Geoscience Reference
In-Depth Information
From Eq. (
10.1
), together with the equation of continuity, we can derive Fick's
second law, otherwise known simply as the diffusion equation; in one-dimensional
form, this is given by
2
c
ox
2
:
o
c
ot
¼ D
o
ð
10
:
2
Þ
This equation relates the temporal concentration of a diffusing chemical to its
location in space. In real soil and aquifer materials, the diffusion coefficient can be
affected by the temperature and properties of the solid matrix, such as mineral
composition (which affects sorption, a process that can be difficult to separate from
diffusion), bulk density, and critically, water content.
Advection (or convection) is the process by which chemicals are transported by
the average (or bulk) water velocity. Thus, the advective flux, J
adv
, is described
simply by
;
J
adv
¼ qc ¼
cK
ð
h
Þ
dh
ð
h
Þ
dx
ð
10
:
3
Þ
where q denotes the specific discharge given by Darcy's law (Eq. 9.1). As for
Eq. (
10.1
), the advective flux is given in units of mass per area per time, that is, the
mass of chemical passing through a unit cross-sectional area of porous material per
unit time. Combining the diffusive and advective fluxes in Eqs. (
10.1
) and (
10.3
),
using, e.g., a volumetric mass conservation, leads to the (one-dimensional form of
the) advection-diffusion equation:
o
ð
hc
Þ
ot
;
¼
o
ox
ð
vhc
Þþ
o
D
h
oc
ox
ð
10
:
4
Þ
ox
where v denotes the average water velocity.
Clearly, particularly, because of the complex topology of the three-dimensional
network of pores that make up soils and aquifer materials, large-scale (larger than
the pore scale) contaminant spreading also is driven by local velocity fluctuations
around q, so that mechanical dispersion must be considered as well. Mechanical
dispersion accounts for the local variations in water flux about the average,
advective flux. It can be considered an artifact of upscaling, that is, of the use of a
continuum-level averaged (''homogenized'') transport equation. Considering
transport processes at the local or pore-scale level, this parameter often is
neglected, which is well justified if the advective flux is characterized at a suffi-
ciently high resolution.
The traditional, advection-dispersion equation, a generalization of Eq. (
10.4
),
is then written in one-dimensional form, as
;
o
ð
hc
Þ
ot
¼
o
ox
ð
vhc
Þþ
o
D
k
h
oc
ox
ð
10
:
5
Þ
ox
