank Stephanie O'Hannesin, Marilyn Bisgould, and Nadia Bahar for their help in the preparation of the manuscript.

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