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model description added to handbook
This commit is contained in:
Bernd Flemisch
Andreas Lauser
parent
2968790d66
commit
06e0654d5d
1228
doc/handbook/EPS/masstransfer.eps
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doc/handbook/EPS/masstransfer.eps
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@ -52,6 +52,20 @@ dumux-handbook.html: $(DOCSOURCE) dune.cfg tex4ht.env
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$(T4HT) dumux-handbook.tex
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$(TEX) dumux-handbook.tex
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all:
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cat ../doxygen/latex/classDune_1_1OnePBoxModel.tex | ../../util/scripts/extractDetailedDescription.py >ModelDescriptions/1pboxmodel.tex
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cat ../doxygen/latex/classDune_1_1OnePTwoCBoxModel.tex | ../../util/scripts/extractDetailedDescription.py >ModelDescriptions/1p2cboxmodel.tex
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cat ../doxygen/latex/classDune_1_1RichardsBoxModel.tex | ../../util/scripts/extractDetailedDescription.py >ModelDescriptions/richardsboxmodel.tex
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cat ../doxygen/latex/classDune_1_1TwoPBoxModel.tex | ../../util/scripts/extractDetailedDescription.py >ModelDescriptions/2pboxmodel.tex
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cat ../doxygen/latex/classDune_1_1TwoPNIBoxModel.tex | ../../util/scripts/extractDetailedDescription.py >ModelDescriptions/2pniboxmodel.tex
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cat ../doxygen/latex/classDune_1_1TwoPTwoCBoxModel.tex | ../../util/scripts/extractDetailedDescription.py >ModelDescriptions/2p2cboxmodel.tex
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cat ../doxygen/latex/classDune_1_1TwoPTwoCNIBoxModel.tex | ../../util/scripts/extractDetailedDescription.py >ModelDescriptions/2p2cniboxmodel.tex
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latex dumux-handbook.tex
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bibtex dumux-handbook
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latex dumux-handbook.tex
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latex dumux-handbook.tex
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dvipdf dumux-handbook
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#dist-hook:
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# sed $(srcdir)/Makefile.dist.am -e 's/Makefile\.dist/Makefile/g' > $(distdir)/Makefile.am
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# sed $(srcdir)/Makefile.dist.in -e 's/Makefile\.dist/Makefile/g' > $(distdir)/Makefile.in
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@ -61,3 +75,4 @@ EXTRA_TEXINPUTS:=$(top_srcdir)
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include $(top_srcdir)/am/global-rules
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include $(top_srcdir)/am/webstuff
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include $(top_srcdir)/am/latex
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||||
1
doc/handbook/ModelDescriptions/1p2cboxmodel.tex
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1
doc/handbook/ModelDescriptions/1p2cboxmodel.tex
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@ -0,0 +1 @@
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Here comes the detailed documentation.
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1
doc/handbook/ModelDescriptions/1pboxmodel.tex
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1
doc/handbook/ModelDescriptions/1pboxmodel.tex
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@ -0,0 +1 @@
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Here comes the detailed documentation.
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3
doc/handbook/ModelDescriptions/2p2cboxmodel.tex
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3
doc/handbook/ModelDescriptions/2p2cboxmodel.tex
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@ -0,0 +1,3 @@
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This implements an isothermal two phase two component model.
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Depending on the value of the \char`\"{}Formulation\char`\"{} property, the primary variables are either \$p\_\-w\$ and \$S\_\-n;X\$ or \$p\_\-n\$ or \$S\_\-w;X\$. By default they are \$p\_\-w\$ and \$S\_\-n\$
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1
doc/handbook/ModelDescriptions/2p2cniboxmodel.tex
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1
doc/handbook/ModelDescriptions/2p2cniboxmodel.tex
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@ -0,0 +1 @@
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This implements a non-isothermal two-phase two-component model with Pw and Sn/X as primary unknowns. You can use Pn and Sw/X as primary variables if you set the Formulation property to pNsW.
