.. _example-phreeqc-manual-18:

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.

:ref:`example-phreeqc-manual-16` ends on the difficulty that several
different reaction sets can explain the same water. This example is the
answer to it.


Dedolomitization
----------------

.. raw:: html
   :file: study_18 - Inverse Modeling of the Madison Aquifer.html

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.