128
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
2
realization is the promulgation in the US in 1980 of the law
known as the Comprehensive Environmental Response,
Compensation, and Liability Act (CERCLA), also known
as Superfund, that authorized the US EPA to respond to
releases, or threatened releases, of hazardous substances
that may endanger public health, welfare, or the
environment, and also enabled the US EPA to force parties
responsible for environmental contamination to clean up
such contamination and/or to reimburse the Superfund for
response or remediation costs incurred by the government.
Thus, the burgeoning field of environmental geotechnics
began to address technical issues related not only to the
prevention of contamination resulting from disposal of new
waste, but also to the remediation or clean up of existing
contamination resulting from improper disposal practices
in the past.
Because of the experience of geotechnical engineers in
using compacted clays for applications such as the low
permeability cores of engineered earthen dams (e.g.,
Mitchell et al. 1965), geotechnical engineers immediately
became involved and identified with the design and use of
CCLs as engineered barriers for disposal of new wastes.
However, the early emphasis in the use of CCLs as barriers
for waste containment focused primarily on the physical
and mechanical properties of the CCLs, such as
minimizing the hydraulic conductivity,
k
h
, of the CCL in
order to reduce the rate of seepage of contaminated liquids
(e.g., leachates),
v
, through the CCLs resulting from the
application of a hydraulic gradient,
i
h
, in accordance with
Darcy's law (i.e.,
v
=
k
h
·i
h
). The realization of the need to
consider the chemical properties of the contaminants as
well as the potential detrimental impacts resulting from the
physico-chemical interactions between the liquids being
contained and the soils used to contain the liquids was
more gradual, and has developed over an extended time
frame. In particular, beginning in the late 1970s to early
1980s, diffusion became recognized as a potentially
important process in assessing contaminant migration
through low permeability barriers in waste containment
applications. This recognition led to a progressively greater
understanding of the role diffusion plays in a wide variety
of applications in environmental geotechnics, including
applications in both waste containment and remediation.
Thus, the objective of this paper is to provide an overview
of the role diffusion plays in the field of environmental
geotechnics.
2 WHAT IS DIFFUSION?
Diffusion is a fundamental, irreversible process whereby
random molecular motions result in the net transport of a
chemical species (e.g., ion, molecule, compound,
radionuclide, etc.) from a region of higher chemical
potential to a region of lower chemical potential (Quigley
et al. 1987, Shackelford and Daniel 1991a, Shackelford
and Moore 2013). Since chemical potential is directly
related to chemical concentration, diffusion is more
commonly described as the net transport of a chemical
species due to a gradient in the concentration of the
chemical species.
The mass flux of a chemical species in a porous
medium due to diffusion can be described by Fick's first
law, which for one-dimensional diffusion may be written
as follows (e.g. Shackelford and Daniel 1991a,
Shackelford and Rowe 1998):
*
d
c
a o c
J nD i
n D i
(1)
where
J
d
is the diffusive mass flux, or the rate of change in
mass of the chemical species per unit cross sectional area
perpendicular to the direction of diffusion [ML
-2
T
-1
; M =
units of mass, L = units of length, and T = units of time],
n
is the total porosity of the porous medium,
D
*
is the
effective diffusion coefficient [L
2
T
-1
],
a
(< 1) is the
apparent tortuosity factor [-],
D
o
is the aqueous-phase or
free solution (without porous medium) diffusion
coefficient [L
2
T
-1
], and
i
c
is the concentration gradient in
the direction of diffusion [-], which is positive when
directed towards decreasing solute concentration. The
apparent tortuosity factor,
a
, represents the product of the
actual matrix tortuosity factor representing the geometry of
the interconnected pores,
m
(< 1), and the restrictive
tortuosity factor,
r
, as follows (Malusis and Shackelford
2002a, Shackelford and Moore 2013):
a m r
(2)
where
r
represents the product of all other factors that may
be effective in reducing the diffusive mass flux of a
chemical species, such as ion exclusion. In essence,
r
represents the ratio of the effective to total porosities, or
(Shackelford and Moore 2013):
e
r
n
n
(3)
where
n
e
≤
n
such that
r
≤ 1.The recognition of an
effective porosity takes into account the possibility that
that there may be pores that are not interconnected or are
inaccessible to specific solutes such that only a fraction of
the pore space may be available for diffusion (Shackelford
and Moore 2013).
Fick's second law governing transient one-dimensional
diffusion of chemical species subject to first-order linear
decay in porous media can be written as follows (e.g.,
Shackelford and Daniel 1991a, Shackelford and Rowe
1998, Shackelford and Moore 2013):
* 2
2
2
2
a
d
C D C
C
C D
C
t
R x
x
(4)
where
C
is solute concentration [ML
-3
],
R
d
is the
dimensionless retardation factor,
D
a
(=
D
*
/
R
d
) is the
apparent diffusion coefficient [L
2
T
-1
], and
is the decay
constant [T
-1
]. For chemical species subjected to first-order
decay (e.g., radionuclides),
is inversely related to the half
life of the chemical species,
t
1/2
, such that
decreases as
t
1/2
increases. For this reason, the decay term in Eq. 4 can
be (and often is) ignored without any significant loss in
accuracy for chemical species with half lives that are
considerably longer than the time frame being considered
for diffusion (Shackelford and Moore 2013).
The retardation factor in Eq. 4 accounts for linear,
reversible, and instantaneous sorption of a chemical
species, and represents the ratio of the total mass of
chemical species per unit total volume of porous medium
relative to the aqueous-phase mass of chemical species per
unit total volume of porous medium. For water saturated
porous media,
R
d
may be expressed as follows:
1
d
d
d
R
K
n
(5)
