Equilibrium Models in Hydro and Geochemistry
In the last decades, a large variety of models and computer programs has been developed for applications in hydro and geochemistry. From a thermodynamic point of view they solve the following task:
Given:  temperature T, pressure P, and total concentrations of elements^{1} 
Find:  concentrations c_{i} (or molar amounts n_{i}) of chemical species in aqueous and other phases in equilibrium state 
In all calculations partial/local equilibrium is assumed, therefore they are also termed “thermodynamic equilibrium models” (in contrast to kinetic models).
Classification Schema
Despite the huge diversity we encounter in hydro/geochemistry models and software today, all these approaches fall into two major categories:
 models based on LMA – Law of Mass Action
 models based on GEM – Gibbs Energy Minimization
LMA. The LMA approach is common, and available in many computer programs, such as PhreeqC, Minteq^{2}, Wateq4f^{3}, EQ3/6^{4}, Chess/JChess^{5}, The Geochemist’s Workbench^{6} and others. The program aqion, that relies on the numerical solver of PhreeqC, belongs to the LMA category as well.
GEM. The GEM approach, represented by codes such as Solgasmix^{7}, ChemSage^{8}, FactSage^{9}, Selektor^{10} or GEMSPSI^{11}, is less common, but is becoming increasingly popular.
The mathematical treatment of the equilibrium speciation problem yields nonlinear equation systems that can only be solved iteratively (such as root finding with NewtonRaphson method and other techniques). Thus, the application of numerical computer programs is a natural consequence.
Remarks & References

They are called ‘master species’. ↩

Allison, J.D., D.S. Brown, K.J. NovoGradac: MINTEQA2/ProdefA2, A Geochemical Assessment Model for Environmental Systems, Version 3.0, User’s Manual, EPA/600/391/021, March 1991 ↩

Ball J.W. and D.K. Nordstrom: WATEQ4F – User’s manual with revised thermodynamic data base and test cases for calculating speciation of major, trace and redox elements in natural waters, USGS OpenFile Report 90129, 185 p, 1991. ↩

Wolery, T.J., S.A. Daveler: EQ6, A Computer Program for Reaction Path Modeling of Aqueous Geochemical Systems: Theoretical Manual, User’s Guide, and Related Documentation (Version 7.0), Lawrence Livermore National Laboratory UCRLMA110662 PT IV, Oct 1992 ↩

Jan van der Lee, L. DeWindt: CHESS Tutorial and Cookbook, User’s Manual, Ecole Nationale Superieure des Mines de Paris Report LHM/RD/02/13, Centre d’Informatique Geologique, Apr 2002 ↩

Bethke, C.M., S. Yeakel: The Geochemist’s Workbench User’s Guides, Version 10.0. Aqueous Solutions LLC, Champaign, 2014 ↩

Eriksson. G, Chem. Scr., 8, p 100103, 1975 ↩

Eriksson, G., K. Hack: CHEMSAGE – A computer program for the calculation of complex chemical equilibria. Metallurgical Transactions B b21, 10131023, 1990 ↩

Bale, C.W., P. Chartrand, S.A. Degterov, G. Eriksson, K. Hack, R. Ben Mahfoud, J. Melançon, A.D. Pelton, S. Petersen: FactSage Thermochemical Software and Databases, published by Elsevier Science Ltd, Calphad, Vol.26, No.2, pp. 189228, 2002 ↩

Karpov I.K., K.V. Chudnenko, D.A. Kulik, O.V. Avchenko, V.A. Bychinskii: Minimization of Gibbs free energy in geochemical systems by convex programming, Geochem. International, 39, 11081119 2001 ↩

Kulik, D., U. Berner, E. Curti: Modeling chemical equilibrium partitioning with the GEMSPSI code. PSI Scientific Report 2003 / Volume IV, Nuclear Energy and Safety. Villigen, Paul Scherrer Institut: 109122, 2004 ↩