DIFFUSION IN SATURATED SOIL . I: BACKGROUND

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

ABSTRACT: Recent studies suggest that diffusion may be an important, if not dominant, mechanism of contaminant transport through waste containment bar­riers. This paper represents the first of two papers pertaining to the measurement of diffusion coefficients of inorganic chemicals diffusing in saturated soil. In this paper, both steady-state and transient equations describing the diffusive transport of inorganic chemicals are presented. Several factors affecting diffusion coeffi­cients are identified. A method for measuring diffusion coefficients for compacted clay soil is described. The definition for the diffusion coefficient for diffusion in soil (known as the effective diffusion coefficient, D*) is shown to vary widely. In general, variations in the definition of D* result from consideration of the different factors that influence diffusion of solutes in soil and the different ways of including the volumetric water content in the governing equations. As a result of the variation in the definition of D*, errors in interpretation and comparison of D* values can result if the appropriate definition for D* is not used.

INTRODUCTION

The design of earthen barriers for the containment of buried wastes tra­ditionally has been based on the assumption that the hydraulic conductivity controls the rate of leachate migration. However, recent field studies have indicated that diffusion is the controlling mechanism of solute transport in many fine-grained soils [e.g., Goodall and Quigley (1977), Desaulniers et al. (1981, 1982, 1984, 1986), Crooks and Quigley (1984), Quigley and Rowe (1986), Quigley et al. (1987), and Johnson et al. (1989)]. As a result, it is becoming necessary to evaluate the migration of chemicals through earthen barriers due to diffusion.

While the measurement of the hydraulic conductivity of fine-grained soils is relatively common practice for geotechnical engineers, the measurement of diffusion coefficients is not. In fact, the concept of diffusion may be unfamiliar to many geotechnical engineers. In addition, the literature abounds with a wide variation in the terminology associated with the study of dif­fusion in soils. Variable terminology can lead to considerable confusion, and an enormous amount of time can be spent in attempting to sort out the de­tails.

This paper is the first of two papers describing the process of diffusion in soils. The intent of this paper is to familiarize the geotechnical engineer with background information required for the measurement and evaluation of diffusion coefficients for use with the design of waste containment bar­riers. The specific objectives of this paper are to present the equations used to describe diffusion of solutes in soil, to discuss the factors affecting dif­fusion coefficients, to clarify some of the variability in the terminology as-

'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 April 26, 1990. This paper is part of the Journal of Geotechnical Engineering, Vol. 117, No. 3, March, 1991. ©ASCE, ISSN 0733-9410/91/0003-0467/$1.00 + $.15 per page. Paper No. 25602.

467

sociated with the study of diffusion in soils, and to describe methods of measurement.

STEADY-STATE DIFFUSION

Diffusion in Free Solution Diffusion of a chemical or chemical species in solution (i.e., a solute)

typically is assumed to occur in response to a concentration gradient in ac­cordance with Fick's first law which, for one dimension, may be written as:

dc J = -D0 - (1)

dx where J = the mass flux, c = the concentration of the solute in the liquid phase, x = the direction of transport, and D0 = the "free-solution" diffusion coefficient. However, several investigators [e.g., Robinson and Stokes (1959), Quigley et al. (1987), Daniel and Shackelford (1987), and Shackelford (1988, 1989)] have noted that there is a more fundamental basis for diffusive trans­port than the empirical Fick's first law. This fundamental basis, which takes the driving force for the solute ions or molecules as the gradient in the chem­ical potential of the chemical species, results in a number of expressions that help to provide insight into the factors affecting the free-solution diffusion coefficient, D0. One of these expressions is the Nernst-Einstein equation (Jost 1960), or

uRT dc J = (2)

N 8x and, by comparison with Eq. 1, the expression for the free-solution diffusion coefficient at infinite dilution [i.e., sufficient dilution such that solutes (ions, molecules) do not interact with each other in solution] becomes

uRT D^~w (3)

where R = the universal gas constant (8.134 J mor'KT1), T = the absolute temperature, N = Avogadro's number (6.022 x 1023 mol-1), and u = the absolute mobility of a particle. The absolute mobility of a particle is the limiting velocity attained under a unit force which, in the aforementioned case, is the gradient in the chemical potential of the diffusing chemical spe­cies (Robinson and Stokes 1959).

