Multiphase multicomponent ow in porous media with general ... · Multiphase multicomponent ow in...

Multiphase multicomponent flow inporous media with general reactions:

efficient problem formulations,conservative discretizations, and

convergence analysis

Mehrphasen-Mehrkomponenten-Fluss in porosen Medien

mit allgemeinen chemischen Reaktionen: effiziente

Problemformulierungen, massenerhaltende

Diskretisierungen und Konvergenzordnungsanalyse

Der Naturwissenschaftlichen Fakultatder Friedrich–Alexander–Universitat Erlangen–Nurnberg

zurErlangung des Doktorgrades Dr. rer. nat.

vorgelegt vonFabian Brunner

aus Weiden i.d.OPf.

Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultatder Friedrich–Alexander–Universitat Erlangen–Nurnberg

Tag der mundlichen Prufung: 22. Dezember 2015

Vorsitzender des Promotionsorgans: Prof. Dr. Jorn Wilms

Gutachter: Prof. Dr. Peter Knabner

Prof. Dr. Florin A. Radu

Contents

Danksagung (German) 11

Zusammenfassung (German) 13

I Efficient formulations and numerical approaches formultiphase-multicomponent flow in porous media withgeneral chemical reaction systems 19

1 Introduction 20

1.1 Current state of research . . . . . . . . . . . . . . . . . . . . . . . 22

1.2 Objective of this work . . . . . . . . . . . . . . . . . . . . . . . . 23

1.3 Overview over this work . . . . . . . . . . . . . . . . . . . . . . . 25

2 Mathematical model 27

2.1 Conservation equations . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 Darcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.2 Diffusive and dispersive fluxes . . . . . . . . . . . . . . . . 31

2.2.3 Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.4 Capillary pressure law . . . . . . . . . . . . . . . . . . . . 33

2.2.5 Relative permeabilities . . . . . . . . . . . . . . . . . . . . 34

2.3 Chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.1 Kinetic reactions according to the law of mass action . . . 36

2.3.2 Equilibrium reactions according to the law of mass action . 37

2.3.3 Equilibrium mineral reactions . . . . . . . . . . . . . . . . 38

2.3.4 Interphase mass exchange . . . . . . . . . . . . . . . . . . 41

2.4 Reactive multiphase multicomponent model . . . . . . . . . . . . 42

3

4 CONTENTS

3 The reduction scheme 47

3.1 Transformation of the system of equations . . . . . . . . . . . . . 48

3.2 Choice of primary variables . . . . . . . . . . . . . . . . . . . . . 54

3.2.1 Existing approaches for phase (dis-)appearance . . . . . . 56

3.2.2 Persistent primary variables for our model . . . . . . . . . 61

3.3 The resulting nonlinear system . . . . . . . . . . . . . . . . . . . . 65

3.4 Variants and special cases . . . . . . . . . . . . . . . . . . . . . . 66

3.4.1 No additional transformed variables . . . . . . . . . . . . . 66

3.4.2 No extended capillary pressure variable . . . . . . . . . . . 69

3.4.3 Gas and liquid phase pressures as local unknowns . . . . . 72

3.4.4 Two-phase two-component flow without reactions . . . . . 73

4 Resolution function 75

4.1 Local resolution function . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Global resolution function . . . . . . . . . . . . . . . . . . . . . . 82

4.2.1 General statements . . . . . . . . . . . . . . . . . . . . . . 82

4.2.2 Existence of a global resolution function . . . . . . . . . . 86

4.3 The Semismooth Newton method . . . . . . . . . . . . . . . . . . 96

5 Discretization 99

5.1 Initial and boundary conditions . . . . . . . . . . . . . . . . . . . 101

5.2 Discretization with linear finite elements . . . . . . . . . . . . . . 102

5.3 FV stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.4 Implicit elimination . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 The numerical framework 113

6.1 The M++ toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2 Adaptive time stepping . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3 The global Newton solver . . . . . . . . . . . . . . . . . . . . . . 115

6.4 Special numerical treatment . . . . . . . . . . . . . . . . . . . . . 118

6.4.1 Updating the global Newton step . . . . . . . . . . . . . . 118

6.4.2 Gas phase appearance and disappearance . . . . . . . . . . 122

6.4.3 The local problems . . . . . . . . . . . . . . . . . . . . . . 123

6.4.4 Kinetic mineral reactions . . . . . . . . . . . . . . . . . . . 128

7 Numerical results 131

7.1 The MoMaS benchmark on multiphase flow . . . . . . . . . . . . 131

7.1.1 Parameters and setup . . . . . . . . . . . . . . . . . . . . . 132

CONTENTS 5

7.1.2 Comparison of GIA and SIA . . . . . . . . . . . . . . . . . 136

7.2 Influence of different retention curves . . . . . . . . . . . . . . . . 141

7.3 CO2 sequestration . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.3.1 CO2 injection without chemical reactions . . . . . . . . . . 152

7.3.2 CO2 injection with chemical reactions . . . . . . . . . . . . 157

II Analysis of robust mixed hybrid finite element dis-cretizations for advection–diffusion problems 166

1 Introduction 167

1.1 Current state of research . . . . . . . . . . . . . . . . . . . . . . . 167

1.2 Objective of this work . . . . . . . . . . . . . . . . . . . . . . . . 168

1.3 Overview over this work . . . . . . . . . . . . . . . . . . . . . . . 169

1.4 Notations and assumptions . . . . . . . . . . . . . . . . . . . . . . 170

2 Continuous mixed variational formulation 172

3 MHFE schemes based on the RT0 element 175

3.1 Approximation spaces and projection operators . . . . . . . . . . 175

3.2 A new class of Euler-implicit MHFE schemes . . . . . . . . . . . . 177

3.2.1 The mixed finite element schemes . . . . . . . . . . . . . . 177

3.2.2 Static condensation . . . . . . . . . . . . . . . . . . . . . . 181

3.3 Error analysis of the fully discrete problem . . . . . . . . . . . . . 182

3.4 Choice of the weight function . . . . . . . . . . . . . . . . . . . . 193

3.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

4 MHFE schemes based on the BDM1 element 198

4.1 Suboptimal convergence of the standard scheme . . . . . . . . . . 199

4.2 The modified hybrid scheme . . . . . . . . . . . . . . . . . . . . . 200

4.3 Optimal convergence of the modified scheme . . . . . . . . . . . . 205

4.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Conclusion 208

Bibliography 211

List of Figures

Part I 19

1.1 CO2 trapping mechanisms [MDdC+05, p. 208] . . . . . . . . . . . 21

2.1 Typical shape of capillary pressure and relative permeability curves 34

5.1 Control volume Ωj associated with aj . . . . . . . . . . . . . . . 105

5.2 Intersection of a control volume Ωj with a triangle T . . . . . . . 107

6.1 Update of the time step size . . . . . . . . . . . . . . . . . . . . . 115

6.2 The adaptive time stepping algorithm . . . . . . . . . . . . . . . . 116

6.3 The damped Newton algorithm . . . . . . . . . . . . . . . . . . . 117

6.4 Algorithm for employing the global Newton update using startingvalue search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.5 Evolution of pc at Γin . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.6 Detailed algorithm for assembling the local defect . . . . . . . . . 127

7.1 Computational domain for the MoMaS benchmark problem . . . . 132

7.2 Evolution of gas phase saturation and pressures at Γin . . . . . . . 136

7.3 Schematic of the sequential iterative approach for the MoMaSbenchmark problem . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.4 Evolution of the time step size for SIA and GIA for εabs = 10−10 . 140

7.5 Retention curves of the modified MoMaS benchmark problem . . 142

7.6 Computational mesh for the modified MoMaS benchmark problem 143

7.7 Evolution of gas phase saturation and pressures at Γin for the easytest case (top) and the hard test case (bottom) . . . . . . . . . . 146

7.8 Mass density of CO2 as a function of pressure . . . . . . . . . . . 151

7.9 Solubility of CO2 as a function of pressure . . . . . . . . . . . . . 151

7.10 Viscosity of CO2 as a function of pressure . . . . . . . . . . . . . 152

7.11 Computational domain for the CO2 injection scenario . . . . . . . 153

6

LIST OF FIGURES 7

7.12 Gas phase saturation during the CO2 injection process after 7, 20and 65 days . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.13 Computational domain for the mineral trapping scenario . . . . . 160

7.14 Results after 7 days . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.15 Results after 20 days . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.16 Results after 85 days . . . . . . . . . . . . . . . . . . . . . . . . . 165

Part II 167

3.1 Scalar unknowns and Lagrange multiplier associated with the com-mon edge of two adjacent triangles K1 and K2 . . . . . . . . . . . 195

4.1 Degrees of freedom of the local BDM1 space . . . . . . . . . . . . 200

List of Tables

Part I 19

7.1 Parameters of the MoMaS benchmark problem (test case 1) . . . 135

7.2 Performance of GIA and SIA for the MoMaS benchmark . . . . . 141

7.3 Parameters of the modified benchmark problem . . . . . . . . . . 144

7.4 Relative L2-errors and experimental orders of convergence (EOC)for the easy test case (α1 = 5 · 10−7) at t = 105 years. . . . . . . . 146

7.5 Relative L2-errors and experimental orders of convergence (EOC)for the hard test case (α2 = 5 · 10−4) at t = 105 years. . . . . . . . 147

