15 Conceptual Models and Computer Models
528
The model is now finished and the essential
margin values compiled in the upper frame of
Table 15.5 are added. The switch in G3 is then set
to zero. Now the key F9 (re-calculation of the
worksheet) is repeatedly pressed until the figures
in the worksheet do not (significantly) change,
i.e., until a (practically) steady state is reached.
Depending on the speed of the computer and the
spreadsheet software used, this can take some few
minutes, so that one can hold the F9-key down
during that time.
At this point it needs to be clearly stated that
only the final iterative steady state of the model
provides a correct result, because the diffusive
flux is calculated according to Fick’s First Law of
Diffusion in the columns (D) or (G) which is only
valid when stationary conditions are provided.
Even if time intervals of specific length are
employed in the process of iteration, it is not
permitted, by no means at all, to draw conclusions
on the time necessary for reaching the steady
state, since the columns (D) and (G), and consequently (F) and (I) as well, contain false information relative to each of the respective nonsteady states.
The length of the time intervals (cell C6) is
only a mechanism used for iteration, whereby too
short intervals will result in an unnecessarily long
pressing of the key F9, whereas too large intervals
would lead to escalating oscillations within the
model (in our example after about dt=2.3E6). Only
after the steady state is reached will this ‘false
way’ of the iteration become ‘consigned to oblivion’, and the result will be correct in all columns.
Non-steady states and intervals prior to the development of particular concentration patterns can
only be managed by applying Fick’s Second Law
of Diffusion, either in form of analytical solutions
(cf. Section 15.2) or in form of numerical solutions
(cf. Sections 15.3.2 and 15.3.3).
The concentration-dependent boundary conditions assigned to the model’s upper-boundary
limits (unspecified but constant concentrations in
the cells F11 and I11) are relatively unproblematic
since they represent a constant concentration in
the bottom water zone. This is an imperative
prerequisite for assuming a steady state in pore
waters from superficial sediments. The condition
for the lower boundary of the profile is somewhat
more problematical. In the above example, the
concentrations of both oxygen and nitrate were
set to zero. In the model, this is tolerable as long
as the value 0.0, constituting the lower boundary
(within the limits) of the model, already results
from the reactions, and as long as the set value 0.0
will not exert any practical influence on the lower
parts of the profile. This ‘zero concentration
boundary condition’ hence constitutes a ‘zero flux
boundary condition’. This also implies that the
boundaries of the model must be extended downward until this state is sufficiently assumed. Selecting a zero concentration and, consequently a
‘zero flux boundary condition’, makes sense in
this example, but it must not always be an
adequate boundary condition. In the following
example, we will consider the case in which a
value is chosen which clearly deviates from 0.0
and thus indirectly provides a steady diffusive
flux across the lower boundary.
Modeling the Reaction Sulfate/Methane
Niewöhner et al. (1998) describe profiles of pore
water which were obtained from upwelling areas
off the shores of Namibia. In these pore-water
samples, sulfate reacted with methane in a ratio of
1:1. A similar example has already been introduced
in Chapter 3 (Figure 3.31, also cf. Chapter 5), in
context with problems occurring in methane analytics. In this example, the measured values will
now be presented in Figure 15.6 together with the
result of a model designed according to the
‘Press-F9- method’.
In principle, the worksheet for the modeling of
Figure 15.6 does not differ very much from the one
described previously, so that it does not demand
special description. The essential features of this
model are:
The decomposition of methane and sulfate (cf.
Chapter 8) occur for both reactants in specific
depths at identical rates which were entered into
the spreadsheet. The adjustment to the measured
profiles was committed only by these (microbial)
decomposition rates. The decomposition parameter is set to 0.0 in all other cells, so that a diffusion controlled transport occurs with a constant
concentration gradient.
The concentration of the upper boundary
condition for the model is again provided by the
appropriate bottom water concentrations: sulfate
= 28 mmoles/l; methane = 0.0 mmoles/l.