When Eq. 3 is combined with expressions relating the absolute mobility to the limiting ionic equivalent conductivity (Robinson and Stokes 1959) and to the viscous resistance of the solvent molecules, i.e., Stokes Law (Bird et al. 1960), two additional expressions for D0 result:

RTk0 D° = J^ W

and

RT

°o = T^T (5)

468

TABLE 1. Self-Diffusion Coefficients for Representative Ions at Infinite Dilution in Water at 25° C

Anion

(1)

o i r F~

cr B r

r HCO3_ NO3_"

sor cof

— — — — ^ — — — — — —

A> x 1010 (m2/s) (2)

52.8 14.7 20.3 20.8 20.4 11.8 19.0 10.6 9.22 — — — — — — — .— — .— —

Cation (3)

H+

Li+

Na+

K+

Rb+

Cs+

Be2+

Mg2+

Ca2+

Sx2* Ba2+

Pb2+

Cu2+

Fe2+» Cd2+* Zn2+

Ni2+* Fe3+* Cr 3 ^ Al3+"

D0 x 1010 (m2/s) (4)

93.1 10.3 13.3 19.6 20.7 20.5

5.98 7.05 7.92 7.90 8.46 9.25 7.13 7.19 7.17 7.02 6.79 6.07 5.94 5.95

"Values from Li and Gregory (1974).

where F = the Faraday (96,490 Coulombs/equivalent), \z\ = the absolute value of the ionic valence, \ 0

= the limiting ionic conductivity, T| = the absolute viscosity of the solution, and r = the molecular or hydrated ionic radius. The limiting ionic conductivity is the conductivity of an aqueous solution containing the specified ion at infinite dilution. Eqs. 4 and 5 com­monly are referred to as the Nernst and the Einstein-Stokes equations, re­spectively, and indicate that D0 is affected by several factors, including the temperature and viscosity of the solution, and the radius and valence of the diffusing chemical species. Based on X,0 values from Robinson and Stokes (1959), the D0 values for several ions have been calculated using Eq. 4 and the results are shown in Table 1. Similar tables can be found in Li and Gregory (1974), Lerman (1979), and Quigley et al. (1987).

The values of D0 reported in Table 1 should be considered to be the max­imum values attainable under ideal conditions (i.e., molecular scale, infinite dilution). Under nonideal conditions (e.g., macroscopic scale, concentrated solutions), a number of effects, negligible for ideal conditions, become im­portant. For example, when two oppositely charged ions are diffusing in solution, an electrical potential gradient is set up between the ions (Robinson and Stokes 1959). Due to this electrical potential gradient, the slower mov­ing ion speeds up while the faster-moving ion slows down, the overall result being that both ions migrate at the same speed. This electrical potential effect is responsible, in part, for the differences between the simple electrolyte diffusion values shown in Table 2 and their respective self-diffusion coef­ficients given in Table 1. Other effects responsible for the difference in DQ

469

TABLE 2. Limiting Free-Solution Diffusion Coefficients for Representative Sim­ple Electrolytes at 25° C [after Robinson and Stokes (1959)]

Electrolyte

(1)

HC1 HBr LiCl LiBr NaCl NaBr Nal KC1 KBr KI CsCl CaCl2

BaCl2

D0 x 1010 (ma/s) (2)

33.36 34.00 13.66 13.77 16.10 16.25 16.14 19.93 20.16 19.99 20.44 13.35 13.85

values under nonideal conditions include solute-solute and solute-solvent in­teractions [e.g., see Robinson and Stokes (1959)].