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1
doc/handbook/ModelDescriptions/2pboxmodel.tex
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1
doc/handbook/ModelDescriptions/2pboxmodel.tex
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@ -0,0 +1 @@
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\doxyref{TwoPBoxModel}{p.}{classDune_1_1TwoPBoxModel} describes the box discretization of an isothermal twophase flow model. The underlying equations are obtained after inserting Darcy's law into the mass balance equation for each phase, yielding \begin{align*} \phi \frac{\partial (\varrho_{\text{mass,w}} S_\text{w})}{\partial t} -\Div \left( \lambda_\text{w} \varrho_{\text{mass,w}} K \left(\grad p_\text{w} - \varrho_{\text{mass,w}}\boldsymbol{g} \right)\right) - q_\text{w} &= 0, \\ \phi \frac{\partial (\varrho_{\text{mass,n}} S_\text{n})}{\partial t} - \Div \left( \lambda_\text{n} \varrho_{\text{mass,n}} K\left( \grad p_\text{n} - \varrho_{\text{mass,n}}\boldsymbol{g} \right)\right) - q_\text{n} &= 0. \end{align*} You can pick the formulation by setting the \char`\"{}Formulation\char`\"{} property. The default is $p_\text{w}$-$S_\text{n}$.
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1
doc/handbook/ModelDescriptions/2pniboxmodel.tex
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1
doc/handbook/ModelDescriptions/2pniboxmodel.tex
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@ -0,0 +1 @@
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This implements a non-isothermal two-phase model with Pw and Sn as primary unknowns. You can also use Pn and Sw as primary variables if you set the Formulation property to pNsW.
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1
doc/handbook/ModelDescriptions/richardsboxmodel.tex
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1
doc/handbook/ModelDescriptions/richardsboxmodel.tex
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@ -0,0 +1 @@
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Here comes the detailed documentation.
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@ -481,3 +481,426 @@
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||||
number = {3},
|
||||
}
|
||||
|
||||
|
||||
@InProceedings{A3:aavatsmark:1994,
|
||||
author = {Aavatsmark, I. and Barkve, T. and B{\o}e, {\O}. and
|
||||
Mannseth, T.},
|
||||
title = {Discretization on non-orthogonal, curvilinear grids
|
||||
for multiphase flow},
|
||||
booktitle = {Proc.\ of the 4th European Conf.\ on the Mathematics
|
||||
of Oil Recovery},
|
||||
year = {1994},
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address = {Norway}
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||||
}
|
||||
|
||||
@Article{A3:aavatsmark:1996,
|
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author = {Aavatsmark, I. and Barkve, T. and B{\o}e, {\O}. and
|
||||
Mannseth, T.},
|
||||
title = {Discretization on Non-Orthogonal, Quadrilateral
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journal = {Journal of Computational Physics},
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||||
volume = {127},
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pages = {2--14}
|
||||
}
|
||||
|
||||
@Article{A3:aavatsmark:2002,
|
||||
author = {Aavatsmark, Ivar},
|
||||
title = {An introduction to multipoint flux approximations
|
||||
for quadrilateral grids},
|
||||
journal = {Computational Geosciences},
|
||||
year = {2002},
|
||||
volume = {6},
|
||||
pages = {405--432}
|
||||
}
|
||||
|
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@article{A3:Braun:2002,
|
||||
author = {Braun, C. and Helmig, R. and Manthey, S.},
|
||||
title = {Determination of constitutive relationships for
|
||||
two-phase flow processes in heterogeneous porous
|
||||
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|
||||
permeability-saturation-relationship},
|
||||
journal = {Journal of Contaminant Hydrology},
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year = {2005}
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||||
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||||
@InProceedings{A3:edwards:1994,
|
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author = {Edwards, M. G. and Rogers, C. F.},
|
||||
title = {A flux continuous scheme for the full tensor
|
||||
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|
||||
booktitle = {Proc.\ of the 4th European Conf.\ on the Mathematics
|
||||
of Oil Recovery},
|
||||
year = {1994},
|
||||
address = {Norway}
|
||||
}
|
||||
|
||||
@Book{A3:whitaker:1999,
|
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author = {Whitaker, Stephen},
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title = {The Method of Volume Averaging},
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||||
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||||
volume = {13},
|
||||
series = {Theory and Applications of Transport in Porous Media},
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address = {Dordrecht}
|
||||
}
|
||||
|
||||
@Book{A3:helmig:1997,
|
||||
author = {Helmig, R.},
|
||||
title = {Multiphase Flow and Transport Processes in the
|
||||
Subsurface --- A Contribution to the Modeling of
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||||
Hydrosystems},
|
||||
publisher = {Springer Verlag},
|
||||
year = {1997}
|
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}
|
||||
|
||||
@PhdThesis{A3:emmert:1997,
|
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author = {Emmert, Martin},
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title = {Numerische Simulation von isothermen/nichtisothermen
|
||||
Mehrphasenprozessen unter Ber\"ucksichtigung der
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||||
Ver\"anderung der Fluideigenschaften},
|
||||
school = {Institut f\"ur Wasserbau, Universit\"at Stuttgart},
|
||||
year = {1997}
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||||
