From d233a9df297157ff0b5d062a23d1fe6be5a7bc60 Mon Sep 17 00:00:00 2001 From: Keurfon Luu Date: Fri, 22 Sep 2023 12:49:59 -0700 Subject: [PATCH] add documentation for tough3 --- doc/requirements.txt | 3 +- doc/source/_static/css/custom.css | 9 + doc/source/_static/logo.png | Bin 0 -> 21569 bytes doc/source/conf.py | 20 +- doc/source/guide/index.rst | 1 + doc/source/guide/installation.rst | 12 + doc/source/index.rst | 1 + doc/source/references.bib | 1055 ++++++++ doc/source/tough3/eos/eco2n.rst | 646 +++++ doc/source/tough3/eos/eos1.rst | 78 + doc/source/tough3/eos/eos2.rst | 53 + doc/source/tough3/eos/eos3.rst | 59 + doc/source/tough3/eos/eos4.rst | 84 + doc/source/tough3/eos/eos5.rst | 96 + doc/source/tough3/eos/eos7.rst | 163 ++ doc/source/tough3/eos/eos7r.rst | 278 ++ doc/source/tough3/eos/eos8.rst | 176 ++ doc/source/tough3/eos/eos9.rst | 116 + doc/source/tough3/eos/ewasg.rst | 247 ++ doc/source/tough3/eos/index.rst | 60 + doc/source/tough3/figures/figure_1.png | Bin 0 -> 80462 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doc/source/references.bib create mode 100644 doc/source/tough3/eos/eco2n.rst create mode 100644 doc/source/tough3/eos/eos1.rst create mode 100644 doc/source/tough3/eos/eos2.rst create mode 100644 doc/source/tough3/eos/eos3.rst create mode 100644 doc/source/tough3/eos/eos4.rst create mode 100644 doc/source/tough3/eos/eos5.rst create mode 100644 doc/source/tough3/eos/eos7.rst create mode 100644 doc/source/tough3/eos/eos7r.rst create mode 100644 doc/source/tough3/eos/eos8.rst create mode 100644 doc/source/tough3/eos/eos9.rst create mode 100644 doc/source/tough3/eos/ewasg.rst create mode 100644 doc/source/tough3/eos/index.rst create mode 100644 doc/source/tough3/figures/figure_1.png create mode 100644 doc/source/tough3/figures/figure_10.png create mode 100644 doc/source/tough3/figures/figure_2.png create mode 100644 doc/source/tough3/figures/figure_8.png create mode 100644 doc/source/tough3/figures/figure_9.png create mode 100644 doc/source/tough3/figures/figure_eco2n_1.png create mode 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}sx@m+3e^=^=TW$~aq)i72SZeZo z@i&HDyKz2h23zKzxmeaR6y8py$(PI?PP}-w8n{uKSoa!YB8I2v%XK)QiEW}7O*^NZ z)1WI!Y!IzfvDgxo358h`oDII{^&V>Si%obAq&$zv=B5#P?21ww9*N{b%rprUt=Rw0 zMCZl8?o002Lgd4xO(W#Aaa#0juLKQfH4&!o(%Ef5 znV_c%amemt-&(YgaYjB7S4+|D)|(MLJmjS(+-qiPthoh;AHKdO4vS6{)^tiUBumi~ yQv^Z1`jfm1@>6nmQh-^v&&NXi-#(bSXXZFho4QFPj7!s8wbb^irYRpg_x}Lurp1u} literal 0 HcmV?d00001 diff --git a/doc/source/conf.py b/doc/source/conf.py index c63e2143..e0e2fdca 100644 --- a/doc/source/conf.py +++ b/doc/source/conf.py @@ -42,8 +42,9 @@ "sphinx.ext.githubpages", "sphinx.ext.napoleon", "sphinxarg.ext", - # "sphinxcontrib.bibtex", + "sphinxcontrib.bibtex", "sphinx_gallery.gen_gallery", + "sphinx_subfigure", ] # Sphinx Gallery settings @@ -85,11 +86,14 @@ # napoleon_use_rtype = True # Numfig settings -numfig = False +numfig = True numfig_format = { "figure": "Figure %s", } +# BibTeX +bibtex_bibfiles = ["references.bib"] + # Add any paths that contain templates here, relative to this directory. templates_path = [ "_templates", @@ -120,13 +124,19 @@ # a list of builtin themes. # html_theme = "pydata_sphinx_theme" +html_logo = "_static/logo.png" html_theme_path = [ "_themes", ] +html_context = { + "default_mode": "light", +} # Add any paths that contain custom static files (such as style sheets) here, # relative to this directory. They are copied after the builtin static files, # so a file named "default.css" will overwrite the builtin "default.css". -# html_static_path = [ -# "_static", -# ] \ No newline at end of file +html_static_path = ["_static"] + +html_css_files = [ + "css/custom.css", +] \ No newline at end of file diff --git a/doc/source/guide/index.rst b/doc/source/guide/index.rst index 02ac76f0..dec9affd 100644 --- a/doc/source/guide/index.rst +++ b/doc/source/guide/index.rst @@ -5,4 +5,5 @@ User Guide :titlesonly: :maxdepth: 2 + installation input diff --git a/doc/source/guide/installation.rst b/doc/source/guide/installation.rst new file mode 100644 index 00000000..28e9afd8 --- /dev/null +++ b/doc/source/guide/installation.rst @@ -0,0 +1,12 @@ +Installation +============ + +**toughio** only requires a Python distribution to be installed. If you don't have any yet, we recommend to install the `Anaconda Distribution `_ with the default options. + +To install **toughio** and enable all its features, simply run the command in a terminal (e.g., Windows Terminal, Anaconda Prompt) or within an IPython shell: + +.. code-block:: + + pip install toughio[full] + +An IDE such as Spyder (automatically installed by Anaconda) or `VSCode `_ is strongly recommended to use **toughio**. \ No newline at end of file diff --git a/doc/source/index.rst b/doc/source/index.rst index 3519019a..b81c9435 100644 --- a/doc/source/index.rst +++ b/doc/source/index.rst @@ -8,3 +8,4 @@ guide/index examples/index api/index + tough3/index diff --git a/doc/source/references.bib b/doc/source/references.bib new file mode 100644 index 00000000..c7476b68 --- /dev/null +++ b/doc/source/references.bib @@ -0,0 +1,1055 @@ +@book{altunin1975thermophysical, + title={Thermophysical properties of carbon dioxide}, + author={Altunin, VV}, + year={1975}, + publisher={Wellingborough, Collets} +} + +@inproceedings{andersen1992accurate, + title={An accurate PVT model for geothermal fluids as represented by H2O-NaCl-CO2}, + author={Andersen, G and Probst, A and Murray, L and Butler, S}, + booktitle={Mixtures, Proceedings 17th Workshop on Geothermal Reservoir Engineering}, + year={1992}, + organization={Citeseer} +} + +@article{atkinson1980behavior, + title={Behavior of the Bagnore steam/CO2 geothermal reservoir, Italy}, + author={Atkinson, PG and Celati, R and Corsi, R and Kucuk, F}, + journal={Society of Petroleum Engineers Journal}, + volume={20}, + number={04}, + pages={228--238}, + year={1980}, + publisher={OnePetro} +} + +@techreport{aunzo1991wellbore, + title={Wellbore models GWELL, GWNACL, and HOLA user's guide}, + author={Aunzo, Zosimo P and Bjornsson, G and Bodvarsson, GS}, + year={1991}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@article{aziz1979petroleum, + title={Petroleum reservoir simulation. 1979}, + author={Aziz, Khalid and Settari, Antonin}, + journal={Applied Science Publ. Ltd., London, UK}, + year={1979} +} + +@article{balay2016petsc, + title={PETSc users manual (Tech. Rep. ANL-95/11-Revision 3.7)}, + author={Balay, Satish and Abhyankar, Shrirang and Adams, Mark F and Brown, Jed and Brune, Peter and Buschelman, Kris and Dalcin, Lisandro and Eijkhout, Victor and Gropp, William D and Kaushik, Dinesh and others}, + journal={Argonne National Laboratory}, + year={2016} +} + +@article{battistelli1997simulator, + title={The simulator TOUGH2/EWASG for modelling geothermal reservoirs with brines and non-condensible gas}, + author={Battistelli, Alfredo and Calore, Claudio and Pruess, Karsten}, + journal={Geothermics}, + volume={26}, + number={4}, + pages={437--464}, + year={1997}, + publisher={Elsevier} +} + +@article{battistelli2004modeling, + title={Modeling biodegradation of organic contaminants under multiphase conditions with TMVOCBio}, + author={Battistelli, Alfredo}, + journal={Vadose Zone Journal}, + volume={3}, + number={3}, + pages={875--883}, + year={2004}, + publisher={Soil Science Society of America} +} + +@inproceedings{battistelli2012improving, + title={Improving the treatment of saline brines in EWASG for the simulation of hydrothermal systems}, + author={Battistelli, Alfredo}, + booktitle={Proceedings of the TOUGH Symposium}, + pages={192--200}, + year={2012} +} + +@book{brooks1965hydraulic, + title={Hydraulic properties of porous media}, + author={Brooks, Royal Harvard}, + year={1965}, + publisher={Colorado State University} +} + +@book{carlslaw1959conduction, + title={Conduction of Heat in Solids, Oxford}, + author={Carlslaw, HS and Jaeger, JC}, + year={1959}, + publisher={Oxford University Press} +} + +@article{cass1984enhancement, + title={Enhancement of thermal water vapor diffusion in soil}, + author={Cass, A and Campbell, GS and Jones, TL}, + journal={Soil Science Society of America Journal}, + volume={48}, + number={1}, + pages={25--32}, + year={1984}, + publisher={Wiley Online Library} +} + +@techreport{charbeneau2007distribution, + title={Distribution and Recovery Model (LDRM) Volume 1: Distribution and Recovery of Petroleum Hydrocarbon Liquids in Porous Media}, + author={Charbeneau, Randall}, + year={2007}, + institution={American Petroleum Institute API} +} + +@inproceedings{coats1977geothermal, + title={Geothermal reservoir modelling}, + author={Coats, KH}, + booktitle={SPE Annual Fall Technical Conference and Exhibition}, + year={1977}, + organization={OnePetro} +} + +@techreport{coats1982effects, + title={Effects of grid type and difference scheme on pattern steamflood simulation results}, + author={Coats, KH and Ramesh, AB}, + year={1982}, + institution={Intercomp Resource Dev. and Engr. 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McGraw-Hill}, + author={Dean, John A}, + journal={New York}, + year={1999} +} + +@techreport{de1986quantitative, + title={Quantitative hydrogeology}, + author={De Marsily, Ghislain}, + year={1986}, + institution={Paris School of Mines, Fontainebleau} +} + +@article{doughty2007modeling, + title={Modeling geologic storage of carbon dioxide: comparison of non-hysteretic and hysteretic characteristic curves}, + author={Doughty, Christine}, + journal={Energy Conversion and Management}, + volume={48}, + number={6}, + pages={1768--1781}, + year={2007}, + publisher={Elsevier} +} + +@techreport{doughty2013user, + title={User's Guide for Hysteretic Capillary Pressure and Relative Permeability Functions in TOUGH2 Earth Sciences Division Lawrence Berkeley National Laboratory}, + author={Doughty, C}, + year={2013}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@article{driesner2007system1, + title={The system H2O--NaCl. Part I: Correlation formulae for phase relations in temperature--pressure--composition space from 0 to 1000 C, 0 to 5000 bar, and 0 to 1 XNaCl}, + author={Driesner, Thomas and Heinrich, Christoph A}, + journal={Geochimica et Cosmochimica Acta}, + volume={71}, + number={20}, + pages={4880--4901}, + year={2007}, + publisher={Elsevier} +} + +@article{driesner2007system2, + title={The system H2O--NaCl. Part II: Correlations for molar volume, enthalpy, and isobaric heat capacity from 0 to 1000 C, 1 to 5000 bar, and 0 to 1 XNaCl}, + author={Driesner, Thomas}, + journal={Geochimica et Cosmochimica Acta}, + volume={71}, + number={20}, + pages={4902--4919}, + year={2007}, + publisher={Elsevier} +} + +@article{edlefsen1943thermodynamics, + title={Thermodynamics of soil moisture}, + author={Edlefsen, N and Anderson, Alfred and others}, + journal={Hilgardia}, + volume={15}, + number={2}, + pages={31--298}, + year={1943}, + publisher={University of California, Agriculture and Natural Resources} +} + +@techreport{edwards1966trump, + title={TRUMP: A computer program for transient and steady-state temperature distributions in multidimensional systems}, + author={Edwards, Arthur L}, + year={1966}, + institution={Lawrence Radiation Lab., Univ. of California, Livermore} +} + +@misc{evans1982atomic, + title={The atomic nucleus. Reprint edition}, + author={Evans, RD}, + year={1982}, + publisher={Robert E. Krieger Publ. Co., Malabar, FL} +} + +@article{fatt1959effect, + title={Effect of fractional wettability on multiphase flow through porous media}, + author={Fatt, Irving and Klikoff, Waldemar A}, + journal={Journal of Petroleum Technology}, + volume={11}, + number={10}, + pages={71--76}, + year={1959}, + publisher={OnePetro} +} + +@article{faust1985transport, + title={Transport of immiscible fluids within and below the unsaturated zone: A numerical model}, + author={Faust, Charles R}, + journal={Water Resources Research}, + volume={21}, + number={4}, + pages={587--596}, + year={1985}, + publisher={Wiley Online Library} +} + +@article{finley1982swift, + title={Swift Self-Teaching Curriculum}, + author={Finley, NC and Reeves, M}, + journal={Report SAND81-0410, Sandia National Laboratory, Albuquerque, NM}, + year={1982} +} + +@techreport{garcia2001density, + title={Density of aqueous solutions of CO2}, + author={Garcia, Julio E}, + year={2001}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@book{garcia2003fluid, + title={Fluid dynamics of carbon dioxide disposal into saline aquifers}, + author={Garcia, Julio Enrique}, + year={2003}, + publisher={University of California, Berkeley} +} + +@article{ghezzehei2004modeling, + title={Modeling coupled evaporation and seepage in ventilated cavities}, + author={Ghezzehei, TA and Trautz, RC and Finsterle, S and Cook, PJ and Ahlers, CF}, + journal={Vadose Zone Journal}, + volume={3}, + number={3}, + pages={806--818}, + year={2004}, + publisher={Wiley Online Library} +} + +@article{grant1977permeability, + title={Permeability Reduction Factors at Wairakei, paper present at AICHE ASME Heat Transfer Conference}, + author={Grant, MA}, + journal={Am. Inst. of Chem. Eng., Am. Soc. of Mech. Eng., Salt Lake City, Utah, 1977}, + year={1977} +} + +@article{hadgu1990multi, + title={A multi-purpose wellbore simulator}, + author={Hadgu, T}, + journal={Geothermal Research Council. 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Applications}, + author={Narasimhan, Thiruppudaimarudhur N and Witherspoon, PA}, + journal={Water Resources Research}, + volume={14}, + number={6}, + pages={1017--1034}, + year={1978}, + publisher={Wiley Online Library} +} + +@techreport{oldenburg1995eos7r, + title={EOS7R: Radionuclide transport for TOUGH2}, + author={Oldenburg, Curtis M and Pruess, Karsten}, + year={1995}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@techreport{oldenburg2004eos7c, + title={EOS7C Version 1.0: TOUGH2 Module for Carbon Dioxide or Nitrogen inNatural Gas (Methane) Reservoirs}, + author={Oldenburg, Curtis M and Moridis, George J and Spycher, Nicholas and Pruess, Karsten}, + year={2004}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@techreport{oldenburg2015eos7ca, + title={EOS7CA Version 1.0: TOUGH2 module for gas migration in shallow subsurface porous media systems}, + author={Oldenburg, Curtis M}, + year={2015}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@article{o1985fluid, + title={Fluid and heat flow in gas-rich geothermal reservoirs}, + author={O'Sullivan, MJ and Bodvarsson, GS and Pruess, K and Blakeley, MR}, + journal={Society of Petroleum Engineers Journal}, + volume={25}, + number={02}, + pages={215--226}, + year={1985}, + publisher={OnePetro} +} + +@article{pan2014t2well, + title={T2Well—an integrated wellbore--reservoir simulator}, + author={Pan, Lehua and Oldenburg, Curtis M}, + journal={Computers \& Geosciences}, + volume={65}, + pages={46--55}, + year={2014}, + publisher={Elsevier} +} + +@article{pan2015eco2n, + title={ECO2N V2. 0: A TOUGH2 fluid property module for mixtures of water, NaCl, and CO2}, + author={Pan, Lehua and Spycher, Nicolas and Doughty, Christine and Pruess, Karsten}, + journal={Scientific report LBNL-6930E}, + year={2015} +} + +@article{parker1987model, + title={A model for hysteretic constitutive relations governing multiphase flow: 1. Saturation-pressure relations}, + author={Parker, JC and Lenhard, RJ}, + journal={Water Resources Research}, + volume={23}, + number={12}, + pages={2187--2196}, + year={1987}, + publisher={Wiley Online Library} +} + +@article{parker1987parametric, + title={A parametric model for constitutive properties governing multiphase flow in porous media}, + author={Parker, JC and Lenhard, RJ and Kuppusamy, T}, + journal={Water Resources Research}, + volume={23}, + number={4}, + pages={618--624}, + year={1987}, + publisher={Wiley Online Library} +} + +@techreport{patterson2012simple, + title={A Simple History-Dependent Nonwetting-Phase Trapping Model for the TOUGH Simulators}, + author={Patterson, Christopher G and Falta, Ronald W}, + year={2012}, + institution={Illinois State Geological Survey} +} + +@book{peaceman2000fundamentals, + title={Fundamentals of numerical reservoir simulation}, + author={Peaceman, Donald W}, + year={2000}, + publisher={Elsevier} +} + +@article{peaceman1983interpretation, + title={Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability}, + author={Peaceman, Donald W}, + journal={Society of Petroleum Engineers Journal}, + volume={23}, + number={03}, + pages={531--543}, + year={1983}, + publisher={OnePetro} +} + +@techreport{phillips1981technical, + title={A technical databook for geothermal energy utilization}, + author={Phillips, SL}, + year={1981}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@article{pickens1979finite, + title={Finite-element analysis of the transport of water and solutes in title-drained soils}, + author={Pickens, John F and Gillham, Robert W and Cameron, Douglas R}, + journal={Journal of Hydrology}, + volume={40}, + number={3-4}, + pages={243--264}, + year={1979}, + publisher={Elsevier} +} + +@article{potter1977volumetric, + title={The Volumetric Properties of Aqueous Sodium Chloride Solutions}, + author={Potter, JM and Brown, David L}, + journal={US Geological Survey, Bulletin}, + year={1977} +} + +@book{prausnitz1998molecular, + title={Molecular thermodynamics of fluid-phase equilibria}, + author={Prausnitz, John M and Lichtenthaler, Rudiger N and De Azevedo, Edmundo Gomes}, + year={1998}, + publisher={Pearson Education} +} + +@techreport{pritchett1981baca, + title={Baca geothermal demonstration project. Equation-of-state for water-carbon dioxide mixtures: Implications for Baca reservoir}, + author={Pritchett, JW and Rice, MH and Riney, TD}, + year={1981}, + institution={Systems, Science and Software, La Jolla, CA (USA)} +} + +@techreport{pruess1983gminc, + title={GMINC: A mesh generator for flow simulations in fractured reservoirs}, + author={Pruess, Karsten}, + year={1983}, + institution={Lawrence Berkeley Lab., CA (USA)} +} + +@techreport{pruess1987tough, + title={TOUGH user's guide}, + author={Pruess, Karsten}, + year={1987}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@techreport{pruess1991tough2, + title={TOUGH2: A general-purpose numerical simulator for multiphase nonisothermal flows}, + author={Pruess, K}, + year={1991}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@techreport{pruess1999tough2, + title={TOUGH2 user's guide version 2}, + author={Pruess, Karsten and Oldenburg, Curtis M and Moridis, GJ}, + year={1999}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@techreport{pruess2011eco2m, + title={ECO2M: a TOUGH2 fluid property module for mixtures of water, NaCl, and CO2, including super-and sub-critical conditions, and phase change between liquid and gaseous CO2}, + author={Pruess, Karsten}, + year={2011}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@article{pruess2004tough, + title={The TOUGH codes—A family of simulation tools for multiphase flow and transport processes in permeable media}, + author={Pruess, Karsten}, + journal={Vadose zone journal}, + volume={3}, + number={3}, + pages={738--746}, + year={2004}, + publisher={Wiley Online Library} +} + +@book{pruess2005eco2n, + title={ECO2N: A TOUGH2 fluid property module for mixtures of water, NaCl, and CO2}, + author={Pruess, Karsten}, + year={2005}, + publisher={eScholarship, University of California} +} + +@article{pruess1982fluid, + title={On fluid reserves and the production of superheated steam from fractured, vapor-dominated geothermal reservoirs}, + author={Pruess, KARSTEN and Narasimhan, TN}, + journal={Journal of Geophysical Research: Solid Earth}, + volume={87}, + number={B11}, + pages={9329--9339}, + year={1982}, + publisher={Wiley Online Library} +} + +@inproceedings{pruess1983seven, + title={A seven-point finite difference method for improved grid orientation performance in pattern steamfloods}, + author={Pruess, Karsten and Bodvarsson, FS}, + booktitle={SPE Reservoir Simulation Symposium}, + year={1983}, + organization={OnePetro} +} + +@article{pruess1985practical, + title={A practical method for modeling fluid and heat flow in fractured porous media}, + author={Pruess, Karsten and Narasimhan, TN}, + journal={Society of Petroleum Engineers Journal}, + volume={25}, + number={01}, + pages={14--26}, + year={1985}, + publisher={OnePetro} +} + +@techreport{pruess2002tmvoc, + title={TMVOC, a numerical simulator for three-phase non-isothermal flows of multicomponent hydrocarbon mixtures in saturated-unsaturated heterogeneous media}, + author={Pruess, Karsten and Battistelli, Alfredo}, + year={2002}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@techreport{pruess2002intercomparison, + title={Intercomparison of numerical simulation codes for geologic disposal of CO2}, + author={Pruess, Karsten and Garcia, Julio and Kovscek, Tony and Oldenburg, Curt and Rutqvist, Jonny and Steefel, Carl and Xu, Tianfu}, + year={2002}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@article{pruess2004code, + title={Code intercomparison builds confidence in numerical simulation models for geologic disposal of CO2}, + author={Pruess, Karsten and Garc{\'\i}a, Julio and Kovscek, Tony and Oldenburg, Curt and Rutqvist, Jonny and Steefel, Carl and Xu, Tianfu}, + journal={Energy}, + volume={29}, + number={9-10}, + pages={1431--1444}, + year={2004}, + publisher={Elsevier} +} + +@article{reeves1986theory, + title={Theory and implementation of SWIFT II, the Sandia waste-isolation flow and transport model for fractured media}, + author={Reeves, M and Ward, DS and Johns, ND and Cranwell, RM}, + journal={Rep. SAND83-1159. Sandia Natl Lab., Albuquerque, NM}, + year={1986} +} + +@article{richards1931capillary, + title={Capillary conduction of liquids through porous mediums}, + author={Richards, Lorenzo Adolph}, + journal={Physics}, + volume={1}, + number={5}, + pages={318--333}, + year={1931}, + publisher={American Institute of Physics} +} + +@article{sleijpen1993bicgstab, + title={BiCGstab (ell) for linear equations involving unsymmetric matrices with complex spectrum}, + author={Sleijpen, Gerard LG and Fokkema, Diederik R}, + journal={Electronic Transactions on Numerical Analysis.