DIFFUSION IN SATURATED SOIL . I I : RESULTS

FOR COMPACTED CLAY

By Charles D. Shackelford,1 Associate Member, ASCE, and David E. Daniel,2 Member, ASCE

ABSTRACT: The effective diffusion coefficients, D*, of three anions (Br~, Cl~, and I~) and three cations (Cd2+, K+, and Zn2+) diffusing in two compacted clay soils, kaolinite and Lufkin clay, are measured. The ions are contained in a sim-ulated waste leachate. The effects of molding water content and method of com-paction on the measured D* values are evaluated for kaolinite. The calculated O* values varied between 4 X 10~10 m2/s and 2 X 10~9 m2/s and, based on the results for chloride diffusion in kaolinite, are relatively insensitive to molding water con-tent and compaction method. The measured D* values for Cl~ and Br~ in kaolinite are in excellent agreement with previous studies, but the D* values for the cations are relatively high. High D* values for the cations are attributed to nonlinear ad-sorption behavior at relatively high concentrations and to the possibility of chem-ical precipitation of the heavy metal species (Cd2+ and Zn2+). Also, D* values determined from reservoir concentrations typically are higher than D* values de-termined from soil concentration profiles.

INTRODUCTION

A companion paper (Shackelford and Daniel 1991) described the background and some methods for measuring the effective diffusion coefficients (D*) of in-organic contaminants diffusing in compacted clay soil. With respect to the design of waste containment barriers, several factors might influence the effective dif-fusion coefficient for compacted clay soil. For example, Mitchell et al. (1965) showed that the molding water content and method of compaction have a sig-nificant impact on the hydraulic conductivity of compacted clay. In addition, the transport of reactive solutes is affected by the adsorption behavior of soil. Since clay mineralogy affects the adsorption characteristics of clay soils [e.g., see Grimm (1953)], the soil mineralogy is expected to affect the diffusive transport of re-active solutes in compacted clay soil.

The purpose of the present paper is to present test data aimed at investigating the effects of molding water content and method of compaction on measured D* values for inorganic chemical species diffusing in compacted clay soil.

MATERIALS AND METHODS

Soils Two soils were used for this study: kaolinite and Lufkin clay. Kaolinite was

used because it is commercially available and because it has a relatively low cation exchange capacity (CEC). Lufkin clay is a naturally occurring smectitic soil. The CEC of Lufkin clay is five times greater than that of kaolinite due to the smectitic content of the Lufkin clay. Both soils have been used previously in laboratory hydraulic conductivity studies (Foreman and Daniel 1986). The properties of the soils are presented in Table 1.

'Asst. Prof., Dept. of Civ. Engrg., Colorado State Univ., Fort Collins, CO 80523. 2Assoc. Prof., Dept. of Civ. Engrg., Univ. of Texas, Austin, TX 78712. Note. Discussion open until August 1, 1991. Separate discussions should be sub-

mitted for the individual papers in this symposium. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on Au-gust 22, 1990. This paper is part of the Journal of Geotechnical Engineering, Vol. 117, No. 3, March, 1991. ©ASCE, ISSN 0733-9410/91/0003-0485/$1.00 + $.15 per page. Paper No. 25603.

485

TABLE 1. Physical and Chemical Properties of Soils

Property

(D Dominant clay mineral Specific gravity of solids Optimum water content (g/g) Maximum dry density (kN/m3) Liquid limit (g/g) Plasticity index (g/g) Particle size distribution

Silt and clay (

TABLE 2. Characteristics of Simulated Waste Leachate

Parameter (1)

Simulated Leachate

"Desired" values"

(2)

"Actual" values

(3)

Actual Inorganic Leachatesb

Representative range

(4)

Probable extremes

(5)

(a) Metals Concentration (mg/L)

Cadmium (Cd) Calcium (Ca) Potassium (K) Zinc (Zn)

562 (256) 200 (176) 391 (384) 327 (261)

546-602 185-208 369-407 301-374

0-2 100-3,000 200-2,000

0-100

0-17 5-4,800 3-3,770 0-1,000

(6) Nonmetals (ligands) Concentration (mg/L)

Bromine (Br) Chlorine (CI) Iodine (I) Sulfate (S02~)

799 (698) 355 (295)

1,269 (1,269) 480 (318)

645-1,012 331-448

1,089-1,567 442-500

_ 30-2,800

— 0-1,280

— 0-3,000

— 0-1,826

(c) Electrical Conductivity (EC)

^mhos/cm at 25° C pH

3,090-3,950 3.7-6.7

3,000-17,000 4-9 _

"Values in parentheses represent concentrations of principal ionic species: bromide (Br~); chloride (CI ); iodide (I ); free sulfate (SOl -); cadmium (Cd2+); calcium (Ca2+); potassium (K+); and zinc (Zn2+), for pH range of 3.0 to 6.8 at temperature of 25° C

Compilation based on data presented by Griffin et al. (1976), Freeze and Cherry (1979), and Daniel and Liljestrand (1984).

concentration of the principal ionic species is less than the corresponding total concentration due to the effects of complexation and/or incomplete dissociation of the salt. For example, chlorine (CI) exists as free chloride (Cl~) and com-plexed chloride (CdCl+ and ZnCl+) in the simulated leachate. The mobilities of these three species of chlorine undoubtedly are different. Nonetheless, it is com-mon practice to refer to the effective diffusion coefficients as being those of the principal ionic species when the values presented might actually encompass the mobilities of all of the chemical species present. This practice has been followed in this study.

The pH of the simulated leachate is reported as a range of values in Table 2. The pH of the leachate was adjusted to that of the soil solution before the start of the diffusion test in order to minimize the effects of pH on the sorption char-acteristics of the soil (Frost and Griffin 1977; "Batch-type" 1987).

Batch Equilibrium Tests Batch equilibrium tests were performed to determine the adsorption charac-

teristics of the soils with respect to the specified ions. Competition between the ions for the exchange sites of the soils was accounted for by using the simulated leachate instead of individual ion solutions.

A 1:4 soil: solution ratio (by weight), which is the highest recommended ratio ("Batch-type" 1987) was used in the batch equilibrium tests. This 1:4 ratio was maintained by adding 50 g (oven-dried basis) of soil and 200 g of solution into each of eight different 250-ml Erlenmeyer flasks. All flasks were stoppered, placed in an end-over-end rotary mixer, and mixed for 48 hours at a temperature of 23° C ± 2° C. At the end of the mixing period, samples of the soil-solution slurries from the flasks were poured into 50-ml centrifuge tubes, sealed, and centrifuged. The supernatant from each tube was then analyzed for equilibrium concentrations using ion chromatography (IC) for the nonmetals and flame atomic absorption spectroscopy (AA) for the metals.

