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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Geochemical modeling</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p><b>Geochemical modeling</b> or <b>theoretical geochemistry</b> is the practice of using <a href="Chemical_thermodynamics" title="Chemical thermodynamics">chemical thermodynamics</a>, <a href="Chemical_kinetics" title="Chemical kinetics">chemical kinetics</a>, or both, to analyze the <a href="Chemical_reactions" class="mw-redirect" title="Chemical reactions">chemical reactions</a> that affect <a href="Geology" title="Geology">geologic systems</a>, commonly with the aid of a computer. It is used in high-temperature <a href="Geochemistry" title="Geochemistry">geochemistry</a> to simulate reactions occurring deep in the Earth's interior, in <a href="Magma" title="Magma">magma</a>, for instance, or to model low-temperature reactions in <a href="Aqueous_solutions" class="mw-redirect" title="Aqueous solutions">aqueous solutions</a> near the Earth's surface, the subject of this article.
</p>
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<div class="mw-heading mw-heading2"><h2 id="Applications_to_aqueous_systems">Applications to aqueous systems</h2></div>
<p>Geochemical modeling is used in a variety of fields, including environmental <a href="Environmental_protection" title="Environmental protection">protection</a> and <a href="Environmental_remediation" title="Environmental remediation">remediation</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> the <a href="Petroleum_industry" title="Petroleum industry">petroleum industry</a>, and <a href="Economic_geology" title="Economic geology">economic geology</a>.<sup id="cite_ref-GBRM_2-0" class="reference"><a href="#cite_note-GBRM-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Models can be constructed, for example, to understand the composition of natural waters; the mobility and breakdown of <a href="Pollution" title="Pollution">contaminants</a> in flowing <a href="Groundwater" title="Groundwater">groundwater</a> or <a href="Surface_water" title="Surface water">surface water</a>; the <a href="Ion_speciation" title="Ion speciation">ion speciation</a> of <a href="Plant_nutrients_in_soil" title="Plant nutrients in soil">plant nutrients in soil</a> and of <a href="Heavy_metals" title="Heavy metals">regulated metals</a> in <a href="Waste_management" title="Waste management">stored solid wastes</a>; the formation and dissolution of rocks and <a href="Minerals" class="mw-redirect" title="Minerals">minerals</a> in geologic formations in response to injection of industrial wastes, steam, or <a href="Carbon_sequestration" title="Carbon sequestration">carbon dioxide</a>; the dissolution of carbon dioxide in seawater and its effect on <a href="Ocean_acidification" title="Ocean acidification">ocean acidification</a>; and the generation of <a href="Acid_mine_drainage" title="Acid mine drainage">acidic waters</a> and leaching of metals from mine wastes.
</p><p>For instance, geochemical modeling of aqueous systems also indicates that, under underground conditions for large ammonia geo-storage, there is typically no significant change in the rock mass. Injecting ammonia into deep porous rocks at specific temperature and pressure conditions has a negligible dissolution effect. This insight, obtained through geochemical simulation and modeling and shows its potential for the research area.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Development_of_geochemical_modeling">Development of geochemical modeling</h2></div>
<p><a href="Robert_Garrels" title="Robert Garrels">Garrels</a> and Thompson (1962) first applied chemical modeling to geochemistry in 25 °C and one atmosphere total pressure. Their calculation, computed by hand, is now known as an <i>equilibrium model</i>, which predicts species distributions, mineral saturation states, and gas fugacities from measurements of bulk solution composition. By removing small aliquots of <a href="Solvent" title="Solvent">solvent</a> water from an equilibrated spring water and repeatedly recalculating the species distribution, Garrels and Mackenzie (1967) simulated the reactions that occur as spring water evaporated.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This coupling of mass transfer with an equilibrium model, known as a <i>reaction path model</i>, enabled geochemists to simulate reaction processes.
