Geoscience Reference
In-Depth Information
''homogeneous,'' 20-cm-to-meter-length flow cells have been shown to display
''anomalous'' (non-Fickian) early time arrivals and late time tails (Levy and
Berkowitz
2003
; Cortis et al.
2004
). Detailed analysis shows that the motion and
spreading of these chemical plumes are characterized by distinct temporal scaling;
that is, the time dependence of the spatial moments does not correspond to the
normal (or Gaussian) distribution that would otherwise lead to Fickian transport.
To account for the effect of a sufficiently broad, statistical distribution of het-
erogeneities on the overall transport, we can consider a probabilistic approach that
will generate a probability density function in space (s) and time (t), w(s, t),
describing key features of the transport. The effects of multiscale heterogeneities
on contaminant transport patterns are significant, and consideration only of the
mean transport behavior, such as the spatial moments of the concentration dis-
tribution, is not sufficient. The continuous time random walk (CTRW) approach is
a physically based method that has been advanced recently as an effective means
to quantify contaminant transport. The interested reader is referred to a detailed
review of this approach (Berkowitz et al.
2006
).
A variety of specific mathematical formulations of the CTRW approach have
been considered to date, and network models have also been applied (Bijeljic and
Blunt
2006
). A key result in development of the CTRW approach is a transport
equation that represents a strong generalization of the advection-dispersion
equation. As shown by Berkowitz et al. (
2006
), an extremely broad range of
transport patterns can be described with the (ensemble-averaged) equation
uc
ð
s
;
u
Þ
c
o
ð
s
Þ
¼
M
ð
u
Þ
½v
w
r
c
ð
s
;
u
Þ
D
w
:
rr
c
ð
s
;
u
Þ
;
ð
10
:
8
Þ
where the Laplace transform of a function f(t) is denoted by f
ð
u
Þ
;
c
o
ð
s
Þ
represents
the initial condition, and the ''memory function'' is defined as
w
ð
u
Þ
1
w
ð
u
Þ
;
M
ð
u
Þ
tu
ð
10
:
9
Þ
where t denotes a characteristic time. Here, it is important to recognize that the
''transport velocity'' v
w
is distinct from the ''average water velocity'' v, whereas in
the classical advection-dispersion picture, these velocities are identical. Similarly,
the ''dispersion,'' D
w
, has a different physical interpretation than in the usual
advection-dispersion equation definition. More important, a key feature of
Eq. (
10.8
) is that it encompasses various common models, such as multirate and
mobile-immobile transport equations (Eq.
10.7
) for specific examples of
M
ð
u
Þð
or w
ð
s
;
t
ÞÞ
, together with other simplifications. For example, under perfectly
homogeneous conditions, M
ð
u
Þ
¼1, and Eq. (
10.8
) is formally equivalent to the
classical advection-dispersion equation.
The CTRW approach accounts naturally for transport in preferential pathways,
with mass transfer to ''stagnant'' and slow flow regions; the CTRW can account