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||||
|
||||
@Book{A3:class:2001,
|
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author = {Class, Holger},
|
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title = {Theorie und numerische Modellierung nichtisothermer
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|
||||
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volume = {105},
|
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series = {Mitteilungsheft}
|
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||||
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|
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||||
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|
||||
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|
||||
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|
||||
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|
||||
series = {Lecture Notes in Physics, Chen, Ewing and Shi (eds.)},
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||||
pages = {52--71}
|
||||
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||||
|
||||
@Book{A3:aziz:1979,
|
||||
author = {Aziz, K. and Settari, A.},
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||||
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|
||||
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|
||||
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||||
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||||
|
||||
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|
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||||
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|
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||||
|
||||
@MastersThesis{A3:grass:2005,
|
||||
author = {Grass, Christoph},
|
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title = {Untersuchung von Randbedingungen bei der numerischen
|
||||
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|
||||
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|
||||
school = {Institut f\"ur Wasserbau, Universit\"at Stuttgart},
|
||||
year = {2005},
|
||||
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|
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author = {Bastian, P. and Birken, K. and Lang, S. and
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||||
H. and Wieners, C.},
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
}
|
||||
|
||||
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|
||||
author = {Lancaster, Peter},
|
||||
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|
||||
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||||
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|
||||
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|
||||
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||||
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|
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author = {\"Olmann, Ulrich},
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||||
title = {Behandlung anisotroper Mobilit\"aten als Resultat
|
||||
von Upscalingverfahren mittels
|
||||
Mehrpunktflu{\ss}approximationen},
|
||||
school = {Institut f\"ur Wasserbau, Universit\"at Stuttgart},
|
||||
year = {to be published 2006}
|
||||
}
|
||||
|
||||
@InBook{A3:sfb404:2003,
|
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author = {Helmig, R. and Class, H. and Jakobs, H. and
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||||
Bierlinski, A. and \"Olmann, U.},
|
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title = {Arbeits- und Ergebnisbericht 2003},
|
||||
chapter = {A3},
|
||||
publisher = {SFB 404},
|
||||
year = {2003},
|
||||
month = {May},
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||||
pages = {69--98}
|
||||
}
|
||||
|
||||
@Article{A3:coats:1974,
|
||||
author = {Coats, K.H. and Chieh Chu, W.D.G. and Marcum, B.E.},
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||||
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|
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journal = {Society of Petroleum Engineers Journal},
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||||
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||||
month = {December}
|
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}
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||||
|
||||
@Article{A3:falta:1992,
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author = {Falta, R.W. and Pruess, K. and Javandel, I. and
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||||
Witherspoon, P.A.},
|
||||
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||||
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|
||||
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||||
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title = {Numerical Simulation of Nonisothermal Multiphase
|
||||
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||||
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||||
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||||
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||||
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||||
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||||
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|
||||
author = {Class, H. and Helmig, R.},
|
||||
title = {Numerical Simulation of Nonisothermal Multiphase
|
||||
Multicomponent Processes in Porous Media --
|
||||
2. Applications for the Injection of Steam and Air},
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||||
journal = {Advances in Water Resources},
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||||
year = {2002},
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||||
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||||
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||||
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||||