</p><p><a href="Harold_C._Helgeson" class="mw-redirect" title="Harold C. Helgeson">Helgeson</a> (1968) introduced the first computer program to solve equilibrium and reaction path models,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> which he and coworkers used to model geological processes like <a href="Weathering" title="Weathering">weathering</a>, sediment <a href="Diagenesis" title="Diagenesis">diagenesis</a>, <a href="Evaporation" title="Evaporation">evaporation</a>, <a href="Hydrothermal_alteration" class="mw-redirect" title="Hydrothermal alteration">hydrothermal alteration</a>, and <a href="Ore_genesis" title="Ore genesis">ore deposition</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Later developments in geochemical modeling included reformulating the governing equations, first as <a href="Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">ordinary differential equations</a>, then later as <a href="Algebraic_equations" class="mw-redirect" title="Algebraic equations">algebraic equations</a>. Additionally, <a href="Component_(thermodynamics)" title="Component (thermodynamics)">chemical components</a> came to be represented in models by aqueous species, minerals, and gases, rather than by the elements and electrons which make up the species, simplifying the governing equations and their numerical solution.<sup id="cite_ref-GBRM_2-1" class="reference"><a href="#cite_note-GBRM-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>Recent improvements in the power of personal computers and <a href="#Software_programs_in_common_use">modeling software</a> have made geochemical models more accessible and more flexible in their implementation.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Geochemists are now able to construct on their laptops complex reaction path or <a href="Reactive_transport_modeling_in_porous_media" title="Reactive transport modeling in porous media">reactive transport models</a> which previously would have required a supercomputer.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Setting_up_a_geochemical_model">Setting up a geochemical model</h2></div>
<p>An aqueous system is uniquely defined by its chemical composition, <a href="Temperature" title="Temperature">temperature</a>, and <a href="Pressure" title="Pressure">pressure</a>.<sup id="cite_ref-Anderson_9-0" class="reference"><a href="#cite_note-Anderson-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Creating geochemical models of such systems begins by choosing the basis, the set of <a href="Chemical_species" title="Chemical species">aqueous species</a>, <a href="Minerals" class="mw-redirect" title="Minerals">minerals</a>, and <a href="Gases" class="mw-redirect" title="Gases">gases</a> which are used to write chemical reactions and express composition. The number of basis entries required equals the number of <a href="Component_(thermodynamics)" title="Component (thermodynamics)">components</a> in the system, which is fixed by the <a href="Gibbs'_phase_rule" class="mw-redirect" title="Gibbs' phase rule">phase rule</a> of thermodynamics. Typically, the basis is composed of water, each mineral in equilibrium with the system, each gas at known <a href="Fugacity" title="Fugacity">fugacity</a>, and important aqueous species. Once the basis is defined, a modeler can solve for the <a href="Chemical_equilibrium" title="Chemical equilibrium">equilibrium state</a>, which is described by <a href="Law_of_mass_action" title="Law of mass action">mass action</a> and mass balance equations for each component.<sup id="cite_ref-GBRM_2-2" class="reference"><a href="#cite_note-GBRM-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>In finding the equilibrium state, a geochemical modeler solves for the distribution of mass of all species, minerals, and gases which can be formed from the basis. This includes the <a href="Activity_(chemistry)" class="mw-redirect" title="Activity (chemistry)">activity</a>, <a href="Activity_coefficient" title="Activity coefficient">activity coefficient</a>, and <a href="Concentration" title="Concentration">concentration</a> of aqueous species, the <a href="Solution_(chemistry)" title="Solution (chemistry)">saturation</a> state of minerals, and the fugacity of gases. Minerals with a saturation index (log Q/K) equal to zero are said to be in equilibrium with the fluid. Those with positive saturation indices are termed <a href="Supersaturation" title="Supersaturation">supersaturated</a>, indicating they are favored to precipitate from solution. A mineral is undersaturated if its saturation index is negative, indicating that it is favored to dissolve.<sup id="cite_ref-Anderson_9-1" class="reference"><a href="#cite_note-Anderson-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>