The results of the chemical analyses were plotted as adsorption isotherms, or sorbed concentration, q, versus dissolved equilibrium concentration, c, for each ion. The sorbed concentrations were determined using the following equation

487

where C0 = the initial concentration of the specified ion in the flask, V = the volume of the solution, and ms = the soil mass.

Diffusion Tests

Sample Preparation The test specimens of clay were prepared by mixing air-dried soil with stan-

dard water (0.005M CaS04 solution) until the desired moisture content was ob-tained. After hydration, the soil was compacted into 102-mm-diameter molds, and the molds were assembled into the diffusion cells shown schematically in Fig. 1. A buret was used to provide volume change readings during both soaking and diffusion periods. Further details of the diffusion system are provided by Shackelford (1988) and Shackelford et al. (1989).

Soaking Stage The soil samples were soaked with standard water prior to the start of the

diffusion tests to minimize mass transport due to suction in the soil. The soils were soaked until the volume change readings from the buret (Fig. 1) were neg-ligible, indicating that equilibrium between the solution in the reservoir and the compacted soil had been established. After equilibrium was established, the soaking solution was withdrawn from the system, the cell was disassembled, and the soil, which had swelled, was trimmed flush with the top of the mold. After trimming, the cell was reassembled and the soaking solution reintroduced into the system so that equilibrium could be reestablished. The entire soaking pro-cedure lasted between 17 and 160 days depending primarily on the size of the mold (see Fig. 1). Further details of the soaking procedures are provided by Shackelford (1988) and Shackelford et al. (1989).

Diffusion Stage The diffusion stage of the tests was initiated by draining the soaking solution

through the sample port in the reservoir. Next, the pH of the leachate was ad-justed to that of the soaking solution by titrating the leachate with 0.1 M sulfuric acid (H2S04). The volume of sulfuric acid added to the leachate usually was 2.0 ml. Finally, samples of the simulated leachate were recovered for chemical analysis of the specified ions, and the leachate was introduced into the diffusion system. The diffusion tests were performed at tem-peratures ranging between 21° C and 25° C.

After the diffusion test was set up, the leachate concentration in the reservoir was monitored periodically by withdrawing samples through the sample port (Fig. 1). The leachate samples were analyzed for the specified ions to determine the variation in reservoir concentration with time.

Sectioning and Extraction Stage Upon completion of the diffusion stage of the test, which lasted from 30 to

109 days, the last reservoir samples were taken and the diffusion cell was dis-assembled. The final weight of the compaction mold plus the soil was measured. The soil samples were extruded and sectioned into slices approximately 0.254 cm (0.1 in.) in thickness to provide: (1) A distribution of water contents existing in the sample; and (2) a concentration profile of the specified cations for use in determinations of mass balance and effective diffusion coefficients. The water content of each slice of soil was determined by oven drying.

Ethylenediaminetetraacetate (EDTA) has been shown by Farrah and Pickering

488

Top Cap

116 mm or

58 mm

4 Tie Rods @ 90°

\ Leachate s \ Reservoir ^ Reservoir

Sample Port

O-Ring

Bottom Cap

?0-mL Buret

Stand PVC Tube

(b)

FIG. 1. Diffusion Test Apparatus: (a) Diffusion Cell; (b) Diffusion System

(1978) to be an effective chelating agent for the extraction of heavy metals (e.g., Cd2+ and Zn2+) from the exchange sites of clay minerals. However, Bohn et al. (1979) indicate that EDTA is a much less effective chelating agent for mon-ovalent cations, such as potassium. Since the primary emphasis of the study was to measure the effective diffusion coefficients of the heavy metal cations, ex-traction of the potassium cations was of secondary importance.

Initially, a one millimolar (1 mM) solution of H4EDTA was used as the cation-extracting solution. The pH of the solution was. approximately 2.8. However, the mass balance calculations for the initial tests witht kaolinite indicated poor efficiencies with respect to extraction of the cations (Cd2+, Zn2+, and K+). In

489

an attempt to improve the cation extraction efficiencies, the strength of the H4EDTA solution was increased to 5 mM and the pH was adjusted to 7.0 by titration with 1.0 M sodium hydroxide (NaOH) for the remaining tests with kaolinite as well as for the tests with Lufkin clay.

Initially, it was thought that the H4EDTA solution extracts could be used to determine the anion concentrations as well as the cation concentrations. How-ever, the H4EDTA solution was found to interfere with the ion chromatographic determination of the chloride and bromide concentrations. Therefore, a separate analysis using deionized, distilled water (DDW) as the extracting solution for the anion concentrations was made [e.g., see Gillham et al. (1984)]. The sep-arate analysis required that two centrifuge tubes be used—one for anions and one for cations.

Soil from each soil slice from the sectioning stage was added to a centrifuge tube with the appropriate extracting solution. The centrifuge tubes were sealed, placed in a rotary, end-over-end mixer, and mixed at 30 rpm for at least 48 hours. The tubes were then removed from the mixer and centrifuged for 30 min at 3,000-4,000 rpm. Finally, the supernatant from the centrifuge tubes was ana-lyzed for the individual ion concentrations (i.e., IC analysis for ion concentra-tions and AA analysis for cation concentrations).

The laboratory-measured concentrations are less than those existing in the soil due to dilution by the extracting solution. The total concentration of each chem-ical species, c', existing in the soil at the time the diffusion cell was disassem-bled can be estimated by multiplying the measured concentration, 'cm, by the inverse of the dilution factor, or

(WA C=C'\TJ (2) where Wsol = the weight of the extracting solution in the centrifuge tube and Ww = the weight of the water in the soil at the time of soil sectioning. Eq. 2 assumes that the densities of the extracting solution and water are equal. The concentration c' represents the total mass (adsorbed plus liquid phase) of the chemical species in terms of the pore water in the soil, assuming the extracting solution is 100% efficient, and is the concentration to be used to determine mass balances.

The soluble (nonadsorbed) concentration of the chemical species, c, in the pore water of the soil can be estimated by dividing the total concentration by the retardation factor, Rd, or

For nonadsorbing tracers, the retardation factor is 1.0.

DATA ANALYSIS

The effective diffusion coefficient (D*) for solutes diffusing in soil was de-fined in Shackelford and Daniel (1991) as

D* =D0ja (4)

where DQ = a free solution diffusion coefficient and ia = the apparent tortuosity factor. Two different analyses were used to determine D* values. The first anal-ysis utilized the reservoir concentrations in conjunction with a closed-form so-lution. The second analysis utilized the concentrations determined from the soil sectioning and extraction procedure with a semianalytical solution in the form of the computer program, POLLUTE (Rowe et al. 1985a).