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||||
@TechReport{A3:forsyth:1993,
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||||
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||||
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||||
address = {University of Waterloo},
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||||
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||||
author = {Bielinski, A.},
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||||
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||||
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||||
school = {Institut f\"ur Wasserbau, Universit\"at Stuttgart},
|
||||
year = {2006}
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||||
}
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||||
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||||
@Article{A3:nordbotten:2005a,
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||||
author = {Nordbotten, J.M. and Celia, M.A. and Bachu, S.},
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||||
title = {Injection and Storage of {CO$_2$} in Deep Saline
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||||
Aquifers: Analytical Solution for {CO$_2$} Plume
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||||
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||||
journal = {Transport in Porous Media},
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||||
year = {2005},
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||||
volume = {58(3)},
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||||
pages = {339--360}
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||||
}
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||||
|
||||
@Article{A3:nordbotten:2005b,
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||||
author = {Nordbotten, J.M. and Celia, M.A. and Bachu, S. and
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||||
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||||
title = {Semi-Analytical Solution for {CO$_2$} Leakage
|
||||
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||||
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||||
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||||
volume = {39(2)},
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||||
pages = {602--611}
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||||
}
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||||
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||||
@Article{A3:acosta:2006,
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||||
author = {Acosta, M. and Merten, C. and Eigenberger, G. and
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||||
Class, H. and Helmig, R. and Thoben, B. and
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||||
M\"uller-Steinhagen, H.},
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||||
title = {Modeling non-isothermal two-phase multicomponent
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||||
flow in the cathode of PEM fuel cells},
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||||
journal = {Journal of Power Sources},
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||||
year = {2006},
|
||||
pages = {in print}
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||||
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||||
|
||||
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||||
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||||
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||||
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|
||||
Parameter Identification of Multiphase Flow and
|
||||
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|
||||
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|
||||
Approaches -- Proceedings of the International
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||||
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||||
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|
||||
19. September 2003, Bauhaus-Universit\"at Weimar)}},
|
||||
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||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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|
||||
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||||
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||||
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|
||||
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||||
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||||
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|
||||
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||||
|
||||
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||||
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||||
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||||
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||||
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|
||||
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||||
|
||||
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|
||||
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||||
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||||
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||||
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|
||||
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|
||||
|
||||
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|
||||
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||||
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||||
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||||
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||||
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||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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|
||||
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||||
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|
||||