7.6 Control parameters for the performance test . . . . . . . . . . . . 148

7.7 Benchmark performance results for the easy test case (α1 = 5 · 10−4)148

7.8 Benchmark performance results for the hard test case (α2 = 5 ·10−4)149

7.9 Computational results reported by Uni Heidelberg (R. de Cuvelandand P. Bastian) for the easy test case (α1 = 5 · 10−7) . . . . . . . 149

7.10 Computational results reported by Uni Heidelberg (R. de Cuvelandand P. Bastian) for the hard test case (α2 = 5 · 10−4) . . . . . . . 149

7.11 Parameters for the CO2 injection scenario . . . . . . . . . . . . . 154

7.12 Comparison of two different choices of primary variables for theCO2 injection scenario . . . . . . . . . . . . . . . . . . . . . . . . 155

7.13 Parameters for the mineral trapping scenario . . . . . . . . . . . . 162

Part II 167

3.1 Numerical results for the upwind-mixed hybrid method . . . . . . 197

3.2 Numerical results for the upwind-mixed method [Daw98] . . . . . 197

4.1 BDM1 basis functions on the reference triangle K . . . . . . . . . 204

8

LIST OF TABLES 9

4.2 L2-errors and experimental orders of convergence (EOC) for themodified BDM1 scheme . . . . . . . . . . . . . . . . . . . . . . . 207

4.3 L2-errors and experimental orders of convergence (EOC) for theclassical BDM1 scheme . . . . . . . . . . . . . . . . . . . . . . . . 207

Acronyms

AE algebraic equation

CCS carbon capture and storage

CPU central processing unit

DAE differential algebraic equation

DPO distributed point object

EOS equation of state

FE finite element

FV finite volume

GIA global implicit approach

LFEM linear finite element method

MHFEM mixed hybrid finite element method

MPI message passing interface

ODE ordinary differential equation

PDE partial differential equation

SIA sequential iterative approach

SNIA sequential non-iterative approach

REV representative elementary volume

10

Danksagung

Mit der Fertigstellung der vorliegenden Dissertationsschrift mochte ich die Gele-

genheit nutzen, allen Personen zu danken, die mich in den vergangenen Jahren

unterstutzt und begleitet haben.

Mein großter Dank gilt dabei meinem Betreuer und Doktorvater Herrn Prof. Dr.

Peter Knabner fur die Moglichkeit, dieses interessante und spannende Promotions-

vorhaben in seiner Arbeitsgruppe umzusetzen. Vor allem danke ich ihm fur seine

zahlreichen inhaltlichen Impulse, die fachliche Beratung und Forderung sowie fur

das in mich gesetzte Vertrauen.

Ebenso gilt mein außerordentlicher Dank Herrn Prof. Dr. Florin A. Radu, der

mir wertvolle Unterstutzung zuteil werden ließ und sich stets geduldig, diskus-

sionsbereit und offen fur alle Fragen zeigte. Als Mitbetreuer des zweiten Teils der

vorliegenden Arbeit war er maßgeblich an der Entstehung der bereits publizierten

Beitrage [BRBK12, BBKR13, BRK14] beteiligt. Besonders mochte ich ihm fur

die Moglichkeit danken, zwei mehrwochige Forschungsaufenthalte in seiner Ar-

beitsgruppe an der Universitat Bergen in Norwegen zu verbringen sowie fur die

in diesem Zusammenhang entgegengebrachte Gastfreundschaft.

Herrn Dr. Joachim Hoffmann danke ich fur seine umfassende Diskussionsbereit-

schaft, die fachlichen Hilfestellungen und die Unterstutzung bei der Einarbeitung

in das Softwarepaket M++. Besonders hervorheben mochte ich weiterhin Herrn

Prof. Dr. Serge Krautle, dessen Tur mir immer offen stand und der mich bei der

Einarbeitung in das Reduktionsverfahren maßgeblich unterstutzte, sowie Herrn

Prof. Dr. Markus Bause, der bereits im Studium mein Interesse an Numerischer

Mathematik geweckt hatte und mir wahrend der Promotion als Mentor wertvolle

Impulse gab. Herrn Dr. Julian Fischer gilt mein Dank fur viele interessante

fachliche Diskussionen und die gute Zusammenarbeit beim Verfassen des Artikels

[BFK15].

11

12 DANKSAGUNG

Mit der Fertigstellung dieser Arbeit bin ich nun seit fast zehn Jahren als Hilfs-

kraft und wissenschaftlicher Mitarbeiter am AM1 tatig. Ich danke allen Kol-

leginnen und Kollegen fur die schone Zeit und das wunderbare Arbeitsklima. Bei

der Bewaltigung von burokratischen und technischen Problemen waren stets die

Sekretarinnen Frau Astrid Bigott und Frau Cornelia Weber sowie die Systemad-

ministratoren Herr Dr. Alexander Prechtel, Herr Dr. Fabian Klingbeil und Herr

Balthasar Reuter mit Rat und Tat und unermudlichem Einsatz zur Stelle.

Fur die vielen zielfuhrenden Diskussionen und die entgegengebrachte Hilfsbereit-

schaft danke ich ferner Herrn Dr. Vadym Aizinger, Herrn Tobias Elbinger, Herrn

Dr. Florian Frank, Herrn Dr. Matthias Herz, Frau Dr. Estelle Marchand, Frau

Dr. Nadja Ray, Herrn Dr. Raphael Schulz und Herrn Dr. Nicolae Suciu. Ganz

besonders mochte ich meine Burokollegen Herrn Dr. Torsten Muller und Herrn

Markus Gahn hervorheben und ihnen fur das freundschaftliche Arbeitsklima

danken, das sich auch uber den Buroalltag hinaus fortsetzte.

Schließlich danke ich meinen Freunden, meinen Eltern und meiner Familie fur

ihre Verbundenheit, Geduld und kontinuierliche Unterstutzung.

Titel, Zusammenfassung und

Aufbau der Arbeit

Mehrphasen-Mehrkomponenten-Fluss in porosen Medien

mit allgemeinen chemischen Reaktionen: effiziente

Problemformulierungen, massenerhaltende

Diskretisierungen und Konvergenzordnungsanalyse

Zusammenfassung

Die vorliegende Dissertationsschrift beschaftigt sich mit der numerischen Simula-

tion von reaktiven Transportprozessen in porosen Medien, wie sie in verschiedenen

Anwendungen in den Geowissenschaften auftreten.

Teil 1

Im ersten Teil der Arbeit wird dabei ein Mehrphasen-Mehrkomponentenmodell

zugrunde gelegt, welches verschiedene physikalische Phanomene wie das konkur-

rierende Fließen einer Gas- und einer Wasserphase im Porenraum, den Massen-

austausch dieser Phasen, den advektiven und diffusiven Transport der einzel-

nen Komponenten der Fluidphasen sowie chemische Reaktionen der Komponen-

ten berucksichtigt, welche sowohl kinetische als auch Gleichgewichtsreaktionen

sein konnen. Die mathematischen Modellgleichungen setzen sich aus partiellen

Differentialgleichungen, gewohnlichen Differentialgleichungen und algebraischen

Gleichungen zusammen und zeichnen sich durch eine starke nichtlineare Kop-

plung aus.

Fur die Entwicklung eines effizienten numerischen Losers wird das mathematische

13

14 ZUSAMMENFASSUNG

Modell zunachst einer Variablentransformation unterzogen und in ein dazu aquiv-

alentes System uberfuhrt, wodurch die Elimination der im System auftretenden

Gleichgewichtsreaktionsraten ermoglicht wird. Dazu wird das Reduktionsver-

fahren aus [KK05, KK07, HKK12] verwendet und auf den Fall dreier vorliegen-

der Phasen (Feststoffphase, Flussigphase und Gasphase) erweitert. Auf Grund

der Kopplung von Fließ- und Transportgeschehen findet, im Gegensatz zu den

o. g. Arbeiten, in denen das Stromungsfeld als bekannt vorausgesetzt wird, eine

Entkopplung von partiellen Differentialgleichungen infolge der Variablentrans-

formation durch das Reduktionsverfahren nicht mehr statt, da diese allesamt

vom zunachst unbekannten Druck der Flussigphase abhangen. Eine Reduktion

der Zahl der Unbekannten ist jedoch durch die implizite Auflosung der alge-

braischen Gleichungen, die aus chemischen Gleichgewichtsbedingungen und der

Diskretisierung von gewohnlichen Differentialgleichungen resultieren und gewis-

sermaßen in die verbleibenden Gleichungen substituiert werden, moglich. Ein

weiterer Vorzug des Reduktionsverfahrens liegt in der Tatsache begrundet, dass

alle im System auftretenden Transportoperatoren linear in den transformierten

Variablen sind.

Die Existenz einer Auflosungsfunktion wird durch zwei verschiedene Methoden

gezeigt. Fur den Fall, dass kinetische Reaktionen vorliegen, wird die Existenz

einer lokalen Auflosungsfunktion im Sinne des Satzes uber die implizite Funk-

tion hergeleitet, wobei eine Version des Satzes fur stuckweise glatte Funktio-

nen verwendet wird. Dies ist notwendig, da die Gleichgewichtsbedingungen, die

aus den Mineralreaktionen und dem Massenaustausch der Phasen resultieren,

nicht glatt sind. Fur den Fall, dass keine kinetischen Reaktionen vorliegen, wird

weiterhin die Existenz einer globalen Auflosungsfunktion nachgewiesen, indem

die Aquivalenz der algebraischen Gleichgewichtsbedingungen zu einem konvexen

Minimierungsproblem mit Nebenbedingungen gezeigt wird.