</p><p>Geochemical modelers commonly create reaction path models to understand how systems respond to changes in composition, temperature, or pressure. By configuring the manner in which mass and heat transfer are specified (i.e., open or closed systems), models can be used to represent a variety of geochemical processes. Reaction paths can assume chemical equilibrium, or they can incorporate kinetic rate laws to calculate the timing of reactions. In order to predict the distribution in space and time of the chemical reactions that occur along a flowpath, geochemical models are increasingly being coupled with <a href="Groundwater_model" title="Groundwater model">hydrologic models</a> of mass and heat transport to form <a href="Reactive_transport_modeling_in_porous_media" title="Reactive transport modeling in porous media">reactive transport models</a>.<sup id="cite_ref-GBRM_2-3" class="reference"><a href="#cite_note-GBRM-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Specialized geochemical modeling programs that are designed as cross-linkable re-entrant software objects enable construction of reactive transport models of any flow configuration.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Types_of_reactions">Types of reactions</h2></div>
<p>Geochemical models are capable of simulating many different types of <a href="Chemical_reaction" title="Chemical reaction">reactions</a>. Included among them are:
</p>
<ul><li><a href="Acid-base_reactions" class="mw-redirect" title="Acid-base reactions">Acid-base reactions</a></li>
<li><a href="Coordination_complex" title="Coordination complex">Aqueous complexation</a></li>
<li>Mineral <a href="Dissolution_(chemistry)" class="mw-redirect" title="Dissolution (chemistry)">dissolution</a> and <a href="Precipitation_(chemistry)" title="Precipitation (chemistry)">precipitation</a>, including <a href="Ostwald_ripening" title="Ostwald ripening">Ostwald ripening</a></li>
<li>Reduction and oxidation (<a href="Redox" title="Redox">redox</a>) reactions, including those catalyzed by <a href="Enzyme" title="Enzyme">enzymes</a>, <a href="Reactions_on_surfaces" title="Reactions on surfaces">surfaces</a>, and <a href="Microorganism" title="Microorganism">microorganisms</a></li>
<li><a href="Adsorption" title="Adsorption">Sorption</a>, <a href="Ion_exchange" title="Ion exchange">ion exchange</a>, and surface complexation</li>
<li>Gas dissolution and <a href="Degasification" class="mw-redirect" title="Degasification">exsolution</a></li>
<li><a href="Stable_isotope" class="mw-redirect" title="Stable isotope">Stable isotope fractionation</a></li>
<li><a href="Radioactive_decay" title="Radioactive decay">Radioactive decay</a></li></ul>
<p>Simple <a href="Phase_diagrams" class="mw-redirect" title="Phase diagrams">phase diagrams</a> or plots are commonly used to illustrate such geochemical reactions. Eh-pH <a href="Pourbaix_diagram" title="Pourbaix diagram">(Pourbaix)</a> diagrams, for example, are a special type of activity diagram which represent acid-base and redox chemistry graphically.
</p>
<div class="mw-heading mw-heading2"><h2 id="Uncertainties_in_geochemical_modelling">Uncertainties in geochemical modelling</h2></div>
<p>Various sources can contribute to a range of simulation results. The range of the simulation results is defined as model uncertainty. One of the most important sources not possible to quantify is the conceptual model, which is developed and defined by the modeller. Further sources are the parameterization of the model regarding the hydraulic (only when simulating transport) and mineralogical properties.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The parameters used for the geochemical simulations can also contribute to model uncertainty. These are the applied thermodynamic database and the parameters for the kinetic minerals dissolution.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Differences in the thermodynamic data (i.e. equilibrium constants, parameters for temperature correction, activity equations and coefficients) can result in large uncertainties. Furthermore, the large spans of experimentally derived rate constants for minerals dissolution rate laws can cause large variations in simulation results. Despite this is well-known, uncertainties are not frequently considered when conducting geochemical modelling.<sup id="cite_ref-:0_13-0" class="reference"><a href="#cite_note-:0-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>
</p><p>Reducing uncertainties can be achieved by comparison of simulation results with experimental data, although experimental data does not exist at every temperature-pressure condition and for every chemical system.<sup id="cite_ref-:0_13-1" class="reference"><a href="#cite_note-:0-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Although such a comparison or calibration can not be conducted consequently the geochemical codes and thermodynamic databases are state-of-the-art and the most useful tools for predicting geochemical processes.