}, + volume={1}, + pages={11--32}, + year={1993}, + publisher={Kent State University} +} + +@article{span1996new, + title={A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 800 MPa}, + author={Span, Roland and Wagner, Wolfgang}, + journal={Journal of physical and chemical reference data}, + volume={25}, + number={6}, + pages={1509--1596}, + year={1996}, + publisher={American Institute of Physics for the National Institute of Standards and~…} +} + +@article{spycher1988fugacity, + title={Fugacity coefficients of H2, CO2, CH4, H2O and of H2O-CO2-CH4 mixtures: A virial equation treatment for moderate pressures and temperatures applicable to calculations of hydrothermal boiling}, + author={Spycher, Nicolas F and Reed, Mark H}, + journal={Geochimica et Cosmochimica Acta}, + volume={52}, + number={3}, + pages={739--749}, + year={1988}, + publisher={Elsevier} +} + +@article{spycher2003co2, + title={CO2-H2O mixtures in the geological sequestration of CO2. I. Assessment and calculation of mutual solubilities from 12 to 100 C and up to 600 bar}, + author={Spycher, Nicolas and Pruess, Karsten and Ennis-King, Jonathan}, + journal={Geochimica et cosmochimica acta}, + volume={67}, + number={16}, + pages={3015--3031}, + year={2003}, + publisher={Elsevier} +} + +@article{spycher2005co2, + title={CO2-H2O mixtures in the geological sequestration of CO2. II. Partitioning in chloride brines at 12--100 C and up to 600 bar}, + author={Spycher, Nicolas and Pruess, Karsten}, + journal={Geochimica et Cosmochimica Acta}, + volume={69}, + number={13}, + pages={3309--3320}, + year={2005}, + publisher={Elsevier} +} + +@article{stauffer2014joule, + title={Joule--Thomson effects on the flow of liquid water}, + author={Stauffer, Philip H and Lewis, KC and Stein, Joshua S and Travis, Bryan J and Lichtner, Peter and Zyvoloski, George}, + journal={Transport in porous media}, + volume={105}, + number={3}, + pages={471--485}, + year={2014}, + publisher={Springer} +} + +@article{stone1970probability, + title={Probability model for estimating three-phase relative permeability}, + author={Stone, HL}, + journal={Journal of petroleum technology}, + volume={22}, + number={02}, + pages={214--218}, + year={1970}, + publisher={OnePetro} +} + +@techreport{sutton1977boiling, + title={Boiling Curves at Broadlands Field, New Zealand}, + author={Sutton, FM and McNabb, A}, + year={1977}, + institution={CNRS} +} + +@article{thomas1981principles, + title={Principles of hydrocarbon reservoir simulation}, + author={Thomas, Gordon W}, + year={1981}, + publisher={IHRDC, Boston, MA} +} + +@techreport{tsang1990further, + title={Further modeling studies of gas movement and moisture migration at Yucca Mountain, Nevada}, + author={Tsang, YW and Pruess, K}, + year={1990}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@article{tuminaro1999official, + title={Official Aztec user's guide: Version 2.1}, + author={Tuminaro, Ray S and Heroux, Mike and Hutchinson, Scott A and Shadid, John N}, + journal={Sandia National Laboratories, Albuquerque, NM}, + volume={87185}, + year={1999} +} + +@article{udell1985heat, + title={Heat transfer in porous media considering phase change and capillarity—the heat pipe effect}, + author={Udell, Kent S}, + journal={International Journal of Heat and Mass Transfer}, + volume={28}, + number={2}, + pages={485--495}, + year={1985}, + publisher={Elsevier} +} + +@article{van1992bi, + title={Bi-cgstab: A fast and smoothly converging variant of bicg in the presence of rounding errors}, + author={van der Vorst, HA}, + journal={SIAM J. Sci. Statist. Comput}, + volume={13}, + pages={631--644}, + year={1992} +} + +@article{van1980closed, + title={A closed-form equation for predicting the hydraulic conductivity of unsaturated soils}, + author={Van Genuchten, M Th}, + journal={Soil science society of America journal}, + volume={44}, + number={5}, + pages={892--898}, + year={1980}, + publisher={Wiley Online Library} +} + +@book{vargaftik1975tables, + title={Tables on the thermophysical properties of liquids and gases}, + author={Vargaftik, Natan B}, + year={1975}, + publisher={Hemisphere Pub. Corp.} +} + +@incollection{vaughn1987analysis, + title={Analysis of permeability reduction during flow of heated, aqueous fluid through Westerly Granite}, + author={Vaughn, PJ}, + booktitle={Coupled processes associated with nuclear waste repositories}, + year={1987} +} + +@techreport{verma1985study, + title={A study of two-phase concurrent flow of steam and water in an unconsolidated porous medium}, + author={Verma, AK}, + year={1985}, + institution={Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States)} +} + +@article{verma1988thermohydrological, + title={Thermohydrological conditions and silica redistribution near high-level nuclear wastes emplaced in saturated geological formations}, + author={Verma, A and Pruess, K}, + journal={Journal of Geophysical Research: Solid Earth}, + volume={93}, + number={B2}, + pages={1159--1173}, + year={1988}, + publisher={Wiley Online Library} +} + +@article{vinsome1980simple, + title={A simple method for predicting cap and base rock heat losses in'thermal reservoir simulators}, + author={Vinsome, PKW and Westerveld, J}, + journal={Journal of Canadian Petroleum Technology}, + volume={19}, + number={03}, + year={1980}, + publisher={OnePetro} +} + +@techreport{walker1981studies, + title={Studies of heat transfer and water migration in soils. Final report}, + author={Walker, Wynn R and Sabey, James D and Hampton, Duana R}, + year={1981}, + institution={Colorado State Univ., Fort Collins (USA). Solar Energy Applications Lab.} +} + +@article{warren1963behavior, + title={The behavior of naturally fractured reservoirs}, + author={Warren, JE and Root, P Jj}, + journal={Society of Petroleum Engineers Journal}, + volume={3}, + number={03}, + pages={245--255}, + year={1963}, + publisher={OnePetro} +} + +@techreport{webb1996gas, + title={Gas-phase diffusion in porous media: Evaluation of an advective-dispersive formulation and the dusty-gas model including comparison to data for binary mixtures}, + author={Webb, Stephen W}, + year={1996}, + institution={Sandia National Lab.(SNL-NM), Albuquerque, NM (United States)} +} + +@techreport{webb1998review, + title={Review of enhanced vapor diffusion in porous media}, + author={Webb, Stephen W and Ho, Clifford K}, + year={1998}, + institution={Sandia National Lab.(SNL-NM), Albuquerque, NM (United States)} +} + +@article{webb1998enhanced, + title={Enhanced Vapor Diffusion in Porous Media-LDRD Final Report}, + author={Webb, SW and Ho, CK}, + journal={Sandia National Laboratories Report SAND98-2772, Albuquerque, NM}, + year={1998} +} + +@article{wu1993buckley, + title={Buckley-Leverett flow in composite porous media}, + author={Wu, Yu-Shu and Pruess, Karsten and Chen, ZX}, + journal={SPE Advanced Technology Series}, + volume={1}, + number={02}, + pages={36--42}, + year={1993}, + publisher={OnePetro} +} + +@article{yanosik1979nine, + title={A nine-point, finite-difference reservoir simulator for realistic prediction of adverse mobility ratio displacements}, + author={Yanosik, JL and McCracken, TA}, + journal={Society of Petroleum Engineers Journal}, + volume={19}, + number={04}, + pages={253--262}, + year={1979}, + publisher={OnePetro} +} + +@techreport{zhang2008user, + title={User's guide for TOUGH2-MP-a massively parallel version of the TOUGH2 code}, + author={Zhang, Keni and Wu, Yu-Shu and Pruess, Karsten and others}, + year={2008}, + institution={Ernest Orlando Lawrence Berkeley National Laboratory} +} + +@article{zhang2011time, + title={A time-convolution approach for modeling heat exchange between a wellbore and surrounding formation}, + author={Zhang, Yingqi and Pan, Lehua and Pruess, Karsten and Finsterle, Stefan}, + journal={Geothermics}, + volume={40}, + number={4}, + pages={261--266}, + year={2011}, + publisher={Elsevier} +} \ No newline at end of file diff --git a/doc/source/tough3/eos/eco2n.rst b/doc/source/tough3/eos/eco2n.rst new file mode 100644 index 00000000..1fc8eb37 --- /dev/null +++ b/doc/source/tough3/eos/eco2n.rst @@ -0,0 +1,646 @@ +.. _eco2n: + +ECO2N +===== + +Injection of CO\ :sub:`2` into saline formations has been proposed as a means whereby emissions of heat-trapping greenhouse gases into the atmosphere may be reduced. +Such injection would induce coupled processes of multiphase fluid flow, heat transfer, chemical reactions, and mechanical deformation. +Several groups have developed simulation models for subsets of these processes. +The present report describes a fluid property module "ECO2N" for the general-purpose reservoir simulator TOUGH2 (:cite:label:`pruess1999tough2, pruess2004tough`), that can be used to model non-isothermal multiphase flow in the system H\ :sub:`2`\O - NaCl - CO\ :sub:`2`. TOUGH2/ECO2N represents fluids as +consisting of two phases: a water-rich aqueous phase, herein often referred to as "liquid", and a CO\ :sub:`2`-rich phase referred to as "gas". +In addition, solid salt may also be present. +The only chemical reactions modeled by ECO2N include equilibrium phase partitioning of water and carbon dioxide between the liquid and gaseous phases, and precipitation and dissolution of solid salt. +The partitioning of H\ :sub:`2`\O and CO\ :sub:`2` between liquid and gas phases is modeled as a function of temperature, pressure, and salinity, using the recently developed correlations of :cite:label:`spycher2003co2`. +Dissolution and precipitation of salt is treated by means of local equilibrium solubility. +Associated changes in fluid porosity and permeability may also be modeled. +All phases - gas, liquid, solid - may appear or disappear in any grid block during the course of a simulation. +Thermodynamic conditions covered include a temperature range from ambient to 100˚C (approximately), pressures up to 600 bar, and salinity from zero to fully saturated. +These parameter ranges should be adequate for most conditions encountered during disposal of CO\ :sub:`2` into deep saline aquifers. + + +Theoretical Background +---------------------- + +Fluid Phases and Thermodynamic Variables in the System Water-NaCl-CO\ :sub:`2` +****************************************************************************** + +In the two-component system water-CO\ :sub:`2`, at temperatures above the freezing point of water and not considering hydrate phases, three different fluid phases may be present: an aqueous phase that is mostly water but may contain some dissolved CO\ :sub:`2`, a liquid CO\ :sub:`2`-rich phase that may contain some water, and a gaseous CO\ :sub:`2`-rich phase that also may contain some water. +Altogether there may be seven different phase combinations (:numref:`fig:eco2n.1`). +If NaCl ("salt") is added as a third fluid component, the number of possible phase combinations doubles, as in each of the seven phase combinations depicted in :numref:`fig:eco2n.1` there may or may not be an additional phase consisting of solid salt. +Liquid and gaseous CO\ :sub:`2` may coexist along the saturated vapor pressure curve of CO\ :sub:`2`, which ends at the critical point (:math:`T_{crit}`, :math:`P_{crit}`) = (31.04˚C, 73.82 bar; :cite:label:`vargaftik1975tables`), see :numref:`fig:eco2n.2`. +At supercritical temperatures or pressures there is just a single CO\ :sub:`2`-rich phase. + +.. figure:: ../figures/figure_eco2n_1.png + :name: fig:eco2n.1 + :width: 50% + + Possible phase combinations in the system water-CO\ :sub:`2`. The phase designations are a - aqueous, l - liquid CO\ :sub:`2`, g - gaseous CO\ :sub:`2`. Separate liquid and gas phases exist only at subcritical conditions. + +The present version of ECO2N can only represent a limited subset of the phase conditions depicted in :numref:`fig:eco2n.1`. +Thermophysical properties are accurately calculated for gaseous as well as for liquid CO\ :sub:`2`, but no distinction between gaseous and liquid CO\ :sub:`2` phases is made in the treatment of flow, and no phase change between liquid and gaseous CO\ :sub:`2` is treated. +Accordingly, of the seven phase combinations shown in :numref:`fig:eco2n.1`, ECO2N can represent the ones numbered 1 (single-phase aqueous with or without dissolved CO\ :sub:`2` and salt), 2 and 3 (a single CO\ :sub:`2`-rich phase that may be either liquid or gaseous CO\ :sub:`2`, and may include dissolved water), and 4 and 5 (two-phase conditions consisting of an aqueous and a single CO\ :sub:`2`-rich phase, with no distinction being made as to whether the CO\ :sub:`2`-rich phase is liquid or gas). +ECO2N cannot represent conditions 6 (two-phase mixture of liquid and gaseous CO\ :sub:`2`) and 7 (three-phase). +All sub- and super-critical CO\ :sub:`2` is considered as a single non-wetting phase, that will henceforth be referred to as "gas". +ECO2N may be applied to sub- as well as super-critical temperature and pressure conditions, but applications that involve sub-critical conditions are limited to systems in which there is no change of phase between liquid and gaseous CO\ :sub:`2`, and in which no mixtures of liquid and gaseous CO\ :sub:`2` occur. + +.. figure:: ../figures/figure_eco2n_2.png + :name: fig:eco2n.2 + :width: 75% + + Phase states of CO\ :sub:`2`. + +Numerical modeling of the flow of brine and CO\ :sub:`2` requires a coupling of the phase behavior of water-salt-CO\ :sub:`2` mixtures with multiphase flow simulation techniques. +Among the various issues raised by such coupling is the choice of notation. There are long-established notational conventions in both fields, which may lead to conflicts and misunderstandings when they are combined. +In an effort to avoid confusion, we will briefly discuss notational issues pertaining to partitioning of CO\ :sub:`2` between an aqueous and a gaseous phase. + +Phase partitioning is usually described in terms of mole fractions of the two components, which are denoted by :math:`x` and :math:`y`, respectively, where :math:`x_1` = :math:`x_{H_2O}` and :math:`x_2` = :math:`x_{CO_2}` specify mole fractions in the aqueous phase, while :math:`y_1` = :math:`y_{H_2O}` and :math:`y_2` = :math:`y_{CO_2}` give mole fractions in the gas phase (:cite:label:`prausnitz1998molecular, spycher2003co2`). +We follow this notation, except that we add the subscript "eq" to emphasize that these mole fractions pertain to equilibrium partitioning of water and CO\ :sub:`2` between co-existing aqueous and gas phases. +Accordingly, we denote the various mole fractions pertaining to equilibrium phase partitioning as :math:`x_{1, eq}`, :math:`x_{2, eq}`, :math:`y_{1, eq}`, and :math:`y_{2, eq}`, while the corresponding mass fractions are denoted using upper-case X and Y. +Mass fractions corresponding to single-phase conditions, where water and CO\ :sub:`2` concentrations are not constrained by phase equilibrium relations, are denoted by :math:`X_1` (for water) and :math:`X_2` (for CO\ :sub:`2`) in the aqueous phase, and by :math:`Y_1` and :math:`Y_2` in the gas phase. + +In the numerical simulation of brine-CO\ :sub:`2` flows, we will be concerned with the fundamental thermodynamic variables that characterize the brine-CO\ :sub:`2` system, and their change with time in different subdomains (grid blocks) of the flow system. +Four "primary variables" are required to define the state of water-NaCl-CO\ :sub:`2` mixtures, which according to conventional TOUGH2 useage are denoted by :math:`X_1`, :math:`X_2`, :math:`X_3`, and :math:`X_4`. +A summary of the fluid components and phases modeled by ECO2N, and the choice of primary thermodynamic variables, appears in :numref:`tab:eco2n.1`. +Different variables are used for different phase conditions, but two of the four primary variables are the same, regardless of the number and nature of phases present. +This includes the first primary variable :math:`X_1`, denoting pressure, and the fourth primary variable :math:`X_4` which is temperature. +The second primary variable pertains to salt and is denoted :math:`X_{sm}` rather than :math:`X_2` to avoid confusion with :math:`X_2`, the CO\ :sub:`2` mass fraction in the liquid phase. +Depending upon whether or not a precipitated salt phase is present, the variable :math:`X_{sm}` has different meaning. +When no solid salt is present, :math:`X_{sm}` denotes :math:`X_s`, the salt mass fraction referred to the two-component system water-salt. +When solid salt is present, :math:`X_s` is no longer an independent variable, as it is determined by the equilibrium solubility of NaCl, which is a function of temperature. +In the presence of solid salt, for reasons that are explained below, we use as second primary variable the quantity "solid saturation plus ten", :math:`X_{sm}` = :math:`S_s` + 10. +Here, :math:`S_s` is defined in analogy to fluid saturations and denotes the fraction of void space occupied by solid salt. + +The physical range of both :math:`X_s` and :math:`S_s` is (0, 1); the reason for defining :math:`X_{sm}` by adding a number 10 to :math:`S_s` is to enable