527
yet prevalent oxygen concentration. This does not
mean to say that this must always be so. As for
the model, any other kinetic, or, in general terms,
any other value could be assigned to the respective cells. As for the first order kinetics example,
the line E12 reads in Excel
®
notation as follows:
E12=$I$3*F12
Cells E13 to E60 are copied from this cell accordingly, cell E61 remains empty. Column (F)
contains O 2 -concentrations, whereby cell F11
contains the upper boundary condition via a
constant bottom water concentration. Cell F61
contains the concentration at the lower boundary
- which in this particular case is equivalent to 0.0.
The cell F12 is calculated from cell F12 (hence
from its own previous value) as well as from the
cells D11, D12, and E12, each multiplied with the
length of the time interval. Written in Excel
®
notation this reads:
F12=IF($G$3;$D$6;F12+(D11*$C$6)(D12*$C$6)-(E12*$C$6))
(The function ‘IF’ refers to the English version of
Excel
®
, e.g. for the German version this function
would be ‘WENN’.)
Cells F13 to F60 are copied from this cell accordingly. Cell F61 contains the lower boundary
conditions (in this case 0.0) which will be further
dealt with later. It appears of practical importance
at this point of calculation that there is the
option of stopping the calculation procedure by
means of a switch function. If the switch cell G3
assumes zero value the calculation will be performed. If the cell contains the value 1, then all
concentrations will be adjusted to the value
present in cell D6 (concentration in bottom
water). Designing the structure of the worksheet,
the switch should be set to a value of 1, in order
to avoid unnecessary error messages due to references to not yet properly filled cells. Additionally, it is easy to rebuild the worksheet
whenever ill-chosen conditions pertaining to the
iteration have led to extreme values and/or oscillations.
Column (G) contains nitrate concentrations
analogous to column (D) used for oxygen. In this
example, the same diffusion coefficients were used
to calculate the fluxes. Here, different individually
adjusted coefficients can certainly also be used.
Again, cell G11 contains the flux rate directed from
the bottom water to the sediment and reads written in Excel
®
notation:
G11=B11*C11*((I11-I12)/$A$6)
Cells G12 to G60 are copied from this cell
accordingly, whereas cell G61 remains empty.
Column (H) contains the decomposition mediated
by nitrate which can be set voluntarily, as has
been the case in oxygen mediated decomposition.
In this example, a decomposition of 0.0 has been
pre-determined for the cells H12 to H31, as well as
the assumption of first-order kinetics in cells H32
to H60. Here, too, it is not intended to claim that
this allocation must necessarily be as shown in
the example. As for the model, any other type of
kinetics would be possible as well as any values
set otherwise . Hence, cell H32 reads in Excel
®
notation:
H32=$I$6*I32
Cells H33 to H60 are copied from this cell accordingly. Cell H61 remains empty.
Analogous to column (F), column (I) this time
contains the calculated nitrate concentrations which
again are connected with loops to the columns (G)
and (H). However, in the case of nitrate, there is also
some release into the pore water fraction due to the
oxygen mediated decomposition of organic substances. Thus, each cell belonging to column (I) is
calculated from its previous value, from the
diffusive import from above, from the diffusive
export downwards, from the import from the
oxidation of organic material mediated by oxygen
(here, selective C:N ratio), as well as from the
decomposition mediated by nitrate. For cell I12,
this reads in Excel
®
notation as follows:
(G12*$C$6)+((1/$H$3)*E12*$C$6)(H12*$C$6))
(The function ‘IF’ refers to the English version of
Excel
®
, e.g. for the German version this function
would be ‘WENN’.)
Cells I13 to I60 are copied from this cell accordingly. Cell I11 contains the bottom water concentration as a fixed boundary condition, whereas cell
I61 assumes the value of 0.0 as the concentration
at the lower boundary. In this column it is also
recommended to link the calculation performance
to the operational ‘switch’ of cell G3.