490

Closed-Form Solution After introduction of the leachate into the reservoir at time zero (f = 0), mass

transport of the chemical constituents in the leachate occurs via molecular dif-fusion from the reservoir into the soil. The diffusive mass transport results in a decrease in the constituent concentrations in the reservoir as a function of time. Since the bottom of the cell is closed during the diffusion stage of the test, none of the mass entering the soil at the soil-reservoir interface can exit the soil at the bottom of the cell. Based on these considerations, the initial and boundary conditions for the diffusion cell are

c = 0 at 0 < x < a (t = 0) (5a)

c = c0 at a < x < a + L (t = 0) (5b)

dc — = 0 at x = 0 (t > 0) (5c)

dx

and

R„y+ ( - ) ( — ) = L c ° atx = a ( f > 0 ) (6)

where a = the length of the diffusion cell, L = the effective length of the res-ervoir, 8 = the volumetric water content of the soil, and y = defined as the amount of the free (nonadsorbed) solute per unit of soil contained between the planes at x = 0 (i.e., base of the soil) and at any distance, x, within the soil, or

Jo y(x,t) = 9 I c(x,t)dx (7)

Jo The effective length of the reservoir was determined by dividing the total volume of leachate introduced into the apparatus (reservoir, top cap, PVC tube, and buret) by the cross-sectional area, A, which was constant.

One solution to Fick's second law for simultaneous diffusion and adsorption in soil with the aforementioned initial and boundary conditions is (Wilson 1948; Crank 1975)

- = " + Y 2a ex (~D*q2j) (8) c0 1 + a £ , 1 + a + a

2ql ^ V Rda2 J

where c, = the concentration of a given solute in the reservoir at any time t after the start of diffusion and the qm = the nonzero positive roots given by

tan qm = -aqm (9)

and a = a dimensionless coefficient given by the following relation

L a = (10)

QRda The complete derivation for Eq. 8 for a saturated soil as well as the roots to Eq. 9 are given by Shackelford (1988). Mott and Nye (1968) used a slightly different form of Eq. 8 to determine the self diffusion coefficients (D*/Rd) of strontium in a saturated clay soil.

Semi-Analytical Solution In addition to the analytical solution represented by Eq. 8, a semianalytical

solution for contaminant transport in soil, POLLUTE (Rowe et al. 1985a), was

491

used to analyze the measured concentration profiles existing in the soil at the end of the diffusion test. The purposes for using POLLUTE to calculate effective diffusion coefficients were to provide: (1) An independent check on the calcu-lated D* values from the closed-form solution; and (2) an assessment of the relative merits of the use of reservoir concentration data versus concentration data from soil extractions. The theory for the derivation of the semianalytical solution implemented by the computer program POLLUTE is described by Rowe and Booker (1984, 1985). The use of the theory to determine D* values in the laboratory is described by Rowe et al. (1985b; 1988).

DIFFUSION TEST PROGRAM

The testing program is outlined in Table 3. At least two replicates were con-ducted for each test series. Five series of diffusion tests were performed with kaolinite to determine the effect of molding water content on the effective dif-fusion coefficients. The standard Proctor compaction method (ASTM D698) was used to compact kaolinite samples at five different water contents. Two addi-tional series of diffusion tests using kaolinite samples compacted by kneading and static compaction methods were performed to determine the effects of com-paction method on the effective diffusion coefficients. All of the samples in these tests were compacted at a water content slightly wet of optimum moisture content based on the standard Proctor procedure. Additional detail on the different com-paction methods used in this study is provided by Shackelford (1988).

One series of diffusion tests was performed with Lufkin clay. The Lufkin clay samples were compacted at a water content slightly greater than optimum mois-ture content using the standard Proctor procedure.

In addition, two control tests were performed to assess the possibility that the diffusion apparatus could act as a source/sink for the chemical species in the leachate. The control tests were performed by introducing leachate into a dif-fusion apparatus without soil, and monitoring the change in concentration of the specified ions as a function of time.

RESULTS

Final Properties of Soils The properties of the soil samples after diffusion used for determination of

the effective diffusion coefficients are presented in Table 4. The water contents for the clay samples represent weighted averages of the water contents deter-mined from each soil slice. Typical variations in water content with depth are shown for selected samples in Fig. 2.

The final, overall water contents and dry unit weights of the kaolinite samples have been plotted together with the original values in Fig. 3 to illustrate the

TABLE 3. Diffusion Test Program Test

series

(D 1 2 3 4 5 6 7 8

Soil (2)

Kaolinite Kaolinite Kaolinite Kaolinite Kaolinite Kaolinite Kaolinite Lufkin clay

Compaction method

(3)

Standard Proctor Standard Proctor Standard Proctor Standard Proctor Standard Proctor Kneading Static Standard Proctor

Molding water content (%)

(4)

24 27 30 33 36 33 33 22

Number of tests

(5)

3 3 3 3 3 2 2 2

492

TABLE 4. Final Soil Properties Used for Effective Diffusion Coefficient Analyses

Test number

(D

Water content,8

w (%) (2)

Total porosity,

n (3)

Degree of saturation,

s, (%) (4)

Volumetric water

content, 9 (5)

Dry (bulk) density,

Pd , (kN/m3)

(6)

(a) Test series 1

1 2 3

51.6 (44.1-61.4) 51.1 (46.3-59.3) 54.5 (50.6-59.9)

0.59 0.59 0.60

96.4 95.0 97.0

0.57 0.56 0.58

10.72 10.68 10.41

(b) Test scries 2

1 2 3

49.1 (42.7-58.7) 49.5 (44.1-61.1) 49.1 (45.1-55.7)

0.58 0.58 0.57

95.1 94.4 97.8

0.55 0.55 0.56

10.95 10.84 11.12

(c) Test series 3

1 2 3

44.6 (38.9-52.1) 43.7 (36.0-54.5) 39.5 (35.4-43.0)

0.56 0.55 0.51

94.3 95.7

102.0b

0.53 0.53 0.52

11.49 11.73 12.79

(d) Test series 4

1 2 3

43.1 (35.5-62.6) 41.7(34.6-58.1) 39.9 (37.9-45.2)

0.54 0.54 0.52

96.2 95.2 97.9

0.52 0.51 0.51

11.85 11.99 12.45

(e) Test series 5

1 2 3

41.0 (37.4-57.5) 39.8 (36.4-53.5) 41.4 (39.7-44.5)

0.54 0.53 0.54

91.3 93.2 94.6

0.49 0.49 0.51

11.84 12.16 12.00

(/) Test series 6

1 2

43.9 (40.9-57.8) 42.6 (38.1-55.0)

0.57 0.54

88.9 96.0

0.S1 0.52

11.22 11.90

Cs) Test series 7

1 2

42.1 (37.8-65.3) 43.2 (38.2-60.4)

0.55 0.54

92.0 96.4

0.51 0.52

11.71 11.85

(h) Test series 8

1 2

28.8 (25.0-37.7) 27.8 (24.7-35.9)

0.47 0.45

86.2 90.7

0.41 0.41

13.87 14.43

"Values given are weighted averages based on water content of each soil slice and thickness of soil section; values in parentheses represent range of water contents from all soil slices.

bCalculated value is in error; 100% used.

effect of swelling on the compacted soil samples. In all cases, the final water content and degrees of saturation were higher than the respective initial values, and the final dry unit weights were lower.

