15 Conceptual Models and Computer Models
538
concentration of 40 µmol l
-1
which is required
in order to obtain saturation in the pore water
fraction relative to the mineral hydroxyapatite
which is known as a primary precipitate of
phosphorite deposits.
• Phosphate release rates of sulfur bacteria ranging from 0.024 to 0.08 pmol l
-1
cell
-1
had to be
postulated. In case of higher release rates,
calcium which is required for the precipitation
of hydroxyapatite [Ca 5 OH(PO 4 ) 3 ] could not
have been supplied fast enough by means of
diffusion. In case of a lower release rate, diffusion would have caused a distinctly broader
spreading of the concentration peak in the
pore water fraction than the peak measured. A
phosphate release rate of 0.011 to 0.028 pmol l
-1
cell
-1
for the bacteria was found in laboratory
experiments. This range is very close to the
postulated values, both ranges even overlap
to certain extent. That the bacteria did not
produce the highest release rates of the model
may be explained with the fact that they simply
did not appreciate their state of ‘captivity’ as
much as their life in the wild. A precipitation
rate applicable to hydroxyapatite emerged
almost incidentally, which was also verified in
laboratory experiments.
• Anticipating the fastest release rate of 0.08
pmol l
-1
cell
-1
it will take only three months
until the contents measured in the sediment
are reached. In case of the lowest rate of 0.024
pmol l
-1
cell
-1
it will take 14 months. Both
values are plausible also against the background of sedimentation rates on the shelf.
• The phosphorite layers are quite well protected
after being embedded in the sediment, for concentrations produced by the dissolution of these
layers need a very long time to return to the sediment surface by diffusion. Even after 100 modeled years, the phosphate contents in the sediment will be only slightly lower than they were at
the time of their development. And 100 years on
the shelf is plenty of time during which everything from further and deeper deposition to erosion of the sediment can happen.
15.3.3 Two-Step Models for Combined
Complex Transport/Reaction
Processes
The main problem encountered in modeling
combinations of advective/dispersive and/or
diffusive transport, on the one hand, and any
(bio)geochemical reaction, on the other hand, is
that at least two totally different processes interact simultaneously on the same object:
Physical transport including diffusion, advection and dispersion, when reactions are absent, is
one part of the problem. In their entirety, these
processes are quantitatively well understood in
model concepts. They are also applicable, without
raising principle problems, in analytical and numerical solutions to the general partial differential
equations of material transport.
Geochemical and (bio)geochemical processes
which are independent of transport processes
present the other part of the problem. Here as well,
there is a number of far advanced model concepts.
The models belonging to the PHREEQE or
PHREEQC - type may be referred to in this respect
(cf. Section 15.1.1 and 15.1.2). Other fields are
object of the various chapters in this book.
First, attempts were made to develop models
that could accomplish the coupling of both
process groups in one single, either analytical or
numerical procedure, although these often bore
strong limitations for the application to geochemical reactions. Attempts now more often
foresee solutions in two steps. This group of
methods is referred to either as “operator splitting”, or as the “two-step method”. Boudreau
(1997) mentions this procedure only briefly at the
close of his book and describes it as ‘an apparently crazy idea that works rather well in practice’. The first versions of such models, which
couple physical transport to geochemical reactions in groundwater are already more than 20
years old (e.g. Schulz and Reardon 1983).
Application examples for the ‘two-step method’ in
modeling diagenetic processes in marine sediments over the recent years were published by
Hamer and Sieger (1994), Van Cappellen and Wang
(1995), Wang and Van Cappellen (1996), Hensen et
al. (1997), Sieger (1993), Landenberger et al. (1997),
Landenberger (1998), Adler et al. (2000a), Adler et
al. (2001), Wenzhöfer et al. (2001), Pfeifer et al.
(2002), Haese et al. (2003), Hensen et al. (2003).
The model CoTReM (Column Transport and
Reaction Model) (Adler et al. 2000b) is primarily
designed for its application to complex early
diagenetic reactions in marine sediments. In the
one-dimensional transport part of the model (as a
continuance of the model DISPER by Flühler and
Jury 1983) the model’s territory is divided up into
a variable number of REVs (representative
elementary volumes). These REVs may differ in
538
concentration of 40 µmol l
-1
which is required
in order to obtain saturation in the pore water
fraction relative to the mineral hydroxyapatite
which is known as a primary precipitate of
phosphorite deposits.