Eine besondere Schwierigkeit bei der numerischen Losung des globalen nicht-

linearen Systems resultiert aus der Tatsache, dass die Gasphase wahrend der

Simulation in Teilen des Gebiets entstehen oder verschwinden kann, und dass

sich Mineralien bilden oder vollstandig auflosen konnen. Um diese Problematik

anzugehen, werden mit Hilfe der Auflosungsfunktion persistente Primarvariablen

definiert, welche in der Lage sind, das System in jedem Zustand zu beschreiben,

unabhangig davon, ob und in welchen Teilen des Rechengebiets die Gasphase bzw.

Mineralien vorliegen. Dadurch kann ein Wechsel der Primarvariablen wahrend

ZUSAMMENFASSUNG 15

der Simulation vermieden werden.

Bei den durchgefuhrten numerischen Testrechnungen hat sich hinsichtlich der

Wahl der Primarvariablen die Verwendung eines erweiterten Gasdrucks in Kom-

bination mit einem erweiterten Kapillardruck als vorteilhaft fur die Konvergenz

des globalen nichtlinearen Losers erwiesen. Dies liegt darin begrundet, dass einer-

seits einige Koeffizientenfunktionen direkt vom Gasdruck abhangen, andererseits

durch die beiden globalen Druckvariablen keine Nichtlinearitat in den Druckgradi-

enten mehr im System vorliegt. Die Verwendung des erweiterten Kapillardrucks

als Primarvariable fuhrt zusatzlich dazu, dass die Gassattigung nur von dieser

globalen Unbekannten abhangt und dass die Auswertung der Auflosungsfunk-

tion, die das Losen eines nichtlinearen Gleichungssystems erfordert, in mehreren

sequentiellen Schritten erfolgen kann, da die nichtlinearen lokalen Probleme in

mehrere Teilprobleme zerfallen.

Das Auswerten der Auflosungsfunktion ist notwendig, sobald sich die primaren

Variablen geandert haben – also beispielsweise nach dem Erhalt einer neuen

Iterierten fur die Primarvariablen im Rahmen einer Fixpunktiteration. Die dazu

erforderlichen Berechnungen konnen lokal, d.h. in jedem Gitterpunkt separat,

durchgefuhrt werden, weshalb sich diese Elimination auf der Ebene des nichtlin-

earen Losers gut fur parallele Rechnungen eignet. Die aus Gleichungen und Ungle-

ichungen bestehenden Mineralgleichgewichtsbedingungen werden als aquivalentes

Komplementaritatsproblem formuliert und mit Hilfe der Minimumfunktion in

eine Gleichung uberfuhrt. Die resultierenden Gleichungen sind hinreichend glatt,

um das sog. Semismooth Newton-Verfahren zur numerischen Losung anzuwen-

den, welches lokal quadratisch konvergiert. Dieses Konvergenzverhalten konnte

in den durchgefuhrten numerischen Testrechnungen bestatigt werden.

Zum Losen der lokalen Probleme sind weiterhin spezielle Techniken erforderlich,

die im Rahmen dieser Arbeit auf das vorliegende Mehrphasenproblem ubertragen

und angewendet wurden. Dazu gehort eine Modifizierung der lokalen Gleichun-

gen, um die Logarithmen der Konzentrationen als Unbekannte verwenden zu

konnen, was sich als vorteilhaft fur die Konditionszahl der resultierenden Glei-

chungssysteme erweist. Weiterhin muss sichergestellt werden, dass der Newton-

Loser keine Updates fur die transformierten Variablen zulasst, die zu negativen

Konzentrationen fuhren. Durch das Losen von linearen Optimierungsproblemen

wird erreicht, dass ein Newton-Schritt nur so gering wie moglich modifiziert wird,

um physikalische Konzentrationswerte sicherzustellen.

16 ZUSAMMENFASSUNG

Wahrend in der Literatur bisher Splitting-Verfahren zur Losung von gekoppel-

ten Fließ- und Transportproblemen bevorzugt werden, liefert die vorliegende Ar-

beit einen Beitrag zur Entwicklung von global impliziten Verfahren fur reaktive

Mehrphasen-Mehrkomponentenprobleme. Das globale nichtlineare Gleichungssys-

tem wird dabei in jedem Zeitschritt mit Hilfe des Newton-Verfahrens gelost, was

sich in einem numerischen Test als effizienter erwiesen hat als ein iteratives Split-

tingverfahren.

Die in dieser Arbeit enthaltenen numerischen Beispiele belegen weiterhin, dass der

entwickelte Loser das Phanomen verschwindender Phasen erfasst und in der Lage

ist, allgemeine reaktive Systeme zu behandeln. Die Konvergenz der verwende-

ten linearen Finite-Elemente-Diskretisierung, erweitert um eine Finite-Volumen-

Stabilisierung fur advektionsdominierte Probleme, wurde in numerischen Kon-

vergenztests untersucht. Im Rahmen eines Bencharkproblems war die erreichte

Genauigkeit vergleichbar mit der Genauigkeit, die von anderen Gruppen erzielt

wurde.

Zusammenfassend wird die Effizienz des numerischen Losers durch die Variablen-

transformation, die Verwendung der Auflosungsfunktion, die Wahl geeigneter

Primarvariablen, die Verwendung des Newton-Verfahrens fur die globalen und

lokalen Probleme sowie die Implementierung in einer parallelen Finite-Elemente-

Bibliothek gewahrleistet.

Teil II

Der zweite Teil dieser Arbeit beschaftigt sich mit der Analyse von hybriden

gemischten Finite-Elemente-Verfahren zur Diskretisierung einer zeitabhangigen

Advektions-Diffusions-Gleichung, die als Modellgleichung fur eine Vielzahl von

Anwendungen in den Natur- und Ingenieurwissenschaften dient. Gemischte Ver-

fahren sind auf Grund ihrer Eigenschaft des lokalen Massenerhalts in den An-

wendungen stark verbreitet.

Fur das Raviart–Thomas-Element niedrigster Ordnung wird eine neue Klasse von

Verfahren untersucht, die auf einer Verallgemeinerung der Diskretisierung des

advektiven Flusses mit Hilfe eines Gewichts basiert, welches von den zellweise

konstanten Approximationen der skalaren Unbekannten und von den Lagrange-

multiplikatoren, die im Rahmen des Hybridisierungsprozesses eingefuhrt werden,

abhangen darf. Als Spezialfall ergeben sich das Standard-Verfahren sowie ein

Upwind-Verfahren, welches sich fur advektionsdominierte Probleme eignet.

ZUSAMMENFASSUNG 17

Bemerkenswert ist dabei, dass das Verfahren fur jede zulassige Wahl des Gewichts

lokal bleibt, sodass die Elimination von Unbekannten durch statische Konden-

sation stets moglich ist. Dies steht im Gegensatz zu gewohnlichen Upwind-

Verfahren, bei welchen ublicherweise Information der Nachbarzelle zur Diskreti-

sierung des advektiven Terms herangezogen wird, und die daher nicht lokal sind.

Im Rahmen der Konvergenzanalyse wird gezeigt, dass fur jede zulassige Wahl

des Gewichts das Verfahren in der L2-Norm mit optimaler erster Ordnung in

der skalaren Variable und der Flussvariable konvergiert. Dies wird durch nu-

merische Tests veranschaulicht, die weiterhin belegen, dass mit dem hybrid-

gemischten Upwind-Verfahren die gleiche Genauigkeit wie mit dem Standard-

Upwind-Verfahren erreicht werden kann, wahrend durch die statische Kondensa-

tion beinahe 50% an Rechenzeit eingespart wird.

Verwendet man dagegen dasBDM1 Element zur Diskretisierung einer Advektions-

Diffusions-Gleichung, so fuhrt die Verwendung der Lagrange-Multiplikatoren bei

der Diskretisierung des advektiven Flusses sogar zu einer Erhohung der Konver-

genzordnung fur den totalen Massenfluss bestehend aus diffusivem und advek-

tivem Fluss. Wahrend das Standardverfahren lediglich suboptimale Konvergenz

erster Ordnung liefert, kann durch die in dieser Arbeit vorgeschlagene Methode

die optimale Konvergenzordnung fur die Flussvariable ohne eine Erhohung des

algorithmischen Aufwands wiederhergestellt werden.

Aufbau der Arbeit

Im zweiten Kapitel des ersten Teils der vorliegenden Arbeit werden die einzel-

nen Bestandteile des behandelten Mehrphasen-Mehrkomponenten-Modells, fur

welches ein numerischer Loser entwickelt wurde, vorgestellt. Im dritten Kapitel

erfolgt die Transformation des Gleichungssystems durch das Reduktionsverfahren

und die Auswahl geeigneter persistenter Primarvariablen. Mit Hilfe der implizit

definierten Auflosungsfunktion, deren Existenz im vierten Kapitel gezeigt wird,

konnen die Sekundarvariablen stets aus den Primarvariablen bestimmt werden.