Types of Diffusion In addition to the previously described factors, the value of D0 depends

on the type of diffusion. There are essentially four different types of dif­fusion (Robinson and Stokes 1959; Li and Gregory 1974; Lerman 1979; Shackelford 1988, 1989): (1) Self-diffusion; (2) tracer diffusion; (3) salt dif­fusion; and (4) counterdiffusion or interdiffusion. The four different types of diffusion are represented schematically in Fig. 1. In the illustration, so­dium chloride (NaCl) and/or potassium chloride (KC1) are assumed to be the diffusion compounds.

In true self-diffusion, the initial system would contain two half-cells, each with equal concentrations of NaCl, but without any isotopically different species. In such a system, the movement of the molecules would truly be random, but the motion of the molecules could never be traced. Therefore, the true self-diffusion system is approximated by the introduction of the iso-topic (tracer) species, depicted schematically in Fig. 1(a). In this case, each half-cell of the system initially contains an equal concentration of sodium chloride (NaCl). However, in one half-cell, a small amount of the sodium, Na+, has been replaced by its isotope, 22Na+. When the two half-cells are connected, diffusion of both Na+ and its isotope, 22Na+, occurs, but in op­posite directions, owing to the small concentration gradients of each species. Since the concentration gradient is extremely small, the movements of the radioactive "tracer" ions (22Na+) and the Na+ ions are not tied to that of the ions of opposite signs (i.e., Cl~), and the tracer ions may be considered to be moving relative to a stationary background of nondiffusing ions (Rob­inson and Stokes 1959). This movement of the tracer ions is termed "self-diffusion," and the diffusion coefficient describing it is called the "self-dif­fusion coefficient.")

Tracer diffusion is the same as self-diffusion except the isotopic species is of a different element. For example, consider a system of two half-cells,

470

Diffusion Prevented Diffusion Allowed

22 NaCl

+ NaCl

NaCl (a)

42

KCI 4-

NaCl NaCl ( b )

NaCl Water (c)

NaCl KCI ( d )

FIG. 1. Diffusion Cells for Different Diffusion Systems: (a) Self-Diffusion; (h) Tracer Diffusion; (c) Salt Diffusion; (d) Counterdiffusion [after Shackelford (1988)]

each containing equal concentrations of NaCl. If a small amount of Na+ in one of the half-cells is replaced by an equal amount of a radioisotope of a different element, say 42K+, and the two half-cells are connected, the dif­fusion of the 42K+ may be traced [Fig. 1(b)]. In this case, the diffusion of 42K+ is termed "tracer diffusion" to distinguish it from self-diffusion. At infinite dilution, the tracer diffusion and self-diffusion coefficients are the same.

Salt diffusion is illustrated by Fig. 1(c). In this case, one half-cell contains a sodium chloride solution whereas the other half-cell contains only the sol­vent. When diffusion is allowed, both the Na+ and the Cl~ ions diffuse in the same direction.

Counterdiffusion or interdiffusion describes the process whereby different ions are diffusing against, or in opposite directions to, each other. A system describing such a process is shown in Fig. 1(d). In this system, two half-cells with equal concentrations of sodium chloride (NaCl) and potassium chloride (KCI) are joined together resulting in the diffusion of Na+ and K+

ions in opposite directions. This same process applies to any system in which concentration gradients are established in opposite directions. Equations for counter diffusion coefficients can be found in Robinson and Stokes (1959), Jost (1960), Helfferich (1962), Olsen et al. (1965), Li and Gregory (1974), Lerman (1979), Low (1981), and Shackelford (1988, 1989).

In reality, both self- and tracer diffusion are iimiting cases of counterdif­fusion, and salt diffusion and counterdiffusion usually occur simultaneously

471

Effective Length, L

FIG. 2. Concept of Effective Length in Transport through Soil [after Shackelford (1988)]

in most systems. The case of salt diffusion [Fig. 1(c)] best represents most practical field problems involving containment of waste by earthen barriers.

