Fluid-Phase Transitions in a Multiphasic Model of
CO2 Sequestration into Deep Aquifers:
A Fully Coupled Analysis of
Transport Phenomena and Solid Deformation
Von der Fakulta¨t Bau- und Umweltingenieurwissenschaften
der Universita¨t Stuttgart zur Erlangung der Wu¨rde
eines Doktor-Ingenieurs (Dr.-Ing.)
genehmigte Abhandlung
vorgelegt von
Dipl.-Ing. Kai Klaus Ha¨berle
aus
Langenau
Hauptberichter: Prof. Dr.-Ing. Dr. h. c. Wolfgang Ehlers
Mitberichter: Prof. Dr.-Ing. Rainer Helmig
Prof. Dr.-Ing. Tim Ricken
Tag der mu¨ndlichen Pru¨fung: 26. Juli 2017
Institut fu¨r Mechanik (Bauwesen) der Universita¨t Stuttgart
Lehrstuhl fu¨r Kontinuumsmechanik
Prof. Dr.-Ing. W. Ehlers
2017
Report No. II-34
Institut fu¨r Mechanik (Bauwesen)
Lehrstuhl fu¨r Kontinuumsmechanik
Universita¨t Stuttgart, Germany, 2017
Editor:
Prof. Dr.-Ing. Dr. h. c. W. Ehlers
c© Kai Ha¨berle
Institut fu¨r Mechanik (Bauwesen)
Lehrstuhl fu¨r Kontinuumsmechanik
Universita¨t Stuttgart
Pfaffenwaldring 7
70569 Stuttgart, Germany
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical, photocopy-
ing, recording, scanning or otherwise, without the permission in writing of the author.
ISBN 978 – 3 – 937399 – 34 – 8
(D 93 – Dissertation, Universita¨t Stuttgart)
Acknowledgements
The content of this doctoral thesis was developed during the years 2010 and 2017 when I
was working as a research assistant at the Institute of Applied Mechanics (Civil Engineer-
ing), Chair of Continuum Mechanics, at the University of Stuttgart. This work would not
have been possible without the help and wisdom of a multitude of people, whom I would
like to express my deepest gratitude.
First of all, I would like to sincerely thank Professor Wolfgang Ehlers for giving me the
opportunity to conduct the work presented here. I really appreciate that he believed in
me to find a solution for the given task, although I encountered a few dead ends along
the way. His rigorous method and broad expertise were a corner stone in preparing this
dissertation. I also have to thank Professor Rainer Helmig, not only for evaluating my
thesis, but also for introducing me to the world of numerical simulations and porous media.
Furthermore, I thank my third supervisor Professor Tim Ricken for his time reading this
work and for the valuable discussions at various conferences.
Very special thanks go to Dr.-Ing. Michael Sprenger, since he dragged me to this institute
and, thus, initialised my research career. He is not only a great friend and travel partner,
but also contributed to this work by posing critical questions. I will always remember the
great and friendly atmosphere at the institute. I believe that the helpful cooperation and
various activities outside of the institute were something special. In particular, I would like
to thank Dr.-Ing. Maik Schenke for enduring my IT and programming related questions
and for the skiing trips, Dr.-Ing. Arndt Wagner for knowing everything about regulatory
things and mechanics, Dr.-Ing. David Koch for discussing thermodynamics, Dr.-Ing. Said
Jamei for his help with the COMMAS course, my office-mates Dr.-Ing. Irina Komarova,
Dr.-Ing. Joffrey Mabuma and Patrick Schro¨der. Furthermore, I want to thank Sami
Bidier, Chenyi Luo, Davina Fink, Christian Bleiler, Lukas Eurich, Professor Oliver Ro¨hrle,
Dr.-Ing. Thomas Heidlauf, Ekin Altan, Mylena Mordhorst, Andreas Hessenthaler, Dr.-
Ing. Okan Avci, Arzu Avci, Dr.-Ing. Uwe Rempler, Dr.-Ing. Nils Karajan, Dr.-Ing. Yousef
Heider, Professor Bernd Markert as the supervisor of my diploma thesis, and all the others
from the institute. I owe special thanks to Nicole Karich for her unwavering helpfulness.
I am also grateful to the people of the graduate school NUPUS, for the financial support
and as a source of profound knowledge regarding porous media topics. For proofreading I
appreciate the time that was sacrificed by Dr.-Ing. Arndt Wagner, Dr.-Ing. Maik Schenke
and Dr.-Ing. Said Jamei.
This leads me to my clever brother Dr. rer. nat. Thomas Ha¨berle who proofread the whole
thesis, as well as the presentation, and always gave wise and constructive remarks. I will
not forget that he used so much of his precious time for me despite finishing his own
thesis. I deeply thank my parents for their never ending support in all regards and their
believe in me. Finally, I would like to express my heartfelt gratitude to my beloved wife
Cornelia who had to endure most shortcomings during this long time, but nonetheless
always encouraged me. The upcoming birth of our lovely daughter Carlotta in 2016 was
a great incentive to conclude the work and now she is the centre of our lives.
Stuttgart, August 2017 Kai Ha¨berle
4In jeder Schwierigkeit
lebt die Mo¨glichkeit.
Albert Einstein (1879–1955)
Contents
Deutschsprachige Zusammenfassung I
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
Stand der Forschung, Zielsetzung . . . . . . . . . . . . . . . . . . . . . . . . . . II
Gliederung der Arbeit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V
Nomenclature IX
Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX
Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X
Selected acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVI
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Scope, aims and state of the art . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Carbon-dioxide capture and storage 7
2.1 Properties of CO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 What is CCS? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Geological storage options . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Storage costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Storage mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.4 Storage safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Numerical simulation of CO2 storage . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Tasks and difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Theoretical fundamentals of multiphasic and multicomponent modelling 21
3.1 Theory of Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Kinematical relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Motion functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
I
II Contents
3.2.2 Deformation and strain measures . . . . . . . . . . . . . . . . . . . 25
3.2.3 Deformation rates and velocity gradient . . . . . . . . . . . . . . . 27
3.2.4 Stress measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Balance relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.1 Specific balance equations . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Singular surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.1 Kinematics of a body with a singular surface . . . . . . . . . . . . . 35
3.4.2 Balance relations for a body with a singular surface . . . . . . . . . 35
4 Thermodynamic theory of fluids 39
4.1 Phase behaviour of a single substance . . . . . . . . . . . . . . . . . . . . . 40
4.1.1 Chemical potential and first-order phase transitions . . . . . . . . . 49
4.1.2 Clausius-Clapeyron equation . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Vaporisation enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Specific heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Shear viscosity and thermal conductivity . . . . . . . . . . . . . . . . . . . 54
4.4.1 Effective shear viscosity . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.2 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5 Constitutive settings 57
5.1 A priori assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Adaptation of balance relations . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.1 Definition of a thermoelastic solid . . . . . . . . . . . . . . . . . . . 59
5.2.2 Mass balances of the fluid phases . . . . . . . . . . . . . . . . . . . 61
5.2.3 Momentum balance of the overall aggregate . . . . . . . . . . . . . 62
5.2.4 Energy balance of the overall aggregate . . . . . . . . . . . . . . . . 63
5.2.5 Adaptation of the entropy inequality . . . . . . . . . . . . . . . . . 64
5.3 Determination of constitutive relations . . . . . . . . . . . . . . . . . . . . 67
5.3.1 The basic thermodynamical principles . . . . . . . . . . . . . . . . 67
5.3.2 Exploitation of the entropy inequality . . . . . . . . . . . . . . . . . 68
5.3.3 Constitutive relations of the solid constituent . . . . . . . . . . . . 74
5.3.4 Constitutive relations of the fluid constituents . . . . . . . . . . . . 75
5.3.5 Constitutive relations of the overall aggregate . . . . . . . . . . . . 85
Contents III
5.4 Phase transition between the gaseous and liquid phases of a single substance 86
5.4.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4.2 Development of the constitutive relation for the mass-production
term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4.3 Switching criterion for the mass transition . . . . . . . . . . . . . . 95
5.5 Governing balance relations in the strong form . . . . . . . . . . . . . . . . 96
6 Numerical treatment 99
6.1 Finite-element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1.1 Weak formulations of the governing equations . . . . . . . . . . . . 102
6.2 Discretisation procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2.1 Spatial discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2.2 Temporal discretisation . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Numerical examples 111
7.1 CO2 sequestration into a deep aquifer . . . . . . . . . . . . . . . . . . . . . 111
7.1.1 Injection into a reservoir with an inclined cap-rock layer . . . . . . 112
7.2 Phase transition and mass transfer during evaporation and condensation . 115
7.2.1 Evaporation around a hot pipe . . . . . . . . . . . . . . . . . . . . 115
7.2.2 Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8 Summary and Outlook 123
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A Selected relations of tensor calculus 127
A.1 Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
A.2 Tensor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B Thermodynamical supplements and specific evaluations 131
B.1 Thermodynamic potentials and Legendre transformation . . . . . . . . . . 131
B.2 Maxwell relations and fundamental relations . . . . . . . . . . . . . . . . . 132
B.3 Derivation of the Maxwell criterion . . . . . . . . . . . . . . . . . . . . . . 132
B.4 Derivation of overall mass balance . . . . . . . . . . . . . . . . . . . . . . . 133
IV Contents
B.5 Weak forms of the effective fluid mass-balance relations . . . . . . . . . . . 134
B.6 Treatment of the stress terms of the overall energy balance . . . . . . . . . 134
B.7 Simplification of the direct momentum and mass production terms in the
energy balance of the overall aggregate . . . . . . . . . . . . . . . . . . . . 135
B.8 Calculation of different versions of the liquid momentum production . . . . 136
B.9 Justifying the assumption of an overall temperature for all constituents . . 136
B.10 Real solutions of a cubic equation in case of the van-der-Waals EOS . . . . 138
Bibliography 141
Curriculum Vitae 155
Deutschsprachige Zusammenfassung
Motivation
Es ist allgemein akzeptiert, dass anthropogen emittierte Treibhausgase zur globalen Erd-
erwa¨rmung beitragen. Die Folge davon sind steigende Meeresspiegel, welche tief liegende
Wohngebiete bedrohen, ha¨ufigere und heftigere Unwetter (z. B. Wirbelstu¨rme, Hochwas-
ser), Du¨rren und viele weitere negative Erscheinungen. Deshalb ist es no¨tig, die globale
Erderwa¨rmung zu stoppen oder wenigstens zu mindern, indem die Emissionen von Treib-
hausgasen herabgesetzt werden. Von den verschiedenen Treibhausgasen ist Kohlendioxid,
CO2, das Wichtigste, da es in großen Mengen durch menschliche Aktivita¨ten, zum Beispiel
in der Energieproduktion aus fossilen Brennstoffen, freigesetzt wird. In diesem Zusammen-
hang bietet die Technik des Carbon-Dioxide Capture and Storage (CCS) (frei u¨bersetzt:
Kohlendioxid-Abfangung und -Speicherung) eine Mo¨glichkeit, die CO2-Emissionen in die
Atmospha¨re zu reduzieren. Ende des zwanzigsten Jahrhunderts galt CCS daher als die
herausragende Technik, um die globale Erwa¨rmung aufzuhalten, und es entstand ein neues
Wissenschaftsfeld, in dem sich Wissenschaftler und Ingenieure aus verschiedensten Fach-
richtungen um die Lo¨sung offener Fragen ku¨mmerten. Heutzutage hat CCS einiges an
Popularita¨t eingebu¨ßt, was zum einen am o¨ffentlichen Widerstand liegt, der sich aus der
Angst vor CO2-Leckagen begru¨ndet, und zum anderen am fehlenden o¨konomischen An-
reiz. Urspru¨nglich sollte dieser durch die sogenannten CO2-Zertifikate erzeugt werden.
Diese Maßnahme verfehlt im Moment jedoch ihre Wirkung, weil sich zu viele und zu
gu¨nstige Zertifikate im Umlauf befinden. Wenn man sich aber bei der Wahl von mo¨gli-
chen Speichern auf Offshore-Lagersta¨tten oder auf unbesiedelte Regionen beschra¨nkt und
das Konzept der CO2-Zertifikate justiert, kann CCS als eine wichtige Bru¨ckentechnologie
fu¨r den U¨bergang von fossilen zu erneuerbaren Energien angesehen werden.
Die CO2-Emissionen werden hauptsa¨chlich bei der Verbrennung von fossilen Brennstoffen
in der Energieproduktion, im Verkehrswesens und in der Industrie erzeugt. Hiervon eignen
sich vor allem Punktquellen, wie zum Beispiel Kraftwerke, zur effektiven Abtrennung des
CO2 aus dem Abgasstrom. Das abgetrennte CO2 kann anschließend zum Beispiel in tief
liegenden Reservoirs im Untergrund verpresst werden. Geeignete geologische Formationen
fu¨r die Speicherung bilden existierende und entleerte Erdo¨l- oder Erdgaslagersta¨tten, fu¨r
eine wirtschaftliche Gewinnung zu tief liegende Kohleflo¨tze und Salzwasseraquifere. Diese
verschiedenen Speichersta¨tten haben alle ihre Vor- und Nachteile in Bezug auf die CO2-
Speicherung, als da wa¨ren: die verbesserte O¨l- und Gasfo¨rderung durch eingepresstes CO2
in aktive Erdo¨l- oder Erdgaslagersta¨tten, die bereits existierende logistische Anschließung
der Erdo¨l- oder Erdgaslagersta¨tten, das Speichervolumen, welches bei Salzwasseraquifere
am gro¨ßten ist, und die Sicherheit der Speichersta¨tte, d. h. der langzeitige Verbleib des
CO2 im Untergrund. Der letzte Punkt ist nur dann gewa¨hrleistet, wenn eine Leckage des
CO2 ausgeschlossen werden kann. Dies wird vor allem durch eine geeignete Deckschicht
aus undurchla¨ssigem Gestein garantiert, welche die aufwa¨rtsgerichtete Ausbreitung des
CO2 verhindert. Es kann jedoch durch den hohen Einpressdruck oder durch seismische
I
II Deutschsprachige Zusammenfassung
Aktivita¨ten zur Rissbildung in der Deckschicht und daraufhin zur Leckage kommen.
Um solche Ereignisse vorhersagen zu ko¨nnen, bieten numerische Simulationen die Mo¨glich-
keit, ein Modell des Injektionsprozesses zu erstellen, das die thermodynamischen A¨nderun-
gen des CO2 und die Verformungen der Reservoirgesteine beru¨cksichtigt. Dies stellt das
angestrebte Ziel der vorliegenden Arbeit dar. Die physikalischen, chemischen und thermo-
dynamischen Prozesse, die bei der CO2-Speicherung auftreten, sind jedoch hoch komplex
und deshalb immer noch Gegenstand aktueller Forschung. Da es im Moment nicht mo¨glich
ist, das komplette Spektrum dieser Prozesse in einem einzigen numerischen Model rea-
lita¨tsgetreu wiedergeben zu ko¨nnen, wird im Rahmen dieser Arbeit speziell auf das me-
chanischen Verhalten der Geologie und die Phasentransformation des eingepressten CO2
eingegangen.
Stand der Forschung, Zielsetzung
Der komplette Prozess des CCS beinhaltet eine Vielzahl verschiedenster Techniken, die
sich grob in drei Teile aufspalten lassen: die chemische Abtrennung des CO2 vom Abluft-
strom, die Verflu¨ssigung und den Transports zum Speicherort, sowie die Sequestrierung
selbst. Hier soll im Weiteren nur der letzte Teil na¨her betrachtet werden. Neben Tief-
seespeicherung, mineraler Karbonisierung und industriellem Gebrauch des CO2 wird die
geologische Speicherung in tief liegenden, unterirdischen Reservoirs als effektivste Spei-
chermethode angesehen, wie in dem Special Report on CCS des Intergovernmental Panel
on Climate Change, Metz et al. [119], dargestellt wird.
Die Machbarkeit der Untergrundspeicherung in industriellem Maßstab konnte bereits in
verschiedenen Projekten besta¨tigt werden, z. B. bei der Sequestrierung von CO2 im Sleip-
ner Gasfeld in der Nordsee und bei den Projekten in Weyburn, Kanada, und In Salah,
Algerien. Jedoch wird bei den meisten CCS-Projekten das eingepresste CO2 zur Verbes-
serung der Erdo¨l- oder Erdgasgewinnung (EOR fu¨r
”
enhanced oil recovery“) eingesetzt.
Dies senkt zwar die Kosten fu¨r die CO2-Speicherung, aber die zusa¨tzlich gewonnenen
Kohlenwasserstoffe werden wiederum hauptsa¨chlich zur Energieerzeugung eingesetzt und
mindern so den Nutzen der Verpressung. Darum kann nur die Speicherung in leeren La-
gersta¨tten oder in Salzwasseraquiferen als o¨kologisch sinnvoll betrachtet werden. Letztere
sind weltweit zu finden und bieten genu¨gend Speichervolumen, um die angestrebte Menge
an CO2 aufzunehmen, damit die globale Erderwa¨rmung aufgehalten werden kann, verglei-
che Metz et al. [119]. Aus diesem Grund wird im Folgenden nur diese Speicherart weiter
betrachtet.
Als Voraussetzung zur Bestimmung der Speicherkapazita¨t und der Beurteilung der Sicher-
heit ist es essentiell das Verhalten des CO2 im Reservoir wa¨hrend und nach Beendigung
des Einpressens zu kennen. Die hierfu¨r existierenden U¨berwachungstechniken, wie zum
Beispiel Bohrlochmessung und seismische Untersuchung, sind jedoch nur begrenzt in der
Lage, genu¨gend Informationen u¨ber die komplexen Mechanismen, die zwischen dem CO2
und dem Gestein ablaufen, zu liefern. Deshalb sind numerische Studien von ho¨chster Be-
deutung, um ein Versta¨ndnis fu¨r die physikalischen Prozesse bei der CO2-Speicherung
gewinnen zu ko¨nnen. In der Literatur finden sich hierzu eine Vielzahl an Vero¨ffentlichun-
Deutschsprachige Zusammenfassung III
gen. Die fru¨hesten Werke zur CO2-Speicherung, welche sich zuna¨chst aus der Simulation
der Kohlenwasserstoffgewinnung entwickelt haben, stammen von der Gruppe um Pruess
& Garc´ıa [46, 141, 144] und weitere Arbeiten werden in Bielinski [15] genannt. Verschie-
dene Techniken zur effizienten Simulation in Form analytischer oder semi-analytischer
Modelle wurden von Kang et al. [93], Nordbotten et al. [129], Celia et al. [34] entwickelt.
Hierzu za¨hlen auch die vertikal-reduzierten Modelle von Court et al. [40] und Gasda et
al. [70]. Nordbotten und Celia [34, 127, 128, 143] vero¨ffentlichten mehrere Publikationen
u¨ber großskalige Simulationen, inklusive einer Abscha¨tzung des Ausmaßes von eventu-
ellen Leckagen als Anhaltspunkt fu¨r Entscheidungstra¨ger. Weiterhin wurden die Spei-
cherkapazita¨ten verschiedener Reservoire von Bradshaw et al. [28] und Kopp et al. [99]
beurteilt. Auf spezielle Fragestellungen zu verschiedenen physikalischen Parametern, z. B.
relative Permeabilita¨t, temperaturabha¨ngiger Kapillardruck, Mineralisierung, verbesser-
te Einschlusstechniken oder geomechanische Parameter, wurde von Juanes et al. [91],
Plug & Bruining [136], Kumar et al. [103], Ebigbo [47] und Rutqvist [154–156] eingegan-
gen. Benchmark-Studien und Vergleiche verschiedener Simulationssoftware ko¨nnen etwa
in Pruess et al. [142] und Class et al. [37] gefunden werden. An diesem Punkt muss
erwa¨hnt werden, dass die Phasentransformation des CO2 in den genannten Publikationen
nur wenig beru¨cksichtigt wurde. Falls doch, so geschieht dies mit Hilfe von tabellarischen
Stoffwerten. Dadurch wird jedoch eine explizite Beschreibung des Massentransfers zwi-
schen den Fluidphasen beim Phasenu¨bergang nicht gewa¨hrleistet.
Phasentransformationen zwischen den Aggregatszusta¨nden Gas, Flu¨ssigkeit und Feststoff
sind ubiquita¨r vorkommende physikalische Prozesse, die zum Beispiel bei der Trocknung
oder dem Gefrieren auftreten. Diese Transformationen treten nicht nur in gut erforschten,
sogenannten
”
offenen Systemen“auf, sondern auch innerhalb poro¨ser Materialien, etwa im
Sandstein von CO2-Reservoiren. Der Mechanismus der Phasentransformation in poro¨sen
Materialien wurde bisher jedoch nur spa¨rlich untersucht. Außer bei der CO2-Speicherung
spielen Phasentransformationen auch bei anderen geologischen Aktivita¨ten (Dampfinjek-
tion fu¨r EOR, Erdsanierung, Geothermie), in der Nahrungsmittelindustrie (Trocknungs-
und Backprozesse) und in vielen weiteren Bereichen eine wichtige Rolle. Die vorgestellte
Arbeit beschra¨nkt sich auf die Betrachtung von Phasentransformationen erster Ordnung
(Gas-zu-Flu¨ssigkeit oder Flu¨ssigkeit-zu-Gas) einer einzelnen Substanz, z. B. CO2. Diese
Art der Phasentransformation muss klar von U¨berga¨ngen zwischen Mischungen, wie zum
Beispiel der Verdampfung von Wasser in die Umgebungsluft, unterschieden werden. Des
Weiteren soll in der vorliegenden Arbeit auch der U¨bergang zwischen den superkritischen
und gasfo¨rmigen Zusta¨nden beru¨cksichtigt werden. Im Gegensatz zur Beschreibung von
Phasentransformationen zwischen Gas und Flu¨ssigkeit, wobei fu¨r jede Phase eine eigene
Massenbilanz eingefu¨hrt wird, genu¨gt hier die Verwendung einer einzelnen beschreiben-
den Massenbilanz fu¨r beide Phasen zusammen, da der U¨bergang zwischen superkritischem
Zustand und Gas (und auch zwischen superkritischem Zustand und Flu¨ssigkeit) kontinu-
ierlich verla¨uft.
Nach Kenntnis des Autors geht die erste Publikation u¨ber Simulationen von Phasen-
transformationen in poro¨sen Medien auf Lykov [112] im Jahr 1974 zuru¨ck. Nachfolgend
wurden weitere Arbeiten von der Gruppe um Be´net [35, 108, 109, 153], von Hassanizadeh
und Gray [75, 81, 125, 126] und von Bedeaux [14] zu diesem Thema erstellt. Als Beispiele
IV Deutschsprachige Zusammenfassung
zur Anwendung dieser Modelle auf reelle Probleme lassen sich im Hinblick auf Trocknung
Kowalski [101] und zum Brotbackprozess Huang et al. [87] nennen.
Die bereits erwa¨hnten Artikel u¨ber Phasentransformationen in poro¨sen Medien betrachten
in der Regel die Feststoffmatrix als starr. Sollen die Deformationen des Feststoffs beru¨ck-
sichtigt werden, so bietet es sich an, auf das bewa¨hrte Konzept der Theorie Poro¨ser Medien
(TPM) zuru¨ckzugreifen. Dieser Ansatz stellt eine ideale Grundstruktur zur Beschreibung
von Mehrphasen- und Mehrkomponentenkontinua dar. Dies beinhaltet auch die mo¨gliche
Einbeziehung verschiedenen Deformationsverhaltens (elastisch, viskoelastisch, elastisch-
plastisch) und vera¨nderbarer Poreninhalte von misch- oder nichtmischbaren Fluiden. Die
Grundlagen zur TPM werden ausfu¨hrlich in den Publikationen von de Boer [20], de Bo-
er & Ehlers [22], Bowen [26, 27], der Arbeitsgruppe um Ehlers [50, 54–56, 59, 60, 180]
oder Schrefler et al. [160, 161] dargelegt. Die Zweckma¨ßigkeit der TPM konnte bereits
in vielen Anwendungen aus verschiedensten Bereichen gezeigt werden. So haben Markert
[115], Avci [8] und Ehlers et al. [57] die Theorien der Elasto-Plastizita¨t und der Elasto-
Viskoplastizita¨t auf Fragestellungen der Materialwissenschaften und der Geomechanik
angewendet. Zur Behandlung von Problemen der Biomechanik wurden von Karajan et
al. [95], Ricken und seinen Mitarbeitern [147, 150], sowie Wagner [176] den existierenden
Modellen chemische, elektrische und biologische Eigenschaften hinzugefu¨gt. Die nume-
rische Simulation biologischer Phasentransformationen in Mu¨lldeponien wurde von der
Forschungsgruppe um Ricken [151, 152] durchgefu¨hrt.
Die Entwicklung einer thermodynamisch konsistenten Beschreibung von Phasentransfor-
mationen im Rahmen der TPM beginnt mit den Arbeiten von de Boer und Mitarbeitern
[19, 21, 23] und setzt sich fort mit den Beitra¨gen von, z. B. Kowalski [101] und Ghadiani
[71], sowie einem Artikel von Ehlers & Graf [59]. Gefrier- und Schmelzprozesse wurden
ausfu¨hrlich, unter anderem, von Kruschwitz und Bluhm [102] und der Gruppe um Ricken
und Bluhm [17, 148, 149] untersucht. In all diesen Arbeiten wurde die beno¨tigte konsti-
tuierende Beziehung fu¨r die Beschreibung des Massentransfers zwischen den Fluidphasen
durch Auswertung der Entropieungleichung bestimmt. Außerdem handelt es sich bei den
betrachteten Phasentransformationen immer um U¨berga¨nge zwischen Mischungen, d. h.
dass die einzelnen Phasen nicht nur aus einer reinen Fluidsubstanz bestehen, sondern ein
Gemisch verschiedener Stoffe, z. B. Wasserdampf und Luft in der Gasphase, darstellen.
Im Rahmen dieser Arbeit soll nun ein Model entwickelt werden, das die Phasentrans-
formation zwischen den reinen Phasen einer einzigen Fluidsubstanz beschreibt. In die-
sem Zusammenhang stellte sich heraus, dass die Massentransferbeziehung hergeleitet aus
der Entropieungleichung fu¨r diesen Fall nicht zielfu¨hrend ist. Darum wird hier nun eine
massefreie, glatte singula¨re Fla¨che zur Beschreibung der Phasengrenzfla¨che zwischen den
flu¨ssigen und gasfo¨rmigen Phasen eingefu¨hrt. Dadurch treten zusa¨tzlich Sprungterme fu¨r
einzelne Funktionen in den Bilanzgleichungen auf. Durch Auswertung dieser Sprungbedin-
gungen kann dann eine neue konstituierende Bedingung fu¨r den Massentransfer gefunden
werden. Dieser Ansatz wurde bereits von Jamet [89], Juric & Tryggvason [92], Morland
et al. [120, 121], Tanguy et al. [170] und Wang & Oberlack [177] verfolgt.
Diese Vorgehensweise wird daraufhin in die TPM eingebettet, vgl. Ehlers and Ha¨berle [61].
Die resultierende Beziehung fu¨r den Massentransfer steht in Abha¨ngigkeit des Verha¨ltnisses
der verfu¨gbaren Energie zur latenten Verdampfungswa¨rme. Da die singula¨re Fla¨che auf
Deutschsprachige Zusammenfassung V
der Mikroskala eingefu¨hrt wurde, ist auch der Massentransfer auf dieser Skala definiert.
Fu¨r die Hochskalierung auf die makroskopische Kontinuumsskala wird an dieser Stelle
die sogenannte Grenzschichtfla¨che als spezifische Dichte der Grenzfla¨che bezu¨glich des
betrachteten Volumens eingefu¨hrt. Die Bestimmung der Grenzschichtfla¨che als Funkti-
on der Sa¨ttigung basiert auf den Ergebnissen von Hassanizadeh und seinen Mitarbeitern
[81, 90, 124, 125], sowie der Dissertation von Graf [74]. Die in dieser Arbeit verwende-
te Anwendung der Grenzschichtfla¨che besitzt A¨hnlichkeiten mit der Vero¨ffentlichung von
Nuske et al. [130]. Das Prinzip der Grenzschichtfla¨che wurde auch in den Publikationen
von Miller et al. [41, 76], der Gruppe um Celia [83, 145], sowie von Gladkikh & Bryant
[72] und Oostrom et al. [132] diskutiert.
Um das thermodynamische Verhalten des CO2 mo¨glichst realistisch modellieren zu ko¨nnen,
mu¨ssen weitere bestimmende Beziehungen gefunden werden. Zuna¨chst wird das kompres-
sible Verhalten, das heißt die Abha¨ngigkeit zwischen Druck, Temperatur und Dichte,
durch eine geeignete Zustandsgleichung beschrieben. Dafu¨r werden die Vor- und Nachtei-
le der Ansa¨tze von van der Waals, Peng und Robinson, sowie Soave, Redlich und Kwong
im Hinblick auf ihre Exaktheit und ihre thermodynamische Konsistenz diskutiert. Einen
U¨berblick u¨ber verschiedene Zustandsgleichungen verschaffen die Bu¨cher von Ott und
Boerio-Goates [134] sowie von Poling et al. [137].
Weitere Konstitutivgleichungen bezeichnen die spezifische Wa¨rme, die Verdampfungsen-
thalpie, die Scherviskosita¨t und die thermische Leitfa¨higkeit. Diese Formulierungen sind
gro¨ßtenteils der sehr ergiebigen Literatur zu diesem Thema entnommen, wobei an dieser
Stelle nur die hier verwendeten Quellen genannt werden sollen: Lemmon et al. [104], Potter
und Somerton [139], Lewis & Randall [105], Pitzer et al. [135], Fenghour et al. [66] und
Vesovic et al. [175].
Mit den vorliegenden thermodynamischen Eigenschaften ist es dann mo¨glich, ein dreipha-
siges Model (Feststoff, Wasser und CO2, oder Feststoff, flu¨ssiges CO2 und gasfo¨rmiges
CO2) zur Beschreibung der CO2-Verpressung in tiefe Reservoire zu erstellen. Wie be-
reits erwa¨hnt, geschieht dies basierend auf dem kontinuummechanischen Rahmenwerk
der TPM. Durch Auswertung der Entropieungleichung werden die restlichen Konstitu-
tivbedingungen bestimmt, na¨mlich die Darcy -Sickergeschwindigkeit, die Kapillardruck-
Sa¨ttigungsbeziehung und die thermoelastische Beschreibung des Feststoffdeformations-
verhaltens der verschiedenen Gesteinsschichten. Dadurch la¨sst sich letztendlich die CO2-
Sequestrierung unter Beru¨cksichtigung von Phasentransformation, kompressiblen Fluid-
phasen und thermisch-elastisch verformbarer Feststoffmatrix in einem vollgekoppelten
Verfahren simulieren. Numerische Studien basierend auf diesem Model sollten es ermo¨gli-
chen, bereits wa¨hrend der Planungsphase die Machbarkeit einer CO2-Speicherung ab-
zuwa¨gen und etwaige Sicherheitsrisiken ausschließen zu ko¨nnen.
Gliederung der Arbeit
Zur Einleitung werden inKapitel 1 die Motivation, der Stand der Forschung und die Ziele
dieser Arbeit aufgefu¨hrt. Daraufhin folgt in Kapitel 2 eine kurze Zusammenfassung u¨ber
die Technik des
”
Carbon-Dioxid Capture and Storage“ (CCS). Es werden die technischen,
VI Deutschsprachige Zusammenfassung
o¨kologischen, o¨konomischen und sicherheitsrelevanten Aspekte diskutiert, wobei im Spe-
ziellen auf die CO2-Speicherung in tief liegenden Aquiferen eingegangen wird. Außerdem
wird dargelegt, wie numerische Simulationen helfen ko¨nnen, die wa¨hrend der Speicherung
auftretenden Prozesse verstehen und vorhersagen zu ko¨nnen.
Anschließend werden inKapitel 3 die theoretischen Grundlagen, die fu¨r die Beschreibung
eines dreiphasigen poro¨sen Mediums beno¨tigt werden, im Rahmen der Theorie Poro¨ser
Medien (TPM) eingefu¨hrt. Dies beinhaltet die Kinematik der u¨berlagerten Konstituieren-
den, die Spannungs- und Dehnungsmaße sowie die Bilanzrelationen, welche sowohl fu¨r die
einzelnen Konstituierenden als auch fu¨r den Gesamtko¨per formuliert werden. Zusa¨tzlich
wird hier das Prinzip der singula¨ren Fla¨chen vorgestellt, inklusive der daraus entstehen-
den A¨nderungen in den Bilanzrelationen. Diese singula¨ren Fla¨chen werden spa¨ter in der
Herleitung der Massentransferrelation fu¨r die Phasentransformation wieder beno¨tigt.
Eine detaillierte Ero¨rterung der Thermodynamik von Fluiden folgt in Kapitel 4. Dies
betrifft vor allem die Beschreibung des Phasenverhaltens. Hierbei werden die Vor- und
Nachteile verschiedener Zustandsgleichungen miteinander verglichen, wobei das Augen-
merk speziell auf den Bereich der Transformation zwischen den flu¨ssigen und gasfo¨rmigen
Phasen gerichtet ist. In diesem Kapitel werden außerdem die thermodynamischen Rela-
tionen fu¨r die spezifische Wa¨rme, die Verdampfungsenthalpie, die Scherviskosita¨t und die
thermische Leitfa¨higkeit pra¨sentiert. Soweit mo¨glich, wird hierbei eine stoffunabha¨ngige
Formulierung der einzelnen Funktionen angestrebt.
Kapitel 5 beinhaltet die Identifikation der beno¨tigten Konstitutivbeziehungen, wobei die
jeweiligen Bilanzrelationen an das gegebene Problem angepasst werden. Dazu geho¨rt auch
die ausfu¨hrliche Auswertung der Entropieungleichung als Voraussetzung fu¨r eine thermo-
dynamisch konsistente Formulierung der Konstitutivrelationen. Im Weiteren wird hier
auch die Beziehung fu¨r den Massentransfer hergeleitet, basierend auf der Einfu¨hrung der
singula¨ren Fla¨che fu¨r die Phasengrenzschicht und der Auswertung der daraus resultieren-
den Sprungbedingungen. In diesem Teil wird zusa¨tzlich die sogenannte Grenzschichtfla¨che
vorgestellt, welche als Abbildungsoperator zwischen der Mikro- und Makroskala fungiert.
Das Kapitel endet mit der Aufstellung der maßgeblichen Bilanzrelationen in ihrer starken
Form.
Die numerische Realisierung wird in Kapitel 6 erkla¨rt. Zur na¨herungsweisen Lo¨sung me-
chanisch dominierter Anfangsrandwertprobleme bietet sich die Finite-Elemente-Methode
(FEM) an. Dafu¨r mu¨ssen die schwachen Formen der Bilanzrelationen ra¨umlich und zeitlich
diskretisiert werden. Als geeigneter numerischer Finite-Elemente-Lo¨ser wird das hausei-
gene Programm PANDAS verwendet, wobei eine monolithische Berechnungsstrategie zum
Einsatz kommt.
Mit Hilfe der numerischen Beispiele in Kapitel 7 sollen die Einsatzmo¨glichkeiten des
entwickelten Modells pra¨sentiert werden. Zuna¨chst wird die CO2-Injektion in ein wasser-
gefu¨lltes Reservoir, welches eine geneigte Deckschicht besitzt, simuliert. Hierdurch sollen
die Ausbreitung des CO2 in Folge des Injektionsdrucks und der Auftriebskra¨fte sowie die
Phasenumwandlung von der superkritischen in die gasfo¨rmige Phase dargestellt werden.
Außerdem werden die Feststoffverformungen auf Grund des Einpressdrucks analysiert. In
den beiden anderen numerischen Beispielen wird die Verdampfung und Kondensation von
CO2 ausfu¨hrlicher betrachtet. Dabei wird vor allem auf das Verhalten des Massentrans-
Deutschsprachige Zusammenfassung VII
fers zwischen der flu¨ssigen und der gasfo¨rmigen Phase Wert gelegt. Auch hier werden die
Feststoffverformungen beru¨cksichtigt.
Schlussendlich wird die Arbeit inKapitel 8 zusammengefasst und mo¨gliche Verbesserun-
gen des Modells, sowie weitere interessante Aspekte in einem Ausblick aufgelistet.
Als Erga¨nzung werden die in dieser Arbeit beno¨tigten mathematischen Relationen der
Tensorrechnung im Anhang A zusammengefasst. Zusa¨tzliche thermodynamische Ge-
setzma¨ßigkeiten und la¨ngliche Herleitungen verschiedener Beziehungen werden im
Anhang B aufgefu¨hrt.

Nomenclature
The notation in this article follows the conventions that are commonly used in modern
tensor calculus, such as in the text books of Ehlers [51], or de Boer [18]. The symbols used
in the context of porous-media theories adhere to the established nomenclature given by,
e. g., de Boer [20] and Ehlers [54, 56].
Conventions
Kernel conventions
( · ) place holder for arbitrary quantities
s, t, . . . or σ, τ, . . . scalars (0th-order tensors)
s, t, . . . or σ, τ , . . . vectors (1st-order tensors)
S,T, . . . or Ψ,Φ, . . . 2nd-order tensors
Index and suffix conventions
i, j,m, n, . . . indices as super- or subscripts range from 1 toN , where N = 3
in the usual 3-d space of our physical experience
( · )α subscripts indicate kinematical quantities of a constituent
within porous-media or mixture theories
( · )α superscripts indicate the belonging of non-kinematical quan-
tities to a constituent within mixture theories
( · )βR effective non-kinematical quantity belonging to a fluid con-
stituent
·
( · )= d( · )/dt material time derivative following the motion of a constituent
α with the solid and fluid constituents α = {S, L,G}
( · )′α = dα( · )/dt material time derivative following the motion of a constituent
α with the solid and fluid constituents α = {S, L,G}
d( · ) differential operator
∂( · ) partial derivative operator
( · )0 initial value of a non-kinematical quantity
( · )α0α initial value of a non-kinematical quantity with respect to the
referential configuration of a constituent
( · )F , ( · )F quantities of the fluid constituents
( · )T , ( · )−1 transposed and inverse forms of a tensor
( · )sym , ( · )skew symmetric and skew-symmetric parts of a tensor
( · )Γ , ( · )Γ quantities of the interface Γ
IX
X Nomenclature
( · )+ , ( · )− quantity belonging to the pore gas (B+ = BG) or pore liquid
(B− = BL)
J( · )K jump related value of the discontinuity surface Γ
( · )crit values at the critical point
( · )r reduced values, usually with respect to critical values
( · )FM , ( · )FM quantities of the fluid matter under consideration
( · )m , ( · )θ purely mechanical and purely thermal parts of a quantity as-
sociated with thermoelastic solid kinematics
( · )αE extra (effective) quantities of a constituent ϕ
α
( · )αE dis., ( · )
α
Emech. dissipative and purely mechanical parts of extra quantities
¯( · ) prescribed quantities (boundary conditions)
δ( · ) test functions of the respective degrees of freedom
( · )h spatially discretised quantities
( · )n, ( · )n+1 discretised quantities in time
Symbols
Greek letters
Symbol Unit Description
α constituent identifier in super- and subscript, i. e., , α =
{S, L,G}
αS [ 1/K ] coefficient of thermal expansion of ϕS
β fluid constituent identifier (here: β = {L,G})
β ′ fluid constituent identifier complementary to β (e. g., β = L
and β ′ = G)
Γ interface between the fluid phases
δji Kronecker symbol or Kronecker delta
ε, εα [ J/kg ] mass specific internal energy of ϕ and ϕα
εˆα [ J/m3 s ] volume specific direct energy production of ϕα
ǫtol [ - ] predefined tolerance used in the Newton solver
ζα [ J/kg ] mass specific enthalpy (Gibbs energy) of ϕα
ζˆα [ J/Km3 s ] volume specific direct entropy production of ϕα
∆ζvap [ J/kg ] mass specific latent heat or enthalpy of evaporation
η, ηα [ J/Kkg ] mass specific entropy of ϕ and ϕα
ηˆ, ηˆα [ J/Km3 s ] volume specific total entropy production of ϕ and ϕα
θ, θα [ K ] absolute temperature of ϕ and ϕα
θeq [ K ] equilibrium temperature
Nomenclature XI
θβ∗ [ K ] reduced temperature in the derivation of the shear viscosity
∆θ [ K ] temperature variation
∆θSF [ K ] temperature difference between ϕ
S and ϕF
∆gθ [ K/km ] geothermal temperature gradient
ϑ [ ◦ ] contact angle of the fluid interface with the solid
κ [ - ] exponent governing the deformation dependency of KS
κ, κβ overall and partial mass-transfer coefficients of ϕβ
λ [ - ] pore-size distribution index for Brooks & Corey law
λS [ N/m2 ] 2nd Lame´ constant of ϕS
µβ, ∆µβ [ J/mol ] chemical potential of ϕβ and difference in chemical potentials
µβR [ N s/m2 ] effective dynamic fluid viscosity of ϕβ
µβR0 , ∆µ
βR,
µβRc
[ N s/m2 ] ideal, excess and critical parts of the effective fluid viscosity
of ϕβ
µS [ N/m2 ] 1st Lame´ constant of ϕS
ξβ [ J/kg ] mass-specific Gibbs free energy (free enthalpy) of ϕβ
ξi local coordinates of a reference element
π [ - ] circle constant
ρ [ kg/m3 ] density of the overall aggregate ϕ
ρα, ραR [ kg/m3 ] partial and effective (realistic) density of ϕα
ρβF [ kg/m
3 ] partial pore density of the fluid phases ϕβ
ρˆα [ kg/m3 s ] volume-specific mass production of ϕα
ˆ̺βΓ [ kg/m
2 s ] area-specific interfacial mass transfer of ϕβ
σ, σα scalar-valued supply terms of mechanical quantities
σ∗ auxiliary term in the derivation of the shear viscosity
ση, σ
α
η [ J/Km
3 s ] volume specific external entropy supply of ϕ and ϕα
σs [ N/m ] surface tension of the fluid-fluid interface
τ ∗ auxiliary term in the derivation of the thermal conductivity
Υ arbitrary field function (steady and steady differentiable)
ϕ, ϕα entire aggregate model and particular constituent α
φju global basis function of a degree of freedom
ψ, ψα [ J/kg ] mass-specific Helmholtz free energy of ϕα
Ψ, Ψα [ ·/m3 ] volume-specific densities of scalar mechanical quantities
Ψˆ, Ψˆα [ ·/m3 ] volume-specific productions of scalar mechanical quantities
ωβ [− ] acentric factor
Ω, ∂Ω spatial domain and boundary of the aggregate body B
∂ΩB , ∂Ωu domain boundary and domain boundary of a primary variable
∂ΩuD Dirichlet boundary with essential boundary conditions for u
∂Ω
(·)
N Neumann boundary with natural boundary conditions
XII Nomenclature
Ωe, Ω
h a finite element and the discretised finite element domain
Ωξe reference finite element described in local coordinates
ξ local coordinates of a reference element
σ, σα vector-valued supply terms of mechanical quantities
Υ arbitrary field function (steady and steady differentiable)
φ, φα general vector-valued mechanical quantities
φη, φ
α
η [ J/Km
3 s ] entropy efflux vector of ϕ and ϕα
φju global basis function of a degree of freedom
χα, χ
−1
α motion and inverse motion function of the constituents ϕ
α
Ψ, Ψα [ ·/m3 ] volume-specific densities of vectorial mechanical quantities
Ψˆ, Ψˆ
α
[ ·/m3 ] volume-specific productions of vectorial mechanical quantities
εS [ - ] linearised contravariant Green-Lagrangean solid strain tensor
σS [ N/m2 ] linearised 2nd Piola-Kirchhoff stress tensor of ϕS
τα [ N/m2 ] Kirchhoff stress tensor of ϕα
Φ, Φα general tensor-valued mechanical quantities
Latin letters
Symbol Unit Description
a [m5/kg s2 ] cohesion pressure, coefficient in the cubic EOS
aΓ [ 1/m ] interfacial area, surface density of the interface
aSF [ 1/m ] interfacial area between the solid and the fluid in a two-phasic
medium
a∆ζ , b∆ζ , fitting parameters for the vaporisation enthalpy
c∆ζ, d∆ζ
ai, bi, ci, di,
dij
coefficients for the calculation of the shear viscosity and ther-
mal conductivity
da [m2 ] current area element
daΓREV [m
2 ] current area element of the interface Γ specific in the REV
Ac, Bc, Cc,
Dc, Qc, Rc,
Sc, Tc
auxiliary terms for the solution of a cubic equation
AA, BA, CA empirical parameters of the Antoine equation
AGL, AΓ [m
2 ] gas-liquid contact area in the volume-equivalent sphere, where
AGL = AΓ
ASG, ASL [m2 ] solid-gas and solid-liquid contact areas of the volume-
equivalent sphere
Aε, Bε auxiliary terms in the evaluation of the temperature assimi-
lation
Nomenclature XIII
dAα [m
2 ] reference area element of ϕα
b [m3/kg ] co-volume, coefficient in the cubic EOS
Bv, Cv, Dv virial coefficients of the virial EOS explicit in specific volume
Bp, Cp, Dp virial coefficients of the virial EOS explicit in pressure
c [ - ] temperature-dependent correction factor in the cubic EOS
cαRV [ J/kgK ] specific heat capacities for constant volume of ϕ
α
cint [ - ] auxiliary term in the derivation of the thermal conductivity
d spatial dimension of the physical problem
d10, d50 [m ] diameter of granular soil represented by 10% of the mass and
medial grain diameter
Dmass [ kg/m s ] local mass-transfer coefficient of ϕ
Dβmass [ kg/m s ] local mass-transfer coefficient of ϕ
β
e [ - ] Euler’s number, mathematical constant
eˆα [ J/m3 s ] volume specific total energy production of ϕα
E number of non-overlapping finite elements Ωe
E∗ finite elements attached to a respective node P j
f function identifier or integration constant
g [m/s2 ] scalar value of the gravitational force vector
g, h integration constants
hd [m ] macroscopic entry-pressure head
hβ [m ] filling height of volume-equivalent sphere of ϕβ
HαR [W/Km ] effective isotropic thermal conductivity of ϕα
HβR0 , H
βR
c ,
∆HβR
[W/Km ] ideal, critical and excess parts of the effective thermal con-
ductivity of ϕβ
Jα, J¯ [ - ] Jacobian determinant of ϕ
α and of a reference element Ωξe
kS [ N/m2 ] solid compression modulus
kβr [ - ] relative permeability factor of ϕ
β
kεSF [W/m
2K ] surface-specific heat-exchange coefficient
KG integration points for the Gaußian quadrature scheme
KS [m2 ] isotropic (deformation-dependent) permeability of ϕS
mα [ kg ] mass of ϕα
mSθ [ N/Km
2 ] stress-temperature modulus
dmα [ kg ] local mass element of ϕα
Mβ [ kg/mol ] molar mass of ϕβ
nα [ - ] volume fraction of ϕα
nF [ - ] porosity, total fluid volume fraction
nβm [ - ] number of moles of ϕ
β
N , Ne number of nodal points for Ω
h and Ωe
XIV Nomenclature
pβR, pRvap [ N/m2 ] effective pore pressure of ϕ
β and vapour pressure
pc, pd, p
FR [ N/m2 ] capillary pressure, bubbling or entry pressure and overall pore
pressure
p0 [ N/m2 ] standard atmospheric pressure
P j nodal point in a finite element Ωe of the set N
q¯ [ J/m2 s ] heat efflux over the boundary
r, rα [ J/kg s ] mass-specific external heat supply of ϕ and ϕα
r˜ [m ] radius of an idealised pore
r˜F [m ] radius of the volume-equivalent sphere representing a pore
r˜S [m ] radius of a characteristic spherical solid particle
rH [ - ] coefficient in the derivation of the thermal conductivity
R [ J/molK ] universal gas constant
Rα [ J/kgK ] specific gas constant of ϕα
R2 [ - ] error measure
sβ, sβres [ - ] saturation and residual saturation of ϕ
β
sLeff [ - ] effective liquid saturation
t [ s ] time
u adjustment parameter for vdW-EOS
uS2 [m ] vertical part of the displacement vector
v¯F , v¯L [ kg/m2 ] volumetric effluxes for ϕF and for ϕL over the boundary
vβR [m3/kg ] specific volume of ϕβ
∆v [m3/kg ] difference of specific volumes
dv, dvα [m3 ] current volume element of ϕ and ϕα
dvξ [m
3 ] current volume element of the reference element Ωξe
V , V α [m3 ] overall volume of B and partial volume of Bα
V βR [m3 ] effective volume of ϕβ
dVα [m
3 ] reference volume element of ϕα
w adjustment parameter for vdW-EOS
wk [ - ] weight for the Gaußian quadrature scheme
xi global coordinates
∆xβ [m ] film thickness
y constant part of a differential equation
Z, Zβ [ - ] general compressibility factor and compressibility factor of ϕβ
da [m2 ] oriented current area element
dAα [m
2 ] oriented reference area element of ϕα
b, bα [m/s2 ] mass specific body force vector
dα [m/s ] diffusion velocity vector of ϕ
α
fα [ N ] volume force vector acting on B from a distance
Nomenclature XV
g [m/s2 ] constant gravitation vector with | g | = g = 9.81m/s2
hˆ
α
[ N/m2 ] volume-specific total angular momentum production of ϕα
j, jβ, jβ
′
[ kg/m2 s ] total mass flux and local mass fluxes due to phase transition
kα, kα
F
, kα
N
, [ N ] total, external and contact force elements of ϕα
m [ - ] reference outward-oriented unit-surface normal vector
mˆα [ N/m2 ] volume-specific direct angular momentum production of ϕα
n [ - ] current outward-oriented unit-surface normal vector
pˆα [ N/m3 ] volume-specific direct momentum production of ϕα
q, qα [ J/m2 s ] total heat influx vector and heat influx vector of ϕα
sˆα [ N/m3 ] volume-specific total momentum production of ϕα
t¯ [ N/m2 ] external load vector acting on the boundary
tα [ N/m2 ] contact force vector per surface acting on S
uS [m ] solid displacement vector
vα [m/s ] velocity vector of ϕ
α, vα =
′
xα
wβ [m/s ] fluid seepage velocity vector of ϕ
β
wβΓ [m/s ] relative velocity vector of the fluid phases ϕ
β with respect to
Γ
x [m ] current position vector of ϕ
x˙,
′
xα [m/s ] velocity vector of the aggregate ϕ and the constituent ϕ
α
x¨,
′′
xα [m/s
2 ] acceleration vector of the aggregate ϕ and the constituent ϕα
dx [m ] current line element
Xα [m ] reference position vector at time t0
dXS [m ] reference line element of the solid
Aα [ - ] contravariant Almansian strain tensor of ϕ
α
Bα [ - ] covariant left Cauchy-Green deformation tensor of ϕ
α
Cα [ - ] contravariant right Cauchy-Green deformation tensor of ϕ
α
Dα [ ·/s ] symmetric deformation velocity tensor of ϕα
Eα [ - ] contravariant Green-Lagrangean strain tensor of ϕ
α
Fα [ - ] material deformation gradient of ϕ
α
Hα,HαR [W/Km ] partial and effective heat-conduction tensors of ϕα
HS [ - ] solid displacement gradient
I [ - ] identity tensor (2nd-order fundamental tensor)
Kβ [m/s ] tensor of hydraulic conductivity of ϕβ
Kβr [m/s ] tensor of relative permeability of ϕ
β
KS [m2 ] intrinsic (deformation-dependent) permeability tensor of ϕS
Lα [ ·/s ] spatial velocity gradients of ϕα
XVI Nomenclature
Pα [ N/m2 ] 1st Piola-Kirchhoff stress tensor of ϕα
Sα [ N/m2 ] 2nd Piola-Kirchhoff stress tensor of ϕα
Sβf [ N s/m
4 ] second-order friction tensor of ϕβ
T, Tα [ N/m2 ] overall and partial Cauchy (true) stress tensors of ϕ and ϕα
Wα [ ·/s ] skew-symmetric spin tensor of ϕα
Calligraphic letters
Symbol Unit Description
Au ansatz (trial) functions of the primary variables
B, Bα aggregate body and partial constituent body
C critical point in the phase-diagram
D dissipative part in the entropy inequality
Gu weak formulation of a governing equation related to a DOF
H1(Ω) Sobolev space
N set of all nodes for the FE discretisation
O origin of a coordinate system
P [ N/m2 ] Lagrangean multiplier
P, Pα material points of ϕ and ϕα
R set of response functions
S, Sα surface of the aggregate and constituent body
T triple point in the phase-diagram
T u test (weighting) functions of the primary variables
V set of independent process variables
V1 subset of independent process variables
f generalised vector of external forces
F vector containing the global and local system of equations
DFkn+1 global residual tangent
Gu abstract vector containing the weak formulations
k generalised stiffness vector
M generalised mass matrix
u , u1, u2 abstract vectors containing the set of the primary variables
y abstract vector containing all nodal DOF
Selected acronyms
Symbol Description
Nomenclature XVII
2-d two-dimensional
3-d three-dimensional
BC boundary condition
Ca2+ calcium ion
CaCO3 calcium carbonate
CCS carbon-dioxide capture and storage
CH4 methane
CO2 carbon dioxide
CO2−3 carbonate
CSP corresponding state principle
DOF degree of freedom
EOR enhanced oil recovery
EOS equation of state
FDM finite-difference method
FEM finite-element method
FVM finite-volume method
GG greenhouse gases
HCO−3 hydrogen carbonate
H2O water
H3O
+ hydronium, aqueous cation
IBVP initial boundary value problem
IPCC Intergovernmental Panel on Climate Change
LG coexisting liquid-gas region
NIST National Institute of Standards and Technology
N2O nitrous oxide
O3 ozone
PANDAS porous media adaptive nonlinear finite element solver based
on differential algebraic systems
PDE partial differential equation
PR-EOS Peng-Robinson equation of state
REV representative elementary volume
SC supercritical region
SG coexisting solid-gas region
SRK-EOS Soave-Redlich-Kwong equation of state
TPM Theory of Porous Media
vdW-EOS van-der-Waals equation of state
XFEM extended finite-element method

Chapter 1: Introduction
1.1 Motivation
It is a widely accepted fact that anthropogenically emitted greenhouse gases (GG) con-
tribute to the global warming. In turn, this might cause rising sea-levels that threat
low lying residential areas, and initiate more severe weather conditions (e. g., hurricanes,
floods, tornadoes), droughts and other issues. For these reasons, it is necessary to stop or
reduce the global warming by cutting the emissions of GG. In this context, carbon diox-
ide, CO2, is one of the most important GG since it is emitted in high amounts by human
activities, e. g., in energy production by fossil-fuel power plants. As a countermeasure,
carbon-dioxide capture and storage (CCS) was invented to reduce CO2 emissions into the
atmosphere. At the end of the 20th century, the latter was deemed to help mitigate the
problem of global warming and, thus, entailed the initiation of a new field of research,
where scientists and engineers from various fields collaborated in solving open questions
of CCS. Nowadays, the promising character of CCS has somewhat reduced, which is, on
the one hand, due to public resistance caused by the fear of leakage of CO2 and, on the
other hand, due to the missing economical pressure caused by the ineffectiveness of the
CO2 certificates that should put a price on CO2 emissions, but are too cheap and too
numerous. However, by concentrating CCS to offshore projects and uninhabited onshore
regions, by improving the communication with the public and by adjusting the concept
of the CO2 certificates, CCS can still be regarded as a promising technique that bridges
the gap until alternative renewable sources can fully replace fossil fuels by reducing the
exhaust of greenhouse gases.