Die Diskretisierung des nach der Reduktion verbliebenen globalen nichtlinearen

Gleichungssystems mit dem impliziten Euler-Verfahren in der Zeit und einem

linearen Finite-Elemente-Verfahren im Ort ist Gegenstand des funften Kapi-

tels. Die einzelnen Komponenten des entwickelten numerischen Losers werden

im sechsten Kapitel genauer vorgestellt. Dies umfasst die adaptive Zeitschritt-

18 ZUSAMMENFASSUNG

weitensteuerung, die Funktionsweise des globalen Newton-Losers sowie spezielle

Techniken, die bei der numerischen Auswertung der Auflosungsfunktion zur An-

wendung kommen. Der erste Teil wird durch die Prasentation der numerischen

Testrechnungen im siebten Kapitel abgeschlossen. Diese umfassen Benchmarkrech-

nungen zur Ausbreitung von Wasserstoff in unterirdischen Speicherstatten fur

langlebige radioaktive Abfalle sowie ein numerisches Beispiel zur Mineralbildung

im Rahmen der unterirdischen Speicherung von CO2.

Der zweite Teil der Arbeit ist in vier Kapitel unterteilt. Nach dem einleitenden

Kapitel erfolgt im zweiten Kapitel die Formulierung eines kontinuierlichen ge-

mischten Variationsproblems fur die zu betrachtende Modellgleichung, welches die

Grundlage fur die im Anschluss betrachteten Diskretisierungsverfahren bildet. Im

dritten Kapitel wird eine Klasse von hybrid-gemischten Diskretisierungsverfahren

fur Advektions-Diffusions-Gleichungen analysiert, die auf dem Raviart–Thomas-

Element niedrigster Ordnung basiert und die Lagrange-Multiplikatoren fur die

Diskretisierung des advektiven Flussanteils heranzieht. Die Anwendung derselben

Idee auf Diskretisierungen mit dem BDM1 Element bildet den Kern des vierten

Kapitels, in dem gezeigt wird, dass dadurch die optimale Konvergenzordnung

zweiter Ordnung fur die Flussvariable wiederhergestellt werden kann, die durch

das Standardverfahren nicht erreicht wird. Letzteres approximiert den Fluss nur

mit suboptimaler Genauigkeit erster Ordnung in der L2-Norm.

Bereits publizierte Beitrage

Die wesentlichen Inhalte des zweiten Teils der Dissertationsschrift konnten be-

reits in den Artikeln [BRK14], [BBKR13] und [BRBK12] veroffentlicht werden.

Die Koautoren Florin. A. Radu, Markus Bause und Peter Knabner haben im

Rahmen ihrer Betreuungstatigkeit zum Entstehen der o.g. Arbeiten beigetragen.

Ein weiterer Artikel, in dem ein Konvergenzbeweis fur das im vierten Kapitel des

zweiten Teils vorgestellte numerische Verfahren gefuhrt wird, wurde in Zusamme-

narbeit mit Julian Fischer erarbeitet und befindet sich derzeit in Begutachtung.

Part I

Efficient formulations and

numerical approaches for

multiphase-multicomponent flow

in porous media with general

chemical reaction systems

19

Chapter 1

Introduction

Since the 1950s, when Charles David Keeling started permanent measurements

at the Hawaii Mauna Loa Observatory, the average concentration of CO2 in the

atmosphere has continuously increased, which is widely recognized as one of the

major causes for global warming and climate change. The increase of greenhouse

gases in the atmosphere is mainly due to human activities, such as the combustion

of fossil fuels, and the reduction of emissions has become a dominating topic

in global climate politics during the last two decades. In 2010, the parties of

the Cancun Climate Change Conference agreed upon that deep cuts in global

greenhouse gas emissions were necessary to meet the long-term goal of a limitation

of future global warming to below 2.0 C relative to preindustrial level [oCC11].

Despite of the effort put into the expansion of renewable energies, about 80% of

the energy consumed worldwide are produced by fossil fuels nowadays [IEA14],

and the lowering of this rate will proceed only slowly, while the global energy

demand is expected to grow further.

In order to meet the reduction targets, geoengineering techniques are discussed

as a bridging technology for the next few decades, until renewable energies have

been pushed sufficiently forward and a low carbon economy has been established.

The potentially most promising approach to mitigate carbon dioxide emissions

is the so-called Carbon Capture and Storage Technology (CCS), which prevents

CO2 produced by industrial sites (e. g., fossil fuel power plants) from being re-

leased into the atmosphere. This is accomplished by capturing and storing it at

a suitable place where it is prevented from reentering into the atmosphere, for

example in a deep geological formation with an overlying caprock. In such a sit-

20

21

208 IPCC Special Report on Carbon dioxide Capture and Storage

Paterson, 2003), although appropriate reservoir engineering can

accelerate or modify solubility trapping (Keith et al., 2005).

5.2.2 CO2 storage mechanisms in geological formations

The effectiveness of geological storage depends on a

combination of physical and geochemical trapping mechanisms

(Figure 5.9). The most effective storage sites are those where

CO2 is immobile because it is trapped permanently under a

thick, low-permeability seal or is converted to solid minerals

or is adsorbed on the surfaces of coal micropores or through a

combination of physical and chemical trapping mechanisms.

5.2.2.1 Physical trapping: stratigraphic and structural

Initially, physical trapping of CO2 below low-permeability seals

(caprocks), such as very-low-permeability shale or salt beds,

is the principal means to store CO2 in geological formations

(Figure 5.3). In some high latitude areas, shallow gas hydrates

may conceivably act as a seal. Sedimentary basins have such

closed, physically bound traps or structures, which are occupied

mainly by saline water, oil and gas. Structural traps include

those formed by folded or fractured rocks. Faults can act as

permeability barriers in some circumstances and as preferential

pathways for luid low in other circumstances (Salvi et al., 2000).

Stratigraphic traps are formed by changes in rock type caused

by variation in the setting where the rocks were deposited. Both

of these types of traps are suitable for CO2 storage, although,

as discussed in Section 5.5, care must be taken not to exceed

the allowable overpressure to avoid fracturing the caprock or

re-activating faults (Streit et al., 2005).

5.2.2.2 Physical trapping: hydrodynamic

Hydrodynamic trapping can occur in saline formations that do

not have a closed trap, but where luids migrate very slowly over long distances. When CO

2 is injected into a formation, it

displaces saline formation water and then migrates buoyantly

upwards, because it is less dense than the water. When it reaches

the top of the formation, it continues to migrate as a separate

phase until it is trapped as residual CO2 saturation or in local

structural or stratigraphic traps within the sealing formation.

In the longer term, signiicant quantities of CO2 dissolve in

the formation water and then migrate with the groundwater.

Where the distance from the deep injection site to the end of the

overlying impermeable formation is hundreds of kilometres,

the time scale for luid to reach the surface from the deep basin can be millions of years (Bachu et al., 1994).

5.2.2.3 Geochemical trapping

Carbon dioxide in the subsurface can undergo a sequence of

geochemical interactions with the rock and formation water that

will further increase storage capacity and effectiveness. First,

when CO2 dissolves in formation water, a process commonly

called solubility trapping occurs. The primary beneit of solubility trapping is that once CO

2 is dissolved, it no longer

exists as a separate phase, thereby eliminating the buoyant

forces that drive it upwards. Next, it will form ionic species as

the rock dissolves, accompanied by a rise in the pH. Finally,

some fraction may be converted to stable carbonate minerals

(mineral trapping), the most permanent form of geological

storage (Gunter et al., 1993). Mineral trapping is believed to

be comparatively slow, potentially taking a thousand years

or longer. Nevertheless, the permanence of mineral storage,

combined with the potentially large storage capacity present in

some geological settings, makes this a desirable feature of long-

term storage.

Dissolution of CO2 in formation waters can be represented by

the chemical reaction:

CO2 (g) + H

2O ↔ H

2CO

3 ↔ HCO

3

– + H+ ↔ CO3

2– + 2H+

The CO2 solubility in formation water decreases as temperature

and salinity increase. Dissolution is rapid when formation water

and CO2 share the same pore space, but once the formation

luid is saturated with CO2, the rate slows and is controlled by

diffusion and convection rates.

CO2 dissolved in water produces a weak acid, which reacts

with the sodium and potassium basic silicate or calcium,

magnesium and iron carbonate or silicate minerals in the

reservoir or formation to form bicarbonate ions by chemical

reactions approximating to:

3 K-feldspar + 2H2O + 2CO

2 ↔ Muscovite + 6 Quartz + 2K

+

+ 2HCO3

–

Figure 5.9 Storage security depends on a combination of physical and

geochemical trapping. Over time, the physical process of residual CO2

trapping and geochemical processes of solubility trapping and mineral

trapping increase.

Figure 1.1: CO2 trapping mechanisms [MDdC+05, p. 208]

uation, storage security is provided by different trapping mechanisms. Initially,

structural trapping below the caprock is dominant, which represents a physical

barrier and prevents the injected gas to migrate upwards by buoyant forces. Once

the CO2 starts to move through the pore space, a certain amount remains in the

pores as disconnected, immobile droplets, which is referred to as residual trap-

ping. Another process that occurs when CO2 dissolves into the brine is solubility

trapping, leading to an increase of density of the formation water. Hence it will

migrate downwards, which further enhances storage effectiveness and capacity.

