Environmental Engineering Reference
In-Depth Information
against the sedimentation direction. The reason is the shrinking size of the pore
space due to compaction processes. Holzbecher (
2002
) showed that in such
a situation the velocity
v
, the sedimentation velocity (burial rate/area)
v
bur
and the
final porosity in the compacted sediment
y
1
are connected by the formula (see
Sidebar 9.1):
v ¼
y
1
y
v
bur
(9.11)
thus determines the local velocity.
Then the general solution for variable parameters
The depth-dependent porosity profile
vðxÞ
0
@
1
A
exp
ð
x
1
JðxÞ
vðxÞ
DðxÞ
cðxÞ¼c
in
þ c
1
c
in
ð
Þ
with
JðxÞ¼
dx
(9.12)
1
JðLÞ
0
has to be taken into account. Only for special cases of parameter dependencies an
analytical solution can be derived, as the integral may be difficult to evaluate. For
that reason numerical methods have to be applied to achieve approximate solutions.
This can be done using MATLAB
. As an example, we assume a porosity profile
which changes exponentially between a high value
®
y
0
at the sediment-water
interface and the final value
y
1
:
yðxÞ¼y
1
þðy
0
y
1
Þ
exp
að Þ
(9.13)
Solutions with MATLAB
®
for the variable parameter situation are given below.
Sulphate Reduction
Dealing with sulphate reduction in aquatic sediments, Berner (
1964
) introduces
a simple model on organic matter. Organic carbon is utilized by bacteria in the
biochemical degradation process. It is assumed that '
the rate of metabolism of the
bacteria is directly proportional to the concentration of utilizable organic matter
'.
Taking into account the sedimentation velocity
w
, which is derived from the burial
rate, he solves the differential equation for the steady state of organic matter in the
solid phase:
w
@
@z
c
org;s
lc
org;s
¼
0
(9.14)
in the physical unit [time
-1
]. Equation
(
9.14
) is a differential equation for the concentration of organic matter in the solid
phase
c
org,s
. Such an equation was already treated in chap. 4 (equation (4.6)). The
equation reflects that under steady state conditions there is a balance between
degradation of organic matter and sedimentation. It can be derived from the general
with degradation constant of organic matter
l