the presence or absence of solid salt to be recognized simply from the numerical value of the second primary variable. +As had been mentioned above, the salt concentration variable :math:`X_s` is defined with respect to the two-component system H\ :sub:`2`\O - NaCl. +This choice makes the salt concentration variable independent of CO\ :sub:`2` concentration, which simplifies the calculation of the partitioning of the H\ :sub:`2`\O and CO\ :sub:`2` components between the aqueous and gas phases (see below). +In the three-component system H\ :sub:`2`\O - NaCl - CO\ :sub:`2`, the total salt mass fraction in the aqueous phase will for given :math:`X_s` of course depend on CO\ :sub:`2` concentration. +Salt mass fraction in the two-component system H\ :sub:`2`\O - NaCl can be expressed in terms of salt molality (moles m of salt per kg of water) as follows + +.. math:: + :label: eq:eco2n.1 + + X_s = \frac{m M_{NaCl}}{1000 + m M_{NaCl}} + +Here :math:`M_{NaCl}` = 58.448 is the molecular weight of NaCl, and the number 1000 appears in the denominator because molality is defined as moles per 1000 g of water. +For convenience we also list the inverse of Eq. :math:numref:`eq:eco2n.1`. + +.. math:: + :label: eq:eco2n.2 + + m = \frac{1000 \frac{X_s}{M_{NaCl}}}{1 - X_s} + +The third primary variable X3 is CO\ :sub:`2` mass fraction (:math:`X_2`) for single-phase conditions (only aqueous, or only gas) and is "gas saturation plus ten" (:math:`S_g` + 10) for two-phase (aqueous and gas) conditions. +The reason for adding 10 to :math:`S_g` is analogous to the conventions adopted for the second primary variable, namely, to be able to distinguish single-phase conditions (0 ≤ :math:`X_3` ≤ 1) from two-phase conditions (10 ≤ :math:`X_3` ≤ 11). +In single-phase conditions, the CO\ :sub:`2` concentration variable :math:`X_2` is "free", i.e., it can vary continuously within certain parameter ranges, while in two-phase aqueous-gas conditions, :math:`X_2` has a fixed value :math:`X_{2, eq}` that is a function of temperature, pressure, and salinity (see below). +Accordingly, for single-phase conditions :math:`X_2` is included among the independent primary variables (= :math:`X_3`), while for two-phase conditions, :math:`X_2` becomes a "secondary" parameter that is dependent upon primary variables (:math:`T`, :math:`P`, :math:`X_s`). +"Switching" primary variables according to phase conditions present provides a very robust and stable technique for dealing with changing phase compositions. + +Initialization of a simulation with TOUGH2/ECO2N would normally be made with the internally used primary variables as listed in :numref:`tab:eco2n.1`. +For convenience of the user, additional choices are available for initializing a flow problem. + +.. list-table:: Summary of ECO2N. + :name: tab:eco2n.1 + :widths: 1 3 + :align: center + + * - Components + - | #1: water + | #2: NaCl + | #3: CO\ :sub:`2` + * - Parameter choices + - | (``NK``, ``NEQ``, ``NPH``, ``NB``) = + | (3, 4, 3, 6) water, air, oil, nonisothermal (default) + | (3, 3, 3, 6) water, air, oil, isothermal + | Molecular diffusion can be modeled by setting ``NB`` = 8 + * - Primary variables + - | Single-phase conditions (only aqueous, or only gas)*: + | (:math:`P`, :math:`X_{sm}`, :math:`X_3`, :math:`T`): (pressure, salt mass fraction in two-component water-salt system or solid saturation plus ten, CO\ :sub:`2` mass fraction in the aqueous phase or in the gas phase, in the three-component system water-salt-CO\ :sub:`2`, temperature) + | Two-phase conditions (aqueous and gas)*: + | (:math:`P`, :math:`X_{sm}` + 10, :math:`S_g`, :math:`T`): (pressure, salt mass fraction in two-component water-salt system or solid saturation plus ten, gas phase saturation, temperature) + +.. note:: + + \* When discussing fluid phase conditions, we refer to the potentially mobile (aqueous and gas) phases only; in all cases solid salt may precipitate or dissolve, adding another active phase to the system. + + +Phase Composition +^^^^^^^^^^^^^^^^^ + +The partitioning of H\ :sub:`2`\O and CO\ :sub:`2` among co-existing aqueous and gas phases is calculated from a slightly modified version of the correlations developed in (:cite:label:`spycher2005co2`). +These correlations were derived from the requirement that chemical potentials of all components must be equal in different phases. +For two-phase conditions, they predict the equilibrium composition of liquid (aqueous) and gas (CO\ :sub:`2`-rich) phases as functions of temperature, pressure, and salinity, and are valid in the temperature range 12˚C ≤ :math:`T` ≤ 110˚C, for pressures up to 600 bar, and salinity up to saturated NaCl brines. +In the indicated parameter range, mutual solubilities of H\ :sub:`2`\O and CO\ :sub:`2` are calculated with an accuracy typically within experimental uncertainties. +The modification made in ECO2N is that CO\ :sub:`2` molar volumes are calculated using a tabular EOS based on Altunin's correlation (:cite:label:`altunin1975thermophysical`), instead of the Redlich-Kwong equation of state used in (:cite:label:`spycher2005co2`). +This was done to maintain consistency with the temperature and pressure conditions for phase change between liquid and gaseous conditions used elsewhere in ECO2N. +Altunin's correlations yield slightly different molar volumes than the Redlich-Kwong EOS whose parameters were fitted by :cite:label:`spycher2005co2` to obtain the best overall match between observed and predicted CO\ :sub:`2` concentrations in the aqueous phase. +The (small) differences in Altunin's molar volumes cause predictions for the mutual solubility of water and CO\ :sub:`2` to be somewhat different also. However, the differences are generally small, see :numref:`fig:eco2n.3` to :numref:`fig:eco2n.5`. + +.. figure:: ../figures/figure_eco2n_3.png + :name: fig:eco2n.3 + :width: 75% + + Dissolved CO\ :sub:`2` mass fractions in two-phase system at :math:`T` = 30˚C for pure water (0m) and 4-molal NaCl brine. Lines represent the original correlation of :cite:label:`spycher2005co2` that uses a Redlich-Kwong EOS for molar volume of CO\ :sub:`2`. Symbols represent data calculated by ECO2N in which the molar volume of CO\ :sub:`2` is obtained from the correlations of :cite:label:`altunin1975thermophysical`. + +.. figure:: ../figures/figure_eco2n_4.png + :name: fig:eco2n.4 + :width: 75% + + H\ :sub:`2`\O mass fractions in gas in two-phase system at :math:`T` = 30˚C for pure water (0m) and 4-molal NaCl brine. Lines represent the original correlation of :cite:label:`spycher2005co2` that uses a Redlich-Kwong EOS for molar volume of CO\ :sub:`2`. Symbols represent data calculated by ECO2N in which the molar volume of CO\ :sub:`2` is obtained from the correlations of :cite:label:`altunin1975thermophysical`. + +.. figure:: ../figures/figure_eco2n_5.png + :name: fig:eco2n.5 + :width: 75% + + Concentration of water in gas and CO\ :sub:`2` in the liquid (aqueous) phase at (:math:`T`, :math:`P`) = (45˚C, 216.18 bar), for salinities ranging from zero to fully saturated. Lines were calculated from the correlation of :cite:label:`spycher2005co2` that uses a Redlich-Kwong EOS for molar volume of CO\ :sub:`2`. Symbols represent data calculated by ECO2N from a modified correlation in which the molar volume of CO\ :sub:`2` is obtained from the correlations of :cite:label:`altunin1975thermophysical`. + +For conditions of interest to geologic dispoal of CO\ :sub:`2`, equilibrium between aqueous and gas phases corresponds to a dissolved CO\ :sub:`2` mass fraction in the aqueous phase, :math:`X_{2, eq}`, on the order of a few percent, while the mass fraction of water in the gas phase, :math:`Y_{1, eq}`, is a fraction of a percent, so that gas phase CO\ :sub:`2` mass fraction :math:`Y_{2, eq}` = 1 - :math:`Y_{1, eq}` is larger than 0.99. +The relationship between CO\ :sub:`2` mass fraction :math:`X_3` and phase composition of the fluid mixture is as follows (see :numref:`fig:eco2n.6`): + +- :math:`X_3` < :math:`X_{2, eq}` corresponds to single-phase liquid conditions; +- :math:`X_3` > :math:`X_{2, eq}` corresponds to single-phase gas; +- intermediate values :math:`X_{2, eq}` ≤ :math:`X_3` ≤ :math:`Y_{2, eq}` correspond to two-phase conditions with different proportions of aqueous and gas phases. + +Dissolved NaCl concentrations may for typical sequestration conditions range as high as 6.25 molal. +This corresponds to mass fractions of up to :math:`X_{sm}` = 26.7% in the two-component system water-salt. +Phase conditions as a function of :math:`X_{sm}` are as follows: + +- :math:`X_{sm}` ≤ :math:`XEQ` corresponds to dissolved salt only; +- :math:`X_{sm}` > :math:`XEQ` corresponds to conditions of a saturated NaCl brine and solid salt. + +.. figure:: ../figures/figure_eco2n_6.png + :name: fig:eco2n.6 + :width: 75% + + CO\ :sub:`2` phase partitioning in the system H\ :sub:`2`\O - NaCl - CO\ :sub:`2`. The CO\ :sub:`2` mass fraction in brine-CO\ :sub:`2` mixtures can vary in the range from 0 (no CO\ :sub:`2`) to 1 (no brine). :math:`X_{2, eq}` and :math:`Y_{2, eq}` denote, respectively, the CO\ :sub:`2` mass fractions in aqueous and gas phases corresponding to equilibrium phase partitioning in two-phase conditions. Mass fractions less than :math:`X_{2, eq}` correspond to conditions in which only an aqueous phase is present, while mass fractions larger than :math:`Y_{2, eq}` correspond to single-phase gas conditions. Mass fractions intermediate between :math:`X_{2, eq}` and :math:`Y_{2, eq}` correspond to two-phase conditions with different proportions of aqueous and gas phases. + + +Phase Change +^^^^^^^^^^^^ + +In single-phase (aqueous or gas) conditions, the third primary variable X3 is the CO\ :sub:`2` mass fraction in that phase. +In single-phase aqueous conditions, we must have :math:`X_3` ≤ :math:`X_{2, eq}`, while in single-phase gas conditions, we must have :math:`X_3` ≥ :math:`Y_{2, eq}`. +The possibility of phase change is evaluated during a simulation by monitoring :math:`X_3` in each grid block. +The criteria for phase change from single-phase to two-phase conditions may be written as follow: + +- single-phase aqueous conditions: a transition to two-phase conditions (evolution of a gas phase) will occur when :math:`X_3` > :math:`X_{2, eq}`; +- single-phase gas conditions: a transition to two-phase conditions (evolution of an aqueous phase) will occur when :math:`X_3` < :math:`Y_{2, eq}` = 1 - :math:`Y_{1, eq}`. + +When two-phase conditions evolve in a previously single-phase grid block, the third primary variable is switched to :math:`X_3` = :math:`S_g` + 10. +If the transition occurred from single-phase liquid conditions, the starting value of :math:`S_g` is chosen as 10\ :sup:`-6`; if the transition occurred from single-phase gas, the starting value is chosen as 1 - 10\ :sup:`-6`. + +In two-phase conditions, the third primary variable is :math:`X_3` = :math:`S_g` + 10. +For two-phase conditions to persist, :math:`X_3` must remain in the range (10, 11 - :math:`S_s`). +Transitions to single-phase conditions are recognized as follows: + +- if :math:`X_3` < 10 (i.e., :math:`S_g` < 0): gas phase disappears; make a transition to single-phase liquid conditions; +- if :math:`X_3` > 11 - :math:`S_s` (i.e., :math:`S_g` > 1 - :math:`S_s`): liquid phase disappears; make a transition to single-phase gas conditions. + +Phase change involving (dis-)appearance of solid salt is recognized as follows. +When no solid salt is present, the second primary variable :math:`X_{sm}` is the concentration (mass fraction referred to total water plus salt) of dissolved salt in the aqueous phase. +The possibility of precipitation starting is evaluated by comparing :math:`X_{sm}` with :math:`XEQ`, the equilibrium solubility of NaCl at prevailing temperature. +If :math:`X_{sm}` ≤ :math:`XEQ` no precipitation occurs, whereas for :math:`X_{sm}` > :math:`XEQ` precipitation starts. +In the latter case, variable :math:`X_{sm}` is switched to :math:`S_s` + 10, where solid saturation :math:`S_s` is initialized with a small non-zero value (10\ :sup:`-6`). +If a solid phase is present, the variable :math:`X_{sm}` = :math:`S_s` + 10 is monitored. +Solid phase disappears if :math:`X_{sm}` < 10, in which case primary variable :math:`X_{sm}` is switched to salt concentration, and is initialized as slightly below saturation, :math:`X_{sm}` = :math:`XEQ` - 10\ :sup:`-6`. + + +Conversion of Units +^^^^^^^^^^^^^^^^^^^ + +The :cite:label:`spycher2005co2` model for phase partitioning in the system H\ :sub:`2`\O-NaCl-CO\ :sub:`2` is formulated in molar quantities (mole fractions and molalities), while TOUGH2/ECO2N describes phase compositions in terms of mass fractions. +This section presents the equations and parameters needed for conversion between the two sets of units. +The conversion between various concentration variables (mole fractions, molalities, mass fractions) does not depend upon whether or not concentrations correspond to equilibrium between liquid and gas phases; accordingly, the +relations given below are valid regardless of the magnitude of concentrations. + +Let us consider an aqueous phase with dissolved NaCl and CO\ :sub:`2`. For a solution that is m-molal in NaCl and n-molal in CO\ :sub:`2`, total mass per kg of water is + +.. math:: + :label: eq:eco2n.3 + + M = 1000 (g H_2 O) + m M_{NaCl} (g NaCl) + n M_{CO_2} (g CO_2) + +where :math:`M_{NaCl}` and M_{CO_2} are the molecular weights of NaCl and CO\ :sub:`2`, respectively (see :numref:`tab:eco2n.2`). +Assuming NaCl to be completely dissociated, the total moles per kg of water are + +.. math:: + :label: eq:eco2n.4 + + m_T = \frac{1000}{M_{H_2 O}} + 2m + n + +The :cite:label:`spycher2005co2` correlations provide CO\ :sub:`2` mole fraction :math:`x_2` in the aqueous phase and H\ :sub:`2`\O mole fraction :math:`y_1` in the gas phase as functions of temperature, pressure, and salt concentration (molality). +For a CO\ :sub:`2` mole fraction x2 we have :math:`n` = :math:`x_2 m_T` from which, using Eq. :math:numref:`eq:eco2n.4`, we obtain + +.. math:: + :label: eq:eco2n.5 + + n = \frac{x_2 \left( 2 m + \frac{1000}{M_{H_2 O}} \right)}{1 - x_2} + +CO\ :sub:`2` mass fraction :math:`X_2` in the aqueous phase is obtained by dividing the CO\ :sub:`2` mass in :math:`n` moles by the total mass + +.. math:: + :label: eq:eco2n.6 + + X_2 = \frac{n M_{CO_2}}{1000 + m M_{NaCl} + n M_{CO_2}} + +Water mass fraction :math:`Y_1` in the CO\ :sub:`2`-rich phase is simply + +.. math:: + :label: eq:eco2n.7 + + Y_1 = \frac{y1 M_{H_2 O}}{y1 M_{H_2 O} + \left( 1 - y_1 \right) M_{CO_2}} + +The molecular weights of the various species are listed in :numref:`tab:eco2n.2` (:cite:label:`evans1982atomic`). + +.. list-table:: Molecular weights in the system H\ :sub:`2`\O-NaCl-CO\ :sub:`2`. + :name: tab:eco2n.2 + :header-rows: 1 + :align: center + + * - Species + - Mol. weight + * - H\ :sub:`2`\O + - 18.016 + * - Na + - 22.991 + * - Cl + - 35.457 + * - NaCl + - 58.448 + * - CO\ :sub:`2` + - 44.0 + + +Thermophysical Properties of Water-NaCl-CO\ :sub:`2` Mixtures +************************************************************* + +Thermophysical properties needed to model the flow of water-salt-CO\ :sub:`2` mixtures in porous media include density, viscosity, and specific enthalpy of the fluid phases as functions of temperature, pressure, and composition, and partitioning of components among the fluid phases. +Many of the needed parameters are obtained from the same correlations as were used in the EWASG property module of TOUGH2 (:cite:label:`battistelli1997simulator`). +EWASG was developed for geothermal applications, and consequently considered conditions of elevated temperatures > 100˚C, and modest CO\ :sub:`2` partial pressures of order 1-10 bar. +The present ECO2N module targets the opposite end of the temperature and pressure range, namely, modest temperatures below 110˚C, and high CO\ :sub:`2` pressures up to several hundred bar. + +Water properties in TOUGH2/ECO2N are calculated, as in other members of the TOUGH family of codes, from the steam table equations as given by the International Formulation Committee (:cite:label:`ifc1967formulation`). +Properties of pure CO\ :sub:`2` are obtained from correlations developed by :cite:label:`altunin1975thermophysical`. +We began using Altunin's correlations in 1999 when a computer program implementingthem was conveniently made available to us by Victor Malkovsky of the Institute of Geology of Ore Deposits, Petrography, Mineralogy and Geochemistry (IGEM) of the Russian Academy of Sciences, Moscow. +Altunin's correlations were subsequently extensively cross-checked against experimental data and alternative PVT formulations, such as :cite:label:`span1996new`. +They were found to be very accurate (:cite:`garcia2003fluid`), so there is no need to change to a different formulation. + +Altunin's correlations are not used directly in the code, but are used ahead of a TOUGH2/ECO2N simulation to tabulate density, viscosity, and specific enthalpy of pure CO\ :sub:`2` on a regular grid of (:math:`T`, :math:`P`)-values. +These tabular data are provided to the ECO2N module in a file called +"CO2TAB", and property values are obtained during the simulation by means of bivariate interpolation. +:numref:`fig:eco2n.7` shows the manner in which CO\ :sub:`2` properties are tabulated, intentionally showing a coarse (:math:`T`, :math:`P`)-grid so that pertinent features of the tabulation may be better seen. +(For actual calculations, we use finer grid spacings; the CO2TAB data file distributed with ECO2N covers the range 3.04 ˚C ≤ :math:`T` ≤ 103.04 ˚C with :math:`\Delta T` = 2˚C and 1 bar ≤ :math:`P` ≤ 600 bar with :math:`\Delta T` ≤ 4 bar in most cases. +The ECO2N distribution includes a utility program for generating CO2TAB files if users desire a different (:math:`T`, :math:`P`)-range or different increments). +As shown in :numref:`fig:eco2n.7`, the tabulation is made in such a way that for sub-critical conditions the saturation line is given by diagonals of the interpolation quadrangles. +On the saturation line, two sets of data are provided, for liquid and gaseous CO\ :sub:`2`, respectively, and in quadrangles that include points on both sides of the saturation line, points on the "wrong" side are excluded from the interpolation. +This scheme provides for an efficient and accurate determination of thermophysical properties of CO\ :sub:`2`. + +.. figure:: ../figures/figure_eco2n_7.png + :name: fig:eco2n.7 + :width: 75% + + Schematic of the temperature-pressure tabulation of CO\ :sub:`2` properties. The saturation line (dashed) is given by the diagonals of interpolation rectangles. + +An earlier version of ECO2N explicitly associated partial pressures of water (vapor) and CO\ :sub:`2` with the gas phase, and calculated CO\ :sub:`2` dissolution in the aqueous phase from the CO\ :sub:`2` partial pressure, using an extended version of Henry's law (:cite:label:`pruess2002intercomparison`). +The present version uses a methodology for calculating mutual solubilities of water and CO\ :sub:`2` (:cite:label:`spycher2005co2`) that is much more accurate, but has a drawback insofar as no partial pressures are associated with the individual fluid components. +This makes it less straightforward to calculate thermophysical properties of the gas phase in terms of individual fluid components. +We are primarily interested in the behavior of water-salt-CO\ :sub:`2` mixtures at moderate temperatures, :math:`T` < 100˚C, say, where water vapor pressure is a negligibly small fraction of total