Batch Equilibrium Tests The adsorption isotherms for the cations are presented in Fig. 4. All of the

adsorption isotherms are nonlinear. Lufkin clay adsorbed about three times as much of a given cation as kaolinite. This is not surprising since the Lufkin clay consists, in part, of smectitic clay minerals with a higher CEC.

Initially, anion adsorption (especially Cl~ and S04"), as well as cation ad-sorption, was expected to be operative in the soils, especially for the kaolinite which has a pH-dependent adsorption capacity (Bohn et al. 1979). However, the anion adsorption that could be discerned from the results of the batch equilibrium tests was nil.

Because the adsorption isotherms are nonlinear, secant lines were used to es-timate the retardation factors. The secant lines, were based on the following formulation for the partition coefficient, Kp [e.g., see Davidson et al. (1976), Rao (1974), and "Batch-type" (1987)]

493

Water Content, w(%)

.0 10 20 30 40 50 60

. - . 1 .

10,1 11.1

Molding I Water . Content <

I

Test Series Teat No. 1

Water Content, w(%)

10 20 30 40 SO 60

Water Content, w(%)

0 10 20 30 40 50 60

£ . 1.0 Content

0I—. 1 , 1 • 1 , 1 , ! , 1

0 100 200 300 400 500 600

Equilibrium Concentration, c

TABLE 5. Mean Effective Diffusion Coefficients Based on Reservoir Concentra-tions

Test (D

Mean Effective Diffusion Coefficients, D* x 1010 mz/s

CI " (2)

Br" (3)

I" (4)

K+

(5) Cd 2 + ^

(6) Zn2 +

(7)

(a) Test series 1

1 2 3 Weighted mean Standard deviation

4.5 (6) 4.6 (6) 7.0 (1) 4.7 0.7

6.7 (6) 6.3 (6) 5.1 (1) 6.4 0.4

9.5 (7) 7.6 (7) 6.7 (1) 8,4 1.0

14.0 (6) 15.5 (5) 12.5 (1) 14.5 0.8

7.3 (7) 7.9 (7) 5.8 (1) 7.5 0.5

8.5 (7) 9.8 (7) 5.2 (1) 8.9 1.2

(6) Test series 2

1 2 3 Weighted mean Standard deviation

10.4 (5) 10.0 (5) 6.5 (1) 9.9 1.1

7.5 (3) 10.5 (3) 5.5 (1) 8.4 2.0

9.5 (4) 8.1 (4)

11.2 (1) 9.1 1.0

11.1 (4) 12.3 (4) 12.1 (1) 11.7 0.6

5.5 (5) 6.1 (5) 6.1 (1) 5.8 0.3

8.1 (5) 10.0 (5) 5.9 (1) 8.8 1.3

(c) Test series 3

1 2 3 Weighted mean Standard deviation

5.5 (5) 7.4 (6) 6.0 (1) 6.5 0.9

4.6 (5) 5.3 (6) 3.9 (1) 4.9 0.4

3.4 (6) 2.5 (6)

10.3 (1) 3.5 2.0

15.4 (5) 12.4 (6) 12.3 (1) 13.6 1.5

8.1 (5) 7.6 (7) 5.5 (1) 7.6 0.7

9.1 (6) 10.4 (7) 4.1 (1) 9.4 1.6

(rf) Test series 4

1 2 3 Weighted mean Standard deviation

8.0 (4) 6.1 (4) 8.7 (1) 7.2 1.0

8.7 (4) 5.3 (4) 8.3 (1) 7.2 1.7

17.6 (2) 4.2 (4) 0.15(1) 7.5 6.6

14.5 (4) 13.6 (4) 12.9 (1) 13.9 0.6

4.9 (4) 4.4 (4) 5.8 (1) 4.8 0.4

8.5 (4) 10.5 (4) 5.9 (1) 9.1 1.5

(e) Test series 5

1 2 3 Weighted mean Standard deviation

9.1 (6) 13.8 (4) 6.9 (1)

10.6 2.5

9.0 (6) 10.4 (4) 4.2 (1) 9.1 1.7

4.9 (7) 8.6 (5) 0.7 (1) 6.0 2.3

14.1 (6) 15.4 (4) 18.4 (1) 15.0

1.2

7.4 (7) 5.8 (5) 5.8 (1) 6.7 0.8

9.8 (7) 9.3 (5) 5.9 (1) 9.3 1.0

(/) Test series 6

1 2 Weighted mean Standard deviation

10.6 (4) 8.9 (6) 9.6 0.8

10.6 (4) 9.4 (6) 9.9 0.6

18.4 (4) 12.2 (6) 14.7 3.0

14.7 (4) 17.4 (7) 16.4 1.3

6.3 (5) 8.0 (7) 7.3 0.8

10.2 (5) 10.4 (7) 10.3 0.1

(g) Test series 7

1 2 Weighted mean Standard deviation

7.1 (6) 16.4 (3) 10.2 4.4

4.7 (6) 8.2 (3) 5.9 1.7

3.5 (6) 13.8 (3) 6.9 4.9

20.2 (7) 13.3 (4) 17.7 3.3

7.7 (7) 5.2 (4) 6.8 1.2

8.5 (7) 7.7 (5) 8.2 0.4

(A) Test series 8

1 2 Weighted mean Standard deviation

4.7 (4) 4.7 (4) 4.7 0.0

21.9 (3) 15.5 (4) 18.2 3.2

5.8 (3) 4.7 (3) 5.3 0.5

19.6 (4) 19.5 (4) 19.6 0.1

10.4 (3) 9.6 (3)

10.0 0.4

25.8 (4) 25.1 (4) 25.4 0.3

Note: Values in parentheses represent number of reservoir samples upon which corresponding mean D* value is based.

10(a)]. Effective diffusion coefficients are not reported for I~ due to problems with chemical analysis or for K+ due to problems with poor extraction. The chemical analysis problems associated with the IC analysis for iodide concen-trations included broad-based peaks requiring long periods (&40 min) for com-plete ion chromatographic analysis, baseline fluctuations, and severe tailing of the iodide peaks. Also, for the kaolinite samples, only the third test in test series 1-5 was analyzed using POLLUTE due to the relatively poor extraction effi-ciencies for the other kaolinite samples (see Table 7).