15.3
Numerical Solutions for Diagenetic Models
yet prevalent oxygen concentration. This does not
mean to say that this must always be so. As for
the model, any other kinetic, or, in general terms,
any other value could be assigned to the respective cells. As for the first order kinetics example,
the line E12 reads in Excel
®
notation as follows:
E12=$I$3*F12
Cells E13 to E60 are copied from this cell accordingly, cell E61 remains empty. Column (F)
contains O 2 -concentrations, whereby cell F11
contains the upper boundary condition via a
constant bottom water concentration. Cell F61
contains the concentration at the lower boundary
- which in this particular case is equivalent to 0.0.
The cell F12 is calculated from cell F12 (hence
from its own previous value) as well as from the
cells D11, D12, and E12, each multiplied with the
length of the time interval. Written in Excel
®
notation this reads:
F12=IF($G$3;$D$6;F12+(D11*$C$6)(D12*$C$6)-(E12*$C$6))
(The function ‘IF’ refers to the English version of
Excel
®
, e.g. for the German version this function
would be ‘WENN’.)
Cells F13 to F60 are copied from this cell accordingly. Cell F61 contains the lower boundary
conditions (in this case 0.0) which will be further
dealt with later. It appears of practical importance
at this point of calculation that there is the
option of stopping the calculation procedure by
means of a switch function. If the switch cell G3
assumes zero value the calculation will be performed. If the cell contains the value 1, then all
concentrations will be adjusted to the value
present in cell D6 (concentration in bottom
water). Designing the structure of the worksheet,
the switch should be set to a value of 1, in order
to avoid unnecessary error messages due to references to not yet properly filled cells. Additionally, it is easy to rebuild the worksheet
whenever ill-chosen conditions pertaining to the
iteration have led to extreme values and/or oscillations.
Column (G) contains nitrate concentrations
analogous to column (D) used for oxygen. In this
example, the same diffusion coefficients were used
to calculate the fluxes. Here, different individually
adjusted coefficients can certainly also be used.
Again, cell G11 contains the flux rate directed from
the bottom water to the sediment and reads written in Excel
®
notation:
G11=B11*C11*((I11-I12)/$A$6)
Cells G12 to G60 are copied from this cell
accordingly, whereas cell G61 remains empty.
Column (H) contains the decomposition mediated
by nitrate which can be set voluntarily, as has
been the case in oxygen mediated decomposition.
In this example, a decomposition of 0.0 has been
pre-determined for the cells H12 to H31, as well as
the assumption of first-order kinetics in cells H32
to H60. Here, too, it is not intended to claim that
this allocation must necessarily be as shown in
the example. As for the model, any other type of
kinetics would be possible as well as any values
set otherwise . Hence, cell H32 reads in Excel
®
notation:
H32=$I$6*I32
Cells H33 to H60 are copied from this cell accordingly. Cell H61 remains empty.
Analogous to column (F), column (I) this time
contains the calculated nitrate concentrations which
again are connected with loops to the columns (G)
and (H). However, in the case of nitrate, there is also
some release into the pore water fraction due to the
oxygen mediated decomposition of organic substances. Thus, each cell belonging to column (I) is
calculated from its previous value, from the
diffusive import from above, from the diffusive
export downwards, from the import from the
oxidation of organic material mediated by oxygen
(here, selective C:N ratio), as well as from the
decomposition mediated by nitrate. For cell I12,
this reads in Excel
®
notation as follows:
(G12*$C$6)+((1/$H$3)*E12*$C$6)(H12*$C$6))
(The function ‘IF’ refers to the English version of
Excel
®
, e.g. for the German version this function
would be ‘WENN’.)
Cells I13 to I60 are copied from this cell accordingly. Cell I11 contains the bottom water concentration as a fixed boundary condition, whereas cell
I61 assumes the value of 0.0 as the concentration
at the lower boundary. In this column it is also
recommended to link the calculation performance
to the operational ‘switch’ of cell G3.
15.3
Numerical Solutions for Diagenetic Models