pressure. +Under these conditions the amount of water present in the CO\ :sub:`2`-rich phase, henceforth referred to as "gas", is small. +Accordingly, we approximate density, viscosity, and specific enthalpy of the gas phase by the corresponding properties of pure CO\ :sub:`2`, without water present. + + +Density +^^^^^^^ + +Brine density :math:`\rho_b` for the binary system water-salt is calculated as in :cite:label:`battistelli1997simulator` from the correlations of :cite:label:`haas1976physical` and :cite:label:`andersen1992accurate`. +The calculation starts from aqueous phase density without salinity at vapor-saturated conditions, which is obtained from the correlations given by the International Formulation Committee (:cite:label:`ifc1967formulation`). +Corrections are then applied to account for effects of salinity and pressure. +The density of aqueous phase with dissolved CO\ :sub:`2` is calculated assuming additivity of the volumes of brine and dissolved CO\ :sub:`2`. + +.. math:: + :label: eq:eco2n.8 + + \frac{1}{\rho_{aq}} = \frac{1 - X_2}{\rho_b} + \frac{X_2}{\rho_{CO_2}} + +where :math:`X_2` is the mass fraction of CO\ :sub:`2` in the aqueous phase. +Partial density of dissolved CO\ :sub:`2`, :math:`\rho_{CO_2}`, is calculated as a function of temperature from the correlation for molar volume of dissolved CO\ :sub:`2` at infinite dilution developed by :cite:label:`garcia2001density`. + +.. math:: + :label: eq:eco2n.9 + + V_{\phi} = a + b T + c T^2 + d T^3 + +In Eq. :math:numref:`eq:eco2n.9`, molar volume of CO\ :sub:`2` is in units of cm\ :sup:`3` per gram-mole, temperature :math:`T` is in ˚C, and :math:`a`, :math:`b`, :math:`c`, and :math:`d` are fit parameters given in :numref:`tab:eco2n.3`. + +.. list-table:: Parameters for molar volume of dissolved CO\ :sub:`2` :math:numref:`eq:eco2n.9`. + :name: tab:eco2n.3 + :header-rows: 1 + :align: center + + * - Parameter + - Value + * - a + - 37.51 + * - b + - -9.585 × 10\ :sup:`-2` + * - c + - 8.740 × 10\ :sup:`-4` + * - d + - -5.044 × 10\ :sup:`-7` + +Partial density of dissolved CO\ :sub:`2` in units of kg/m\ :sup:`3` is then + +.. math:: + :label: eq:eco2n.10 + + \rho_{CO_2} = \frac{M_{CO_2}}{V_{\phi}} \times 10^{-3} + +where :math:`M_{CO_2}` = 44.0 is the molecular weight of CO\ :sub:`2`. + +Dissolved CO\ :sub:`2` amounts at most to a few percent of total aqueous density. Accordingly, dissolved CO\ :sub:`2` is always dilute, regardless of total fluid pressure. +It is then permissible to neglect the pressure dependence of partial density of dissolved CO\ :sub:`2`, and to use the density corresponding to infinite dilution. + +As had been mentioned above, the density of the CO\ :sub:`2`-rich (gas) phase is obtained by neglecting effects of water, and approximating the density by that of pure CO\ :sub:`2` at the same temperature and pressure conditions. +Density is obtained through bivariate interpolation from a tabulation of CO\ :sub:`2` densities as function of temperature and pressure, that is based on the correlations developed by :cite:label:`altunin1975thermophysical`. + + +Viscosity +^^^^^^^^^ + +Brine viscosity is obtained as in EWASG from a correlation presented by :cite:label:`phillips1981technical`, that reproduces experimental data in the temperature range from 10-350˚C for salinities up to 5 molal and pressures up to 500 bar within 2%. +No allowance is made for dependence of brine viscosity on the concentration of dissolved CO\ :sub:`2`. +Viscosity of the CO\ :sub:`2`-rich phase is approximated as being equal to pure CO\ :sub:`2`, and is obtained through tabular interpolation from the correlations of :cite:label:`altunin1975thermophysical`. + + +Specific Enthalpy +^^^^^^^^^^^^^^^^^ + +Specific enthalpy of brine is calculated from the correlations developed by :cite:label:`lorenz2000analytische`, which are valid for all salt concentrations in the temperature range from 25˚C ≤ :math:`T` ≤ 300˚C. +The enthalpy of aqueous phase with dissolved CO\ :sub:`2` is obtained by adding the enthalpies of the CO\ :sub:`2` and brine (pseudo-) components, and accounting for the enthalpy of dissolution of CO\ :sub:`2`. + +.. math:: + :label: eq:eco2n.11 + + h_{aq} = \left( 1 - X_2 \right) h_b + X_2 h_{CO_2, aq} + +:math:`h_{CO_2, aq} = h_{CO_2} + h_{dis}` is the specific enthalpy of aqueous (dissolved) CO\ :sub:`2`, which includes heat of dissolution effects that are a function of temperature and salinity. +For gas-like (low pressure) CO\ :sub:`2`, the specific enthalpy of dissolved CO\ :sub:`2` is + +.. math:: + :label: eq:eco2n.12 + + h_{CO_2, aq} (T, P, X_s) = h_{CO_2, g} (T, P) + h_{dis, g} (T, X_s) + +where :math:`h_{dis, g}` is obtained as in :cite:label:`battistelli1997simulator` from an equation due to :cite:label:`himmelblau1959partial`. +For geologic sequestration we are primarily interested in liquid-like (high-pressure) CO\ :sub:`2`, for which the specific enthalpy of dissolved CO\ :sub:`2` may be written + +.. math:: + :label: eq:eco2n.13 + + h_{CO_2, aq} (T, P, X_s) = h_{CO_2, l} (T, P) + h_{dis, l} (T, X_s) + +Here :math:`h_{dis, l}` is the specific heat of dissolution for liquid-like CO\ :sub:`2`. +Along the CO\ :sub:`2` saturation line, liquid and gaseous CO\ :sub:`2` phases may co-exist, and the expressions Eqs. :math:numref:`eq:eco2n.12` and :math:numref:`eq:eco2n.13` must be equal there. +We obtain + +.. math:: + :label: eq:eco2n.14 + + h_{dis, l} (T, X_s) = h_{dis, g} (T, X_s) + h_{CO_2, gl} (T) + +where :math:`h_{CO_2, gl} (T) = h_{CO_2, g} (T, P_s) - h_{CO_2, l} (T, P_s)` is the specific enthalpy of vaporization of CO\ :sub:`2`, and :math:`P_s = P_s (T)` is the saturated vapor pressure of CO\ :sub:`2` at temperature :math:`T`. +Depending upon whether CO\ :sub:`2` is in gas or liquid conditions, we use Eq. :math:numref:`eq:eco2n.12` or :math:numref:`eq:eco2n.13` in Eq. :math:numref:`eq:eco2n.11` to calculate the specific enthalpy of dissolved CO\ :sub:`2`. +At the temperatures of interest here, :math:`h_{dis, g}` is a negative quantity, so that dissolution of low-pressure CO\ :sub:`2` is accompanied by an increase in temperature. +:math:`h_{CO_2, gl}` is a positive quantity, which will reduce or cancel out the heat-of-dissolution effects. +This indicates that dissolution of liquid CO\ :sub:`2` will produce less temperature increase than dissolution of gaseous CO\ :sub:`2`, and may even cause a temperature decline if :math:`h_{CO_2, gl}` is sufficiently large. + +Application of Eq. :math:numref:`eq:eco2n.11` is straightforward for single-phase gas and two-phase conditions, where :math:`h_{CO_2}` is obtained as a function of temperature and pressure through bivariate interpolation from a tabulation of Altunin's correlation (:cite:label:`altunin1975thermophysical`). +A complication arises in evaluating :math:`h_{CO_2}` for single-phase aqueous conditions. +We make the assumption that :math:`h_{CO_2} (P, X_s, X_2, T)` for single-phase liquid is identical to the value in a two-phase system with the same composition of the aqueous phase. +To determine :math:`h_{CO_2}`, it is then necessary to invert the :cite:label:`spycher2005co2` phase partitioning relation :math:`X_2 = X_2 (P, T, X_s)`, in order to obtain the pressure :math:`P` in a two-phase aqueous-gas system that would correspond to a dissolved CO\ :sub:`2` mass fraction :math:`X_2` in the aqueous phase, :math:`P = P(X_2, X_s, T)`. +The inversion is accomplished by Newtonian iteration, using a starting guess :math:`P_0` for :math:`P` that is obtained from Henry's law. +Specific enthalpy of gaseous CO\ :sub:`2` in the two-phase system is then calculated as :math:`h_{CO_2} = h_{CO_2} (T, P)`, and specific enthalpy of dissolved CO\ :sub:`2` is :math:`h_{CO_2} + h_{dis}`. + + +Description +----------- + +Initialization Choices +********************** + +Flow problems in TOUGH2/ECO2N will generally be initialized with the primary +thermodynamic variables as used in the code, but some additional choices are available for the convenience of users. +The internally used variables are (:math:`P`, :math:`X_{sm}`, :math:`X_3`, :math:`T`) for grid blocks in single-phase (liquid or gas) conditions and (:math:`P`, :math:`X_{sm}`, :math:`S_g` + 10, :math:`T`) for two-phase (liquid and gas) grid blocks (see :numref:`tab:eco2n.1`). +Here :math:`X_3` is the mass fraction of CO\ :sub:`2` in the fluid. +As had been discussed above, for conditions of interest to geologic sequestration of CO\ :sub:`2`, :math:`X_3` is restricted to small values 0 ≤ :math:`X_3` ≤ :math:`X_{2, eq}` (a few percent) for single-phase liquid conditions, or to values near 1 (:math:`Y_{2, eq}` ≤ :math:`X_3` ≤ 1, with :math:`Y_{2, eq}` > 0.99 typically) for single-phase gas (:numref:`fig:eco2n.6`). +Intermediate values :math:`X_{2, eq}` < :math:`X_3` < :math:`Y_{2, eq}` correspond to two-phase conditions, and thus should be initialized by specifying :math:`S_g` + 10 as third primary variable. +As a convenience to users, ECO2N allows initial conditions to be specified in the full range 0 ≤ :math:`X_3` ≤ 1. +During the initialization phase of a simulation, a check is made whether :math:`X_3` is in fact within the range of mass fractions that correspond to single-phase (liquid or gas) conditions. +If this is found not to be the case, the conditions are recognized as being two-phase, and the corresponding gas saturation is calculated from the phase equilibrium constraint. + +.. math:: + :label: eq:eco2n.15 + + X_3 \left( S_l \rho_l + S_g \rho_g \right) = S_l \rho_l X_{2, eq} + S_g \rho_g Y_{2, eq} + +Using :math:`S_l = 1 - S_g - S_s`, with Ss the "solid saturation" (fraction of pore space occupied by solid salt), we obtain + +.. math:: + :label: eq:eco2n.16 + + S_g = A \left( 1 - S_s \right) + +and the third primary variable is reset internally to :math:`X_3` = :math:`S_g` + 10. +Here the parameter :math:`A` is given by + +.. math:: + :label: eq:eco2n.17 + + A = \frac{\left( X_3 - X_{2, eq} \right) \rho_l}{\left( X_3 - X_{2, eq} \right) \rho_l + \left( Y_{2, eq} - X_3 \right) \rho_g} + +Users may think of specifying single-phase liquid (aqueous) conditions by setting :math:`X_3` = 10 (corresponding to :math:`S_g` = 0), and single-phase gas conditions by setting :math:`X_3` = 11 - :math:`S_s` (corresponding to :math:`Sl` = 0). +Strictly speaking this is not permissible, because two-phase initialization requires that both :math:`S_g` > 0 and :math:`S_l` > 0. +Single-phase states should instead be initialized by specifying primary +variable :math:`X_3` as CO\ :sub:`2` mass fraction. +However, as a user convenience, ECO2N accepts initialization of single-phase liquid conditions by specifying :math:`X_3` = 10 (:math:`S_g` = 0). +Such specification will be converted internally to two-phase in the initialization phase by adding a small number (10\ :sup:`-11`) to the third +primary variable, changing conditions to two-phase with a small gas saturation :math:`S_g` = 10\ :sup:`-11`. + +Salt concentration or saturation of solid salt, if present, is characterized in ECO2N by means of the second primary variable :math:`X_{sm}`. +When no solid phase is present, :math:`X_{sm}` denotes :math:`X_s`, the mass fraction of NaCl referred to the two-component system water-NaCl. +This is restricted to the range 0 ≤ :math:`X_{sm}` ≤ :math:`XEQ`, where :math:`XEQ = XEQ(T)` is the solubility of salt. +For :math:`X_{sm}` > 10 this variable means :math:`S_s` + 10, solid saturation plus 10. +Users also have the option to specify salt concentration by means of molality :math:`m` by assigning :math:`X_{sm} = -m`. +Such specification will in the initialization phase be internally converted to :math:`X_s` by using Eq. :math:numref:`eq:eco2n.1`. +When salt concentration (as a fraction of total H\ :sub:`2`\O + NaCl mass) exceeds :math:`XEQ`, this corresponds to conditions in which solid salt will be present in addition to dissolved salt in the aqueous phase. +Such states should be initialized with a second primary variable :math:`X_{sm}` = :math:`S_s` + 10. +However, ECO2N accepts initialization with :math:`X_{sm}` > :math:`XEQ`, recognizes this as corresponding to presence of solid salt, and converts the second primary variable internally to the appropriate solid saturation that will result in total salt mass fraction in the binary system water-salt being equal to :math:`X_{sm}`. +The conversion starts from the following equation. + +.. math:: + :label: eq:eco2n.18 + + X_{sm} = \frac{XEQ \times S_l \rho_l \left( 1 - X_2 \right) + S_s \rho_s}{S_l \rho_l \left( 1 - X_2 \right) + S_s \rho_s} + +where the numerator gives the total salt mass per unit volume, in liquid and solid phases, while the denominator gives the total mass of salt plus water. +Substituting :math:`S_l = 1 - S_g - S_s`, this can be solved for :math:`S_s` to yield + +.. math:: + :label: eq:eco2n.19 + + S_s = \frac{B \left( 1 - S_g \right)}{1 + B} + +where the parameter :math:`B` is given by + +.. math:: + :label: eq:eco2n.20 + + B = \frac{\left( X_{sm} - XEQ \right) \rho_l \left( 1 - X_2 \right)}{\rho_s \left( 1 - X_{sm} \right)} + +The most general conditions arise when both the second and third primary variables are initialized as mass fractions, nominally corresponding to single-phase fluid conditions with no solid phase present, but both mass fractions being in the range corresponding to two-phase fluid conditions +with precipitated salt. +Under these conditions, Eqs. :math:numref:`eq:eco2n.16` and :math:numref:`eq:eco2n.19` are solved simultaneously in ECO2N for :math:`S_s` and :math:`S_g`, yielding + +.. math:: + :label: eq:eco2n.21 + + S_g = \frac{A}{1 + B - A \times B} + +.. math:: + :label: eq:eco2n.22 + + S_s = \frac{B \times \left( 1 - A \right)}{1 + B - A \times B} + +Then both second and third primary variables are converted to phase saturations, :math:`S_s` + 10 and :math:`S_g` + 10, respectively. + + +Permeability Change from Precipitation and Dissolution of Salt +************************************************************** + +ECO2N offers several choices for the functional dependence of relative change in permeability, :math:`\frac{k}{k_0}`, on relative change in active flow porosity. + +.. math:: + :label: eq:eco2n.23 + + \frac{k}{k_0} = f \left( \frac{\phi_f}{\phi_0} \right) \equiv f \left( 1 - S_s \right) + +The simplest model that can capture the converging-diverging nature of natural pore channels consists of alternating segments of capillary tubes with larger and smaller radii, respectively; see :numref:`fig:eco2n.8`. +While in straight capillary tube models permeability remains finite as long as porosity is non-zero, in models of tubes with different radii in series, permeability is reduced to zero at a finite porosity. + +.. subfigure:: AB + :name: fig:eco2n.8 + :layout-sm: A|B + :subcaptions: above + :gap: 32px + + .. image:: ../figures/pore_conceptual_model.png + :alt: Conceptual model + + .. image:: ../figures/pore_tubes_in_series.png + :alt: Tubes-in-series + + Model for converging-diverging pore channels. + +From the tubes-in-series model shown in :numref:`fig:eco2n.8`, the following relationship can be derived (:cite:label:`verma1988thermohydrological`) + +.. math:: + :label: eq:eco2n.24 + + \frac{k}{k_0} = \theta^2 \frac{1 - \Gamma + \frac{\Gamma}{\omega^2}}{1 - \Gamma + \Gamma \left( \frac{\theta}{\theta + \omega - 1} \right)^2} + +Here + +.. math:: + :label: eq:eco2n.25 + + \theta = \frac{1 - S_s - \phi_r}{1 - \phi_r} + +depends on the fraction :math:`1 - S_s` of original pore space that remains available to fluids, and on a parameter :math:`\phi_r`, which denotes the fraction of original porosity at which permeability is reduced to zero. +:math:`\Gamma` is the fractional length of the pore bodies, and the parameter :math:`\omega` is given by + +.. math:: + :label: eq:eco2n.26 + + \omega = 1 + \frac{\frac{1}{\Gamma}}{\frac{1}{\phi_r} - 1} + +Therefore, Eq. :math:numref:`eq:eco2n.24` has only two independent geometric parameters that need to be specified, :math:`\phi_r` and :math:`\Gamma`. +As an example, :numref:`fig:eco2n.9` shows the permeability reduction factor from Eq. :math:numref:`eq:eco2n.24`, plotted against :math:`\frac{\phi}{\phi_0} \equiv \left( 1 - S_s \right)`, for parameters of :math:`\phi_r` = :math:`\Gamma` = 0.8. + +.. figure:: ../figures/porosity_permeability.png + :name: fig:eco2n.9 + :width: 75% + + Porosity-permeability relationship for tubes-in-series model, after :cite:label:`verma1988thermohydrological`. + +For parallel-plate fracture segments of different aperture in series, a relationship similar to Eq. :math:numref:`eq:eco2n.24` is obtained, the only difference being that the exponent 2 is replaced everywhere by 3 (:cite:label:`verma1988thermohydrological`). +If only straight capillary tubes of uniform radius are considered, we have :math:`\phi_r` = 0, :math:`\Gamma` = 0, and Eq. :math:numref:`eq:ewasg.2` simplifies to + +.. math:: + :label: eq:eco2n.27 + + \frac{k}{k_0} = \left( 1 - S_s \right)^2 + + +Selections +---------- + +Various options for ECO2N can be selected through parameter specifications in data block **SELEC**. +Default choices corresponding to various selection parameters set equal to zero provide the most comprehensive thermophysical property model. +Certain functional dependencies can be turned off or replaced by simpler and less accurate models, see below. +These options are offered to enable users to identify the role of different effects in a flow problem, and to facilitate comparison with other simulation programs that may not include full dependencies of thermophysical properties. + +.. list-table:: Record **SELEC.1**. + :name: tab:eco2n.selec.1 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - ``IE(1)`` + - I5 + - set equal to 1, to read one additional data record (a larger value with more data records is acceptable, but only one additional record will be used by ECO2N). + * - ``IE(11)`` + - I5 + - | selects dependence of permeability on the fraction :math:`\frac{\phi_f}{\phi_0} = \left( 1 - S_s \right)` of original pore space that remains available to fluids: + | 0: permeability does not vary with :math:`\phi_f`. + | 1: :math:`\frac{k}{k_0} = \left( 1 - S_s \right)^{\gamma}`, with :math:`\gamma` = ``FE(1)``. + | 2: fractures in series, i.e., Eq. :math:numref:`eq:eco2n.24` with exponent 2 everywhere replaced by 3. + | 3: tubes-in-series, i.e., Eq. :math:numref:`eq:eco2n.24`. + * - ``IE(12)`` + - I5 + - | allows choice of model for water solubility in CO\ :sub:`2`: + | 0: after Spycher and Pruess (2005). + | 1: evaporation model; i.e., water density in the CO\ :sub:`2`-rich phase is calculated as density of saturated water vapor at prevailing temperature and salinity. + * - ``IE(13)`` + - I5 + - | allows choice of dependence of brine density on dissolved CO\ :sub:`2`: + | 0: brine density varies with dissolved CO\ :sub:`2` concentration, according to García's (2001) correlation for temperature dependence of molar volume of dissolved CO\ :sub:`2`. + | 1: brine density is independent of CO\ :sub:`2` concentration. + * - ``IE(14)`` + - I5 + - | allows choice of treatment of thermophysical properties as a function of salinity: + | 0: full dependence. + | 1: no salinity dependence of thermophysical properties (except for brine enthalpy; salt solubility constraints are maintained). + * - ``IE(15)`` + - I5 + - | allows choice of correlation for brine enthalpy at saturated vapor pressure: + | 0: after Lorenz et al. (2000). + | 1: after Michaelides (1981). + | 2: after Miller (1978). + +.. list-table:: Record **SELEC.2**. + :name: tab:eco2n.selec.2 