The CO2 emissions are mainly caused by the combustion of fossil fuels for energy produc-
tion, for transportation and in industrial processes. Thereof, especially point-like sources
such as power plants are suitable for the segregation of the emitted CO2 from the exhaust
air. This CO2 can then be stored, for example, in deep underground reservoirs. Suitable
geological formations for this purpose are depleted oil or gas fields, coal beds or deep
saline aquifers. All of these storage methods have advantages and disadvantages con-
cerning different factors, for instance, enhanced recovery of oil or gas, accessibility of the
storage site, storage volume, and the safety of the reservoir. The latter is only guaranteed
if leakage of CO2 can be excluded. This is considered fulfilled, if the reservoir is sealed
by an almost impermeable cap-rock layer, to prevent the CO2 from upwards migration.
However, due to the increasing pressure in the reservoir during injection or due to seis-
mic movements, it could still be possible that crack development in the cap-rock layer is
initiated and leakage occurs. In order to predict such events beforehand, numerical sim-
ulations can provide means to model the injection process including the accompanying
thermodynamical changes of the CO2 and the deformations of surrounding rock matrix.
Hence, this poses the objective of this monograph. The physical, chemical and thermo-
dynamical processes during CO2 injection are highly complex and, therefore, in the scope
of ongoing research. Since it is not possible at the moment to reproduce the whole set of
1
2 Chapter 1: Introduction
processes in a realistic manner in one single numerical model, the investigations in this
monograph will be concentrated on the mechanical behaviour of the geological formation
and the phase transition of the injected CO2.
1.2 Scope, aims and state of the art
The whole process of CCS incorporates a great variety of technically challenging steps,
which can be separated into the chemical segregation of the CO2 from the exhaust air,
the liquefaction and transport to the storage site and the sequestration of the CO2. Here,
only the last part is discussed further. In this context, the geological storage of the CO2
in underground reservoirs is regarded as the most effective storage method apart from
deep ocean storage, mineral carbonation, or industrial usage, cf. Special Report on CCS
by the Intergovernmental Panel on Climate Change, Metz et al. [119].
Existing geological CO2 storage projects that can confirm the feasibility of underground
storage on an industrial scale, i. e., at least 1 MtCO2/a, are, for example, the sequestration
at the Sleipner gas field in the North Sea, the Weyburn project in Canada and the In
Salah project in Algeria. However, as for the first CO2-injection process in the early 1970s,
also in present CCS projects the CO2 is mainly used for enhanced oil recovery (EOR).
In this regard, the usage of the CO2 lessens of course the costs for CCS, but does not
really improve the greenhouse-gas problem, since the additionally produced oil or gas will
also be used mainly for energy production. Thus, only the injection into depleted oil or
gas fields and into saline aquifers can be considered as ecologically worthwhile, whereas
in this contribution, the latter technique shall be examined further. Deep saline aquifers
are distributed world-wide and supposed to provide enough storage space for the aspired
amount of CO2 to reduce global warming, cf. Metz et al. [119].
The knowledge of the movement and behaviour of the CO2 during and after injection
is essential for the storage-capacity estimation and to assess safety issues. Since in situ
monitoring techniques, e. g. borehole measurements and seismic surveys, do not provide
sufficient information of the complex mechanisms acting between the injected CO2 and
the solid matrix, numerical studies are of paramount importance for a detailed under-
standing of the involved physical processes. The literature contains a vast amount of
contributions on the numerical simulation of underground CO2 storage. This science has
been developed from the knowledge in predicting and simulating hydrocarbon production
and early works can be found by Pruess & Garc´ıa and co-workers [46, 141, 144], as well as
Bielinski [15] and references therein. Different aspects for an efficient simulation have been
investigated, for example, concerning analytical or semi-analytical models by Kang et al.
[93], Nordbotten et al. [129], Celia et al. [34], or regarding vertical-equilibrium models by
cf. Court et al. [40] and Gasda et al. [70]. Nordbotten and Celia [34, 127, 128, 143] have
provided several publications on large-scale simulations, including predictions of potential
leakage amounts for policy makers. Moreover, Bradshaw et al. [28] and Kopp et al. [99]
estimated the storage capacities of various possible reservoirs. Specific physical questions
concerning, e. g., relative permeability, temperature-dependent capillary pressure, miner-
alisation, enhanced trapping mechanism, or geomechanics, have been studied by Juanes
1.2 Scope, aims and state of the art 3
et al. [91], Plug & Bruining [136], Kumar et al. [103], Ebigbo [47], and Rutqvist [154–156].
Benchmark studies and code comparisons can be found, for instance, by Pruess et al. [142]
and Class et al. [37]. However, the phase transition of the CO2 has not gained much of
interest in these contributions. When the phase transition was considered in these models,
usually look-up tables were utilised for the thermodynamics properties. The drawback of
these tables is the negligence of an explicit description of the mass transfer between the
fluid phases.
Transitions between gas, liquid and solid phases are important physical processes that
appear everywhere in the environment. These processes, as for example drying or freezing,
do not only occur in well investigated “open systems”, but also in porous media, such as
CO2 reservoirs. However, the mechanism of phase transitions in porous media has been
only scarcely investigated so far. Apart from CO2 sequestration, this kind of processes
are also important in other geological activities (e. g., steam injection for enhanced oil
recovery, soil remediation, geothermal-energy production), in food industries (drying and
baking processes), and in many more. In this work, the consideration is restricted to first-
order phase transitions, i. e., “gas-into-liquid” or “liquid-into-gas”, of a single substance,
e. g., CO2. These specific phase transitions must be clearly distinguished from phase
transitions between mixtures of various substances, such as water evaporating into air,
since the thermodynamics are rather different. A detailed explanantion of these differences
will be provided in this work. Furthermore, the phase change between the supercritical and
the gaseous state is regarded. This phase change is modelled using a single mass-balance
relation for both fluid phases, which is in contrast to gas-liquid phase transitions, where
for each phase a distinct mass balance is applied. This is possible, since the transition
between supercritical and gas (and also supercritical and liquid) is continuous.
To the authors’ knowledge, the earliest work on the treatment of phase transitions in
a porous aggregate goes back to Lykov [112] in 1974. Further articles in this direction
have been presented by the group around Be´net [35, 108, 109, 153], by Hassanizadeh
and Gray [75, 81, 125, 126] or by Bedeaux [14]. Examples of applying these models to
actual physical problems are the simulation of drying processes by Kowalski [101] or the
numerical investigation of bread-baking by Huang et al. [87].
The previously mentioned articles consider phase-transition processes in porous media,
where the solid matrix is usually idealised as a rigid body. If solid deformations shall
be included into the model, it is helpful to proceed from the well-founded concept of the
Theory of Porous Media (TPM). This approach provides an ideal framework for the mod-
elling of multiphasic and multicomponent continua including arbitrary solid deformations
based on elasticity, viscoelasticity or elasto-plasticity, as well as an arbitrary pore content
of either miscible or immiscible fluids. The reader who is interested in the basics of the
TPM is referred to, e. g., the publications of de Boer [20], de Boer & Ehlers [22], Bowen
[26, 27], Ehlers and coworkers [50, 54–56, 59, 60, 180] or Schrefler et al. [160, 161]. The
TPM has proven its usefulness in various applications of different fields. With focus on
the exact description of the solid deformations, the theories of elasto-plasticity or elasto-
viscoplasticity have been applied to topics of material sciences or of geomechanics, e. g.,
Markert [115], Avci [8], Ehlers et al. [57]. Concerning biomechanical questions, chem-
ical, electrical and biological properties were added to the portfolio by Karajan et al.
4 Chapter 1: Introduction
[95], Ricken and coworkers [147, 150] and Wagner [176], to name a few. The problem of
biological phase transitions within landfills has also been tackled by the research group
among Ricken [151, 152].
The development of a thermodynamically consistent description of phase-transition pro-
cesses in porous media based on the TPM starts with contributions by de Boer and
coworkers [19, 21, 23] and continues with the works of, e. g., Kowalski [101] and Ghadiani
[71], and an article by Ehlers & Graf [59]. Extensive investigations of freezing and melting
processes have been conducted, for instance, by Kruschwitz and Bluhm [102] and the group
around Bluhm and Ricken [17, 148, 149]. In all of these works, the constitutive relation
for the mass transfer between the two fluid phases was derived from the exploitation of
the entropy inequality. Moreover, the phase transition takes place between mixtures of
different fluid matters.
Here, it is intended to model the phase transition between pure phases of the same fluid
substance. It appears that the constitutive mass-transfer relation derived from the entropy
inequality for the phase transition between mixtures is not valid in this case. Instead,
an immaterial, smooth singular surface is introduced at the interface that separates the
liquid phase from the gas phase. The introduction of this singular surface in turn involves
the definition of jump conditions for the respective functions and balance relations. By
exploitation of these jump conditions, it is possible to find a constitutive relation for the
mass transfer. This approach has been already applied by different groups, for example,
Jamet [89], Juric & Tryggvason [92], Morland et al. [120, 121], Tanguy et al. [170] and
Wang & Oberlack [177].
In this monograph, the jump conditions evolving at the interface between the two fluid
phases are evaluated within the framework of the TPM, cf. Ehlers and Ha¨berle [61].
The resulting relation defines the mass transfer as a function of the ration between the
energy provided for the phase transition and the latent heat of evaporation. Since the
singular surface and therewith also the mass transfer are formulated on the microscale,
it is necessary to find a suitable upscaling relation that relates the mass transfer on the
microscale to the mass production on the continuum macroscale. In this regard, the
interfacial area, i. e., the specific density of the internal phase-transition surface with
respect to the considered volume, is introduced as a mapping function. The derivation of
this interfacial area is based on the achievements by Hassanizadeh and coworkers [81, 90,
124, 125], as well as on the dissertation of Graf [74] and bears similarity in its usage with
the work of Nuske et al. [130]. The principle of interfacial areas is also subject of the
contributions by Miller and coworkers [41, 76], Celia and coworkers [83, 145], Gladkikh &
Bryant [72] and Oostrom et al. [132], amongst others.
For a preferably realistic representation of the thermodynamic behaviour of the CO2
further constitutive relations must be added to the model. Hereby, the compressibility of
the CO2 is governed by a suitable equation of state (EOS) that represents the relationship
between pressure, temperature and density. In this context, different relations from van
der Waals, from Peng and Robinson, as well as from Soave, Redlich and Kwong are
discussed with regard to their accuracy in describing the phase behaviour and with respect
to the thermodynamical consistency for implementation into the TPM framework. An
extensive survey of the various EOS is provided, for instance, in the books of Ott and
1.3 Outline of this thesis 5
Boerio-Goates [134] and Poling et al. [137].
Furthermore, constitutive relations for the specific heat, the vaporisation enthalpy, the
shear viscosity and the thermal conductivity are defined, which resort to different con-
tributions, wherein the determination of the respective relations are based on collected
data and various extents of empiricism. Due to the large number of publications in this
field of research only these that have been used in this monograph shall be mentioned
here: Lemmon et al. [104], Potter and Somerton [139], Lewis & Randall [105], Pitzer et
al. [135], Fenghour et al. [66] and Vesovic et al. [175].
With these thermodynamical properties at hand, a triphasic model (solid, water and
CO2, or solid, liquid CO2 and gaseous CO2) for the description of CO2 injection into
a deep reservoir is formulated in the continuum-mechanical framework of the TPM. In
this regard, further constitutive relations apart from the thermodynamical ones must
be obtained form the evaluation of the entropy inequality. In particular, these are a
Darcy -like relation for the seepage velocity linked to the creeping flow, the capillary-
pressure-saturation relation for the interaction between the constituents due to frictional
forces, and the thermoelastic formulation for the deformation behaviour of the porous
reservoir rock and the impermeable cap rock. Thus, the model is able to simulate CO2
injection into a deep reservoir, while considering in detail the phase-transition process,
the compressibility of the involved fluid phases and the thermoelasticity of the solid rock
matrix in a fully coupled way. Consequently, a tool is provided that allows to conduct
numerical studies of CO2 storage in advance of the actual injection in order to provide
decision support in view of the safety issues and feasibility of a contemplated reservoir.
1.3 Outline of this thesis
The introductory Chapter 1 is followed by a brief overview of the specifics of carbon-
dioxide capture and storage (CCS) in Chapter 2. Therein, the technical, environmental,
economical and safety aspects are discussed, where special attention is turned on the stor-
age of CO2 in deep underground aquifers. Furthermore, it is pointed out how numerical
simulations can help in understanding the processes in the reservoir.
Subsequently, in Chapter 3, the theoretical fundamentals needed for the description of a
triphasic porous medium within the Theory of Porous Media (TPM) are explained. This
includes the kinematics of superimposed constituents, the strain and stress measures, as
well as the balance relations, both for the particular constituents and the overall aggregate.
Additionally, singular surfaces are introduced here together with the changes imposed on
the balance relations. These singular surfaces are a precondition for the later description
of the phase-transition process, which occurs across the gas-liquid interface.
A detailed discussion of the thermodynamics of fluids follows in Chapter 4. This con-
cerns especially the description of the phase behaviour, where the advantages and disad-
vantages of different equations of state (EOS) are pondered. In this context, the specifics
of the phase-transition process between the liquid and gaseous phases are of most inter-
est. Moreover, the thermodynamics and the thereon based descriptive relations for the
specific heat, the vaporisation enthalpy, the shear viscosity and the thermal conductivity
6 Chapter 1: Introduction
are presented in this chapter. As far as possible, a general formulation is sought that is
independent of the specific fluid substance.
In Chapter 5, the required constitutive setting is identified from adapting the respective
balance relations, i. e., the mass, momentum and energy balances for the solid material,
for the fluid phases and for the overall aggregate, to the problem at hand. In this regard,
a sophisticated evaluation of the entropy inequality is conducted to derive the consti-
tutive relations in a thermodynamically consistent manner. Additionally, this chapter
comprehends the thorough derivation of the mass-transfer term, which is developed by
introducing a separating interface between the two fluid phases gas and liquid and by
evaluating the jump conditions at this singularity. This latter part also involves the in-
troduction of the so-called interfacial area as a mapping operator between the micro- and
continuum-scale. Furthermore, a switching criterion for the mass transition is formulated.
The chapter ends with the presentation of the final strong forms of the governing balance
relations.
The numerical realisation is explained in Chapter 6. It is customary to solve mechanical
dominated initial boundary value problems (IBVP) approximately with the finite-element
method (FEM). In this connection, the weak formulations of the governing balance re-
lations are spatially and temporally discretised. As a suitable finite-element tool, the
in-house code PANDAS1 is chosen, wherein an monolithic solution strategy is embedded.
By the numerical examples in Chapter 7, the capabilities of the derived model are illus-
trated. At first, the model is applied to simulate the injection of CO2 into a water-filled
reservoir with an inclined cap-rock layer. Hereby, it is intended to show the migration of
the CO2 due to the injection pressure and buoyancy forces and the phase change from the
supercritical phase to the gaseous phase. Furthermore, the solid displacement in conse-
quence of the injection pressure is observed. In the other two examples, the evaporation
and condensation processes between liquid and gaseous CO2 are regarded, while concen-
trating especially on the mass transfer between the two fluid phases. Here, also the solid
displacement is accounted for.
Finally, conclusions are given in Chapter 8 and possible enhancements of the presented
model as well as further interesting aspects are collected in the outlook.
Mathematical relations used throughout this work, specifically on tensor calculus, are
added in Appendix A. Additional thermodynamic supplements and lengthy derivations
of certain relations are stored in Appendix B.
1Porous Media Adaptive Nonlinear Finite Element Solver based on Differential Algebraic Systems
Chapter 2: Carbon-dioxide capture and
storage
It is common sense that greenhouse gases (GG) are the cause for the global warming
problem. These gases absorb and emit infrared radiation coming from the earth’s surface
(caused by solar irradiation) back to earth, and thus, hindering the radiation from emitting
into space. This leads to a net increase in energy in the earth’s atmosphere and on the
surface.
The main GG are water (H2O), carbon dioxide (CO2), methane (CH4), nitrous oxide
(N2O) and ozone (O3). From this group, water is the most important GG, being respon-
sible for about 60% of the global warming effect. However, it is related to the so-called
“natural” GG, since the water in the atmosphere corresponds to the natural water cycle.
CO2 is regarded as the most important “artificial” GG, because the anthropogenically
emitted amount of CO2 is much higher compared to the emissions of the other gases.
In this chapter, the specifics of CO2 storage in underground reservoirs and the accom-
panying difficulties are explained. For a better understanding of the process of CO2
sequestration and its requirements, first the properties of CO2 are provided, then the
method of CCS is explained, followed by an overview of different storage options and
their advantages and disadvantages. Last, the tasks that arise from open questions con-
cerning the storage size and the safety of the reservoir are collected. These tasks are then
further classified into the ones that are possible to solve by numerical simulations of CCS.
This serves as the motivation for the development a continuum-mechanical model of CO2
injection.
2.1 Properties of CO2
Carbon dioxide is a vital molecule for the life on earth, since it is essential for the respi-
ratory process of most to all organisms1. Photosynthetic organisms, such as plants, algae
and cyanobacteria, consume CO2 together with water and light energy to synthesize car-
bohydrates, which are then build into their bodies. As a byproduct of the photosynthesis,
oxygen is generated that can be respirated in the metabolism of aerobic organisms, e. g.,
in the gills of fish, in the trachea of insects or in the lungs of mammals, reptiles and birds.
During the respiratory process, which represents the opposite process of the photosynthe-
sis, CO2 is produced again and the CO2 cycle is closed.
Other sources of CO2 are the metabolism of microorganisms, e. g., the fermentation of
sugars during wine or beer production, and the combustion of materials composed of car-
bohydrates, e. g., wood and fossil fuels, such as coal, peat, oil or natural gas. Furthermore,
CO2 is discharged at volcanic-active sites, e. g., volcanoes, hot springs or geysers. Another
1Apart from exotic organisms such as sulfur-based organisms that live around black smokers in the
deep sea.
7
8 Chapter 2: Carbon-dioxide capture and storage
natural source of CO2 represents the dissolution of carbonate rocks into water.
CO2 is widely used in technical applications, for example, as a shielding gas for welding
and fire extinguishers or as a pressurised gas for air guns. In the petroleum industry, it
is injected for enhanced oil recovery (EOR), it is used as a refrigerant, for example, in
automobile air-conditioning systems and also in its solid form, the so-called dry-ice, CO2
is applied for cooling applications or as an abrasive in dry-ice blasting. Furthermore, CO2
finds usage in the food and beverage industry for the carbonation of drinking water, beer
or sparkling wine. Here, also the supercritical state of CO2 finds benefit in the so-called
supercritical drying process, where it acts as a non-toxic solvent for lipophilic organic
materials, e. g., in the decaffeination of coffee, cf. McHugh & Krukonis [118].
At atmospheric conditions (p0 = 0.1MPa), CO2 in its gaseous state is as a trace gas of the
ambient air, with a concentration of about 0.04 % per volume (400 ppm, measured in the
year 20152), cf. Figure 2.1. CO2 is not classified as a toxic gas, but acts as an asphyxiant
gas, where the effects on humans can be already felt at concentrations of 1000 ppm. At
concentrations of 7% to 10% (70 000 ppm to 100 000 ppm), life threatening conditions are
reached. This asphyxiant effect is also applied in butcheries to stun animals.
C
O
2
co
n
ce
n
tr
a
ti
o
n
[p
p
m
]
C
O
2
co
n
ce
n
tr
a
ti
o
n
[p
p
m
]
full record ending January 11, 2016
thousands of years ago
0100
200
200
300
300
400
400
400
500600700800
150
250
350
350
310
320
330
340
360
370
380
390
1960 1970 1980 1990 2000 2010
Figure 2.1: (left): CO2 concentration measured from 1958 to January 2016 at Mauna Loa
Observatory (USA). This curve is also known as the Keeling curve [96]. The yearly oscillations
originate from the increase of vegetation in spring and summer (decrease in CO2 concentration)
and decrease of vegetation in autumn and winter (increase of CO2 concentration) in the northern
hemisphere, since most of the landmass is located there.
(right): CO2 concentrations from 800 000 years ago until today. Measurements of the CO2
concentrations are taken from ice-cores. It clearly shows that today’s CO2 concentration has
not been encountered for a really long time. (Source: http://keelingcurve.ucsd.edu/).
Figure 2.1 shows that today’s CO2 concentration has not been encountered for almost
800 000 years. Keeling [96] also states that the increase in CO2 concentration matches
the amount of combusted fossil fuels starting with the industrial age. Consequently, these
curves indicate that this strong increase in CO2 concentration over the last fifty years has
an anthropogenic background. Besides its gaseous occurrence in air, CO2 is soluble in
water and, thus, is pervasive in all natural water distributions, both liquid and ice.
2Parts per million (ppm) is a volumetric concentration measure for highly diluted materials, which
denotes one part per 1 000 000 parts.
2.1 Properties of CO2 9
Physical characteristics of CO2
The required conditions and features of an eligible reservoir for a safe storage of CO2,
explained in the next Section 2.2, are dictated by the physical and thermodynamical
properties and characteristics of the CO2. As its name implies, carbon-dioxide consists
of one carbon and two oxygen atoms, which form a linear molecule without an electrical
dipole3. A couple of physical and thermodynamical properties are listed in Table 2.1.
Considering the sequestration process, it is obvious that a high density and low viscosity
Table 2.1: Selected physical and thermodynamical properties of CO2.
molar mass: 44.01 g/mol
critical pressure: 7.374 MPa
critical temperature: 304.12 K
critical density: 468.19 kg/m3
of the CO2 are favourable in order to store as much CO2 as possible in a given reservoir
and, thus, lower the costs of storage. Since the temperature within a reservoir changes
only slightly over depth, usually ∆θ ≈ 25.0K/km, cf. Rutqvist [155], the density of the
CO2 in the reservoir depends mainly on the increasing pressure with depth. In Figure 2.2,
the relation between the density of CO2 and the injection-depth is illustrated. It shows
D
ep
th
[m
]
Density [kg/m3]
ground level
critical depth
gaseous CO2
supercritical CO2
0
0 200
500
400
1500
600
1000
1000
2000
2500
800
Figure 2.2: Sketch of the depth of a reservoir versus the density of CO2 illustrating the different
phase behaviours.
that below a certain critical depth, at around 800m, the density increases dramatically
from somewhere below 200 kg/m3 to over 700 kg/m3, which indicates the phase change
from gaseous to supercritical CO2. Thus, injection should be conducted below this critical
depth in order to provide that a larger amount of CO2 can be stored per unit available
3The water, H2O, molecule on the other hand, which also consists of three atoms, two hydrogen and
one oxygen, exhibits an ankle of 104.45◦ and has a strong electrical dipole. This is the reason for several
special properties of water, such as the density anomaly (i. e., the solid phase is lighter than the liquid
phase and the highest density of water exists at around 4 ◦C), highest specific heat capacity, highest
surface tension and highest latent heat of all fluids.
10 Chapter 2: Carbon-dioxide capture and storage
pore volume, cf., e. g., Bachu [10]. Furthermore, supercritical CO2 has a lower viscosity
than liquid CO2, which presents another advantage for the injection and storage of CO2
in subsurface reservoirs, also mentioned by Celia et al. [34].
In natural systems, however, it is not guaranteed that the pressure and temperature
conditions within the reservoir remain constant over time. In case of changing conditions,
the thermodynamical state of CO2 can alter from the supercritical or the liquid to the
gaseous state. Due to the previously mentioned density behaviour, such a phase change
is undesirable. Also, a transition to a lower density leads to an increase in pressure,
implying a greater stress on the sealing cap-rock layer and causing deformations in the
solid skeleton. In the worst case, fractures could be induced leading to leakage of the
injected CO2. Therefore, it is important to know when and why phase transitions occur.
In this regard, in situ experiments are not feasible to find concluding answers and are
also too expensive. Hence, the numerical simulation provides a potent, cheap and more
feasible tool to understand the phase change and other processes in CO2 storage.
Besides the position of the reservoir in depths lower than 800 m, several other requirements
have to be fulfilled to guarantee a safe storage of the CO2. These are discussed in the
following Section 2.2.
2.2 What is CCS?
The following description of term carbon-dioxide capture and storage (CCS) is a brief
roundup of the extensive discussion presented in the report of the Intergovernmental
Panel on Climate Change (IPCC) [119]. CCS incorporates all means necessary to reduce
the amount of CO2 in the atmosphere, which is supposed to be one of the major GG
causing global warming. Besides the already discussed CO2, there are several other GG
with even greater global warming potential4, e. g., methane, CH4, or nitrous oxide, N2O.
However, the anthropogenic emitted amount of CO2 is much higher compared to the
other gases and, thus, still has the greatest effect on global warming, which is measured
by the so-called radiative forcing5, cf. Table 2.2. The sources of CO2 emissions have been
illustrated before. A hint that these emissions really are the cause for the increasing CO2
concentration in the atmosphere is given by the Keeling curve in Figure 2.1. The best
method to reduce these emissions is of course to use alternative energy sources, which
in case of electricity production could be solar-, wind-, or hydro-power, different means
of vehicle propulsion, or the enhancement of biological sinks. If not possible, increasing
4Measured relative to the same mass of CO2, with the global warming potential of CO2 being equal
to 1.
5Radiative forcing is a measure for the change in energy of the earth-atmosphere system caused by
climate-affecting species. Because these species change the balance between incoming solar radiation and
outgoing infrared radiation within the earths atmosphere that controls the earth’s surface temperature,
the term radiative is applied here. Radiative forcing is usually given in W/m2 and expresses the “rate
of energy change per unit area of the globe, measured at the top of the atmosphere”, cf. Myhre et al.
[123]. Positive radiative forcing leads to an increase in the energy of the earth-atmosphere system, i. e.,
to a warming of the system. In contrast, a negative radiative forcing decreases the energy and leads to a
cooling of the system.
2.2 What is CCS? 11
Table 2.2: Comparison of global warming potential for a 100-year time horizon in relation to
CO2, changes in atmospheric concentrations from 2005 until 2011 and radiative forcing in 2011
of CO2, CH4 and N2O, taken from Myhre et al. [123].
species
global warming potential concentration change radiative forcing
100-year horizon 2005 - 2011 in 2011
CO2 1 +12ppm 1.82
CH4 34 +29ppb 0.48
N2O 298 +5ppb 0.17
the efficiency of already existing technology should be the second step to reduce CO2
emissions. Finally, the last possibility to mitigate climate change is CCS, which could
be an important strategy to bridge the gap until conventional, fossil-energy production
can be replaced by alternative, renewable resources. In the report of the IPCC [119] it is
stated that many scenarios show a domination of primary energy production by fossil fuels
until the middle of the 20th century. Therefore, CCS could be one part of the portfolio
to at least stabilise the amount of CO2 in the atmosphere. A relatively new approach, to
even achieve a net removal of atmospheric CO2, is the combination of bio-energy power
plants with CCS.
The method of CCS comprises the capture of CO2 at point-like sources (i. e., fossil-fuel
power-plants, CO2-emitting industries, oil and gas processing, refineries, cement plants
and steel mills), compression of the separated CO2 to its supercritical state, transporta-
tion by pipeline or ship to the storage side and a set of different monitoring and verifica-
tion technologies to control the safety, efficacy and spreading of the injected CO2 in the
reservoir, cf. e. g., Herzog [85]. Possible storage methods are: injection into underground
reservoirs, deep seafloor ocean storage and industrial fixation of CO2 into inorganic car-
bonates. Since this thesis is only concerned about the geological storage process, the other
storage methods and the capture and transportation will not be explained any further and
the interested reader is referred to the IPCC Special Report on Carbon Dioxide Capture
and Storage [119].
To get an impression on how many CCS projects have been accomplished so far and how
much CO2 is sequestered, it is resorted to a report by the Global CCS Institute [73]. It
states that 75 large-scale geological CO2 storage projects existed in the year 2012, that
captured at least 800 000 tonnes of CO2 annually per power plant and 400 000 tonnes of
CO2 annually per industrial facility.
2.2.1 Geological storage options
Promising geological formations for the underground storage of CO2 are depleted oil or
gas reservoirs, deep saline aquifers and unmineable coal beds, cf. Figure 2.3. These forma-
tions can both be onshore and offshore, whereas recent cancellations of onshore projects
due to public resistance (e. g., in 2011 Vattenfall discontinues the pilot power-plant in
Ja¨nschwalde, Germany) propose less feasibility. A common characteristic of geological
12 Chapter 2: Carbon-dioxide capture and storage
Figure 2.3: Overview of geological storage options for CO2, IPCC report [119].
reservoirs is a sealing of the permeable storage layer by an almost impermeable cap-rock
layer to prevent the CO2 from upwards migration. The various geological storage options
have different advantages or disadvantages concerning their storage volume, accessibility
and safety, which are discussed more thoroughly in the following.
The most obvious candidates for CO2 storage are depleted oil and gas reservoirs, since
they already demonstrated their ability to trap fluids lighter than water over a long time.
Usually, an extensive amount of data about the geological structure, physical properties
of the rock and the storage volume is available from the previous gas or oil production.
Furthermore, computer models have been developed to predict the hydrocarbon move-
ment, improve the enhanced oil/gas recovery and to estimate the recovery factor6 of a
reservoir, which can be directly applied to the CO2 storage.
Table 2.3: Storage capacities of oil and gas fields and deep saline formations [119].
reservoir type
lower estimate of upper estimate of
storage capacity storage capacity
(GtCO2) (GtCO2)
oil and gas fields 675 900
deep saline formations 1000 uncertain, ∼ 104
In case of non-depleted fields with a decreasing production rates, the CO2 injection can
6The recovery factor refers to the ratio between the amount of oil that can be produced from a reservoir
and the amount of oil initially in place.
2.2 What is CCS? 13
be combined with the enhancement of oil recovery. By flooding the oil reservoir with
CO2, the recovery factor can be risen to an average of 13.2%, furthermore reducing the
cost of CO2 storage by the produced hydrocarbons. However, about half of the injected
CO2 returns with the oil and must be separated and re-injected. Moreover, the physical
conditions of the reservoir and the oil must meet certain criteria to allow for an efficient
CO2-driven oil recovery. For example, very thick, homogeneous reservoir layers are not
suitable for EOR, since the lower density of the CO2 with respect to the reservoir oil leads
to a movement of the CO2 along the top of the reservoir layer without pushing the oil
towards the production well.
Another drawback of these developed reservoirs is the high number of wells that are often
not thoroughly documented. If these wells are not properly sealed, they pose a risk of
leakage by jeopardising the integrity of the cap-rock and its confining character.
The second kind of storage formations with an even greater supposed storage capacity,
cf. Table 2.3, are deep saline aquifers. These consist of sedimentary rocks containing
brine with a high salt concentration and, therefore, are unsuitable for drinking-water
production. The only imaginable usage of this salt water is for health spas and geothermal
energy. A prominent example for CO2 storage in a saline formation is the Sleipner Project
in the North Sea, cf. IPCC report [123], where CO2 is removed from the produced natural
gas and injected back underground.
The unknown geological properties and a missing infrastructure are disadvantages of saline
aquifers compared to the depleted hydrocarbon reservoirs. A further problem, which has
now gained more interest is the displacement of the saline water due to the increase in
reservoir pressure during the injection of CO2, e. g., Ott et al. [133]. The question hereby
is, if this displaced brine could contaminate neighbouring drinking water reservoirs.
The other storage options pictured in Figure 2.3, i. e., unmineable coal seems, enhanced
methane recovery from coal beds and storage in basalts, oil shales or cavities, are of minor
importance and will not be discussed at this point. More information to these storage
options is provided in the IPCC report [119].
2.2.2 Storage costs
It is obvious that CCS requires additional energy for the capture, transportation and
storage processes. Thus, the costs of the produced electricity increase, since the efficiency
of the energy production is reduced. The largest part of these additional costs stems from
the capture process that is technically the most complicated part of the three partial
processes. An overview of the range of total costs, the mitigated CO2 amount, and
the increased fuel requirement, is depicted in Table 2.4, both for a coal power plant
and a natural gas power plant. Therein, the avoided amount of CO2 is calculated by
comparing the emissions of a regular plant without CCS with the emissions of a plant
with capture and storage, which means that the energy requirements for capture and
storage are already included in the avoided emissions. The mitigation costs in Table 2.4
show that CCS systems can only become economically feasible, if the prices for carbon
emissions reach about 25−30 US$/tCO2, cf. [119], if regulatory mandates are established
14 Chapter 2: Carbon-dioxide capture and storage
Table 2.4: Total costs for electricity for a power plant with capture and geological storage in
case of a coal and a natural gas power plant. Data taken from the IPCC report [119].
pulverised coal natural gas
power plant power plant
increased fuel requirement [%] 24− 40 11− 22
CO2 avoided [%] 81− 88 83− 88
electricity cost increase [%] 43− 91 37− 85
mitigation cost [US$/tCO2 avoided] 30− 71 38− 91
for the use of CCS, e. g., on new facilities, or if direct or indirect financial support is
provided by the policy makers. Otherwise, CCS is not competitive with CO2-free energy
technologies, such as nuclear power or renewable energy sources. The former of these two,
at least in the opinion of the author, does not pose a reasonable solution to the global
warming problem, but initiates further difficulties, e. g., the storage of the nuclear waste.
2.2.3 Storage mechanisms
Prior to the discussion of risks and security of CO2 storage, the mechanisms of CO2 storage
must be understood. The migration of the CO2 is affected by the pressure difference
between the injection pressure and the reservoir pressure, the differences in density and
viscosity between the CO2 and the in situ fluids, and the formation heterogeneities. The
pressure difference depends on the rate of injection, the permeability and the geological
geometry of the reservoir.
In this context, different trapping mechanisms can be distinguished, cf. Figure 2.4, and
are introduced in the following. When injecting CO2 into the reservoir, it displaces the
in-situ formation fluids in the reservoir, such as saline water, and collects under the im-
permeable cap-rock layer due to barycentric forces. This is usually identified as structural
& stratigraphical trapping. The second trapping mechanism is called residual trapping and
refers the CO2 that is trapped in the pore space by capillary forces, cf. Class [36]. The
mobility7 contrast between the CO2 and water or oil can cause viscous fingering (natural
convection), intensively investigated by Homsy [86]. This is a positive effect, since it en-
hances the diffusion and dissolution processes of the CO2 in the formation fluids and, thus,
is defined as the third trapping mechanism, namely the solubility trapping. The CO2 will
not only interact with the fluids in place, but also with the porous rock. Adsorption and
finally mineralisation can guarantee a longterm fixation of the CO2 in the subsurface. In
particular, during the dissolution process in water, the CO2 forms acidic ions (hydrogen
carbonate , HCO−3 , and carbonate CO
2−
3 ), which react in the next step with the calcium,
magnesium, clays or feldspars of the surrounding rock matrix, forming stable carbonate or
7The mobility is usually defined as the ratio between the relative permeability and the shear viscosity
of a constituent.
2.2 What is CCS? 15
silicate minerals8. This so-called mineral trapping is the last and most stable mechanism.
All these trapping mechanisms are acting on different time scales, i. e., from the start of
the injection, the mainly active trapping mechanism changes, cf. Figure 2.4. This wide
0
1 10 100
100
1 000 10 000
T
ra
p
p
in
g
co
n
tr
ib
u
ti
o
n
[%
]
Time since injection stopped [years]
structural & stratigraphical
residual CO2
solubility
mineral
trapping
trapping
trapping
trapping
increasing storage security
Figure 2.4: Time dependence of different trapping mechanisms in CO2 storage, cf. IPCC report
[119]. The four distinguishable mechanisms replace each other over time, whereas the trapping
security increases as well.
range of important time scales severely increases the complexity of the system and, thus,
makes the understanding and modelling of all the involved processes, needed for a realistic
simulation of CO2 injection and storage, a task that is not easily achieved. Therefore,
numerical simulations usually concentrate only on specific times of the storage process,
i. e., either on the early times around the injection itself, or on the longterm behaviour
of the CO2 in the reservoir after the injection has ended. Here in this work, the focus is
restricted to the early times around injection, which are the most interesting in terms of
high pressures and temperature differences.
2.2.4 Storage safety
Safety issues are an important aspect in CCS. Risks arising from CO2 storage can be
classified into three groups:
8The specific dissolution processes of CO2 in water are:
CO2 + 2H2O⇄ HCO
−
3 +H3O
+ and HCO−3 +H2O⇄ CO
2−
3 +H3O
+ ,
and the mineralisation with, e. g., calcium ions Ca2+, reads:
Ca2+ +CO2−3 → CaCO3 ↓ .
16 Chapter 2: Carbon-dioxide capture and storage
1. Elevated gas phase concentrations in the shallow subsurface and near-surface envi-
ronment.
2. Dissolved CO2 in the groundwater.
3. Displacement of fluids by the injected CO2.
The first kind of risks evolves when CO2 still exists as a separate phase, i. e., it is not yet
dissolved into the brine and is hindered from upwards migration by the cap-rock layer. In
this early stage, the storage safety is not very high, since the CO2 can migrate through
faults, fractures, or poorly sealed wells towards the surface. In this regard, the crack
development in the cap-rock layer can be caused by seismic movements, or due to the
increasing pressure in the reservoir during the injection itself. In cases where the leaking
CO2 reaches the atmosphere, the risk of asphyxiation of humans and animals exists for
sudden and large amounts of CO2 releases or at conditions, where the surface topography
enables the accumulation of CO2 in shallow depressions due to its higher density than air.
However, this latter risk is rather small, since surface winds will dilute the leaking CO2
quickly. Figure 2.4 indicates how the continuous change in trapping mechanisms increases
the storage security over time. The residual trapping already improves the storage safety
by locking a part of the CO2 in capillary traps. When the CO2 dissolves into the brine,
the solution becomes heavier than the surrounding brine and sinks to the bottom of the
reservoir layer, away from the cap-rock layer. Subsequently, in this stage the CO2 is less
affected by potential damages of the cap-rock layer. The safest stage is reached when
most of the CO2 is minerally bound. However, it is supposed to take several thousands
of years until this last process is finished.
