Environmental Engineering Reference
In-Depth Information
pumped for drinking water or any other purposes, the contaminants can return back
to the ground surface with the potential to produce hazards in the antroposphere.
The time scale for such a return can differ substantially. In the vicinity of surface
water bodies the residence time in the subsurface compartments may be as short as
several days or weeks. When deeper groundwater layers are involved, the time scale
of residence is surely to be counted in years. In several regions fossil groundwater is
pumped which was recharged several 1,000s of years before. In comparison to the
ambient state without any anthropogenic influence, the natural residence time
within the subsurface may be significantly shortened by pumping wells.
It was already mentioned in Sidebar 11.1 that in situations in which the wall
friction is a relevant process a linear relationship between flux or velocity on one
side and the pressure or head gradient on the other side can be expected. The
relationship was derived from the Navier-Stokes (
11.1
) and (
11.4
) for one-
dimensional pipe flow. A similar situation is given for fluid flow within the pore
space of a porous matrix or porous medium. Pour water movements in sediments,
seepage through the soil, or groundwater flow in aquifers are environmental
systems that fall into that category.
The mentioned proportionality between flux and pressure drop for porous media
is stated in
Darcy's Law
. In 1856, Henry Darcy
10
was the first who published such
a proportionality law. He has performed a series of experiments in metal columns
filled with sand. An experimental set-up, which in many parts resembles the
original facility, is sketched in Fig.
11.7
.
Driven by a pressure gradient water flows from the inlet of the column to the
outlet. The pressure is kept constant if the two piezometer pipes are connected with
water reservoirs of constant height. The height difference
D
h
between both reser-
voir levels is taken as a measure for the pressure difference. The finding from the
Darcy experiment is stated mathematically as follows:
Q=A /
D
h=L
(11.11)
where
Q
denotes the volumetric flow rate,
A
the cross-sectional area,
L
the length of
the column and
D
h
the head gradient. In the 150 years that passed since the first
publication, Darcy's law has been confirmed to be valid for a huge variety of porous
media and for wide ranges of velocities and scales. In terms of the Reynolds-
number the limiting value is
Re ¼
10 (Bear
1972
). For higher values of
Re
a generalized formulation has to be used, but in most subsurface aqueous environ-
ments the velocities are so low that Re stays below the threshold.
Nowadays it is common to formulate the law as an explicit equation for the
Darcy velocity in an infinitesimal scale of the three-dimensional space:
y
v
¼K
f
rh
(11.12)
10
Henry Darcy (1803-1858), French engineer.