• Phosphate release rates of sulfur bacteria ranging from 0.024 to 0.08 pmol l
-1
cell
-1
had to be
postulated. In case of higher release rates,
calcium which is required for the precipitation
of hydroxyapatite [Ca 5 OH(PO 4 ) 3 ] could not
have been supplied fast enough by means of
diffusion. In case of a lower release rate, diffusion would have caused a distinctly broader
spreading of the concentration peak in the
pore water fraction than the peak measured. A
phosphate release rate of 0.011 to 0.028 pmol l
-1
cell
-1
for the bacteria was found in laboratory
experiments. This range is very close to the
postulated values, both ranges even overlap
to certain extent. That the bacteria did not
produce the highest release rates of the model
may be explained with the fact that they simply
did not appreciate their state of ‘captivity’ as
much as their life in the wild. A precipitation
rate applicable to hydroxyapatite emerged
almost incidentally, which was also verified in
laboratory experiments.
• Anticipating the fastest release rate of 0.08
pmol l
-1
cell
-1
it will take only three months
until the contents measured in the sediment
are reached. In case of the lowest rate of 0.024
pmol l
-1
cell
-1
it will take 14 months. Both
values are plausible also against the background of sedimentation rates on the shelf.
• The phosphorite layers are quite well protected
after being embedded in the sediment, for concentrations produced by the dissolution of these
layers need a very long time to return to the sediment surface by diffusion. Even after 100 modeled years, the phosphate contents in the sediment will be only slightly lower than they were at
the time of their development. And 100 years on
the shelf is plenty of time during which everything from further and deeper deposition to erosion of the sediment can happen.
15.3.3 Two-Step Models for Combined
Complex Transport/Reaction
Processes
The main problem encountered in modeling
combinations of advective/dispersive and/or
diffusive transport, on the one hand, and any
(bio)geochemical reaction, on the other hand, is
that at least two totally different processes interact simultaneously on the same object:
Physical transport including diffusion, advection and dispersion, when reactions are absent, is
one part of the problem. In their entirety, these
processes are quantitatively well understood in
model concepts. They are also applicable, without
raising principle problems, in analytical and numerical solutions to the general partial differential
equations of material transport.
Geochemical and (bio)geochemical processes
which are independent of transport processes
present the other part of the problem. Here as well,
there is a number of far advanced model concepts.
The models belonging to the PHREEQE or
PHREEQC - type may be referred to in this respect
(cf. Section 15.1.1 and 15.1.2). Other fields are
object of the various chapters in this book.
First, attempts were made to develop models
that could accomplish the coupling of both
process groups in one single, either analytical or
numerical procedure, although these often bore
strong limitations for the application to geochemical reactions. Attempts now more often
foresee solutions in two steps. This group of
methods is referred to either as “operator splitting”, or as the “two-step method”. Boudreau
(1997) mentions this procedure only briefly at the
close of his book and describes it as ‘an apparently crazy idea that works rather well in practice’. The first versions of such models, which
couple physical transport to geochemical reactions in groundwater are already more than 20
years old (e.g. Schulz and Reardon 1983).
Application examples for the ‘two-step method’ in
modeling diagenetic processes in marine sediments over the recent years were published by
Hamer and Sieger (1994), Van Cappellen and Wang
(1995), Wang and Van Cappellen (1996), Hensen et
al. (1997), Sieger (1993), Landenberger et al. (1997),
Landenberger (1998), Adler et al. (2000a), Adler et
al. (2001), Wenzhöfer et al. (2001), Pfeifer et al.
(2002), Haese et al. (2003), Hensen et al. (2003).
The model CoTReM (Column Transport and
Reaction Model) (Adler et al. 2000b) is primarily
designed for its application to complex early
diagenetic reactions in marine sediments. In the
one-dimensional transport part of the model (as a
continuance of the model DISPER by Flühler and
Jury 1983) the model’s territory is divided up into
a variable number of REVs (representative
elementary volumes). These REVs may differ in