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - ``FE(1)`` + - E10.4 + - parameter :math:`\gamma` (for ``IE(11)`` = 1); parameter :math:`\phi_r` (for ``IE(11)`` = 2, 3). + * - ``FE(2)`` + - E10.4 + - parameter :math:`\Gamma` (for ``IE(11)`` = 2, 3). diff --git a/doc/source/tough3/eos/eos1.rst b/doc/source/tough3/eos/eos1.rst new file mode 100644 index 00000000..16ad1c1d --- /dev/null +++ b/doc/source/tough3/eos/eos1.rst @@ -0,0 +1,78 @@ +.. _eos1: + +EOS1 +==== + +This is the most basic EOS module, providing a description of pure water in its liquid, vapor, and two-phase states. +All water properties (density, specific enthalpy, viscosity, saturated vapor pressure) are calculated from the steam table equations as given by the :cite:label:`ifc1967formulation`. +The formulation includes subregion 1 (subcooled water below T=350˚C), subregion 2 (superheated steam), and subregion 6 (saturation line up to T=350˚C). +In these regions, density and internal energy are represented within experimental accuracy. +Viscosity of liquid water and steam are represented to within 2.5% by correlations given in the same reference. +For details of the formulation, its accuracy and range of validity, refer to the original publication. +Vapor pressure lowering from capillary and adsorption effects is neglected; thus, in two-phase conditions vapor pressure is equal to saturated vapor pressure of bulk liquid. + + +Specifications +-------------- + +A summary of EOS1 specifications and parameters is given in :numref:`tab:eos1`. +The default parameter settings are (``NK``, ``NEQ``, ``NPH``, ``NB``) = (1, 2, 2, 6) for solving mass and energy balances for a single water component. +The option ``NEQ`` = 1 is available for running single-phase flow problems that involve only liquid water, or only superheated steam, under constant temperature conditions. +EOS1 also has a capability for representing two waters of identical physical properties, which contain different trace constituents. +This option can be invoked by specifying (``NK``, ``NEQ``, ``NPH``, ``NB``) = (2, 3, 2, 6) in data block **MULTI**. +With this option, two water mass balances will be set up, allowing separate tracking of the two components. +For example, one could specify the water initially present in the flow system as "water 1", while water being injected is specified as "water 2". +When the two waters option is chosen, molecular diffusion can also be modeled by setting parameter ``NB`` equal to 8. + +The primary variables are (:math:`P`, :math:`T``) for single-phase points, (:math:`P_g`, :math:`S_g`) for two-phase points. +For the convenience of the user it is possible to initialize two-phase points as (:math:`T`, :math:`S_g`); when the numerical value of the first primary variable is less than 374.15, this will automatically be taken to indicate that this represents temperature rather than pressure, and will cause variables to be converted internally from (:math:`T`, :math:`S_g`) to (:math:`P_{sat}(T)`, :math:`S_g`) prior to execution. +When the two waters option is used, primary variables are (:math:`P`, :math:`T`, :math:`X`) for single-phase points, and (:math:`P_g`, :math:`S_g`, :math:`X`) for two-phase points, where :math:`X` is the mass fraction of "water 2" present. +All thermophysical properties (density, specific enthalpy, viscosity) are assumed independent of the component mixture; i.e., independent of mass fraction :math:`X`. This approximation is applicable for problems in which the identity of different waters is distinguished by the presence of different trace constituents, which occur in concentrations low enough to not appreciably affect the thermophysical properties. + +.. list-table:: Summary of EOS1. + :name: tab:eos1 + :widths: 1 3 + :align: center + + * - Components + - | #1: water + | #2: "water 2" (optional) + * - Parameter choices + - | (``NK``, ``NEQ``, ``NPH``, ``NB``) = + | (1, 2, 2, 6) one water component, nonisothermal (default) + | (1, 1, 2, 6) only liquid, or only vapor; isothermal + | (2, 3, 2, 6) two-waters, nonisothermal* + | Molecular diffusion can be modeled by setting ``NK`` = 2, ``NB`` = 8 + * - Primary variables + - | Single-phase conditions: + | (:math:`P`, :math:`T`, :math:`[X]`): (pressure, temperature, [mass fraction of water 2]†) + | Two-phase conditions: + | (:math:`P_g`, :math:`S_g`, :math:`[X]`): (gas phase pressure, gas saturation, [mass fraction of water 2]†) + + +.. note:: + + | \* two waters cannot be run in isothermal mode, because in this case temperature is not the last primary variable. + | † optional, for ``NK`` = 2 only. + +The phase diagnostic procedures are as follows. +When initializing a problem, each grid block has two primary variables (:math:`X1`, :math:`X2`). +The meaning of :math:`X2` depends on its numerical value: for :math:`X2` > 1.5, :math:`X2` is taken to be temperature in ˚C, otherwise it is gas saturation. +Although physically saturation is restricted to the range :math:`0 \le S \le 1`, it is necessary to allow saturations to exceed 1 during the Newton-Raphson iteration. +If :math:`X2` is temperature, this indicates that single-phase conditions are present; specifically, for :math:`P` (= :math:`X1`) :math:`\gt P_{sat}(T)` we have single-phase liquid, otherwise we have single-phase steam. +Subsequent to initialization, the phase condition is identified simply based on the value of :math:`S_g` as stored in the *PAR* array: + +- 0: single-phase liquid +- 1: single-phase vapor +- :math:`0 \lt S_g \lt 1`: two-phase + + +Phase change is recognized as follows. +For single-phase points the temperature (second primary variable) is monitored, and the corresponding saturation pressure is compared with actual fluid pressure :math:`P`. +For a vapor (liquid) point to remain vapor (liquid), we require that :math:`P \lt P_{sat}` (:math:`P \gt P_{sat}`); if this requirement is violated, a transition to two-phase conditions takes place. +The primary variables are then switched to (:math:`P_g`, :math:`S_g`), and these are initialized as :math:`P_g = P_{sat}(T)`, :math:`S_g` = 0.999999 if the point was in the vapor region, and :math:`S_g` = 10\ :sup:`-6` if it was in the liquid region. +For two-phase points :math:`S_g` is monitored; we require that :math:`0 \lt S_g \lt 1` for a point to remain two-phase. +If :math:`S_g \lt 0` this indicates disappearance of the gas phase; the primary variables are then switched to (:math:`P`, :math:`T`), and the point is initialized as single-phase liquid, with :math:`T` taken from the last Newton-Raphson iteration, and :math:`P` = 1.000001 × :math:`P_{sat}(T)`. +For :math:`S_g \gt 1` the liquid phase disappears; again the primary variables are switched to (:math:`P`, :math:`T`), and the point is initialized as single-phase vapor, with :math:`T` taken from the last Newton-Raphson iteration, and :math:`P` = 0.999999 × :math:`P_{sat}(T)`. +Note that in these phase transitions we preserve temperature rather than pressure from the last iteration. +This is preferable because in most flow problems temperature tends to be more slowly varying than pressure. diff --git a/doc/source/tough3/eos/eos2.rst b/doc/source/tough3/eos/eos2.rst new file mode 100644 index 00000000..fef1d157 --- /dev/null +++ b/doc/source/tough3/eos/eos2.rst @@ -0,0 +1,53 @@ +.. _eos2: + +EOS2 +==== + +This is an updated version of the fluid property module originally developed by :cite:label:`o1985fluid` for describing fluids in gas-rich geothermal reservoirs, which may contain CO\ :sub:`2` mass fractions ranging from a few percent to occasionally 80% or more (:cite:label:`atkinson1980behavior`). +EOS2 accounts for non-ideal behavior of gaseous CO\ :sub:`2`, and dissolution of CO\ :sub:`2` in the aqueous phase with heat of solution effects. +According to Henry's law, the partial pressure of a non-condensible gas (NCG) in the gas phase is proportional to the mole fraction of dissolved NCG in the aqueous phase + +.. math:: + :label: eq:eos2.1 + + P_{NCG} = K_h x_{aq}^{NCG} + +The Henry's law coefficient :math:`K_h` for dissolution of CO\ :sub:`2` in water is strongly dependent on temperature. +The correlation used in the previous release of EOS2 had been developed by :cite:label:`o1985fluid` for the elevated temperature conditions encountered in geothermal reservoirs. +It is accurate to within a few percent of experimental data in the temperature range of 40˚C ≤ :math:`T` ≤ 330˚C, but becomes rather inaccurate at lower temperatures and even goes to negative values for :math:`T` < 30˚C (see :numref:`fig:eos2.1`). +Subroutine *HENRY* in EOS2 was replaced with a new routine that uses the correlation developed by :cite:label:`battistelli1997simulator`, which is accurate for 0˚C ≤ :math:`T` ≤ 350˚C. +The Battistelli et al. formulation agrees well with another correlation that was developed by S. White (1996, private communication). + +.. figure:: ../figures/figure_eos2_1.png + :name: fig:eos2.1 + + Henry's law coefficients for dissolution of CO\ :sub:`2` in water. + +The viscosity of vapor - CO\ :sub:`2` mixtures is described with a formulation due to :cite:label:`pritchett1981baca`; other thermophysical property correlations are based on the model of :cite:label:`sutton1977boiling`. + + +Specifications +-------------- + +A summary of EOS2 specifications and parameters is given in :numref:`tab:eos2`. +A more detailed description and application to geothermal reservoir problems are given in the paper by :cite:label:`o1985fluid`. +EOS2 has the parameter settings of (``NK``, ``NEQ``, ``NPH``, ``NB``) = (2, 3, 2, 6) for solving mass and energy balances for water and CO\ :sub:`2`. +Molecular diffusion can also be modeled by setting parameter ``NB`` equal to 8. + +.. list-table:: Summary of EOS2. + :name: tab:eos2 + :widths: 1 3 + :align: center + + * - Components + - | #1: water + | #2: CO\ :sub:`2` + * - Parameter choices + - | (``NK``, ``NEQ``, ``NPH``, ``NB``) = + | (2, 3, 2, 6) + | Molecular diffusion can be modeled by setting ``NB`` = 8 + * - Primary variables + - | Single-phase conditions: + | (:math:`P`, :math:`T`, :math:`P_{CO_2}`): (pressure, temperature, CO\ :sub:`2` partial pressure) + | Two-phase conditions: + | (:math:`P_g`, :math:`S_g`, :math:`P_{CO_2}`): (gas phase pressure, gas saturation, CO\ :sub:`2` partial pressure) diff --git a/doc/source/tough3/eos/eos3.rst b/doc/source/tough3/eos/eos3.rst new file mode 100644 index 00000000..8b58edf6 --- /dev/null +++ b/doc/source/tough3/eos/eos3.rst @@ -0,0 +1,59 @@ +.. _eos3: + +EOS3 +==== + +This module is an adaptation of the EOS module of the TOUGH simulator, and implements the same thermophysical properties model (:cite:label:`pruess1987tough`). +All water properties are represented by the steam table equations as given by the :cite:label:`ifc1967formulation`. +Air is approximated as an ideal gas, and additivity is assumed for air and vapor partial pressures in the gas phase, :math:`P_g = P_a + P_v`. +The viscosity of air-vapor mixtures is computed from a formulation given by :cite:label:`hirschfelder1964molecular`. +The solubility of air in liquid water is represented by Henry's law, as follows. + +.. math:: + :label: eq:eos3.1 + + P_a = K_h x_{aq}^a + +where :math:`K_h` is Henry's constant and :math:`x_{aq}^a` is air mole fraction in the aqueous phase. +Henry's constant for air dissolution in water is a slowly varying function of temperature, varying from 6.7 x 10:\ :sup:`9` Pa at 20˚C to 1.0 x 10\ :sup:`10` Pa at 60˚C and 1.1 x 10\ :sup:`10` Pa at 100˚C (:cite:label:`loomis1928solubilities`). +Because air solubility is small, this kind of variation is not expected to cause significant effects, and a constant :math:`P_a` = 10\ :sup:`10` Pa was adopted. + + +Specifications +-------------- + +A summary of EOS3 specifications and parameters is given in :numref:`tab:eos3`. +The default parameter settings are (``NK``, ``NEQ``, ``NPH``, ``NB``) = (2, 3, 2, 6). +The option ``NEQ`` = 2 is available for constant temperature conditions. +The choice of primary thermodynamic variables is (:math:`P`, :math:`X`, :math:`T`) for single-phase, (:math:`P_g`, :math:`S_g` + 10, :math:`T`) for two-phase conditions. +The rationale for the seemingly bizarre choice of :math:`S_g` + 10 as a primary variable is as follows. +As an option, we wish to be able to run isothermal two-phase flow problems with the specification ``NEQ`` = ``NK``, so that the then superfluous heat balance equation needs not be engaged. +This requires that temperature :math:`T` be the third primary variable. +The logical choice of primary variables would then appear to be (:math:`P`, :math:`X`, :math:`T`) for single-phase and (:math:`P_g`, :math:`S_g`, :math:`T`) for two-phase conditions. +However, both :math:`X` and :math:`S_g` vary over the range (0, 1), so that this would not allow a distinction of single-phase from two-phase conditions solely from the numerical range of primary variables. +By taking the second primary variable for two-phase conditions to be :math:`X2` = 10 + :math:`S_g`, the range of that variable is shifted to the interval (10, 11), and a distinction between single and two-phase conditions can be easily made. +As a convenience to users, we retain the capability to optionally initialize flow problems with TOUGH-style primary variables by setting ``MOP(19)`` = 1. +In TOUGH we have (:math:`P`, :math:`T`, :math:`X`) for single-phase conditions, (:math:`P_g`, :math:`S_g`, :math:`T`) for two-phase conditions. + +.. list-table:: Summary of EOS3. + :name: tab:eos3 + :widths: 1 3 + :align: center + + * - Components + - | #1: water + | #2: air + * - Parameter choices + - | (``NK``, ``NEQ``, ``NPH``, ``NB``) = + | (2, 3, 2, 6) water and air, nonisothermal (default) + | (2, 2, 2, 6) water and air, isothermal + | Molecular diffusion can be modeled by setting ``NB`` = 8 + * - Primary variables* + - | Single-phase conditions: + | (:math:`P`, :math:`X`, :math:`T`): (pressure, air mass fraction, temperature) + | Two-phase conditions: + | (:math:`P_g`, :math:`S_g` + 10, :math:`T`): (gas phase pressure, gas saturation plus 10, temperature) + +.. note:: + + \* By setting ``MOP(19)`` = 1, initialization can be made with TOUGH-style variables (:math:`P`, :math:`T`, :math:`X`) for single-phase, (:math:`P_g`, :math:`S_g`, :math:`T`) for two-phase. diff --git a/doc/source/tough3/eos/eos4.rst b/doc/source/tough3/eos/eos4.rst new file mode 100644 index 00000000..0a26acc6 --- /dev/null +++ b/doc/source/tough3/eos/eos4.rst @@ -0,0 +1,84 @@ +.. _eos4: + +EOS4 +==== + +This module is an adaptation of the :ref:`eos3` module, and implements vapor pressure lowering effects. +Vapor pressure :math:`P_v` is expressed by Kelvin's equation, +Eq. :math:numref:`eq:eos4.1` (:cite:label:`edlefsen1943thermodynamics`); it is a function not only of temperature, but depends also on capillary pressure, which in turn is a function of saturation. + +.. math:: + :label: eq:eos4.1 + + P_v (T, S_l) = f_{VPL} (T, S_l) P_{sat} (T) + +where + +.. math:: + :label: eq:eos4.2 + + f_{VPL} = \exp \left( \frac{M_w P_{cl}(S_l)}{\rho_l R \left( T + 273.15 \right)} \right) + +is the vapor pressure lowering factor. +:math:`P_{sat}` is the saturated vapor pressure of bulk aqueous phase, :math:`T` is temperature, :math:`S_l` is the liquid (water) saturation, :math:`P_{cl}` is the capillary pressure (i.e., the difference between aqueous and gas phase pressures), :math:`M_w` is the molecular weight of water, :math:`rho_l` is the liquid density, and :math:`R` is the universal gas constant. + + +Specifications +-------------- + +A summary of EOS4 specifications and parameters is given in :numref:`tab:eos4`. +The default parameter settings are (``NK``, ``NEQ``, ``NPH``, ``NB``) = (2, 3, 2, 6). +The option for single-component mode of water only, no air (``NK`` = 1) is available for nonisothermal conditions (``NEQ`` = 2). +The specification of thermophysical properties in this EOS differs from :ref:`eos3` in that provision is made for vapor pressure lowering effects. +The primary variables are (:math:`P`, :math:`T`, :math:`P_a`) for single-phase conditions and (:math:`P_g`, :math:`S_g`, :math:`P_a`) for two-phase conditions, where :math:`P_a` is the partial pressure of air. +Note that in two-phase conditions temperature is not among the primary variables. +It is implicitly determined from the relationship :math:`P_g - P_a = P_v`, with :math:`P_v = P_v(T, S_l)` as given in Eq. :math:numref:`eq:eos4.1`. + +It would be possible to use other sets of primary variables; in particular, temperature could be used also in two-phase conditions. +Our test calculations for a number of examples indicated, however, that the choice (:math:`P_g`, :math:`S_g`, :math:`P_a`) usually leads to better convergence behavior than the choice (:math:`P_g`, :math:`S_g`, :math:`T`). +The reason for the numerically inferior behavior of the latter set is in the air mass balance. +With the variables (:math:`P_g`, :math:`S_g`, :math:`T`), the amount of air present in a grid block becomes controlled by the difference between total gas pressure :math:`P_g` and effective vapor pressure :math:`P_v = P_{sat}(T) f_{VPL}(T, S_l)` (Eq. :math:numref:`eq:eos4.1`), which can be subject to very severe numerical cancellation. +From the applications viewpoint, however, initialization of a flow problem with the set (:math:`P_g`, :math:`S_g`, :math:`T`) may be much more physical and convenient. +EOS4 allows the initialization of two-phase points as (:math:`P_g`, :math:`S_g`, :math:`T`); this capability can be selected by specifying ``MOP(19)`` = 1. +When using ``MOP(19)`` = 1, the second primary variable upon initialization can also be relative humidity :math:`RH`, expressed as a fraction (0 < :math:`RH` ≤ 1); this choice is made by entering the second primary variable as a negative number, which will serve as a flag to indicate that it means (negative of) relative humidity, and will be internally converted to saturation in the initialization phase. +The conversion consists of iteratively solving Kelvin's equation for given :math:`RH = f_{VPL}(T, S_l)` for :math:`S_l`. +Users need to beware that relative humidity specifications must be within the range that is feasible for the capillary pressure functions used. +If maximum capillary pressures are not strong enough to accommodate a chosen value of :math:`RH`, a diagnostic will be printed and execution will terminate. + +.. list-table:: Summary of EOS4. + :name: tab:eos4 + :widths: 1 3 + :align: center + + * - Components + - | #1: water + | #2: air + * - Parameter choices + - | (``NK``, ``NEQ``, ``NPH``, ``NB``) = + | (2, 3, 2, 6) water and air, nonisothermal (default) + | (1, 2, 2, 6) water only (no air), nonisothermal + | ``MOP(20)`` = 1 optionally suppresses vapor pressure lowering effects + | Molecular diffusion can be modeled by setting ``NB`` = 8 + * - Primary variables*† + - | Single-phase conditions: + | (:math:`P_g`, :math:`S_g`, :math:`P_a`): (pressure, temperature, air partial pressure) + | Two-phase conditions: + | :math:`P_g`, :math:`S_g`, :math:`P_a`: (gas phase pressure, gas saturation, air partial pressure) + +.. note:: + + | \* By setting ``MOP(19)`` = 1, initialization of two-phase conditions can be made with (:math:`P_g`, :math:`S_g`, :math:`T`), or with (:math:`P_g`, :math:`-RH`, :math:`T`), where :math:`RH` is relative humidity (0 < :math:`RH` ≤ 1). + | † By setting ``MOP(19)`` = 2, initialization can be made with EOS3-style variables (:math:`P`, :math:`X`, :math:`T`) for single-phase (:math:`X` is air mass fraction), (:math:`P_g`, :math:`S_g` + 10, :math:`T`) for