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|
||||
|
||||
@Misc{A3:oelmann:2006b,
|
||||
author = {\"Olmann, U. and Aavatsmark, I. and Helmig, R.},
|
||||
title = {{Buckley-Leverett heterogen --- Konstruktion der
|
||||
L\"osung mit der Charakteristikenmethode}},
|
||||
howpublished = {Preprint-Reihe des SFB404},
|
||||
note = {2006/05},
|
||||
month = {March},
|
||||
year = {2006}
|
||||
}
|
||||
|
||||
@book{A3:Stauffer:1984,
|
||||
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|
||||
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|
||||
publisher = {Taylor \& Francis},
|
||||
year = {1994}
|
||||
}
|
||||
|
||||
@article{A3:King:1996,
|
||||
author = {King, P. R.},
|
||||
title = {Upscaling Permeability: Error Analysis for
|
||||
Renormalisation},
|
||||
journal = {Transport in Porous Media},
|
||||
volume = {23},
|
||||
pages = {337--354},
|
||||
year = {1996}
|
||||
}
|
||||
|
||||
@article{A3:Williams:1989,
|
||||
author = {Williams, J. K.},
|
||||
title = {Simple Renormalisation Schemes for Calculating
|
||||
Effective Properties of Heterogeneous Reservoirs},
|
||||
journal = {1st European Conference on the Mathematics of Oil
|
||||
Recovery, Cambridge, UK, July 1989},
|
||||
year = {1989}
|
||||
}
|
||||
|
||||
@article{A3:Wen:1996,
|
||||
author = {Wen, X. H. and G\'{o}mez-Hern\'{a}ndez, J. J.},
|
||||
title = {Upscaling hydraulic conductivities in heterogenous
|
||||
media: An overview},
|
||||
journal = {Journal of Hydrology},
|
||||
volume = {183},
|
||||
pages = {ix--xxxii},
|
||||
year = {1996}
|
||||
}
|
||||
|
||||
@InProceedings{A3:HelmigEtAl:2006,
|
||||
author = {Helmig, R. and Miller, C. T. and Jakobs, H. and
|
||||
Class, H. and Hilpert, M. and Kees, C. E. and
|
||||
Niessner, J.},
|
||||
title = {{Multiphase Flow and Transport Modeling in
|
||||
Heterogeneous Porous Media}},
|
||||
booktitle = {Progress in Industrial Mathematics at ECMI 2004},
|
||||
pages = {449--488},
|
||||
year = {2006},
|
||||
editor = {Di Bucchianico, A. and Mattheij, R. M. M. and
|
||||
Peletier, M. A.},
|
||||
address = {Eindhoven University of Technology},
|
||||
month = {6},
|
||||
publisher = {Springer-Verlag},
|
||||
type = {Plenary lecture},
|
||||
note = {3-540-28072-3}
|
||||
}
|
||||
|
||||
@Unpublished{A3:nordbotten:2005c,
|
||||
author = {Nordbotten, J. M. and Aavatsmark, I. and Eigestad,
|
||||
G. T.},
|
||||
title = {Monotonicity of Control Volume Methods},
|
||||
note = {submitted to Numerische Mathematik},
|
||||
year = {2005}
|
||||
}
|
||||
|
||||
@Article{A3:bastian:1999,
|
||||
author = {Bastian, P. and Helmig, R.},
|
||||
title = {Efficient Fully-Coupled Solution Techniques for Two
|
||||
Phase Flow in Porous Media. Parallel Multigrid
|
||||
Solution and Large Scale Computations},
|
||||
journal = {Advances in Water Resources},
|
||||
year = {1999}
|
||||
}
|
||||
|
||||
@Article{A3:deneef:1997,
|
||||
author = {De Neef, M. and Molenaar, J.},
|
||||
title = {Analysis of DNAPL infiltration in a Medium with a
|
||||
low-permeable lens},
|
||||
journal = {Computational Geosciences},
|
||||
year = {1997},
|
||||
volume = {1},
|
||||
pages = {191-214}
|
||||
}
|
||||
|
||||
@InProceedings{A3:allan:1998,
|
||||
author = {Allan, J. and Ewing, J. and Helmig, R. and Braun, J.},
|
||||
title = {Scale effects in multiphase flow modeling},
|
||||
booktitle = {1. International conference on remediation of
|
||||
chlorinated and recalcitrant compounds},
|
||||
year = {1998},
|
||||
editor = {Wickramanayake, G.B. and Hinchee, R.E.},
|
||||
address = {Monterey, California, USA},
|
||||
month = {18th--21st of may},
|
||||
publisher = {Battelle Press, Columbus, OH, USA}
|
||||
}
|
||||
|
||||
|
||||
@ -20,6 +20,24 @@
|
||||
|
||||
\newcommand{\Dune}{{\sf\bfseries DUNE}}
|
||||
\newcommand{\Dumux}{DuMu$^\text{x}$ }
|
||||
\newcommand{\doxyref}[3]{\textnormal{#1}}
|
||||
\newenvironment{CompactList}
|
||||
{\begin{list}{}{
|
||||
\setlength{\leftmargin}{0.5cm}
|
||||
\setlength{\itemsep}{0pt}
|
||||
\setlength{\parsep}{0pt}
|
||||
\setlength{\topsep}{0pt}
|
||||
\renewcommand{\makelabel}{\hfill}}}
|
||||
{\end{list}}
|
||||
\newenvironment{CompactItemize}
|
||||
{
|
||||
\begin{itemize}
|
||||
\setlength{\itemsep}{-3pt}
|
||||
\setlength{\parsep}{0pt}
|
||||
\setlength{\topsep}{0pt}
|
||||
\setlength{\partopsep}{0pt}
|
||||
}
|
||||
{\end{itemize}}
|
||||
|
||||
%The theorems
|
||||
\theorembodyfont{\upshape}
|
||||
@ -29,6 +47,10 @@
|
||||
\newtheorem{lst}{Listing}
|
||||
\newtheorem{warn}[exc]{Warning}
|
||||
|
||||
\DeclareMathOperator{\grad}{grad}
|
||||
\DeclareMathOperator{\curl}{curl}
|
||||
\DeclareMathOperator{\Div}{div}
|
||||
|
||||
\pagestyle{scrheadings}
|
||||
|
||||
\title{\Dumux Handbook}
|
||||
@ -61,6 +83,7 @@ Universit\"at Stuttgart, Paffenwaldring 61, D-70569 Stuttgart, Germany}\\
|
||||
\input{intro}
|
||||
\input{getting-started}
|
||||
\input{tutorial}
|
||||
\input{models}
|
||||
|
||||
\bibliographystyle{plain}
|
||||
\bibliography{dumux-handbook}
|
||||
|
||||
150
doc/handbook/models.tex
Normal file
150
doc/handbook/models.tex
Normal file
@ -0,0 +1,150 @@
|
||||
\chapter[Models]{Physical and numerical models}
|
||||
|
||||
\section{Physical and mathematical description}
|
||||
|
||||
Characteristic of compositional multiphase models is that the phases
|
||||
are not only matter of a single chemical substance. Instead, their
|
||||
composition in general includes several species, and for the mass transfer,
|
||||
the component behavior is quite different from the phase behavior. In the following, we
|
||||
give some basic definitions and assumptions that are required for the
|
||||
formulation of the model concept below. As an example, we take a
|
||||
three-phase three-component system water-NAPL-gas
|
||||
\cite{A3:class:2002a}. The modification for other multicomponent
|
||||
systems is straightforward and can be found, e.\ g., in
|
||||
\cite{A3:bielinski:2006,A3:acosta:2006}.