Finally, the safest of all trapping mechanisms is mineral trapping, denoting the

fact that the dissolved CO2 may be transformed into a mineral by geochemical

reactions, which is the most secure stage of CO2 trapping.

Although CO2 sequestration has been an emerging field of research during the

past years and a number of demonstration projects has been started worldwide,

its long term risks are hardly predictable, and the public acceptance and support

is low in many countries. Mathematical models and numerical simulations can be

a valuable tool to predict the long term evolution and assess risks that potentially

emanate from a CCS site, e. g., leakage due to unplugged wells, faults, fractures,

or an insufficiently impermeable caprock.

In the same manner as storage security increases with time and different trap-

ping mechanisms (cf. Figure 1.1), the complexity of mathematical models that

are needed to describe the relevant physical processes grows. While a simple two-

22 CHAPTER 1. INTRODUCTION

phase model is sufficient to reproduce structural trapping, a miscible multiphase

multicomponent model including geochemical effects must be considered to simu-

late mineral trapping. The mathematical equations modeling all these processes

are far too complicated to be solved analytically, and the need for accurate and

reliable numerical methods to obtain approximate solutions is well recognized.

Based on simulations, predictions about the long-term fate of the injected CO2

are possible.

1.1 Current state of research

In a miscible multiphase multicomponent flow model, the dominant physical pro-

cesses are strongly coupled. For example, fluid properties like viscosities and den-

sities are influenced by the pressures and the composition of the phases, which in

turn influences the flow regime and may trigger density driven flow. On the other

hand, the composition of the phases is heavily affected by geochemical reactions.

In the corresponding mathematical model, these complex interactions are re-

flected by coupling terms in the governing equations, which are of quasilinear

type and exhibit strong nonlinearities resulting from nonlinear coefficient func-

tions. This represents a challenge in the design and implementation of robust

and efficient numerical solvers since large systems of nonlinear equations must be

solved if implicit time stepping methods are used.

In an attempt to tackle this problem, most existing reactive multiphase multicom-

ponent flow simulators are based on splitting approaches to treat the nonlinear

coupling, for example TOUGHREACT [XSS+12], PFLOTRAN [MLLH07] and

STOMP [WBM+12]. This means that the computations of one time step are split

into a flow problem and a reactive transport problem, with the relevant physical

quantities being updated after each of these subproblems has been solved.

If this is done non-iteratively (sequential non-iterative approach, SNIA), depend-

ing on the specific problem, there may be heavy restrictions on the time step

size due to stability problems, and the accuracy of the numerical solution may be

poor as a result of splitting errors [VM92]. To eliminate these errors, the subprob-

lems can be solved alternately (sequential iterative approach, SIA) until a certain

tolerance has been reached. An advantage of splitting approaches is that they

are easy to implement and offer flexibility regarding numerical solution strate-

1.2. OBJECTIVE OF THIS WORK 23

gies, e. g., discretization. Moreover, customized codes developed separately for

the specific subproblems (e. g., a reactive transport solver and a multiphase flow

code) can be easily merged using a splitting approach. The approximation that

is obtained from SIA after the iteration has converged corresponds to a solution

of the global nonlinear problem up to some prescribed tolerance. However, many

iteration steps and small time steps may be necessary to establish convergence

[SCA00, HKK12].

For single phase reactive transport problems, the global implicit approach (GIA)

has become more popular in recent years, see, e. g., [dDEK09, dDE10, AK10].

Although GIA requires most computational resources per time step, it avoids

the potential drawbacks of splitting methods and is usually considered to be the

most stable solution method. In the GDR MoMaS reactive transport benchmark

[CKK10], the method of Krautle, Knabner and Hoffmann [KK05, KK07, HKK12]

worked to solve all test cases accurately while being the most efficient of all ap-

proaches [CHK+10]. It is based on a model-preserving transformation of the

system of equations with the help of a reduction scheme, which eliminates the un-

known equilibrium reaction rates and reduces the number of nonlinearly coupled

equations by decoupling a certain number of equations. Moreover, the nonlinear

algebraic equations are used to define a resolution function in order to further

reduce the computational cost. This resembles the so-called direct substitution

approach. Recently, the global implicit approach has also been applied to coupled

multiphase flow and reactive transport problems, cf. [FDT12, SVG+13].

From an implementation point of view, not only the strong nonlinear coupling

terms and the presence of a large number of chemical species may be challenging

but also the possibility of local appearance and disappearance of phases, which

requires special numerical treatment. During the past years, many different meth-

ods were proposed in the literature to deal with this problem. A review of some

recent approaches is given in Section 3.2.1.

1.2 Objective of this work

The goal of the first part of this thesis is to extend the work of Hoffmann, Krautle

and Knabner [HKK12] to the case of multiple fluid phases and thus to contribute

to the development of global implicit solvers for miscible reactive multiphase


multicomponent flow in porous media. For this purpose, we consider a mathe-

matical model including three phases (gas, liquid and solid) consisting of multiple

components, which may participate in chemical reactions (equilibrium or kinetic

reactions, homogeneous or heterogeneous). Moreover, the transfer of mass across

the phases is taken into account. Due to these complex interactions, the model

is strongly nonlinear. While in this work we focus on the injection and storage of

CO2 into deep saline aquifers [Bie06, NC12], it should be noted that multiphase

multicomponent flow models are of importance for a wide range of applications

arising in several fields of environmental engineering and reservoir engineering in

the subsurface, e. g., enhanced oil recovery [CGCM74], groundwater protection

and remediation [CHB02], or hydrogen migration in the vicinity of radioactive

waste repositories [BJS09].

After transforming the system of equations using the reduction scheme, it is

possible to reduce the size of the global problem by eliminating variables locally

with the help of the chemical equilibrium laws, which define a nonlinear and

possibly nonsmooth resolution function. The proof of existence of this resolution

function using techniques from the field of optimization represents one of the

main issues of the first part of this work.

The implementation of the resulting formulation in the parallel finite element

toolbox M++ [Wie, Wie05, Wie10] represents another main issue of this work. By

the choice of our primary variables based on the resolution function and extended

phase pressures, it is ensured that the formulation remains valid if the gas phase

appears or disappears. The same holds for the precipitation and dissolution of

minerals. In order to deal with realistic chemical problems, certain additional

variables proposed in [Hof10] are used, and special techniques to evaluate the

resolution function are implemented.

Altogether, the efficiency of our solver is provided by the transformation of the

system of equations, the use of the resolution function, the selection of appropriate

primary variables, the use of Newton’s method as a nonlinear solver for the

global and local problems, and the implementation in a parallel finite element

library. Different numerical examples related to nuclear waste management and

CO2 sequestration demonstrate that our method is efficient and produces accurate

results.

1.3. OVERVIEW OVER THIS WORK 25

1.3 Overview over this work

The multiphase multicomponent model will be introduced in Chapter 2. It in-

cludes the exchange of mass between three phases (gas, liquid, solid) and the

reactive transport of the components of the mobile phases due to advection, dif-

fusion and dispersion. In particular, the different types of chemical reactions

are introduced and constitutive relationships for the phase densities, capillary

pressure and relative permeabilities are given.

In Chapter 3, the reduction scheme of Krautle, Knabner and Hoffmann, cf.

[KK05, KK07, HKK12], is applied to our model, which means that linear combi-

nations of equations are taken to eliminate the equilibrium reaction rates and that

a linear variable transformation of the original variables is performed. Moreover,

we give an overview over existing approaches to handle the problem of vanishing

phases, and we specify our choice of primary and secondary variables. Finally, the

size of the transformed system is reduced by resolving the local equations result-

ing from chemical equilibrium laws and discretized ODEs in terms of a resolution

function. The chapter ends with a presentation and discussion of alternative for-

mulations based on different choices of primary variables, and a consideration of

the special case of two-phase two-component flow.

In Chapter 4, the existence of the resolution function is proven in two different

ways. In Section 4.1, the existence of a local resolution function in terms of the

implicit function theorem is given for the full reaction system. Since they con-

tain complementarity constraints, the equilibrium conditions related to mineral

reactions and the interphase mass exchange are only piecewise smooth. Conse-

quently, the assumptions of an implicit function theorem for piecewise smooth

functions must to be verified. In the absence of kinetic reactions, we can prove

the existence of a global resolution function, which represents a stronger result.

This is accomplished by showing that the algebraic equations are equivalent to

the KKT system of a convex minimization problem, cf. Section 4.2. Numerically,

the nonlinear resolution function must be evaluated by some iterative method.

This is referred to as local problem, while solving the remaining coupled non-

linear system is called the global problem. In Section 4.3, we describe how the

Semismooth Newton method can be applied to solve the local problems.

In Chapter 5, the discretization of the transformed system using a linear finite

element method is illustrated for a model problem. For advection–dominated


problems, an upwind-weighted scheme is used, which is based on a finite volume

approximation of the advective term. At the end of this chapter, a method to

evaluate the derivative of the resolution function is presented, cf. Section 5.4.