Diffusion in Soil Solutes diffuse at slower rates in soil than in free solution because the

pathways for migration are more tortuous in soil. Also, diffusive mass fluxes are less in soil than in free solution because solid particles in soil occupy some of the cross-sectional area. These effects are illustrated schematically in Fig. 2.

Effect of Reduction in Cross-Sectional Area of Flow Due to the reduced cross-sectional area of flow in soil, the concentration

of the diffusing species, c, is the concentration in the liquid phase of the pore space. Since fluxes are defined with respect to the total cross-sectional area, Eq. 1 must be modified for diffusion in soil as follows

dc J = -D 09 —

dx (6)

where 6 = the volumetric water content defined as follows

B = nSr (7)

where n = the total soil porosity and Sr = the degree of saturation of the soil, expressed as a decimal. Therefore, the maximum flux for liquid phase diffusion will occur when the soil is saturated (Sr= 1.0), all other conditions in Eq. 6 being equal.

Effect of Tortuous Pathway The tortuosity of the soil usually is accounted for by including a tortuosity

factor, T, in Eq. 6 as follows [e.g., Porter et al. (1960), Olsen and Kemper (1968), and Bear (1972)]

dc J = - D 0 T 6 —

dx

Typical values of T (discussed later) are < 1.

(8)

472

Other Effects Additional factors, not included in Eq. 8, tend to reduce the rate of dif­

fusive transport of solutes in soil. Kemper et al. (1964) incorporated a "flu­idity" or "mobility" factor, a, into Eq. 8 to account for the increased vis­cosity of the water adjacent to the clay mineral surfaces relative to that of the bulk water. In addition, Porter et al. (1960) and van Schaik and Kemper (1966) added a factor, 7, to account for exclusion of anions from the smaller pores of the soil. Anion exclusion can result in compacted clays and shales when clay particles are squeezed so close together that the diffuse double layer of ions associated with the particles occupies much of the remaining pore space (Freeze and Cherry 1979; Drever 1982). The process is also known as salt filtering, ultrafiltration, or membrane filtration (McKelvey and Milne 1962; Hanshaw and Coplen 1973; Mitchell 1976; Freeze and Cherry 1979; Drever 1982). Berner (1971) states that anion exclusion may occur in natural deposits when the average porosity of the soil has been reduced to 0.3. An­ion exclusion may also be operative in highly unsaturated soils and in rel­atively small pores where the available cross-sectional area of flow is re­duced (Olsen and Kemper 1968).

Eq. 8 can be modified to account for these additional effects, or

dc J = -DOTOCYG — (9)

dx

Since in most cases it is difficult, if not impossible, to separate the effects of geometry (T), fluidity (a), and anion exclusion (7) in soil diffusion stud­ies, it seems best to define a single factor that accounts for all of them. Nye (1979) has done this by defining the "impedance fac tor , " / , as

ft = Ta7 (10)

Olsen et al. (1965) included the volumetric moisture content, 6, into the definition of the "tortuosity factor" and called it the "transmission factor," t„ or

tr = T0178 (11)

When Eq. 9 is written in terms of an impedance factor (Eq. 10) or a trans­mission factor (Eq. 11), the following equations result, respectively

dc J = -D0ffi - (12)

dx

dc J = -D0t, — (13)

dx

The similarity in the forms of Eqs. 8, 12, and 13 should be noted. Due to this similarity, many researchers report tortuosity factors, T, when they may in effect be measuring impedance factors,/-, or transmission factors, tr. For this reason, it seems more appropriate to define an "apparent tortuosity fac­tor," Ta, in which is included not only the actual, geometric tortuosity, T, but also all other factors, which may be inherent in its measurement, in-

473

eluding solute-solute and solute-solvent interactions. Since the volumetric water content, 6, can be determined independently of all other factors, Fick's first law describing the diffusion of a chemical species in soil is more con­veniently expressed as:

dc J = - £>0T„e — (14)

dx

Effective Diffusion Coefficient At present, tortuosity factors cannot be measured independently. There­

fore, it is convenient to define an effective diffusion coefficient, D*, as fol­lows