|
||||
|
||||
\subsection{Basic Definitions and Assumptions for the Compositional
|
||||
Model Concept}
|
||||
\textbf{Components:}
|
||||
The term {\it component} stands for constituents of the phases which
|
||||
can be associated with a unique chemical species, or, more generally, with
|
||||
a group of species exploiting similar physical behavior. In this work, we
|
||||
assume a water-gas-NAPL system composed of the phases water (subscript
|
||||
$\text{w}$), gas ($\text{g}$), and NAPL ($\text{n}$). These phases are
|
||||
composed of the components water (superscript $\text{w}$), air
|
||||
($\text{a}$), and the organic contaminant ($\text{c}$) (see Fig.\
|
||||
\ref{A3:fig:mundwtrans}).
|
||||
%
|
||||
\begin{figure}[hbt]
|
||||
\centering
|
||||
\includegraphics[width=0.7\linewidth]{EPS/masstransfer}
|
||||
\caption{Mass and energy transfer between the phases}
|
||||
\label{A3:fig:mundwtrans}
|
||||
\end{figure}
|
||||
|
||||
\textbf{Equilibrium:}
|
||||
For the nonisothermal multiphase processes in porous media under
|
||||
consideration, we state that the assumption of local thermal
|
||||
equilibrium is valid since flow velocities are small. We neglect
|
||||
chemical reactions and biological decomposition and assume chemical
|
||||
equilibrium. Mechanical equilibrium is not valid in a porous medium,
|
||||
since discontinuities in pressure can occur across a fluid-fluid
|
||||
interface due to capillary effects.
|
||||
|
||||
\textbf{Notation:} The index $\alpha \in \{\text{w}, \text{n}, \text{g}\}$ refers
|
||||
to the phase, while the index $\kappa \in \{\text{w}, \text{a}, \text{c}\}$ refers
|
||||
to the component. \\
|
||||
\begin{tabular}{llll}
|
||||
$p_\alpha$ & phase pressure & $\phi$ & porosity \\
|
||||
$T$ & temperature & $K$ & absolute permeability tensor \\
|
||||
$S_\alpha$ & phase saturation & $\tau$ & tortuosity \\
|
||||
$x_\alpha^\kappa$ & mole fraction of component $\kappa$ in phase $\alpha$ & $\boldsymbol{g}$ & gravitational acceleration \\
|
||||
$X_\alpha^\kappa$ & mass fraction of component $\kappa$ in phase $\alpha$ & $q^\kappa$ & volume source term \\
|
||||
$\varrho_{\text{mol},\alpha}$ & molar density of phase $\alpha$ & $u_\alpha$ & specific internal energy \\
|
||||
$\varrho_{\text{mass},\alpha}$ & mass density of phase $\alpha$ & $h_\alpha$ & specific enthalpy \\
|
||||
$k_{\text{r}\alpha}$ & relative permeability & $c_\text{s}$ & specific heat enthalpy \\
|
||||
$\mu_\alpha$ & phase viscosity & $\lambda_\text{pm}$ & heat conductivity \\
|
||||
$D_\alpha^\kappa$ & diffusivity of component $\kappa$ in phase $\alpha$ & $q^h$ & heat source term
|
||||
\end{tabular}
|
||||
|
||||
|
||||
\subsection{Balance Equations}
|
||||
For the balance equations for multicomponent systems, it is in many
|
||||
cases convenient to use a molar formulation of the continuity
|
||||
equation. Considering the mass conservation for each component allows
|
||||
us to drop source/sink terms for describing the mass transfer between
|
||||
phases. Then, the
|
||||
molar mass balance can be written as:
|
||||
%
|
||||
\begin{eqnarray}
|
||||
\label{A3:eqmass1}
|
||||
&& \phi \frac{\partial (\sum_\alpha \varrho_{\text{mol}, \alpha}
|
||||
x_\alpha^\kappa S_\alpha )}{\partial t} \nonumber
|
||||
- \sum\limits_\alpha \Div \left( \frac{k_{\text{r}
|
||||
\alpha}}{\mu_\alpha} \varrho_{\text{mol}, \alpha}
|
||||
x_\alpha^\kappa K (\grad p_\alpha -
|
||||
\varrho_{\text{mass}, \alpha} \boldsymbol{g}) \right) \nonumber \\
|
||||
%
|
||||
\nonumber \\
|
||||
%
|
||||
&& - \sum\limits_\alpha \Div \left( \tau \phi S_\alpha D_\alpha^\kappa \varrho_{\text{mol},
|
||||
\alpha} \grad x_\alpha^\kappa \right) \nonumber
|
||||
- q^\kappa = 0, \qquad \kappa \in \{\text{w,a,c}\}.
