DIFFUSION IN SATURATED S O I L . 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.
Assoc. Prof., Dept. of Civ. Engrg., Univ. of Texas, Austin, TX 78712.
Note. Discussion open until August 1, 1991. Separate discussions should be submitted 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.
2
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
(3)
^~w
where R = the universal gas constant (8.134 J mor'KT 1 ), 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:
D
RTk0
° = J^
W
and
°o =
RT
T^T
(5)
468
TABLE 1. Self-Diffusion Coefficients for Representative Ions at Infinite Dilution
in Water at 25° C
Anion
(1)
oir
F~
cr
Br
rHCO _
3
NO3_"
sor
cof
—
—
—
—
^
—
—
—
—
—
—
A> x 1010 (m 2 /s)
(2)
Cation
(3)
D 0 x 10 10 (m 2 /s)
(4)
52.8
14.7
20.3
20.8
20.4
11.8
19.0
10.6
9.22
H+
Li +
Na +
K+
Rb +
Cs +
Be 2+
Mg 2+
Ca2+
Sx2*
Ba 2+
Pb 2+
Cu 2+
Fe2+»
Cd2+*
Zn 2+
Ni 2+ *
Fe 3+ *
Cr3^
Al 3+ "
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)]
D 0 x 10 10 (m a /s)
(2)
Electrolyte
(1)
HC1
HBr
LiCl
LiBr
NaCl
NaBr
Nal
KC1
KBr
KI
CsCl
CaCl2
BaCl2
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 isotopic (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 "selfdiffusion," 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
Diffusion
Prevented
Allowed
22
NaCl
+
NaCl
NaCl
(a)
NaCl
(b)
NaCl
Water
(c)
NaCl
KCI
(d)
42
KCI
4-
NaCl
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
42 +
K 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 halfcells 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 0 9 —
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
(8)
J = -D0T6 —
dx
Typical values of T (discussed later) are < 1.
472
Other Effects
Additional factors, not included in Eq. 8, tend to reduce the rate of diffusive 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 f a c t o r , " / , 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
J = -D0ffi
dc
dx
(12)
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, in473
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 —
dx
(14)
Effective Diffusion Coefficient
At present, tortuosity factors cannot be measured independently. Therefore, 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 —
dx
(16)
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* —
dx
(17)
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
Saturated or
unsaturated
(2)
Soil(s)
(1)
T„ Values8
(3)
Reference
(4)
(a) MC1 Tracer
Bentonite: sand mixtures
50% sand:bentonite mixture
Bentonite .sand mixtures
Saturated
Saturated
Saturated
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
Silty clay loam; sandy loam
Sandy loam
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
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
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.
(1988)]
Definitions of Effective Diffusion Coefficient, D* [after Shackelford
Reference
Definition
d)
D* = D 0 T
D* = D0C4T
D*
= D06T
D* = D0Tory8
D* = D 0 T7
D* = A,Tae
(2)
Gillham e t al. (1984)
Li and Gregory (1974)
Berner (1971); D r e v e r (1982)
K e m p e r et al. (1964); Olsen and K e m p e r (1968); N y e (1979)
Porter e t al. (1960)
van Schaik a n d K e m p e r (1966)
Note: D* = effective diffusion coefficient of chemical species; D0 = free-solution diffusion coefficient of chemical species; T = tortuosity factor; a = fluidity or viscosity
factor; y = negative adsorption or anion exclusion factor; and 9 = volumetric water content.
A g a i n , t h e r e a d e r m u s t b e v e r y careful in i n t e r p r e t i n g t h e literature t o d e t e r m i n e w h e t h e r E q . 1 6 , 1 7 , o r 18 h a s b e e n u s e d .
TRANSIENT DIFFUSION
Nonreactive Solutes
Fick's first law describes steady-state diffusive flux of solutes. For time475
dependent (transient) transport of nonreactive solutes in soil, Fick's second
law is assumed to apply, or
dc
3c
(20)
— ==: I)* —1
dt
dx
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 adsorptiondesorption (sorption) reactions and radioactive decay are routinely modeled.
Dissolution, precipitation, oxidation-reduction, and ion pairing or complexation 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. 2 1 , 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
(23)
Rd dx2
where Rd = the "retardation factor," or
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 concentration 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
Dt =D
d2c
(26)
jr ^ ^
where D* is an "apparent diffusion coefficient" (Li and Gregory 1974), also
known as "the effective diffusion coefficient of the reactive solute, Ds" (Gillham 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 fluxcontrolled boundary conditions because incorrect results are obtained due to
477
c = Equilibrium Concentration
q = Sorbed Concentration
(a)
(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
Concentration in Soil
Depth in
Soil
Column
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.
APPENDIX.
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