The numerical framework is presented in Chapter 6 and includes a description of

the time stepping method (Section 6.2), the global Newton solver (Section 6.3)

and special numerical treatment that is necessary to deal with realistic problems

(Section 6.4). For example, the logarithms of the concentrations are used as un-

knowns in the local problems, which requires to enlarge this system by additional

equations. By solving linear optimization problems and using a local backtracking

line search strategy based on Armijo’s rule, it is ensured that the global Newton

solver does not produce iterates of the transformed variables corresponding to

nonphysical negative concentrations. Moreover, feasible starting values for the

local Newton iterations are provided. The section ends with a description of the

treatment of kinetic mineral reactions.

Chapter 7 contains the results of our numerical experiments. First, a recent

benchmark focusing on the appearance and disappearance of the gas phase dur-

ing hydrogen injection into a deep geological repository of nuclear waste is re-

computed, cf. [BGS13]. This admits a comparison of our results with the results

of other groups and demonstrates the ability of the numerical model to deal with

vanishing phases. The same test problem is used to compare our global implicit

solver with an iterative splitting method, where the problem is decomposed into

two subproblems that are solved alternately until global convergence is obtained.

To further analyze the behavior of the nonlinear solver and the time stepping

method, a slightly different version of this benchmark problem was considered

together with a group from the University of Heidelberg. The results of this

comparison are presented in Section 7.2. Finally, in Section 7.3 we present two

numerical tests related to the injection and storage of CO2 in a deep geologic for-

mation. With the help of these examples, we are able to analyze the influence of

the choice of primary variables on the convergence of the global nonlinear solver,

and we demonstrate the ability of our numerical solver to handle the strong non-

linear coupling of flow, transport, chemical reactions and mass transfer across

phases using the global implicit approach.

Chapter 2

Mathematical model

In this chapter, the mathematical model of interest is presented in detail. It

consists of a system of partial differential equations (PDE), ordinary differential

equations (ODE) and algebraic equations (AE) modeling the flow of two partially

miscible fluid phases through a porous medium on a macroscopic length scale,

including capillary effects, compressibility of phases, interphase mass exchange,

and chemical reactions. The variables and physical quantities of this model are

obtained from averaging over a representative elementary volume (REV) on the

micro scale [Bea72], resulting in a continuum description on the macro scale.

A porous medium (e. g., rock, soil) is a body composed of a solid skeletal material

and the remaining (connected) pore space. It is characterized by its porosity φ,

which is the ratio between the volume of the pore space within a given REV

and the volume of the REV itself. In this work, we assume that the underlying

porous medium is rigid, i. e., the porosity is a function of space only and remains

constant over time.

The pore space may be filled by one or two fluid phases: a liquid phase (denoted

by the subscript `) and a gas phase (denoted by the subscript g). While we

require that the liquid phase is always present throughout the domain, the gas

phase may appear or disappear locally. The ratio between the volume of phase

α ∈ `, g and the total volume of pore space in a given REV is defined as sα, the

saturation of phase α. It follows directly from this definition that the saturations

sum up to one. Hence, it is sufficient to consider only sg as an independent

variable.

27

28 CHAPTER 2. MATHEMATICAL MODEL

A phase can be composed of multiple components relating to chemical substances.

The liquid phase and the solid phase (subscript s) may consist of an arbitrary

number of I` and Is components, respectively, whereas the gas phase is assumed

to consist of one component only. This assumption is justified if there is one dom-

inant component after a gas injection process. It should be noted, however, that

this simplification does not represent a restriction of our method, which can be

extended to the case of multiple components in the gas phase. The chemical reac-

tions may be homogeneous reactions between the components of the liquid phase

or heterogeneous reactions between the components of the liquid phase and the

solid matrix, the latter being sorption reactions or mineral reactions. The trans-

port of the components of the fluid phases is subject to advection, diffusion and

dispersion. For simplicity, our model does not include thermal effects by assum-

ing a constant temperature throughout the domain. Non-isothermal conditions

could be easily incorporated, however, by assuming a geothermal gradient.

2.1 Conservation equations

On a macroscopic level, the temporal and spatial distribution of the chemical

components in a multiphase multicomponent system are governed by component

mass balance equations [Hel97]. Let Ω ⊂ Rd, d ∈ 2, 3, be a bounded domain

and let Tend > 0 denote some fixed final time. Then, the mass balance equation

for component i in phase α ∈ `, g reads

∂t(φsαciα) +∇ · (qαciα + jiα) = f iα in Ω× (0, Tend) , i ∈ 1, . . . , Iα , (2.1)

where:

Iα number of components in phase α [–] ,

φ porosity [–] ,

sα saturation of phase α [–] ,

ciα molar concentration of comp. i in phase α [molm3 ] ,

qα Darcy flux of phase α [ms] ,

jiα Diffusive/dispersive flux [ molm2·s ] ,

f iα source/sink term [ molm3·s ] .

2.1. CONSERVATION EQUATIONS 29

The first term on the left hand side of (2.1) is an accumulation term, whereas the

other terms represent advective and diffusive/dispersive transport, respectively.

The source terms on the right hand side model production rates due to chemical

reactions, cf. Section 2.3.

Since the components of the solid phase are immobile, the corresponding mass

balance equations do not include transport terms. They are given by

∂t(φs`cis) = f is in Ω× (0, Tend) , i ∈ 1, . . . , Is , (2.2)

where cis denotes the concentration of the i-th immobile species of the solid phase.

In this work, the concentrations of immobile species are defined as the amount

of substance per liquid volume, cf. [YT89, Hof10]. This is possible as long as

the liquid phase does not vanish. Alternatively, one could use the amount of

substance per volume bulk or per pore surface as a unit for the immobile concen-

trations. The advantage of the choice taken here is that linear combinations of

liquid and immobile concentrations can be taken without including a conversion

factor.

Throughout this work, let the liquid concentrations be represented by the vector

c` ∈ RI` and the solid concentrations by the vector cs ∈ RIs . Moreover, let

the first Is,nmin entries of cs represent the concentrations of nonminerals (sorbed

species), followed by Jmin minerals:

c` = (c1` , . . . , c

I`` )T , cs =

cs,nmin

cs,min

= (c1s, . . . , c

Is,nmins , cIs,nmin+1

s , . . . , cIss )T ,

where cs,nmin ∈ RIs,nmin and cs,min ∈ RJmin . Note that in our model, the number

of equilibrium minerals is equal to the number of equilibrium mineral reactions,

cf. Section 2.3.3. For later use, we also define the global concentration vector

c = (c1, . . . , cI`+Is)T ∈ RI`+Is ,


which has the block structure

c =

c`cs

=

cnmin

cmin

=

c`

cs,nmin

cs,min

,

where cmin = cs,min denotes the vector of all mineral concentrations, and cnmin

represents the vector of all nonmineral concentrations,

cnmin =

c`

cs,nmin

∈ RInmin , Inmin = I` + Is,nmin .

The entries c1, . . . , cI`+Is of c are given by

c` =: (c1, . . . , cI`)T ∈ RI` ,

cs,nmin =: (cI`+1, . . . , cI`+Is,nmin)T ∈ RIs,nmin ,

cs,min =: (cI`+Is,nmin+1, . . . , cI`+Is) ∈ RJmin .

Note that since the gas phase consists of one constituent only by assumption,

the molar concentration of this component equals its molar density ρmol,g and is

therefore not considered as an independent variable. The concentration of the

dissolved gas as a component of the liquid phase is assumed to be the first entry

in the concentration vector c`.

2.2 Constitutive relations

The governing equations (2.1)–(2.2) are complemented with constitutive rela-

tions representing the functional dependence of rock and fluid properties (e. g.,

velocities, densities, viscosities, capillary pressure, relative permeability) from

thermophysical and chemical variables.

2.2. CONSTITUTIVE RELATIONS 31

2.2.1 Darcy’s law

The movement of fluids in the subsurface is typically slow and can be described

by Darcy’s Law, which represents conservation of momentum and is obtained

from microscopic momentum balance by upscaling. In its generalized form for

multiphase flow, it states that the phase volumetric flow rates are given by

qα = −K krαµα

(∇pα − ρmass,αg) , (2.3)

where:

K intrinsic permeability tensor of the porous medium [m2] ,

krα relative permeability of phase α [–] ,

µα viscosity of phase α [Pa·s] ,

pα phase pressure of phase α [Pa] ,

ρmass,α mass density of phase α [ kgm3 ] ,

g vector of gravitational acceleration [ ms2

] .

Note that relative permeability is often represented as a nonlinear function of the

gas phase saturation, which introduces a nonlinearity in (2.3). Moreover, there

may be a nonlinear relationship between µg and pg, cf. Figure 7.10.