D*=D0Ta (15)

When Eq. 15 is substituted into Eq. 14, Fick's first law for diffusion in soil becomes

J = -D*9 — (16) dx

Eq. 16 can be utilized to determine effective diffusion coefficients of chem­ical species, D*, diffusing in soil from experimental results. After D* is determined, the apparent tortuosity factor can be calculated from Eq. 15 using an appropriate value for the free-solution diffusion coefficient. Some typical values for Ta reported in the literature are presented in Table 3.

There are several definitions for the effective diffusion coefficient, D*, besides that of Eq. 15. Some of these definitions for D* are reported in Table 4. Of particular importance is the fact that some investigators have included 6 in the definition of D* while others (including the writers) have not. Cau­tion should be exercised when interpreting the effective diffusion coefficient data of various researchers. Errors in interpretation of 50% or more can re­sult if the appropriate definition of D* is not used.

Definition of Concentration A few investigators [e.g., Porter et al. (I960)] have defined the solute

concentration in terms of the total volume of soil (i.e., c' = 9c) and rewritten Eq. 16 in terms of this modified concentration as follows

J = -D* — (17) dx

where D* is as defined in Eq. 15. When the definition of D* includes the volumetric water content (i.e., D* = Z>oT„6), Eq. 17 is written as follows [e.g., Porter et al. (I960)]

J= -Df— (18) dx

where

„. D* D$ = — (19)

474

TABLE 3. Representative Apparent Tortuosity Factors Taken from Literature

Soil(s)

(1)

Saturated or unsaturated

(2) T„ Values8

(3) Reference

(4)

(a) MC1 Tracer

Bentonite: sand mixtures 50% sand:bentonite mixture Bentonite .sand mixtures

Saturated Saturated Saturated

0.59-0.84 0.08-0.12 0.04-0.49

Gillham et al. (1984) Gillham et al. (1985) Johnston et al. (1984)

(6) Cl~ Tracer

Sandy loam Sand Silty clay loam Clay Silt loam Silty clay loam; sandy loam Silty clay Clay

Unsaturated Unsaturated Unsaturated Unsaturated Unsaturated Saturated Saturated Saturated

0.21-0.35* 0.025-0.29* 0.064-0.26* 0.091-0.28* 0.031-0.57*

0.08-0.22* 0.13-0.30* 0.28-0.31*

Barraclough and Nye (1979) Porter et al. (1960) Porter et al. (1960) Porter et al. (1960) Warncke and Barber (1972) Barraclough and Tinker (1981) Crooks and Quigley (1984) Rowe et al. (1988)

(e) Br Tracer

Silty clay loam; sandy loam Sandy loam

Saturated Saturated

0.19-0.30* 0.25-0.35*

Barraclough and Tinker (1981) Barraclough and Tinker (1982)

(</) 3H Tracer

Bentonite:sand mixtures Bentonite: sand mixtures

Saturated Saturated

0.33-0.70 0.01-0.22

Gillham et al. (1984) Johnston et al. (1984)

"Values were calculated using appropriate D0 value from Table 1 with D* value taken from ref­erence.

TABLE 4. Definitions of Effective Diffusion Coefficient, D* [after Shackelford

(1988)]

Definition

d) D* = D0T D* = D0C4T

D* = D 0 6 T

D* = D0Tory8

D* = D0T7

D* = A,Tae

Reference

(2)

Gillham et al. (1984)

Li and Gregory (1974)

Berner (1971); Drever (1982)

Kemper et al. (1964); Olsen and Kemper (1968); Nye (1979)

Porter et al. (1960)

van Schaik and Kemper (1966)

Note: D* = effective diffusion coefficient of chemical species; D0 = free-solution dif­

fusion coefficient of chemical species; T = tortuosity factor; a = fluidity or viscosity

factor; y = negative adsorption or anion exclusion factor; and 9 = volumetric water con­

tent.