</p>
<div class="mw-heading mw-heading2"><h2 id="Software_programs_in_common_use">Software programs in common use</h2></div>
<ul><li><a href="Aqion" title="Aqion">Aquion</a></li>
<li><a rel="nofollow" class="external text" href="https://www.eawag.ch/en/department/surf/projects/chemeql/">ChemEQL</a><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="https://chemplugin.gwb.com/">ChemPlugin</a></li>
<li><a rel="nofollow" class="external text" href="http://chess.geosciences.ensmp.fr/home/view?set_language=en">CHESS</a>,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140719094056/http://hr.geosciences.ensmp.fr/modelisation/hytec">HYTEC</a></li>
<li>CHILLER,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> CHIM-XPT</li>
<li><a rel="nofollow" class="external text" href="https://bitbucket.org/crunchflow/crunchtope-dev/wiki/Home">CrunchFlow</a><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="http://www.wipp.energy.gov/library/CRA/2009_CRA/references/Others%5CWolery_1992_EQ36_A_Software_Package_for_Geochemical_Modeling_of_Aqueous_Systems_ERMS241375.pdf">EQ3/EQ6</a><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="http://gems.web.psi.ch/">GEMS-PSI</a><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="http://www.plantmineralnutrition.net/software/geochem_ez/index.html">GEOCHEM-EZ</a><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li>
<li><a href="The_Geochemist's_Workbench" title="The Geochemist's Workbench">The Geochemist's Workbench</a><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="https://gibbsstudio.io">GibbsStudio</a></li>
<li><a rel="nofollow" class="external text" href="http://community.gwb.com/community_overview.php">GWB Community Edition</a></li>
<li><a rel="nofollow" class="external text" href="http://www.stormwater.ucf.edu/research/yeh_publications/HYDROGEOCHEM_5-0.pdf">HYDROGEOCHEM</a><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="http://www.mineql.com/">MINEQL+</a><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="https://archive.today/20121212100732/http://www.epa.gov/ceampubl/mmedia/minteq/">MINTEQA2</a><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li>
<li>ORCHESTRA<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="https://wwwbrr.cr.usgs.gov/projects/GWC_coupled/phreeqc/">PHREEQC</a><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="http://reaktoro.org/">Reaktoro</a><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="http://www.telusplanet.net/public/geogams/products/default.html">SOLMINEQ.88, GAMSPATH.99</a><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="https://tough.lbl.gov/software/toughreact/">TOUGHREACT</a><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="https://vminteq.lwr.kth.se/">Visual MINTEQ</a><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="https://wwwbrr.cr.usgs.gov/projects/GWC_chemtherm/software.htm">WATEQ4F</a><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="https://worm-portal.asu.edu/">Water-Organic-Rock-Microbe (WORM) Portal</a><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup></li>
<li><a rel="nofollow" class="external text" href="http://www.ceh.ac.uk/products/software/wham/index.html">WHAM</a><sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup></li></ul>
<p>The <a rel="nofollow" class="external text" href="https://water.usgs.gov/software/lists/geochemical">USGS website</a> provides free access to many of the software listed above. <sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Chemical_thermodynamics" title="Chemical thermodynamics">Chemical thermodynamics</a></li>
<li><a href="Chemical_kinetics" title="Chemical kinetics">Chemical kinetics</a></li>
<li><a href="Geochemistry" title="Geochemistry">Geochemistry</a></li>
<li><a href="Geomicrobiology" title="Geomicrobiology">Geomicrobiology</a></li>
<li><a href="Hydrogeology" title="Hydrogeology">Hydrogeology</a></li>
<li><a href="Groundwater_model" title="Groundwater model">Groundwater model</a></li>
<li><a href="Reactive_transport_modeling_in_porous_media" title="Reactive transport modeling in porous media">Reactive transport model</a></li>
<li><a href="Reservoir_simulation" title="Reservoir simulation">Reservoir simulation</a></li>
<li><a href="Chemical_process_modeling" title="Chemical process modeling">Chemical process modeling</a></li>
<li><a href="Chemical_transport_model" title="Chemical transport model">Chemical transport model</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>
<ul><li>Appelo, C.A.J. and D. Postma, 2005, Geochemistry, Groundwater, and Pollution. Taylor & Francis, 683 pp. <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0415364287</bdi></li>
<li>Bethke, C.M., 2008, Geochemical and Biogeochemical Reaction Modeling. Cambridge University Press, 547 pp. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0521875547</bdi></li>
<li>Merkel, B.J., B. Planer-Friedrich, and D.K. Nordstrom, 2008, Groundwater Geochemistry: A Practical Guide to Modeling of Natural and Contaminated Aquatic Systems. Springer, 242 pp. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3540746676</bdi></li>
<li>Oelkers, E.H. and J. Schott (eds.), 2009, Thermodynamics and Kinetics of Water-Rock Interaction. <i>Reviews in Mineralogy and Geochemistry</i> <b>70</b>, 569 pp. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-939950-84-3</bdi></li>
<li>Zhu, C. and G. Anderson, 2002, Environmental Applications of Geochemical Modeling. Cambridge University Press, 300 pp. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0521005777</bdi></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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