129
Honour Lectures /
Conférences honorifiques
3
where
d
is the dry density of the solid phase, or mass of
solids per unit total volume of solids [ML
-3
], and
K
d
is the
distribution coefficient [L
3
M
-1
], which relates the solid-
phase concentration,
C
s
, expressed as the sorbed mass of
the chemical species per unit mass of the solid phase [MM
-
1
], to the aqueous-phase concentration,
C
, of the chemical
species (i.e., assuming linear, reversible, and instantaneous
sorption), or
K
d
=
C
s
/
C
. As a result, for sorbing chemical
species,
K
d
> 0 such that
R
d
> 1, whereas for nonsorbing
chemical species,
K
d
= 0 (i.e.,
C
s
= 0) such that
R
d
= 1.
Thus,
D
a
as given by Eq. 4 represents a lumped effective
diffusion coefficient that includes the effect of attenuation
via
R
d
. For this reason,
D
a
also has been referred to as the
effective diffusion coefficient of a reactive chemical
species (Shackelford and Daniel 1991a). For water
unsaturated porous media, the total porosity,
n
, in Eq. 5 is
replaced by the volumetric water content,
w
, where
w
=
nS
w
and
S
w
is the degree of water saturation (0
≤
S
w
≤ 1).
Since the notation for the various diffusion coefficients
defined herein may not match the notation used by others
(e.g.,
D
*
as defined herein also is commonly designated as
D
e
), caution should be exercised in terms of understanding
the basis for the definition of the various diffusion
coefficients when interpreting values extracted from the
published literature. Unless indicated otherwise, the default
definition of the diffusion coefficient used herein is that
corresponding to
D
*
. For liquid-phase diffusion of aqueous
soluble chemical species in saturated porous media, values
of
D
*
generally fall within range 10
-9
m
2
/s >
D
*
> 10
-11
m
2
/s, with lower values of
D
*
being associated with finer
textured and/or denser soils (Shackelford and Daniel
1991a, Shackelford 1991). Since
a
< 1, the upper limit on
D
*
of 10
-9
m
2
/s is dictated by the
D
o
values, which
generally ranges from about 1 to 2 x 10
-9
m
2
/s for most
aqueous soluble chemical species, except for those
involving H
+
or OH
-
, in which case
D
o
is approximately 2
to 4 times higher (Shackelford and Daniel 1991a). Values
of
D
*
< 10
-11
m
2
/s are possible in situations involving
bentonite-based containment barriers, such as highly
compacted bentonite buffers for high-level radioactive
waste disposal, primarily as a result of ion exclusion
resulting from the existence of semipermeable membrane
behavior such that
r
< 1 (e.g., Malusis and Shackelford
2002a, Shackelford and Moore 2013). Liquid-phase values
of
D
*
for unsaturated porous media generally decrease with
decreasing
w
or
S
w
and can be several orders of magnitude
lower than the respective values at full water saturation
(Shackelford 1991). Finally, values of
D
a
for reactive
chemical species (e.g., heavy metal cations) typically range
from one to several orders of magnitude lower than the
corresponding
D
*
values due to attenuation mechanisms
(e.g., sorption, ion exchange, precipitation, etc.), i.e.,
R
d
> 1.
3 WHEN IS DIFFUSION SIGNIFICANT?
Following the approach of Shackelford (1988), the
significance of diffusion on the migration of aqueous
soluble chemical species, or solutes, through porous media
can be illustrated with the aid of solute breakthrough
curves, or BTCs, representing the temporal variation in the
concentration of a given chemical species at the effluent
end of a column of porous medium. As depicted
schematically in Fig. 1a, BTCs can be measured in the
laboratory for a column of a porous medium of length
L
by
(a) establishing steady-state seepage conditions, (b)
continuously introducing at the influent end of the column
a chemical solution containing a known chemical species
at a concentration
C
o
, and (c) monitoring the concentration
of the same chemical species emanating from the column
as a function of time, or
C
(
L
,
t
) (Shackelford 1993, 1994,
1995, Shackelford and Redmond 1995). Because the
source concentration,
C
o
, is constant, the BTCs typically
are presented in the form of dimensionless relative
concentration,
C
(
L
,
t
)/
C
o
, versus elapsed time. The time
required for the solute to migrate from the influent end to
the effluent end of the column is referred to as the
"breakthrough time" or the "transit time."
For example, consider the three BTCs depicted in Fig.
1b for the case of a low permeability clay (
k
h
= 5 x 10
-10
m/s) contained within a column of length 0.91 m and at a
porosity of 0.5, and subjected to an applied hydraulic
gradient,
i
h
, of 1.33. The chemical solution serving as the
permeant liquid contains a nonreactive solute at a constant
concentration of
C
o
and is assumed to be sufficiently dilute
such that no adverse interactions between the clay and the
solution result in any changes in
k
h
during the test.
The BTC in Fig. 1b labeled "pure advection" represents
the case commonly referred to as "piston" or "plug" flow,
whereby the breakthrough time is the time predicted in the
absence of any dispersive spreading of the solute front
using the seepage velocity,
v
s
, in accordance with Darcy's
law (i.e.,
t
=
L
/
v
s
=
nL
/
k
h
i
h
). Under purely advective
(hydraulic) transport conditions, 21.8 yr would be required
for the solute to completely break through the effluent end
of the column (i.e.,
C
(
L
,
t
)/
C
o
= 1) in the absence of any
dispersive spreading of the solute front, owing to the very
low seepage rate.
The BTC in Fig. 1b labeled "advection plus mechanical
dispersion" represents the spreading effect on the solute
front primarily due to mechanical (advective) dispersion
(i.e., diffusive dispersion is assumed negligible), which is
the case commonly depicted in groundwater hydrology
textbooks because the primary concern pertains to
contaminant migration within aquifers, or coarse-grained,
water-bearing strata subjected to relatively high seepage
velocities. The BTC for this case, as well as that for the
next case, was generated using a commonly applied
analytical model to the advective-dispersive solute
transport equation developed by Ogata and Banks (1961)
for the stated conditions of the column test (e.g.,
Shackelford 1990). In this case, the dispersive spreading
of the solute front is attributed to variations in the pore-
scale velocity profiles at the column scale and
heterogeneities in hydraulic conductivity at the field scale
(e.g., Shackelford 1993). Due to this spreading effect of the
solute front, there are an infinite number of possible
breakthrough times depending on the value of
C
(
L
,
t
)/
C
o
used to define the breakthrough time. However, the typical
practice is to evaluate the breakthrough time at a relative
concentration of 0.5, which is the time at which the BTCs
for pure advection and advection plus mechanical
dispersion intersect.
The BTC in Fig. 1b labeled "advection plus diffusion"
is the true BTC for this column, as this BTC reflects the
situation when the seepage velocity is sufficiently low such
that the effect of diffusion is not masked by the effects of
advection and mechanical dispersion. The spreading effect
is still noticeable in this BTC, but this BTC is displaced to
the left of the previous two BTCs, resulting in a
breakthrough time at
C
(
L
,
t
)/
C
o
of 0.5 of 14.8 yr, which is
considerably less than the value of 21.8 yr for the two
previous cases where diffusion is ignored. Thus, failure to
include the diffusion as a transport process under the