The second risk of CO2 dissolving into drinking water was already briefly mentioned
before. It describes the contamination of valuable drinking water by contaminants, either
brought into the subsurface together with the injected CO2 or by altering the groundwater
chemistry, i. e., the formation of carbonic acids when the CO2 dissolves into the drinking
water. This carbonated water might react with the rock matrix and mobilise possible
hazardous contaminants. Furthermore, concerning offshore storage sites, leaking CO2
might dissolve into the sea water, creating possible harmful environments for marine life.
Similar concerns regard the last group of risks. Hereby, the injected CO2 displaces the
saline water, which could then again migrate or leak into more shallow formations and
contaminate valuable drinking-water reservoirs.
Possible treatments or assessments of these risks imply the active reservoir management,
for example, by withdrawing of formation water to lower the pressure in the reservoir, and
of course detailed measurement, monitoring, and verification of injection data, in order
to quickly detect any leakage out of the reservoir layer.
Obviously, the prediction of these risks and the probability estimations are an important
part of the planning process of CCS. Besides the information from the few already existing
storage sites and geological measurements, numerical simulations can contribute a major
part to gain data for the risk assessment.
2.3 Numerical simulation of CO2 storage 17
2.3 Numerical simulation of CO2 storage
In order to predict the storage capacity, to estimate a suitable injection pressure, and
to classify the safety of a storage site, numerical simulations of the CO2 injection are of
great importance. Their advantage lies in the low costs and comparatively minor effort
in relation to expensive on-site tests. However, numerics are at the moment not able to
handle the full complexity of the storage process, e. g., regarding the wide span in time
and space scales, the multitude in physical and chemical effects and the heterogeneity of
the subsurface. Hence, reasonable assumptions and simplifications have to be made to
allow for meaningful simulation results within adequate simulation times.
2.3.1 State of the art
The numerical simulation of CO2 storage is relatively new, starting around the beginning
of the 21st century. Due to the close relation to the petroleum industry, it is only log-
ical that the first sequestration simulations were conducted using existing hydrocarbon
production simulators, mainly for the understanding of enhanced oil or gas recovery, cf.
Bielinski [15] and the references therein. Further early works originate from, e. g., Pruess
& Garc´ıa and co-workers [46, 141, 144], including the development of the well-known
FEM-simulator TOUGH2, or the group around Ennis-King [65], looking, for example, at
convective mixing in the long-term of the injection process.
Benchmark studies and code comparisons were initiated and executed by Pruess et al.
[142] and Class et al. [37]. These showed that most of the discrepancies in the results
between the numerical simulations conducted with different codes, originate from uncer-
tainties in material parameters, diverse definitions of the boundary conditions and errors
or various behaviour induced by the gridding.
Analytical or semi-analytical models were derived by, Kang et al. [93], Nordbotten et al.
[129], Celia et al. [34]. The latter two include these models in a quantitative estimation
of CO2 leakage from geological storage at leaky wells.
Since the vertical extent of a subsurface reservoir is usually much smaller than the lat-
eral extent, also the horizontal movement of CO2 is in many scenarios more important.
Therefore, so-called vertical-equilibrium models are developed to reduce numerical costs
and to allow for the simulation of models with a larger horizontal dimensions, cf. Court
et al. [40] and Gasda et al. [70].
The group around Nordbotten and Celia established a number of models to efficiently
simulate CO2 sequestration with regard to reliable predictions for policy makers. These
include the long-term evolution of the CO2, the leakage from geological storage sites, and
large-scale simulations, cf. e. g., [34, 127, 143]. Nordbotten et al. [128] also state that
simulations alone cannot give satisfactory answers to CO2 storage problems, but must be
complemented by monitoring and history matching9 during the injection process. In this
9The term history matching refers to the process of adjusting the parameters and boundary conditions
of a model by comparing the measured data with the data gained from numerical simulations for the
same time span.
18 Chapter 2: Carbon-dioxide capture and storage
context, one can also mention the works of Bradshaw et al. [28] and Kopp et al. [99] on
storage-capacity estimations.
It was already mentioned in the previous sections that the CO2 sequestration incorporates
many different processes and difficulties. Thus, a multiplicity of researches have been
conducted on specific topics to understand these processes. To give an impression of the
broadness of this field, a few shall be mentioned here: Juanes et al. [91] investigated the
impact of relative permeability on CO2 sequestration, the temperature dependencies of
capillary pressure in CCS were modelled by Plug & Bruining [136], the long-term storage
mechanism of mineralisation was, for example, considered by Kumar et al. [103], Ebigbo
modelled enhanced trapping mechanisms due to biofilm growth [47], and Rutqvist [154–
156] looked at the geomechanical impact of CO2 sequestration, especially, at one of the
presently three main injection sites at In Salah, Algeria.
This is by no means an exhaustive listing of all the work done in the field of CO2 seques-
tration, but shall give a brief overview. For a more extensive collection, the interested
reader is referred to, e. g., the IPCC report [119] or the dissertation by Bielinski [15].
2.3.2 Tasks and difficulties
Although extensive research has been conducted in CCS, open questions still remain.
These are related to all parts of the CCS process, e. g., improving the efficiency of the
CO2 capture, which still contributes most to the costs of CCS, cf. Bruckner et al. [32].
However, in the framework of this monograph, only questions concerning the storage of
CO2 are discussed. Basically, these are usually related to the operational safety and
long-term integrity of a CO2 storage site:
• the effects of the pressure build-up due to injection, which can cause various un-
wanted situations:
– uplift of the overlying rock, e. g., at the injection site In Salah, Algeria [156],
– leakage of the injected CO2 through cracks in the sealing cap-rock layer or
re-opened faults, induced by the high pressure,
– initiation of seismic events,
– far-field pressure effects, i. e., the displacement of the saline formation water,
which might lead to a contamination of nearby drinking water reservoirs.
• a sudden pressure increase in the reservoir, which can also be caused by a phase
change of the injected CO2 due to changing reservoir conditions (temperature or
pressure), causing the before mentioned situations. In particular, the transition
from the high density supercritical or liquid phase to the low density gaseous phase
has to be regarded in this context.
• the long-term integrity of wells, where leakage along the well-casing can occur if
the carbonic acid, originating from the dissolution of CO2 in the formation water,
harms the cement of the well-casing and makes it porous.
2.3 Numerical simulation of CO2 storage 19
A collection of further open questions is given, e. g., by Komarova [98].
These points represent the background for the following derivation of a multiphasic model,
which is able to handle phase-transition processes under various thermodynamical condi-
tions together with solid deformations. In this context, numerical models can definitely
help to understand the physical processes occurring in the reservoir. In combination with
geological and geophysical data, it should then become possible to form a judgment of
the suitability of a reservoir for CO2 sequestration.

Chapter 3: Theoretical fundamentals of
multiphasic and multicomponent modelling
This chapter introduces the theoretical fundamentals for the modelling of multiphasic
and multicomponent systems, e. g., CO2 and water percolating a porous rock. First of all,
the Theory of Porous Media (TPM) is outlined as the basic concept for the continuum-
mechanical description of miscible and immiscible constituents, including the definitions
of volume fractions and densities. Subsequently, kinematical relations are defined for the
superimposed constituents, passing on to presenting relevant deformation and strain mea-
sures. Furthermore, a brief overview of the stress state is given. It follows an elaborate
introduction of general balance relations and their adaptation to both, the overall aggre-
gate and the specific constituents. Finally, the concept of singular surfaces is provided,
together with corresponding kinematical and balance relations. This last part is a prereq-
uisite for the later description of the phase-transition process at the interface between the
two fluid phases. Therewith, a model of a thermo-elastic porous solid percolated by two
immiscible, compressible and interacting fluid phases is established later in Chapter 5.
3.1 Theory of Porous Media
For a detailed and realistic model of a CO2 injection and storage process, the multi-
phasic and multicomponent structure of the reservoir and the percolating fluids have to
be mathematically acknowledged. In this regard, a macroscopic approach embedded in
a continuum theory has proven to be a suitable model for such a complex and hetero-
geneous medium as subsurface rock. This was shown, for example, by the petroleum
industry and their simulations of oil and gas recovery, e. g., Coats et al. [38]. Moreover,
due to the scarce knowledge of the exact underground conditions, where only statistical
information can be obtained by geophysical measurements (e. g., seismic surveys, ground-
radar, core samples), such a homogenised macroscopic model provides a more reasonable
approach than microscopic models or singlephase material models. A comparison of the
macroscopic approach, i. e., the TPM, with the classical continuum-mechanical approach
of singlephasic materials is presented in Ehlers [56]. Furthermore, the TPM provides
a profound basement not only for the description of soil formations, but also for other
natural systems and materials, such as biological systems, cf. Ehlers [55].
In combining the classical Theory of Mixtures (e. g., Truesdell & Toupin [173] and Bowen
[25]) with the concept of volume fractions (cf. Biot [16], Bowen [26, 27] and Ehlers [56]),
the modelling of miscible constituents in a macroscopic model is enhanced with the mi-
croscopic information of the inner composition of the system. Thus, since the structure of
the pores is usually not known, a statistical homogenisation over a representative volume
element provides a basis for a continuum-mechanical description, cf. Figure 3.1. Formu-
lating the basic idea of the TPM, a multiphasic aggregate ϕ consisting of constituents ϕα
is statistically distributed over the representative elementary volume (REV), leading to a
21
22 Chapter 3: Theoretical fundamentals of multiphasic and multicomponent modelling
O
x
B
P
microscale
macroscale
solid material
fluid interface
non-wetting fluid
wetting fluid
homogenised model
concept of volume fractions
REV
dvS
dvG
dvL
dv
Figure 3.1: The point P of a porous sandstone, filled with two fluid phases ϕL and ϕG, is
located in the current configuration at position x. From the idealistic model of an REV on the
microscale the development of a macroscopic, multiphasic modelling approach follows, consisting
of the homogenised model and the concept of volume fractions.
model of superimposed and interacting continua
ϕ =
⋃
α
ϕα = ϕS ∪ ϕF , α ∈ {S, L, G} , and ϕF =
⋃
β
ϕβ , β ∈ {L, G} . (3.1)
Hereby, the porous solid constituent is indicated by S and is percolated by the fluid
phases F , which again can be subdivided into the liquid phase L and the gaseous phase
G. The constituents ϕα are usually identified as the differentiable parts which build
the multiphasic aggregate. It is not essential that these constituents just represent the
different phases, they can also be distinguished as the constituents of a solution. However,
in this monograph diffusion processes are neglected and, thus, it suffices to deal with a
classification of the constituents as phases.
The summation over the partial volumes of the immiscible phases V α provides the total
volume V of the overall aggregate B at its current configuration:
V =
∫
B
dv =
∑
α
V α . (3.2)
Herein, the partial volumes can be further split via
V α =
∫
B
nα(x) dv =:
∫
B
dvα → nα = dv
α
dv
, with
∑
α
nα = 1 , (3.3)
where nα is the volume fraction at a certain position x, defined as the ratio between the
volume element dvα of a constituent and the volume element dv of the overall aggregate.
Furthermore, it is assumed that the regarded system contains no vacant spaces, which is
guaranteed by the closure condition (3.3)3.
3.2 Kinematical relations 23
When considering multiphasic aggregates, where two fluids have different wetting be-
haviours, it is practical to introduce the saturation sβ , which is defined as the volume
fraction of the pore fluids ϕβ with respect to the pore space:
sβ =
nβ
nF
, where nF =
∑
β
nβ and
∑
β
sβ = 1 . (3.4)
Herein, the volume fraction of the pore space nF is also known as the so-called porosity.
Assuming that the individual phases are immiscible and occupy separate volumes within
the overall medium, two different densities can be defined:
ραR =
dmα
dvα
, ρα =
dmα
dv
and ρα = nα ραR . (3.5)
Therein, the effective (or realistic) density ραR represents the real material density of ϕα
at a local point, the partial density ρα relates the local mass to the bulk volume of the
overall porous medium and dmα is the local mass.
The relation between the two density definitions (3.5)3 implies the two possible reasons
for a change in the partial density ρα:
1. The volume fraction nα can change due to, e. g., deformations of the solid matrix,
explained later in the evaluation of the solid mass balance in Section 5.2.1.
2. The material can be mechanically compressible, i. e., ραR = ραR(pβR, θα), whereas
pβR is the specific pressure and θα is the absolute temperature of the constituent
ϕα.
3.2 Kinematical relations
The formulation of motion functions for a superimposed continuum are based on classi-
cal continuum mechanics of singlephasic materials, e. g., Altenbach [4] or Haupt [82]. A
complete overview of these for multiphasic materials is presented, e. g., by Ehlers [54, 56].
Here, only the required kinematical relations for a triphasic mixture of a thermo-elastic
solid material and two immiscible fluid phases are provided. For more complex kinemati-
cal formulations of material deformations, e. g., visco-plasticity and miscible components,
the interested reader is referred to, e. g., Avci [8] or Wagner [176].
3.2.1 Motion functions
Subsequently, a body B with its surface S is regarded that is composed of a multiphasic
mixture consisting of the three constituents ϕS, ϕL and ϕG, as illustrated in Figure 3.2.
Following the basic concept of the TPM, each constituent ϕα is represented by its own
motion function
x = χα(Xα, t) , (3.6)
24 Chapter 3: Theoretical fundamentals of multiphasic and multicomponent modelling
due to the superposition of the interacting continua over the REV. This implies that each
spatial point x is simultaneously occupied by material points Pα of all constituents ϕα, cf.
Figure 3.2. Assuming a unique motion function χα, each material point Pα in the current
PL
PG
PS PS , PL, PG
t0
t
B SB0
S0
χS(t)
χL(t)
χG(t)
O
XS
XL
XG
x
uS
Figure 3.2: Motion of a multiphasic body B with surface S between the reference configuration
(time t0) and the current configuration (time t). Motion functions χα connect the reference
positions Xα of a point ϕ
α to the current position x.
configuration at time t must have a unique reference position Xα at time t0. Because of
the requested uniqueness of these motion functions at all times t, Equation (3.6) can be
inverted to identify the reference position Xα at time t0:
Xα = χ
−1
α (x, t) , if Jα := det
∂χα(Xα, t)
∂Xα
6= 0 . (3.7)
The uniqueness of the inverse motion function requires the non-singularity of the Jacobian
determinant Jα, cf. (3.7)2. From (3.6) and (3.7) it can be concluded that each constituent
has its own velocity and acceleration fields, which are given in the Lagrangean description
as:
′
xα =
∂χα(Xα, t)
∂t
,
′′
xα =
∂2χα(Xα, t)
∂t2
. (3.8)
In Equation (3.8), the material time derivative ( · )′α was used
1, which follows the motion
of the constituent ϕα. Thus, due to the conventions made in the TPM, the total material
time derivative of an arbitrary, steady and sufficiently differentiable scalar function Υ or
vector function Υ depends on the motion of the overall aggregate ϕ
Υ˙ =
d
dt
Υ(x, t) =
∂Υ
∂t
+ gradΥ · x˙ ,
Υ˙ =
d
dt
Υ(x, t) =
∂Υ
∂t
+ (gradΥ) x˙ ,
(3.9)
or on the motion of the constituent ϕα
(Υ)′α =
dα
dt
Υ(x, t) =
∂Υ
∂t
+ gradΥ · ′xα ,
(Υ)′α =
dα
dt
Υ(x, t) =
∂Υ
∂t
+ (gradΥ)
′
xα .
(3.10)
1Throughout this monograph, ( · ) is used as a placeholder for either arbitrary scalar or vectorial
quantities, depending on the context of its use.
3.2 Kinematical relations 25
In the latter four equations, the differential operator grad ( · ) denotes the gradient with
respect to the current position x. This is in contrast to the partial derivative with respect
to the reference position Xα of the constituent ϕ
α, which is written as Grad α( · ). Finally,
the velocity of the overall medium can be derived from a mass average over all constituents
leading to the local barycentric velocity:
x˙ =
1
ρ
∑
α
ρα
′
xα , (3.11)
wherein, ˙( · ) is the material time derivative with respect to the overall aggregate ϕ. Due
to the fact that the constituents ϕα can move independently from the overall aggregate
ϕ, a diffusion velocity2 is introduced via
dα :=
′
xα − x˙ , with
∑
α
ρα dα =
∑
α
ρα
′
xα − x˙
∑
α
ρα = 0 . (3.12)
In continuum mechanics, the solid phase ϕS is usually described by a Lagrangean setting,
which leads to the introduction of the displacement vector uS as follows:
uS = x−XS . (3.13)
In contrast to the solid phase, it is more practical to use a modified Eulerian setting for
the pore fluids ϕβ. Therefore, the seepage velocity wβ is defined as
wβ :=
′
xβ − ′xS , (3.14)
describing the fluid motions with respect to the deforming solid matrix.
Due to the Lagrangean formulation of the continuum-mechanical model with respect to
the solid deformation vector uS, it is of great help to transform the material time deriva-
tives ( · )′β, to the motion of the solid constituent. The required transformation operator
is derived by subtracting (3.10)1 of the considered constituent from that of the solid
constituent and by further using (3.14), viz.:
( · )′β = ( · )
′
S + grad ( · ) ·wβ . (3.15)
3.2.2 Deformation and strain measures
The material deformation gradient Fα is introduced as the basic deformation measure,
which is essential in continuum mechanics to describe motion changes of the constituents
ϕα. Fα and its inverse form are given by
Fα =
∂χα(Xα, t)
∂Xα
=
∂x
∂Xα
= Gradαx ,
F−1α =
∂χ−1α (x, t)
∂x
=
∂Xα
∂x
= gradXα .
(3.16)
2This diffusion velocity must not be confused with the pore-diffusion velocity, which describes the
relative motion of a miscible component to the motion of the fluid mixture.
26 Chapter 3: Theoretical fundamentals of multiphasic and multicomponent modelling
The requirement of uniqueness of the motion function χα was already mentioned in (3.7),
and it follows that the Jacobian is restricted to positive values, viz.:
Jα = detFα > 0 if detFα(t0) = 1 , for Fα(t0) =
∂Xα
∂Xα
= GradαXα = I , (3.17)
wherein I is the second order identity tensor, and the definition of the determinant detFα
is provided in Appendix A. With this at hand, the mapping characteristic of the material
deformation gradient can be applied to do a push-forward transformation of a local line
element from the reference configuration into the current configuration via
dx = Fα dXα ↔ dXα = F−1α dx , (3.18)
where also the pull-back transformation, i. e., from the current to the reference configura-
tion, is presented and requires the existence of the inverse deformation gradient (3.16)2.
In an analogue way, the mapping can also be adopted to area and volume elements
da = (cofFα) dAα and dv = (detFα) dVα , (3.19)
where the referential configuration quantities dAα and dVα of the area and volume ele-
ments are mapped to the current configuration quantities da and dv, respectively. The
vectorial form of the area elements originates from a definite surface orientation, indicated
in the following by the normal vector n for the current configuration and the normal vec-
tor m for the referential configuration. Thus, the vectorial area elements can be written
as da = n da and dAα = mα dAα. The calculation of the cofactor cofFα, appearing in
(3.19), is given in Appendix A.
So far, the formulation of the deformation gradient is valid for all kinds of constituents,
such as solid, liquid or gas. However, since the fluid motion is described relative to
the deforming solid matrix (i. e., motions of the pore fluids ϕβ are given in a modified
Eulerian setting), the fluid deformation gradients Fβ are not needed in the following, but
are indirectly included by the seepage velocity wβ (3.14). Hence, the following discussions
of the finite kinematical relations are presented only for the solid constituent. By taking
the solid displacement vector uS as the primary variable for the solid constituent, it is
customary to write the solid deformation tensor as
FS =
∂(XS + uS)
∂XS
= I+GradS uS , (3.20)
where (3.6) and (3.13) were applied. The inverse of the solid material deformation gradient
then reads
F−1S =
∂XS
∂x
= gradXS = I− graduS . (3.21)
Next, deformation measures are derived from the square of the line elements (3.18) in
both, referential and current configurations:
dx · dx = dXS · (FTS FS) dXS =: dXS ·CS dXS → CS = FTS FS ,
dXS · dXS = dx · (FT−1S F−1S︸ ︷︷ ︸
(FS F
T
S )
−1
) dx =: dx ·BS dx → BS = FS FTS . (3.22)
3.2 Kinematical relations 27
Therein, the right, CS, and the left, BS, Cauchy -Green deformation tensor of the solid ma-
trix were introduced. Furthermore, strain measures are found by subtracting the squares
of the current and referential line elements:
dx · dx− dXS · dXS = dXS ·CS dXS − dXS · dXS =
= dXS · (CS − I) dXS → ES = 12 (CS − I) ,
dx · dx− dXS · dXS = dx · dx− dx ·B−1S dx =
= dx · (I−B−1S ) dx → AS = 12 (I−B−1S )
(3.23)
with the Green -Lagrangean strain tensor ES and the Almansian strain tensor AS. The
factor 1
2
dates back to the common convention that the strain tensor should be reducible to
the one-dimensional Hookean elasticity law in case of linearisation. The Green -Lagrangean
and the Almansian strain tensor can be converted into each other by AS = F
T−1
S ES F
−1
S .
Further existing strain measures are not important in the context of this monograph, but
can be found in, e. g., Truesdell & Noll [172], or Ogden [131], also cf. Ehlers [50].
3.2.3 Deformation rates and velocity gradient
To describe the temporal change of the previously established deformation and strain
measures, the material velocity gradient (Fα)
′
α is introduced as the deformation rate
relating to the reference state via
(Fα)
′
α =
dα
dt
Fα =
∂
′
xα(Xα, t)
∂Xα
= GradS
′
xα (3.24)
and the spatial velocity gradient Lα of the current state:
Lα = (Fα)
′
αF
−1
α = grad
′
xα and Lα · I = div ′xα . (3.25)
The spatial velocity gradient can be additively decomposed into its symmetric part Dα
and its skew-symmetric part Wα, as shown in the following:
Lα =
1
2
(Lα + L
T
α) +
1
2
(Lα − LTα) = Lsymα + Lskwα =: Dα +Wα . (3.26)
3.2.4 Stress measures
The deformations and strains discussed before are caused by forces k acting on the body
B. These total forces can also be given separately for each constituent ϕα via kα and split
into external forces kα
F
acting from a distance onto the volume of the body B and into
contact forces kαN acting at the near vicinity on the surface of the body S:
kα = kα
F
+ kα
N
=
∫
B
fα dv +
∫
S
tα da . (3.27)
28 Chapter 3: Theoretical fundamentals of multiphasic and multicomponent modelling
Hereby, fα is the constituent- and volume-specific force acting from a distance and tα
is the contact force per surface area acting on the constituent ϕα. With the a priori
assumption that the external body force fα is proportional to the partial density ρα for
all constituents ϕα and with the usual interpretation of the body force as the gravitation3
g, it follows
fα =: ρα bα = ρα g , (3.28)
wherein, bα is the external volume force per mass unit.
The contact force per surface area tα = tα(x, t, n) is also called the traction and is a
function of the current position x, the time t and of the outward-oriented, normal surface
vector n, which was already introduced in (3.19). In order to find a stress measure that
is independent of this surface normal vector, Cauchy’s theorem is applied and yields
tα(x, t, n) = [Tα(x, t)]n . (3.29)
The stress measure Tα is the partial Cauchy stress tensor of the constituent ϕα, also
known as the true stress tensor. Proceeding from there, the surface force element dkα
N
can be written as
dkα
N
= tα da = Tα n da = Tα da . (3.30)
Up to this point, the stress measures were completely formulated in the current configura-
tion. Naturally, different alternative stress measures can be found by transporting both,
the stress tensor itself and the area element back into the referential configuration. The
sequential pull-back transport of the contact force then leads to the following expressions:
Tα da = (detFα)T
α (detFα)
−1 da =: τ α da¯α =
= (detFα)T
αFT−1α (cofFα)
−1 da =: Pα dAα =
= (detFα)F
−1
α T
αFT−1α (cofFα)
−1 da =: Sα dAα .
(3.31)
The Kirchhoff stress tensor τα = (detFα)T
α is derived by applying (3.19)2 in order
to weight the area element da in the current configuration with the volumetric change,
via da¯α =
dVα
dv
da. The pull-back transformation of the area element da to the referen-
tial configuration by dAα = (cofFα)
−1 da leads to the first Piola-Kirchhoff stress ten-
sor Pα = (detFα)T
αFT−1α . This two-field tensor relates forces specified in the current
configuration to geometrical quantities given in the referential configuration. A further
pull-back transport of the first Piola-Kirchhoff stress tensor finally results in the second
Piola-Kirchhoff stress tensor Sα = F−1α P
α, which completely lives in the referential config-
uration. For completeness, the relations between the Kirchhoff and both Piola-Kirchhoff
stress tensors are deduced from (3.31) and read:
Pα = τ αFT−1α and τ
α = PαFTα ,
Sα = F−1α τ
αFT−1α and τ
α = Fα S
αFTα .
(3.32)
3Other external forces are, for example magnetism.
3.3 Balance relations 29
Throughout the present work, the assumption of small deformations is made. Thus, a
geometrically linear theory is exerted with Fα ≈ I, for both, push-forward and pull-back
operations. Hence, the above developed relations between the stress tensors show that
in the linear case, the current and the referential configurations of the stress tensors are
approximately equal and are given by the linear stress tensor σα, viz.:
Tα ≈ τα ≈ Pα ≈ Sα ≈: σα . (3.33)
3.3 Balance relations
Balance relations are the basic instrument to describe the change of a physical quantity
over time. In this fundamental principle, the temporal change of the physical quantity in a
body B is balanced with the flow (influx or efflux) over the body’s surface S, the external
source (supply) and the production of the physical quantity inside the body B, cf. Figure
3.2. It is possible to integrate all balance relations into a master balance formulation,
from which the particular balance relations can be axiomatically determined by inserting
the respective quantities. Since this procedure has already been discussed in a number
of contributions, e. g., Ehlers [54, 55] and citations therein, only a brief summary is given
here.
In case of multi-phasic continua, which are the subject of this monograph, the formulation
of the balance relations follows the works on the classical Theory of Mixtures, e. g., by
Bowen [24, 25], Kelly [97], Truesdell & Toupin [173], and is especially based on the three
well-known metaphysical principles, introduced by Truesdell [171]:✬
✫
✩
✪
1. All properties of the mixture must be mathematical consequences
of properties of the constituents.
2. So as to describe the motion of a constituent, we may in imagination
isolate it from the rest of the mixture, provided we allow properly
for the actions of the other constituents upon it.
3. The motion of the mixture is governed by the same equations as is
a single body.
These three arguments state that the balance relations of the mixture result from the
summation of the balance relations of the constituents. Thus, firstly, the global master
balance relations for the constituents shall be presented here, according to Ehlers [52],
both for scalar Ψα and vectorial quantities Ψα, respectively:
dα
dt
∫
B
Ψα dv =
∫
S
φα · n da +
∫
B
σα dv +
∫
B
Ψˆα dv ,
dα
dt
∫
B
Ψα dv =
∫
S
Φα n da +
∫
B
σα dv +
∫
B
Ψˆ
α
dv .
(3.34)
30 Chapter 3: Theoretical fundamentals of multiphasic and multicomponent modelling
The physical quantities in (3.34) are itemised in the following: Ψα andΨα are the volume-
specific densities of the physical quantity to be balanced, σα, σα are the volume densities of
the supply acting on the body (external source) and Ψˆα, Ψˆ
α
represent the total production
due to interactions between the constituents ϕα (intrinsic supply). Furthermore, φα, Φα
are the surface densities, describing the efflux of the physical quantity at the outer surface
S of B. All these quantities are identified for the respective balance relations according to
Table 3.1. The coefficients in Table 3.1 that have not been explained yet are: the Cauchy
Table 3.1: Specific physical quantities of the modified constituent master balances.
Balance Ψα, Ψα φα, Φα σα, σα Ψˆα, Ψˆ
α
mass ρα 0 0 ρˆα
momentum ρα
′
xα T
α ρα bα sˆα
m.o.m. x× (ρα ′xα) x×Tα x× (ρα bα) hˆα
energy ρα (εα + 1
2
′
xα · ′xα) (Tα)T ′xα − qα ρα ( ′xα · bα + rα) eˆα
entropy ρα ηα φαη σ
α
η ηˆ
α
stress tensor Tα, the body force bα, the internal energy εα, the heat influx qα, the external
heat supply rα, the entropy ηα, the entropy efflux φα, the external entropy supply σαη and
the total production terms for mass ρˆα, momentum sˆα, moment of momentum hˆ
α
, energy
eˆα and entropy ηˆα. Hereby, the total production terms follow the restrictions:∑
α
ρˆα = 0 ,
∑
α
sˆα = 0 ,
∑
α
hˆ
α
= 0 ,
∑
α
eˆα = 0 ,
∑
α
ηˆα ≥ 0 . (3.35)
Therewith, it is guaranteed that the summation over all respective constituent balances
results in the relations of the overall aggregate. The mass production ρˆα can be understood
as the mass that is transferred from one constituent to the other, either due to chemical
reactions, or due to phase-transition processes, whereas this work is only interested in the
latter one. In mixture theories, it is customary to split the total production terms into
direct parts and parts including productions terms from the lower balance relations, viz.:
sˆα = pˆα + ρˆα
′
xα ,
hˆ
α
= mˆα + x× sˆα ,
eˆα = εˆα + pˆα · ′xα + ρˆα (εα + 12
′
xα · ′xα) ,
ηˆα = ζˆα + ρˆα ηα .
(3.36)
Herein, pˆα denotes the direct momentum production, mˆα corresponds to the direct mo-
ment of momentum production, εˆα is the direct part of the energy production and ζˆα is
the direct entropy production.
The global representation of the master balances in (3.34) can be transferred into the
3.3 Balance relations 31
local form by presuming the steadiness and the steady differentiability of (3.34):
(Ψα)′α + Ψ
α div
′
xα = divφ
α + σα + Ψˆα ,
(Ψα)′α + Ψ
α div
′
xα = divΦ
α + σα + Ψˆ
α
.
(3.37)
Note in passing that (3.37) is valid at each material point Pα of the body B.
Following the first and third metaphysical principles posted by Truesdell, the summation
of the constituent balance relations leads to the mixture balance relations, where the
global form reads:
d
dt
∫
B
Ψdv =
∫
S
φ · n da +
∫
B
σ dv +
∫
B
Ψˆ dv ,
d
dt
∫
B
Ψ dv =
∫
S
Φnda +
∫
B
σ dv +
∫
B
Ψˆ dv .
(3.38)
The physical quantities in (3.38) have the same meaning as in the constituent balance
relations, cf. Table 3.2. However, production terms cannot be encountered in the mixture
Table 3.2: Specific physical quantities of the modified mixture master balances.
Balance Ψ, Ψ φ, Φ σ, σ Ψˆ, Ψˆ
mass ρ 0 0 0
momentum ρ x˙ T ρ x˙ 0
m.o.m. x× (ρ x˙) x×T x× (ρb) 0
energy ρ (ε+ 1
2
x˙ · x˙) (T)T x˙− q ρ (x˙ · b+ r) 0
entropy ρ η φη ση ηˆ ≥ 0
balance relations for the fact of (3.35), except for the total entropy production ηˆ.
In analogy to the local form of the constituent balance relation (3.37), the local mixture
balance relation follows as
Ψ˙ + Ψdiv x˙ = divφ + σ + Ψˆ ,
Ψ˙ + Ψdiv x˙ = divΦ + σ + Ψˆ ,
(3.39)
wherein ˙( · ) represents the material time derivative d/dt of the respective quantity. By
comparison of the constituent formulations with the mixture formulations, the result of
Truesdell’s first principle becomes apparent in each field of the balance relations:
Ψ =
∑
α
Ψα , φ · n =
∑
α
(φα −Ψα dα) · n , σ =
∑
α
σα , Ψˆ =
∑
α
Ψˆα ,
Ψ =
∑
α
Ψα , Φn =
∑
α
(Φα −Ψα ⊗ dα)n , σ =
∑
α
σα , Ψˆ =
∑
α
Ψˆ
α
.
(3.40)
32 Chapter 3: Theoretical fundamentals of multiphasic and multicomponent modelling
3.3.1 Specific balance equations
The specific balance equations for both, the mixture ϕ and the constituents ϕα, are
found by inserting the physical quantities of the Tables 3.1 and 3.2 into the respective
master balance equations (3.37) and (3.39), in this case in the local form. Since the
balance equations fashion a hierarchical structure, where “higher” equations build upon
the “lower” ones, the lowest relation should be mentioned first.
Mass balance
The mass balance states that in a closed system the mass of the overall aggregate ϕ or of
a constituent ϕα must always remain constant:
mixture : ρ˙ + ρ div x˙ = 0 ,
constituent : (ρα)′α + ρ
α div
′
xα = ρˆ
α .
(3.41)
The summation of (3.41)2 over all constituents ϕ
α, while using relations (3.9), (3.11) and
(3.35)1, yields (3.41)1, which verifies Truesdell’s first principle.
Momentum balance
The mixture and constituent formulations of the momentum balance follow next in the
sequence of balance relations:
mixture : ρ x¨ = divT + ρb ,
constituent : ρα
′′
xα = divT
α + ρα bα + pˆα .
(3.42)
The momentum balance is essential to describe the motions of the mixture or constituents,
respectively.
Moment of momentum balance
The balance of moment of momentum is given as
mixture : 0 = I × T ,
constituent : 0 = I × Tα + mˆα .
(3.43)
Thereof, the relation for the overall aggregate (3.43)1 can be rewritten into T = T
T ,
which represents the symmetry of the overall Cauchy stress tensor. In case of non-polar
materials (Cauchy or Boltzmann continua) that exhibit symmetric stresses on the mi-
croscale, the symmetry can be shown for the macroscopic stresses as well, cf. Ehlers [54]
and Hassanizadeh & Gray [79]. Thus, the moment of momentum balance of a constituent
(3.43)2 becomes for non-polar materials
Tα = (Tα)T → mˆα ≡ 0 . (3.44)
3.3 Balance relations 33
Since this monograph deals only with non-polar materials, it shall only be mentioned here
that the polar theory is applied for so-called Cosserat continua and more information on
this topic can be found in, e. g., Scholz [159], Diebels [44, 45] and Ehlers [54].
Energy balance
The next balance relation is the energy balance, also known as the first law of thermo-
dynamics. The energy balance plays a major role in governing heat-transport processes
and relating the different storage forms of energy, i. e., the internal energy, the kinetic
energy, the external mechanical power and the non-mechanical power. Therefore, it is of
special interest in this monograph, concerning the main topic of fluid-phase transition.
The formulations of the energy balances for the mixture and the specific constituents
read:
mixture : ρ ε˙ = T · L − divq + ρ r ,
constituent : ρα (εα)′α = T
α · Lα − divqα + ρα rα + εˆα .
(3.45)
Entropy balance
Finally, the entropy balance is given as
mixture : ρ η˙ = − div
(q
θ
)
+
ρ r
θ
+ ηˆ ,
constituent : ρα (ηα)′α = − div
(qα
θα
)
+
ρα rα
θα
+ ζˆα ,
(3.46)
wherein, the entropy efflux φη and the entropy supply ση were replaced based on a priori
constitutive assumptions, which can be found in Ehlers [52–56] and citations therein, and
are formulated as
φη = −
q
θ
and ση =
ρ r
θ
, or φαη = −
qα
θα
and σαη =
ρα rα
θα
(3.47)
with θ and θα being the mixture and constituent temperatures, respectively. The sum-
mation of (3.46)2 over all constituents ϕ
α together with the restriction of the entropy
production (3.35)5, demonstrates that the total entropy production ηˆ is never negative,
cf. Table 3.2:
ηˆ =
∑
α
ηˆα =
∑
α
[
ρα (ηα)′α + ρˆ
α ηα + div
(qα
θα
)
− ρ
α rα
θα
]
≥ 0 . (3.48)
This means that the entropy of a closed system can only increase, but never decrease,
which is mathematically captured in the famous second law of thermodynamics. Since the
entropy is a physical quantity which is hardly determinable, a better applicable relation
can be derived by incorporating both the energy balance (3.45)2 and the definition of the
Helmholtz free energy by the Legendre transformation (B.5),
ψα := εα − θα ηα , (3.49)
34 Chapter 3: Theoretical fundamentals of multiphasic and multicomponent modelling
into (3.48), yielding the so-called inequality
∑
α
1
θα
{
Tα · Lα − ρα [ (ψα)′α + (θα)′α ηα ]− pˆα ·
′
xα−
− ρˆα (ψα + 1
2
′
xα · ′xα)− q
α
θα
· grad θα + eˆα
}
≥ 0 .
(3.50)
The formulations of constitutive relations for the mechanical and thermodynamical be-
haviour of multiphasic materials, which will be presented in Chapter 5, starts from evalu-
ating this Clausius-Duhem entropy inequality to ensure the thermodynamical consistency
of the formulation.
The balance relations in (3.41), (3.42), (3.43), (3.45) and (3.46) have been presented both
for the overall aggregate ϕ and for the individual constituents ϕα. The two formulations
are linked together by the sum over all constituents with respect to the barycentric motion
of the overall aggregate, shown already in general in (3.40) and which dictates the usage
of (3.11) and (3.12). The resulting restrictions between the mixture and constituent
formulations are:
ρ =
∑
α
ρα , ρb =
∑
α
ρα bα ,
ρ x˙ =
∑
α
ρα
′
xα , ρ x¨ =
∑
α
[ ρα
′′
xα − div (ρα dα ⊗ dα) + ρˆα ′xα ] ,
T =
∑
α
(Tα − ρα dα ⊗ dα) , q =
∑
α
[qα − (Tα)T dα + ρα εα dα + 12 (dα · dα)dα ] ,
ρ r =
∑
α
ρα (rα + bα · dα) , ρ ε =
∑
α
ρα (εα + 1
2
dα · dα) .
(3.51)
3.4 Singular surfaces
In Section 5.4.2 a constitutive relation for the mass production term ρˆα is developed based
on the microscopic behaviour at the interface between the liquid and gaseous phases. This
interface is mathematically described by a singular surface Γ, where discontinuities or
jumps exist in the physical quantities. For this purpose, the balance relations introduced
in Section 3.3 have to be adapted to include possible discontinuities. These are formulated
here in a general form according to the work of Mahnkopf [113], who applied singular
surfaces for the description of shear-band localisation within the TPM and derives the
required compatibility conditions based on Hadamards Lemma [78]. To avoid too much
repetition of the work done by Mahnkopf, only a brief summary of the additional changes
in the kinematics and balance relations is presented here.
3.4 Singular surfaces 35
3.4.1 Kinematics of a body with a singular surface
The derivation of the kinematical relations for a singular surface is oriented at the works of
Truesdell & Toupin [173] and Kosinski [100], and described here within the TPM for mul-
tiphasic continua, based on, e. g., Markert et al. [117] and Ehlers & Ha¨berle [61]. Consider
a triphasic aggregate B = ⋃α Bα with boundary surface S, where α = {S, L, G}. By
introducing a separating and immaterial smooth and local surface indicating the interface
Γ, the body B is locally separated into two parts given by B+ and B−, cf. Figure 3.3. The
nΓn
+
Γ
n−Γ
n−
n+
Γ
Γ
+
−
BS
B+
B−
S+
S−O
xΓ
Figure 3.3: Local microstructural interface Γ dividing the body B into the partitions B+ and
B−.
local body itself and its total surface are then given by B = B+ ∪ B− and S = S+ ∪ S−,
whereas S± ∪ Γ yields the entire surface of the body parts B±.
In this regard, a scalar-valued function Ψ(x, t), which is continuous in B+ and B−, and
which experiences a jump of Ψ over the interface Γ, is defined as the difference between
its values in B+ and B−, viz.: JΨ K := Ψ+ −Ψ− , (3.52)
wherein the jump operator J · K is introduced. The orientation of Γ at B+ and B− is given
by the outward-oriented surface normals n+Γ and n
−
Γ yielding
n+Γ = −n−Γ , where n+Γ =: nΓ , n−Γ = −nΓ . (3.53)
Regarding the kinematics of the singular surface shown in Figure 3.3, the inner surface Γ
with the unit normal vector nΓ pointing from B+ to B−, is allowed to propagate through
B and its velocity, as well as the relative velocity of a constituent ϕα with respect to the
moving interface Γ, are defined as
′
xΓ= vΓ , wαΓ =
′
xα− ′xΓ . (3.54)
3.4.2 Balance relations for a body with a singular surface
In case that B is intersected by a singular surface Γ, one has to derive balance equations
for the partitions B+ and B− with the respective external surfaces S(B+) = S+ ∪ Γ and
S(B−) = S− ∪ Γ. The derivation starts from the global master balance relations (3.34),
which are still valid for the partitions B+ and B−, if the partitions do not contain any
36 Chapter 3: Theoretical fundamentals of multiphasic and multicomponent modelling
further discontinuous surfaces themselves. Thus, the respective master balances for the
two partitions read, cf. right picture in Figure 3.3:
in B− : dα
dt
∫
B−
Ψα dv =
∫
S−
φα · n− da+
∫
Γ
(φα)− · n−Γ da +
∫
B−
(σα + Ψˆα) dv ,
in B+ : dα
dt
∫
B+
Ψα dv =
∫
S+
φα · n+ da+
∫
Γ
(φα)+ · n+Γ da +
∫
B+
(σα + Ψˆα) dv ,
(3.55)
provided that the field functions Ψα, φα and σα are continuous in B− and B+, as well as
on the surface Γ. Please notice in passing that in (3.55) the surface integral for the flux
φα ·n has been split into a part for the outer boundary of the partition (S− and S+) and
a part for the interface Γ. The summation of the balances of the two partitions (3.55)1,2
yields the global balance relation for a body containing a singular surface Γ:
dα
dt
∫
B
Ψα dv =
∫
B\Γ
(σα + Ψˆα) dv +
∫
S
φα · n da +
∫
Γ
Jφα K · nΓ da . (3.56)
In an analogous way, this form can be derived for vectorial physical quantities Ψα, Φα
and σα:
dα
dt
∫
B
Ψα dv =
∫
B\Γ
(σα + Ψˆ
α
) dv +
∫
S
Φα n da+
∫
Γ
JΦα K nΓ da . (3.57)
Proceeding to the local form of the balance relations, in a first step, the modified Reynolds
transport theorem has to be deduced, following Mahnkopf [113]:
dα
dt
∫
B
Ψα dv =
∫
B\Γ
∂Ψα
∂t
dv +
∫
S
Ψα
′
xα · n da+
∫
Γ
r
Ψα (
′
xα − ′xΓ)
z
· nΓ da ,
dα
dt
∫
B
Ψα dv =
∫
B\Γ
∂Ψα
∂t
dv +
∫
S
(Ψα ⊗ ′xα)n da+
∫
Γ
r
Ψα ⊗ ( ′xα − ′xΓ)
z
nΓ da .
(3.58)
Herein, the left side stands for the material change of the field function Ψα in relation to
a material and timely variable volume element dv. This equals on the right side the local
change of Ψα based on a fixed (in time and space) volume element dv, the flux of Ψα over
the constant surface element da and the inner flux of Ψα over the interface Γ for
′
xα 6= ′xΓ.
Furthermore, the formulation for a vectorial physical quantity Ψα is provided in (3.58)2.
Next, the insertion of the modified Reynolds transport theorem (3.58)1 into the global
3.4 Singular surfaces 37
balance relations (3.56) and (3.57) yields:∫
B\Γ
∂Ψα
∂t
dv +
∫
S
Ψα
′
xα · n da+
∫
Γ
r
Ψα (
′
xα − ′xΓ)
z
· nΓ da =
=
∫
B\Γ
(σα + Ψˆα) dv +
∫
S
φα · n da+
∫
Γ
Jφα K · nΓ da ,
∫
B\Γ
∂Ψα
∂t
dv +
∫
S
(Ψα ⊗ ′xα)n da+
∫
Γ
r
Ψα ⊗ ( ′xα − ′xΓ)
z
nΓ da =
=
∫
B\Γ
(σα + Ψˆ
α
) dv +
∫
S
Φα nda +
∫
Γ
JΦα K nΓ da .
(3.59)
Applying the Gauß ian integral theorem to the integrals over the outer surface S, and using
the relations for the total material time derivative (3.10) and for the relative interface
velocity wαΓ (3.54)2, it follows from (3.59), also compare, e. g., Alts and Hutter [5]:∫
B\Γ
[ (Ψα)′α +Ψ
α div
′
xα ] dv +
∫
Γ
JΨαwαΓ K · nΓ da =
∫
B\Γ
(divφα + σα + Ψˆα) dv+
+
∫
Γ
Jφα K · nΓ da ,∫
B\Γ
[ (Ψα)′α +Ψ
α div
′
xα ] dv +
∫
Γ
JΨα ⊗wαΓ K nΓ da =
∫
B\Γ
(divΦα + σα + Ψˆα) dv+
+
∫
Γ
JΦα K nΓ) da .
(3.60)
Since these equations have to hold at the same time as (3.34), one obtains after sorting
the integral terms for the body B and the singular surface Γ:
(Ψα)′α +Ψ
α div
′
xα = divφ
α + σα + Ψˆα ,
(Ψα)′α +Ψ
α div
′
xα = divΦ
α + σα + Ψˆ
α
 ∀ x ∈ B\Γ , (3.61)
and JΨαwαΓ − φα K · nΓ = 0 ,
JΨα ⊗wαΓ −Φα K nΓ = 0
 ∀ x = xΓ ∈ Γ (3.62)
substituting (3.37). Note that the local balances (3.61) are unchanged in comparison
with (3.37), but are accompanied by jump conditions (3.62), describing the jump of the
physical quantities across Γ.
Combining (3.61) and (3.62) with the specific physical quantities of Table 3.1, one obtains
the local balance equations and jump conditions for mass, momentum and energy:
38 Chapter 3: Theoretical fundamentals of multiphasic and multicomponent modelling
• mass:
(ρα)′α + ρ
α div
′
xα = ρˆ
α ∀ x ∈ B\Γ ,
J ραwαΓ K · nΓ = 0 ∀ x = xΓ ∈ Γ ,
(3.63)
• momentum:
ρα
′′
xα = divT
α + ρα b+ pˆα ∀ x ∈ B\Γ ,r
ρα
′
xα ⊗wαΓ −Tα
z
nΓ = 0 ∀ x = xΓ ∈ Γ ,
(3.64)
• energy:
ρα (εα)′α = T
α · Lα − divqα + ρα rα + εˆα ∀ x ∈ B\Γ ,r
ρα (εα + 1
2
′
xα · ′xα)wαΓ − (Tα)T ′xα + qα
z
· nΓ = 0 ∀ x = xΓ ∈ Γ .
