Radial flow takes place in a heterogeneous porous formation of
random and stationary
log-conductivity Y(x), characterized by the mean
〈Y〉, the variance σ2Y
and the two-
point autocorrelation ρY which in turn has
finite and different horizontal and vertical
integral scales, I and Iv,
respectively. The steady flow is driven by a head difference
between a fully penetrating well and an outer boundary, the mean velocity
U being
radial. A tracer is injected for a short time through the well envelope
and the thin
plume spreads due to advection by the random velocity field and to pore-scale
dispersion. Transport is characterized by the mean front
r=R(t) and by the second
spatial moment of the plume Srr. Under ergodic
conditions, i.e. for a well length much
larger than the vertical integral scale, Srr
is equal to the radial fluid trajectory variance
Xrr.
The aim of the study is to determine Xrr(t)
for a given heterogeneous structure
and for given pore-scale dispersivities. The problem is more complex than
the similar
one for mean uniform flow. To simplify it, the well is replaced by a line
source, the
domain is assumed to be infinite and a first-order approximation in
σ2Y is adopted. The
solution is still difficult, being expressed with the aid of a few quadratures.
It is found,
however, that it can be derived quite accurately for a sufficiently small
anisotropy
ratio e=Iv/I by
retaining only one term of the velocity two-point covariance. This
major simplification leads to simple calculations and even to analytical
solutions in
the absence of pore-scale dispersion.
To compare the results with those prevailing in homogeneous media, apparent
and
equivalent macrodispersivities are defined for convenience.
The major difference between transport in radial and uniform flow is
that the
asymptotic, large-time, apparent macrodispersivity in the former is smaller
by a
factor of 3 than in the latter. For a three-dimensional point source the
reduction is by
a factor of 5. This effect is explained by the rapid change of the mean
velocity during
the period in which the velocities of two particles injected at the source
become
uncorrelated.
In contrast, the equivalent macrodispersivity tends to its value in
uniform flow far
from the well, where the flow is slowly varying in space.