Analysis The weighted-mean D* values from reservoir concentrations (Table 5) and the

496

600

500

400,

300

200

1

w s--ir •

(a)

®

ci-

c 0 (mg/L) 381~ D* x 10l0m2/s 4.5 Observed A Theoretical —

571

7.3

20 40 60 80 100

Br • K+

c 0 (mg/L) 923 377 D* x 10l0m2/s 6.7 14.0 Observed A B Theoretical

Zr,2+

c0 (mg/L) D* x 10l0in2/s Observed Theoretical -

1178 308

9.5 8.5

20 40 60 80 100 Time (Days)

FIG. 5. Reservoir Concentration versus Time for Diffusion Test with Kaolinite (Test Series 1, Test No. 1): (a) Chloride and Cadmium; (h) Bromide and Potassium; (c) Iodide and Zinc

D* values determined from the final concentration profiles in the soil (Table 6) are compared in Fig. 11. The comparison varies from excellent for anions dif-fusing in kaolinite to poor for the results with Lufkin clay, with the tendency for D* from the reservoir concentrations to be greater than D* from the final concentration profiles.

The D* values determined for anions diffusion in kaolinite were almost all in the range from 4 to 10 X 1CT10 m2 /s , regardless of the method used to calculate D*. Values of D* reported for Cl~ diffusing in saturated, natural clays, silty clays, and sand-bentonite mixtures are almost all in the same range (Clarke and Graham 1968; Barraclough and Tinker 1981, 1982; Desaulniers et al. 1981; Crooks and Quigley 1984; and Gillham et al. 1984). Except for the D* values for CI" and Br~ determined from the final concentration profiles in Lufkin clay (Table 6), the D* values measured in this study for anions compare favorably to each other and to values reported elsewhere. Aside from experimental error, the poor results from the final concentration profiles of Cl_ and Br~ in Lufkin clay may be due, in part, to the inefficiency of the extracting solution (DDW) in removing the complexed anions (e.g., CdCl+, CdBr+, ZnCl+, and ZnBr+) from the diffuse double layer [e.g., see Shackelford (1988) and Shackelford et al. (1989)], al-though the results from the batch equilibrium tests did not indicate that this is the case. In any event, it was impossible to determine the extent of this effect since the distribution of complexed species with depth in the soil is not known.

In addition, high background concentrations of Cl~ (Table 1) may have con-tributed to the poor results for Lufkin clay. No attempt was made to determine the background distribution of ion concentrations in the soil. The background

497

CI- Cd*+

c„ (mg/L) 397 566

D* xlOlOm 2 / s 9.1 7.4 Observed A H Theoretical

\> 20 40 60 80 100

B r - K+

c0 (mg/L) 953 ~3~7? D* x l o l O m 2 / s 9.0 14.1 Observed A H Theoretical

0 20 40 60 80 100

I- Zn 2 +

Co (mg/L) 1163 308 D* X l 0 l 0 r a 2 / s 4.9 9.8 Observed A B Theoretical .

"0 20 40 60 80 100

Time (Days)

FIG. 6. Reservoir Concentration versus Time for Diffusion Test with Kaolinite (Test Series 5, Test No. 1): (a) Chloride and Cadmium; (b) Bromide and Potassium; (c) Iodide and Zinc

ion concentrations reported in Table 1 were measured on saturated soil extracts that do not represent the conditions in the soil during the diffusion test. Some of the background ions in the soil undoubtedly diffused into the reservoir during the soaking stage of the tests and subsequently were removed when the soaking solution was replaced by the simulated waste leachate. Therefore, the back-ground concentrations of the ions in the soil samples are unknown. Nonetheless, the high background concentration reported for CI" in Table 1 for Lufkin clay does provide an indication that a significant amount of Cl~ may have been pres-ent in the Lufkin clay before the start of the diffusion stage of the test and, therefore, may have biased the test results. High background concentrations of Cl_ would affect the determination of D* values using the analytical solution (Eq. 8) since the initial condition for Eq. 8 is based on the assumption that the background concentrations of the diffusing ions in the soil are zero.

The D* values for the cations computed from the reservoir concentrations (Ta-ble 5) appear to be high and, in some cases, may appear to be greater than the corresponding free-solution diffusion coefficients, D0. However, the free-solu-tion diffusion coefficients for the ions diffusing in the simulated leachate solution were not measured and, therefore, are unknown. The D0 value that typically is assumed to apply is the self-diffusion coefficient at infinite dilution. However, the use of self-diffusion coefficients at infinite dilution clearly is incorrect in this case for at least two reasons: (1) The concentrations of the ions in the leachate solution do not represent the concentrations of the ions at infinite dilution; and (2) the use of the self-diffusion coefficient, D0, at infinite dilution does not rec-ognize the requirement for electroneutrality in solution of all diffusing ions

Note: Data Polnta Hot Considered In Analysta Are Circled

600r

1000,

800

600

400,

200

V , • »Vlr A

-C

A (b)

(*

498

600 i

500

400,

300

200 _. mo

Note: Data Points Not Considered In Analysis Are Circled

I

^ " ^ ^ U ^ * - a » j < * ^

.

A

,

(b)

Br"

K+

e 0 (mg/L) 882 379 D* x 10l0m2/s 21.9 19.6 Observed A B Theoretical

20 40 60 80 100

1600,

1200

800

400,

v. A

*— "

® ~

(c) z„2*

I- Zn2+

c 0 (mg/L) 1528 314 D* x 10lOra2/s 5.8 25.8 Observed A a Theoretical

0 20 40 60 80 100

Time (Days)

FIG. 7. Reservoir Concentration versus Time for Diffusion Test with Lufkin Clay (Test Series 8, Test No. 1): (a) Chloride and Cadmium; (o) Bromide and Potassium; (c) Iodide and Zinc

Equllbrium Concentration, c (mg/L)

0

.0

.0

.0

.0

.0

) 200 400 6

•/ f I

•I " / ' cr I / Br"

. I ,.

i I («>

300 450

CI- Br -

c0 (mg/L) 33t~~~64S D* xl0N>m2/s 6.0 6.S Observed H 9 Theoretical

Cd2+ Zn2+

c0 (mg/L) 602~"325" D' xl0tm2/s 4.2 4.2 Observed • B Theoretical

Duration of Test = 30 Days

FIG. 8. Concentration Versus Depth in Soil for Kaolinite (Test Series 1, Test No. 3): (a) Chloride and Bromide; (b) Cadmium and Zinc

499

Equllbrlum Concentration, c (mg/L)

a. a

o.o

1.0

2.0

3.0

4.0

5.0

400 _ 600

/ 7

' c r '

1 1 j • / 1 / 1 /

m

(a)

ci- Br "

c0 (mg/L) 346 661 D* x 10">ml/s 4.4 4-8 Observed • a Theoretical

Cd2* Zn2+

c0 (mg/L) 602 ~~32S~ D* x lO^mVs 3.2 3.5 Observed • a Theorelical

Duration of Test = 30 Days

FIG. 9. Concentration versus Depth in Soil for Kaollnite (Test Series 5, Test No. 3): (a) Chloride and Bromide; (D) Cadmium and Zinc

Equllbrlum Concentration, c (mg/L)

200 400 600 800 100 200 300 400

(iua

o m c

! a

0.0

1.0

2 0

3.0

4.0

5.0

B. a i •

/ / , 1 • /

1 / a 1 / »r-

f • / 1 /

. 1 / y ! /V.f^V,.