(3.65)
These are the final local forms of the balance equations, which will be evaluated later in
Chapter 5 to derive constitutive relations, e. g., for the mass production term ρˆα, and to
formulate the governing balance relations.
Therewith, the theoretical fundamentals are completed and one can proceed to the next
Chapter 4, where the thermodynamics of fluids are examined more closely.
Chapter 4: Thermodynamic theory of fluids
Obviously, the main focus of this work, i. e., the phase transition of fluids, can only be
handled by understanding the thermodynamic theory of fluids. The physical properties
of gases and liquids are dependent on the physical interactions between the molecules of a
substance. Since the direct modelling of this molecular behaviour is difficult and too ex-
pensive for an engineering approach in large systems, the thermodynamic properties shall
be derived here by correlations and semi-empirical functions, i. e., the so-called equations
of state (EOS).
In this context, the probably most well-known correlation is based on Avogadro’s hypoth-
esis [9]:
✎
✍
☞
✌
Equal volumes of all gases, at the same temperature and pressure, have
the same number of molecules.
This correlation between the three physical properties effective fluid pressure pβR, effective
fluid volume V βR, and temperature θβ, is mathematically formulated in the ideal-gas law:
pβR V βR = nβmRθ
β , (4.1)
where nβm is the number of moles and R = 8.314 J/molK is the universal gas constant.
Hence, the pβR-V βR-θβ-relation defines the thermodynamical conditions. As the name
of the ideal gas law already suggests, it is only valid for ideal gases. An ideal gas is
distinguished by randomly moving, point-like molecules, which only interact with each
other by elastic collisions. To describe real gases and also liquids, this equation has to
be extended, which will be presented in Section 4.1 together with a detailed discussion of
the phase-transition process from the thermodynamical point of view.
In the subsequent sections, the relations for further thermodynamic properties are intro-
duced. In particular, these are the vaporisation enthalpy, the specific heat capacity, the
shear viscosity and the thermal conductivity. These properties are required for a realistic
model of non-isothermal fluid flow through porous media.
Although the main topic of this monograph is the CO2 sequestration and the thermody-
namical behaviour of this material under changing reservoir conditions, it is intended in
the following, to formulate the model and the thermodynamic theory not specifically for
CO2, but in general for arbitrary substances. Hence, the goal is to develop a model that
is capable of describing different substances by only changing the respective parameters
for the fluid in question, e. g., CO2, or water, and so on.
39
40 Chapter 4: Thermodynamic theory of fluids
4.1 Phase behaviour of a single substance
Each substance can inherit different physical states, i. e. solid, liquid, gas or supercritical1.
These states are defined by the pressure pβR, the temperature θβ and the specific volume
vβR (or the density ρβR as the inverse form). The relationship between these variables and
how they define the different states or phases of a substance are visualised in the phase-
diagram in Figure 4.1. The regions denoted by S, L and G represent the solid, liquid and
θβ
pβR
ρβR
ρβR = ρβR
crit
pβR = pβR
crit
θβ =
θ
β
crit
vβR
S
SG
SC
L
G
LG
C
T
Figure 4.1: 3-d phase diagram of the pβR-θβ-ρβR (or -vβR) relationship. The dashed lines
indicate the lines of constant critical values of the three variables. The character abbreviations
stand for the regions with specific phase behaviour: S: solid, L: liquid, G: gas, SG: solid-gas,
LG: liquid-gas, SC: supercritical, C: critical point and T : triple point.
gaseous phases, respectively. Below the blue line, which is called the vaporisation curve,
are the two-phase regions LG and SG, where two phases, liquid and gas or solid and
gas, coexist. The vapour-pressure curve comprises also the critical point C, which marks
the end of the two-phase region for increasing pressures, as well as the triple point T ,
that is the point of three coexisting phases solid, liquid and gas. The supercritical region
SC is reached for pressures and temperatures above the critical values of the respective
material. In Figure 4.1, the thin black lines and the dashed lines mark lines of constant
variables, whereas the latter stand for critical values of pressure pβRcrit, temperature θ
β
crit
and density ρβRcrit. Please note in passing that the critical isotherm θ
β
crit exhibits a saddle
point at the critical point C.
A projection of the 3-d phase diagram into the pressure-temperature plane reveals the 2-d
phase diagram in Figure 4.2(a), showing the vaporisation curve (blue). During the phase
transition from liquid to gas, under isothermal conditions and driven by a decreasing
pressure, the actual phase transition appears in the pβR-θβ-diagram, Figure 4.2(a), as a
1Sometimes, also plasma is regarded as a separate physical state. Plasma is generated by heating
gas or applying a strong electromagnetic field to gas. It is characterised by its charged ions, making it
electrically conductive.
4.1 Phase behaviour of a single substance 41
single point on the vaporisation curve, whereas in the pβR-vβR-diagram, Figure 4.2(b),
an isobar connects the points of the liquid and gaseous phases on the vaporisation curve.
These two points indicate the liquid specific volume vLR and the gaseous specific volume
vGR, which are in equilibrium during phase transition. It is also possible to change between
the gas and liquid phases without crossing the two-phase region. This can be accomplished
by making a detour around the critical point through the supercritical region. In this
regard, it is important to point out that changes between liquid and supercritical, as
well as between gas and supercritical are continuous, i. e., without jumps in the density,
unlike the phase transition through the two-phase region. This feature is adopted, e. g.,
in extraction processes, called supercritical fluid extraction, such as the decaffeination of
coffee. For further information on this topic, the interested reader is referred to Brunner
& Budich [33] and citations therein.
θβ
θβ =const.
θβ =const.θβ =const.
pβRpβRpβR
ζβvβR ∆ζvapvLR vGR
va
po
ris
at
ion
cu
rv
e
C C C
LL LG G G
LG LG
(a) (b) (c)
Figure 4.2: 2-d phase diagrams, where in all of them the same isotherm is highlighted (dark green:
liquid, green: two-phase region, light green: gas). (a) pβR-θβ-diagram with the vaporisation
curve (blue), (b) pβR-vβR-diagram with the two-phase region LG, (c) pβR-ζβ-diagram with the
vaporisation enthalpy ∆ζvap. The dotted line connects the point or lines of phase transition in
all three diagrams.
Furthermore, the rightmost diagram, Figure 4.2(c), shows the relation between the ef-
fective pressure pβR and the enthalpy ζβ. The difference between the liquid and gas
enthalpies, ∆ζvap = ζ
G − ζL, expresses the vaporisation enthalpy. This enthalpy differ-
ence will be discussed in more detail later in Section 4.1.2.
Ideal gas law
The first attempt to describe the relationship between pressure, temperature and volume
builds upon findings by Boyle and Mariotte:✎
✍
☞
✌
The pressure is inversely proportional to the volume for isothermal pro-
cesses.
by Gay-Lussac and Charles:
42 Chapter 4: Thermodynamic theory of fluids
☛
✡
✟
✠The volume is proportional to the temperature for isobaric processes.
and Avogadro:☛
✡
✟
✠The gas constant is identical for all ideal gases.
These statements condense into the so-called ideal-gas law, as already mentioned in (4.1)
in the introduction to this chapter, cf., e. g., Lewis & Randall [105]:
pβR vβR = Rβ θβ , (4.2)
which takes into account the linear relationship between pressure, temperature and specific
volume and where Rβ is the specific gas constant related to the considered substance by
dividing the universal gas constant R with the respective molar mass Mβ , Rβ = R/Mβ.
Furthermore, (4.1) and (4.2) are connected by the definition of the specific volume vβR =
V βR/mβ and the relation between the massmβ and the number of moles nβm,m
β = nβmM
β .
Ideal-gas behaviour can only be assumed for a gas at very low pressure and/or high
temperature, where the interactions between the molecules of the gas can be neglected.
To account for the deviations from ideal-gas behaviour, the dimensionless compressibility
factor Zβ has been introduced via
Zβ ≡ p
βR vβR
Rβ θβ
. (4.3)
The compressibility factor is equal to one for an ideal gas, little less for real gases and
much less for liquids. In case of very high temperatures and pressures, it can be slightly
larger than one. Due to the dimensionless formulation of the compressibility factor, it is
usually given as a function of reduced temperature and pressure or volume, e. g., Poling
et al. [137]:
Zβ = fpβRr (p
βR
r , θ
β
r ) = fvβRr (v
βR
r , θ
β
r ) . (4.4)
The dimensionless or reduced variables are derived by relating the actual quantities to
characteristic properties, where usually the critical properties (pβRcrit, θ
β
crit, ρ
βR
crit) of a sub-
stance are taken:
pβRr =
pβR
pβRcrit
, θβr =
θβ
θβcrit
, vβRr =
vβR
vβRcrit
. (4.5)
This allows for a substance-independent description of the pβR-vβR-θβ-relation, also called
the corresponding states principle (CSP). This principle was proposed by van der Waals
on the assumption that intermolecular forces dictate equilibrium properties, such as the
critical properties. These forces are universal for many fluids, except for molecules with
strong polarisations or subjected to hydrogen-bonds. In other words, most substances
obey the same form of EOS, which results in a general compressibility factor
Z =
pβRr v
βR
r
Rβ θβr
, (4.6)
4.1 Phase behaviour of a single substance 43
θβr = 1.0
θβr = 1.1
θβr = 1.2
θβr = 1.3
θβr = 1.5
θβr = 2.0
pβRr
Z
β
0
0.1
0.2
0.3
0.4
0.5
0.5
0.6
0.7
0.8
0.9
1.0
1.0
1.1
1.5 2.5 3.5 4.5 5.5 6.52.0 3.0 4.0 5.0 6.0 7.0
Legend
Methane
Ethylene
Ethane
Propane
n-Butane
n-Heptane
Isopentane
Nitrogen
CarbonDioxide
Water
Average curve
Figure 4.3: Graph of the compressibility factor Zβ versus the reduced pressure pβRr at different
reduced temperatures θβr for several gases. Additionally, an average curve derived from data
of hydrocarbons is given. It shows that in terms of reduced variables, all gases obey the same
EOS. Figure was taken from Su [169].
when using reduced variables as illustrated in Figure 4.3. Therefore, CSP is often the
basis for the development of correlations between different substances, not only for the
determination of the EOS, but also for other thermodynamic properties, e. g., viscosity.
For further information on the topic of the application of CSP, the interested reader is
referred to Lewis & Randall [105], or Ott & Boerio-Goates [134].
Virial equation of state
The ideal gas law is the lowest level CSP and works only for simple fluids, such as inert
gases. For other substances, the EOS must be expanded. This can be accomplished
based on the kinetic theory of gases, which takes into account the non-zero size of the
molecules and describes them as hard spheres. In this regard, semi-empirical functions
are defined for the mutual attractive and repulsive behaviour between these hard spheres.
One approach to describe real gases on this basis is the so-called virial equation of state,
which uses a power series in either 1/vβR, such that:
pβR vβR = Rβ θβ
[
1 +
Bv(θ
β)
vβR
+
Cv(θ
β)
(vβR)2
+
Dv(θ
β)
(vβR)3
+ . . .
]
, (4.7)
44 Chapter 4: Thermodynamic theory of fluids
or in pβR
pβR vβR = Rβ θβ +Bp(θ
β) pβR + Cp(θ
β) (pβR)2 +Dp(θ
β) (pβR)3 + . . . . (4.8)
Therein, Bv, Cv, Dv are the virial coefficients of the virial EOS explicit in specific volume,
and Bp, Cp, Dp are the virial coefficients of the virial EOS explicit in pressure. These coef-
ficients are only functions of the temperature. In case of (4.7), they roughly represent the
interaction potential between the molecules, i. e., pair-wise interaction in Bv, interaction
between three molecules in Cv and so on. However, these calculations become difficult
and inaccurate for higher order terms and must be usually derived experimentally. The
first two coefficients of (4.7) and (4.8) are related via
Bp = Bv , and Cp =
Cv − B2v
Rβ θβ
. (4.9)
In case of low pressures, equation (4.8) is truncated after the second term and solved
explicitly for the volume:
vβR =
Rβ θβ
pβR
+Bp . (4.10)
Cubic equation of state
The next type of EOS, which shall be discussed here, is the cubic equation of state. These
equations are able to represent the pressure as a function of temperature and volume or
density for both, gas and liquid phases at the same time. To achieve this with only one set
of parameters, the equation requires at least a cubic formulation in the volume / density.
The simplest formulation of this kind was proposed by van der Waals (vdW-EOS), who
derived an extension of the ideal gas law (4.2), e. g., Abbott & van Ness [1]:
pβR =
Rβ θβ
vβR − b −
a
(vβR)2
. (4.11)
Herein, the constant a is a measure for the attractive forces, also called the cohesion pres-
sure, and the constant b refers to the non-zero volume of the molecules and is sometimes
denoted as the co-volume. To achieve better representation of the real phase behaviour,
the parameters a and b in (4.11) can be calculated from the compressibility factor (4.6),
which yields a representation depending on the critical temperatures θβcrit and the critical
pressures pβRcrit:
a =
27 (Rβ θβcrit)
2
64 pβRcrit
, b =
Rβ θβcrit
8 pβRcrit
. (4.12)
This semi-empirical equation gives reasonable results for conditions which are not too close
to the critical point and not at too high temperatures and pressures. In other words, the
vdW-EOS is not very accurate for the liquid phase, cf. Ott & Boerio-Goates [134].
Since in this work, CO2 is the mainly investigated substance, the two most prominent
cubic EOS for the description of this substance shall be mentioned here. These are the
Peng-Robinson equation (PR-EOS) and the Soave-Redlich-Kwong equation (SRK-EOS).
4.1 Phase behaviour of a single substance 45
The PR-EOS is given here with respect to the density, but can easily be written in the
form of the specific volume:
pβR =
Rβ θβ ρβR
1− b ρβR −
a c (ρβR)2
1 + 2 b ρβR − (ρβR)2 . (4.13)
The coefficients a, b and c are calculated as follows:
a = 0.45724
(Rβ θβcrit)
2
pβRcrit
,
b = 0.0778
Rβ θβcrit
pβRcrit
,
c =
[
1 + (0.37464 + 1.54226ωβ − 0.26992 (ωβ)2)(1−
√
θβ
θβcrit
)
]2
.
(4.14)
Therein, ωβ is the acentric factor, which is a CSP parameter for phase characterisation,
based on the polarisation of the substance in question. The estimation for the acentric
factor proposed by Lewis and Randall [105],
ωβ ≡ − log10
(
pRvap
pβRcrit
)
− 1.0 , at θβr = 0.7 , (4.15)
is calculated for a vapour pressure pRvap at θ
β
r = θ
β/θβcrit = 0.7. The latter can be deter-
mined, for example, with the Antoine equation, cf. (4.19). In case of CO2, the acentric
factor is found as ωCO2 = 0.225.
An EOS with a similarly good descriptive behaviour as the PR-EOS is the Soave-Redlich-
Kwong equation SRK-EOS, Soave [166]:
pβR =
Rβ θβ ρβR
1− b ρβR −
a c (ρβR)2
1 + b ρβR
, (4.16)
where the coefficients a, b and c are given as
a = 0.42747
(Rβ θβcrit)
2
pβRcrit
,
b = 0.08664
Rβ θβcrit
pβRcrit
,
c =
[
1 + (0.48 + 1.574ωβ − 0.176 (ωβ)2)(1−
√
θβ
θβcrit
)
]2
.
(4.17)
Beside these three presented cubic EOS, a long list of slightly different definitions of cubic
EOS exist, which will not all be mentioned here. A good overview of them is presented,
46 Chapter 4: Thermodynamic theory of fluids
e. g., in Poling et al. [137]. The different EOS are compared in their prediction of pβR-θβ-
ρβR-properties using the CSP in the work of Ott & Boerio-Goates [134].
In this monograph, it is intended to implement a holistic calculation of the thermody-
namical properties, i. e., the Helmholtz free energy, the internal energy and the entropy,
in the numerical model, that are needed, for example, in the energy balance. These cal-
culations build upon the relation between the Helmholtz free energy with the quantities
pβR, θβ , and ρβR, which are again dependent on the EOS. Therefore, it is necessary that
the EOS is good-natured, since it is applied in building the derivatives or integrals with
respect to temperature and pressure during the determination of the Helmholtz free en-
ergies. More precisely, it became obvious that the second terms in the PR-EOS (4.13)
and in the SRK-EOS (4.16) jeopardise the calculations, because of the square-root in the
temperature, appearing in the coefficient c. Thus, it would not be possible to find a closed
formulation for the Helmholtz free energies. This is due to the fact that EOS are usually
only phenomenologically motivated, without caring for their thermodynamic consistency,
which was already acknowledged by Lewis and Randall [105]. Thus, many EOS are not
suitable for a closed formulation. Consequently, it was decided to chose the vdW-EOS
as the governing EOS in the development of the model, since (4.11) does not contain the
coefficient c. The principle of deriving the Helmholtz free energies based on the cubic EOS
is discussed in Section 5.3.4.
The drawback in the choice of the vdW-EOS is the poor accuracy in the description of the
liquid phase, as was already explained before. In order to illustrate this discrepancy, water
is selected as a fluid with well-known phase behaviour. Figure 4.4 depicts several isotherms
in a pressure-density diagram for water. Therein, the dashed lines were calculated from the
p
β
R
[M
P
a]
p
β
R
[M
P
a]
p
β
R
[M
P
a]
ρβR
[
kg
m3
]
ρβR
[
kg
m3
]
ρβR
[
kg
m3
]
0
0
0
0
0 200 400 600 800
1000
1000
1020980
20
30
10
−20
−30
−10
1
0.2
0.2
0.2
0.150.15
0.10.1
0.4 0.6 0.8
0.050.05
373.15 K
647.3 K
300 K
Figure 4.4: Middle: Pressure-density diagram of water calculated for ambient (300K), boiling
(373.15 K) and critical (647.3K) temperatures from the original vdW-EOS (4.11) (dashed lines)
and from the modified vdW-EOS (4.18) (solid lines). For water, the parameters of the modified
vdW-EOS were found as u = 1.34 and w = 2.23, by adjusting to two points: first, at boiling
temperature, the Maxwell criterion must provide a density of 0.598 kg/m3 for the gaseous phase
at ambient pressure (0.1013MPa), cf. black dot in the left picture, and second, the blue isotherm
should yield a density of 996.56 kg/m3 at ambient pressure (0.1013MPa), cf. black dot depicted
in the right picture. The positions of the black dots were taken form the NIST Chemistry
WebBook Lemmon et al. [104]. The brown line marks the standard atmospheric pressure at
pβR = 0.1013MPa.
original vdW-EOS. It clearly shows that the density at standard conditions (θβ = 300K
4.1 Phase behaviour of a single substance 47
and pLR = 1.0 · 105Pa) is not around the expected 1000 kg/m3, but much smaller at ca.
500 kg/m3. By adding two parameters u and w to (4.11) for adjustment and replacing
vβR by 1/ρβR via
pβR = u
Rβ θβ (ρ
βR
w
)
1− b (ρβR
w
)
− a (ρβR
w
)2
, (4.18)
this discrepancy can be removed. The results of this modified vdW-EOS are shown
in Figure 4.4 by the solid isotherms, where in the right picture the zoom of the 300K
isotherm at ambient pressure pβR = 0.1MPa reveals the improved reproduction of the real
behaviour of the water. However, these parameters cause the vdW-EOS to be incorrect
at other conditions, for example, the critical isotherm (for water at 647.3K) is no longer
marked by a saddle point at the critical pressure (for water at 22.09MPa), cf. red solid
isotherm in Figure 4.4. Even so, these adjustment parameters will be used in the following,
since the temperature changes are limited to ∆θmax. = 120K and the considered conditions
stay away from the critical temperature2. Thus, it is possible to determine realistic
densities, also in the case of phase change, which can be seen in the left picture in Figure
4.4, where the green curves compare the boiling temperature of water calculated with
the original and the modified vdW-EOS. Applying the Maxwell criterion (introduced in
the next section) to the boiling temperature isotherm (for water 373, 15K) shows that
the dashed green line does not yield a vapour density close to one at ambient pressure,
whereas for the solid green line, calculated with the adjustment parameters, the Maxwell
criterion calculates the correct vapour density.
When dealing with CO2, one also has to mention another kind of EOS, which is often
applied. This is the Span and Wagner formulation, cf. [167], a non-analytic EOS derived
by an optimization strategy based rigorously on vast amounts of experimental data. This
model consists of about forty different terms and coefficients, depending on the considered
substance. Due to this complicated formulation, the Span and Wagner EOS turns out
to be too expensive to handle the problems discussed in this monograph. Furthermore,
this model cannot be differentiated or integrated for the determination of other thermo-
dynamic functions, which jeopardises the self-imposed restrictions in this monograph of
deriving a consistent thermodynamical model.
Maxwell criterion
Calculating the pressures using the vdW-EOS (4.11) for a given set of temperatures and
densities yields the phase-diagram presented in Figure 4.5. Therein, the red line depicts
the isotherm at the critical temperature of CO2, θ
CO2
crit = 304.21K with zero gradient at
the critical point (black dot in Figure 4.5). One can see that for higher temperatures and
pressures above the critical values, there is a unique dependence between the pressure and
the density, whereas for lower temperatures, i. e., in the two-phase region, one pressure
is connected to three density values. However, the latter cannot be observed in reality.
2This problem constellation is the reason for the vast amount of existing EOS, which try to solve these
issues by adding additional terms and parameters to the vdW-EOS. However, in many cases there is no
thermodynamical motivation for these additional terms.
48 Chapter 4: Thermodynamic theory of fluids
p
β
R
[M
P
a]
ρβR
[
kg
m3
]0 0 100 200 300 400 500 600 700 800
2
4
6
8
10
33
0
K 320
K
310 K
304
.21
K
300 K
290 K
280 K
270
K
260
K
250
K
Figure 4.5: Pressure versus density for CO2 calculated for different temperatures from the vdW-
EOS (4.11). The red line represents the isotherm at the critical temperature θCO2crit = 304.21K
and the black dot indicates the critical point.
Experiments reveal that the temperature and the pressure are constant during evaporation
or condensation. Thus, the isotherm crossing the two-phase region must be replaced by
a straight line according to the so-called Maxwell criterion, e. g., Ott & Boerio-Goates
[134]. To execute this criterion, the isotherm must be cut by an isobar such that the
emerging regions (grey) between the isotherm and the isobar have equal areas, illustrated
exemplary in Figure 4.6. This specific isobar is also sometimes named the tie-line, since it
connects the coexisting gaseous and liquid densities in the pressure-density diagram. The
pβR
ρβRρ
GR ρLR
pRvap
stable gas
metastable gas
stable liquid
metastable liquid
instable
instable
two-phase region
C
Figure 4.6: Identification of the gas-liquid equilibrium exemplary for one isotherm using the
Maxwell criterion, which states that the grey indicated regions must have equal areas. Further-
more, the border of the two-phase region (blue) is sketched that connects all saturation densities
for all θβ < θβcrit. The maximum of the two-phase region is given by the critical point C.
constant pressure value of the tie-line is called the vapour pressure pRvap. The derivation
of the Maxwell criterion from an equilibrium observation is given in Appendix B.3.
4.1 Phase behaviour of a single substance 49
Antoine equation
The Antoine equation provides a phenomenological description of the relation between
pRvap and θ
β during vapour-liquid equilibrium, e. g., Antoine [7] or Abbott & van Ness [1]
log10(p
R
vap) = AA −
BA
(θβ − 273.15 + CA) , (4.19)
where AA, BA and CA are material parameters and 273.15 has to be subtracted, since the
original equation was formulated for temperatures in ◦C. More information to the correla-
tion and extrapolation of vapour pressures, as well as tabulated values of the parameters
AA, BA and CA for various materials, can be found in Poling et al. [137].
By collecting the liquid and gaseous densities of all isotherms that satisfy the Maxwell
criterion, the border of the two-phase region (blue) in Figure 4.6 is found. This border
is also called the saturated-liquid line on the liquid side and saturated-vapour line on the
gaseous side.
Additionally, one can identify states of varying stability on the isotherm in Figure 4.6.
The stable states lie outside of the two-phase region and indicate gas or liquid in their
single phase. The metastable states, between the intersection points of the isotherm with
the border of the two-phase region and the local maximum/minimum of the isotherm,
are called superheated and subcooled conditions, referring to the metastable liquid and
metastable gas, respectively. These states can be observed in experiments only in undis-
turbed conditions3. What cannot be experienced in reality are the instable states, i. e.,
the remaining section of the isotherm.
4.1.1 Chemical potential and first-order phase transitions
Regarding equilibrium assumptions, e. g., during the gas-liquid phase transition, three
transient processes can be observed before equilibrium is reached: first, heat is transferred
until the whole systems has the same temperature, second, dilatations and compressions
occur to reach equal pressures (except for mechanical constraints), and third, mass is
transported to achieve chemical equilibrium. Mathematically, this can be described by
the total differential of the Gibbs free enthalpy ξβ,
dξβ =
(∂ξβ
∂θβ
)
pβR, nβm
dθβ︸ ︷︷ ︸
thermal
+
( ∂ξβ
∂pβR
)
θβ , nβm
dpβR︸ ︷︷ ︸
mechanical
+
∑
i
( ∂ξβ
∂nim
)
θβ , pβR, nj 6=im
dnim︸ ︷︷ ︸
chemical
!
= 0 , (4.20)
where nβm are the moles of material ϕ
β and use was made of the equilibrium criterion
for the total derivative of the Gibbs free enthalpy, dξβ = 0, cf. Poling et al. [137]. For
thermal and mechanical equilibrium, i. e., equal temperatures and pressures, respectively,
3Subcooled vapours are used, for example, in the cloud chamber, also known as theWilson chamber, as
a particle detector for ionizing radiation. If an ion crosses the chamber, it acts as a condensation nucleus,
which initiates phase change from the subcooled vapour to liquid, making the ion-crossing visible, cf. Das
Gupta & Ghosh [43].
50 Chapter 4: Thermodynamic theory of fluids
the first and second terms in (4.20) vanish. Consequently, the third term must refer to
the chemical equilibrium. At this point, the chemical potential µβ is introduced as the
change in Gibbs free enthalpy ξβ per change of moles nβm at constant temperature and
pressure via
µβ =
( ∂ξβ
∂nβm
)
θβ , pβR
. (4.21)
In case of a system that consists of a single pure material, where two phases ϕL and ϕG
coexist, the total free-enthalpy change at constant temperature and pressure yields
dξβ
θβ , p
= dξL + dξG = µL dnLm + µ
G dnGm = 0 . (4.22)
Assuming that the system is closed, i. e., no mass is added from the outside, dnLm = −dnGm,
it follows
(µL − µG) dnGm = 0 and dnGm > 0 . (4.23)
Finally, it can be concluded that at equilibrium, the chemical potentials of the two involved
phases must be equal, viz.:
µL = µG . (4.24)
Of course, this directly implies the non-equilibrium conditions, when the chemical poten-
tials are not equal, i. e., µL ≶ µG. These latter conditions lead to spontaneous transitions
from the phase with high chemical potential to the phase with low chemical potential,
until equilibrium is reached again. Figure 4.7 illustrates this relationship between the
phase transition and the chemical potential in dependence of the temperature.
µβ
µL
µG
θβθeq
Figure 4.7: Illustration of the chemical potential µβ versus the temperature θβ. At the equilib-
rium temperature θeq, equilibrium exists, i. e., µ
L = µG. For higher temperatures θβ > θeq and
µL > µG phase ϕL changes to phase ϕG, whereas for lower temperatures θβ < θeq and µ
G > µL
phase ϕG changes to phase ϕL. In imitation of Ott & Boerio-Goates [134].
Phase transitions can be grouped into first-order phase transitions and second-order phase
transitions. The former correspond to transitions, where, although the difference of the
chemical potentials at the equilibrium temperature θeq is zero ∆µ = µ
L − µG = 0, cf.
Figure 4.7, the Gibbs free enthalpy changes its slope there. This means that the first
derivative of the chemical potential with respect to an intensive parameter (e. g., pβR),
4.1 Phase behaviour of a single substance 51
which represents the corresponding extensive parameter (e. g., vβR), is not continuous:( ∂ξβ
∂pβR
)
θeq
= vβR 6= 0 . (4.25)
From this jump in the first derivative of the free enthalpy, the name “first-order phase
transitions” is deduced.
Second-order phase transitions denote continuous phase transitions, e. g., liquid to gas at
the critical point, where also the first derivatives become equal to zero and only the second
derivative of the potential has a jump at the point of phase transition. This second group
is important for the treatment of superfluid helium and transitions from paramagnetic to
ferromagnetic conditions, which are not discussed in this monograph, since here the focus
is restricted to first-order phase transitions.
4.1.2 Clausius-Clapeyron equation
In Figure 4.2(a) the circle indicates the coexistence point of the gaseous and liquid phase
on the vaporisation curve. As mentioned before, temperature and pressure are constant
during phase change and equal for both phases: θL = θG = θeq and p
LR = pGR = pRvap.
From the definition (4.24) of the chemical equilibrium derived in the previous Section
4.1.1, it was found for the chemical potentials µL and µG at equilibrium:
µL(pLR, θβ) = µG(pGR, θβ) . (4.26)
Taking the derivative of (4.26) with respect to pRvap and θ
β, leads to(∂µL
∂θβ
)
pRvap
dθβ +
( ∂µL
∂pRvap
)
θβ
dpRvap =
(∂µG
∂θβ
)
pRvap
dθβ +
( ∂µG
∂pRvap
)
θβ
dpRvap . (4.27)
From the fundamental relations of the chemical potential, cf. Appendix B.2, it follows for
its partial derivatives(∂µβ
∂θβ
)
pRvap
= −ηβ , and
( ∂µβ
∂pRvap
)
θβ
= vβR , (4.28)
where ηβ is the entropy of ϕβ. Applying these relations to (4.27), yields
−ηL dθβ + vLR dpRvap = −ηG dθβ + vGR dpRvap . (4.29)
Subsequently, a separation of variables leads to
(ηG − ηL) dθβ = (vGR − vLR) dpRvap (4.30)
and by application of dηβ = dζβ/θβ, cf. Appendix B.2, one gets
dpRvap
dθβ
=
ηG − ηL
vGR − vLR =
ζG − ζL
θβ (vGR − vLR) . (4.31)
52 Chapter 4: Thermodynamic theory of fluids
Furthermore, with the definition of the vaporisation enthalpy or latent heat ∆ζvap :=
ζG − ζL and the difference of specific volume ∆v = vGR − vLR, the most-known form of
the Clausius-Clapeyron equation is derived:
dpRvap
dθβ
=
∆ζvap
θβ ∆v
. (4.32)
This is a relation for the vaporisation curve in Figure 4.2(a), while the specific-volume
difference and the vaporisation enthalpy are indicated in Figures 4.2(b) and 4.2(c), re-
spectively. The Clausius-Clapeyron equation is therefore also the basis for the formulation
of the Antoine equation (4.19), where a detailed discussion is given, for example, by Graf
[74].
4.2 Vaporisation enthalpy
The enthalpy of vaporisation, ∆ζvap, is the difference between the enthalpies of the sat-
urated gas and liquid phases at the same temperature and pressure. In case of vapori-
sation, it delineates the energy required to destroy the intermolecular bonds within the
liquid phase. In the other direction, i. e., condensation, the same amount of energy is
released when these bonds are restored. A frequently used synonym for the vaporisation
enthalpy is latent heat of vaporisation. To estimate the vaporisation enthalpy, one can
resort to the CSP, which was mentioned earlier. Pitzer et al. [135] developed from the
Clausius-Clapeyron equation (4.32) an expression depending on the critical temperature
θβcrit, the acentric factor ω
β and the molar mass Mβ of the respective substance:
∆ζvap = 7.08RM
β
[
θβcrit (1− θβr )0.354 + 10.95ωβ (1− θβr )0.456
]
. (4.33)
This equation is valid for reduced temperatures between 0.6 < θβr < 1.0.
The vaporisation enthalpy or latent heat ∆ζvap is calculated in this work by fitting an
exponential function to the CO2 vaporisation data taken from Potter & Somerton [139]
and displayed in Figure 4.8:
∆ζvap =
1000.0 a∆ζ
[c∆ζ (θβ − 273.15) + d∆ζ] b∆ζ (4.34)
with the fitting parameters a∆ζ , b∆ζ , c∆ζ and d∆ζ.
4.3 Specific heat capacity
The heat capacity is a measure for the amount of heat needed to change the temperature
of a substance by 1K and which is an extensive quantity with the unit J/K. However, it
is convenient to use the mass-specific formulation in J/(kgK) as an intensive property.
In classical thermodynamics, it is common to look at a heating process under various cir-
cumstances, i. e., either at constant density (respectively volume) or at constant pressure,
4.3 Specific heat capacity 53
θβ [◦C]
∆
ζ v
a
p
[k
J
k
g
]
fitted curve
measured data
0
0 5 10 15 20 25
50
100
150
200
250
30 35
Figure 4.8: Vaporisation enthalpy of CO2. Data are taken from Potter & Somerton [139], and
the parameters for the fitted curve (4.34) are: a∆ζ = 239.90, b∆ζ = −0.40, c∆ζ = −0.030 and
d∆ζ = 0.92.
cf. Lewis & Randall [105]. In this work, only the first formulation is needed:
cβRV =
∂εβ
∂θβ
∣∣∣∣
ρβR=const.
. (4.35)
The specific internal energy εβ, found in (4.35), is a function of the extensive state variables
entropy ηβ and strain Eβ, namely ε
β = εβ(ηβ, Eβ). In case of non-viscous fluids, the
original dependency of the specific internal energy on the strain changes to a dependency
on the volume, respectively the density, via εβ = εβ(ηβ, ρβR). With the conjugated
intensive quantity θβ for the entropy, it follows
∂εβ
∂ηβ
∣∣∣∣
Eβ or ρβR
= θβ . (4.36)
Hence, if (4.36) is used in (4.35), one finds for the specific heat capacity at constant
volume
cβRV =
∂εβ
∂ηβ
∂ηβ
∂θβ
= θβ
∂ηβ
∂θβ
. (4.37)
Potter and Somerton [139] state in their work that the specific heat capacity of gases in-
creases slowly with increasing temperature and, thus, can be taken as constant. However,
Figure 4.9, which shows the specific heat capacity for CO2, indicates that this is only true
for higher temperatures (θβ > 320K). Close to the temperature, where phase transition
occurs (in this case at pβR = 4.5 · 106Pa and θβ = 283.13K), the change in the specific
heat capacity cannot be neglected. For a continuous approximation of the whole curve,
i. e., both, liquid and gas, a hyperbolic tangent function is fitted to the data obtained
from the NIST Chemistry WebBook Lemmon et al. [104], cf. Figure 4.9. The resulting
function reads
cβRV = −75.351 tanh [ 1.695 (θβ − 289.985) ] + 871.396 with [cβRV ] =
J
kg K
(4.38)
54 Chapter 4: Thermodynamic theory of fluids
θβ [K]
cβ
R
V
[
J
k
g
K
]
liquid
gaseous
fit
1000
950
900
850
800
750
240 260 280 300 320 340 360 380 400 420 440 460 480 500
Figure 4.9: Specific heat capacity of CO2 for constant density and at a pressure of
pβR = 4.5 · 106 Pa. The jump at θβ = 283.13 K coincides with the phase transition between
liquid and gaseous CO2. The green lines, both for liquid and gas, were derived with the NIST
Chemistry WebBook by Lemmon et al. [104]. The red line shows a continuous fit using a
hyperbolic tangent function (4.38) with an error of R2 = 0.915.
and has an error of R2 = 0.915. Since the heat capacity is only weakly dependent on
the pressure, it can be argued to chose a single reference pressure, cf. Poling et al. [137].
Please note in passing that in the derivation of the Helmholtz free energies of the fluid
constituents in Section 5.3.4, the liquid and gaseous specific heat capacities are taken as
constant for the integration procedures.
4.4 Shear viscosity and thermal conductivity
The empirical relations for the calculations of the effective shear (dynamic) viscosity µβR
and the effective thermal conductivity HβR for the fluid phases are gathered from the
works of Fenghour et al. [66] and Vesovic et al. [175], which were developed solely for
CO2. For both parameters, the formulations are composed of a part for the zero-density
limit (i. e., ideal-gas behaviour), an excess part (added for non-ideal temperature-pressure
conditions) and a third enhancement part for the critical region (accounting for the strong
divergence of parameters close to the critical point). Hence, the structure of the universal
formulation for both, the viscosity and conductivity, looks like this:
(·) = (·)0︸︷︷︸
id. gas
+∆(·)︸︷︷︸
excess
+∆(·)c︸ ︷︷ ︸
crit.
. (4.39)
Formulations and empirical relations that are valid for other substances besides CO2 are
presented and compared, e. g., in Poling et al. [137].
4.4.1 Effective shear viscosity
As mentioned before, the shear viscosity is decomposed into three parts:
µβR(ρβR, θβ) =
[
µβR0 (θ
β) + ∆µβR(ρβR, θβ) + ∆µβRc (ρ
βR, θβ)
] · 10−6 , [µβR] = Pa s ,
(4.40)
where µβR0 (θ
β) is the effective shear viscosity for a density close to zero, i. e., ideal gas,
the excess viscosity ∆µβR(ρβR, θβ) stands for the increase in viscosity for higher densities
4.4 Shear viscosity and thermal conductivity 55
and the critical enhancement ∆µβRc (ρ
βR, θβ) accounts for the divergence of the viscosity
around the critical point. In contrast to the original publication, (4.40) has been multiplied
by 10−6. This is due to the fact that Fenghour et al. [66] have based their shear-viscosity
formulation on µPa s instead of Pa s, what is required in this work.
According to Fenghour et al. [66], the zero-density viscosity is given as:
µβR0 (θ
β) =
1.00697
√
θβ
eσ∗(θβ)
with σ∗(θβ) =
m∑
i=1
ai (ln
θβ
251.196K
)i and [µβR0 ] = µPa s .
(4.41)
The coefficients ai are listed in Table 4.1 on the left with m = 4.
Table 4.1: The coefficients ai for the formulation of the zero-density viscosity (left) and the
coefficients dij for the formulation of the excess viscosity (right), both only valid for CO2,
Fenghour et al. [66].
i ai
0 0.235156
1 −0.491566
2 5.211155 · 10−2
3 5.347906 · 10−2
4 −1.537105 · 10−2
ij dij
11 0.4071119 · 10−2
21 0.7198037 · 10−4
64 0.2411697 · 10−16
81 0.2971072 · 10−22
82 −0.1627888 · 10−22
The excess viscosity ∆µβR(ρβR, θβ) is the real-gas/fluid correction factor at higher densi-
ties. Due to the lack of a sufficient theory for the determination of this factor, an empirical
equation in the form of a power-series expansion of the density provides adequate accu-
racy:
∆µβR(ρβR, θβ) = d11 ρ
βR + d21 (ρ
βR)2 +
d64 (ρ
βR)6
(θβ∗)3
+ d81 (ρ
βR)8 +
d82 (ρ
βR)8
θβ∗
, (4.42)
where θβ∗ = θβ/251.196K and the constants dij are depicted in the right-hand side of
Table 4.1. The excess viscosity in (4.42) is formulated in [∆µβR] = µPa s.
Finally, it remains to define the critical enhancement ∆µβRc (ρ
βR, θβ). Fenghour et al.
[66] state in their work that the deviation of the shear viscosity in narrow vicinity to the
critical point is rather small. Since in this monograph conditions close to the critical point
are avoided, this term is omitted in the following. For a detailed discussion on the critical
enhancement, the interested reader is referred to Vesovic et al. [175].
4.4.2 Thermal conductivity
Analogous to the kinematic viscosity, the partitioning function (4.39) is applied for the
thermal conductivity, viz.:
HβR(ρβR, θβ) = [HβR0 (θ
β) + ∆HβR(ρβR, θβ) + ∆HβRc (ρ
βR, θβ) ] · 10−3 , [HβR] = W
mK
,
(4.43)
56 Chapter 4: Thermodynamic theory of fluids
where the original formulation has been multiplied by 10−3, since Vesovic et al. [175]
have calculated the thermal conductivity in mW/mK, whereas here W/mK is needed.
In (4.43), the zero-density part of the thermal conductivity is given via
HβR0 (θ
β) =
0.475598
√
θβ (1 + (rH)
2)
τ ∗(θβ)
with τ ∗(θβ) =
7∑
i=0
bi
(θβ∗)i
and [HβR0 ] =
mW
mK
.
(4.44)
The coefficient rH in (4.44) is expressed as
rH =
√
2
5
cint and cint = 1.0 + e
−183.5
θβ
5∑
i=1
ci (
θβ
100
)2−i . (4.45)
Furthermore, the coefficients bi and ci in (4.44) and (4.45)2 are taken from Table 4.2.
Table 4.2: The coefficients bi and ci for the formulation of the zero-density thermal viscosity and
the coefficients di for the formulation of the excess thermal conductivity for CO2, cf. Vesovic et
al. [175].
i bi ci di
0 0.4226159
1 0.6280115 2.387869 · 10−2 2.447164 · 10−2
2 −0.5387661 4.350794 8.705605 · 10−5
3 0.6735941 −10.33404 −6.547950 · 10−8
4 0.0 7.981590 6.594919 · 10−11
5 0.0 −1.940558
6 −0.4362677
7 0.2255388
The excess thermal conductivity ∆HβR(ρβR), as the correction factor for higher densities,
is again determined from a polynomial in the density:
∆HβR(ρβR) =
4∑
i=1
di (ρ
βR)i and [∆HβR] =
mW
mK
, (4.46)
where the coefficients di are listed in Table 4.2.
Opposite to the kinematic viscosity, the change in thermal conductivity around the crit-
ical point is significant and cannot be neglected. However, since this region will not be
touched in the simulations presented in this monograph, it is decided to omit the critical
enhancement part in the calculation of the kinematic viscosity as well. The formulation
of this critical factor can be found in Vesovic et al. [175].
Chapter 5: Constitutive settings
In Chapter 3, the general framework for the description of a multiphasic model was
presented without any specification to a particular system. In the present chapter, this
framework shall be adapted to the problem of CO2 sequestration. It was illustrated in
Chapter 2 that the modelling of CO2 sequestration into a deep aquifer is a numerically
challenging task due to the considered range in time and length scales, the multiphasic
composition, and the process complexity with respect to thermodynamics and chemistry.
As the early time span of the injection process is most critical in terms of high pressures,
including the unwanted consequences of uplift, crack formation, induced seismicity and
displacement of saline formation-water, which was elucidated in Section 2.3.2, this stage
is of particular interest for policy makers regarding the storage safety, cf. Celia et al. [34].
Hence, the customised model should be able to describe the solid deformations and the
phase transitions between the fluid phases, which are induced either by the rising pore
pressures during the injection stage or by changing temperature conditions. However, it
is not intended to include dissolution or mineralisation, since these effects only appear
over a greater time span and can therefore be treated separately.
At first, in Section 5.1, a priori assumptions are made for the simplification of the prob-
lem at hand, which is of course accomplished without losing the important characteristics
that distinguish the CO2 storage problem. In Section 5.2, the basic balance equations
derived in Chapter 3 are adopted to these assumptions. This section also incorporates a
comparison between the number of governing equations and the number of unknowns that
are needed for an exhaustive description of the motion and temperature states. The com-
parison displays a superior number of unknowns, which is distinguished as the so-called
closure problem. The surplus unknowns have to be defined by additional constitutive
equations that are derived in Section 5.3 for the involved constituents by evaluating the
entropy inequality in order to ensure the thermodynamic consistent modelling procedure.
These constitutive equations solve the closure problem by adding the required number
of equations to the set of governing equations and, thus, even out the superior number
of unknowns. Furthermore, the particular thermodynamical and physical behaviours of
the involved constituents in the process of CO2 storage in the subsurface are represented
by the constitutive equations. Since the phase transition process is of paramount impor-
tance in this work, the derivation of the corresponding constitutive equations is regarded
in detail in Section 5.4. The final Section 5.5 contains the collection of the strong forms
of the governing balance relations, while applying the results of the previous evaluations.
These strong forms build the basis for the numerical simulations.
5.1 A priori assumptions
The triphasic model consists of an inert (ρˆS ≡ 0), thermoelastic solid phase ϕS and two
immiscible and compressible pore-fluid phases ϕL and ϕG. The following derivations will
57
58 Chapter 5: Constitutive settings
be done for the case, where one single matter can adopt two different phases. More
specifically, the two fluid phases belong to one single substance, e. g., CO2. Hence, the
two fluid phases are ϕL (liquid CO2) and ϕ
G (gaseous CO2). However, as in Chapter 7
a numerical example of CO2 being injected into a water-filled reservoir is presented, two
different fluid matters are considered, i. e., water (ϕL) and CO2 (ϕ
G). The minor but
necessary changes in the formulations of the model for the second case will be explained
in detail in the introduction to the particular numerical example.
The flow and heat transport processes in CO2 reservoirs are supposed to be slow, which
justifies the assumption of quasi-static conditions and, thus, the acceleration terms of the
respective constituents can be neglected:
′′
xα ≡ 0 . (5.1)
In connection to the assumption of slow processes (quasi-static), which also excludes
chemical reactions originating from sudden temperature variations, it is postulated that
a local thermodynamical equilibrium exists. Consequently, a common temperature for all
constituents can be assumed:
θα = θ . (5.2)
The single temperature formulation requires only the consideration of the mixture energy
balance of the overall aggregate in the set of the governing balance equations, instead of
three separate energy balances for the individual constituents. This is further justified
by an evaluation of the temperature equalisation between a fluid and a solid constituent,
see Appendix B.9. In this context, it is important to point out that the assumption of
local thermal equilibrium between the constituents (θα = θ) is rather strong, but does
not imply global thermal equilibrium, meaning the temperature can change spatially.
Still, non-thermal equilibrium conditions trigger the mass transfer during phase transition
through, for instance, a heat flux
∑
α
qα.
Finally, the body forces are assumed to be governed by constant and homogeneous grav-
itational forces for all constituents, i. e., bα = g.
5.2 Adaptation of balance relations
The balance relations introduced by (3.63)1, (3.64)1 and (3.65)1 are altered with the
previously introduced a priori assumptions. In particular, the adapted mass, momentum
and energy balances comprise the set of the governing balance relations:
(ρS)′S + ρ
S div
′
xS = 0 ,
(ρβ)′β + ρ
β div
′
xβ = ρˆ
β , β = {L, G} ,
0 = divTα + ρα g + pˆα , α = {S, L, G} ,
ρα (εα)′α = T
α · Lα − divqα + ρα rα + εˆα , α = {S, L, G} .
(5.3)
5.2 Adaptation of balance relations 59
Note in passing that the solid mass balance (5.3)1 reduces to a volume balance, i. e., (n
S)′S+
nS div
′
xS = 0, for ρ
SR = const. However, this only holds for materially incompressible
solid constituents under isothermal conditions, which is not the case in this monograph.