Aga in , the reader mus t b e very careful in interpreting the literature to de­

termine whether Eq . 16, 17, or 18 has been used.

TRANSIENT DIFFUSION

Nonreactive Solutes Fick's first law describes steady-state diffusive flux of solutes. For time-

475

dependent (transient) transport of nonreactive solutes in soil, Fick's second law is assumed to apply, or

dc — =: dt

= I)* 3 c — dx1

(20)

where the effective diffusion coefficient is as defined in Eq. 15. Solutions to Eq. 20 are provided in standard texts on diffusion [e.g., Barrer (1951), Jost (1960), and Crank (1975)] and heat transfer [e.g., Carslaw and Jaeger (1959)]. The application of Eq. 20 to problems of practical interest is illus­trated by Daniel and Shackelford (1988), Quigley et al. (1987), Rowe et al. (1985), and Shackelford (1989).

Reactive Solutes Nonreactive solutes are those chemical species that are not subject to

chemical and/or biochemical reactions. The transport of solutes that are sub­ject to chemical and/or biochemical reactions, known as "reactive solutes," can differ substantially from the transport of nonreactive solutes.

Of the numerous types of chemical and/or biochemical reactions that can affect contaminant concentrations during transport in soil, only adsorption-desorption (sorption) reactions and radioactive decay are routinely modeled. Dissolution, precipitation, oxidation-reduction, and ion pairing or complex-ation reactions typically are not modeled. For nondecaying, reactive solutes subject to reversible sorption reactions during diffusive transport in soil, Eq. 20 may be modified as follows [e.g., see Bear (1972) and Freeze and Cherry (1979)]

dc ^ d2c dq' — = £»* — - — (21) dt dx2 dt

where q' = the sorbed concentration of the chemical species expressed in terms of the mass of sorbed species per unit volume of voids (i.e., occupied by the liquid phase), or

«' = 7 « (22)

where q = the sorbed concentration expressed as the mass of solute sorbed per mass of soil and pd = the dry (bulk) density of the soil. When Eq. 22 is differentiated with respect to time, and substituted into Eq. 21 , and the resulting expression is rearranged, Fick's second law for reactive solutes subject to reversible sorption reactions during diffusive transport in soil be­comes (Bear 1972; Freeze and Cherry 1979):

dc _ D* d2c

dt Rd dx2

where Rd = the "retardation factor," or

(23)

Rd=l+-KD (24)

476

and

KP = — • (25) dc

where Kp = the "partition coefficient." When the q versus c relationship is linear, Kp is termed the "distribution coefficient, Kd." Otherwise, Kp is a function of the equilibrium concentration in the porewater of the soil. For nonreactive solutes, Kp = 0, Rd = 1.0, and Eq. 23 reduces to Eq. 20. Sorp­tion reactions of interest to geotechnical engineers may include desorption as well as adsorption.

Adsorption Isotherms A plot of the mass of solute sorbed per mass of soil, q, versus the con­

centration of the solute in solution, c, is called an "adsorption isotherm." Adsorption isotherms typically are determined in the laboratory by perform­ing batch-equilibrium tests. The procedure for batch-equilibrium tests con­sists of mixing a known amount of soil and chemical solution in a prede­termined mixing ratio for a specified time (usually 24 or 48 hours) and at a constant temperature [usually 20° C (68° F) or 25° C (77° F)]. The total concentration of the chemical species of interest in the solution is measured before the solution is added to the soil and the equilibrium concentration is measured after the mixing is completed. The sorbed concentration is deter­mined by taking the difference between the two measured concentrations. The relationship between the sorbed and equilibrium concentration is estab­lished by repeating the procedure several times with different, initial con­centrations of the solute. More details concerning batch-equilibrium test pro­cedures can be found in ASTM standards ES-10-85 and D4319.