/

' / (a)

CI- l l r -

Co (roB'M 367 882

D* x 18l«ra2/3 1.8 ? Observed 0 a Theorelical

0.0

1.0

2.0

3.0

4.0

5.0

—T»—T 7 ^ - T • ,y •

/ /

/ /«» / • / • /

/

/ (b)

Cd*+ Zn2+

n * i l 0 l 0 m * / i 4.0 2.8

Theorelical

Duration ol Test = 76 Days

FIG. 10. Concentration versus Depth in Soli for Lufkin Clay (Test Series 8, Test No. 1): (a) Chloride and Bromide; (b) Cadmium and Zinc

TABLE 6. Effective Diffusion Coefficients Calculated from Final Concentration Profiles in Soil

Test series

(D 1 2 3 4 5 8 8

Test number

(2)

3 3 3 3 3 1 2

Effective Diffusion Coefficient, D* x 101C

CI" (3)

6.0 5.5 5.5 4.5 4.4 1.8 1.5

Br" (4)

6.5 6.0 5.8 6.1 4.8

—B —"

Cd2+

(5)

4.2 3.5 3.2 3.5 3.2 4.0 3.0

m2/s

Zn2 +

(6)

4.2 4.5 4.5 3.5 3.5 2.8 1.5

"No value reported since theoretical concentration profile using reasonable value for D* could not be fit to measured con-centration profile.

500

TABLE 7. Mass Balances

Soil (1)

Kaolinite Kaolinite Kaolinite Kaolinite Kaolinite Kaolinite Kaolinite Kaolinite Kaolinite Kaolinite Kaolinite Kaolinite Lufkin Lufkin Lufldn Lufkin Lufkin Lufkin

Test duration (days)

(2)

87-109 87-109 87-109 87-109 87-109 87-109

30 30 30 30 30 30 76 76 76 76 76 76

Number of tests

(3)

14 14 14 14 14 14 5° 5C

5C

5C

5C

5C

2 2 2 2 2 2

Extracting solution3

(4)

1 mM H4EDTA at pH = 2.8 1 niM RiEDTA at pH = 2.8 1 mM H4EDTA at pH = 2.8 DDW DDW DDW 5 mM H4EDTA at pH = 7.0 5 mM H4EDTA at pH = 7.0 5 mM H4EDTA at pH = 7.0 DDW DDW DDW 5 mM H4EDTA at pH = 7.0 5 mM H4EDTA at pH = 7.0 5 mM H4EDTA at pH = 7.0 DDW DDW DDW

Metal or nonmetal

(5)

K Cd Zn CI Br I K Cd Zn CI Br I K Cd Zn CI Br I

Mass Balances6 (%)

Range (6)

35.9-59.1 35.5-76.2 46.0-76.5

(-)39.2-58.1 47.2-76.2

(-)26.7-68.1 60.3-70.3 14.9-30.4

(-J5.0-289.3 8.8-23.8

(-J15.8-16.0 (-J255-61.3

85.9, 86.4 35.3, 48.4 43.6, 52.2 45.5, 47.9 78.1, 78.4 72.2, 84.9

Mean ± standard deviation (7)

50.5 ± 7.0 59.9 ± 20.0 58.5 ± 8.0 28.6 ± 21.8 59.3 ± 7.0 23.0 ± 32.1 65.5 ± 3.5 23.5 ± 5.1 11.9 ± 11.9 16.6 ± 5.7

(-)1.6 ± 11.8 (-)40.9 ± 137

86.2 ± 0.3 41.9 ± 6.6 47.9 ± 4.3 46.9 ± 1.1 78.3 ± 0.2 78.6 ± 6.4

aH4EDTA represents ethylenediaminetetraacetic acid used for metals extraction; DDW represents deionized distilled water for nonmetals extraction.

Values represent percent difference (PD) in mass, where PD = (MR - MS)/MR X 100%, MR = change in solute mass in reservoir, and Ms = mass of solute in soil at end of test.

Represents third test in test series 1-5.

(Shackelford and Daniel 1991). Due to this requirement for electroneutrality, the free-solution diffusion coefficients for Cd2+ and Zn2+ in the concentrated leach-ate solution probably are higher, and the free-solution diffusion coefficients for the anions (Cl~, Br", and I - ) lower, than the corresponding self-diffusion coef-ficients at infinite dilution [e.g., see Shackelford and Daniel (1991)]. Therefore, use of free-solution diffusion coefficients at infinite dilution for Cd2+ and Zn2+

in the simulated leachate solution is incorrect and probably is unconservative. In most cases, the agreement between theoretical and measured concentration-

versus-time profiles for the cations is excellent (e.g., see Figs. 5, 6, and 7). This excellent agreement may appear somewhat perplexing in view of the rel-atively high D* values reported for the cations in Table 5. However, it should

D* (x 10-10 m2s) from Profile Concentrations

FIG. 11. Effective Diffusion Coefficients (£>*) Based on Reservoir Concentra-tions versus D* Based on Concentration Profiles

501

be recognized that back-calculation of D* values for cations using the analytical solution, Eq. 8, requires that the retardation factor, Rd, be known a priori. As a result, the value determined for D* is totally dependent upon the value used for Rd, i.e., a relatively high estimate of Rd will result in a relatively low estimate of D*, and vice versa. In theory, there are an infinite number of combinations of D* and Rd that will provide identical concentration-versus-time profiles based on analytical solutions. Therefore, based on the results of the tests in this study, a relatively good match between theoretically and experimentally determined concentration-versus-time profiles does not necessarily mean that accurate D* values have been determined. The estimate of D* is only as good as the estimate for Rd.