The other balance relations are the two fluid mass balances (5.3)2, the momentum bal-
ances (5.3)3 and the energy balances (5.3)4. Comparing this set of balance relations with
the relations introduced in Section 3.3.1, the attentive reader recognises the missing en-
tropy inequality, since the entropy inequality does not serve as a governing equation.
Instead, it reveals constraints to be imposed on the constitutive relations for the sake
of a thermodynamically consistent modelling procedure. These constitutive relations are
needed to solve the closure problem.
The closure problem arises as the system of governing equations in (5.3) incorporates
more unknowns than equations. In numbers, the three constituents lead to three mass
balance equations, nine momentum balance equations (or three vector-valued equations)
and three energy balance equations summing up to 15 equations. These stand adversely
to the set of unknowns: ρα (1 ·3), ′xα (3 ·3), ρˆβ (1 ·2), Tα (6 ·3), g (3 ·1), pˆα (3 ·3), εα (1 ·
3), Lα (6 · 3), qα (3 · 3), rα (1 · 3) and εˆα (1 · 3), which makes a total of 80 unknowns1. It
is now easy to conclude that besides the equations in (5.3), further constitutive relations
are required to determine the 65 open fields, consisting of the densities, velocities, stress
tensors, internal energies, heat fluxes and production terms.
The procedure of generating sound constitutive equations for a multiphasic, multicom-
ponent model, which in this case concerns the CO2 storage, follows the line of action of
Ehlers [54, 55]. It can be subdivided into several successive steps. At first, the governing
balance relations (5.3) are adapted. Next, the application of the basic thermodynamic
principles retrieves the dependencies of the Helmholtz free energies ψα, which are then
used to, finally, gain the missing constitutive relations exploiting the entropy inequality.
5.2.1 Definition of a thermoelastic solid
In this monograph, an inert, thermoelastic, porous solid material is regarded following
the line of Ehlers and Ha¨berle [61]. Its describing mass balance was already presented
in (5.3)1. Due to the thermoelastic behaviour, it is not enough to just use this mass
balance for the computation of the solid volume fraction nS, as it was done, e. g., by
Wagner [176]. Here, besides the solid volume fraction, also the solid partial and material
densities, ρS and ρSR, are determined from the solid mass balance. To do this, at first, the
thermomechanical behaviour of the solid constituent must be discussed, which basically
depends on a multiplicative split of the total solid deformation gradient FS into a purely
mechanical and a purely thermal part:
FS = FSmFSθ . (5.4)
While FS only depends on the solid displacement uS through the displacement gradient
HS = GradS uS, the thermal part FSθ is described by a constitutive assumption following
1Herein, the first number in the bracket refers to a scalar (1), a vectorial (3), or a tensorial quantity
(6) (for symmetric tensors) and the second number to the involved constituents.
60 Chapter 5: Constitutive settings
Lu & Pister [110], viz.:
FS = I+GradS uS = I+HS ,
FSθ = (detFSθ)
1/3 I with detFSθ = exp (3α
S∆θ) ,
FSm = FS F
−1
Sθ .
(5.5)
Therein, αS is the coefficient of thermal expansion, and ∆θ = θ − θ0 is the temperature
variation with respect to the initial temperature θ0. Furthermore, (5.4) is reorganised for
the purely mechanical part FSm.
The multiplicative decomposition of the deformation gradient is associated with the ex-
istence of an intermediate configuration and an additive split of the strain tensors in the
solid reference, intermediate and current configuration, similar to elasto-plasticity, cf.,
for example, Ehlers [50]. One obtains, for instance, the following decomposition of the
Green-Lagrangean strain in the solid reference configuration:
ES =
1
2
(FTS FS − I) = ESm + ESθ ,
ESθ =
1
2
(FTSθ FSθ − I) ,
ESm = ES − ESθ .
(5.6)
Hence, it is possible to describe a stress-free strain condition at the reference state, where
the initial strain state is governed by the initial temperature θ0.
Proceeding from the small-strain assumption, a formal linearisation of the above strain
measures around the reference state is suitable and yields for the linearised Green-
Lagrangean solid strains
εS := linES =
1
2
(FS + F
T
S )− I = 12 (HS +HTS ) ,
εSθ := linESθ =
1
2
(FSθ + F
T
Sθ)− I ,
εSm = εS − εSθ .
(5.7)
Furthermore, if the temperature variation is such that detFSθ is approximately equal to
lin (detFSθ), a formal linearisation of (detFSθ)
1/3 around ∆θ = 0 provides
lin (detFSθ)
1/3 = 1 + αS∆θ . (5.8)
Given this result, the thermal part of the deformation gradient (5.5)2 and the thermal
strain (5.7)2 read
lin (FSθ) = (1 + α
S∆θ) I ,
εSθ = α
S∆θ I .
(5.9)
By integration of the solid mass balance (5.3)1 over time t, one obtains
ρS = ρS0S (detFS)
−1 , (5.10)
5.2 Adaptation of balance relations 61
where ρS0S is the partial solid density in the solid reference configuration at t = t0. Splitting
the partial density into the effective density and the volume fraction yields by use of (5.4)
nS ρSR = nS0S ρ
SR
0S (detFSm)
−1 (detFSθ)
−1 (5.11)
with nS0S and ρ
SR
0S being the solid volume fraction and the effective density at t = t0,
respectively.
As explained before, ρSR = ρSR0S is constant at constant temperatures for materially in-
compressible solids. Consequently, variations in ρSR can only be initiated by temperature
variations. Thus, it is obvious that (5.11) can be split as follows, see Ghadiani [71],
ρSR = ρSR0S (detFSθ)
−1 = ρS0S exp (−3αS∆θ) ,
nS = nS0S (detFSm)
−1 = nS0S (detFS)
−1 exp (3αS∆θ) ,
(5.12)
where (5.5)3 has been used. Linearising (detFS)
−1, (detFSθ)
−1 and (detFSm)
−1 via
lin (detFS)
−1 = 1−DivS uS ,
lin (detFSm)
−1 = 1−DivS uS + 3αS∆θ ,
lin (detFSθ)
−1 = 1− 3αS∆θ ,
(5.13)
wherein DivS ( · ) is the divergence operator corresponding to GradS ( · ), the following
relations for the solid densities and the volume fractions are obtained:
ρS = ρS0S (1− DivS uS) ,
nS = nS0S (1−DivS uS + 3αS∆θ) ,
ρSR = ρSR0S (1− 3αS∆θ) .
(5.14)
Summarising the results of the present section, the evaluation of the solid volume balance
towards a thermoelastic material behaviour by applying a multiplicative split of the solid
deformation gradient, led to comprehensive formulations for the solid density relations and
the solid volume fraction. This is also justified by the fact that in solid mechanics, ρS is
given solely by ρS0S and FS, as far as ρˆ
S = 0. The latter is a consequence of the integration
of (5.3)1. Because of this procedure, the solid volume balance is already inherently added
and is therefore removed from the set of governing equations.
5.2.2 Mass balances of the fluid phases
The mass balance of the fluid phase ϕβ was displayed in (5.3)2, where β = {L, G}.
Therein, the partial density ρβ is replaced by its relation to the material density ρβR
(3.5)3, viz.:
nβ (ρβR)′β + (n
β)′β ρ
βR + nβ ρβR div
′
xβ = ρˆ
β . (5.15)
62 Chapter 5: Constitutive settings
For a formulation of the material time derivatives solely with respect to the solid motion
( · )′S, the time derivative ( · )
′
β in (5.15) must be transcribed by the help of (3.15), which
yields
(nβ ρβR)′S + div (n
β ρβRwβ) + n
β ρβR div (uS)
′
S = ρˆ
β . (5.16)
Hereby, the definitions of the seepage velocity (3.14), the solid displacement (3.13) and
the divergence theorem (cf. Appendix A) were used.
In this context, the coupling between the two fluid mass balances should be addressed.
This coupling is ensured by the mass-production terms ρˆβ . From the sum over all con-
stituents of the mass-production term (3.35)1 and the preliminary assumption of an inert
solid material, i. e., ρˆS ≡ 0, it follows:
ρˆL = −ρˆG . (5.17)
In the case, where the fluid phases are composed of water (ϕL) and CO2 (ϕ
G), no mass
exchange between the two fluid phases is considered and, hence, the mass-production term
on the right-hand side of (5.16) vanishes, ρˆβ = 0. Additionally, it is assumed in this case
that the water phase is incompressible for θL = const., hence, ρLR = const., and, thus,
the water mass balance becomes the water volume balance after dividing (5.16) by ρLR:
(nL)′S + div (n
LwL) + n
L div (uS)
′
S = 0 . (5.18)
5.2.3 Momentum balance of the overall aggregate
The summation of the constituent momentum balances (5.3)3 over all constituents ϕ
α
yields the mixture or overall momentum balance in its quasi-static formulation
0 =
∑
α
[ divTα + ρα g + pˆα ] . (5.19)
Prior to this, one has to go back to the restrictions between the overall aggregate and
the individual constituent formulations given in (3.51) to derive the particular terms for
the mixture formulation of the momentum balance. Specifically, the overall Cauchy stress
tensor T is given by
T =
∑
α
(Tα−ρα dα⊗dα) = TS+TL+TG−ρS dS⊗dS−ρL dL⊗dL−ρG dG⊗dG , (5.20)
and the density of the overall aggregate ρ reads
ρ =
∑
α
ρα = nS ρSR + nL ρLR + nG ρGR , (5.21)
where the definition of partial densities (3.5) has been exploited. The sum over the
momentum productions pˆα is treated by the help of (3.14), (3.35)1, (3.35)2, (3.36)1 and
the previously found relation ρˆL = −ρˆG (5.17) via∑
α
pˆα =
∑
α
(sˆα − ρˆα ′xα) = ρˆL (wG −wL) . (5.22)
5.2 Adaptation of balance relations 63
Although quasi-static problems were assumed, which allows for the negligence of the con-
stituent accelerations
′′
xα ≡ 0, the same cannot be concluded directly for the barycentric
acceleration x¨. The reason therefor is found in (3.51)4:
x¨ =
1
ρ
∑
α
[ ρα
′′
xα − div (ρα dα ⊗ dα) + ρˆβ ′xβ ] . (5.23)
This equation shows that even for vanishing constituent accelerations, the terms including
the diffusion velocities dα and the mass production ρˆ
β do remain. Inserting (5.20), (5.21)
and (5.23) into the momentum balance of the overall aggregate (3.42), the divergence of
the diffusion velocity terms vanishes and only the Cauchy stress tensors remain. The final
result of the mixture momentum balance after these manipulation reads:
0 =
∑
α
(divTα) + ρg + ρˆL (wG −wL) , (5.24)
where the body force was replaced by the gravitational acceleration and the mass pro-
ductions ρˆβ were replaced with the help of (3.14).
5.2.4 Energy balance of the overall aggregate
In case that all components share the same temperature, as it is assumed in the present
consideration, cf. Section 5.1, the energy balances of the constituents have to be summed
up towards the total energy balance for the computation of the temperature change. In
this case, the direct energy production εˆα has to be substituted by (3.36)3, where the sum
over all constituents of eˆα vanishes according to (3.35)3, viz.:∑
α
ρα (εα)′α =
∑
α
[Tα · Lα − divqα + ρα rα − pˆα · ′xα − ρˆα (εα + 12
′
xα · ′xα) ] . (5.25)
Thus, no constitutive equation is needed for the heat exchange εˆα between the different
constituents. For more information to εˆα, please refer toGhadiani [71]. However, frictional
effects induced by pˆα · ′xα and energetic quantities transported during the phase-transition
process through ρˆα also lead to mechanical and non-mechanical energy exchanges and heat
transfers. Therewith, the energy balance of the overall aggregate (5.25) becomes:
ρS (εS)′S + ρ
L (εL)′L + ρ
G (εG)′G = T
S · LS +TL · LL +TG · LG−
−
∑
α
divqα +
∑
α
ρα rα − pˆS · ′xS − pˆL · ′xL − pˆG · ′xG−
− ρˆS (εS + 1
2
′
xS · ′xS)− ρˆL (εL + 12
′
xL · ′xL)− ρˆG (εG + 12
′
xG · ′xG) .
(5.26)
An additional simplification of (5.26) can be achieved by combining the direct momentum
and mass production terms in a similar manner as in the derivation of (5.22). The detailed
64 Chapter 5: Constitutive settings
procedure is presented in Appendix B.7. The implementation of these simplifications into
(5.26) yields:
ρS (εS)′S + ρ
L (εL)′L + ρ
G (εG)′G = T
S · LS +TL · LL +TG · LG −
∑
α
divqα+
+
∑
α
ρα rα − pˆL ·wL − pˆG ·wG + ρˆL [ εG − εL + 12 (wG ·wG −wL ·wL) ] ,
(5.27)
which is the form of the mixture energy balance used in the following.
5.2.5 Adaptation of the entropy inequality
It was mentioned before that the derivation of the required constitutive relations for
the constituents ϕα is based on the evaluation of the entropy inequality. The latter was
introduced in its general form as the Clausius-Duhem inequality, see (3.50). Incorporating
therein the assumptions of equal constituent temperatures θα = θ, as well as the sum of
the total energy productions (3.35)4, (3.50) recasts to:
1
θ
{
TS · LS +TL · LL +TG · LG − ρS [ (ψS)′S + (θ)′S ηS ]− ρL [ (ψL)′L + (θ)′L ηL ]−
− ρG [ (ψG)′G + (θ)′G ηG ]− pˆS ·
′
xS − pˆL · ′xL − pˆG · ′xG − ρˆS (ψS + 12
′
xS · ′xS)−
− ρˆL (ψL + 1
2
′
xL · ′xL)− ρˆG (ψG + 12
′
xG · ′xG)− 1
θ
∑
α
qα · grad θ
}
≥ 0 .
(5.28)
Next, one can again merge the direct momentum and mass production terms, pˆα and ρˆβ ,
following the procedures presented above for the momentum and energy balances:
TS · LS +TL · LL +TG · LG − ρS [ (ψS)′S + (θ)′S ηS ]− ρL [ (ψL)′L + (θ)′L ηL ]−
− ρG [ (ψG)′G + (θ)′G ηG ]− pˆL ·wL − pˆG ·wG+
+ ρˆL [ψG − ψL + 1
2
(wG ·wG −wL ·wL) ]− 1
θ
∑
α
qα · grad θ ≥ 0 .
(5.29)
Herein, in addition, the whole equation was multiplied with the absolute temperature,
where θ > 0 K.
Saturation constraint:
By incorporating the saturation constraint (3.3)2 into the entropy inequality, the mechan-
ical constraint of a constant mass of the overall aggregate is added to the restrictions
of the thermodynamical process. However, not the saturation constraint (3.3)2 itself is
applied, but its material time derivative with respect to the solid motion:
(nS)′S + (n
L)′S + (n
G)′S = 0 , (5.30)
5.2 Adaptation of balance relations 65
where
(nS)′S =
1
ρSR
[−nS (ρSR)′S − nS ρSR div
′
xS ] ,
(nL)′S =
1
ρLR
[ ρˆL − nL (ρLR)′L − nL ρLR div
′
xL ]− gradnL ·wL ,
(nG)′S =
1
ρGR
[ ρˆG − nG (ρGR)′G − nG ρGR div
′
xG ]− gradnG ·wG .
(5.31)
Therein, the material time derivatives of the volume fractions with respect to the solid
constituent ϕS were transformed by utilising the mass balance equations of the respective
constituents ϕα (5.3)1,2 and the relation (3.15). The material time derivative of the effec-
tive solid density ρSR in (5.31)1 has to be defined according to its material behaviour, cf.
(5.14)3. In particular, the effective density of the materially incompressible solid material
solely depends on the temperature θ. Consequently, its material time derivative reads
ρSR = ρSR(θ) → (ρSR)′S =
dρSR
dθ
(θ)′S . (5.32)
Subsequently, relations (5.31) and (5.32) are inserted into (5.30), where the latter is
multiplied with a so-called Lagrangean multiplier2 P, cf. Liu [107]:
P
{ 1
ρSR
[nS
dρSR
dθ
(θ)′S + n
S ρSR div
′
xS ] +
+
1
ρLR
[−ρˆL + nL (ρLR)′L + nL ρLR div
′
xL ] + gradn
L ·wL+
+
1
ρGR
[−ρˆG + nG (ρGR)′G + nG ρGR div
′
xG ] + gradn
G ·wG
}
= 0 .
(5.33)
2For further information on the principle of Lagrange multipliers please refer to, e. g., Zinatbakhsh
[182].
66 Chapter 5: Constitutive settings
The usage of a Lagrangean multiplier allows for an incorporation of the saturation con-
straint into the entropy inequality (5.29), which then yields the following formulation:
(TS + P nS I︸ ︷︷ ︸
TSE
) · LS − ρS (ψS)′S − ρS (ηS − P
1
(ρSR)2
dρSR
dθ︸ ︷︷ ︸
ηSE
) (θ)′S +
+ (TL + P nL I︸ ︷︷ ︸
TLE
) · LL − ρL (ψL)′L − ρL ηL (θ)′L + P
nL
ρLR
(ρLR)′L+
+ (TG + P nG I︸ ︷︷ ︸
TGE
) · LG − ρG (ψG)′G − ρG ηG (θ)′G + P
nG
ρGR
(ρGR)′G−
− (pˆL − P gradnL︸ ︷︷ ︸
pˆLE
) ·wL − (pˆG − P gradnG︸ ︷︷ ︸
pˆGE
) ·wG − 1
θ
∑
α
qα · grad θ+
+ ρˆL [ψL − ψG + 1
2
(wG ·wG −wL ·wL) + P 1
ρLR
− P 1
ρGR
] ≥ 0 .
(5.34)
Herein, the relation (3.25) in the form
div
′
xα = I · grad ′xα = I · Lβ (5.35)
was applied. Furthermore, the so-called extra (or effective) quantities are introduced
following the work of Truesdell & Noll [172]. Therewith, the partial stresses, TS, TL and
TG, the partial entropy, ηS, and the total momentum productions, pˆL and pˆG, are split
into their extra part and a part containing the Lagrange multiplier P:
TSE := T
S + P nS I , TLE := TL + P nL I , TGE := TG + P nG I ,
ηSE := η
S − P 1
(ρSR)2
dρSR
dθ
, pˆLE := pˆ
L − P gradnL , pˆGE := pˆG − P gradnG .
(5.36)
By including these extra terms into (5.34), one obtains the final form of the entropy
inequality, as it will be used in the constitutive modelling procedure:
TSE · LS − ρS (ψS)′S − ρS ηSE (θ)′S +TLE · LL − ρL (ψL)′L − ρL ηL (θ)′L + P
nL
ρLR
(ρLR)′L+
+TGE · LG − ρG (ψG)′G − ρG ηG (θ)′G + P
nG
ρGR
(ρGR)′G − pˆLE ·wL − pˆGE ·wG+
+ ρˆL [ψL − ψG + 1
2
(wG ·wG −wL ·wL) + P 1
ρLR
− P 1
ρGR
]− 1
θ
∑
α
qα · grad θ ≥ 0 .
(5.37)
5.3 Determination of constitutive relations 67
5.3 Determination of constitutive relations
The set of governing balance relations for the considered CO2 sequestration model in
(5.3) has to be closed by constitutive relations in order to account for the open fields
described in Section 5.2. Reasonable and thermodynamically consistent constitutive rela-
tions are hereby found by taking into account the basic thermodynamic principles. These
basic principles are the same as in classical continuum mechanics of single-phasic mate-
rials, presented for example by Truesdell [171], Truesdell & Noll [172] and Truesdell &
Toupin [173]. The particular principles of determinism, equipresence, local action, mate-
rial frame indifference and dissipation are introduced and discussed in the following. With
these principles it is possible to obtain constitutive relations, which describe the physical
behaviour of the constituents and the interactions between themselves in the considered
CO2 sequestration model.
5.3.1 The basic thermodynamical principles
Following, e. g., Ehlers [56], the quantities in the entropy inequality (5.37), which cannot
be directly determined from the given motion and temperature fields x = χα(Xα, t) and
θ = θ(x, t), and from the balance relations, are collected in the set of undetermined
response functions R, viz.:
R = {ψα, TαE, ηSE, ηβ, pβE , ρˆβ , qα } . (5.38)
The principle of determinism now states that these response functions must be defined
by the constitutive relations. The response functions R(V) may depend on the whole set
of process or constitutive variables V, both at the initial and at the current state of the
system (principle of equipresence). In the general case of a multiphasic material, this set
is given as, cf. Ehlers [56]:
V = { θα, grad θα, nα, gradnα, ραR, grad ραR, Fα, gradFα, ′xα, grad ′xα, Xα } .
(5.39)
Remembering the preliminary assumptions from Section 5.1, the general set V can be
reduced to a subset V1 of the process variables
V1 = { θα, sL, ρLR, ρGR, FS, wβ, Dα } , where V1 ⊂ V . (5.40)
Herein, the volume fractions nα were replaced solely by the liquid saturation sL by evalu-
ation of the expressions in (3.3)2, (3.4) and (5.12)2. Furthermore, based on the principle
of frame indifference that states that the response functions should be independent of the
position of the observer, cf., e. g., Bowen [26], de Boer and Ehlers [22], Ehlers [56], as well
as Wagner [176], the velocities
′
xα are replaced by the seepage velocities wβ according to
(3.14) and the gradients of the fluid velocities are substituted by the symmetric parts of
the velocity gradients Dβ. Moreover, due to the assumption of a homogeneous distribu-
tion of all material quantities in the reference configuration, the reference position vector
Xα can be removed from (5.40). Based on the principle of local action, the gradients of
68 Chapter 5: Constitutive settings
the process variables can be omitted in (5.40), since only the action at and close to the
material point Pα is of interest, cf. e. g., Ehlers [56], Bowen [26, 27] or Truesdell & Noll
[172].
From the principle of phase separation, i. e., each constituent ϕα only depends on its own
process variables, it follows that the subset of process variables V1 can be distinguished
into parts corresponding only to the specific constituents ϕα, compare with Ehlers [49].
Therefore, it follows for the dependencies of the Helmholtz free energies ψα in case of the
given CO2 sequestration model
ψS = ψS(FS, θ) → (ψS)′S =
∂ψS
∂FS
FTS ·DS +
∂ψS
∂θ
(θ)′S ,
ψL = ψL(ρLR, θ, sL) → (ψL)′L =
∂ψL
∂ρLR
(ρLR)′L +
∂ψL
∂θ
(θ)′L +
∂ψL
∂sL
(sL)′L ,
ψG = ψG(ρGR, θ) → (ψG)′G =
∂ψG
∂ρGR
(ρGR)′G +
∂ψG
∂θ
(θ)′G ,
(5.41)
where the dependency of ψL on both ρLR and sL takes into account the existence of the
two fluid phases and their distribution in the pore space.
The last principle, in particular the principle of dissipation, is applied in the following
Section 5.3.2, where the entropy inequality is evaluated to finally derive the required
constitutive relations.
5.3.2 Exploitation of the entropy inequality
To implement the time derivative of the liquid free energy (5.41)2 into the modified entropy
inequality (5.37), the derivative of the liquid saturation (sL)′L has to be modified by using
the definition of the saturations (3.4)1, viz.:
(sL)′L =
1
nF
[ (nL)′L − sL (nF )′L ] , (5.42)
where the derivative of the fluid volume fraction can be altered with the transformation
relation (3.15) towards
(nF )′L = (1− nS)′L = −(nS)′L = −(nS)′S − gradnS ·wL . (5.43)
Furthermore, (nL)′L is replaced in (5.42) with the help of the liquid mass balance (5.15)
(nL)′L = −
nL
ρLR
(ρLR)′L − nL div
′
xL +
ρˆL
ρLR
. (5.44)
In the same manner, (nS)′S in (5.43) is substituted by the mass balance of the solid
constituent (5.3)1, where, in addition, the dependence of the effective solid density on the
temperature (5.32) is accounted for:
(nS)′S = −
1
ρSR
[
nS (ρSR)′S + ρ
S div
′
xS
]
= − 1
ρSR
[
nS
∂ρSR
∂θ
(θ)′S + ρ
SDS · I
]
. (5.45)
5.3 Determination of constitutive relations 69
The combination of (5.42), (5.43), (5.44) and (5.45) yields the time derivative of the liquid
saturation:
(sL)′L =
1
nF
[
− n
L
ρLR
(ρLR)′L − nLDL · I+
ρˆL
ρLR
−
− sL n
S
ρSR
∂ρSR
∂θ
(θ)′S − sL nSDS · I+ sLgradnS ·wL
]
.
(5.46)
Herein and in (5.45), the relation div
′
xα = Lα · I = Dα · I was used.
Inserting the material time derivatives of the Helmholtz free energies (5.41) into the mod-
ified entropy inequality (5.37), thereby replacing (sL)′L in (5.41)2 by (5.46) and sorting
the terms, yields:(
TSE + s
L n
S
nF
ρL
∂ψL
∂sL
I︸ ︷︷ ︸
TSEmech.
−ρS ∂ψ
S
∂FS
FTS
)
·DS − ρS
(
ηSE − sL
ρLR
(ρSR)2
∂ρSR
∂θ
∂ψL
∂sL︸ ︷︷ ︸
ηSEmech.
+
∂ψS
∂θ
)
(θ)′S +
+
(
TLE + s
L ρL
∂ψL
∂sL
I︸ ︷︷ ︸
TLE dis.
)
·DL − ρL
(
ηL +
∂ψL
∂θ
)
(θ)′L +T
G
E ·DG − ρG
(
ηG +
∂ψG
∂θ
)
(θ)′G+
+
(
P
nL
ρLR
+ sL nL
∂ψL
∂sL
− ρL ∂ψ
L
∂ρLR
)
(ρLR)′L +
(
P
nG
ρGR
− ρG ∂ψ
G
∂ρGR
)
(ρGR)′G+
+ ρˆL
[
ψG − ψL + P 1
ρGR
− P 1
ρLR
+ sL
∂ψL
∂sL
+ 1
2
(wG ·wG −wL ·wL)
]
−
−
(
pˆLE +
sL
nF
ρL
∂ψL
∂sL
gradnS︸ ︷︷ ︸
pˆLE dis.
)
·wL − pˆGE ·wG −
1
θ
∑
α
qα · grad θ ≥ 0 .
(5.47)
Following the evaluation procedure of Coleman & Noll [39], the resulting factors in (5.47)
in front of the process variables Dα, (θ)
′
α, (ρ
βR)′β, ρˆ
L, wβ and q
α must fulfill the entropy
inequality on their own, which ensures the satisfaction of (5.47) for an arbitrary choice of
the free parameters. Wagner [176] argues that the first part in (5.47) can be recognized as
a non-dissipative (equilibrium) part in case of an elastic solid behaviour. In this regard,
equilibrium is achieved if the expression in parentheses in front of the free parameter DS
vanishes. Thus, the solid mechanical extra-stress tensor reads
TSEmech. = T
S
E + s
L n
S
nF
ρL
∂ψL
∂sL
I = ρS
∂ψS
∂FS
FTS . (5.48)
In analogy to the solid mechanical extra-stress tensor, the solid mechanical extra entropy
is defined via
ηSEmech. = η
S
E − sL
ρLR
(ρSR)2
∂ρSR
∂θ
∂ψL
∂sL
= −∂ψ
S
∂θ
. (5.49)
70 Chapter 5: Constitutive settings
Further constraints are found from (5.47) in case of equilibrium assumptions for the fluid
entropies ηβ and the Lagrange multiplier P:
ηL = −∂ψ
L
∂θ
,
ηG = −∂ψ
G
∂θ
,
P = (ρLR)2
∂ψL
∂ρLR
− sL ρLR ∂ψ
L
∂sL
,
P = (ρGR)2
∂ψG
∂ρGR
.
(5.50)
The Lagrange multiplier P in (5.50)4 is distinguished as the effective gas pressure pGR,
viz.:
P = (ρGR)2
∂ψG
∂ρGR
=: pGR . (5.51)
In this context, it is helpful to introduce the capillary pressure pc as the difference between
the effective fluid pressure of the non-wetting fluid, here pGR, and the effective fluid
pressure of the wetting fluid, in this case pLR, cf. Brooks & Corey [31] or Graf [74]:
pc = pGR − pLR . (5.52)
Subsequently, the definition (5.51) is inserted into (5.50)3 and a comparison with (5.52)
yields
P = pGR = (ρLR)2
∂ψL
∂ρLR
− sL ρLR ∂ψ
L
∂sL
= pLR + pc . (5.53)
Hence, the following relations between the liquid effective pressure pLR and the capil-
lary pressure pc together with the derivatives of the liquid Helmholtz free energy ψL are
determined:
pLR := (ρLR)2
∂ψL
∂ρLR
,
pc := − sL ρLR ∂ψ
L
∂sL
.
(5.54)
Dividing (5.51) by ρGR and dividing (5.50)3 by ρ
LR after insertion of (5.54)1 yields
P
ρGR
=
pGR
ρGR
and
P
ρLR
=
pLR
ρLR
− sL ∂ψ
L
∂sL
, (5.55)
which will be needed in the following derivations.
Next, the attention is drawn to the dissipative terms of the entropy inequality (5.47),
which are greater than zero. For convenience, the following two abbreviations for the
liquid extra stress TLE dis. and the liquid extra momentum production pˆ
L
E dis. are postulated:
TLE dis. = T
L
E + s
L ρL
∂ψL
∂sL
I ,
pˆLE dis. = pˆ
L
E +
sL
nF
ρL
∂ψL
∂sL
gradnS .
(5.56)
5.3 Determination of constitutive relations 71
Furthermore, following Dalton’s law3 the overall pore pressure pFR is introduced as the
sum of the effective fluid pressures pβR, weighted by the respective saturations sβ via
pFR = sL pLR + sG pGR . (5.57)
With the previously found relations for the Lagrange multiplier P = pGR and for the
capillary pressure pc, cf. (5.54)2, and with the formulation of the overall pore pressure
pFR (5.57), a second look is taken on the extra terms. Starting with the solid mechanical
extra stress TSEmech., the relations (5.36)1 and (5.48) are combined to retrieve the overall
solid Cauchy stress tensor:
TS = TSEmech. − P nS I− sL
nS
nF
ρL
∂ψL
∂sL
I . (5.58)
Replacing the Lagrange multiplier P and the term including the derivative of the liquid
Helmholtz free energy by (5.54)2 and using the relations for the capillary pressure (5.52),
the partial density (3.5)3, the saturations (3.4), as well as the overall pore pressure (5.57),
yields:
TS = TSEmech. − pFR nS I . (5.59)
This clearly shows that loads imposed on a fluid filled porous body do not stress the solid
material alone, but are also absorbed by the fluids inside the pore space.
Continuing with the fluid extra stress tensors from (5.36)2, (5.56)1 and (5.36)3 the same
replacements are used as in the derivation of the overall Cauchy stress tensor. Conse-
quently, the fluid stresses are given by
TL = TLE dis. − P nL I− sL ρL
∂ψL
∂sL
I → TL = TLE dis. − pLR nL I ,
TG = TGE − P nL I → TG = TGE − pGR nG I .
(5.60)
Next, the solid mechanical extra entropy given in (5.49) and (5.36)4 is reformulated to
obtain the solid entropy. Hereby, (5.55)2 is used, which relates the Lagrange parameter
P , the liquid pressure pLR and the liquid saturation sL :
ηS = ηSEmech. + P
1
(ρSR)2
dρSR
dθ
+ sL
ρLR
(ρSR)2
∂ρSR
∂θ
∂ψL
∂sL
,
→ ηS = ηSEmech. +
pLR
(ρSR)2
∂ρSR
∂θ
.
(5.61)
Thereafter, the fluid extra momentum productions are investigated. The relations (5.36)5,
(5.56)2 and (5.36)5 are processed in the same way as the other extra terms before, yielding
3The original definition of Dalton’s law was established only for ideal gases, cf., e. g., Class [36] or Lewis
& Randall [105]. However, it is still valid here (also for real gases and liquids), since the formulation of pFR
is based on the TPM, more exactly, on the principle of superimposed, statistically distributed continua,
where the interaction between the constituents is controlled by production terms and the porous medium
together with its percolating fluid constituents is treated as a mixture. Thus, the effective pressures pβR
act similar as in an ideal gas, where it is assumed that the different molecules do not “see” each other.
72 Chapter 5: Constitutive settings
for the liquid momentum production
pˆL = pˆLE dis. + P gradnL −
sL
nF
ρL
∂ψL
∂sL
gradnS
→ pˆL = pˆLE dis. + pLR gradnL + pc nF grad sL .
(5.62)
The result of (5.62) can be varied by the relations of the volume fractions, saturations
and capillary pressure as shown in Appendix B.8. In case of the gaseous constituent, the
fluid extra momentum production reads
pˆG = pˆGE + P gradnG → pˆG = pˆGE + pGR gradnG . (5.63)
At this point, the attention is drawn to the discussion of the specific heat capacities cαRV .
Starting with the fluid constituents, the relations for the entropies ηβ, found in (5.50)1
and (5.50)2, can be combined with the definition of the specific heat capacities c
βR
V of the
fluid constituents, see Section 4.3. This yields a formulation for the specific heat capacity
with respect to the Helmholtz free energy ψβ for the individual fluid constituents:
cβRV = θ
∂ηβ
∂θ
= −θ∂
2ψβ
∂θ2
. (5.64)
A similar relation is obtained for the specific heat capacity of the solid constituent. How-
ever, within the derivation, care has to be taken on the solid entropy ηS of (5.61), which
includes the derivative of the solid density with respect to the temperature as an addi-
tional term. In this context, it is postulated that the solid internal energy depends on the
mechanical extra entropy via εS = εS(ηSEmech., ES). Hence, it follows for the solid specific
heat capacity:
cSV =
∂εS
∂θ
=
∂εS
∂ηSEmech.
∂ηSEmech.
∂θ
. (5.65)
The Legendre transformation for the internal energy εS(ηSEmech., ES), cf. Appendix (B.5),
yields
εS = ψS + θ ηSEmech. . (5.66)
Thus, the derivative of the internal energy with respect to the mechanical extra entropy
reads
∂εS
∂ηSEmech.
= θ . (5.67)
With the usage of (5.49), the specific heat capacity of the solid constituent is then found
as
cSV = θ
∂ηSEmech.
∂θ
= −θ ∂
∂θ
(
∂ψS
∂θ
) = −θ ∂
2ψS
∂θ2
. (5.68)
Dissipation inequality
With the help of the abbreviations of the extra terms (5.36) and the relations found in
(5.55), the remaining dissipative parts of the entropy inequality (5.47) can be collected in
5.3 Determination of constitutive relations 73
the so-called dissipation inequality D
D = TLE dis. · LL +TGE · LG − pˆLE dis. ·wL − pˆGE ·wG −
1
θ
∑
α
qα · grad θ +
+ ρˆL
[
ψG − ψL + p
GR
ρGR
− p
LR
ρLR
+ 1
2
(wG ·wG −wL ·wL)
]
≥ 0 .
(5.69)
According to Ehlers et al. [58], who carried out a dimensional analysis, the fluid extra
stress tensors TLE dis. and T
G
E , which describe the frictional stresses, can be neglected in
comparison with the fluid extra momentum productions. Thus,
TLE dis. ≈ 0 , and TGE ≈ 0 . (5.70)
Consequently, the dissipation inequality reduces to
D = − pˆLE dis. ·wL − pˆGE ·wG −
1
θ
∑
α
qα · grad θ +
+ ρˆL
[
ψG − ψL + p
GR
ρGR
− p
LR
ρLR
+ 1
2
(wG ·wG −wL ·wL)
]
︸ ︷︷ ︸
Dmass transition
≥ 0 . (5.71)
To ensure the positive definiteness of (5.71), all terms by themselves must be positive.
For the given problem, a possible way to guarantee this is to require a proportionality
(negative or positive, depending on the sign of the term) between the variables of each
term, leading to a quadratic formulation that assures the positiveness. At first, this
is performed for the third term in (5.71), where the heat flux and the gradient of the
temperature4 should be proportional to each other, i. e., qα ∝ grad θ5. From this, the
well-known Fourier’s law can be developed, which will be shown in Section 5.3.5. In
the same way, proportionalities are defined between the extra momentum productions
of the fluids and the seepage velocities, namely pˆLE dis. ∝ wL and pˆGE ∝ wG. In turn,
these relations will result in the well-known Darcy’s law, which describes the motion of
percolating fluids in a porous medium, due to a pressure gradient. The detailed derivation
will be presented in Section 5.3.4.
A discussion of the liquid mass production ρˆL in the dissipation inequality (5.71) is ad-
dressed in what follows. To begin with, the definition of the chemical potential
µβ = ψβ +
pβR
ρβR
, (5.72)
as it was introduced in Section 4.1.1, will be incorporated in the mass-transition term of
the dissipation inequality, i. e.,
Dmass transition = ρˆL
[
µG − µL + 1
2
(wG ·wG −wL ·wL)
]
. (5.73)
4The division by θ can be omitted in this procedure, since the absolute temperature is always greater
than zero, i. e., θ > 0 K.
5Please keep in mind that this holds for a single common temperature θ = θα.
74 Chapter 5: Constitutive settings
Herein, the quadratic seepage-velocity terms wβ ·wβ can be neglected, since kinetic effects
are usually smaller than thermal ones, cf. Morland & Gray [120]. In consequence, only
the difference in the chemical potentials remains. In order to ensure positive definiteness
of the dissipation inequality, a proportional relation between the mass production ρˆL and
the difference between the chemical potentials is proposed, i. e., ρˆL ∝ (µG − µL). This
relationship will be discussed further in Section 5.4.
5.3.3 Constitutive relations of the solid constituent
The effective and partial densities ρSR and ρS, as well as the solid volume fraction nS
were already defined in Section 5.2.1 by evaluating the solid volume balance with respect
to a thermoelastic solid material. Therein, the solid deformation gradient FS was mul-
tiplicatively split into a purely mechanical and a purely thermal part to account for the
dependence of the solid deformation, both on the displacement uS and the temperature
θ. Furthermore, the Green-Lagrangean deformation tensor ES and after linearisation also
the total strain tensor εS were additively decomposed into mechanical and thermal parts,
cf. (5.6)3 and (5.7)3, discussed in Section 5.2.1. Consequently, an additively splitting
procedure must also be exercised on the solid Helmholtz energy ψS, in order to find the
conjugated variables. Following Ehlers and Ha¨berle [61] and proceeding from the small
strain assumption, the solid free energy can be given as the sum of a purely mechanical
part ψSm and a purely thermal part ψ
S
θ in the geometrically linearised setting
ρS0S ψ
S (εSm, θ) = ρ
S
0S ψ
S
m (εSm) + ρ
S
0S ψ
S
θ (θ) . (5.74)
Subsequently, formulations for the linearised solid extra stress σSEmech and the solid en-
tropy ηS can be obtained from the Helmholtz free energy by exploiting the conjugated
variable pairs {σSEmech, εSm} and {ηSEmech, θ}.
Assuming an elastic material governed by Hooke’s law, the mechanical part in (5.74) is
given by
ρS0S ψ
S
m (εSm) = µ
S εSm · εSm + 12 λS (εSm · I )2 , (5.75)
where µS and λS are the Lame´ constants. The thermal part is found from the specific
heat capacity at constant volume, cf. (5.68),
cSV = − θ
∂2ψSθ
∂θ2
. (5.76)
Substituting εSm by εS − εSθ from (5.7)3 and (5.9)2 and integrating (5.76) together with
the side conditions ψSθ (θ0) = 0 and ∂ψ
S
θ /∂θ(θ0) = 0, one obtains
ρS0S ψ
S
m = µ
S (εS · εS) + 12 λS (εS · I )2 − 3 kS αS ∆θ (εS · I ) + 12 kS (3αS∆θ)2 ,
ρS0S ψ
S
θ = − 12 kS (3αS∆θ)2 − ρS0S cSV (θ ln
θ
θ0
−∆θ) ,
(5.77)
5.3 Determination of constitutive relations 75
where kS = 2
3
µS + λS is the compression modulus. A summation of both relations in
(5.77) yields the free energy of a linear thermoelastic solid skeleton:
ρS0S ψ
S = µS (εS · εS) + 12 λS (εS · I )2 +mSθ ∆θ (εS · I )− ρS0S cSV (θ ln
θ
θ0
−∆θ) .
(5.78)
Therein, mSθ = −3 kS αS is introduced as the stress-temperature modulus.
Proceeding from the basic constitutive relations derived in Section 5.3.2, the second Piola-
Kirchhoff stress SSEmech. is given by
SSEmech. = ρ
S
0S
∂ψS
∂ES
= detFS F
−1
S T
S
Emech. F
T−1
S . (5.79)
Under small-strain conditions, where TSEmech. ≈ SSEmech. ≈ σSEmech. and ES ≈ εS, the first
partial derivative of ρS0S ψ
S with respect to εS yields the solid extra stress σ
S
Emech. :
σSEmech. = 2µ
S εS + λ
S (εS · I ) I+mSθ ∆θ I . (5.80)
Note in passing that (5.80) can also be found directly from (5.75), when (5.75) is differ-
entiated with respect to εSm, which is then substituted by εS − εSθ.
Given the above, the solid mechanical extra entropy is obtained on the basis of (5.49) and
(5.78), viz.:
ηSEmech. = −
∂ψS
∂θ
= − 1
ρS0S
mSθ (εS · I ) + cSV ln
θ
θ0
. (5.81)
Finally, the internal energy of the solid constituent is found by use of the Legendre trans-
formation, cf. (B.5),
εα = ψα + θ ηα . (5.82)
After multiplication with the initial solid density ρS0S and with (5.78) and (5.61) together
with (5.81), the solid internal energy6 reads
ρS0S ε
S = µS (εS · εS) + 1
2
λS (εS · I )2 −mSθ θ0 (εS · I) + ρS0S cSV ∆θ + ρS0S θ
pLR
(ρSR)2
∂ρSR
∂θ
.
(5.83)
5.3.4 Constitutive relations of the fluid constituents
After having found the required constitutive relations of the solid constituent, the eval-
uation of the entropy inequality with respect to the fluid constituents follows. Herein,
the distribution of the percolating fluid phases in the pore space, their respective motion,
the pβR-θ-ρβR behaviour and the thermodynamical quantities, i. e., the Helmholtz free en-
ergy and the entropy, are investigated. The parts concerning the porosity, the capillary
pressure, the saturation and the seepage velocity are of particular interest in groundwater
engineering, e. g., Helmig [84]. The preliminaries for the part, where the thermodynamical
constitutive relations are identified, have already been discussed in Chapter 4.
6Please note in passing that the internal energy is actually a function of the entropy, whereas here it
is given with respect to its conjugated variable the temperature θ. This can be justified with regard to
the measurability, which is simple for the temperature, but more or less impossible in case of the entropy.
76 Chapter 5: Constitutive settings
Porosity and capillary-pressure-saturation relation
To start with, the volume fraction of the overall pore fluid nF , also denoted as the porosity,
was introduced in (3.4)2 and is related to the solid volume fraction n
S by (3.3)2. In
addition, an insertion of the relations for the solid volume fraction (5.12)2 and (5.13)2
yields
nF = 1− nS = 1− nS0S (1− DivS uS + 3αS∆θ) . (5.84)
Further considerations have to be made for the determination of distributive quantities of
the specific fluid constituents, namely nL and nG, or sL and sG. Note that both the fluid
volume fractions and the saturations nβ and sβ, respectively, are inversely proportional
related to each other, cf. (3.4)1. The introduction of the saturation is motivated by the
fact that the distribution of the two fluid constituents in the pore space is only a function
of the pore-fluid pressures pLR and pGR. More precisely, the saturations are a function
of the difference between these pressures, denoted as the capillary pressure, cf. (5.52),
pc = pGR − pLR. A prominent formulation of this relationship is provided by the Brooks
and Corey law7 [31], given by
sLeff =
(pd
pc
)λ
. (5.85)
Therein, sLeff is the effective saturation, and the material parameters pd and λ are the
entry pressure8 and the pore-size distribution index, respectively. In this context, it has
to be mentioned that capillary-pressure-saturation relations are usually regarded as time-
independent, cf. Morrow [122]. Equation (5.85) basically accounts in a homogenised sense
for the surface tension, where the latter acts on the microscale between the wetting and
non-wetting fluids, cf. Figure 5.1. The surface tension σs originates from inter-molecular
microscale
macroscalenon-wetting fluid
wetting fluid
homogenisation
idealised pore
REV
pGR
pLR
r˜
ϑ
σs
p
c
[1
0
5
P
a]
sL [−]
0
0
1
1
2
3
4
0.2 0.4 0.6 0.8
Figure 5.1: Depiction of the surface tension in a capillary tube with the different pressures of
the two fluid phases and the macroscopic equivalent approximation by Brooks & Corey for an
entry pressure of pd = 2.0 · 104 Pa and for two pore-size distribution parameters λ = 2.3 (red)
and λ = 0.6 (green). The residual saturations were chosen to sLres = s
G
res = 0.1.
forces in the involved fluids that tries to minimise the surface area between the two
7A further well-known capillary-pressure-saturation relation is the law by van Genuchten [174], which
in contrast to the Brooks and Corey law exhibits a continuous behaviour around the entry pressure pd. For
more information on the van Genuchten law and its relation to the Brooks and Corey law, the interested
reader is referred to, e. g., Class [36].
8Also sometimes called “bubbling pressure” or “threshold pressure”.
5.3 Determination of constitutive relations 77
fluids and between the fluids and the solid. In consequence, the surface tension causes
a discontinuity in the pressure field across the interface, where the pressure difference
between the low pressure of the wetting fluid and the high pressure of the non-wetting
side depends on the curvature of the interface. The Laplace equation formulates the
relation between the pressure difference and the surface tension σs via
pGR − pLR = 2 σs cosϑ
r˜
= pc . (5.86)
Herein, r˜ is the radius of an idealised pore or the meniscus formed by the wetting fluid.
Moreover, the surface tension σs is the tangential force at the contact line between the fluid
interface and the solid material, and the contact angle ϑ between the fluid interface and
the solid, measured through the wetting fluid, is an indicator for the so-called wettability
of a fluid. Hereby, a contact angle of 0◦ ≤ ϑ < 90◦ identifies the wetting fluid (here ϕL),
an angle of 90◦ < ϑ ≤ 180◦ defines the non-wetting fluid (here ϕG), and for ϑ = 90◦
no capillary forces exist. For further information on this topic, e. g., the temperature
dependence of the surface tension, please refer to Bear [13] or Class [36]. Equation (5.86)
shows that the capillarity effect is more pronounced in small pores, tubes or channels with
a small radius r¯, as it usually is the case in porous media, cf. Figure 5.1.
At this point, the introduction of the phrases imbibition and drainage might be helpful.
