18 - Inverse Modeling of the Madison Aquifer¶
The third and most constrained of the inverse models: the Madison aquifer in Montana, with isotopes added to the mole balance.
16 - Inverse Modelling ends on the difficulty that several different reaction sets can explain the same water. This example is the answer to it.
Dedolomitization¶
TITLE Example 18.--Inverse modeling of Madison aquifer
SOLUTION 1 Recharge number 3
units mmol/kgw
temp 9.9
pe 0.
pH 7.55
Ca 1.2
Mg 1.01
Na 0.02
K 0.02
Fe(2) 0.001
Cl 0.02
S(6) 0.16
S(-2) 0
C(4) 4.30
-i 13C -7.0 1.4
-i 34S 9.7 0.9
SOLUTION 2 Mysse
units mmol/kgw
temp 63.
pH 6.61
pe 0.
redox S(6)/S(-2)
Ca 11.28
Mg 4.54
Na 31.89
K 2.54
Fe(2) 0.0004
Cl 17.85
S(6) 19.86
S(-2) 0.26
C(4) 6.87
-i 13C -2.3 0.2
-i 34S(6) 16.3 1.5
-i 34S(-2) -22.1 7
INVERSE_MODELING 1
-solutions 1 2
-uncertainty 0.05
-range
-isotopes
13C
34S
-balances
Fe(2) 1.0
ph 0.1
-phases
Dolomite dis 13C 3.0 2
Calcite pre 13C -1.5 1
Anhydrite dis 34S 13.5 2
CH2O dis 13C -25.0 5
Goethite
Pyrite pre 34S -22. 2
CaX2 pre
Ca.75Mg.25X2 pre
MgX2 pre
NaX
Halite
Sylvite
PHASES
Sylvite
KCl = K+ + Cl-
-log_k 0.0
CH2O
CH2O + H2O = CO2 + 4H+ + 4e-
-log_k 0.0
EXCHANGE_SPECIES
0.75Ca+2 + 0.25Mg+2 + 2X- = Ca.75Mg.25X2
log_k 0.0
END
The process being quantified is dedolomitization, and it is driven by gypsum:
gypsum dissolves, adding calcium and sulfate;
the added calcium pushes calcite to precipitate;
removing carbonate lets dolomite dissolve to replace it;
which adds more calcium and magnesium, and the cycle continues.
The net effect is dolomite and gypsum dissolving while calcite precipitates, driven by the common ion. It is a textbook carbonate-aquifer process, and the mole-balance question is how far it has gone at each point along the flow path.
Isotopes as extra constraints¶
Carbon isotopes are included in the balance alongside the elements.
This is what sharpens the result. Each carbon-bearing phase has its own isotopic signature – dolomite, calcite, soil CO₂ and organic carbon differ in δ¹³C – so requiring the isotopes to balance as well as the carbon rules out reaction sets that balance the elements but would need carbon from the wrong source.
Mole balance alone cannot distinguish two carbon sources that contribute the same number of moles. Isotopes can, because they are a second, independent accounting of the same atoms. Each additional constraint of this kind cuts the number of models that survive.
That is the general principle, and it is why isotope data earns its cost in this kind of study: the limiting factor in inverse modelling is almost never the solver, it is the number of independent constraints available.
Source¶
Parkhurst, D. L. and Appelo, C. A. J. (2013). Description of input and examples for PHREEQC version 3. U.S. Geological Survey Techniques and Methods, book 6, chapter A43. This is Example 18 of that manual.
Plummer, L. N., Busby, J. F., Lee, R. W. and Hanshaw, B. B. (1990). Geochemical modeling of the Madison aquifer in parts of Montana, Wyoming, and South Dakota. Water Resources Research 26, 1981-2014. The dedolomitization study this reproduces is theirs.