2.2.2 Diffusive and dispersive fluxes

While the advective flux describes the transport of components of a fluid phase

α ∈ l, g due to the movement of the fluid phase, the diffusive/dispersive flux

models the movement of the components within a phase in response to concen-

tration gradients. It is composed of molecular diffusion and mechanical disper-

sion, which is caused by microscopic variations in velocity as a consequence of

fluid viscosity, variations in the pore size and the path lengths of flow channels

[Fet93]. Following Fick’s law, the diffusive/dispersive flux of component i in

phase α ∈ `, g reads

jiα = −Diα ρmol,α∇χiα , (2.4)


where χiα = ciαρmol,α

is the mole fraction of component i in phase α, and Diα

is a tensor of second order representing hydrodynamic dispersion and molecular

diffusion. Following Bear [Bea72] and using the approach of Millington and Quirk

[MQ61] for molecular diffusion, the diffusion–dispersion tensor can be expressed

as

Diα = αT |qα|I + (αL − αT )

qα ⊗ qα|qα|

+ φ43 s

103α D

idiff,αI , (2.5)

where αT and αL are the transversal and longitudinal dispersion coefficient, re-

spectively, and Didiff,α denotes the molecular diffusion coefficient associated with

the i-th component of phase α. For later use, let us assume that the molecular dif-

fusion coefficients of all components of the liquid phase coincide. This assumption

is necessary to employ the reduction scheme, and it is justified since mechanical

dispersion typically dominates molecular diffusion. In this case, the diffusion–

dispersion tensor is species-independent, and we shall denote it by D`. Note that

the Darcy velocity qα and the saturation sα are unknowns in the system, which

induces a strong nonlinearity in the diffusion tensor.

2.2.3 Densities

At a constant temperature T , the density of the gas phase may be represented

as a function of the gas pressure,

ρmol,g = fg(pg) , (2.6)

where fg stands for a generic compressibility law. A simple choice for such a

compressibility law fg is the ideal gas law,

fg(pg) = C(T )pg ,

which will be employed in Section 7.1 to simulate hydrogen migration in geolog-

ical radioactive waste repositories. When CO2 sequestration in deep geological

formations is considered, however, different approaches must be used since the

injection takes place at very high pressures at a supercritical state of CO2. In our

numerical examples, the EOS of Duan [DMW92] is used to calculate the density

of the CO2 phase, cf. Section 7.3.

2.2. CONSTITUTIVE RELATIONS 33

The density of the liquid phase may depend on the chemical composition of the

phase. For example, the density of aqueous solutions of CO2 can be as much as

3% higher than the density of pure water, which influences the groundwater flow

regime. In geological sequestration of CO2, the injected CO2 will be concentrated

below an overlying caprock and, after some time, dissolve into the aqueous phase,

causing the heavier CO2 rich water to migrate downward and be displaced by

water with a lower CO2 content, cf. [Gar01]. The molar density of the liquid phase

is modeled as a function of the concentrations of its components, represented by

the generic compressibility law

ρmol,` = f`(c`) . (2.7)

The particular choices of f` and fg are specified for each numerical example in

Chapter 7.

2.2.4 Capillary pressure law

When two fluid phases coexist in a porous medium, they are separated by a sharp

interface causing a discontinuity in fluid properties, e. g., pressure. The pressure

difference is called capillary pressure,

pc := pg − p` . (2.8)

Its magnitude depends on the interfacial tension between the phases and the

curvature of the interface. On the macroscopic level, the van Genuchten–Mualem

model [vG80] is widely used to model capillary pressure. It provides a functional

relationship between capillary pressure and saturation,

pc =1

αVG

(S−1/m`,e − 1)1/n , m = 1− 1

n, (2.9)

where

S`,e =s` − s`,res

1− s`,res − sg,res

denotes the effective liquid saturation and sα,res the residual saturation of phase

α ∈ g, `. The water retention curve is characteristic for different types of


sg

pc(sg)

100

1

0 1

sg

krg(sg)krℓ(sg)

Figure 2.1: Typical shape of capillary pressure and relative permeability curves

soil, and the parameters are obtained by laboratory experiments and parameter

fitting. Alternatively, the model of Brooks and Corey [BC64] can be used, which

reads

pc = pentry · S−1λ

`,e ,

where pentry denotes the entry pressure and λ represents the pore size distribution

index.

2.2.5 Relative permeabilities

If two fluid phases coexist in a porous medium, they interfere each other in

their ability to pass through the pores. This effect is reflected and quantified by

introducing relative permeabilities, which, in the presence of a gas and a liquid

phase, are typically modeled as a function of one of the saturations. The van

Genuchten-Mualem relations for the relative permeabilities of a two-phase gas-

liquid system read

kr` =√S`,e(1− (1− S1/m

`,e )m)2 , (2.10)

krg =√

1− S`,e(1− S1/m`,e )2m , (2.11)

2.3. CHEMICAL REACTIONS 35

with S`,e, m and n defined as above. The Brooks–Corey relations for relative

permeabilities are given by

kr` = S2+3λλ

`,e , (2.12)

krg = (1− S`,e)2(1− S2+λλ

`,e ) . (2.13)

Typical shapes of capillary pressure and relative permeability curves are shown

in Figure 2.1. Note that in the following, we consider pc, kr` and krg as functions

of the gas phase saturation sg.

2.3 Chemical reactions

The chemical reaction system is specified by the stoichiometric matrix

S =

Sg

S`

Ss

∈ R(Ig+I`+Is,J) ,

where J = Jeq +Jkin denotes the total number of chemical reactions consisting of

Jeq equilibrium reactions and Jkin kinetic reactions. Each column of S represents

one reaction, whereas each row stands for a chemical component. The stoichio-

metric coefficient sij describes how component i participates in the reaction j.

Consider, for example, the generic reaction

se1A1 + . . .+ senA

n −→ sp1A1 + . . .+ spnA

n

between the chemical species A1, . . . , An, where sei , spi ∈ N0 for all i ∈ 1, . . . , n.

Note that if sei > 0, Ai is called an educt of the reaction. On the other hand, if

spi > 0, it is a product. Note that it is admitted that sei > 0 and spi > 0, e. g., if

Ai is a catalyst. The column of S associated to this reaction reads

(sp1 − se1, . . . , spn − sen)T .

Besides the stoichiometric matrix, the vector of reaction rates R = (R1, . . . , RJ)T

is used to describe the chemical reactions. It indicates how fast each reaction


proceeds. In our model, at least one component of the liquid phase is involved

in each reaction. Therefore, the reaction rates can be related to the volume of

the liquid phase, i. e., they indicate how many moles per time and volume of

the liquid phase are reacting. Consequently, the factor φs` is included in the

source/sink terms of the conservation equations, which are given by

f iα = φs`

J∑j=1

Sα,ijRj , i = 1, . . . , Iα , α ∈ g, `, s ,

or, in a more compact form,

f =

f g

f `

f s

= φs`

SgR

S`R

SsR

= φs`SR .

In this work, we make the assumption that all chemical reactions are reversible

and may proceed in both directions. Such a pair of reactions is denoted by

se1A1 + . . .+ senA

n ←→ sp1A1 + . . .+ spnA

n ,

and the corresponding rate function will consist of a forward reaction rate and a

backward reaction rate.

2.3.1 Kinetic reactions according to the law of mass action

For a kinetic reaction, the reaction rate is given as a (typically nonlinear) function

of the concentrations of the reacting species. A well-known example of such a

rate function is the so-called kinetic mass action law. Assuming that only homo-

geneous reactions between components of the liquid phase (subscript ’mob’) and

heterogeneous reactions between components of the liquid phase and nonmineral

components of the solid phase (subscript ’sorp’) participate in kinetic reactions

according to the law of mass action, the rate function reads

Rkin,j(c) = kf,j

Inmin∏i=1sij<0

ai−sij − kb,j

Inmin∏i=1sij>0

ai+sij , (2.14)


where kf,j, kb,j > 0 denote the forward and backward rate constant and ai = ai(c)

is the activity of the i-th chemical species, cf. [Bet96]. Usually, the activity is

related to the concentrations by

ai(c) = γi(c)ci ,

where γi denotes the activity coefficient of the i-th species, which accounts for

deviations from the state of an ideal solution. Assuming that the concentrations

are not too large, the approximation γi ≈ 1 is justified, and the activities can

be replaced by the concentrations in (2.14). This assumption is not justified,

however, for the component water, which is the main constituent of the liquid

phase, and for minerals. In both cases a constant activity is assumed, and we

may set ai = 1 by incorporating any activity constant into the rate coefficients.

2.3.2 Equilibrium reactions according to the law of mass

action

The timescales at which chemical reactions take place typically vary over sev-

eral orders of magnitude. Therefore, besides kinetically controlled reactions, we

consider equilibrium reactions, which are running so fast that a state of local

equilibrium can be assumed. Each equilibrium reaction is governed by an alge-

braic equation that holds at every point of the domain and typically depends

nonlinearly on the species concentrations. Requiring the additional algebraic

equation, the corresponding reaction rate becomes an additional unknown of the

system. However, by employing the reduction scheme, we can ensure that each

equilibrium rate appears only in one of the transformed equations, which is then

dropped. If the j-th equilibrium reaction is described by the law of mass action,

the associated equilibrium condition reads

kf,j

I`+Is∏i=1sij<0

ai(c)−sij − kb,j

I`+Is∏i=1sij>0

ai(c)+sij = 0 . (2.15)

Equilibrium mineral reactions and the interphase mass exchange between the

gas phase and the liquid phase require a different treatment, cf. Sections 2.3.3

and 2.3.4. Assuming ideal activities for all involved species and strictly positive


concentrations, (2.15) can be rewritten as

φj(c) := − ln(Kj) +

I`+Is∑i=1

sij ln(ci) = 0 , (2.16)

where Kj > 0 denotes the equilibrium constant of the reaction and depends

only on the quotient of the backward and the forward rate constant. Obviously,

the equilibrium mass action law represents a linear relation if the logarithms

of the concentrations are considered. This is usually done if the equilibrium

constants are very large and if the concentration values vary over several orders

of magnitude.