These originate from the fields of groundwater and petroleum engineering and describe the
displacement of one fluid phase by the other fluid phase. In particular, the displacement
of the non-wetting phase by the wetting phase is named imbibition, whereas the opposite,
in particular the displacement of the non-wetting phase by the wetting phase, is called
drainage.
It is obvious that a porous medium is not composed of pores with just one single diameter,
as in the idealised pore depicted in Figure 5.1 (middle). Rather, a great variety exists,
where for the considered soil material, the Gauß ian distribution of the diameters is a
function of the assortment and packing of the solid particles, cf., e. g., Fredlund et al.
[69]. The plot on the right side in Figure 5.1 shows the capillary-pressure-saturation
relation, on the one hand, for a well-sorted soil (red curve) with a high value for the
pore-size distribution parameter λ = 2.3, for instance of a sand, and on the other hand,
for a poorly sorted soil (green curve) with many different pore sizes resulting in a small
value in λ = 0.6, e. g., of a clayey silt. Obviously, the pore-size distribution is directly
related to the grain-size distribution. More information on this relationship can be found,
for example, in Folk [67].
The porous medium can be understood as a bundle of idealised pores with different
diameters, where, based on (5.86), large diameters correspond to a low capillary pressure
and vice versa. Consequently, for a high wetting-fluid pressure, i. e., low pc, almost all
pores are liquid-filled implying a high saturation of the wetting fluid. This is also depicted
in the diagram in Figure 5.1 (left). The diagram furthermore shows that in the fully liquid-
and in the fully gas-saturated cases residual saturations, sLres and s
G
res, remain. These are
fluid portions that are trapped by capillary forces in the system and are, therefore, no
longer reachable for pressure-induced saturation changes. Accordingly, only the saturation
in between these residual saturations, denoted as the effective saturation sLeff , is free in
78 Chapter 5: Constitutive settings
the sense of replacement and, thus, is calculated from the Brooks & Corey law, see (5.85).
A relation for the effective saturation is given for example by van Genuchten [174] via
sLeff =
sL − sLres
1− sLres − sGres
. (5.87)
With this relation at hand, the entry pressure pd of (5.85) can be explained as follows.
Considering a porous medium fully saturated by the wetting fluid (sLeff = 1.0), the en-
try pressure equals the minimal pressure needed for the non-wetting fluid to enter the
pore space during drainage. Thus, the entry pressure resembles the capillary pressure
associated with the largest pore in the system, cf. (5.86).
Please note in passing that in reality hysteresis effects in the capillary-pressure-saturation
relation are encountered if successive imbibition and drainage processes occur. This is
caused, for example, by the so-called ink-bottle effect or Haines jump, where the changing
pore diameters trigger local instabilities between neighbouring pores, cf. Morrow [122]. A
second cause is the raindrop effect, which describes the different contact angles ϑ in case
of an advancing or receding wetting phase. For more information on this topic, please
refer to, e. g., Bear [13]. Furthermore, Niessner & Hassanizadeh [125], Joekar-Niasar et al.
[90] and Pop et al. [138] commonly suggested that other factors should be included into
the capillary-pressure-saturation relation as well, such as the specific interfacial area. The
intention behind this suggestion is that the interfacial-area concept inherently incorporates
hysteresis. Since in this monograph no alternating imbibition and drainage processes are
considered, the influence of hysteretic effects are omitted. However, in the derivation of
the constitutive relation for the mass-production term ρˆβ in Section 5.4, a similar approach
for the specific interfacial area, as suggested by Niessner & Hassanizadeh [125], is used.
The required properties for the entry pressure pd and the pore-size distribution index λ
can be found for water-CO2 systems, e. g., in Graf [74]. To the best knowledge of the
author, the capillary-pressure-saturation data for pure CO2 systems, where only liquid
and gaseous CO2 exist, has not been determined so far. Thus, the properties of pd and λ
for water-CO2 systems are also applied to the CO2-only system.
Fluid velocities
In Section 3.2.1 it was explained that the pore-fluid motion is given in an Eulerian set-
ting by the seepage velocity wβ, cf. (3.14). Considering a multiphasic system with two
fluids encountering each other in the pore space, the formulation of the seepage velocity
must also incorporate constraints that reflect the mutual obstruction between these two
percolating fluids. In this regard, it is logical to proceed from the extra momentum pro-
duction terms in the dissipation inequality (5.71), to find thermodynamically consistent
descriptions for wβ. Following, for instance, Graf [74] and Wagner [176], the dissipation
inequality can be satisfied by requesting a negative proportionality between the extra
momentum production and the seepage velocity, i. e.,
pˆLE dis. ∝ −wL and pˆGE ∝ −wG , (5.88)
5.3 Determination of constitutive relations 79
which can be reformulated by introducing the so-called friction tensors Sβf of second-order,
cf. Ehlers [56] and Wagner [176], via
pˆLE dis. = −SLf wL and pˆGE = −SGf wG , (5.89)
respectively. A constitutive relation for the positive-definite friction tensor Sβf includes
physical meaningful parameters for the description of the fluid-flow processes, see Ehlers
[55],
Sβf = (n
β)2 ρβR g (Kβr )
−1 , (5.90)
where Kβr is the second-order tensor of the relative permeability and g = |g| is the scalar
value of the gravitational force. Proceeding from the fluid momentum balance (5.3)3 under
the assumption of creeping-flow conditions (
′′
xβ ≈ 0), thereby neglecting the fluid extra
stresses (5.70) and inserting the derived relations for the liquid momentum production
(5.62)3 and for the gas momentum production (5.63), yields for the liquid constituent
0 = divTL + ρL g + pˆL =
= div (−nL pLR I) + ρL g + pˆLE dis. + pLR gradnL + pc nF grad sL =
= −nL grad pLR + nL ρLR g + pc nF grad sL − (nL)2 ρLR g (KLr )−1wL ,
→ nLwL = − K
L
r
ρLR g
[ grad pLR − ρLR g − p
c
sL
grad sL ] ,
(5.91)
and for the gaseous constituent
0 = divTG + ρG g + pˆG =
= div (−nG pGR I) + ρG g + pˆGE dis. + pGR gradnG =
= −nG grad pGR + nG ρGR g − (nG)2 ρGR g (KGr )−1wG ,
→ nGwG = − K
G
r
ρGR g
[ grad pGR − ρGR g ] .
(5.92)
Consequently, after rigorous exploitation of the dissipation inequality, Darcy -like filter
velocities nβ wβ, cf. Darcy [42], are obtained from the fluid momentum balances. However,
the liquid filter velocity (5.91) contains an additional gradient of the liquid saturation in
the right-most term, which does not appear in the original Darcy filter law. In some works,
e. g., Avci [8], Fredlund & Rahardjo [68], Graf [74] and Wagner [176], it is discussed that
the contribution of this liquid saturation gradient can be omitted, since it has usually
negligible effects on wL compared to the other terms in the liquid filter-velocity law.
However, in this monograph the term is kept in the following.
The tensor of relative permeability Kβr in (5.91) and (5.92) can be related to the ten-
sor of hydraulic conductivity Kβ (also called Darcy permeability) and to the intrinsic
permeability tensor KS of the deformable solid skeleton through
Kβr = k
β
r K
β and Kβ =
ρβR g
µβR
KS . (5.93)
80 Chapter 5: Constitutive settings
Therein, µβR indicates the effective shear viscosity of the pore fluids, cf. Section 4.4, and
kβr are the so-called relative permeability factors, which, following Brooks & Corey [31],
read
kLr = (s
L
eff)
2+3λ
λ ,
kGr = (1− sLeff)2 (1− (sLeff)
2+λ
λ ) .
(5.94)
The relative permeability factors are weighting functions for the hydraulic conductivities
and are depicted in Figure 5.2 for two different pore-size distributions, in particular,
λ = 2.3 and λ = 0.6. Since both fluid phases percolate the same pore space, the two fluids
k
β r
[−
]
k
β r
[−
]
kLr [−]kLr [−]
kGr [−]kGr [−]
sL [−]sL [−]
00
00
11
11
0.2
0.2
0.2
0.2
0.40.4
0.40.4
0.60.6
0.60.6
0.80.8
0.80.8
Figure 5.2: Relative permeabilities kβr after Brooks & Corey for a pore-size distribution index
λ = 2.3 (left) and λ = 0.6 (right), corresponding to the pc-sL-curves in Figure 5.1. The solid line
corresponds to the wetting liquid phase and the dashed line to the non-wetting gaseous phase.
The residual saturations are set to sLres = s
G
res = 0.1.
affect each other in their flow behaviour, cf. Helmig [84]. The equations (5.94) already
show that this behaviour is connected to the effective saturation, which can be explained
as follows. If the liquid saturation is equal to the residual saturations, i. e., sL = sLres, the
liquid phase is immobile and, thus, the liquid relative permeability kLr becomes equal to
zero. In contrast, the gas phase “owns” almost the whole pore space and is free to move
and, consequently, its relative permeability is equal to one. This applies of course for the
opposite case as well. Hence, in (5.94) the effective saturation sβeff corresponds only to
saturations, where both fluids are mobile. Furthermore, Figure 5.2 shows that the grade
of sorting of the soil particles (here defined by the pore-size distribution index λ) has
almost no influence on the gaseous phase, whereas the liquid phase exhibits in the case of
a poorly sorted soil a reduced mobility for low liquid saturations. This is due to the fact
that the wetting fluid enters mainly the small pores, which present a stronger hindrance to
flow, cf. Class [36]. Please note in passing that the equations in (5.94) are only empirical
relations, derived from the capillary-pressure-saturation conditions. More information on
the topic of relative permeabilities, for instance the temperature dependency of kβr , can
be found, e. g., in Class [36], or Bear [13] and citations therein.
Considering isotropic permeabilities, the permeability tensor KS reduces to the scalar
quantity KS. According to Markert [115], the following relation for the deformation-
5.3 Determination of constitutive relations 81
dependent isotropic permeability KS can be given
KS =
( 1− nS
1− nS0S
detFSm
)κ
KS0S , (5.95)
where KS0S is the intrinsic solid permeability in the reference configuration and κ > 0 is
an additional material parameter governing the deformation dependency. Furthermore,
note that detFSm can be substituted within the given geometrically linear approach by
the formula (5.13)2, which reads
lin (detFSm) = 1 + DivuS − 3αS∆θ . (5.96)
It is possible, to estimate the intrinsic solid permeability KS0S based on the effective grain
diameter9 d10, cf. Bear [13]. Note that, therein, also other, more or less empirical, relations
are presented for KS0S. For further insights into the different permeability measures, please
refer to, e. g., Eipper [63] or Markert [115].
Finally, by inserting relations (5.93) into (5.91) and (5.92), while considering isotropic
permeabilities, the filter velocities of the fluid phases in their final forms read
nLwL = −k
L
r K
S
µLR
(grad pLR − ρLR g− p
c
sL
grad sL) ,
nGwG = −k
G
r K
S
µGR
(grad pGR − ρGR g) .
(5.97)
Effective fluid densities
Proceeding from the thermodynamical properties of the fluids, the determination of the
effective fluid density by means of an equation of state (EOS) is given in what follows.
The EOS defines a relation between the effective pressure pβR, the temperature θ and the
effective density ρβR, whereas the former two are governed by the mass and the energy
balances, respectively. Therefore, the EOS must be solved for the density. Note that the
principle of the EOS was elaborately discussed in Chapter 4. For reasons of thermody-
namical consistency in the derivation of the Helmholtz free energy ψβ, the van-der-Waals
EOS (vdW-EOS) is applied, whereas in the following the modification parameters u and
w that were introduced in (4.18) are omitted here for the sake of clarity. Thus, the
substitution of the specific volume vβR in (4.11) by the effective density ρβR yields
pβR = Rβ θ
ρβR
1− b ρβR − a (ρ
βR)2 . (5.98)
Therein, Rβ is the specific gas constant of ϕβ and the material constants for the cohesion
pressure a and the co-volume b are functions of the critical temperature θβcrit and the
critical pressure pβRcrit, proceeding from the CSP:
a =
27 (Rβ θβcrit)
2
64 pβRcrit
, b =
Rβ θβcrit
8 pβRcrit
. (5.99)
9The reference diameter d10 implies that 10% per mass of the grains have a smaller diameter than
that specific diameter.
82 Chapter 5: Constitutive settings
In order to retrieve the effective fluid densities from (5.98), one has to solve a cubic
equation. The real roots of a cubic equation can be derived by the mathematical procedure
presented in the Numerical Recipies by Press et al. [140] and is demonstrated in detail
for the vdW-EOS in Appendix B.10. The resulting three real roots correlate to the
intersections of a constant pressure with the respective isotherm, cf. Figure 5.3. After
pβR
ρβR
pLR
pGR
ρGR ρLR
pRvap
θ
Figure 5.3: Example of the derivation of the density ρβR for a given pressure pβR and temperature
θ with the vdW-EOS and the Antoine equation, both for liquid conditions (dark green) and for
gaseous conditions (light green).
ordering the three roots, or densities, by size, the middle value can be discarded, since
it refers to an unnatural state, see Section 4.1. From the remaining two solutions, the
smallest value equals the density of the gas phase and the largest value belongs to the liquid
phase, as shown in Figure 5.3. To choose the correct density for the given temperature
and pressure conditions, the pressure pβR is compared with the vapour pressure pRvap that
was calculated for the given temperature θ with the Antoine equation (4.19). If pβR > pRvap
(dark green in Figure 5.3), the fluid is in its liquid state, cf. Figure 4.2(a), and the final
solution is the liquid density ρLR. In the other case, i. e., pβR < pRvap (light green), the
gaseous density ρGR would be the solution.
Helmholtz free energies of the fluid constituents
Next, the Helmholtz free energies ψβ are derived for the fluid constituents ϕβ. The deriva-
tives of the final form of the free energies must satisfy the equations (5.64), (5.54)1 and
(5.54)2 in case of the liquid constituent and the equations (5.64) and (5.51) for the gaseous
constituent. In this regard, the procedure proposed by Ghadiani [71] and Graf [74] is
executed. At first, the partial derivatives of the Helmholtz free energies are indefinitely
integrated. Then, by properly combining these integrated forms, the integration constants
can be identified and the final relations for ψL and ψG are obtained. In this connection, it
will also be explained, why the simpler vdW-EOS is chosen instead of a more exact EOS
like the Peng-Robinson or Soave-Redlich-Kwong EOS.
Commencing with the liquid constituent ϕL, the derivative of the Helmholtz free energy
5.3 Determination of constitutive relations 83
ψL with respect to the liquid saturation sL (5.54)2 reads
pc = −sL ρLR ∂ψ
L
∂sL
→ ∂ψ
L
∂sL
= − p
c
ρLR
(sL)−1 . (5.100)
By inserting the inverted form of the Brooks & Corey law (5.85) to replace the capillary
pressure,
pc = pd (s
L
eff)
−1/λ = ρLR g hd (s
L
eff)
−1/λ , (5.101)
where the entry pressure is substituted by pd = ρ
LR g hd, g is the norm of the local
gravitational force g and hd is the macroscopic entry-pressure head, one obtains from
(5.100)2
∂ψL
∂sL
= −g hd (sL)− 1+λλ . (5.102)
Hereby, it was assumed that the difference between the liquid saturation sL and the
effective liquid saturation sLeff can be neglected in the derivation of ψ
L. Now, (5.102) is
indefinitely integrated
ψL = g hd λ (s
L)−
1
λ + f , (5.103)
wherein, the integration constant f = f(ρLR, θ) is a function of the remaining dependen-
cies, ρLR and θ. Proceeding from the derivative of the Helmholtz free energy with respect
to the effective liquid density (5.54)1
pLR = (ρLR)2
∂ψL
∂ρLR
→ ∂ψ
L
∂ρLR
=
pLR
(ρLR)2
, (5.104)
a comparison with the derivative of (5.103) with respect to the effective liquid density
yields
∂ψL
∂ρLR
=
∂f
∂ρLR
!
=
pLR
(ρLR)2
. (5.105)
Subsequently, the effective pressure pLR is substituted in (5.105) by the vdW-EOS (5.98),
∂f
∂ρLR
=
RL θ
ρLR − b (ρLR)2 − a . (5.106)
Indefinite integration over ρLR results in
f = RL θ ln
ρLR
1− b ρLR − a ρ
LR + g , (5.107)
where g = g(θ) is a temperature dependent integration constant. Inserting f back into
(5.103) and taking the first and second derivatives with respect to the temperature θ gives
∂ψL
∂θ
= RL
ρLR
1− b ρLR +
∂g
∂θ
,
∂2ψL
∂θ2
=
∂2g
∂θ2
.
(5.108)
84 Chapter 5: Constitutive settings
Remark: At this point, the discrepancy between a more elaborate and exact EOS on the
one hand and a thermodynamically consistent derivation of ψβ on the other hand becomes
obvious. The more exact equations of state, e. g., the PR-EOS (4.13), or the SRK-EOS
(4.16), exhibit a dependency on the square root of the temperature in the term containing
the cohesion pressure a, cf. (4.14)3 and (4.17)3. Due to the square root, this term would
not vanish for the first and second derivatives with respect to the temperature, as it does
in case of the vdW-EOS (5.108). Consequently, it is not possible to find a closed form for
the Helmholtz free energies of the pore fluids with the PR-EOS or SRK-EOS.
Continuing with the formula for the specific heat capacity (5.64) and comparing it with
(5.108)2 yields
cLRV = −θ
∂2ψα
∂θ2
→ ∂
2ψα
∂θ2
= −c
LR
V
θ
!
=
∂2g
∂θ2
. (5.109)
Double integration of (5.109)2 together with the side conditions ψ
β
θ (θ0) = 0 and
∂ψβθ /∂θ(θ0) = 0, cf. (5.77), determines the integration constant g(θ). It reads
∂g
∂θ
= −cLRV ln
θ
θ0
,
g = −cLRV (θ ln
θ
θ0
−∆θ) .
(5.110)
Finally, by assembling the derived integration constants f and g together with (5.103),
the Helmholtz free energy of the liquid phase reads
ψL = g hd λ (s
L)−
1
λ +RL θ ln
ρLR
1− b ρLR − a ρ
LR − cLRV (θ ln
θ
θ0
−∆θ) . (5.111)
It can be shown that this relation of ψL yields again the equations (5.64), (5.54)1 and
(5.54)2 through derivatives with respect to θ, ρ
LR and sL.
In the same way, the gaseous Helmholtz free energy ψG is derived. However, it lacks
the dependency on the saturation, hence, simplifying the calculations. Here, the starting
point is the derivative of ψG with respect to the effective gas density, cf. (5.51)
pGR = (ρGR)2
∂ψG
∂ρGR
→ ∂ψ
G
∂ρGR
=
pGR
(ρGR)2
=
RG θ
ρGR − b (ρGR)2 − a , (5.112)
where the vdW-EOS (5.98) was implemented for pGR. It follows from the indefinite
integration of (5.112) over ρGR
ψG = RG θ ln
ρGR
1− b ρGR − a ρ
GR + h (5.113)
with the integration constant h = h(θ). Before (5.113) can be combined with the specific
heat capacity (5.64), the first and second derivatives with respect to the temperature are
computed, yielding
∂ψG
∂θ
= RG
ρGR
1− b ρGR +
∂h
∂θ
,
∂2ψG
∂θ2
=
∂2h
∂θ2
.
(5.114)
5.3 Determination of constitutive relations 85
Double integration, thereby utilising the specific heat capacity cGRV , yields the integration
constant h
h = −cGRV (θ ln
θ
θ0
−∆θ) , (5.115)
which is afterwards substituted into (5.113) to derive the final form of the Helmholtz free
energy of the gas phase:
ψG = RG θ ln
ρGR
1− b ρGR − a ρ
GR − cGRV (θ ln
θ
θ0
−∆θ) . (5.116)
If the two fluid phases ϕL and ϕG belong to the same substance, the constants a and b
in (5.111) and (5.116) and the specific gas constant Rβ are equal for both phases. How-
ever, the specific heat capacities cLRV and c
GR
V remain different, due to their temperature
dependency, cf. Section 4.3 for details.
Entropies and internal energies
With the fluid Helmholtz free energies at hand, cf. (5.111), (5.116), and applying the
thermodynamic potentials (5.50)1,2, the fluid entropies are obtained:
ηβ = −Rβ ln ρ
βR
1− b ρβR + c
βR
V ln
θ
θ0
. (5.117)
The internal energies of the fluid constituents can then be determined by using the Leg-
endre transformation (B.5) and they read
εL = − a ρLR + cLRV ∆θ + g hd λ (sL)−
1
λ ,
εG = − a ρGR + cGRV ∆θ .
(5.118)
5.3.5 Constitutive relations of the overall aggregate
Next, the constitutive relations for the mixture terms appearing in the mixture momentum
and the mixture energy balances are defined. To begin with, the overall pore pressure
pFR is given by Dalton’s law, which was already mentioned in (5.57)
pFR = sL pLR + sG pGR . (5.119)
Subsequently, the constituent heat influx vectors qα are regarded. On the basis of the
dissipation inequality (5.71)3, one easily concludes to the applicability of Fourier’s law for
the heat influx vectors qα. Thus,
qα = −Hα grad θ , (5.120)
where Hα is the partial heat-conduction tensor, which is related to the effective heat-
conduction tensor HαR by
Hα = nα HαR . (5.121)
86 Chapter 5: Constitutive settings
Moreover, in case of isotropic heat conduction the effective heat-conduction tensor HαR
reduces to
HαR = HαR I . (5.122)
Note that the identification of the isotropic thermal conductivity HαR in terms of its
density and temperature dependency was extensively discussed in Section 4.4.
5.4 Phase transition between the gaseous and liquid
phases of a single substance
A major goal of this monograph is to model the phase-transition process between two
different phases of a single substance, e. g., CO2. The quantities that govern this process
are the so-called mass-production terms ρˆβ, which couple the constituent mass balances
of the respective fluid constituents ϕβ (5.16), but also appear in the mixture momentum
balance (5.24) and the mixture energy balance (5.27). The appearance of ρˆβ in the
latter two equations results from the momentum and the heat being transported by the
transferred mass between the two phases over the interface Γ. The mass production ρˆβ
is the last remaining variable that has to be defined by a thermodynamically consistent,
constitutive relation, which will be accomplished within this chapter.
Before the newly developed derivation of the constitutive relation is presented, a brief
overview of the existing works on this topic is given. Also, the difference in describing
phase-change processes between mixtures and between phases of one single matter shall be
pointed out. Thereafter, the new constitutive law for ρˆβ , which is based on the introduc-
tion of a singular surface for the interface, is discussed thoroughly. Finally, a switching
criterion is specified, which controls the phase-transition process within the numerical
implementation.
5.4.1 State of the art
First articles that investigate phase transitions between percolating pore fluids in non-
deformable, porous aggregates date back to Lykov [112] and continue with the group
around Be´net [35, 108, 109, 153] or Bedeaux [14]. Hassanizadeh and Gray together with
Niessner [75, 81, 125, 126] also looked extensively at heat and mass transfer in porous
media, where in their more recent works the focus lies on the interfacial area. Nuske et al.
[130] also studied the phase-transition process, where the formulation of the interfacial
area is based on the work of Hassanizadeh, Niessner and Gray. These latter works bear a
certain similarity to the model to be presented in this monograph, but are not based on
the TPM and describe the phase-transition process between mixtures and not between
the phases of a single material.
In the literature one can also find contributions with closer connection to this monograph,
i. e., discussions of the phase-transition process in deformable porous media based on the
TPM. The first works in this context are provided by de Boer and coworkers [19, 21, 23].
Basically in all of these works, the mass production term ρˆβ is derived from the dissipative
5.4 Phase transition between the gaseous and liquid phases of a single substance 87
part of the entropy inequality, where the dissipation due to phase transition is recapped
here, cf. (5.73),
Dmass transition = ρˆL
[
µG − µL + 1
2
(wG ·wG −wL ·wL)
]
. (5.123)
To satisfy the inequality constraint, proportionality is demanded between the liquid mass
production and the term in brackets containing the differences in the chemical potentials
and the kinetic energies, i. e.,
ρˆL ∝
[
µG − µL + 1
2
(wG ·wG −wL ·wL)
]
. (5.124)
Furthermore, by introducing the so-called mass-transfer coefficient κ as a positive response
parameter, a constitutive relation for the liquid mass production is found
ρˆL = κ
[
µG − µL + 1
2
(wG ·wG −wL ·wL)
]
. (5.125)
If the difference in the squares of the kinetic energies in (5.125) is neglected, the resulting
relation equals the one found by de Boer & Kowalski [23], viz.:
ρˆL = κ (µG − µL) . (5.126)
The discrepancy to the other works by de Boer [19, 21] originates from the fact that
they did not neglect solid mass production ρˆS and used separate temperatures θα for all
constituents ϕα, which read
ρˆL = −κL
( µL
θL
− µ
S
θS
)
and ρˆG = −κG
( µG
θG
− µ
S
θS
)
, (5.127)
where in [19], also the kinetic-energy term from (5.125) appears. More recent works on
phase transitions using this approach are provided by Ricken & Bluhm and coworkers
[17, 102, 148, 149], however, applied to thawing and freezing processes. Therein, the
response parameter κ is derived by evaluating the energy balance and postulating a
proportionality of κ to the divergence of the mixture heat flux
∑
α divq
α.
The literature also provides examples which descend from the same theory as used in
the present work, i. e., the evaluation of the dissipation inequality in the framework of
the TPM, but are not formulated for phase-transition systems in the thermodynamical
sense. In this regard, Steeb & Diebels [168] present a continuum-mechanical model for
erosion and deposition, growth and atrophy, and remodelling phenomena. The continuum-
mechanical model for bacterial-driven methane oxidation in landfill cover-layers by Ricken
et al. [151, 152] is a further example. This group has also published a work, based
on the same continuum-mechanical principles, for the phenomenological description of
transversely isotropic, saturated, biological tissues including the phenomena of growth
[150]. In this context, the contribution of Kowalski [101] on the simulation of drying
processes shall be mentioned as an example for the actual application of the phase-change
model to real physical problems.
If the works on thermodynamical phase-transition processes are studied thoroughly, it
becomes obvious that the main problem in the constitutive relation (5.127) for the mass
88 Chapter 5: Constitutive settings
production is the definition of the mass-transfer coefficient κ. So far, the only thermody-
namical explanation for κ exists for the freezing/thawing problem, cf. Ricken and Bluhm
[17, 102, 148, 149]. A comparison of equation (5.127) with the classical thermodynamical
description of mass transport during phase transition in a two-fluid phase system, based
on the two-film idea of Lewis & Whitman [106], shows strong similarities between both
approaches.
interface Γinterface Γ
equilibrium, µΓ
equilibrium, µΓ
linearised
homogenisation
∆x→ 0
ρβRρβR
ρβ
′R ρβ
′R
µβµβ
µβ
′
µβ
′
j
jβ
jβ
′
ϕβϕβ
ϕβ
′
ϕβ
′
∆xβ ∆xβ
′
Figure 5.4: Sketch of the two-film principle that describes transition processes between two
phases ϕβ and ϕβ
′
, following the idea of Lewis & Whitman [106].
In the two-film theory, two bulk phases (e. g., ϕβ and ϕβ
′
) with different chemical poten-
tials, µβ and µβ
′
, are divided by an interface Γ, cf. Figure 5.4. Due to the difference in the
chemical potentials a chemical non-equilibrium exists, which induces mass transfer over
the interface. However, locally at the interface equilibrium still remains. Furthermore, it is
supposed that the interface is connected on both sides to boundary layers, which indicate
the regions, where the motion of convection is small compared to the motion in the bulk
phase. In these boundary layers, the phase transition becomes obvious within the changes
in density and chemical potential. For simplicity, stationary films with thicknesses ∆xβ
and ∆xβ
′
are assumed on both sides of the interface and the change in process variables is
linearised, cf. purple dashed line in Figure 5.4. The film thickness couples the film-theory
to a specific problem at hand, for example, to specific flow characteristics, where the flow
velocity parallel to the interface influences the transition. Consequently, the mass fluxes
on either side of the interface depend on the film thicknesses ∆xβ and ∆xβ
′
, the local
mass-transfer coefficients Dβmass and D
β′
mass, and the gradients of the chemical potential
∆µβ and ∆µβ
′
, hence,
jβ = −D
β
mass
Rθ
∆µβ
∆xβ
!
= jβ
′
= −D
β′
mass
Rθ
∆µβ
′
∆xβ′
. (5.128)
Herein, in addition, it was included that the interface is mass-less and cannot store any
mass, thus jβ = jβ
′
. The change in the chemical potential is given as ∆µβ = µβ −µΓ and,
in analogy, as µβ
′
= µΓ − µβ′.
After a homogenisation by taking the limes of the film thicknesses ∆xβ → 0 and ∆xβ′ → 0
and by furthermore assuming that the local mass-transfer coefficients are equal on both
sides of the interface, i. e., Dβmass = D
β′
mass = Dmass, it follows for the mass flux
j = −Dmass
Rθ
(µβ − µβ′) . (5.129)
5.4 Phase transition between the gaseous and liquid phases of a single substance 89
This relation possesses great similarity to the equation for ρˆL (5.127), which was derived
from the dissipation inequality. It seems that the exploitation of the dissipation inequality
is a viable procedure for the determination of ρˆL. Consequently, one could comprise the
terms standing in front of the difference in chemical potential in (5.127) and (5.129) to
find κ, i. e.,
κ =̂
Dmass
Rθ
. (5.130)
At this point, a closer look shall be taken at the local mass-transfer coefficient Dmass in
(5.130). Dmass is a measure for the resistance that fluid molecules of one phase experience
while moving through the boundary layer of the other phase. Hence, Dmass is the common
diffusion coefficient used for the description of diffusion in mixtures, also known as Fick’s
law. However, in this monograph it is intended to find a constitutive relation for the
description of mass transfer between phases of the same fluid substance, which means
that only pure phases but no mixtures are regarded. This implies in turn that boundary
layers on both sides of the interface consist of the same molecules. Thus, no resistance
for the movement of the molecules exists and the diffusion coefficient is equal to zero
Dmass = 0. Consequently, the mass flux j or the mass production ρˆ
L would equal zero,
which shows that this approach is not applicable for the description of the transition
between phases of a single substance. Unfortunately, this issue is not communicated in
most of the previously named articles, where (5.127) is used. Therefore, in the next
section, a new and different way for the determination of a constitutive relation for the
mass-production term in case of single-substance phase transition is presented.
5.4.2 Development of the constitutive relation for the mass-
production term
In order to find a constitutive relation for the mass-production term appearing in the
balance relations as the macroscopic density production ρˆα, cf. (3.63)1 and (3.36), a
closer look is taken at the microstructure of the porous medium, cf. Figure 5.5. The
L
LL
G
G
G
S
n+Γ
n−Γ
n−
n+
Γ
ΓdaΓREV
+
−
B+
B−
S+
S−
ˆ̺LΓ
ˆ̺GΓ
Figure 5.5: Illustrations of the gas-liquid interface Γ in a porous microstructure and the mass
transfer ˆ̺βΓ between the gaseous and liquid phases across the singular surface.
phase-transition process is characterised by a mass transfer over the local interfaces Γ,
90 Chapter 5: Constitutive settings
separating the gas from the liquid. The solid phase remains continuous over Γ, since it is
not affected by this mass exchange. For the description of this process, a bicomponent,
triphasic aggregate of a porous rock and a single fluid matter10 (component) ϕFM is
concerned, where the three phases are given by the porous solid ϕS together with the
liquid phase ϕL and the gas phase ϕG of component ϕFM . Furthermore, the interface is
mathematically represented by a singular surface, cf. Section 3.4. Hence, local jumps in
physical quantities over the singular surface are described by the discontinuous part of
the local mass balance (3.63)2. Since the solid phase is not affected by the jump across
Γ, only the jump of the fluid matter ϕFM has to be observed. Additionally, it is assumed
that ϕFM exists in B+ only in its gaseous phase and in B− only in its liquid phase, cf.
right part of Figure 5.5. The following derivations are taken from the contribution by
Ehlers and Ha¨berle [61] and otherwise.
Applying the mass-jump equation (3.63)2 to ϕ
FM yields
q
ρFM wFMΓ
y · nΓ = (ρFM+w+FMΓ − ρFM−w−FMΓ) · nΓ = 0 . (5.131)
With ρFM+w+FMΓ = ρ
GwGΓ and ρ
FM−w−FMΓ = ρ
LwLΓ, (5.131) becomes
(ρGwGΓ − ρLwLΓ) · nΓ = 0 . (5.132)
Following the work of Whitaker [178], the interfacial mass-transfer ˆ̺βΓ of ϕ
β is specified
through
ˆ̺βΓ := ρ
β wβΓ · nβΓ , (5.133)
such that
ˆ̺GΓ = ρ
FM+w+FMΓ · nFM+Γ = ρGwGΓ · nΓ ,
ˆ̺LΓ = ρ
FM−w−FMΓ · nFM−Γ = − ρLwLΓ · nΓ
and ˆ̺GΓ + ˆ̺
L
Γ = 0 ,
(5.134)
where nFM+Γ = nΓ and n
FM−
Γ = −nΓ have been used. From (5.134)3, it is clearly seen
that, for example during evaporation, mass is removed from the liquid phase and added
to the gaseous phase through ˆ̺GΓ = − ˆ̺LΓ . Since the interfacial mass production ˆ̺βΓ and
the continuum mass production ρˆβ have to be equivalent in the sense that ˆ̺GΓ leaves B+
and generates the density production ρˆL in B− and vice versa, ˆ̺βΓ and ρˆβ can be related
to each other by
ρˆG dv =̂ ˆ̺LΓ daΓ and ρˆ
L dv =̂ ˆ̺GΓ daΓ , (5.135)
where dv is the unit volume of the REV and daΓ is the unit area of the interface in the
REV given by
daΓ =
∫
AREV
daΓREV . (5.136)
10The universal component ϕFM is chosen here, to keep the generality of the model. Hence, it can
be used for different fluid matters by only adapting the respective material parameters. Later in this
monograph, the universal component will be identified with CO2, ϕ
FM = ϕCO2 .
5.4 Phase transition between the gaseous and liquid phases of a single substance 91
Interfacial Area
Based on (5.135) and the work of Niessner and Hassanizadeh [124], the so-called interfacial
area aΓ is introduced as the density of the internal phase-change surfaces measured with
respect to the unit volume of the REV:
aΓ :=
daΓ
dv
. (5.137)
The interfacial area aΓ comprises all menisci separating the liquid and the gaseous phases
in the pore space of the REV, cf. Figure 5.5. In turn, the menisci depend on the surface
tensions of the involved phases and the pore structure, or in other words, on the capillary-
pressure relation, which is given in (5.101) as a function of the effective saturation. This
justifies that aΓ can be assumed as aΓ = aΓ(s
L
eff) or as aΓ = aΓ(s
L), respectively. Finally, it
has to be mentioned that the influence of the common lines on the phase-change process,
i. e., the influence of the contact lines of the interface with the solid material, is neglected
in this study.
The first to incorporate interfacial areas into a macroscopic formulation of multiphase flow
were Hassanizadeh and Gray [80, 81]. It is also claimed by Niessner & Hassanizadeh [125],
Joekar-Niasar et al. [90] and Pop et al. [138] that the inclusion of the interfacial area into
the capillary-pressure-saturation relation removes the hysteresis, which appears otherwise
in the classical pc-sL-relationship during alternating drainage and imbibition processes.
However, the effect of hysteresis is not observed in this monograph, since either imbibition
or drainage cases are regarded in the numerical examples and no alteration between these
two effects is considered.
To calculate the interfacial area, different approaches exist. In [124], Niessner and Has-
sanizadeh present an empirical derivation of the interfacial area aΓ based on data obtained
by Joekar-Niasar et al. [90], using pore-network model tests introduced by, e. g., Sahimi
[157]. Nuske et al. [130] extended this model by including the interfaces between the
fluid phases and the solid phase. Another approach, which is also based on the work of
Niessner and Hassanizadeh, but uses a lattice Boltzmann simulation to derive the mate-
rial parameters, is given by Ahrenholz et al. [3]. In this monograph, use is made of an
approximation by Graf [74], who described the interfacial area between the fluid phases
as a function of the liquid saturation.
The basic idea of the latter contribution is to approximate the pore space by introducing a
sphere with the pore-space equivalent volume V F , which is composed of the fluid volumes
V L and V G that are again given as a function of the filling heights hL and hG, cf. Figure
5.6 (left):
V F = V L + V G = 4
3
π (r˜F )3 ,
V β = 1
3
π (hβ)2 (3 r˜F − hβ) , β = {L, G} .
(5.138)
Furthermore, the surface area between the liquid and gaseous volumes reads
AGL = π hβ (2 r˜F − hβ) , (5.139)
where in (5.138) and (5.139) r˜F is the radius of the volume-equivalent sphere and hβ has
to be taken as the larger value out of hL and hG such that hβ ≥ r˜F .
92 Chapter 5: Constitutive settings
r˜F
hG
hL
ASG
ASL
AGL
ϕG
ϕL
sL [−]
a
Γ
[
1 m
]
0
0 10.2 0.4 0.6 0.8
2000
4000
6000
8000
10000
12000
Figure 5.6: Volume-equivalent sphere of the pore space (left) and interfacial area aΓ(s
L) pre-
sented for d50 = 0.06mm and n
S = 0.6 (right).
Based on (3.4), the saturation sβ is defined by the local ratio of V β over V F . Thus,
sβ =
nβ
nF
=
V β
V F
=
(hβ)2 (3 r˜F − hβ)
4 (r˜F )3
. (5.140)
Proceeding from (5.140), the filling height hβ can be determined as a function of the
saturation sβ and the equivalent pore-fluid radius r˜F :
hβ =
[36864
8910
(sβ)3 − 18432
2970
(sβ)2 +
12084
2970
sβ
]
r˜F ≈
≈ [4.137 (sβ)3 − 6.206 (sβ)2 + 4.069 sβ] r˜F .
(5.141)
Since the distribution, sizes and forms of the solid particles, as well as the tortuosity
and connectivity of the pores are unknown, it is not possible to calculate the exact pore
volume V F . Therefore, an appropriate approximation is needed. Comparing a spherical
pore with radius r˜F with a characteristic, spherical solid particle with radius r˜S yields
V S = nSV
V F = nFV
 such that nFnS = V FV S = (r˜F )3(r˜S)3 and, thus, r˜F = (nFnS )1/3r˜S .
(5.142)
Proceeding from d50 as the medial grain diameter of a granular soil, one ends up with
r˜F = 1
2
(nF
nS
)1/3
d50 . (5.143)
Given the above results, the interfacial area aΓ can be specified. Based on (5.137), (5.138)1
and (5.139) with AΓ = A
GL, aΓ can be obtained as a function of s
β via
aΓ(s
β) =
daΓ
dV
=
AΓ
V
= nF
AGL
V F
=
=
3nF hβ (2 r˜F − hβ)
4 (r˜F )3
.
(5.144)
5.4 Phase transition between the gaseous and liquid phases of a single substance 93
While r˜F , at a certain state of the solid deformation, is a function of d50, h
β depends on
sL, which is given as a function of the capillary pressure pc. As a result, the aΓ-s
L curve,
cf. Figure 5.6 (right), is comparable to the curve found by Joekar-Niasar et al. [90], when
their three-dimensional plane (aΓ-s
L-pc) is cut at a certain value of pc.
Interfacial mass transfer
Given the equation of the interfacial area (5.137), the macroscopic density production ρˆβ
and the interfacial mass-transfer ˆ̺βΓ are related to each other through
ρˆG = aΓ ˆ̺
L
Γ and ρˆ
L = aΓ ˆ̺
G
Γ . (5.145)
After aΓ has now been calculated from (5.144), it is still necessary to determine the
interfacial mass transfer ˆ̺βΓ such that ρˆ
β can be fixed, cf. (5.145). For this purpose, one
proceeds from the energy jump across the singular surface Γ, cf. (3.65)2,
r
ρα (εα + 1
2
′
xα · ′xα)wαΓ −Tα ′xα + qα
z
· nΓ = 0 , (5.146)
where Tα = (Tα)T has been used according to (3.44). Applying (5.146) to the solid
constituent ϕS, one obtains with the aid of (3.44):
r
ρS (εS + 1
2
′
xS · ′xS)wSΓ − (TSE − nS pFR)
′
xS + q
S
z
· nΓ = 0 . (5.147)
Since the solid material is inert and not involved in the phase-transition process, all terms
related to the solid material itself are considered continuous over the singular surface.
Thus, it remains that
nS
q
pFR
y ′
xS · nΓ = 0 . (5.148)
However, since the solid velocity
′
xS is not necessarily perpendicular to the single-surface
normal nΓ, it is obvious to require
q
pFR
y
= 0 . (5.149)
In the next step, (5.146) has to be applied to the fluid component ϕFM . Thus,
r
(εFM + 1
2
′
xFM · ′xFM) ρFM wFMΓ −TFM ′xFM + qFM
z
· nΓ = 0 . (5.150)
Following the same procedure as to obtain (5.132) from (5.131), with the gaseous phase of
ϕFM only existent in B+ and the liquid phase of ϕFM only in B−, the jump in the energy
balance becomes
(εG + 1
2
′
xG · ′xG) ρGwGΓ · nΓ − (TG ′xG − qG) · nΓ−
− (εL + 1
2
′
xL · ′xL) ρLwLΓ · nΓ + (TL ′xL − qL) · nΓ = 0 .
(5.151)
94 Chapter 5: Constitutive settings
Applying (5.134)1, 2 to (5.151) yields
(εG + 1
2
′
xG · ′xG) ˆ̺GΓ − (TG
′
xG − qG) · nΓ+
+ (εL + 1
2
′
xL · ′xL) ˆ̺LΓ + (TL
′
xL − qL) · nΓ = 0 .
(5.152)
Finally, this equation can be solved with the help of (5.134)3, in the form ˆ̺
G
Γ = − ˆ̺LΓ ,
such that
ˆ̺LΓ =
(nG pGR
′
xG − nL pLR ′xL + qG − qL) · nΓ
εG − εL + 1
2
′
xG · ′xG − 12
′
xL · ′xL
, (5.153)
where (5.60)2, 4 together with (5.70) have been used to substitute the partial stresses T
G
and TL.
In (5.152), the difference εG − εL of internal energies can be substituted by the Gibbs
energy (enthalpy) difference ζG − ζL, through a Legendre transformation, cf. (B.5), i. e.,
ζβ = εβ +
pβR
ρβR
→ εβ = ζβ − p
βR
ρβR
. (5.154)
Furthermore, since the effective pore pressure pFR has been found to be jump-free, cf.
(5.149), one can conclude that
q
pFR
y
= pFR+ − pFR− = 0 with
 p
FR+ = sG pGR
pFR− = sL pLR
. (5.155)
Following this, the partial pore pressures sG pGR and sL pLR and, as a result, the partial
pressures nG pGR and nL pLR are equivalent, such that
nG pGR
′
xG − nL pLR ′xL = nL pLR ( ′xG − ′xL) = nL pLR (wG −wL) . (5.156)
Inserting (5.154) and (5.156) into (5.152) finally yields
ˆ̺LΓ =
[nL pLR (wG −wL) + qG − qL ] · nΓ
∆ζvap − pGRρGR + p
LR
ρLR
+ 1
2
′
xG · ′xG − 12
′
xL · ′xL
, (5.157)
where ∆ζvap := ζ
G − ζL is the latent heat or the enthalpy of evaporation, cf. Section 4.2.
Equation (5.157) is often simplified with the argument that differences in mass-specific
pressures p
LR
ρLR
− pGR
ρGR
and in mass-specific kinetic energies 1
2
′
xG · ′xG − 12
′
xL · ′xL are small
in comparison with the latent heat ∆ζvap, cf. Morland & Gray [120]. Following this
argumentation, (5.157) recasts to
ˆ̺LΓ =
[nL pLR (wG −wL) + qG − qL ] · nΓ
∆ζvap
. (5.158)
5.4 Phase transition between the gaseous and liquid phases of a single substance 95
Furthermore, if phase transitions are mainly induced by heat, the pressure-dependent
term in the nominator of (5.158) is negligible compared to the heat conduction and
(5.158) further reduces to
ˆ̺LΓ =
(qG − qL) · nΓ
∆ζvap
. (5.159)
Note that this equation, is well known from classical thermodynamics, e. g., Silhavy [165].
Finally, the interfacial normal nΓ, cf. Figure 3.3, has to be found. For this purpose, use
is made of the fact that the interface is always oriented perpendicular to the gradient of
the fluid densities grad ρβR. Thus, similar to the level-set method, grad ρβR is calculated
and normalised, to provide a simple way for the determination of nΓ.
With (5.145) and (5.159) the constitutive relation for the mass production is provided by
ρˆL = aΓ ˆ̺
G
Γ = −aΓ
(qG − qL) · nΓ
∆ζvap
. (5.160)
For the numerical implementation of the mass production between the fluid phases it is
necessary to define a switching criterion, which controls if phase transition occurs or not.
5.4.3 Switching criterion for the mass transition
In analogy to models for plasticity, where a yield surface restricts the stress domain to
the elastic region, cf., e. g., Ehlers [55], the surfaces in the phase diagram in Figure 4.1
divide the pβR-θ-ρβR regions of the different phases. Since in the present case only the
liquid and gas regions are of interest, it suffices to look at the 2-d phase diagram in
Figure 4.2 (a), where the pressure pβR is depicted over the temperature θ. Therein, the
vaporisation curve can be calculated from the Antoine equation (4.19), which relates the
vapour pressure pRvap to the temperature θ. This vaporisation curve divides the liquid
phase from the gaseous phase and stands in analogy to the yield surface of the theory of
plasticity. By comparing the effective fluid pressure pβR with the vapour pressure pRvap,
it is possible to identify the current phase state, cf. Figure 5.3. Therewith, a switching
criterion is defined that furthermore depends on the direction of the phase transition, i. e.,
evaporation or condensation.
In the evaporation case, initially only liquid exists, sL = 1, and the switching criterion
reads
if sL 6= 0
 p
LR > pRvap then ρˆ
L = 0 ,
pLR ≤ pRvap then ρˆL 6= 0 .
(5.161)
When the transition occurs in the other direction, i. e., from gas to liquid, the switching
criterion for condensation starting from sG = 1 is given by
if sG 6= 0
 p
GR < pRvap then ρˆ
G = 0 ,
pGR ≥ pRvap then ρˆG 6= 0 .
(5.162)
96 Chapter 5: Constitutive settings
Hence, the set of constitutive relations that belongs to the set of governing equations is
completed. The relevant governing balance relations will be recapitulated and formulated
in their strong form in the next section.
5.5 Governing balance relations in the strong form
With the constitutive relations derived in the previous section, the final form of the
governing balance relations for the considered multiphasic system can be defined, in order
to conclude the closure problem. In this regard, the governing balance relations must be
identified from the materially independent mass, momentum and energy balances (5.3).
