Environmental Engineering Reference
In-Depth Information
Darcy's law can be written as:
k
[13.4]
v
=− ∇ +
(
p
ρ
gz
)
w
µ
where
k
is the intrinsic permeability tensor,
µ
is the fluid dynamic viscosity,
p
is the
pore pressure,
z
is the vertical coordinate positively directed upwards,
g
is the
acceleration due to gravity and ρ is the fluid density.
We do not consider the notion of storage, which poses no particular problem for
the study of transfer and retention of pollutants in saturated porous media here.
13.2.2.
Transport laws: advection
A pollutant in a volume concentration
C
(average or homogenized REV value)
in a fluid in motion is carried in this fluid by
advection
: pollutant particles move
with the fluid at the same velocity by clinging to it. If we consider that the flow is
laminar (an approximation that is reasonably verified to the first order in soils), the
velocity of the pollutant is equal to that of the fluid flow
u
. The pollutant advective
flow (average or homogenized REV value) is therefore given by:
v
=
Cu
[13.5]
advection
The effective velocity
u
, or average linear velocity, differs from Darcy's
velocity. Indeed, it is expressed per unit of total area and not per unit area of actual
flow. If we make the assumption that the cross-section of the channels in which the
fluid flows (or surface porosity) is equal to the effective porosity
n
e
(i.e. porosity of
the flow), we get:
v
u
=
w
[13.6]
n
e
13.2.3.
Transport laws: dispersion
As the channels of fluid flow are not straight and of limited size, the actual flow
velocity differs from the effective velocity at point to point, which is a REV space
average (Figure 13.1). On a channel cross-section it varies between a velocity near
to zero at the contact point with soil particles, and a maximum velocity in the middle
of the fairway. The channels are tortuous and cause the pollutant to move in
orthogonal directions to that of the average flow.