16 - Inverse Modelling¶
Every other example in this set runs forward: given a water and some reactions, what comes out? This one runs backwards. Given two waters – an initial and a final – what reactions account for the difference?
That is inverse modelling, and it answers the question a field geochemist actually has, since the two waters are what gets sampled and the reactions are what has to be inferred.
The case is the chemical evolution of spring waters in the Sierra Nevada, from the classic paper of Garrels and Mackenzie (1967): dilute snowmelt reacting with granite.
Mole balance, not a simulation¶
TITLE Example 16.--Inverse modeling of Sierra springs
SOLUTION_SPREAD
-units mmol/L
Number pH Si Ca Mg Na K Alkalinity S(6) Cl
1 6.2 0.273 0.078 0.029 0.134 0.028 0.328 0.01 0.014
2 6.8 0.41 0.26 0.071 0.259 0.04 0.895 0.025 0.03
INVERSE_MODELING 1
-solutions 1 2
-uncertainty 0.025
-range
-phases
Halite
Gypsum
Kaolinite precip
Ca-montmorillonite precip
CO2(g)
Calcite
Chalcedony precip
Biotite dissolve
Plagioclase dissolve
-balances
Ca 0.05 0.025
PHASES
Biotite
KMg3AlSi3O10(OH)2 + 6H+ + 4H2O = K+ + 3Mg+2 + Al(OH)4- + 3H4SiO4
log_k 0.0 # No log_k, Inverse modeling only
Plagioclase
Na0.62Ca0.38Al1.38Si2.62O8 + 5.52 H+ + 2.48H2O = \
0.62Na+ + 0.38Ca+2 + 1.38Al+3 + 2.62H4SiO4
log_k 0.0 # No log_k, inverse modeling only
END
Nothing is simulated here. PHREEQC is given the two analyses and a list of candidate phases, and it solves for the set of mole transfers – so much plagioclase dissolved, so much kaolinite precipitated – that turns the first water into the second while balancing every element, the charge, and the electrons.
Two things follow from that, and both matter.
The analytical uncertainty is part of the problem. Each analysis is given a tolerance, and the solver is allowed to adjust concentrations within it. Without that there would usually be no exact solution, because real analyses do not balance perfectly – so inverse modelling has to be posed as fitting within error rather than as solving equations.
There is usually more than one answer. PHREEQC returns every set of phases that works, not one. A model that accounts for the data is not thereby the explanation: several different reaction paths can produce the same water, and distinguishing them needs evidence from outside the chemistry – the mineralogy actually present, isotopes, what is petrologically plausible.
The results are the mole transfers of each candidate phase in each model found.
What it shows¶
That inverse modelling constrains hypotheses rather than selecting one. Its value is as much in the models it rules out – those that cannot balance within the analytical error – as in those it returns.
17 - Inverse Modelling with Evaporation adds evaporation to the same approach, and 18 - Inverse Modeling of the Madison Aquifer adds isotopes, which is how the ambiguity described above gets narrowed in practice.
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 16 of that manual.
Garrels, R. M. and Mackenzie, F. T. (1967). Origin of the chemical compositions of some springs and lakes. In Equilibrium Concepts in Natural Water Systems, Advances in Chemistry Series 67, American Chemical Society, 222-242. The spring waters and the original interpretation are theirs.