.. _example-phreeqc-manual-16:

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
------------------------------

.. raw:: html
   :file: study_16 - Inverse Modelling.html

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.

:ref:`example-phreeqc-manual-17` adds evaporation to the same approach, and
:ref:`example-phreeqc-manual-18` 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.