In general, sorption isotherms are linear [Fig. 3(a)], concave nonlinear [Fig. 3(b)], or convex nonlinear [Fig. 3(c)] (Melnyk 1985). Nonlinear ad­sorption isotherms for soils typically are of the concave type [Fig. 3(b)]. Eq. 23 can be solved for many initial and boundary conditions if the isotherm is linear; however, solutions for nonlinear isotherms are much more difficult to develop. For this reason, linear adsorption isotherms often are assumed to apply when in fact the experimental data suggest nonlinear adsorption behavior.

Apparent Diffusion Coefficients In some cases, it may seem convenient to rewrite Eq. 23 as follows:

dc „ d2c d2c

jrDt^=D^ (26)

where D* is an "apparent diffusion coefficient" (Li and Gregory 1974), also known as "the effective diffusion coefficient of the reactive solute, Ds" (Gill-ham et al. 1984; and Quigley et al. 1987). The result of this substitution is that only one unknown (D* or Ds) must be solved instead of the two un­knowns (D* and Rd) in Eq. 23.

However, Rowe et al. (1985, 1988) and Rowe (1987) caution against the use of a single coefficient, D* or Ds, for analyzing problems with flux-controlled boundary conditions because incorrect results are obtained due to

477

(a)

c = Equilibrium Concentration q = Sorbed Concentration

(b)

(c)

FIG. 3. General Types of Sorption Isotherms: (a) Linear; (fa) Concave Nonlinear; (c) Convex Nonlinear [after Melnyk (1985)]

the dependency of flux on D* (Eq. 16), not D* or Ds. In addition, the coef­ficient D* or Ds is a function of the sorption characteristics of the soil, whereas D* is not. Therefore, it is meaningless to report a value for D* or Ds without reporting the associated Rd value since different soils have different sorption characteristics.

Finally, an error in interpretation of diffusion data by a factor of R^ will result if D* (or Ds) values from one study are compared with D* values from another study. For this reason, extreme caution should be exercised when interpreting the effective diffusion coefficient data for reactive solutes.

478

EFFECT OF COUPLED FLOW PROCESSES

The equations presented in this paper do not consider the effect of coupled flow processes, i.e., solute transport due to hydraulic, thermal, or electrical gradients [e.g., see Mitchell (1976)]. Coupled flow phenomena could, under certain conditions, act against diffusion in fine-grained soils of high activity and low void ratio, and could affect significantly measured diffusion coef­ficients [e.g., see Olsen (1969)]. These processes may be especially impor-

Sampling Port

-Concentration, c

(a)

DURING TEST

Analytical Solution for One D*

Experimental Point

Elapsed Time

(b)

AFTER TEST

Depth in Soil

Column

Concentration in Soil

Analytical Solution for One D*

Experimental Point

FIG. 4. Illustration of Method for Measuring Effective Diffusion Coefficients: (a) Concentration Profile in Reservoir; (ft) Concentration Profile in Soli [after Rowe et al. (1988)]

479

tant in soils containing bentonite used for waste containment liners [e.g., see Greenberg et al. (1973)].

METHODS OF MEASUREMENT

Several approaches have been used for the measurement of D* [e.g., see Shackelford (1991)]. One technique is to saturate two half-cells of soil with different solutions, place the half-cells together and allow diffusion to occur. After a sufficient period, the apparatus is disassembled, the soil is sectioned to determine the resulting concentration profile within the soil, and the ex­perimental results are curve-fit with an analytical solution to Eq. 20 or Eq. 23 to determine the effective diffusion coefficient [e.g., see Olsen et al. (1965) and Gillham et al. (1984)]. However, this approach may be inap­propriate for compacted clay soils because it is difficult to obtain good con­tact between the half-cells, counterdiffusion may exist when the interest often is in salt diffusion, and it is relatively difficult to saturate low-permeability soils in the two half-cells with leachate.