|
||||
\end{eqnarray}
|
||||
|
||||
In the case of non-isothermal systems, we further have to balance the
|
||||
thermal energy. We assume fully reversible processes, such that entropy
|
||||
is not needed as a model parameter. Furthermore, we neglect
|
||||
dissipative effects and the heat transport due to molecular
|
||||
diffusion. The heat balance can then be
|
||||
formulated as:
|
||||
%
|
||||
\begin{eqnarray}
|
||||
\label{A3:eqenergmak1}
|
||||
&& \phi \frac{\partial \left( \sum_\alpha \varrho_{\text{mass},
|
||||
\alpha} u_\alpha S_\alpha \right)}{\partial t} + \left( 1 -
|
||||
\phi \right) \frac{\partial \varrho_{\text{s}} c_{\text{s}}
|
||||
T}{\partial t} \nonumber
|
||||
- \Div \left( \lambda_{\text{pm}} \grad T \right)
|
||||
\nonumber \\
|
||||
%
|
||||
\nonumber \\
|
||||
%
|
||||
&& - \sum\limits_\alpha \Div \left( \frac{k_{\text{r}
|
||||
\alpha}}{\mu_\alpha} \varrho_{\text{mass}, \alpha} h_\alpha
|
||||
K \left( \grad p_\alpha - \varrho_{\text{mass}, \alpha}
|
||||
\boldsymbol{g} \right) \right) \nonumber
|
||||
- q^h \; = \; 0.
|
||||
\end{eqnarray}
|
||||
|
||||
In order to close the system, supplementary constraints for capillary pressure, saturations and mole
|
||||
fractions are needed, \cite{A3:helmig:1997}.
|
||||
According to the Gibbsian phase rule, the number of degrees of freedom
|
||||
in a non-isothermal multiphase multicomponent system is equal to the
|
||||
number of components plus one. This means we need as many independent
|
||||
unknowns in the system description. The
|
||||
available primary variables are, e.\ g., saturations, mole/mass
|
||||
fractions, temperature, pressures, etc.
|
||||
|
||||
|
||||
|
||||
|
||||
\section{Available models}
|
||||
|
||||
\subsection{Fully coupled models}
|
||||
|
||||
\subsubsection{OnePBoxModel}
|
||||
\input{ModelDescriptions/1pboxmodel}
|
||||
|
||||
\subsubsection{OnePTwoCBoxModel}
|
||||
\input{ModelDescriptions/1p2cboxmodel}
|
||||
|
||||
\subsubsection{RichardsBoxModel}
|
||||
\input{ModelDescriptions/richardsboxmodel}
|
||||
|
||||
\subsubsection{TwoPBoxModel}
|
||||
\input{ModelDescriptions/2pboxmodel}
|
||||
|
||||
\subsubsection{TwoPNIBoxModel}
|
||||
\input{ModelDescriptions/2pniboxmodel}
|
||||
|
||||
\subsubsection{TwoPTwoCBoxModel}
|
||||
\input{ModelDescriptions/2p2cboxmodel}
|
||||
|
||||
\subsubsection{TwoPTwoCNIBoxModel}
|
||||
\input{ModelDescriptions/2p2cniboxmodel}
|
||||
|
||||
|
||||
\subsection{Decoupled models}
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user