Since the evaluation of the solid volume balance in Section 5.2.1 showed that this volume
balance is already incorporated in the comprehensive formulations for the solid volume
fraction nS and the effective solid density ρSR, cf. (5.14), it must not be added as a separate
governing balance relation. Consequently, the remaining equations (5.3)2,3,4, which are
not used otherwise, are selected as the governing balance relations.
Proceeding from Section 5.2.2, the fluid mass-balance equations can be directly applied
as they were presented in (5.16):
(ρL)′S + div (ρ
LwL) + ρ
L div (uS)
′
S = ρˆ
L ,
(ρG)′S + div (ρ
GwG) + ρ
G div (uS)
′
S = ρˆ
G .
(5.163)
In the mixture momentum balance (5.24) from Section 5.2.3 only slight changes concerning
the overall Cauchy stress tensor are made by including (5.59), (5.60) and the definition
of the total pore pressure pFR (5.57), viz.:
0 = divTSEmech. − grad pFR + ρg + ρˆL (wG −wL) . (5.164)
Finally, the energy balance (5.27) is adapted to the derived constitutive relations. Anal-
ogous to the momentum balance, it also remains to process the Cauchy stress tensors
with the help of (5.59), (5.60), (5.57) and Lα = grad
′
xα, where the specific procedure is
presented in Appendix B.6, and which yields∑
α
ρα (εα)′α = T
S
Emech. · grad (uS)′S − pFR div (uS)′S − nL pLR divwL−
−nG pGR divwG −
∑
α
divqα +
∑
α
ρα rα − pˆL ·wL − pˆG ·wG+
+ ρˆL [ εG − εL + 1
2
(wG ·wG −wL ·wL) ] .
(5.165)
Note in passing that the kinetic part of the phase-transition energy, ρˆL[ 1
2
(wG ·wG−wL ·
wL) ], can be neglected under lingering-flow conditions. Furthermore, the assumption of
negligible external heat supplies (
∑
α ρ
α rα = 0), i. e., for example no radiation, is made.
Thus, both terms are dropped in the subsequent weak form of the overall energy balance,
cf. (6.10).
5.5 Governing balance relations in the strong form 97
With the set of the governing balance relations and the corresponding set of constitutive
relations at hand, phase transitions in deformable, porous media can be simulated after
defining the respective initial and boundary values. Hence, the computation of initial-
boundary-value problems (IBVP) is possible.

Chapter 6: Numerical treatment
After the compilation of the model, the logical next step is to apply it to initial-boundary-
value problems (IBVP) and solve these in numerical simulations. Therefore, an appro-
priate solution method for the treatment of partial differential equations (PDE) has to
be chosen. A common discretisation method for IBVP within the field of solid mechanics
is the finite-element method (FEM). This spatial discretisation scheme is then usually
combined with a finite-difference method (FDM) in time to solve the PDE of the model.
Although the problem of CO2 sequestration is mainly governed by the physics of fluids,
where commonly finite-volume methods (FVM) are applied to discretise the fluid mechan-
ics, the model derived in this monograph still uses the FEM by embedding the solid and
fluid mechanics into the framework of the TPM. This provides an elegant way to simulate
deforming, porous solid materials with percolating fluids. For a deeper insight into the
field of FEM, the interested reader is referred to, e. g., Bathe [12], Braess [29], Schwarz
[162] and Zienkiewicz & Taylor [181].
The numerical implementation of the triphasic thermoelastic model is accomplished in
the FE tool PANDAS1. This tool was developed at the Institute of Applied Mechanics
(Chair of Continuum Mechanics) at the University of Stuttgart to solve porous-media
problems concerning a wide field of applications in different materials, e. g., soils, foams,
biological tissues or bones. In particular, Ellsiepen [64], Eipper [63] and Ammann [6]
started this work and established the basis for further enhancements of PANDAS. To
name a few of those enhancements, Acartu¨rk [2] added real chemical mixtures together
with electrically charged materials. In terms of the solid-deformation behaviour, the suite
of tools was extended to visco-elastic materials at large strains by Markert [114] and
Karajan [94] and to elasto-plastic materials by Graf [74] and Avci [8]. The latter two
contributions also introduce threephasic models including capillary-pressure-saturation
relationships and, thus, together with the work on CO2 sequestration by Komarova [98],
build the basis for the numerical implementation in this monograph. The possibility
of modelling discontinuities, e. g., fractures or cracks, was provided by Rempler [146],
by upgrading the governing fields towards the so-called extended finite-element method
(XFEM).
To solve larger problems with a large number of degrees of freedom (DOF), a paral-
lelisation of the computation is inevitable. This can be achieved by a coupling interface
provided by Schenke and Ehlers [158] that links PANDAS to the commercial finite-element
package ABAQUS2 using the user-defined element subroutine of ABAQUS. The latter is
applied in the simulation of the numerical example on evaporation in Section 7.2.1.
Within this chapter, the choice of the set of primary variables is firstly discussed. Then,
boundary conditions (BC), ansatz and test functions are introduced, and the weak for-
1Porous media Adaptive Nonlinear finite-element solver based on Differential Algebraic Systems, see
http://www.get-pandas.com.
2FEM-based commercial simulation program, see http://www.3ds.com/products-services/
simulia/products/abaqus/.
99
100 Chapter 6: Numerical treatment
mulations of the governing balance relations are given as a necessity for the application of
the FEM. Thereafter, the spatial and temporal solution procedures are outlined briefly.
This provides all necessary means for the numerical treatment of the IBVP formulated in
Chapter 7.
6.1 Finite-element method
Selection of primary variables
At the end of Chapter 5, the final forms of the governing balance relations of the model
were presented by the relations (5.163), (5.164) and (5.165). These balance equations are
related to a corresponding set of primary variables. The choice of this latter set depends
on the specifications of the IBVP, i. e., to avoid unwanted constraints implied by the
applied BC, it might be necessary to switch from the effective fluid pressures pLR and pGR
as primary variables to the liquid saturation sL and the total pore pressure pFR. In this
regard, the first set of primary variables that is used within this monograph is collected
in the vector u1. This set contains the solid displacement uS, the effective fluid pressures
pLR and pGR and the temperature θ:
u1 = [uS, p
LR, pGR, θ ]T . (6.1)
Hereby, the solid displacement uS corresponds to the overall momentum balance (5.164),
the effective fluid pressures pLR and pGR are related to the fluid volume balances (5.163)
and the temperature θ belongs to the overall energy balance (5.165).
In a second set u2, the constituent pressures are replaced by the overall pore pressure p
FR
and the liquid saturation sL, viz.:
u2 = [uS, p
FR, sL, θ ]T . (6.2)
In this case, the governing balance relation corresponding to sL is the liquid mass balance
(5.163)1, whereas for p
FR the following overall mass balance has to be derived:
(ρ)′S + ρ div (uS)
′
S + div (ρ
LwL + ρ
GwG) = 0 . (6.3)
A detailed derivation thereof is given in Appendix B.4. If u2 is used as the set of primary
variables, the effective pore pressures pLR and pGR are no longer intrinsically calculated
and, thus, have to be derived with the aid of the capillary pressure (5.52) and Dalton’s
law (5.57) via
pLR = pFR − sG pc and pGR = pFR + sL pc . (6.4)
Note that the numerical solution of the problem may yield values greater than one or
smaller than zero for the liquid saturation sL, when the latter is chosen as a primary vari-
able. These physically unacceptable values also cause non-physical results for the effective
fluid pressures, cf. (6.4). To overcome this problem, the implementation of the second
set of primary variables u2 should be associated with the introduction of appropriate
“if-conditions” to remove non-physical values of sL.
6.1 Finite-element method 101
The change in primary variables can also help to improve the performance of the numerical
computations. Specifically, this alludes to the capillary-pressure-saturation relation, which
can cause numerical oscillations due to steep gradients in its formulations, cf. Figure 5.1.
For more information and a detailed discussion of this problem, please refer to Graf [74]
or Helmig [84].
Below, only the second set of primary variables u2 will be used to illustrate the numerical
treatment of the derived model, to avoid double formulations. Therefore, the index in the
vector u ( · ), indicating the set of primary variables, will be omitted in the following.
Boundary conditions
Next, the BC corresponding to the governing equations have to be investigated. In spe-
cific, these are Dirichlet (essential) boundaries ∂ΩD, which define exact values for the
respective primary variables on the boundary, and Neumann (natural) boundaries ∂ΩN
that formulate the flux corresponding to a primary variable over the boundary. Since
these two kinds of BC cannot be defined simultaneously for one specific primary vari-
able at a certain position, the surface ∂Ω of the body has to be divided, cf. Figure 6.1.
However, overlapping definitions between the primary variables are permitted.
u¯S
n
v¯L v¯
F
q¯
t¯
Ω
∂Ω
∂Ωt¯N
∂Ωv¯
L
N
∂Ωv¯
F
N ∂Ωq¯N
∂Ωu¯SD
∂Ωs¯
L
D ∂Ωp¯
FR
D
∂Ωθ¯D
s¯L
p¯FR
θ¯
Restrictions imposed on the BC:
∂ΩuS = ∂Ω
u¯S
D ∪ ∂Ωt¯N , ∅ = ∂Ωu¯SD ∩ ∂Ωt¯N ,
∂ΩpFR = ∂Ω
p¯FR
D ∪ ∂Ωv¯FN , ∅ = ∂Ωp¯
FR
D ∩ ∂Ωv¯FN ,
∂ΩsL = ∂Ω
s¯L
D ∪ ∂Ωv¯LN , ∅ = ∂Ωs¯LD ∩ ∂Ωv¯LN ,
∂Ωθ = ∂Ω
θ¯
D ∪ ∂Ωq¯N , ∅ = ∂Ωθ¯D ∩ ∂Ωq¯N .
(6.5)
Figure 6.1: The observed domain Ω with the applied BC on the boundary ∂Ω for the set of
primary variables u .
Ansatz and test functions
For the numerical treatment of the IBVP, the ansatz (trial) and test functions correspond-
ing to the set of primary variables (6.2) are chosen from the Sobolev space H1(Ω) to be
square integrable on Ω. The ansatz functions A( · )(t) with t ∈ [t0, T ] read:
AuS (t) := { uS ∈ H1(Ω)d : uS(x, t) = u¯S(x, t) on ∂ΩuSD } ,
ApFR(t) := { pFR ∈ H1(Ω) : pFR(x, t) = p¯FR(x, t) on ∂ΩpFRD } ,
AsL(t) := { sL ∈ H1(Ω) : sL(x, t) = s¯L(x, t) on ∂ΩsLD } ,
Aθ(t) := { θ ∈ H1(Ω) : θ(x, t) = θ¯(x, t) on ∂ΩθD } .
(6.6)
102 Chapter 6: Numerical treatment
The ansatz functions are defined such that they provide the values of the Dirichlet BC
at the boundary ∂ΩD . The test (weighting) functions T ( · )(t) disappear at the Dirichlet
boundaries ∂ΩD, and are given as
T uS := { δuS ∈ H1(Ω)d : δuS(x) = 0 on ∂ΩuSD } ,
T pFR := { δpFR ∈ H1(Ω) : δpFR(x) = 0 on ∂ΩpFRD } ,
T sL := { δsL ∈ H1(Ω) : δsL(x) = 0 on ∂ΩsLD } ,
T θ := { δθ ∈ H1(Ω) : δθ(x) = 0 on ∂ΩθD } .
(6.7)
In (6.6)1 and (6.7)1, the superscript d ∈ {1, 2, 3} denotes the spatial dimension of the
physical problem.
6.1.1 Weak formulations of the governing equations
The governing balance relations (5.163), (5.164) and (5.165) are formulated locally in
their strong form and are, thus, continuously fulfilled at each material point P of the
mixture body B. This system of coupled PDE appears to be too complicated to be solved
numerically. Therefore, it is necessary within the framework of the FEM to transform
these balance relations into an energetic expression that is satisfied in a so-called weak
sense, i. e., it fulfills the relations in an integral sense over the spatial domain Ω. This
transformation from the strong, local form into the weak, global form G( · ) is accomplished
by multiplying the governing balance equations with the independent test functions δuS,
δpFR, δsL and δθ introduced in (6.7), to account for the error made by this approximation
procedure. The integration of the governing balance relations is conducted using the
Gauß ian integral theorem. In this regard, the final weak form of the mixture mass-
balance equation and the mass-balance equation of the constituent ϕL read
GpFR(u , δpFR) =
∫
Ω
{(ρ)′S + ρ div (uS)′S} δpFR dv −
∫
Ω
(ρLwL + ρ
GwG) · grad δpFR dv+
+
∫
∂Ωv¯
F
N
(ρLwL + ρ
GwG) · n︸ ︷︷ ︸
v¯F
δpFR da = 0 ,
GsL(u , δsL) =
∫
Ω
{(ρL)′S + ρL div (uS)′S} δsL dv −
∫
Ω
ρLwL · grad δsL dv−
−
∫
Ω
ρˆL δsL dv +
∫
∂Ωv¯
L
N
ρLwL · n︸ ︷︷ ︸
v¯L
δsL da = 0 .
(6.8)
Therein, arguments of the boundary integrals over ∂Ωv¯
F
N and ∂Ω
v¯L
N denote the volumetric
effluxes of the domain Ω, once for both fluid phases, v¯F = (ρLwL + ρ
GwG) · n, and once
6.1 Finite-element method 103
for the liquid phase solely, v¯L = ρLwL ·n with n being the outward-oriented unit normal
vector, cf. Figure 6.1. These effluxes can be directly defined by Neumann BC.
Proceeding to the next balance equation, the weak form of the mixture momentum balance
is given as
GuS(u , δuS) =
∫
Ω
(TSE − pFR I) · grad δuS dv −
∫
Ω
ρg · δuS dv+
+
∫
Ω
ρˆL (wL −wG) δuS dv −
∫
∂Ωt¯
N
(TSE − pFR I)n︸ ︷︷ ︸
t¯
·δuS da = 0 ,
(6.9)
where t¯ = (TSE − pFR I)n is the external total stress vector on the Neumann boundary
∂Ωt¯N . And finally, the weak form of the mixture energy balance is derived analogously,
viz.:
Gθ(u , δθ) =
∫
Ω
{
ρS (εS)′S + ρ
L (εL)′L + ρ
G (εG)′G −TSEmech. · grad (uS)′S +
+nS pFR div (uS)
′
S + pˆ
L ·wL + pˆG ·wG
}
δθ dv−
−
∫
Ω
(∑
α
qα + nL pLRwL+ n
G pGRwG
)
· grad δθ dv+
+
∫
Ω
ρˆL (εL − εG) δθ dv+
+
∫
∂Ωq¯
N
(∑
α
qα + nL pLRwL + n
G pGRwG
)
· n︸ ︷︷ ︸
q¯
δθ da = 0
(6.10)
with the Neumann BC on ∂Ωq¯N being governed by the heat efflux q¯ = (
∑
α
qα+nL pLRwL+
nG pGRwG) ·n. Furthermore, the time derivatives of the internal energies with respect to
the fluid motions, i. e., (εL)′L and (ε
G)′G in (6.10), are transformed by (3.15) via
(εβ)′β = (ε
β)′S + grad ε
β ·wβ , (6.11)
so that the mixture energy balance is given solely with respect to the motion of the solid
constituent.
A compact formulation of the FE problem is obtained as follows (cf. Ellsiepen [64]):
Find u ∈ Au(t) such that Gu(u , δu) = 0 ∀ δu ∈ ∂Ωu , t ∈ [t0, T ] . (6.12)
Therein, Gu = [GuS , GpFR, GsL, Gθ]T is the system of the governing balance relations in
the weak formulation.
For the sake of completeness, the weak forms of the effective fluid mass balances, corre-
sponding to the fluid partial pressures pLR and pGR of the set of primary variables u1, are
supplemented in Appendix B.5.
104 Chapter 6: Numerical treatment
6.2 Discretisation procedures
6.2.1 Spatial discretisation
As the name already conveys, it is a major principle of the FEM to subdivide the con-
tinuous domain Ω into E non-overlapping finite elements Ωe, as is shown in Figure 6.2.
Each finite element is composed of Ne nodal points named as P
j. The total number of
P j(Ωe) ∈ N
Ωe ∈ Ωh
Ωh
Ω ≈ Ωh =
E⋃
e=1
Ωe ,
N =
N⋃
j=1
P j(Ωe) .
(6.13)
Figure 6.2: Example of the spatial discretisation of a spatial domain Ω.
nodal points is denoted by N . The edges of the elements connecting the nodal points are
straight lines. Furthermore, the nodal points and edges of each element are shared with
the neighbouring elements, except for nodal points and edges belonging to a boundary.
This yields a finite-element mesh that is composed of the set of nodes N . Note that the
continuous domain Ω is only approximated by this spatial discretisation Ωh, because of
the simple geometry of the finite elements. The resulting error is naturally reduced by
increasing the number of elements.
This spatial discretisation also alters the ansatz and test spaces, A( · )(t) and T ( · )(t), in
subdividing them into N discretised ansatz and test spaces, A( · )h(t) and T ( · )h(t). In
detail, this yields for the primary variables u the following discretised ansatz functions
uh:
uS(x, t) ≈ uhS(x, t) = u¯hS(x, t) +
N∑
j=1
φj
uS
(x)ujS(t) ∈ AuS h(t) ,
pFR(x, t) ≈ pFRh(x, t) = p¯FRh(x, t) +
N∑
j=1
φj
pFR
(x) pFRj(t) ∈ ApFR h(t) ,
sL(x, t) ≈ sLh(x, t) = s¯Lh(x, t) +
N∑
j=1
φj
sL
(x) sL j(t) ∈ AsL h(t) ,
θ(x, t) ≈ θh(x, t) = θ¯h(x, t) +
N∑
j=1
φjθ(x) θ
j(t) ∈ Aθ h(t) ,
(6.14)
where {u¯hS, p¯FRh, s¯Lh, θ¯h} are determined by the Dirichlet BC and
{ujS, pFRj , sL j, θj} are the unknown nodal quantities, also called the degrees of free-
dom (DOF) of the system. Analogously, the discretised test functions δuh corresponding
6.2 Discretisation procedures 105
to the primary variables uh read:
δuS(x) ≈ δuhS(x) =
N∑
j=1
φj
uS
(x) δujS ∈ T uS h ,
δpFR(x) ≈ δpFRh(x) =
N∑
j=1
φj
pFR
(x) δpFRj ∈ T pFR h ,
δsL(x) ≈ δsLh(x) =
N∑
j=1
φj
sL
(x) δsL j ∈ T sL h ,
δθ(x) ≈ δθh(x) =
N∑
j=1
φjθ(x) δθ
j ∈ T θ h .
(6.15)
Therein, the same global basis functions {φj
uS
, φj
pFR
, φj
sL
, φjθ} are chosen both for the
ansatz and test functions, according to the well-known Bubnov-Galerkin method3. Please
note in passing that the global basis function of the solid displacement has multiple entries
corresponding to the dimension of the physical problem (in 3-d: φj
uS
= [φjuS1, φ
j
uS2
, φjuS3, ]
T ).
Hence, every scalar-valued DOF {ujS1, ..., ujSd, pFR j , sL j, θj} is assigned to a basis func-
tion φjDOF at each nodal point P
j. Furthermore, the DOF are only time dependent,
whereas the basis functions are determined by the spatial position x. Thus, the values of
the DOF (representing the physical quantities) are mapped by the basis function φjDOF to
any place within the finite elements E∗ belonging to the respective node P j. This can be
mathematically formulated into, cf. Rempler [146]:
φ
j
DOF(x) = 0 if x /∈
⋃
e∈E∗
Ωe and φ
j
DOF(xi) = δ
j
i with
i = j : δ
j
i = 1 ,
i 6= j : δji = 0 ,
(6.16)
where the basis functions φjDOF are normalised by the Kronecker symbol δ
j
i (for {i, j} =
1, ..., N), ensuring that at each nodal position xi, the nodal quantity corresponds to
the approximated value of the DOF (except for Dirichlet boundary nodes). A further
limitation is stipulated for the discretised test functions {δuhS, δpFRh, δsLh, δθh}, which
must fulfill the so-called partition-of-unity principle, i. e., the sum of the basis functions
at each point x ∈ Ωh must be equal to one. This yields a system of DOF×N linearly
independent equations for each DOF.
The gradient of the primary variables gradu appearing in the weak forms of the balance
equations (6.8), (6.9) and (6.10) are calculated by the partial derivative of the ansatz and
3This is also known as the Galerkin method. Contrary to this, the Petrov-Galerkin method is charac-
terised by selecting different global basis functions for the ansatz and test functions. For more information
in this regard and a comparison between the two methods, please refer to, e. g., Ehlers et al. [58] or Helmig
[84].
106 Chapter 6: Numerical treatment
test functions, via:
graduh(x, t) =
Nu∑
i=1
gradφiu(x)u
i(t) ,
grad δuh(x, t) =
Nu∑
i=1
gradφiu(x) δu
i(t) .
(6.17)
The strongly coupled problem arising from the multiphasic model is solved here in a mono-
lithic way4. That means, all independent linear equations corresponding to the primary
variables (DOF) are calculated simultaneously. In principle, different ansatz functions for
the primary variables uh can be chosen. However, this choice influences the stability of the
numerical solutions, i. e., a poor choice could cause oscillations. Acartu¨rk [2] and Graf [74]
explicate that the reason for this performance is found in the governing overall momentum
balance (5.164), specifically in the overall stress. Assuming negligible fluid extra stresses,
the overall stress is composed of the mechanical extra-stress tensor of the solid matrix,
given by (5.80)1, and the effective pore pressure, presented in (5.119). By means of the
definitions of the linearised solid strain (5.7) and the solid deformation gradient (5.5), the
solid extra-stress tensor depends on the gradient of the solid displacement vector uS. In
this regard, a mixed finite-element formulation consisting of quadratic ansatz functions for
the solid displacement uS and linear ansatz functions for the remaining primary variables,
i. e., pore pressure pFR, liquid saturation sL and temperature θ, leads to an approximation
of equal polynomial order with respect to the primary variables. These elements are also
called extended Taylor-Hood elements, cf. Figure 6.3, where a 2-d triangular element (6
nodes) and a 3-d hexahedral element (20 nodes) are exemplary depicted.
displacement uhS
pressure pFRh, saturation sLh
and temperature θh
Figure 6.3: Extended triangular (2-d) and hexahedral (3-d) Taylor-Hood elements.
Mathematically, the mixed finite elements must fulfill the Ladyzhenskaya-Babusˇka-Brezzi
(or inf-sub) condition. More information in this direction can be found in, e. g., Braess
[29], Brezzi & Fortin [30] and Hughes [88].
The final feature of the spatial discretisation is the integration of the weak balance
equations locally at a reference element that is positioned at the local coordinates ξ =
(ξ1, ..., ξd), where usually ξ(·) ∈ [0, 1] or ξ(·) ∈ [−1, 1]. The results of this integration are
mapped to the global coordinates x. The advantage of this approach is the conduction
of complicated integrations on a simple local geometry, where after the results have to be
4It is also possible to solve this model in a decoupled way. Therefore, special methods such as
the operator-splitting method would be required, which are presented, e. g., in Markert et al. [116] or
Zinatbakhsh [182].
6.2 Discretisation procedures 107
transformed to the original, complex physical geometry. The transformation between the
local and global coordinates is given by
x(ξ) =
Ne∑
j=1
φjgeo(ξ)xj =
Ne∑
j=1
φj(ξ)xj , (6.18)
where x(ξ) defines the position of an arbitrary point in the element and is calculated by
utilising the mapping basis function φjgeo(ξ) on the nodal point in the global coordinate
system of the finite element Ωe, see Figure 6.4. This is accomplished for all nodes Ne of
mapping
J¯(ξ)
P j(Ωe)
Ωe
Ωh
Ωξe
ξ1
ξ2ξ3
x1
x2x3
Figure 6.4: Illustration of the mapping procedure between the hexahedral reference element Ωξe
and the actual element Ωe, i. e., from the coordinates ξ to x.
the element Ωe. In this work, an isoparametric geometry transformation is applied, i. e.,
φjgeo(ξ) = φ
j(ξ). This leads to the transformation of the differential line element dx to
dξ by the Jacobi an determinant J¯ , given as
J¯(ξ) = det
(dx(ξ)
dξ
)
. (6.19)
Therewith, the integration of a function f(x) over the finite element Ωe can be reformu-
lated within the local coordinates of the reference element domain Ωξe via∫
Ωe
f(x) dv =
∫
Ωξe
f(x(ξ)) J¯(ξ)dvξ (6.20)
with the incremental volume of the reference element dvξ. By applying the Gauß ian
quadrature scheme to the integral expression in the local element coordinates in (6.20),
the final numerical integration reads
∫
Ωe
f(x) dv =
KG∑
k=1
f(x(ξk)) J¯(ξk)wk . (6.21)
Therein, the function f(x(ξk)) is evaluated by the integration at the Gauß points KG,
which are located in the reference element at ξk, furthermore weighted with the quadrature
weighting factors wk and summed over all Gauß points of the element Ωe. The weighting
factors are hereby chosen according to the number of Gauß points and their position,
where, in turn, the number of Gauß points must be derived according to the integration
order, to get an accurate FEM solution, cf. Wieners et al. [179] for more details.
108 Chapter 6: Numerical treatment
6.2.2 Temporal discretisation
Before the temporal discretisation of the governing equations are discussed, it is convenient
to reformulate the semi-discrete initial-value problem, which is spatially discretised but
still continuous in time, in an abstract manner. This is achieved by collecting all nodal
DOF in a vector y , viz.:
y = [ (u1S, p
FR, 1, sL, 1, θ1), ..., (uNS , p
FR,N , sL,N , θN) ]T . (6.22)
Please be aware that the solid displacement velocity uS contains multiple entries depend-
ing on the regarded spatial dimension d of the IBVP. Since in the weak formulations of
the governing balance relations (6.8), (6.9) and (6.10), the time derivatives are only given
with respect to the solid motion, in the following ( · )′S will be denoted as ( · )
′. Therewith,
the abstract formulation of the semi-discrete IBVP reads
F(t, y , y ′) = [My ′(t) + k(y(t))− f (t) ] != 0 . (6.23)
Herein, the generalised mass matrix is expressed by M , the generalised stiffness vector
is presented by k(y), and f is the generalised vector of external forces, composed of the
Neumann BC.
Proceeding to the temporal discretisation of (6.23), the implicit (or backward) Euler time-
integration method is chosen, which belongs to the general class of Runge-Kutta methods.
In case of 1st-order systems with quasi-static behaviour, large time-steps are desirable for
the simulation. However, this requires an unconditional stability of the time-integration
method, which is guaranteed, for example, by the implicit Euler method, but not by
explicit schemes. Furthermore, this implicit or backward Euler method is a so-called
single-step method, where the unknown values of the next time-step are determined only
from one previous solution. A deeper discussion of the single-step Runge-Kutta methods
and their numerical relevance can be found, e. g., in Rempler [146].
The implicit Euler scheme proceeds from a backward Taylor-series expansion of the tem-
poral discretisation of the vector of unknowns y around time tn+1,
yn = yn+1 −∆t y ′n+1 → y ′n+1 =
1
∆t
(yn+1 − yn) , n = 0, 1, ... , (6.24)
where the time tn refers to the previous time-step and, consequently, time tn+1 denotes
the current time. Furthermore, the actual time-step size ∆t = tn+1 − tn > 0 splits the
total simulation time [t0, T ] into a number of subintervals [tn, tn+1]. Hereby, only the first
expansion term of the Taylor series is kept, while the higher-order terms are neglected.
Next, the time-integration scheme (6.24) is substituted into the previously derived problem
formulation (6.23). It yields
Fn+1(tn+1, yn+1, y
′
n+1(yn+1)) = [M(yn+1) y
′
n+1 + k(yn+1)− fn+1 ] != 0 . (6.25)
This nonlinear set of differential algebraic equations is solved in this monograph by the
well-known Newton-Raphson iteration method. Therefore, the required global or residual
6.2 Discretisation procedures 109
tangent DFkn+1 is numerically calculated according to Acartu¨rk [2], and it reads
DFkn+1 :=
dFkn+1
dykn+1
=
∂Fkn+1
∂y kn+1
+
1
∆t
∂Fkn+1
∂(y ′)kn+1
. (6.26)
The numerical calculation of the residual tangent has the advantage of a simple numerical
implementation, but is numerically costly and leads in some cases to instability of the
numerical solution, which can be avoided by inserting the analytical tangent, cf. Wagner
[176] for more details. With the global tangent of (6.26), a linear system of equations
DFkn+1∆y
k
n+1 = −Fkn+1 (6.27)
has to be solved for the stage increment vector ∆ykn+1 at the current Newton step k.
Therefore, both direct and iterative solvers can be used, where an overview of these
solvers is presented, e. g., in Ellsiepen [64]. In a last step, the stage increment is added to
the solution vector and the next Newton step k + 1 is calculated:
yk+1n+1 = y
k
n+1 +∆y
k
n+1 . (6.28)
This iteration procedure is repeated until the norm of the residuum falls below a pre-
defined tolerance ǫtol,
‖Fk+1n+1 ‖ < ǫtol. (6.29)
Admittedly, this was a rather brief description of the temporal discretisation and the
solution procedure. For more information, the interested reader is referred to the works
of, e. g., Ammann [6], Ellsiepen [64] and Rempler [146].
Thereby, the numerical model is completed and can now be utilised to simulate different
IBVP concerning the sequestration of CO2, as is presented in the next chapter.

Chapter 7: Numerical examples
In this chapter, the previously developed theoretical model is applied to simulate CO2
sequestration and to investigate the problem of phase transition. This is achieved in
conjunction with the numerical implementation introduced in Chapter 6. Furthermore,
the material-parameter set is chosen such that it represents CO2 and water percolating
a porous rock matrix. Therefor, the relations derived in Chapter 4 are drawn on for the
thermodynamical behaviour of the CO2, whereas for the parameters describing flow and
deformation, it is resorted to the works of Graf [74], Class [36, 37] and Rutqvist [154, 155],
which also address either CO2 sequestration directly or related fields, e. g., remediation
of contaminated porous media.
After discretisation, the strongly coupled system of partial differential equations given
by (6.9)-(6.10) is solved monolithically with an unconditionally stable, implicit time-
integration scheme using the finite-element solver PANDAS. For the example presented in
Section 7.2.1 a parallel solution technique is applied, where PANDAS is included into the
commercial FEM-program ABAQUS, cf. Schenke & Ehlers [158].
To gain an impression of what happens during CO2 injection into a deep aquifer, at first,
a numerical example of a fictitious aquifer is simulated to demonstrate the dependence
between the pressure and temperature conditions, the phase behaviour of CO2 and the
solid deformations of the porous rock. Thereafter, the phase-transition process in conse-
quence of changing temperature conditions is examined further, both in the evaporation
and in the condensation case. Hereby, use is made of the derived constitutive relation for
the mass production, cf. Section 5.4, in order to get an understanding of the influence of
the mass transfer between the two fluid phases on the phase-transition process.
7.1 CO2 sequestration into a deep aquifer
The reasons and requirements for injecting CO2 into deep saline aquifers have been dis-
cussed at length in Chapter 2. Therein, it was pointed out why numerical simulations
can help in understanding the physical and thermodynamical processes during CO2 in-
jection and, thereupon, being able to make risk assessments by estimating the safety of
the storage site.
In CO2 sequestration, the CO2 is injected with a high pressure into depths greater than
800m to displace the reservoir water and to ensure the supercritical state of the CO2.
Before the injected CO2 dissolves into the reservoir water, it rises until it reaches the
bottom of the cap-rock layer, due to its lower density compared to that of the water
in place. Hence, a disturbance of the cap-rock layer, e. g., fracture, fault or inclination,
leads to situations where the undissolved CO2 might leak to the surface or ends up at
surrounding physical conditions that lead to the phase change from supercritical to gas.
The latter is accompanied by an expansion of the CO2, which can intensify the stress
on the cap-rock layer, and thus, is an unwanted incident. This phase-change process is
111
112 Chapter 7: Numerical examples
simulated in the following model.
7.1.1 Injection into a reservoir with an inclined cap-rock layer
For the simulation of CO2 injection into a water-filled reservoir, a three-phasic model
(solid S, water L, CO2 G) is considered, where the first set of primary variables u1 from
(6.1) is chosen, consisting of the solid displacement uS, the effective water pressure p
LR
and the effective CO2 pressure p
GR. The temperature is omitted here as a degree of
freedom since a constant temperature distribution is assumed, which is governed by the
geothermal gradient ∆gθ = 25.0K/km. The water is regarded as incompressible, whereas
the CO2 is modelled as compressible by calculating the effective CO2 density ρ
GR by one
single EOS. In this first simulation example, the mass transfer between the liquid and
gaseous CO2 phases is omitted, i. e., ρˆ
α = 0. Since both CO2 phases are modelled by one
EOS alone, the phase change only becomes visible in the density jump.
To simulate phase-change occurrence during CO2 injection into a deep aquifer, the sim-
ulation setup is made up of a section of the subsurface with the dimensions of 500m
width and a height that reaches from the surface to a depth of −850m. The model
consists of two permeable reservoir layers, whereas the lower layer is enclosed between
the impermeable cap- and base-rock layers. The cap-rock layer has an inclination that
shifts its bottom boundary from −700m up to −500m, cf. Figure 7.1 (left). By mod-
elling the whole reservoir body from surface to bottom also the weight of the water-filled
overlying rock is considered in the simulation. The model is confined in horizontal direc-
tion and the vertical displacement is blocked at the bottom boundary. The temperature
is assumed to be constant throughout the whole simulation and has a vertical gradient
of ∆gθ = 25.0K/km, starting with ambient temperature θ = 283.15K at the surface
boundary. Initially, a hydrostatic pressure distribution across the whole model is applied
by defining hydrostatic pressure distributions at the right and left boundaries. The CO2
is injected at the left boundary of the lower reservoir layer over a height of 10m and 5m
above the top of the base-rock layer, cf. blue arrows in Figure 7.1 (left). The CO2 is
injected over a time period from t = 0min to t = 33.3min by setting a Dirichlet BC that
increases the pressure of the CO2 constituent up to p
GR = 7.85MPa. After the injection
is stopped, the movement of the CO2 is only triggered by buoyancy forces.
The material parameters used in this simulation were chosen on the basis of Rutqvist &
Tsang [155], as well as of Graf [74] and are displayed in the table in Figure 7.1. Please
note here that the initial intrinsic permeabilities KS0S and the Lame´ parameters λ
S and
µS of the reservoir layers differ from that of the cap- and base-rock layers.
Concerning the mesh, initially a rather coarse elementation is applied, obeying the large
dimensions of the model setup, cf. Figure 7.1 (left). To account for the moving front of the
CO2 plume as well as for the jump in density at phase change, a spatial mesh-adaptivity
depending on the change in saturation sG and effective CO2 density ρ
GR is exerted, which
significantly improves the numerical performance in these cases.
The results of the simulation are presented in Figure 7.2, where the partial pore density
ρCO2F = s
GρGR of the CO2 phase is plotted for six characteristic times 30.57min, 33.0min,
7.1 CO2 sequestration into a deep aquifer 113
0m
−850m
1
0
m
5
m
5
0
m
5
0
m
5
0
m
5
0
m
8
5
m
6
5
0
m
500m
4
5
0
m
3
0
0
m
x1
x2
θ = 283.15K whole model:
g = 9.81m/s2
nS0S = 0.6
µLR = 6.53 · 10−4Pa s
µGR = 1.57 · 10−5Pa s
ρLR = 1000.0kg/m3
ρSR = 2650.0kg/m3
pd = 2000.0Pa
λ = 1.3
sLres = 0.1
sGres = 0.01
κ = 1.0
∆gθ = 25.0K/km
reservoir-rock layer:
KS0S = 10
−10m2
λS = 1.67 · 109Pa
µS = 2.50 · 109Pa
base-/ cap-rock layer:
KS0S = 10
−15m2
λS = 3.73 · 109Pa
µS = 8.70 · 109Pa
Figure 7.1: Geometry of the model to simulate CO2 injection into a water-filled reservoir with
an inclined cap-rock layer and a base-rock layer (light brown). The red, green and blue dots
mark the places where the vertical displacement is measured for Figure 7.3. The table shows
the applied material parameters with different intrinsic permeabilities and Lame´ parameters for
the reservoir-, base- and cap-rock layers.
34.80min, 39.80min, 42.22min and 66.25min. In chronological order, these show the en-
try of CO2, the spreading due to the injection pressure, further migration due to buoyancy
forces along the bottom of the cap-rock layer after the injection is stopped, phase change
from supercritical to gaseous CO2 when the plume moves up along the inclination, the
arrival at the upper part of the cap-rock layer, and the final stage when almost all CO2 in
its gaseous state is collected under the highest point of the reservoir layer. The point of
phase change between the supercritical and the gaseous phase is clearly indicated by the
sudden decrease in density from 600 kg/m3 to around 150 kg/m3 at a depth of roughly
−675m. It can also be noticed that the plume widens after the phase change, owed to
the increase in specific volume of the gaseous CO2. Additionally, Figure 7.2 displays the
seepage velocity of the CO2 by black vectors. These show that the flow is highest when
pressure or buoyancy forces are strong.
To investigate the solid deformation during the injection process, the vertical displacement
uS2 in the cap-rock layer was monitored at three different positions, cf. red, green and
blue points in Figure 7.1 (left), and plotted over time in Figure 7.3. Admittedly, the
displacements are not large, which is due to the confining overburden pressure, but it
114 Chapter 7: Numerical examples
ρCO2F [
kg
m3
]
0 600
Figure 7.2: Result of the CO2-injection simulation at times 30.57min, 33.65min, 34.80min,
39.80min, 42.22min and 66.25min. The contour plot shows the partial pore density ρCO2F =
sGρGR of the CO2 phase and the black arrows indicate the partial seepage velocity of the CO2
plume, wCO2 F = s
GwG.
is still possible to find a connection between the migrating CO2 plume and the solid
deformations. For a better comparison, the dotted black lines in Figure 7.3 refer to the
six time shots used in Figure 7.2. The first kink in the red curve can be attributed to the
first appearance of CO2 in the reservoir layer. The next apparent event is the shut-down
of the injection at t = 33.3min. Hereby, a short delay between shutting down the injection
and the change in the tangent of the vertical displacement is observed in Figure 7.3. The
green line, which belongs to the second measuring point at the inclination of the cap-rock
layer, shows that as soon as the plume has passed this point the displacement recedes
back to zero. When a steady state is reached and the gaseous CO2 is collected under the
upper part of the cap-rock, the displacement remains constant for t > 60min, cf. blue
line in Figure 7.3.
u
S
2
[m
]
time [min]
Pos. 1
Pos. 2
Pos. 3
0
0 10 20 30 40 50 60 70
−0.0005
0.0005
0.0010
0.0015
0.0020
0.0025
Figure 7.3: Evolution of the vertical displacement uS2 at three discrete positions just above the
lower boundary of the cap-rock layer, cf. Figure 7.1. The dotted, vertical lines indicate the times
used in Figure 7.2.
It becomes obvious that the phase-change process has a significant influence on CO2
7.2 Phase transition and mass transfer during evaporation and condensation 115
sequestration and it is worth to investigate this in more detail. Since the CO2 is rep-
resented by just one mass balance in this current model, the phase transition is only
visible in the change of CO2 density. Thus, this model is not capable of monitoring, for
example, how much mass is transferred between the two phases and the influence of tem-
perature changes on the phase-transition process. Consequently, in the upcoming section,
the model is changed in a way that both fluid mass balances are used for the two CO2
phases, while omitting the water phase. These mass balances are coupled by the derived
constitutive relation for the mass transfer. Furthermore, the energy balance is added to
account for the change in temperature.
7.2 Phase transition and mass transfer during evap-
oration and condensation
This section presents the potential of the derived model of mass transfer during the
phase transition of CO2 in a porous rock by conducting the simulation of evaporation
and condensation in two academic examples. The somewhat constructed idea of the
examples consists of a 2-d model of a CO2-filled reservoir, where the CO2 has completely
displaced the water. This reservoir is intersected by a well containing either high or low
temperature fluids that heat or cool the reservoir, which in turn leads to the evaporation
or condensation of the CO2.
Please note here that in real CO2-sequestration processes also the dissolution of CO2 in
the saline water is of great importance. But since the focus lies here on single-substance
phase transitions between pure phases, this effect is omitted.
The academic character of the example emanates from the fact that it does only demon-
strate a fraction of the model’s capabilities, since, e. g., the thermal induced deformations
of sandstone are admittedly rather small. Furthermore, the example is composed of a
simplified geometry with idealised BC. However, the material parameters and the ther-
modynamical conditions represent a realistic system. For the numerical implementation
of the two examples, the second set of primary variables u2, cf. (6.2), is selected, which
consists of the solid displacement uS, the total pore pressure p
FR, the liquid saturation
sL and the temperature θ.
7.2.1 Evaporation around a hot pipe
This example of a hot pipe, or well intersecting a reservoir that is fully saturated with
liquid CO2 tends to show the evaporation process due to heating. The simulation domain
of 10m x 10m is composed of a thermoelastic porous solid filled initially with liquid
CO2, s
L
0 = 0.99. The liquid phase-state is guaranteed by the initial pore pressure of
pFR0 = 4.5MPa and the initial temperature of θ0 = 270K. To simulate a connection of the
outer boundaries to a surrounding environment that also contains liquid CO2, Dirichlet
BC are applied at these boundaries for the pore pressure and the liquid saturation in the
form of pFR = 4.5MPa and sL = 0.99, respectively. The simulation domain is chosen as a
116 Chapter 7: Numerical examples
quarter of the original setup, by taking advantage of the double symmetry. In this regard,
the domain is confined in normal direction at the mirror boundaries and in horizontal
and vertical direction at the curved boundary of the heating pipe with a radius 2m, cf.
Figure 7.4. At this heating pipe, indicated by the red-coloured part, a Dirichlet BC is
applied, where the temperature is increased from θ = 270K to θ = 295K over 450 s and
held constant thereafter. Furthermore, gravitational forces are omitted in this example.
heating
mirror boundary
mirror boundary
p
F
R
=
4
.5
M
P
a,
s
L
=
0
.9
9
pFR = 4.5MPa, sL = 0.99
pFR0 = 4.5MPa, s
L
0 = 0.99
θ0 = 270K
8m
8
m
2m
2
m
Figure 7.4: Simulation setup of a quarter of the double-symmetric domain (left) and the applied
mesh (right).
The parameters used in this example are plotted in Table 7.1. The solid-material pa-
rameters resemble that of sandstone and were taken from Graf [74] and Rutqvist et al.
[156]. For the fluid phases, i. e., CO2 , the thermodynamic parameters stem from Abbott
& Ness [1], whereas the relations derived in Chapter 4 are applied for the specific heat
capacities cβRV , the vaporisation enthalpy ∆ζvap, the thermal conductivities H
βR and the
shear viscosities µβR.
Concerning the numerical implementation, a parallel solution technique is utilised, where
PANDAS is included into the commercial FEM-program ABAQUS, cf. Schenke & Ehlers
[158].
The results of the simulation are depicted in Figures 7.5 and 7.6 and visualise the evapora-
tion of liquid CO2 due to heating. For each parameter, five snapshots are presented, taken
during the simulation at times 3.3min, 37.3 h, 91.6 h, 155.7 h and 255.7 h. By utilising the
symmetry of the example, four parameters are concentrated into one picture, which also
allows for a good understanding of the coupled effects between the respective parameters.
The first set of pictures in Figure 7.5 shows in the upper left the changes in tempera-
ture θ due to the applied cooling condition, the resulting gaseous mass production ρˆG in
the upper right, the gas saturation sG in the lower left and the partial gas pore-density
ρGF := s
GρGR in the lower right. When the evaporation temperature is reached, phase
transition commences, the gas saturation increases and from the sudden changes in the
partial pore-density of the gas ρGF the proliferation of the gas front becomes obvious. Com-
paring the plots of the gas saturation sG and the gas mass production ρˆG, it can be clearly
7.2 Phase transition and mass transfer during evaporation and condensation 117
Table 7.1: Parameters used for the 2-d simulation of evaporation in a porous sandstone.
initial solid volume fraction nS0S = 0.83
effective densities ρSR0S = 2650kg/m
3
intrinsic permeability, permeability parameter KS0S = 1.3 · 10−10m2, κ = 1
Lame´ parameters µS = 2.5 · 109Pa, λS = 1.67 · 109Pa
thermal expansion coefficient αS = 1.2 · 10−5 1/K
medial grain diameter d50 = 6 · 10−5m
initial temperature θ0 = 270K
Brooks & Corey parameters pd = 4000Pa, λ = 2.6
residual saturations sLres = 0.01, s
G
res = 0.01
thermodynamic parameters for CO2 R
CO2 = 188.91mJ/K, θCO2crit = 304.21K
pCO2Rcrit = 7.38 · 106Pa
Antoine parameters for CO2 AA = 7.8101, BA = 987.44, CA = 290.9
specific solid heat capacity cSV = 700J/kgK
solid thermal conductivity HS = 2000W/mK
observed that the mass transfer only appears in the transition zone where both phases
coexist, i. e., for intermediate saturations. Consequently, after complete evaporation and
sG = 1, the mass production stops.
The second set of pictures in Figure 7.6 contains the liquid saturation sL, the liquid partial
pore-density ρLF , the interfacial area aΓ and the pore pressure p
FR, from left to right and
top to bottom. As expected, these show the decrease in liquid saturation and a change in
liquid density from 950 kg/m3 to 0 kg/m3. According to the saturation dependence of the
interfacial area aΓ, depicted in Figure 5.6 (right), the plot of aΓ in Figure 7.6 indicates
high values for intermediate saturations, which reflects the increase and decrease of the
interfacial area during the phase-transition process. The increase in pore pressure pFR is
caused by the decrease in density, i. e., from a high liquid density of 950 kg/m3 to a low
θ [K]
270 295
ρˆG [ kg
m3s
]
0.0 0.002
sG [−]
1
ρGF [
kg
m3
]
00 180
Figure 7.5: Results of the evaporation simulation at times 3.3min, 37.3 h, 91.6 h, 155.7 h, 255.7 h.
The depicted quantities are: upper-left: temperature θ, upper-right: gaseous mass production
ρˆG, lower-left: gas saturation sG, lower-right: gaseous partial pore-density ρGF = s
GρGR.
118 Chapter 7: Numerical examples
gas density of 180 kg/m3, which is furthermore accompanied by an increase in volume.
Last, a comparison between the different time shots reveals a dilatation of the overall
system, which is also due to the increase in pore pressure.
aΓ [
1
m
]
200 2000
pFR [MPa]
4.5 4.56
sL [−]
1
ρLF [
kg
m3
]
0 0 950
Figure 7.6: Results of the evaporation simulation at times 3.3min, 37.3 h, 91.6 h, 155.7 h, 255.7 h.