A more convenient approach for measuring effective diffusion coefficients in compacted clay soils is based on techniques described by Mott and Nye (1968), Stoessell and Hanor (1975), and Rowe et al. (1988). The concept is illustrated in Fig. 4. Soil is compacted in a mold, soaked to destroy suction that might produce advective mass transport, and then exposed to leachate in a reservoir. The difference in concentration of solutes establishes a con­centration gradient between the reservoir and the compacted soil, and the concentration of solutes in the reservoir decreases with time [Fig. 4(a)]. At the end of the diffusion test, the soil is extruded and sectioned, and the aqueous phase of each section is analyzed to develop a profile of solute concentrations within the soil [Fig. 4(b)]. Effective diffusion coefficient (D*) for a nonreactive solute or the ratio D*/Rd for a reactive solute is calculated from the variation in solute concentration in the reservoir versus time and/ or from the profile of solute concentrations in the soil at the end of the test [e.g., see Rowe et al. (1988); Shackelford (1988, 1991); and Shackelford et al. (1989)].

SUMMARY AND CONCLUSIONS

Equations describing the diffusive transport of inorganic chemicals in free solutions and in soils have been presented. Diffusion in aqueous or free so­lution is described by Fick's first law, which is empirical. Several more fundamental expressions for Fick's first law help to provide insight into the factors affecting the free-solution diffusion coefficient (D0). Some of these factors include the temperature and viscosity of the solution, the radius and valence of the diffusing chemical species, and solute-solute and solute-sol­vent interactions. In addition, the value for D0 depends on the system used to measure it. Four different systems—self-diffusion, tracer diffusion, salt diffusion, and counterdiffusion—were identified in this paper.

Diffusive transport in soil is slower than diffusive transport in free solution because of the reduced cross-sectional area of flow and the more tortuous pathways experienced by solutes diffusing through soil. In addition, solutes may be subject to adsorption reactions that will reduce further their rate of transport.

480

The definition for the effective diffusion coefficient, D*, for diffusion in soil can vary widely. In general, variations in the definition of D* result from the number of factors included in the definition of Fick's first law for diffusion in soil, the most important factor being the volumetric water con­tent (6). In addition, the reference frame for the concentration of the dif­fusing chemical species also can affect the definition of D*. As a result of the variation in the definition of D*, caution should be exercised when in­terpreting and comparing the results of different studies performed to mea­sure D*.

It is suggested that all factors that influence D* of nonreactive solutes be lumped into a single factor known as the "apparent tortuosity factor, T„." Since the volumetric water content, 6, is an independently determined vari­able, it should not be included in the definition of the effective diffusion coefficient, D*.

In general, it is not a good policy to lump the effective diffusion coeffi­cient (£)*) and the retardation factor (Rd) into a single coefficient known as the "apparent diffusion coefficient, D*" or the "effective diffusion coeffi­cient of the reactive solute, Ds" since the value of D* or D„ is a function of the adsorption characteristics of the soil.

Effective diffusion coefficients can be measured in cells in which com­pacted soils are first presoaked to eliminate suction that would cause mass transport via advection and then exposed to a reservoir of leachate and later sectioned to determine the distribution of diffusing solutes at the end of the test. Effective diffusion coefficients can be determined either from the rates of decrease of solute concentrations in the reservoir or from the final con­centration profiles of solutes in the soil.

ACKNOWLEDGMENTS

This project was sponsored by the U.S. Environmental Protection Agency under cooperative agreement CR812630-01. The contents of this article do not necessarily reflect the views of the agency, nor does mention of trade names or commercial products constitute an endorsement or recommendation for use. Appreciation 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 Mssrs. Fred Donath, Geoff Martin, and Hudson Matlock are appreciated. Also, the cooperation of Drs. R. W. Gillham (Uni­versity of Waterloo), D. H. Gray (University of Michigan), R. M. Quigley (University of Western Ontario), and R. K. Rowe (University of Western Ontario) in sharing research findings concerning diffusion is appreciated.

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