The D* values for Cd2+ and Zn2+ computed from reservoir concentrations are approximately twice the values computed from final concentration profiles for the kaolinite samples. The discrepancy between D* values for Cd2+ and Zn2+

determined from reservoir concentrations versus concentration profiles for Lufkin clay is much greater. Precipitation of these heavy metal species may have con-tributed to this difference. In the presence of anaerobic bacteria, the sulfur (S) in sulfate (SO2-) is reduced to sulfide (S2-), which precipitates metal species [e.g., see Middleton and Lawrence (1977), Sawyer and McCarty (1978), Freeze and Cherry (1979), and Kim and Amodeo (1983)].

Mass balances for each ion and each test were calculated to assess the pos-sibility for experimental error as well as unknown concentration sources and/or sinks. The mass balances were calculated by comparing the mass lost from the reservoir solution to the mass in the soil at the end of the diffusion test. In determining the mass in the soil at the end of the diffusion test, no attempt was made to correct for the background mass of the ion in the soil for reasons pre-viously described. The mass balance discrepancies for all tests are reported in Table 7. Based on the results of the control tests, no significant sources/sinks were associated with the diffusion system.

In many cases, a significant portion of the mass balance discrepancies reported in Table 7 may be attributed to natural scatter in the data [e.g., see Br - in Fig. 8(a), Cd2+ in Fig. 9(b), and Cd2+ and Zn2+ in Fig. 10(b). The mass balance discrepancies reported for cadmium and zinc in Table 7 provide further indi-cation that precipitation of heavy metals may have been operative in the diffusion system. Also, based on the dependency of the mass balances on the strength of the extracting solution illustrated in Table 7, mass balance discrepancies for the metal species may be due not only to precipitation from the reservoir solution but also to incomplete extraction of the metal species from the soil sections.

The D* values reported for K+ from reservoir concentrations (Table 5) are higher than expected. The volume readings taken during the course of the dif-fusion test indicated that, in all cases, the volume changes were s ±1 .1% of the initial volume of the leachate in the reservoir. Therefore, mass transport due to suction was negligible. The relatively high D* values for K+ can be attributed, in part, to the use of Rd values that are too low, since the value of D* determined using the analytical solution (Eq. 8) is inversely proportional to the value of Rd. Low Rd values could have resulted from the different conditions set up in the batch equilibrium tests versus those in the diffusion tests (e.g., different soil: solution ratios), and/or from the use of secant Kp values to describe non-linear adsorption behavior.

Effect of Nonlinear Adsorption Behavior In most studies involving the measurement of effective diffusion coefficients

of reactive solutes, sufficiently low concentrations of the solute are used so that the adsorption behavior can be modeled with a linear distribution coefficient,

502

if

' " 20 25 30 35 40

Molding Water Content, w (%)

FIG. 12. Effective Diffusion Coefficients (£>*) for Chloride versus Molding Water Content for Kaolinite

Kd. However, in many practical applications, the solute concentrations will be elevated and the resulting adsorption behavior will be nonlinear. For nonlinear adsorption isotherms, such as the ones shown in Fig. 4, a secant value for Kp will be less than a linear coefficient, Kd, determined from a line drawn tangent to the initial portion of the isotherm. As a result, a retardation factor based on a secant value for Kp will tend to underestimate the retardation of a solute species at low concentrations and to overestimate D*. A detailed mathematical model that accounts for concentration-dependent Kp's would be needed to establish whether D* values determined using secant Kp's are significantly greater than values de-termined using the initial tangent Kd [e.g., see Melnyk (1985)]. Without such analyses, the prudent (conservative) approach is to use a secant Kp, since the tendency is to overestimate D* for the reactive solutes. Therefore, the D* values for the reactive solutes reported in Tables 5 and 6 may be high due to the use of Rd values that were based on secant lines evaluated at an equilibrium con-centration, c0. With respect to this study, the use of Rd values based on secant lines affected the D* values for cadmium the most because cadmium exhibited the greatest degree of nonlinear adsorption behavior (see Fig. 4).

Effect of Compaction Variables

Molding Water Content Mitchell et al. (1965) showed that shifts in molding water content of just a

few percent could produce a change in hydraulic conductivity of several orders of magnitude. One wonders whether D* is similarly sensitive to molding water content. The D* values for chloride, the principal nonreactive tracer, are plotted versus molding water content for kaolinite in Fig. 12. The effective diffusion coefficient is relatively insensitive to molding water content. Variations in Fig. 12 almost certainly reflect experimental scatter rather than any real trend.

Compaction Method Mitchell et al. (1965) also found that the hydraulic conductivity of compacted

clay was sensitive to the method of compaction. The weighted-mean, D* values for chloride from Table 5 for the kaolinite samples compacted wet of optimum water content by standard Proctor, kneading, and static compaction methods are

Weighted Means

Upper Bound of Measured Value

Lower Bound of Measured Values

Jrt-1 L

503

7.2 x 10"10 m2 /s , 9.6 x 10~10 m2 /s , and 10.2 X 10"10 m2 /s , respectively. On the average, the D* values for chloride decrease in the order of static > kneading > standard Proctor. However, the differences between the D* values are rela-tively minor, especially when considering the variability in the test results. Therefore, it may be concluded that compaction method has relatively little effect on laboratory-measured D* values for the conditions imposed in this study.

Tortuosity Factors Apparent tortuosity factors (T„) were calculated in accordance with Eq. 4 using

the results of the diffusion tests and an appropriate value for the free-solution diffusion coefficient, D0. Tortuosity factors usually are based on D* values for CI" since CI" is the most commonly used nonreactive solute. The theoretical maximum D0 value for CI" is on the order of 2.0 x 10"

9 m2/s (Robinson and Stokes 1959). Based on this D0 value and the D* values for CI" listed in Table 5, the minimum T„ values are 0.24 for Lufkin clay and range from 0.24 to 0.53 for kaolinite. These T„ values compare favorably with other Ta values reported for diffusion in saturated fine-grained soils (Shackelford and Daniel 1991).

CONCLUSIONS

Considerable care is required to measure accurate effective diffusion coeffi-cients (D*) of inorganic chemicals diffusing in compacted clay soils under lab-oratory conditions approximating those in the field. Problems with precipitation of metals from the reservoir solutions and incomplete extraction of metals from the soils likely were encountered in this study.

Mass flow due to suction gradients must be minimized in order to perform the type of diffusion test conducted in this study. Volume changes recorded dur-ing the diffusion test indicated that soaking the compacted soils with water prior to the start of a diffusion test was effective in minimizing advective transport during diffusion.

The value of the retardation factor, Rd, affects the determination of effective diffusion coefficients of reactive solutes measured in transient systems. Based on the results of the tests in this study, a relatively good match between theo-retically and experimentally determined concentration-versus-time profiles does not necessarily mean that accurate D* values have been determined. The estimate of D* is only as good as the estimate for Rd.