The depicted quantities are: upper-left: liquid saturation sL, upper-right: liquid partial pore-
density ρLF = s
LρLR, lower-left: interfacial area aΓ, lower-right: pore pressure p
FR.
7.2.2 Condensation
For the example of condensation of CO2 due to cooling, a pipe containing cool liquids
crossing a reservoir is considered. As a representation thereof, a 2-d simulation domain of
10m x 10m is chosen that is composed of a thermoelastic porous solid and filled completely
with gaseous CO2, s
L
0 = 0.01, cf. Figure 7.7 (left). This example refers to the work of
Ehlers and Ha¨berle [61]. The gaseous state of the CO2 is guaranteed by an initial pore
pressure of pFR0 = 4.0MPa and an initial temperature of θ0 = 320K. The same pressure
is applied as Dirichlet BC at the upper boundary in order to simulate an open boundary
with a connection to a surrounding environment that also contains gaseous CO2. Note in
passing that the implied BC of sL = 0.01 at the upper boundary yields pFR ≈ pGR. The
domain is horizontally confined at the left and right boundaries and vertically confined
at the bottom. Moreover, to induce the condensation process at the cool pipe, the blue-
coloured part at the bottom and a width of 2m is subjected to a temperature decrease from
θ = 320K to θ = 200K over 500 s and held constant thereafter, as can be seen in Figure
7.7 (right). For a realistic simulation, the solid parameters resemble that of sandstone
and are recorded in Table 7.2. The thermodynamic parameters of CO2 in Table 7.2 have
been taken from Abbott & Ness [1], and fluid and solid parameters from Graf [74] and
Rutqvist et al. [156]. The effective fluid shear viscosities are determined as a function of
temperature and effective density given by Fenghour et al. [66], cf. Section 4.4.
The results of the simulation are depicted in Figures 7.8-7.12, which visualise the conden-
sation of the gaseous CO2 in consequence of cooling. For each parameter, four snapshots
are presented, taken during the simulation at times 0 h, 17.5 h, 47.0 h and 82.8 h. The
change in temperature due to the applied cooling condition was already depicted in Figure
7.7 (right). Figure 7.8 shows the liquid mass production ρˆL indicating the mass fraction
that is transferred from the gaseous CO2 to the liquid CO2. It can be clearly observed that
the mass transfer only appears in the transition zone where both phases coexist. Note
7.2 Phase transition and mass transfer during evaporation and condensation 119
pFR = 4.0MPa, sL = 0.01
pFR0 = 4.0MPa
sL0 = 0.01
θ0 = 320K
cooling
gaseous CO2
1
0
m
4 m4m
2m
θ [K]
200 320
Figure 7.7: Simulation setup of the condensation example (left) and the temperature distribution
θ at times 0 h, 17.5 h, 47.0 h and 82.8 h (right).
Table 7.2: Parameters used for the 2-d simulation of CO2 condensation in a porous sandstone.
initial solid volume fraction nS0S = 0.9
effective densities ρSR0S = 2650kg/m
3
intrinsic permeability, permeability parameter KS0S = 1.3 · 10−10m2, κ = 1
Lame´ parameters µS = 2.5 · 109Pa, λS = 1.67 · 109Pa
thermal expansion coefficient αS = 1.2 · 10−5 1/K
medial grain diameter d50 = 6.0 · 10−5m
initial temperature θ0 = 320K
Brooks & Corey parameters pd = 2000Pa, λ = 1.3
residual saturations sLres = 0.01, s
G
res = 0.01
thermodynamic parameters for CO2 R
CO2 = 188.91mJ/K, θCO2crit = 304.21K
pCO2Rcrit = 7.38 · 106Pa
Antoine parameters for CO2 AA = 7.8101, BA = 987.44, CA = 290.9
specific heat capacities cSV = 700 J/(kgK), c
LR
V = 933.6J/(kgK)
cGRV = 790.65J/(kgK)
thermal conductivity of solid HS = 2000W/(mK)
thermal conductivity of CO2 H
CO2R = 0.26W/(mK)
that this zone is indicated by the intermediate gas saturations sG and sL, depicted in Fig-
ures 7.10 and 7.11. Thus, after complete transition of the gaseous CO2 to liquid CO2, the
mass production vanishes. The saturation plots in Figures 7.10 and 7.11 also contain the
120 Chapter 7: Numerical examples
ρˆL [ kg
m3s
]
0.00.01
Figure 7.8: Liquid mass production ρˆL at times 0 h, 17.5 h, 47.0 h and 82.8 h.
pFR [MPa]
3.98 4.0
Figure 7.9: Pore pressure pFR together with the solid displacement vector uS (black arrows) at
times 0 h, 17.5 h, 47.0 h and 82.8 h.
7.2 Phase transition and mass transfer during evaporation and condensation 121
sG [−]
0 1
Figure 7.10: Gas saturation sG and gaseous seepage-velocity vectors wG (black arrows) at times
0 h, 17.5 h, 47.0 h and 82.8 h.
sL [−]
0 1
Figure 7.11: Liquid saturation sL and liquid seepage-velocity vectors wL (black arrows) at times
0 h, 17.5 h, 47.0 h and 82.8 h.
122 Chapter 7: Numerical examples
gaseous and liquid seepage-velocity vectors, wG and wL, respectively. It can be seen that
gaseous CO2 is replenished from the open boundary and liquid CO2 is fanning-out along
the transition zone. Finally, Figure 7.12 shows the partial pore densities ρβF := s
βρβR
of the two fluid phases. The figure illustrates the transition from gaseous CO2, with a
density of about 110 kg/m3, to liquid CO2, with a maximum density of 1200 kg/m
3. Con-
sequently, this increase in density causes a drop in pore pressure pFR that again affects
the field of the solid-displacement vectors, represented by the black arrows in Figure 7.9.
As anticipated, the plot exhibits a settlement zone around the cooling region.
ρGF [
kg
m3
]
0 850
ρLF [
kg
m3
]
0 1200
Figure 7.12: Partial pore density of the gaseous phase ρGF = s
GρGR (left) and partial pore density
of the liquid phase ρLF = s
LρLR (right), each at times 0 h, 17.5 h, 47.0 h and 82.8 h.
Chapter 8: Summary and Outlook
8.1 Summary
For the realisation of CO2 sequestration into deep aquifers, it is essential to guarantee the
safety and economic viability of the storage site. From the multitude of issues that concern
the safety of the reservoir, the problems related to the thermodynamical behaviour of the
CO2 and the deformation of the porous rock were further investigated in this monograph.
Numerical simulations have proven to be a helpful tool to predict the processes within the
reservoir during CO2 injection. In this regard, a reasonable model had to be developed
including all physical and thermodynamical effects that govern the sequestration process.
Therefor, an abundant continuum-mechanical model environment was found within the
TPM, which enables the derivation of a multiphasic and multiphysics approach.
To illustrate the effects that the phase transition exerts on the solid deformation and
the flow processes, a triphasic model was considered, consisting of the thermoelastic solid
phase and two compressible fluid phases. The model was formulated such that the two
fluid phases can either represent the immiscible phases of two different substances, e. g.,
water and CO2, or that the two fluid phases correspond to one single fluid matter, for
instance CO2, and describe different phase states, e. g., gas and liquid.
Based on the exploitation of the entropy inequality, a thermodynamically consistent set
of constitutive relations was derived, which specifies the physics of the CO2 sequestra-
tion problem. Concerning the thermoelastic porous-rock matrix, a multiplicative split
of the deformation tensor was applied and the resulting mechanical and thermal parts
were described by a linear elastic (Hookean) law and by a thermal expansion relation,
respectively. The fluid constituents percolating the porous solid were thermodynamically
governed by an EOS to account for the compressible nature of the fluid phases, where
the van-der-Waals equation was chosen. Furthermore, suitable definitions for the specific
heat capacity, the thermal conductivity and the shear viscosity were included for a correct
representation of the thermal behaviour of the fluid phases.
Special attention was paid to the phase-transition process between the gaseous and liq-
uid phases of the fluid substance. In this regard, a microstructural description of the
interface between the two fluid phases was based on the introduction of an immaterial,
smooth singular surface. Thereby, additional jump conditions appeared in the balance
relations of the fluid constituents that account for the jump over the interface. The eval-
uation of these jump conditions resulted in a constitutive relation of the mass transfer
on the microstructure, depending on the heat flux and the latent heat of evaporation.
The upscaling to the continuum-mechanical macroscale was then achieved by considering
the interfacial area, i. e., the volume-specific area of the interface surface as a mapping
function.
The strongly coupled partial differential equations were spatially discretised by mixed
finite elements and an implicit Euler time-integration scheme governed the temporal dis-
123
124 Chapter 8: Summary and Outlook
cretisation. The monolithic solution of the partial differential equations was achieved by
implementation into the in-house FE-program PANDAS. In some of the presented IBVP,
it was also required to use an adaptive mesh refinement in order to account for the jump
in density at the interface between the two fluid phases as well as for the moving CO2
front. The coupling of PANDAS to the commercial FE-program ABAQUS that enables
parallel computation, proved useful in the simulation of large-scale problems.
By the simulation of a CO2 injection into a water-filled deep aquifer with realistic bound-
ary conditions and parameters, it was possible to visualise the solid deformations of the
cap-rock layer induced by the high injection pressure. Additionally, an inclination in the
cap-rock layer showed the effects of buoyancy and the change between the supercritical
and gaseous phases due to varying temperature and pressure conditions. Moreover, this
also exhibited an influence on the solid deformations. In two further numerical examples,
the phase-transition process caused by alternating temperature conditions was examined
in more detail, both for evaporation and condensation. Hereby, the derived constitutive
relation for the mass transfer, stemming from the assumption of a singular surface at the
phase interface, was applied. Hence, a quantitative measure for the exchanged amount of
mass between the fluid phases was established.
8.2 Outlook
At this state, the developed model is capable to describe a triphasic, thermoelastic porous-
material model, e. g., for the simulation of CO2 injection into a sandstone reservoir, in-
cluding an elaborate description of the thermodynamics of the involved fluids, especially
the phase transition. However, since in the model with explicit consideration of the mass
transfer both fluid phases are already occupied by the gaseous and liquid phases of a
single fluid matter, an extension to a four-phasic model, for instance, by adding water as
an additional constituent, would allow for a wider field of applications. In this context,
a further improvement in the sense of a realistic description of a CO2 reservoir could be
achieved by including dissolution effects between the various fluid matters. Within this
mindset of a maximal accurate model for the physical conditions, also a formulation that
accounts for separate temperatures of each constituents, instead of a single temperature,
should be discussed. Of course, this would increase the set of governing equations by the
respective energy balances and additional constitutive relations for the coupling between
the latter. Furthermore, the so far thermoelastic behaviour of the solid material might
be insufficient for a correct description of the solid displacements and, thus, could be
expanded towards an elasto-plastic description, e. g., Avci [8]. While talking about solid
deformation, the fracturing of the cap-rock layer poses a great threat to the safety of a
CO2 storage site. For the formulation of a model, which explicitly describes the crack
propagation, one could either resort to extended finite elements (XFEM), cf. e. g., Rem-
pler [146] and Ha¨berle & Ehlers [77], or to phase-field models, for example Luo & Ehlers
[111], as well as Ehlers & Luo [62].
Since the model, and especially the thermodynamical part, was formulated mostly inde-
pendent of a specific fluid substance (except for the shear viscosity and thermal conduc-
8.2 Outlook 125
tivity), it would be a simple task to apply the model to problems that concern different
fluid matters. This could be achieved by changing the respective material parameters
that govern, for example, the EOS or the capillary-pressure-saturation relation.
Furthermore, questions remain in the proposed model, for example in the definition of the
interfacial area. To this extent, a rather simple approach was applied, but could be easily
substituted by other definitions, e. g., Nuske et al. [130] and Niessner & Hassanizadeh
[125]. The previously mentioned inclusion of dissolution processes would also affect the
phase transition formulation, since in that case, no longer mass transfer between phases
of a single substance is regarded, but the transfer between mixtures. The constitutive
relation can then be directly derived from the entropy inequality, yielding a formulation
similar to the two-film principle, as it was already briefly mentioned in Section 5.4.1.
Moreover, the exploitation of the entropy inequality revealed a problematic relationship
between the thermodynamical consistent formulation of a model and the phenomeno-
logical relations needed therein, for example, in the derivation of the capillary-pressure-
saturation relation or in the choice of an appropriate EOS. From the existing phenomeno-
logical relations, only few have a mathematical form that is compatible with the constitu-
tive modelling procedure, i. e., the Brooks & Corey pc-sL-relation and the van-der-Waals
EOS. However, especially the latter relation has drawbacks in the correct description
of the physical behaviour of the fluid. Thus, it would be necessary to find constitutive
phenomenological relations that satisfy both the thermodynamical consistency and the
descriptive accuracy.
Another issue represents the validation and verification of the developed model. Obvi-
ously, it is almost impossible to survey phase transitions of the CO2 in the deep aquifer,
which makes the comparison of the model with real in situ data unfeasible. However, pore-
scale experiments do exist and can also be observed by advanced imaging devices. In the
context of this monograph, the contributions of Shahraeeni and Or [163, 164] should be
mentioned here, which monitor the phase-transition process within a porous medium by
synchrotron x-ray tomography. This might present a possibility to validate the developed
model, which, in a further step, could be compared to benchmark problems, for example,
by Class et al. [37].

Appendix A: Selected relations of tensor
calculus
An excerpt of important vector and tensor operations is collected within this appendix.
The chosen collection is strongly restricted to relations which are particularly required in
the presented monograph. In general, the relations are extracted from the comprehensive
work of Ehlers [51], which is partly based on the fundamental textbook of de Boer [18].
A.1 Tensor algebra
For the following considerations arbitrary placeholders are introduced, viz.:
{α, β} ∈ R : scalars (zero-order tensors) as rational quantities,
{a,b, c} ∈ V3 : vectors (first-order tensors) of the proper
Euklidian 3-d vector space V3,
{A,B,C} ∈ V3 ⊗ V3 : tensors (of second order) of the corresponding
dyadic product space V3 ⊗ V3.
Collected rules for products of second-order tensors with scalars or vectors:
α (βA) = (αβ)A : associative law
A (α a) = α(Aa) = (αA) a : associative law
(α + β)A = αA+ βA : distributive law
α (A+B) = αA+ αB : distributive law
A (a+ b) = Aa+Ab : distributive law
(A+B) a = Aa+Ba : distributive law
αA = Aα : commutative law
a = Ab : linear mapping
I a = a : I : identical element (linear mapping)
0a = 0 : 0 : zero element (linear mapping)
(A.1)
127
128 Appendix A: Selected relations of tensor calculus
Collected rules for scalar (inner) products of tensors:
(αA) ·B = A · (αB) = α (A ·B) : associative law
A · (B+C) = A ·B+A ·C : distributive law
A ·B = B ·A : commutative law
A ·B = 0 ∀ A , if B ≡ 0
A ·A > 0 ∀ A 6= 0
(A.2)
Collected rules for tensor products of second-order tensors:
α (AB) = (αA)B = A (αB) : associate law
(AB) a = A (Ba) : associate law
(AB)C = A (BC) : associate law
A (B + C) = AB + AC : distributive law
(A + B)C = AC + BC : distributive law
AB 6= BA : no commutative law
IA = AI = A : I : identical element (linear mapping)
0A = A0 = 0 : 0 : zero element (linear mapping)
(A.3)
Collected rules for transposed and inverse second-order tensors:
(a⊗ b)T = (b⊗ a)
(αA)T = αAT
(AB)T = BTAT
a · (Bb) = (BTa) · b
A · (BC) = (BTA) ·C
(A + B)T = AT + BT
A−1 = (detA)−1 (cofA)T
→ A−1 exists if detA 6= 0
AA−1 = A−1A = I
(A−1)T = (AT )−1 =: AT−1
(AB)−1 = B−1A−1
(A.4)
The computation rules of the determinant and the cofactor are given via
detA = 1
6
(A
@
@A) ·A = 1
6
(trA)3 − 1
2
(trA) (AT ·A) + 1
3
(AA)T ·A
cofA = 1
2
A
@
@A , where cofA = 1
2
(aik ano einj ekop) (ej ⊗ ep) =: +ajp (ej ⊗ ep)
can be evaluated using (A.8) and index notation. Thus, the coefficient matrix
+
ajp contains
at each position ( · )jp the corresponding subdeterminant, e. g., +a11 = a22 a33 − a23 a32 .
A.2 Tensor analysis 129
Collected rules for the determinant and the inverse of second-order tensors:
(cofA)T = cofAT
detAT = detA
det (AB) = detA detB
det (αA) = α3 detA
det I = 1
det(cofA) = (detA)2
detA−1 = (detA)−1
det(A+B) = detA + cofA ·B+
+ A · cof B + detB
(A.5)
Collected rules for the trace operator of second-order tensors:
trA = A · I
tr (a⊗ b) = a · b
tr (AB) = tr (BA)
= A ·BT = AT ·B
tr (αA) = α trA
trAT = trA
tr (ABC) = tr (BCA)
= tr (CAB)
(A.6)
The third-order fundamental (Ricci) tensor and the axial vector:
a× b =
3
E (a⊗ b) : where
3
E is the permutation tensor, cf. (A.8)
A×B =
3
E (ABT ) : with the specific case I×C =
3
ECT = 2
A
c
A
c = 1
2
3
ECT : where
A
c is the axial vector of C
(A.7)
In index notation, the properties of the permutation tensor are given, viz.:
3
E = eijk (ei ⊗ ej ⊗ ek) with the “permutation symbol” eijk
eijk =

1 : even permutation
−1 : odd permutation
0 : double indexing
−→

e123 = e231 = e312 = 1
e321 = e213 = e132 = −1
all remaining eijk vanish
(A.8)
A.2 Tensor analysis
The product rule of derivatives of products of functions:
(a⊗ b)′ = a′ ⊗ b + a⊗ b′ and (AB)′ = A′B + AB′ (A.9)
130 Appendix A: Selected relations of tensor calculus
Collected derivatives of tensors and their invariants:
∂A
∂A
= (I⊗ I)
23
T =
4
I
∂AT
∂A
= (I⊗ I)
24
T
∂A−1
∂A
= − (A−1 ⊗AT−1)
23
T
∂ trA
∂A
= I
∂ detA
∂A
= cofA = (detA)AT−1
∂ cofA
∂A
= detA [(AT−1 ⊗AT−1)−
− (AT−1 ⊗AT−1)
24
T ]
(A.10)
Selected computation rules for the gradient and the divergence operators:
grad (αβ) = α gradβ + β gradα
grad (αb) = b⊗ gradα + α gradb
grad (αB) = B⊗ gradα+ α gradB
div (αb) = b · gradα + α divb
div (a⊗ b) = a divb+ (grada)b
div (αB) = B gradα + α divB
div (Ab) = (divAT ) · b+AT · gradb
div
(b
α
)
=
1
α
divb− 1
α2
b · gradα
(A.11)
Appendix B: Thermodynamical
supplements and specific evaluations
B.1 Thermodynamic potentials and Legendre trans-
formation
The fundamental thermodynamic potentials, namely the internal energy ε, the internal
enthalpy ζ , the Helmholtz free energy ψ and the Gibbs free enthalpy ξ, can be formulated
by different variables, depending on the observed process. In solid mechanics usually the
energetically conjugated variable pairs tension and strain { 1
ρ0
S, E} and temperature and
entropy {θ, η} are used, while for the description of fluids the first pair is replaced by
pressure and specific volume {p, v}. Therewith, the potentials read for the two kinds of
observation:
ε(E, η) → ε(v, η) ,
ζ( 1
ρ0
S, η) → ζ(p, η) ,
ψ(E, θ) → ψ(v, θ) ,
ξ( 1
ρ0
S, θ) → ξ(p, θ) .
(B.1)
Please note that superscripts indicating a certain constituent were omitted here.
The Legendre transformations between the conjugated variables allows for a switch be-
tween the thermodynamic potentials. Exemplary, this is presented here for the transfor-
mation between θ and ηSEmech., which yields the definition of the Helmholtz free energy ψ
S
based on the internal energy εS:
ε˙S(ES, η
S
Emech.) =
∂εS
∂ES
· E˙S + ∂ε
S
∂ηSEmech.
η˙SEmech. ,
=
1
ρS0
SS · E˙S + θ η˙SEmech. ,
(B.2)
where
1
ρS0
SS =
∂ε
∂ES
and θ =
∂ε
∂ηSEmech.
(B.3)
were applied. With
θ η˙SEmech. = (θ η
S
Emech.)˙− ηSEmech. θ˙ (B.4)
it follows
(εS − θ ηSEmech.︸ ︷︷ ︸
=: ψ
)˙ =
1
ρ0
SS · E˙S − ηSEmech. θ˙ ,
→ ψS(ES, θ) = εS(ES, ηSEmech.)− ηSEmech. θ .
(B.5)
131
132 Appendix B: Thermodynamical supplements and specific evaluations
For a detailed discussion of the thermodynamical potentials and a summary of the various
Legendre transformations the interested reader is referred to, e. g., Ehlers [48].
B.2 Maxwell relations and fundamental relations
A list of thermodynamic formulas that are a consequence of the relations presented in
B.1 and, which are important for the alternation between different energy expressions is
presented, based on Lewis & Randall [105]:
Maxwell relations energy-function derivatives(∂θ
∂v
)
η
= −
(∂p
∂η
)
v(∂θ
∂p
)
η
=
(∂v
∂η
)
p(∂η
∂v
)
θ
=
(∂p
∂θ
)
v(∂η
∂p
)
θ
= −
(∂v
∂θ
)
p
(∂ε
∂η
)
v
=
(∂ζ
∂η
)
p
= θ(∂ε
∂v
)
η
=
(∂ψ
∂v
)
θ
= −p(∂ζ
∂p
)
η
=
(∂ξ
∂p
)
θ
= v
(∂ξ
∂θ
)
p
=
(∂ψ
∂θ
)
v
= −η
Since these relations are needed for the description of the thermodynamical behaviour
of liquids and gases in Chapter 4, the formulations were written with respect to fluids.
However, analogous formulas for solid constituents can be derived from the relations in
B.1. Superscripts indicating a certain constituent were omitted.
B.3 Derivation of the Maxwell criterion
The Maxwell criterion, briefly introduced in Section 4.1, is derived here based on Baehr
[11]. The coexistence of both phases in the two-phase region requires the local phase equi-
librium between the liquid and gas phase, i. e., the equilibrium of the respective chemical
potentials:
µL = µG . (B.6)
With µβ = ψβ + pβR vβR and ψβ = εβ − θβ ηβ, it follows from (B.6)
εL − θL ηL + pLR vLR = εG − θG ηG + pGR vGR . (B.7)
Furthermore, Figure 4.6 shows that the two phases coexist at the constant vapour pressure
pRvap for a given constant temperature θ, cf. Antoine equation (4.19). This yields for (B.7):
εL − θ ηL + pRvap vLR = εG − θ ηG + pRvap vGR . (B.8)
Thus, it can be written after sorting and applying the Legendre transformation (3.49):
pRvap (v
GR − vLR) = (εL − θ ηL)− (εG − θ ηG) = ψL − ψG . (B.9)
B.4 Derivation of overall mass balance 133
The differential of the Helmholtz free energy
dψβ = −pβR dv − ηβ dθ , (B.10)
becomes for constant temperature θ
dψβ = −pβR(θ, vβR) dv . (B.11)
The integration of (B.11) between vLR and vGR together with (B.9) finally delivers
pRvap (v
GR − vLR) =
vGR∫
vLR
pβR dv . (B.12)
Geometrically, equation (B.12) formulates the determination of the vapour pressure pRvap(θ)
such that the grey indicated regions in Figure 4.6 have the same area.
B.4 Derivation of overall mass balance
The overall mass balance is derived by adding up all constituent mass balances for ϕα,
α = {S, L, G}. This summation follows the constraints presented in Section 3.3. Starting
from the specific balance equations of the constituents
(ρS)′S + ρ
S div
′
xS = 0 ,
(ρL)′L + ρ
L div
′
xL = ρˆ
L ,
(ρG)′G + ρ
G div
′
xG = ρˆ
G ,
(B.13)
the dependencies of the time derivatives of the fluid densities, (ρL)′L and (ρ
G)′G, are
switched from the respective fluid constituents ϕβ to the solid constituent ϕS using (3.15)
and the fluid velocities,
′
xL and
′
xG, are treated by (3.14), which yields in total
(ρS)′S + ρ
S div (uS)
′
S = 0 ,
(ρL)′S + (grad ρ
L) ·wL + ρL divwL + ρL div (uS)′S = ρˆL ,
(ρG)′G + (grad ρ
G) ·wG + ρG divwG + ρG div (uS)′S = ρˆG ,
(B.14)
where in addition
′
xS = (uS)
′
S was used. In the next step, these three equations are
added up under the conditions of
∑
α ρˆ
β = 0,
∑
α ρ
α = ρ and the rule for the divergence
operator, cf. Appendix A, viz.:
(ρ)′S + div (ρ
LwL) + div (ρ
GwG) + ρ div (uS)
′
S = 0 , (B.15)
which can be further summarised to
(ρ)′S + div (ρ
LwL + ρ
GwG) + ρ div (uS)
′
S = 0 . (B.16)
134 Appendix B: Thermodynamical supplements and specific evaluations
B.5 Weak forms of the effective fluid mass-balance
relations
For the first set of primary variables u1 (6.1), the corresponding weak forms of the effective
fluid mass balance relations read
GpLR(u1, δpLR) =
∫
Ω
{(ρL)′S + ρL div (uS)′S} δpLR dv −
∫
Ω
ρLwL · grad δpLR dv−
−
∫
Ω
ρˆL δpLRdv +
∫
Γv¯
L
N
ρLwL · n δpLR da = 0 ,
GpGR(u1, δpGR) =
∫
Ω
{(ρG)′S + ρG div (uS)′S} δpGR dv −
∫
Ω
ρGwG · grad δpGR dv+
−
∫
Ω
ρˆG δpGR dv +
∫
Γv¯
G
N
ρGwG · n δpGR da = 0 .
(B.17)
B.6 Treatment of the stress terms of the overall en-
ergy balance
For the final formulation of the overall energy balance in (5.165) it is necessary to adapt
the terms containing the Cauchy stress tensors in (5.27), i. e.,
TS · LS +TL · LL +TG · LG . (B.18)
Incorporating the definitions of the extra stresses (5.59) and (5.60), as well as Lα =
grad
′
xα, yields
(TSEmech.−nS pFR I) ·grad (uS)′S+(TLE dis.−nL pLR I) ·grad
′
xL+(T
G
E−nG pGR I) ·grad
′
xG .
(B.19)
The extra stress terms of the fluid constituents can be neglected, cf. (5.70), and with
I · grad ( · ) = div ( · ) one finds
TSEmech. · grad (uS)′S − nS pFR div (uS)′S − nL pLR div
′
xL − nG pGR div ′xG . (B.20)
Next, with the relations between the volume fractions and saturations (3.4) and Dalton’s
law for the total pore pressure (5.57), the last three terms in (B.20) are reformulated to
− (1− nF ) pFR div (uS)′S − nL pLR div
′
xL − nG pGR div ′xG =
= − pFR div (uS)′S + (nL pLR + nG pGR) div
′
xS − nL pLR div ′xL − nG pGR div ′xG =
= − pFR div (uS)′S − nL pLR divwL − nG pGR divwG ,
(B.21)
B.7 Simplification of the direct momentum and mass production terms in the energy balance
of the overall aggregate 135
where in addition, the relation for the seepage velocities (3.14) was applied. Hence, the
final formulation of the Cauchy stress tensor terms for the insertion into (5.165), reads
TSEmech. · grad (uS)′S − pFR div (uS)′S − nL pLR divwL − nG pGR divwG . (B.22)
B.7 Simplification of the direct momentum and mass
production terms in the energy balance of the
overall aggregate
It is convenient to combine the direct momentum and mass production terms in the energy
balance of the overall aggregate (5.26):
...− pˆS · ′xS − pˆL · ′xL − pˆG · ′xG−
− ρˆS (εS + 1
2
′
xS · ′xS)− ρˆL (εL + 12
′
xL · ′xL)− ρˆG (εG + 12
′
xG · ′xG) .
(B.23)
Writing (3.36)1 for the three constituents ϕ
S, ϕL and ϕG while using (3.35)2 yields
pˆS + pˆL + pˆG + ρˆS
′
xS + ρˆ
L ′xL + ρˆ
G ′xG = 0 . (B.24)
After scalar multiplication of (B.24) by
′
xS it is added to (B.23), where also the definition
of the seepage velocities (3.14) is exploited:
...− pˆL ·wL − pˆG ·wG + ρˆS ′xS · ′xS + ρˆL ′xL · ′xS + ρˆG ′xG · ′xS −
− ρˆS (εS + 1
2
′
xS · ′xS)− ρˆL (εL + 12
′
xL · ′xL)− ρˆG (εG + 12
′
xG · ′xG) .
(B.25)
With the relation
∑
α ρˆ
α = 0 the third term in (B.25) becomes:
ρˆS
′
xS · ′xS = −ρˆL ′xS · ′xS − ρˆG ′xS · ′xS . (B.26)
Next, the binomial formula is applied for the kinematical terms containing either ρˆL or
ρˆG:
−ρˆL ′xS · ′xS + ρˆL ′xL · ′xS − ρˆL 12
′
xL · ′xL = −ρˆL 12 wL ·wL − ρˆL 12
′
xS · ′xS ,
−ρˆG ′xS · ′xS + ρˆG ′xG · ′xS − ρˆG 12
′
xG · ′xG = −ρˆG 12 wG ·wG − ρˆG 12
′
xS · ′xS ,
(B.27)
where again (3.14) was used. Under the consideration of ρˆS ≡ 0 and ρˆL = −ρˆG the two
equations in (B.27) add up to ρˆL (1
2
wG · wG − 12 wL · wL). Thus, the simplified form of
(B.23) reads:
...− pˆL ·wL − pˆG ·wG + ρˆL [εG − εL + 12 (wG ·wG −wL ·wL)] . (B.28)
136 Appendix B: Thermodynamical supplements and specific evaluations
B.8 Calculation of different versions of the liquid mo-
mentum production
The result of the liquid momentum production (5.62), i. e., the two right-most terms, can
be varied by the relations of the volume fractions (3.3)2, saturations (3.4) and capillary
pressure (5.52), via:
pˆL = pˆLE dis. + P gradnL −
sL
nF
ρL
∂ψL
∂sL
gradnS
→ pˆL = pˆLE dis. + pGR gradnL + pc sL gradnS ,
→ pˆL = pˆLE dis. + pLR gradnL + pc (sG gradnL − sL gradnG) ,
→ pˆL = pˆLE dis. + pLR gradnL + pc nF grad sL .
(B.29)
B.9 Justifying the assumption of an overall temper-
ature for all constituents
Here, the duration for the assimilation between the temperatures of two constituents (with
different initial temperatures) is examined in order to justify the assumption of the usage
of only one single overall temperature for all constituents. Since the numerical examples in
this work consider mostly (except for the CO2 injection into a water filled reservoir exam-
ple, cf. Section 7.1) conditions, where only one single fluid phase exists inside the porous
solid structure, it is chosen here to investigate the temperature assimilation between a
solid phase ϕS (sandstone) and a fluid phase ϕF (CO2).
The starting point for the justification procedure are the constituent energy balances of
the two constituents ϕS and ϕF , cf. (3.45):
ρS (εS)′S −TS · LS + divqS − ρS rS − εˆS = 0 ,
ρF (εF )′F −TF · LF + divqF − ρF rF − εˆF = 0 .
(B.30)
Since this examination is only concerned about the assimilation of the two temperatures θS
and θF , a non deformable (Lα = 0), flow-free (
′
xα = 0), heat-flux-free (q
α = 0), radiance-
free (rα = 0) system with incompressible constituents (ραR = const.) is considered, which
simplifies (B.30) towards:
ρS (εS)′ − εˆS = 0 ,
ρF (εF )′ − εˆF = 0 ,
(B.31)
where also the subscripts for the indication of the observer of the time derivative can be
omitted, due to the negligence of solid deformations. Regarding the first terms in (B.31),
while considering εα = ψα+θα ηα and the relation between the entropy and the Helmholtz
B.9 Justifying the assumption of an overall temperature for all constituents 137
free energy
ηα = −∂ψ
α
∂θα
, (B.32)
yields:
ρα (εα)′ = ρα (ψα + θα ηα)′ = ρα (ψα)′ + ρα (θα)′ ηα + ρα θα (ηα)′ =
= −ρα ∂ψ
α
∂θα
(θα)′ + ρα (θα)′ ηα + ρα θα (ηα)′ =
= −ρα ηα (θα)′ + ρα (θα)′ ηα + ρα θα (ηα)′ =
= ρα θα (ηα)′ = ρα θα
∂ηα
∂θα
(θα)′ = ρα cαRV (θ
α)′ ,
(B.33)
where in the last step the relation for the specific heat was used:
cαRV = θ
α ∂η
α
∂θα
. (B.34)
The second terms in (B.31) symbolise the direct energy productions εˆα that describe the
caloric interactions between the two constituents, governed by the temperature difference
between them. Thus, following the result for the direct energy production gained by
Ghadiani [71] and Graf [74] by exploiting the dissipation inequality, furthermore using
(3.35)4 and (3.36)3 and neglecting the terms with
′
xα and ρˆ
α for a flow-free and non-
reacting system, the constitutive relations for εˆS and εˆF read:
εˆS = −εˆF = −aSF kεSF (θS − θF ) , (B.35)
wherein, aSF is the interfacial area between the solid and the fluid and k
ε
SF represents the
surface-specific heat-exchange coefficient.
Merging (B.33) and (B.35) together in (B.31) and using ρα = nα ραR, yields:
nS ρSR cSV (θ
S)′ + aSF k
ε
SF (θ
F − θF ) = 0 ,
nF ρFR cFRV (θ
F )′ − aSF kεSF (θS − θF ) = 0 .
(B.36)
After sorting the constant parts and replacing them by
Aε = − aSF k
ε
SF
nS ρSR cSV
and Bε =
aSF k
ε
SF
nF ρFR cFRV
, (B.37)
it follows:
(θS)′ = Aε (θS − θF ) ,
(θF )′ = Bε (θS − θF ) .
(B.38)
The decrease of the temperature difference ∆θSF = θ
S − θF over time indicates the
duration of the temperature assimilation. Therewith, the two equations in (B.38) can be
combined to
(∆θSF )
′ = (θS)′ − (θF )′ = (Aε − Bε) (θS − θF ) . (B.39)
138 Appendix B: Thermodynamical supplements and specific evaluations
This is a differential equation of the form ˙( · ) = y ( · ), for which an analytic solution can
be found with the ansatz ( · ) = ( · )0 e
λ t. Applying this approach to (B.39) provides:
∆θSF = (∆θSF )0 exp
{
[Aε − Bε ] t
}
= (∆θSF )0 exp
{[
− aSF k
ε
SF
nS ρSR cSV
− aSF k
ε
SF
nF ρFR cFRV
]
t
}
.
(B.40)
Therein, (∆θSF )0 is the initial temperature difference between the two constituents, which
should decrease over a short time, in order to justify the assumption of a single overall
temperature. The interfacial area aSF is calculated based on the procedure and assump-
tions presented in Section 5.4.2, where the important relations read:
aSF = n
F A
F
V F
= nF
4 π (r˜F )2
4
3
π (r˜F )3
=
3nF
r˜F
and r˜F = 1
2
(nF
nS
)1/3
d50 . (B.41)
The remaining coefficients are taken as: nS = 0.6, nF = 0.4; d50 = 0.00006, k
ε
SF =
10.0W/(m2K), ρSR = 2650.0 kg/m3, ρFR = 900.0 kg/m3, cSV = 700.0 J/(kgK) and c
FR
V =
950.0 J/(kgK), representing sandstone and CO2.
t [s]
∆
θ
S
F
[K
]
0
0 2 4 6 8 10
20
40
60
80
100
Figure B.1: Temperature assimilation of CO2 and sandstone over time, with initial temperature
difference of 100K.
Therewith, (B.40) is plotted over time, for an initial temperature difference of (∆θSF )0 =
100K. Figure B.1 shows that after less than 4 s both constituents have reached equal
temperatures. This short assimilation time clearly justifies the assumption of a single
overall temperature θ = θα for all constituents for the purposes of this monograph.
B.10 Real solutions of a cubic equation in case of the
van-der-Waals EOS
The vdW-EOS introduced in Chapter 4 is a cubic EOS that reads:
pβR =
Rβ θ ρβR
1− b ρβR − a (ρ
βR)2 , (B.42)
B.10 Real solutions of a cubic equation in case of the van-der-Waals EOS 139
where Rβ is the specific gas constant and a and b are material parameters depending
on the critical pressure and temperature of the respective substance (e. g., CO2 ), cf.
Chapter 4. The vdW-EOS is used in this monograph to calculate the effective density
ρβR from the effective pressure pβR and the temperature θ. Hence, it is necessary to find
the real solutions for ρβR by solving the cubic equation (B.42). Here, this is accomplished
based on the procedure presented in Numerical Recipes by Press et al. [140].
In a first step, (B.42) is written with respect to the density as follows
a b︸︷︷︸
Ac
(ρβR)3 −a︸︷︷︸
Bc
(ρβR)2 + (Rβ θ + b pβR)︸ ︷︷ ︸
Cc
ρβR−pβR︸ ︷︷ ︸
Dc
= 0 , (B.43)
where Ac, Bc, Cc and Dc are the coefficients of the cubic equation:
Ac = a b , Bc = −a , Cc = Rβ θ + b pβR , Dc = −pβR . (B.44)
For convenience, two auxiliary variables are defined via:
Qc =
1
9
[ B2c
A2c
− 3 Cc
Ac
]
, and Rc =
1
54
[
2
B3c
A3c
− 9 BcCc
A2c
+ 27
Dc
Ac
]
. (B.45)
Therewith, it is possible to distinguish, if the vdW-EOS has three or only one real solution
for the given pβR and θ, by checking
R2c < Q
3
c
{
true → three real solutions ,
false → one real solution . (B.46)
Therein, the first case corresponds to pressure-temperature conditions within the two-
phase region, where phase transition between gas and liquid occurs, and the second case
relates to all conditions outside this region, with a distinct relationship between pβR-θ-ρβR,
cf. Figure 4.2(b).
The three real solution in the first case are determined by
ρβR1 = −2
√
Qc cos (
Sc
3
) − Bc
3Ac
,
ρβR2 = −2
√
Qc cos (
Sc + 2 π
3
) − Bc
3Ac
,
ρβR3 = −2
√
Qc cos (
Sc − 2 π
3
) − Bc
3Ac
,
(B.47)
where cos( · ) is the cosine operator and Sc is another auxiliary term
Sc = arccos (
Rc√
Q3c
) , (B.48)
with arccos( · ) symbolising the inverse trigonometric function of the cosine. From these
three solutions only the two with the lowest, min{ρβR1 , ρβR2 , ρβR3 }, and highest value
140 Appendix B: Thermodynamical supplements and specific evaluations
max{ρβR1 , ρβR2 , ρβR3 }, have a physical meaning, i. e., they represent the gas and liquid
densities on the border of the two-phase region, cf. Figure 4.2(b). From these two remain-
ing solutions, the required density corresponding to the actual conditions is found with
the help of the Antoine equation and the vapour pressure pRvap, according to the procedure
described in Chapter 4.
Proceeding with the second case, the one real solution of the vdW-EOS is determined
after introducing another auxiliary term:
Tc = sgn (Rc)
[
|Rc|+
√
R2c −Q3c
]1/3
, (B.49)
where the operator sgn ( · ) stands for the signum function and is defined as follows
sgn ( · ) :=

−1 if ( · ) < 0 ,
0 if ( · ) = 0 ,
1 if ( · ) > 0 .
(B.50)
In this regard, the one real solution is given by
ρβR = Tc +
Qc
Tc
− Bc
3Ac
, (B.51)
This density corresponds to thermodynamical states outside of the two-phase region. In
particular, ρβR can represent in this case gas, liquid or supercritical phases.
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Curriculum Vitae
Personal Data:
Name: Kai Klaus Ha¨berle
Date of birth: 9th July, 1982
Place of birth: Langenau, Germany
Nationality: German
Parents: Klaus and Klara Ha¨berle
Siblings: Thomas Ha¨berle
Marital status: married to Cornelia Ha¨berle
Children: Carlotta Ha¨berle
Education:
09/1989 – 07/1993 elementary school “Verbandsgrundschule Heusteig”,
Asselfingen, Germany
09/1993 – 07/2002 secondary school “Robert-Bosch-Gymnasium”,
Langenau, Germany
07/2002 degree: “Allgemeine Hochschulreife” (high school diploma)
10/2003 – 01/2010 studies in environmental engineering at the
University of Stuttgart, Germany
01/2010 degree: “Diplom-Ingenieur (Dipl.-Ing.) Umweltschutztechnik”
Civilian Service:
08/2002 – 05/2003 civilian service, “Kreiskrankenhaus” Langenau, Germany
Education in Foreign Countries:
08/2006 – 06/2007 ERASMUS, Norwegian University of Science and Technology,
Trondheim, Norway
research project, University Centre Svalbard,
Longyearbyen, Spitzbergen, Norway
Internships:
07/2003 – 09/2003 ARGE Donaumoos, Riedheim, Germany
10/2008 – 03/2009 Heidelberger Druckmaschinen AG, Heidelberg, Germany
Professional Occupation:
04/2010 – 03/2016 assistant lecturer and research associate at the
Institute of Applied Mechanics (Civil Engineering) at the
University of Stuttgart, Germany
09/2016 CFD-engineer at Alfred Ka¨rcher GmbH & Co. KG, Winnenden, Germany

Released Report Series
II-1 Gernot Eipper: Theorie und Numerik finiter elastischer Deformationen in fluid-
gesa¨ttigten poro¨sen Festko¨rpern, June 1998.
II-2 Wolfram Volk: Untersuchung des Lokalisierungsverhaltens mikropolarer poro¨ser
Medien mit Hilfe der Cosserat-Theorie, May 1999.
II-3 Peter Ellsiepen: Zeit- und ortsadaptive Verfahren angewandt auf Mehrphasen-
probleme poro¨ser Medien, July 1999.
II-4 Stefan Diebels: Mikropolare Zweiphasenmodelle: Formulierung auf der Basis der
Theorie Poro¨ser Medien, March 2000.
II-5 Dirk Mahnkopf: Lokalisierung fluidgesa¨ttigter poro¨ser Festko¨rper bei finiten elasto-
plastischen Deformationen, March 2000.
II-6 Heiner Mu¨llerscho¨n: Spannungs-Verformungsverhalten granularer Materialien
am Beispiel von Berliner Sand, August 2000.
II-7 Stefan Diebels (Ed.): Zur Beschreibung komplexen Materialverhaltens: Beitra¨ge
anla¨ßlich des 50. Geburtstages von Herrn Prof. Dr.-Ing. Wolfgang Ehlers, August
2001.
II-8 Jack Widjajakusuma: Quantitative Prediction of Effective Material Parameters
of Heterogeneous Materials, June 2002.
II-9 Alexander Droste: Beschreibung und Anwendung eines elastisch-plastischen
Materialmodells mit Scha¨digung fu¨r hochporo¨se Metallscha¨ume, October 2002.
II-10 Peter Blome: Ein Mehrphasen-Stoffmodell fu¨r Bo¨den mit U¨bergang auf Inter-
face-Gesetze, October 2003.
II-11 Martin Ammann: Parallel Finite Element Simulations of Localization Phenomena
in Porous Media, April 2005.
II-12 Bernd Markert: Porous Media Viscoelasticity with Application to Polymeric
Foams, July 2005.
II-13 Saeed Reza Ghadiani: A Multiphasic Continuum Mechanical Model for Design
Investigations of an Effusion-Cooled Rocket Thrust Chamber, September 2005.
II-14 Wolfgang Ehlers & Bernd Markert (Eds.): Proceedings of the 1st GAMM
Seminar on Continuum Biomechanics, September 2005.
II-15 Bernd Scholz: Application of a Micropolar Model to the Localization Phenomena
in Granular Materials: General Model, Sensitivity Analysis and Parameter Opti-
mization, November 2007.
II-16 Wolfgang Ehlers & Nils Karajan (Eds.): Proceedings of the 2nd GAMM
Seminar on Continuum Biomechanics, December 2007.
II-17 Tobias Graf: Multiphasic Flow Processes in Deformable Porous Media under
Consideration of Fluid Phase Transitions, 2008.
II-18 Ayhan Acartu¨rk: Simulation of Charged Hydrated Porous Materials, 2009.
II-19 Nils Karajan: An Extended Biphasic Description of the Inhomogeneous and
Anisotropic Intervertebral Disc, 2009.
II-20 Bernd Markert: Weak or Strong – On Coupled Problems In Continuum Me-
chanics, 2010.
II-21 Wolfgang Ehlers & Bernd Markert (Eds.): Proceedings of the 3rd GAMM
Seminar on Continuum Biomechanics, 2012.
II-22 Wolfgang Ehlers: Poro¨se Medien – ein kontinuumsmechanisches Modell auf
der Basis der Mischungstheorie, 2012.
II-23 Hans-Uwe Rempler: Damage in multi-phasic Materials Computed with the Ex-
tended Finite-Element Method, 2012.
II-24 Irina Komarova: Carbon-Dioxide Storage in the Subsurface: A Fully Coupled
Analysis of Transport Phenomena and Solid Deformation, 2012.
II-25 Yousef Heider: Saturated Porous Media Dynamics with Application to Earth-
quake Engineering, 2012.
II-26 Okan Avci: Coupled Deformation and Flow Processes of Partial Saturated Soil:
Experiments, Model Validation and Numerical Investigations, 2013.
II-27 Arndt Wagner: Extended Modelling of the Multiphasic Human Brain Tissue with
Application to Drug-Infusion Processes, 2014.
II-28 Joffrey Mabuma: Multi-Field Modelling and Simulation of the Human Hip Joint,
2014.
II-29 Robert Krause: Growth, Modelling and Remodelling of Biological Tissue, 2014.
II-30 Seyedmohammad Zinatbakhsh: Coupled Problems in the Mechanics of Multi-
Physics and Multi-Phase Materials, 2015.
II-31 David Koch: Thermomechanical Modelling of Non-isothermal Porous Materials
with Application to Enhanced Geothermal Systems, 2016.
II-32 Maik Schenke: Parallel Simulation of Volume-coupled Multi-field Problems with
Special Application to Soil Dynamics, 2017.
II-33 Steffen Mauthe: Variational Multiphysics Modeling of Diffusion in Elastic Solids
and Hydraulic Fracturing in Porous Media, 2017.