two-phase. + +As a further convenience to users, the choice ``MOP(19)`` = 2 allows EOS4 to be initialized with :ref:`eos3` variables of (:math:`P`, :math:`X`, :math:`T`) for single-phase, (:math:`P_g`, :math:`S_g` + 10, :math:`T`) for two-phase. +This way continuation runs with EOS4 can be made from :ref:`eos3`-generated conditions. +Note that, when using ``MOP(19)`` ≠ 0 options, data block or file INCON must terminate on a blank record. +If ``+++`` is encountered in *INCON*, it is assumed that primary variables are provided in agreement with internal usage; ``MOP(19)`` is then reset to zero and an informative message is printed. + +Vapor pressure lowering effects raise new issues because they make it possible for a liquid phase to exist under conditions where vapor partial pressure and gas phase total pressure are less than the saturation pressure. +What is the appropriate pressure at which liquid phase density, enthalpy and viscosity are to be evaluated? +We believe that a physically plausible choice is to take :math:`P_1 = \max(P_g, P_{sat})`, and this has been implemented in EOS4. +The implementation faces a difficulty, however, because temperature is not among the primary variables in two-phase conditions, so that :math:`P_{sat}` is only implicitly known; moreover, vapor pressure lowering effects are functionally dependent on liquid phase density, which is also a function of temperature. +This leads to a potentially unstable situation with regard to the choice of liquid phase pressure under conditions where :math:`P_g \approx P_{sat}`, which happens to be a common occurrence in boiling regions. In order to avoid this problem we evaluate liquid water density in the Kelvin equation for vapor pressure lowering (Eq. :math:numref:`eq:eos4.1`) always at :math:`P_l = P_{sat}`, which will be an excellent approximation due to the small compressibility of liquid water. +In all accumulation and flow terms the density of liquid water is evaluated at :math:`P_l = \max(P_g, P_{sat})`. +Vapor pressure lowering can be optionally suppressed by setting ``MOP(20)`` = 1. diff --git a/doc/source/tough3/eos/eos5.rst b/doc/source/tough3/eos/eos5.rst new file mode 100644 index 00000000..41801d0f --- /dev/null +++ b/doc/source/tough3/eos/eos5.rst @@ -0,0 +1,96 @@ +.. _eos5: + +EOS5 +==== + +The EOS5 fluid property module is developed to study the behavior of groundwater systems in which hydrogen releases take place. +For instance, in a number of waste disposal projects, corrosive metals are to be emplaced in geologic formations beneath the water table. +These will evolve a mixture of gases, with hydrogen being the chief constituent. +EOS5 is a close "cousin" of :ref:`eos3`, the main difference being that the air component is replaced by hydrogen, with considerably different thermophysical properties (see :numref:`tab:eos5.1` to :numref:`tab:eos5.3`). +The assignment and handling of primary thermodynamic variables in EOS5 is identical to :ref:`eos3` (see :numref:`tab:eos5.4`). +The main differences in the assignment of secondary parameters are as follows. +Density of gaseous hydrogen is computed from the ideal gas law. +Viscosity and water solubility of hydrogen are interpolated from the data given in Table 1. +For temperatures in excess of 25˚C, the solubility at 25˚C is used. + +.. list-table:: Density of hydrogen at :math:`P` = 1 bar. + :name: tab:eos5.1 + :header-rows: 1 + :align: center + + * - Temperature + - Experimental* + - Ideal gas law† + * - 280 K + - 0.086546 kg/m\ :sup:`3` + - 0.08660 kg/m\ :sup:`3` + * - 300 K + - 0.080776 kg/m\ :sup:`3` + - 0.08082 kg/m\ :sup:`3` + +.. list-table:: Viscosity* of hydrogen. + :name: tab:eos5.2 + :header-rows: 1 + :align: center + + * - Pressure + - :math:`T` = 0˚C + - :math:`T` = 100˚C + * - 1 bar + - 8.40 x 10\ :sup:`-6` Pa·s + - 10.33 x 10\ :sup:`-6` Pa·s + * - 100 bar + - 8.57 x 10\ :sup:`-6` Pa·s + - 10.44 x 10\ :sup:`-6` Pa·s + +.. list-table:: Solubility in water at :math:`P` = 1 bar\ :sup:`§`. + :name: tab:eos5.3 + :header-rows: 1 + :align: center + + * - Temperature + - Solubility + * - :math:`T` = 0˚C + - 1.92 x 10\ :sup:`-6` g H\ :sub:`2`/g H\ :sub:`2`\O + * - :math:`T` = 25˚C + - 1.54 x 10\ :sup:`-6` g H\ :sub:`2`/g H\ :sub:`2`\O + +.. note:: + + | \* from :cite:label:`vargaftik1975tables`, p. 39. + | † universal gas constant :math:`R` = 8314.56 J/mol/˚C; molecular weight of hydrogen 2.0160 kg/mol. + | \ :sup:`§` after :cite:label:`dean1999lange`; solubility at different pressures is computed from Henry's law. + + +Specifications +-------------- + +A summary of EOS5 specifications and parameters is given in Table :numref:`tab:eos5.4`. +The default parameter settings are (``NK``, ``NEQ``, ``NPH``, ``NB``) = (2, 3, 2, 6). +The option ``NEQ`` = 2 is available for constant temperature conditions. +The choice of primary thermodynamic variables is (:math:`P`, :math:`X`, :math:`T`) for single-phase, (:math:`P_g`, :math:`S_g` + 10, :math:`T`) for two-phase conditions. +As a convenience to users, we retain the capability to optionally initialize flow problems with TOUGH-style primary variables by setting ``MOP(19)`` = 1. +In TOUGH we have (:math:`P`, :math:`T`, :math:`X`) for single-phase conditions, (:math:`P_g`, :math:`S_g`, :math:`T`) for two-phase conditions. + +.. list-table:: Summary of EOS5. + :name: tab:eos5.4 + :widths: 1 3 + :align: center + + * - Components + - | #1: water + | #2: hydrogen + * - Parameter choices + - | (``NK``, ``NEQ``, ``NPH``, ``NB``) = + | (2, 3, 2, 6) water and hydrogen, nonisothermal (default) + | (2, 2, 2, 6) water and hydrogen, isothermal + | Molecular diffusion can be modeled by setting ``NB`` = 8 + * - Primary variables* + - | Single-phase conditions: + | (:math:`P`, :math:`X`, :math:`T`): (pressure, hydrogen mass fraction, temperature) + | Two-phase conditions: + | (:math:`P_g`, :math:`S_g` + 10, :math:`T`): (gas phase pressure, gas saturation plus 10, temperature) + +.. note:: + + \* By setting ``MOP(19)`` = 1, initialization can be made with TOUGH-style variables (:math:`P`, :math:`T`, :math:`X`) for single-phase, (:math:`P_g`, :math:`S_g`, :math:`T`) for two-phase. diff --git a/doc/source/tough3/eos/eos7.rst b/doc/source/tough3/eos/eos7.rst new file mode 100644 index 00000000..c066867a --- /dev/null +++ b/doc/source/tough3/eos/eos7.rst @@ -0,0 +1,163 @@ +.. _eos7: + +EOS7 +==== + +An extension of the :ref:`eos3` module for water/air mixtures, EOS7 represents the aqueous phase as a mixture of (pure) water and brine. +This approach is very useful for flow problems in which salinity does not reach saturated levels (:cite:label:`reeves1986theory, herbert1988coupled`). +The salinity of the aqueous phase is described by means of the brine mass fraction, Xb, and density and viscosity are interpolated from the values for the water and brine endmembers. +Salinity-dependent air solubility is also taken into account, but no allowance is made for reduction of vapor pressure with salinity. +The brine is modeled as NaCl solution, while the non-condensible gas is air, although the treatment could be adapted, with minor modifications, to other brines and gases. +The representation of the temperature and pressure dependence of thermophysical properties is somewhat more general than that of :cite:label:`reeves1986theory`, retaining the flexibility of the TOUGH3 formulation for nonisothermal processes. + +EOS7 can describe phase conditions ranging from single-phase liquid to two-phase to single-phase gas. +However, the approach of describing variably saline fluids not as mixtures of water and salt but as mixtures of water and brine has specific limitations which need to be considered in applications. +For example, in problems with significant boiling it would be possible for salinity to reach saturated levels, corresponding to brine mass fraction :math:`X_b` = 1 (supposing that the brine component was chosen as fully saturated salt solution). +Upon further boiling solid salt would precipitate, but solubility limits and solids precipitation are not taken into account in the approach used in EOS7 (these can be modeled with the EWASG module, see the addendum for EWASG). +As evaporation continues from a saline aqueous phase, eventually brine mass fraction would increase beyond :math:`X_b` = 1, in which case other mass fractions would become negative, and non-physical results would be obtained. +From a physical viewpoint brine mass fraction in the gas phase should always be equal to zero, but the only way the brine mass balance can be maintained during phase transitions from two-phase to single-phase vapor conditions is by allowing :math:`X_b`, gas to vary freely. +Users need to carefully examine problem setups and results to guard against unphysical results in applications that involve boiling. + +We now briefly summarize the treatment of thermophysical properties in EOS7. +The density of the aqueous phase is calculated from the assumption, shown to be very accurate by :cite:label:`herbert1988coupled`, that fluid volume is conserved when water and brine are mixed. +Mixture density :math:`\rho_m` can then be expressed in terms of water and brine densities as follows + +.. math:: + :label: eq:eos7.1 + + \frac{1}{\rho_m} = \frac{1 - X_b}{\rho_w} + \frac{X_b}{\rho_b} + +where :math:`\rho_w` and :math:`\rho_b` are water and brine density, respectively. +Eq. :math:numref:`eq:eos7.1` applies to densities at fixed pressure and temperature conditions. +In order to achieve a simple approximation for fluid density at variable temperatures and pressures, EOS7 takes compressibility and expansivity of brine to be equal to those of water. +This will provide a reasonable approximation at least for a limited range of temperatures and pressures around the reference conditions (:math:`P_0`, :math:`T_0`). +The default reference brine has a density of 1185.1 kg/m\ :sup:`3` at reference conditions of :math:`P_0` = 1 bar, :math:`T_0` = 25˚C, corresponding to an NaCl solution of 24.98 wt-%, or 5.06 molar (:cite:label:`potter1977volumetric`; cited after :cite:label:`finley1982swift`). +The user may specify different reference conditions and brine densities. +Effects of salinity on the enthalpy of the aqueous phase are ignored. + +Following :cite:label:`herbert1988coupled`, salinity effects on aqueous phase viscosity are modeled with a polynomial correction to the viscosity of pure water. +Mixture viscosity :math:`\mu_m` is represented as follows + +.. math:: + :label: eq:eos7.2 + + \mu_m (P, T, X_b) = \mu_w (P, T) f(X_b) + +where + +.. math:: + :label: eq:eos7.3 + + f(X_b) = 1 + \nu_1 X_b + \nu_2 X_b^2 + \nu_3 X_b^3 + +with default values of :math:`\nu_1` = 0.4819, :math:`\nu_2` = -0.2774, and :math:`\nu_3` = 0.7814. +Different values for the coefficients may be specified by the user. + +Gas (air) dissolution in the aqueous phase is modeled by Henry's law, as follows + +.. math:: + :label: eq:eos7.4 + + P_a = K_h x_{aq}^a + +where :math:`K_h` is Henry's constant and :math:`x_{aq}^a` is air mole fraction in the aqueous phase. +In saline solutions, gases are generally less soluble than in water ("salting out" effect). +For a 5 :math:`N` (molar) NaCl solution, nitrogen solubility is virtually independent of temperature in the range 0˚C ≤ :math:`T` ≤ 100˚C, and corresponds to a Henry's constant of :math:`K_h` = 4.0 x 10\ :sup:`10` Pa (:cite:label:`cygan1991solubility`). + +We retain the value of :math:`K_h` = 10\ :sup:`10` Pa for pure water, and represent air solubility (inverse of Henry's constant) as a linear function of mixture molarity :math:`N_m`, as follows + +.. math:: + :label: eq:eos7.5 + + \frac{1}{K_h} = 1.0 \times 10^{-10} + \frac{N_m}{5} \left( \frac{1}{4.0 \times 10^{10}} - 10^{-10} \right) + + +Specifications +-------------- + +A summary of EOS7 specifications and parameters appears in :numref:`tab:eos7`. +The default parameter settings are (``NK``, ``NEQ``, ``NPH``, ``NB``) = (3, 3, 2, 6). +The ``NK`` = 2 (no air) option may only be used for problems with single-phase liquid conditions throughout. +The primary variables are (:math:`P`, :math:`X_b`, :math:`X`, :math:`T`) for single-phase conditions and (:math:`P`, :math:`X_b`, :math:`S` + 10, :math:`T`) for two-phase conditions, where :math:`X` is air mass fraction. + +.. list-table:: Summary of EOS7. + :name: tab:eos7 + :widths: 1 3 + :align: center + + * - Components + - | #1: water + | #2: brine + | #3: air (optional)* + * - Parameter choices + - | (``NK``, ``NEQ``, ``NPH``, ``NB``) = + | (3, 3, 2, 6) water, brine, air, isothermal (default) + | (3, 4, 2, 6) water, brine, air, nonisothermal + | (2, 2, 2, 6) water, brine, isothermal* + | (2, 3, 2, 6) water, brine, nonisothermal* + | Molecular diffusion can be modeled by setting ``NB`` = 8 + * - Primary variables + - | Single-phase conditions: + | (:math:`P`, :math:`X_b`, :math:`X`, :math:`T`): (pressure, brine mass fraction, air mass fraction, temperature) + | Two-phase conditions: + | (:math:`P`, :math:`X_b`, :math:`S` + 10, :math:`T`): (gas phase pressure, brine mass fraction, gas saturation plus ten, temperature) + +.. note:: + + \* The ``NK`` = 2 (no air) option may only be used for problems with single-phase liquid conditions throughout. + + +Selections +---------- + +Users may specify parameters for reference brine in the TOUGH3 input file by means of an optional data block **SELEC**, as follows + +.. list-table:: Record **SELEC.1**. + :name: tab:eos7.selec.1 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - ``IE(1)`` + - I5 + - set equal to 2, to read two additional data records (a larger value with more additional data records is acceptable, but only the first two will be used by EOS7). + +.. list-table:: Record **SELEC.2**. + :name: tab:eos7.selec.2 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - :math:`P_0`\* + - E10.4 + - reference pressure (Pa). + * - :math:`T_0`\* + - E10.4 + - reference temperature (˚C). + * - :math:`\rho_b`\* + - E10.4 + - brine density at (:math:`P_0`, :math:`T_0`) (kg/m\ :sup:`3`). + +.. note:: + + \* If any of these parameters is entered as zero, default values of :math:`P_0` = 1 bar, :math:`T_0` = 25˚C, :math:`\rho_b` = 1185.1 kg/m\ :sup:`3` will be used. + For :math:`P_0` < 0, brine properties will be assumed identical to water. + +.. list-table:: Record **SELEC.3**. + :name: tab:eos7.selec.3 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - :math:`\nu_1`, :math:`\nu_2`, :math:`\nu_3` + - 3E10.4 + - coefficients for salinity correction of aqueous phase viscosity, see Eq. :math:numref:`eq:eos7.3`. diff --git a/doc/source/tough3/eos/eos7r.rst b/doc/source/tough3/eos/eos7r.rst new file mode 100644 index 00000000..b81337bd --- /dev/null +++ b/doc/source/tough3/eos/eos7r.rst @@ -0,0 +1,278 @@ +.. _eos7r: + +EOS7R +===== + +This is an enhanced version of the :ref:`eos7` module in which two additional mass components have been added, providing radionuclide transport capability for TOUGH3. +These components can undergo decay with user-specified half-life, with radionuclide 1 (Rn1 for short) being the "parent" and Rn2 the "daughter". +The radionuclides are considered water-soluble as well as volatile, but are not allowed to form a separate non-aqueous fluid phase. +Sorption onto the solid grains is also included. +The decaying components are normally referred to as radionuclides, but they may in fact be any trace components that decay, adsorb, and volatilize. +The decay process need not to be radioactive decay, but could be any process that follows a first-order decay law, such as biodegradation. +Stable trace components, such as volatile and water soluble organic chemicals (VOCs), can be modeled simply by setting half-life to very large values. +A detailed description of the mathematical model and numerical implementation used in EOS7R is available in a laboratory report, which also presents a number of illustrative problems, including verification against analytical solutions (:cite:label:`oldenburg1995eos7r`). +Here we highlight the main aspects of the thermophysical properties model. +Note that the current TOUGH3 does not include the dispersion module T2DM, therefore cannot simulate Fickian hydrodynamic dispersion. + +Radionuclide decay is described by + +.. math:: + :label: eq:eos7r.1 + + \frac{dM^{\kappa}}{dt} = -\lambda_{\kappa} M^{\kappa} + +where :math:`M^{\kappa}` is the mass of radionuclide :math:`\kappa` (= Rn1, Rn2) per unit volume, and the decay constant :math:`\lambda_{\kappa}` is related to the half-life by + +.. math:: + :label: eq:eos7r.2 + + T_{1/2} = \frac{\log(2)}{\lambda_{\kappa}} + +Adsorption of radionuclides on the solid grains is modeled as reversible instantaneous linear sorption, so that mass of radionuclide component :math:`\kappa` per unit reservoir volume is given by + +.. math:: + :label: eq:eos7r.3 + + M^{\kappa} = \phi \sum_{\beta} S_{\beta} \rho_{\beta} X_{\beta}^{\kappa} + \left( 1 - \phi \right) \rho_R \rho_{aq} X_{aq}^{\kappa} K_d + +where :math:`K_d` is the aqueous phase distribution coefficient (:cite:label:`de1986quantitative`, p. 256). + +In addition to adsorbing onto solid matrix grains, the radionuclide components may volatilize into the gas phase, if present. +Radionuclides partition between aqueous and gaseous phases according to Henry's law: + +.. math:: + :label: eq:eos7r.4 + + P_a^{\kappa} = K_h^{\kappa} x_{aq}^{\kappa} + +where :math:`P_a^{\kappa}` is the partial pressure in the gas phase of radionuclide :math:`\kappa`, :math:`K_h^{\kappa}` is Henry's constant and :math:`x_{aq}^{\kappa}` is the mole fraction of radionuclide :math:`\kappa` in the aqueous phase. +In EOS7R as in :ref:`eos7`, no solubility constraints are enforced for the brine. +Users need to be aware that there are inherent limitations in the ability of a water-brine mixture model to describe processes that involve significant vaporization. +Unphysical results may be obtained in thermal problems with strong vaporization effects. + +For the gas phase we assume ideal gas law for air and the radionuclides, and additivity of partial pressures. + +.. math:: + :label: eq:eos7r.5 + + P_{gas} = P_{air} + P_{vapor} + P_{Rn1} + P_{Rn2} + +Gas phase density is calculated as the sum of the partial densities of gaseous components, while gas phase viscosity and enthalpy are calculated from the same air-vapor mixing models used in :ref:`eos3`. +Apart from these approximations for gas phase viscosity and enthalpy, there is no restriction to "small" radionuclide concentrations in the gas phase, fully allowing gas phase radionuclide partial pressures. + +The thermophysical properties of the aqueous phase are assumed independent of radionuclide concentrations. +Implicit in this approximation is the assumption that aqueous radionuclide concentrations are small. +Users need to keep this limitation in mind, because EOS7R does not provide any intrinsic constraints on radionuclide concentrations. + + +Specifications +-------------- + +A summary of EOS7R specifications and parameters are given in :numref:`tab:eos7r`. +The default parameter settings are (``NK``, ``NEQ``, ``NPH``, ``NB``) = (5, 5, 2, 8). +The ``NK`` = 4 (no air) option may only be used for problems with single-phase liquid conditions throughout. +The primary variables are (:math:`P`, :math:`X_b`, :math:`X_{Rn1}`, :math:`X_{Rn2}`, :math:`X_{air}`, :math:`T`) for single-phase conditions and (:math:`P`, :math:`X_b`, :math:`X_{Rn1}`, :math:`X_{Rn2}`, :math:`S` + 10, :math:`T`) for two-phase conditions. + +.. list-table:: Summary of EOS7R. + :name: tab:eos7r + :widths: 1 3 + :align: center + + * - Components + - | #1: water + | #2: brine + | #3: Rn1 (radionuclide 1; "parent") + | #4: Rn2 (radionuclide 2; "daughter") + | #5: air (optional)† + * - Parameter choices + - | (``NK``, ``NEQ``, ``NPH``, ``NB``)\ :sup:`§` = + | (5, 5, 2, 8) water, brine, Rn1, Rn2, air, isothermal (default) + | (5, 6, 2, 8) water, brine, Rn1, Rn2, air, nonisothermal + | (4, 4, 2, 8) water, brine, Rn1, Rn2, no air, isothermal + | (4, 5, 2, 8) water, brine, Rn1, Rn2, no air, nonisothermal + | Molecular diffusion can be suppressed by setting ``NB`` = 6 + * - Primary variables\ :sup:`§` + - | Single-phase conditions: + | (:math:`P`, :math:`X_b`, :math:`X_{Rn1}`, :math:`X_{Rn2}`, :math:`X_{air}`, :math:`T`): (pressure, brine mass fraction, mass fraction of Rn1, mass fraction of Rn2, temperature) + | Two-phase conditions: + | (:math:`P`, :math:`X_b`, :math:`X_{Rn1}`, :math:`X_{Rn2}`, :math:`S` + 10, :math:`T`): (gas phase pressure, brine mass fraction, mass fraction of Rn1, mass fraction of Rn2, gas saturation plus ten, temperature)* + +.. note:: + + | † the no air option (``NK`` = 4) may only be used for problems with single-phase liquid conditions throughout. + | \ :sup:`§` parameter NKIN following NB may optionally be set to ``NKIN`` = ``NK`` - 2, in which caseradionuclide mass fractions will be omitted, and initialization will be made from only four :ref:`eos7`-style variables; radionuclide mass fractions will be initialized as zero. + | \* in two-phase conditions, :math:`X_{Rn1}` and :math:`X_{Rn2}` are mass fractions in the aqueous phase. + +The phase change diagnostics are as follows. +For single-phase liquid conditions, Henry's law is used to calculate the partial pressures that the non-condensible gases would have if a gas phase were present. +The total pressure that a gas phase would have if present is then calculated from Eq. :math:numref:`eq:eos7r.5`, using saturated vapor pressure at prevailing temperature. +This is compared with the aqueous phase pressure, and a transition to two-phase conditions is made when :math:`P_{gas}` :math:`P_{gas}` > :math:`P_{aq}`. +In two-phase conditions, the saturation variable is monitored. +A phase transition to single-phase liquid occurs when :math:`S_g` < 0, while for :math:`S_g` > 1 a transition to single-phase gas conditions is made. +For transitions from two-phase to single-phase liquid conditions, liquid pressure is initialized as :math:`P_{aq}` = (1 + 10\ :sup:`-6`) × :math:`P_{gas}`, with :math:`P_{gas}` given by Eq. :math:numref:`eq:eos7r.5`, while for transitions to single-phase gas conditions, pressure is initialized as (1 - 10\ :sup:`-6`) × :math:`P_{gas}`. +For single-phase gas conditions we monitor vapor pressure :math:`P_{vap} = P_{gas} - P_{air} - P_{Rn1} - P_{Rn2}`; a transition to two-phase conditions occurs when :math:`P_{vap}` > :math:`P_{gas}`. + + +Selections +---------- + +Brine and radionuclide properties are specified in the TOUGH3 input file by means of a data block **SELEC**, as follows. +Note that the current version of TOUGH3 does not include the dispersion module T2DM. + +.. list-table:: Record **SELEC.1**. + :name: tab:eos7r.selec.1 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - ``IE(1)`` + - I5 + - set equal to 6 to read six additional records with data for brine and radionuclides, and for hydrodynamic dispersion. The 7 input variables following ``IE(1)`` are for the dispersion module T2DM only and can be left blank if T2DM is not used. + * - ``NGBINP(1)`` + - I5 + - number of grid blocks in X (must always be equal to 1). + * - ``NGBINP(2)`` + - I5 + - number of grid blocks in Y. + * - ``NGBINP(3)`` + - I5 + - number of grid blocks in Z. + * - ``NFBL`` + - I5 + - number of the first ("left") column of grid blocks within the flow domain (defaults to 1 if zero or blank). + * - ``NFBR`` + - I5 + - number of the last ("right") column of grid blocks within the flow domain (defaults to ``NGBINP(2)`` if zero or blank). + * - ``NFBT`` + - I5 + - number of the first ("top") row of grid blocks within the flow domain (defaults to 1 if zero or blank). + * - ``NFBB`` + - I5 + - number of the last ("bottom") row of grid blocks within the flow domain (defaults to ``NGBINP(3)`` if zero or blank). + +.. list-table:: Record **SELEC.2**. + :name: tab:eos7r.selec.2 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - :math:`P_0`\* + - E10.4 + - reference pressure (Pa). + * - :math:`T_0`\* + - E10.4 + - reference temperature (˚C). + * - :math:`\rho_b`\* + - E10.4 + - brine density at (:math:`P_0`, :math:`T_0`) (kg/m\ :sup:`3`). + +.. note:: + + \* If any of these parameters is entered as zero, default values of :math:`P_0` = 1 bar, :math:`T_0` = 25˚C, :math:`\rho_b` = 1185.1 kg/m\ :sup:`3` will be used. + For :math:`P_0` < 0, brine properties will be assumed identical to water. + +.. list-table:: Record **SELEC.3**. + :name: tab:eos7r.selec.3 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - :math:`\nu_1`, :math:`\nu_2`, :math:`\nu_3` + - 3E10.4 + - coefficients for salinity correction of aqueous phase viscosity, following :cite:label:`herbert1988coupled`. + + .. math:: + :label: eq:eos7r.6 + + f(X_b) = 1 + \nu_1 X_b + \nu_2 X_b^2 + \nu_3 X_b^3 + + with default values of :math:`\nu_1` = 0.4819, :math:`\nu_2` = -0.2774, and :math:`\nu_3` = 0.7814. + Different values for the coefficients may be specified by the user. + +.. list-table:: Record **SELEC.4**. + :name: tab:eos7r.selec.4 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - ``ALPHAT`` + - E10.4 + - transverse dispersivity (m). + * - ``ALPHAL`` + - E10.4 + - longitudinal dispersivity (m). + +.. list-table:: Record **SELEC.5**. + :name: tab:eos7r.selec.5 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - ``FDDIAG(NP,NK)`` + - E10.4 + - ``NP`` = 1, 2; ``NK`` = 1, 2, 5 molecular diffusivities in units of m\ :sup:`2`/s; first the three gas phase diffusivities for water, brine, and air; then the three aqueous phase diffusivities for water, brine, and air. If a data block **DIFFU** is present, it will override the diffusivity specifications made in **SELEC**. + +.. list-table:: Record **SELEC.6**. + :name: tab:eos7r.selec.6 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - ``XHALF(3)`` + - E10.4 + - half-life of parent radionuclide (Rn1, component 3) (seconds). + * - ``XMW(3)`` + - E10.4 + - molecular weight of Rn1 (g/mol). + * - ``FDDIAG(NP,3)`` + - E10.4 + - molecular diffusivity of Rn1 in the gas phase in m\ :sup:`2`/s; followed by molecular diffusivity of Rn1 in the aqueous phase. If a data block **DIFFU** is present, it will override the diffusivity specifications made in **SELEC**. + * - + - 20X + - (void) + * - ``HCRN(1)`` + - E10.4 + - inverse Henry's constant :math:`K_h^{-1}` (see Eq. :math:numref:`eq:eos7r.4`) for parent radionuclide Rn1 (Pa\ :sup:`-1`). (The inverse Henry's constant can be thought of as an aqueous phase solubility). + +.. list-table:: Record **SELEC.7**. + :name: tab:eos7r.selec.7 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - ``XHALF(4)`` + - E10.4 + - half-life of parent radionuclide (Rn2, component 4) (seconds). + * - ``XMW(4)`` + - E10.4 + - molecular weight of Rn2 (g/mol). + * - ``FDDIAG(NP,4)`` + - E10.4 + - molecular diffusivity of Rn2 in the gas phase in m\ :sup:`2`/s; followed by molecular diffusivity of Rn2 in the aqueous phase. If a data block **DIFFU** is present, it will override the diffusivity specifications made in **SELEC**. + * - + - 20X + - (void) + * - ``HCRN(1)`` + - E10.4 + - inverse Henry's constant :math:`K_h^{-1}` (see Eq. :math:numref:`eq:eos7r.4`) for daughter radionuclide Rn2 (Pa\ :sup:`-1`). (The inverse Henry's constant can be thought of as an aqueous phase solubility). diff --git a/doc/source/tough3/eos/eos8.rst b/doc/source/tough3/eos/eos8.rst new file mode 100644 index 00000000..cd8987ee --- /dev/null +++ b/doc/source/tough3/eos/eos8.rst @@ -0,0 +1,176 @@ +.. _eos8: + +EOS8 +==== + +EOS8 is an enhanced version of :ref:`eos3`, and is developed from :ref:`eos3` by adding a third non-volatile, insoluble component, called "oil", which can only exist in a separate, immiscible phase. +This module provides a very basic and simple capability for modeling three fluid phases, including gaseous, aqueous, and oleic phases. +EOS8 can represent what in petroleum engineering is often referred to as a "dead oil", meaning a non-aqueous phase liquid that has no volatile or soluble components, so that it is present only in the non-aqueous phase. +The thermophysical property description related to the third (oil) phase is intentionally kept very simple and, although EOS8 may be applicable to some flow problems of practical interest as is, it should be considered a development platform rather than a realistic description of three-phase fluid systems. + +The relative permeabilities of the gas and aqueous phases are considered functions of their respective phase saturations only, :math:`k_{rg} = k_{rg}(Sg)`, :math:`k_{rl} = k_{rl}(S_{aq})`, and are specified through input data in the usual manner. +The oil phase is considered of intermediate wettability, and its relative permeability is described in a schematic fashion that is intended to serve as a template for users. + +.. math:: + :label: eq:eos8.1 + + k_{ro} = \frac{S_o - S_{or}}{1 - S_{or}} + +Here, :math:`S_{or}` is the residual oil saturation. +Oil phase capillary pressure is neglected. +Viscosity, density, and specific enthalpy of the oil phase are described as functions of pressure and temperature through low-order polynomials, with user-specified parameters entered through data block **SELEC**. +More specifically, viscosity is written as + +.. math:: + :label: eq:eos8.2 + + \mu_{oil} = a + b \left( P - P_1 \right) + c \left( P - P_2 \right)^2 + d \left( T - T_1 \right) + e \left( T - T_2 \right)^2 + +Oil density is + +.. math:: + :label: eq:eos8.3 + + \rho_{oil} = \rho_0 + C \left( P - P_0 \right) - E \left( T - T_0 \right) + +where :math:`\rho_0` is oil density at reference pressure and temperature conditions of (:math:`P_0`, :math:`T_0`). +Specific enthalpy of oil is assumed proportional to temperature (normalized to :math:`h_{oil}` = 0 at :math:`T` = 0˚C). + +.. math:: + :label: eq:eos8.4 + + h_{oil} = \zeta T + +All parameters appearing in the oil phase property description in Eqs. :math:numref:`eq:eos8.2`, :math:numref:`eq:eos8.3`, and :math:numref:`eq:eos8.4` are to be provided by the user through data block **SELEC**. +The property correlations are implemented through arithmetic statement functions at the top of EOS8 in a manner that should be transparent to users. +Users may also refer detailed comment statements in the EOS8 source code for instructions of preparing EOS8 input data. + + +Specifications +-------------- + +A summary of EOS8 specifications and parameters is given in :numref:`tab:eos8`. +The default parameter settings are (``NK``, ``NEQ``, ``NPH``, ``NB``) = (3, 3, 3, 6). +The choice of primary thermodynamic variables is (:math:`P`, :math:`X`, :math:`S_o`, :math:`T`) for single-phase, (:math:`P_g`, :math:`S_g` + 10, :math:`S_o`, :math:`T`) for two-phase conditions. +With EOS8 the user can optionally run just two phases (aqueous-gas, without the oil phase), in which case the aqueous-gas process descriptions reduce to those of :ref:`eos3`. + +.. list-table:: Summary of EOS8. + :name: tab:eos8 + :widths: 1 3 + :align: center + + * - Components + - | #1: water + | #2: brine + | #3: oil + * - Parameter choices + - | (``NK``, ``NEQ``, ``NPH``, ``NB``) = + | (3, 3, 3, 6) water, air, oil, isothermal (default) + | (3, 4, 3, 6) water, air, oil, nonisothermal + | (2, 2, 2, 6) water, air, no oil, isothermal + | (2, 3, 2, 6) water, air, no oil, nonisothermal + * - Primary variables + - | Single-phase conditions: + | (:math:`P`, :math:`X`, :math:`S_o`, :math:`T`): (pressure, air mass fraction, oil phase saturation, temperature) + | Two-phase conditions: + | (:math:`P_g`, :math:`S_g` + 10, :math:`S_o`, :math:`T`): (gas phase pressure, gas saturation plus ten, oil phase saturation, temperature) + + +Selections +---------- + +The parameters used in the oil phase property description are specified in the TOUGH3 input file by means of a data block **SELEC**, as follows + +.. list-table:: Record **SELEC.1**. + :name: tab:eos8.selec.1 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - ``IE(1)`` + - I5 + - set equal to 6 to read six additional records with data for brine and radionuclides, and for hydrodynamic dispersion. + +.. list-table:: Record **SELEC.2**. + :name: tab:eos8.selec.2 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - + - X + - (void) + +.. list-table:: Record **SELEC.3**. + :name: tab:eos8.selec.3 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - + - X + - (void) + +.. list-table:: Record **SELEC.4**. + :name: tab:eos8.selec.4 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - :math:`S_{or}` + - E10.4 + - irreducible saturation of oil phase. + * - :math:`a` + - E10.4 + - constant portion in Eq. :math:numref:`eq:eos8.2` for oil phase viscosity. + +.. list-table:: Record **SELEC.5**. + :name: tab:eos8.selec.5 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - :math:`a`, :math:`P_1`, :math:`c`, :math:`P_2`, :math:`d`, :math:`T_1`, :math:`e`, :math:`T_2` + - 8E10.4 + - coefficients for oil phase viscosity in Eq. :math:numref:`eq:eos8.2`. + +.. list-table:: Record **SELEC.6**. + :name: tab:eos8.selec.6 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - :math:`\rho_0`, :math:`C`, :math:`P_0`, :math:`E`, :math:`T_0` + - 5E10.4 + - coefficients for oil density in Eq. :math:numref:`eq:eos8.3`. + +.. list-table:: Record **SELEC.7**. + :name: tab:eos8.selec.7 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - :math:`\zeta` + - 10.4 + - coefficients for oil specific enthalpy in Eq. :math:numref:`eq:eos8.4`. diff --git a/doc/source/tough3/eos/eos9.rst b/doc/source/tough3/eos/eos9.rst new file mode 100644 index 00000000..8236c734 --- /dev/null +++ b/doc/source/tough3/eos/eos9.rst @@ -0,0 +1,116 @@ +.. _eos9: + +EOS9 +==== + +This module considers variably saturated flow of a single aqueous phase, which consists entirely of a single water component, and neglects phase change effects. +The gas phase is treated as a passive bystander at constant pressure, and conditions are assumed to be isothermal. +Thus, no mass balance equation for gas and no heat balance is needed, and only a single water mass balance equation is solved for each grid block. +This is very efficient numerically, making EOS9 the module of choice for problems for which the underlying approximations are applicable. + +Liquid flow in EOS9 is described as follows: + +.. math:: + :label: eq:eos9.1 + + \frac{\partial}{\partial t} \phi \S_l \rho_l = \nabla \cdot \left( k \frac{k_{rl}}{\mu_l} \rho_l \nabla \left( P_l + \rho_l g z \right) \right) + +where :math:`\phi` is porosity, :math:`S_l` is water saturation, :math:`\rho_l` is water density, :math:`k` is absolute permeability, :math:`k_{rl}` is relative permeability to the aqueous phase, :math:`\mu_l` is water viscosity, :math:`P_l` is water pressure, :math:`g` is acceleration of gravity, and :math:`z` is defined positive upward. +Neglecting variations in liquid phase density and viscosity, as is appropriate for (nearly) isothermal conditions, Eq. :math:numref:`eq:eos9.1` simplifies to Richards' equation (:cite:label:`richards1931capillary`). + +.. math:: + :label: eq:eos9.2 + + \frac{\partial \theta}{\partial t} = \nabla \cdot \left( K \nabla h \right) + +where :math:`\theta = \phi S_l` is specific volumetric moisture content, :math:`K = k k_{rl} \frac{\rho_l g}{\mu_l}` is hydraulic conductivity, and :math:`h = z + \frac{P_l}{\rho_l g}` is the hydraulic head. +EOS9 can describe flow under partially saturated (0 < :math:numref:`S_l` < 1) as well as fully saturated conditions, and phase changes between the two. + + +Specifications +-------------- + +With only a single mass balance equation per grid block, there is only a single primary thermodynamic variable. +This is taken to be pressure for single-phase (saturated) conditions, and is water saturation for unsaturated conditions. +A distinction between the two is made simply on the basis of the numerical value of the first (and only) primary variable, :math:`X1`. +If :math:`X1` < 1, this indicates that If :math:`X1` represents water saturation and conditions are unsaturated; if If :math:`X1` is larger than a user-specified gas phase reference pressure (default :math:`P_{gas}` = 1.013 x 10\ :sup:`5` Pa), it is taken to be water pressure, and saturated conditions prevail. +When phase changes between saturated and unsaturated conditions occur, the primary variable is switched, as follows. +The numerical value of :math:`X1` and its change during the Newton-Raphson iteration process is monitored. +If :math:`X1` changes from being smaller than 1 to larger than 1, this indicates attainment of fully saturated conditions. +In that case :math:`X1` is switched to pressure, and is initialized at a pressure slightly in excess of gas phase reference pressure as :math:`X1 = P_{gas}(1 + \varepsilon)`, with :math:`\varepsilon` = 10\ :sup:`-6`. +If :math:`X1` changes from being larger than :math:`P_{gas}` to smaller than :math:`P_{gas}`, this indicates a transition from fully to partially saturated conditions. +:math:`X1` is then switched to saturation, and is initialized as :math:`X1 = 1 - \varepsilon`. +Actually, a transition from fully to partially saturated conditions is made only when :math:`X1` drops below :math:`P_{gas}(1 - \varepsilon)`; test calculations have shown that such a (small) finite-size window for phase change improves numerical stability and efficiency. + +In EOS9, the thermophysical properties of water are taken at default reference conditions of :math:`P` = 1.013 x 10\ :sup:`5` Pa, :math:`T` = 15˚C. +These defaults can be overwritten in a flexible manner by specifying appropriate data in a fictitious **ROCKS** domain "``REFCO``", as follows + +.. list-table:: Domain "``REFCO``". + :name: tab:eos9.1 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - ``DROK`` + - E10.4 + - reference pressure + * - ``POR`` + - E10.4 + - reference temperature + * - ``PER(1)`` + - E10.4 + - liquid density + * - ``PER(2)`` + - E10.4 + - liquid viscosity + * - ``PER(3)`` + - E10.4 + - liquid compressibility + +Note that assignment of thermophysical data through a specially-named domain was set up just as a convenient way of providing floating-point parameters to the code. +No volume elements (grid blocks) should be attached to domain "``REFCO``", as the data in general will not correspond to reasonable hydrogeologic parameters. +The above mentioned defaults will be overwritten for any parameters for which a non-zero entry is provided in "``REFCO``". +This allows the generation of these parameters internally for user-defined (:math:`P`, :math:`T`); it also allows for directly assigning user-desired values as, e.g., :math:`\rho_{liq}` = 1000 kg/m\ :sup:`3`, :math:`\mu_{liq}` = 10\ :sup:`-3` Pa s (:math:`equiv` 1 centipoise), etc. +A summary of EOS9 specifications appears in :numref:`tab:eos9.2`. + +.. list-table:: Summary of EOS9. + :name: tab:eos9.2 + :widths: 1 3 + :align: center + + * - Components + - | #1: water + * - Parameter choices + - | (``NK``, ``NEQ``, ``NPH``, ``NB``) = + | (1, 1, 1, 6) water, isothermal (default, no other choices available) + * - Primary variables*† + - | Saturated conditions: + | (:math:`P_{liq}`): (water pressure: :math:`P_{liq}` ≥ :math:`P_{gas}`) + | Unsaturated conditions: + | (:math:`S_{liq}`): (water