The concentration of the lower boundary
condition for sulfate is 0.0 mmol/l, since it was
decomposed to 0.0 mmol/l long before the lower
boundary of the model had been reached. However, the concentration of the lower boundary
528
The model is now finished and the essential
margin values compiled in the upper frame of
Table 15.5 are added. The switch in G3 is then set
to zero. Now the key F9 (re-calculation of the
worksheet) is repeatedly pressed until the figures
in the worksheet do not (significantly) change,
i.e., until a (practically) steady state is reached.
Depending on the speed of the computer and the
spreadsheet software used, this can take some few
minutes, so that one can hold the F9-key down
during that time.
At this point it needs to be clearly stated that
only the final iterative steady state of the model
provides a correct result, because the diffusive
flux is calculated according to Fick’s First Law of
Diffusion in the columns (D) or (G) which is only
valid when stationary conditions are provided.
Even if time intervals of specific length are
employed in the process of iteration, it is not
permitted, by no means at all, to draw conclusions
on the time necessary for reaching the steady
state, since the columns (D) and (G), and consequently (F) and (I) as well, contain false information relative to each of the respective nonsteady states.
The length of the time intervals (cell C6) is
only a mechanism used for iteration, whereby too
short intervals will result in an unnecessarily long
pressing of the key F9, whereas too large intervals
would lead to escalating oscillations within the
model (in our example after about dt=2.3E6). Only
after the steady state is reached will this ‘false
way’ of the iteration become ‘consigned to oblivion’, and the result will be correct in all columns.
Non-steady states and intervals prior to the development of particular concentration patterns can
only be managed by applying Fick’s Second Law
of Diffusion, either in form of analytical solutions
(cf. Section 15.2) or in form of numerical solutions
(cf. Sections 15.3.2 and 15.3.3).
The concentration-dependent boundary conditions assigned to the model’s upper-boundary
limits (unspecified but constant concentrations in
the cells F11 and I11) are relatively unproblematic
since they represent a constant concentration in
the bottom water zone. This is an imperative
prerequisite for assuming a steady state in pore
waters from superficial sediments. The condition
for the lower boundary of the profile is somewhat
more problematical. In the above example, the
concentrations of both oxygen and nitrate were
set to zero. In the model, this is tolerable as long
as the value 0.0, constituting the lower boundary
(within the limits) of the model, already results
from the reactions, and as long as the set value 0.0
will not exert any practical influence on the lower
parts of the profile. This ‘zero concentration
boundary condition’ hence constitutes a ‘zero flux
boundary condition’. This also implies that the
boundaries of the model must be extended downward until this state is sufficiently assumed. Selecting a zero concentration and, consequently a
‘zero flux boundary condition’, makes sense in
this example, but it must not always be an
adequate boundary condition. In the following
example, we will consider the case in which a
value is chosen which clearly deviates from 0.0
and thus indirectly provides a steady diffusive
flux across the lower boundary.
Modeling the Reaction Sulfate/Methane
Niewöhner et al. (1998) describe profiles of pore
water which were obtained from upwelling areas
off the shores of Namibia. In these pore-water
samples, sulfate reacted with methane in a ratio of
1:1. A similar example has already been introduced
in Chapter 3 (Figure 3.31, also cf. Chapter 5), in
context with problems occurring in methane analytics. In this example, the measured values will
now be presented in Figure 15.6 together with the
result of a model designed according to the
‘Press-F9- method’.
In principle, the worksheet for the modeling of
Figure 15.6 does not differ very much from the one
described previously, so that it does not demand
special description. The essential features of this
model are:
The decomposition of methane and sulfate (cf.
Chapter 8) occur for both reactants in specific
depths at identical rates which were entered into
the spreadsheet. The adjustment to the measured
profiles was committed only by these (microbial)
decomposition rates. The decomposition parameter is set to 0.0 in all other cells, so that a diffusion controlled transport occurs with a constant
concentration gradient.
The concentration of the upper boundary
condition for the model is again provided by the
appropriate bottom water concentrations: sulfate
= 28 mmoles/l; methane = 0.0 mmoles/l.
The concentration of the lower boundary
condition for sulfate is 0.0 mmol/l, since it was
decomposed to 0.0 mmol/l long before the lower
boundary of the model had been reached. However, the concentration of the lower boundary