Analytical solutions for determining D* values of reactive solutes apply for the case of a linear adsorption isotherm (i.e., a constant retardation factor). How-ever, the results of the tests in this study indicate that it may be difficult to determine accurate D* values using analytical solutions at relatively high ion concentrations in which nonlinear adsorption behavior is evident.

Mass balance analyses showed significant differences between the mass of a given chemical species lost from the reservoir versus the mass of the species gained by the soil. For the metal species, precipitation from the reservoir and poor extraction from the soil were likely contributors. Complexation of anions and problems with the chemical analysis (I") may have contributed to the de-ficiencies reported for the mass balances of the nonmetal species.

For the test conditions and soils used in this study, molding water content and method of compaction had little influence on the effective diffusion coefficient (£>*) of chloride, the principal nonreactive solute in the simulated leachate.

The effective diffusion coefficients that were measured fell within the fairly narrow range of 4.0 X 10~10 m2/s to 2.0 X 10"9 m2 /s . In addition, the D* values determined from the rate of decrease in reservoir concentrations typically were higher than the D* values determined from the concentration profiles in the soil. It was clear from this study that chemical factors had a greater effect

504

than did physical factors on the measurement of the D* values of inorganic chemicals diffusing in compacted clay soils.

ACKNOWLEDGMENTS

This project was sponsored by the U.S. Environmental Protection Agency un-der cooperative agreement CR812630-01. The contents of this article do not nec-essarily reflect the views of the agency, nor does mention of trade names or commercial products constitute an endorsement or recommendation for use. Ap-preciation also is extended to the Earth Technology Corporation of Long Beach, California, for financial assistance in support of this work. In particular, the efforts of Messrs. Fred Donath, Geoff Martin, and Hudson Matlock are appre-ciated. Also, the cooperation of Drs. R. W. Gillham (University of Waterloo), D. H. Gray (University of Michigan), and R. M. Quigley and R. K. Rowe (University of Western Ontario) in sharing research findings concerning diffusion is appreciated.

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Clarke, A. L., and Graham, E. R. (1968). "Zinc diffusion and distribution coefficients in soil as affected by soil texture, zinc concentration and pH." Soil Sci., 105(6), 409-418.

Crank, J. (1975). The mathematics of diffusion, 2nd Ed., Clarendon Press, Oxford, En-gland.

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Farrah, H., and Pickering, W. F. (1978). "Extraction of heavy metal ions sorbed on clay." Water Air Soil Pollut., 9(4), 491-498.

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Griffin, R. A., Cartwright, K., Shimp, N. F., Steele, J. D. , Ruch, R. R., White, W. A., Hughes, G. M., and Gilkeson, R. H. (1976). "Attenuation of pollutants in mu-nicipal landfill leachate by clay minerals. Part I—Column leaching and field verifica-tion." Envir. Geology Notes, No. 76, Illinois State Geological Survey, Champaign, 111.

Grimm, R. E. (1953). Clay mineralogy. McGraw-Hill Co., Inc., New York, N.Y. Kim, B. M., and Amodeo, P. A. (1983). "Calcium sulfide process for treatment of metal-

containing wastes." Environ. Prog., 2(3), 175-180. Li, Y-H., and Gregory, S. (1974). "Diffusion of ions in sea water and in deep-sea sed-

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Middleton, A. C , and Lawrence, A. W. (1977). "Kinetics of microbial sulfate reduc-tion." J. Water Pollut. Control Fed., 49(7), 1659-1670.

Mitchell, J. K., Hooper, D. R., and Campanela, R. G. (1965). "Permeability of com-pacted clay." J. Soil Mech. and Found. Engrg. Div., ASCE, 91(4), 41-65.

Mott, C. J. B., and Nye, P. H. (1968). "Contribution of adsorbed strontium to its self-diffusion in a moisture-saturated soil." Soil Sci., 105(1), 18-23.

Page, A. L., Miller, R. H., and Keeney, R. H., eds. (1982). Methods of soil analysis, part 2—chemical and microbiological properties. 2nd Ed., Amer. Society of Agron-omy, Inc., Soil Sci. Society of America, Inc., Madison, Wis.

Rao, P. S. C. (1974). "Pore geometry effects on solute dispersion in aggregated soils and evaluation of a predictive model," thesis provided to the University of Hawaii, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Robinson, R. A., and Stokes, R. H. (1959). Electrolyte solutions, 2nd Ed., Butterworths Scientific Publications, London, England.

Rowe, R. K., and Booker, J. R. (1984). "The analysis of pollutant migration in a non-homogeneous soil." Geotechnique, 34(4), 601-612.

Rowe, R. K., and Booker, J. R. (1985). " 1-D pollutant migration in soils of finite depth." J. Geotech. Engrg., ASCE, 111(4), 479-499.

Rowe, R. K,, Caers, C. J., andBarone, F. (1988). "Laboratory determination of diffusion and distribution coefficients of contaminants using undisturbed clayey soil." Can. Geo-tech. J., 25(1), 108-118.

Rowe, R. K., Caers, C. J., and Crooks, V. E. (1985a). "POLLUTE 3.3, 1-D pollutant migration through a non-homogeneous soil." Report Number SACDA 84-13, Univ. of Western Ontario, London, Ontario, Jul.

Rowe, R. K., Caers, C. J., and Crooks, V. E. (1985b). "Pollutant migration through clay soils." Proc. 11th Int. Conf. on Soil Mech. and Foundation Engrg., International Society for Soil Mechanics and Foundation Engineering, San Francisco, Calif., 1293-1298.

Sawyer, C. N. , and McCarty, P. L. (1978). Chemistry for environmental engineering. McGraw-Hill, New York, N.Y.

Shackelford, C. D. (1988). "Diffusion of inorganic chemical wastes in compacted clay," thesis presented to the University of Texas, at Austin, Tex., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Shackelford, C. D., and Daniel, D. E. (1991). "Diffusion in saturated soil: I. Back-ground." J. Geotech. Engrg., ASCE, 117(3), 467-484.

Shackelford, C D . , Daniel, D. E., and Liljestrand, H. M. (1989). "Diffusion of inorganic chemical species in compacted clay soil." / . Contaminant Hydrology, Amsterdam, the Netherlands, 4(3), 241-273.

Triegel, E. K. (1980). "Attenuation by soils of selected potential contaminants from coal conversion facilities: a literature review." ORNL/TM-7249, Oak Ridge Nat. Lab., Oak Ridge, Tenn.

Wilson, A. H. (1948). "A diffusion problem in which the amount of diffusing substance is finite-I." Philos. Mag., Series 7, 39(288), 48-58.

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