saturation: 0 < :math:`S_{liq}` < 1) + +.. note:: + + | \* The first primary variable may be initialized as :math:`X1` < 0, in which case it will be taken to denote capillary pressure, and will be converted internally to :math:`S_{liq}` in the initialization phase. + | † Reference gas phase pressure, flow system temperature, and (optionally) thermophysical parameters of water density, viscosity, and compressibility may be specified through a fictitious ROCKS domain "``REFCO``". + +In addition to specifying the primary thermodynamic variable on a default, domain, or grid block basis, EOS9 offers alternative ways of initializing flow problems. +The primary variable may be entered as a negative number upon initialization, in which case it will be taken to denote capillary pressure, and will be internally converted to :math:`S_l` in the initialization phase. +EOS9 can also initialize a flow problem with gravity-capillary equilibrium, relative to a user-specified reference elevation :math:`z_{ref}` of the water table. +This type of initialization will be engaged if the user enters a non-zero number in slot ``CWET`` in **ROCKS** domain "``REFCO``", in which case ``CWET`` will be taken to denote the water table elevation :math:`z_{ref}`, in units of meters. +Water pressure at :math:`z_{ref}` is taken equal to reference gas pressure, :math:`P_l(z_{ref}) = P_{gas}`, and is initialized as a function of grid block elevation according to :math:`P(z) = P_{gas} + \left( z_{ref} - z \right) \rho g`. +By convention, the z-axis is assumed to point upward. +In order to use this facility, the z-coordinates (grid block elevations) must be specified in the **ELEME**-data, which will be done automatically if internal *MESH* generation is used. + +In the assignment of gravity-capillary equilibrium as just discussed, water saturations at "sufficiently" high elevations above the water table may end up being smaller than the irreducible water saturation :math:`S_{lr}` specified in the relative permeability function, which may or may not be consistent with the physical behavior of the flow system. +Users may optionally enforce that :math:`S_l` = :math:`S_{lr}` in regions where the capillary pressure function would dictate that :math:`S_{lr}` < :math:`S_{lr}`. +This is accomplished by entering an appropriate parameter in slot ``SPHT`` of **ROCKS** domain "``REFCO``", and works as follows. +The irreducible saturation :math:`S_{lr}` will be taken to be parameter ``RP(int(SPHT))`` of the relative permeability function. +As an example, for the ``IRP`` = 7 relative permeability function, irreducible water saturation is the parameter ``RP(2)``; therefore, for ``IRP`` = 7 the user should specify ``SPHT`` = 2.0 in "``REFCO``" to use this facility. + +EOS9 differs from all of the other EOS modules in that, having only a single primary thermodynamic variable, the first (and here only) primary variable does not necessarily denote pressure. +This necessitates certain other coding adjustments. +For EOS9, the flow terms is assembled differently in subroutine *MULTI*, and the slot #6 of the PAR-array, which normally holds just the capillary pressure, represents total water pressure :math:`P_l = P_{gas} + P_{cap}`. diff --git a/doc/source/tough3/eos/ewasg.rst b/doc/source/tough3/eos/ewasg.rst new file mode 100644 index 00000000..6adfb7f4 --- /dev/null +++ b/doc/source/tough3/eos/ewasg.rst @@ -0,0 +1,247 @@ +.. _ewasg: + +EWASG +===== + +Thermophysical Properties +------------------------- + +The EWASG (WAter-Salt-Gas) fluid property module was developed by :cite:label:`battistelli1997simulator` for modeling geothermal reservoirs with saline fluids and non-condensible gas (NCG). +In contrast to :ref:`eos7`, EWASG describes aqueous fluid of variable salinity not as a mixture of water and brine, but as a mixture of water and NaCl. +This makes it possible to represent temperature-dependent solubility constraints, and to properly describe precipitation and dissolution of salt. +EWASG represents the active system components (water, NaCl, NCG) as three-phase mixtures. +Solid salt is the only active mineral phase, and is treated in complete analogy to fluid phases (aqueous, gas), except that, being immobile, its relative permeability is identically zero. +From mass balances on salt in fluid and solid phases we calculate the volume fraction of precipitated salt in the original pore space :math:`\phi_0`, which is termed "solid saturation", and denoted by :math:`S_s`. +A fraction :math:`\phi_0 S_s` of reservoir volume is occupied by precipitate, while the remaining void space :math:`\phi_f = \phi_0 \left( 1 - S_s \right)` is available for fluid phases. +We refer to :math:`\phi_f` as the "active flow porosity". +The reduction in pore space reduces the permeability of the medium. + +Several choices are available for the non-condensible gas (NCG): CO\ :sub:`2`, air, CH\ :sub:`4`, H\ :sub:`2`, and N\ :sub:`2`. +Gas dissolution in the aqueous phase is described by Henry's law, with coefficients that depend not only on temperature but also on salinity to describe the reduction in NCG solubility with increasing salinity ("salting out"). +The dependence of brine density, enthalpy, viscosity, and vapor pressure on salinity is taken into account, as are vapor pressure-lowering effects from suction pressures (capillary and vapor adsorption effects). +The thermophysical property correlations used in EWASG are accurate for most conditions of interest in geothermal reservoir studies: temperatures in the range from 100 to 350˚C, fluid pressures up to 80 MPa, CO\ :sub:`2` partial pressures up to 10 MPa, and salt mass fraction up to halite saturation. +With the exception of brine enthalpy, thermophysical property correlations are accurate to below 10˚C. +A full discussion of the thermophysical property correlations used and their empirical basis is given in the original paper (:cite:label:`battistelli1997simulator`). + +TOUGH3 also adopts recent improvements of EWASG. +Internally consistent correlations for the water-NaCl mixture properties are included (:cite:label:`battistelli2012improving`), which are developed by :cite:label:`driesner2007system1` and :cite:label:`driesner2007system2`. +These brine correlations are capable to calculate phase properties for temperatures from 0 to 350˚C, pressures from 1 to 100 MPa, and salt mass fraction up to saturation. +In TOUGH3, the brine correlations in :cite:label:`driesner2007system2` are used as the default, unless specified differently by users. + +New options are also added to calculate NCG (CO\ :sub:`2`, CH\ :sub:`4`, and H\ :sub:`2` only) density and fugacity using a virial equation treatment of :cite:label:`spycher1988fugacity`. +This method is available and reliable only for the following temperature and pressure ranges: (1) for CO\ :sub:`2`, 50 - 350˚C and up to 50 MPa, (2) for CH\ :sub:`4`, 16 - 350˚C and up to 50 MPa, and (3) for H\ :sub:`2`, 25 - 600˚C and up to 300 MPa. + + +Permeability Change +------------------- + +As noted above, the relationship between the amount of solid precipitation and the pore space available to the fluid phases is very simple. +The impact of porosity change on formation permeability on the other hand is highly complex. +Laboratory experiments have shown that modest reductions in porosity from chemical precipitation can cause large reductions in permeability (:cite:label:`vaughn1987analysis`). +This is explained by the convergent-divergent nature of natural pore channels, where pore throats can become clogged by precipitation while disconnected void spaces remain in the pore bodies (:cite:label:`verma1988thermohydrological`). +The permeability reduction effects depend not only on the overall reduction of porosity but on details of the pore space geometry and the distribution of precipitate within the pore space. +These may be quite different for different porous media, which makes it difficult to achieve generally applicable, reliable predictions. +EWASG offers several choices for the functional dependence of relative change in permeability, :math:`\frac{k}{k_0}`, on relative change in active flow porosity. + +.. math:: + :label: eq:ewasg.1 + + \frac{k}{k_0} = f \left( \frac{\phi_f}{\phi_0} \right) \equiv f \left( 1 - S_s \right) + +The simplest model that can capture the converging-diverging nature of natural pore channels consists of alternating segments of capillary tubes with larger and smaller radii, respectively; see :numref:`fig:ewasg.1`. +While in straight capillary tube models permeability remains finite as long as porosity is non-zero, in models of tubes with different radii in series, permeability is reduced to zero at a finite porosity. + +.. subfigure:: AB + :name: fig:ewasg.1 + :layout-sm: A|B + :subcaptions: above + :gap: 32px + + .. image:: ../figures/pore_conceptual_model.png + :alt: Conceptual model + + .. image:: ../figures/pore_tubes_in_series.png + :alt: Tubes-in-series + + Model for converging-diverging pore channels. + +From the tubes-in-series model shown in :numref:`fig:ewasg.1`, the following relationship can be derived (Verma and Pruess, 1988) + +.. math:: + :label: eq:ewasg.2 + + \frac{k}{k_0} = \theta^2 \frac{1 - \Gamma + \frac{\Gamma}{\omega^2}}{1 - \Gamma + \Gamma \left( \frac{\theta}{\theta + \omega - 1} \right)^2} + +Here + +.. math:: + :label: eq:ewasg.3 + + \theta = \frac{1 - S_s - \phi_r}{1 - \phi_r} + +depends on the fraction :math:`1 - S_s` of original pore space that remains available to fluids, and on a parameter :math:`\phi_r`, which denotes the fraction of original porosity at which permeability is reduced to zero. +:math:`\Gamma` is the fractional length of the pore bodies, and the parameter :math:`\omega` is given by + +.. math:: + :label: eq:ewasg.4 + + \omega = 1 + \frac{\frac{1}{\Gamma}}{\frac{1}{\phi_r} - 1} + +Therefore, Eq. :math:numref:`eq:ewasg.2` has only two independent geometric parameters that need to be specified, :math:`\phi_r` and :math:`\Gamma`. +As an example, :numref:`fig:ewasg.2` shows the permeability reduction factor from Eq. :math:numref:`eq:ewasg.2`, plotted against :math:`\frac{\phi}{\phi_0} \equiv \left( 1 - S_s \right)`, for parameters of :math:`\phi_r` = :math:`\Gamma` = 0.8. + +.. figure:: ../figures/porosity_permeability.png + :name: fig:ewasg.2 + :width: 75% + + Porosity-permeability relationship for tubes-in-series model, after :cite:label:`verma1988thermohydrological`. + +For parallel-plate fracture segments of different aperture in series, a relationship similar to Eq. :math:numref:`eq:ewasg.2` is obtained, the only difference being that the exponent 2 is replaced everywhere by 3 (:cite:label:`verma1988thermohydrological`). +If only straight capillary tubes of uniform radius are considered, we have :math:`\phi_r` = 0, :math:`\Gamma` = 0, and Eq. :math:numref:`eq:ewasg.2` simplifies to + +.. math:: + :label: eq:ewasg.5 + + \frac{k}{k_0} = \left( 1 - S_s \right)^2 + + +Specifications +-------------- + +A summary of EWASG specifications and parameters appears in :numref:`tab:ewasg`. +The default parameter settings are (``NK``, ``NEQ``, ``NPH``, ``NB``) = (3, 4, 3, 6). +The ``NK`` = 2 (no air) option may only be used for problems with single-phase liquid conditions throughout. +The primary variables are (:math:`P`, :math:`X_{sm}`, :math:`X3`, :math:`T`) for single-phase conditions and (:math:`P`, :math:`X_{sm}`, :math:`S_g` + 10, :math:`T`) for two-phase conditions. + +Primary variable #2 (:math:`X2`) is used for NaCl, and denotes mass fraction :math:`X_s` in the aqueous phase when no solid salt is present, while it is solid saturation plus ten (:math:`S_s` + 10) in the presence of precipitated salt. +The number 10 is added here to be able to determine whether or not a precipitated phase is present from the numerical range of the second primary variable. +Solubility of NaCl in the gas phase is very small at the pressure and temperature conditions considered for EWASG and has been neglected. +During the Newton-Raphson iteration process, possible appearance or disappearance of a solid phase is checked, as follows. +If no solid phase was present at the previous iteration, primary variable :math:`X2` is known to denote salt mass fraction :math:`X_s` in the aqueous phase, and the latest updated value is compared with the equilibrium solubility :math:`XEQ`. +If :math:`S_s` > :math:`XEQ`, precipitation starts, a small solid phase saturation is initialized as :math:`S_s` = 10\ :sup:`-6`, and the second primary variable is switched to :math:`X2` = :math:`S_s` + 10. +If solid salt had been present at the previous iteration, EWASG checks whether :math:`S_s` = :math:`X2` - 10 is still larger than 0. +If not, this indicates that the solid phase disappears; the second primary variable is then switched to dissolved salt mass fraction, and is initialized just below equilibrium solubility as :math:`S_s` = :math:`XEQ` - 10\ :sup:`-6`. + +.. list-table:: Summary of EWASG. + :name: tab:ewasg + :widths: 1 3 + :align: center + + * - Components + - | #1: water + | #2: NaCl + | #3: NCG (CO\ :sub:`2`, air, CH\ :sub:`4`, H\ :sub:`2`, N\ :sub:`2`; optional) + * - Parameter choices + - | (``NK``, ``NEQ``, ``NPH``, ``NB``) = + | (3, 4, 3, 6) water, NaCl, NCG, nonisothermal (default) + | (3, 3, 3, 6) water, NaCl, NCG, isothermal + | (2, 3, 2, 6) water, NaCl, nonisothermal* + | (2, 2, 2, 6) water, NaCl, isothermal* + | Molecular diffusion can be modeled by setting ``NB`` = 8 + * - Primary variables + - | Single-phase conditions (only liquid or only gas): + | (:math:`P`, :math:`X_{sm}`, :math:`X3`, :math:`T`): (pressure, salt mass fraction or solid saturation plus ten, NCG mass fraction, temperature) + | Two-phase conditions: + | (:math:`P`, :math:`X_{sm}`, :math:`S_g` + 10, :math:`T`): (pressure, salt mass fraction or solid saturation plus ten, gas phase saturation plus ten, temperature) + +.. note:: + + | \* the ``NK`` = 2 (no NCG) option may only be used for problems with single-phase liquid conditions throughout. + | † two-phase conditions may be initialized with variables (:math:`T`, :math:`X_{sm}`, :math:`S_g` + 10, :math:`P_{NCG}`), or (:math:`T`, :math:`X_{sm}`, :math:`S_g` + 10, :math:`X3`), where :math:`P_{NCG}` is the partial pressure of NCG, :math:`X3` is mass fraction of NCG in the liquid phase; by convention, EWASG will assume the first primary variable to be pressure if it is larger than 370, otherwise it will be taken to be temperature; if the first primary variable is temperature, the last primary variable will be taken to mean mass fraction of NCG if it is less than 1, otherwise it will be taken to mean NCG partial pressure. + + +Selections +---------- + +Various options for EWASG can be selected through parameter specifications in data block **SELEC**, as follows + +.. list-table:: Record **SELEC.1**. + :name: tab:ewasg.selec.1 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - ``IE(1)`` + - I5 + - set equal to 1, to read one additional data record (a larger value with more data records is acceptable, but only one additional record will be used by EWASG). + * - ``IE(3)`` + - I5 + - | allows choice of brine viscosity calculation: + | 0: after Phillips et al. (1981) (default). + | 1: after Palliser and McKibbin (1998). + | 2: after Mao and Sun (2006). + | 3: after Potter (1978). + * - ``IE(4)`` + - I5 + - | allows choice of correlation for compressed brine density: + | 1: after Andersen et al. (1992) (default = 0). + | 2: Pritchett (1993). + | 3: Brine compressibility equal to water compressibility at the same reduced temperature. + | 4: Brine compressibility equal to water compressibility at the same temperature. + | 5: after Batzle and Wang (1992). + | 6: after Driesner (2007). + * - ``IE(8)`` + - I5 + - | allows choice of NCG density and fugacity calculation: + | 0: original EWASG approach (default). + | 1: NCG density according to Spycher and Reed (1988). Only for CO\ :sub:`2`, CH\ :sub:`4`, and H\ :sub:`2`. + | 2: NCG density and gas-aqueous equilibrium according to Spycher and Reed (1988). Only for CO\ :sub:`2`, CH\ :sub:`4`, and H\ :sub:`2`. + * - ``IE(9)`` + - I5 + - | allows choice of NCG enthalpy calculation in the aqueous phase: + | 0: original EWASG approach (default). + | 1: NCG enthalpy as a function of temperature. + * - ``IE(10)`` + - I5 + - | allows to turn vapor pressure lowering on/off: + | 0: VPL is off. + | 1: VPL is on. + * - ``IE(11)`` + - I5 + - | selects dependence of permeability on the fraction :math:`\frac{\phi_f}{\phi_0} = \left( 1 - S_s \right)` of original pore space that remains available to fluids: + | 0: permeability does not vary with :math:`\phi_f`. + | 1: :math:`\frac{k}{k_0} = \left( 1 - S_s \right)^{\gamma}`, with :math:`\gamma` = ``FE(1)``. + | 2: fractures in series, i.e., Eq. :math:numref:`eq:ewasg.2` with exponent 2 everywhere replaced by 3. + | 3: tubes-in-series, i.e., Eq. :math:numref:`eq:ewasg.2`. + * - ``IE(14)`` + - I5 + - | allows choice of treatment of thermophysical properties as a function of salinity: + | 0: full dependence (default). + | 1: vapor pressure independent of salinity. + | 2: vapor pressure and brine enthalpy independent of salinity. + | 3: no salinity dependence of thermophysical properties (salt solubility constraints are maintained). + * - ``IE(15)`` + - I5 + - | allows choice of correlation for brine enthalpy at saturated vapor pressure: + | 1: after Michaelides (1981). + | 2: after Miller (1978) (obsolete). + | 3: after Phillips et al. (1981). + | 4: after Lorenz et al. (2000) (default = 0). + | 5: after Driesner (2007). + * - ``IE(16)`` + - I5 + - | allows choice of the type of NCG (default for ``IE(16)`` = 0 is CO\ :sub:`2`): + | 1: air. + | 2: CO\ :sub:`2`. + | 3: CH\ :sub:`4`. + | 4: H\ :sub:`2`. + | 5: N\ :sub:`2`. + +.. list-table:: Record **SELEC.2**. + :name: tab:ewasg.selec.2 + :widths: 1 1 6 + :header-rows: 1 + :align: center + + * - Parameter + - Format + - Description + * - ``FE(1)`` + - E10.4 + - parameter :math:`\gamma` (for ``IE(11)`` = 1); parameter :math:`\phi_r` (for ``IE(11)`` = 2, 3). + * - ``FE(2)`` + - E10.4 + - parameter :math:`\Gamma` (for ``IE(11)`` = 2, 3). diff --git a/doc/source/tough3/eos/index.rst b/doc/source/tough3/eos/index.rst new file mode 100644 index 00000000..e1b052cb --- /dev/null +++ b/doc/source/tough3/eos/index.rst @@ -0,0 +1,60 @@ +EOS +=== + +TOUGH3 currently includes all equation-of-state (EOS) modules of the Version 2 of TOUGH2, as well as ECO2N (:cite:label:`pruess2005eco2n,pan2015eco2n`), ECO2M (:cite:label:`pruess2011eco2m`), EOS7C (:cite:label:`oldenburg2004eos7c`), EOS7CA (:cite:label:`oldenburg2015eos7ca`), and TMVOC (:cite:label:`pruess2002tmvoc`). + +.. list-table:: Fluid property modules for TOUGH3. + :name: tab:1 + :widths: 1 3 + :header-rows: 1 + :align: center + + * - Module + - Capabilities + * - :ref:`eos1` + - water, water with tracer + * - :ref:`eos2` + - water, CO\ :sub:`2` + * - :ref:`eos3` + - water, air + * - :ref:`eos4` + - water, air, with vapor pressure lowering + * - :ref:`eos5` + - water, hydrogen + * - :ref:`eos7` + - water, brine, air + * - :ref:`eos7r` + - water, brine, air, parent-daughter radionuclides + * - EOS7C + - water, brine, CO\ :sub:`2` (or N\ :sub:`2`), gas tracer, CH\ :sub:`4` + * - EOS7CA + - water, brine, non-condensible gas, gas tracer, air + * - :ref:`eos8` + - water, "dead" oil, non-condensible gas + * - :ref:`eos9` + - variably-saturated isothermal flow according to Richard's equation + * - :ref:`ewasg` + - water, salt (NaCl), 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