2.3.3 Equilibrium mineral reactions

In addition to kinetic reactions and equilibrium reactions between nonminerals,

we would like to consider equilibrium reactions of the form

s1jA1` + . . .+ sI`jA

I`` ←→M j

min

with A1` , . . . , A

I`` being components of the liquid phase and M j

min being a mineral.

It is assumed that each mineral participates in one and only one mineral reaction,

and that in every mineral reaction, one and only one mineral is involved, cf.

(2.22)–(2.24).

For minerals (pure solids), the assumption of ideal activity is not justified. In-

stead, they are usually assumed to have constant activity, which implies that the

algebraic equation resulting from the mass action law does not depend on the

associated mineral concentration. If the j-th equilibrium reaction is a mineral

reaction and the assumption of ideal activity is justified for all other components

involved in this reaction, we define

ϕj(c`) := − ln(Kj) +

I∑i=1

sij ln(ci`) , (2.17)

where the vector c` = (c1` , . . . , c

I`` ) represents the concentrations associated with

the components A1` , . . . , A

I`` . It should be noted, however, that the equilibrium


equation

ϕj(c`) = 0 (2.18)

is only valid as long as the corresponding mineral is present, which must be

considered in the design of numerical methods. A common approach in the geo-

sciences community is to determine the correct mineral assemblage by some kind

of trial and error strategy [Bet96, CMB02]. Thereby, in each time step an initial

guess is made as to which the mineral is present at a grid point or not, and

the solution is computed under this assumption. If this results in a nonphysical

solution (e. g., a negative mineral concentration), the guess is modified and the

time step is repeated until a physical solution is obtained. Obviously, this solu-

tion strategy significantly increases the computational cost compared to problems

without mineral reactions, which may be a severe limitation, especially if large

reaction systems are considered and a global implicit approach is applied, where

large coupled systems of nonlinear equations have to be solved again and again.

Another strategy of handling equilibrium mineral reactions is to formulate it as

a moving boundary problem [LSO96]. A more appropriate and mathematically

sound approach to deal with mineral reactions was suggested in [Kra08]. It formu-

lates the mineral equilibrium conditions as a complementarity problem unifying

both cases of presence and full dissolution of a mineral. Problems including com-

plementarity conditions are well-known in the field of optimization theory, and

the locally quadratic convergence of a Newton-like solution strategy, the so-called

Semismooth Newton method, can be shown. The Semismooth Newton method

for the particular application of mineral dissolution and precipitation was further

studied in [Kra11], where the complementarity conditions were incorporated into

a general reactive transport model and combined with a reformulation technique

to reduce the size of the fully coupled systems, which were then solved fully

implicitly. The method proved to be efficient and robust and was successfully ap-

plied to challenging benchmark problems [Hof10, CHK+10, HKK10]. Slightly less

general reaction systems were studied in [BKKK11], allowing to prove stronger

theoretical results.

We follow the complementarity approach and treat mineral reactions as comple-

mentarity problems. For that purpose, the equilibrium equation (2.18) is supple-

mented by additional inequalities. The equilibrium condition associated with the


j-th mineral reaction then reads(ϕj(c`) = 0 ∧ cjmin ≥ 0

)∨(ϕj(c`) > 0 ∧ cjmin = 0

),

where cjmin denotes the concentration of that mineral participating in the j-th

equilibrium reaction. We assume that the mineral concentrations are listed in

the vector cs,min in ascending order of the equilibrium reactions, i. e., the first

entry of cs,min corresponds to the mineral associated with the first equilibrium

mineral reaction etc. The above equilibrium condition states that if the mineral

is present, the equation ϕj(c`) = 0 holds. This is called the saturated state.

Conversely, if the mineral is not present, it is only known that ϕj(c`) > 0, which

is referred to as the undersaturated state. This equilibrium condition can be

equivalently written in the form

ϕj(c`)cjmin = 0 ∧ ϕj(c`) ≥ 0 ∧ cjmin ≥ 0 .

A condition of the type

E(c)c = 0 ∧ E(c) ≥ 0 ∧ c ≥ 0

is called a nonlinear complementarity condition. By choosing a function ϕ : R2 →R with the properties

ϕ(a, b) = 0 ⇐⇒ ab = 0 ∧ a ≥ 0 ∧ b ≥ 0 ,

it can be transformed into the equivalent equation

ϕ(E(c), c) = 0 ,

which is completely free of inequalities. Typical representatives of such a com-

plementarity function are the Fischer-Burmeister function ϕFB and the minimum

function ϕmin,

ϕFB(a, b) = a+ b−√a2 + b2 ,

ϕmin(a, b) = mina, b .

Following [Hof10] and [Kra08], we choose the minimum function for our numerical


computations, and the equilibrium condition is rewritten as

φj(c`) := minϕj(c`), cjmin = 0 . (2.19)

The mineral reaction is called active in a point if the minimum is attained in the

second argument of (2.19) in that point and inactive if the minimum is attained

in the first argument. The existence of solutions of problems involving nonsmooth

and nonlinear problems of the above type and the numerical approximation using

the Semismooth Newton method will be discussed in Chapter 4.

2.3.4 Interphase mass exchange

Modeling a partially miscible multiphase multicomponent system requires a de-

scription of the interphase mass transfer between the fluid phases, which can

massively effect the mass and volume of the gas phase and may even lead to

its local appearance or disappearance. As stated above, we assume that the gas

phase consists only of one component, and that the concentration of the liquid

phase associated with this component has the index 1, i. e., its concentration is

represented by the first entry in the concentration vector c`. As gas dissolves in

liquid at a very high dissolution rate, the liquid and gas phases are assumed at

thermodynamic equilibrium, governed by a generic equilibrium equation of the

form

φex(c`, pg) = 0 . (2.20)

An example of such a function φex is Henry’s law, which describes the equilibrium

partitioning of a dilute component in a gas–liquid system. It states that, at a

constant temperature T , the solubility of a gas in a liquid is proportional to its

partial pressure,

φex(c`, pg) = H(T )pg − c1` , (2.21)

where H(T ) is the Henry law constant. However, (2.20) holds only as long as the

gas phase is present because otherwise pg is undefined. In our work, we extend

the definition of pg by requiring (2.20) also in the absence of the gas phase.

To emphasize that its scope has been extended, the extended gas pressure will


henceforth be denoted by pg. It will be ensured that it holds pg = pg whenever

the gas phase exists.

Note that Henry’s law holds only at modest pressures and for low solute mole

fractions. If CO2 sequestration in a deep saline aquifer is considered (taking place

at very high pressures), different approaches must be employed, e. g., the EOS of

Duan and Sun [DS03] or the EOS of Spycher and Pruess [SP05], cf. Figure 7.9.

The particular choice of the solubility law φex will be specified for each numerical

example in Chapter 7.

2.4 Reactive multiphase multicomponent model

Let the chemical reactions be sorted in the following way: the first reaction rep-

resents the equilibrium mass transfer between the gas and the liquid phase and

is denoted with the subscript “ex”. The second type of reactions are equilibrium

reactions between components of the liquid phase (subscript “mob”), followed by

equilibrium sorption reactions (subscript “sorp”) and equilibrium mineral reac-

tions (subscript “min”). Finally, the last type of reactions are kinetic reactions

(subscript “kin”) between the components of the liquid phase and nonmineral

components of the solid phase. The number of reactions of each type is denoted

by Jex, Jmob, Jsorp, Jmin and Jkin. Concerning the numbering of the columns of

the stoichiometric matrix, we assume that the first Jeq := Jex +Jmob +Jsorp +Jmin

columns of S represent the equilibrium reactions and that the last Jkin columns

of S correspond to kinetic reactions. In this work, we will make the assumption

that Jex = 1. The general index Jex is used, however, to indicate an extension

to the case that the gas phase consists of multiple components which participate

in exchange reactions between the gas and the liquid phase. Taking into account

the above numbering, the stoichiometric matrix exhibits the block structure

S = (Seq | Skin ) =

Sg

S`

Ss

=

Sg,eq 0

S`,eq S`,kin

Ss,eq Ss,kin

,

2.4. REACTIVE MULTIPHASE MULTICOMPONENT MODEL 43

where

Sg = (Sg,eq 0) = (−1, 0, . . . , 0) , (2.22)

S` = (S`,eq S`,kin) =(S`,ex S`,mob S`,sorp S`,min S`,kin

), (2.23)

Ss = (Ss,eq Ss,kin) =

0 0 Ss,sorp 0 Ss,kin

0 0 0 IJmin0

. (2.24)

Here, S`,ex = (1, 0, . . . , 0)T ∈ R(I`,Jex), S`,mob ∈ R(I`,Jmob), S`,sorp ∈ R(I`,Jsorp),

S`,min ∈ R(I`,Jmin), S`,kin ∈ R(I`,Jkin), Ss,sorp ∈ R(Is,nmin,Jsorp), Ss,kin ∈ R(Is,nmin,Jkin),

IJmin∈ R(Jmin,Jmin). The structure of the stoichiometric matrix induces the fol-

lowing partitioning of the vector of reaction rates:

R =

Req

Rkin

=

Rex

Rmob

Rsorp

Rmin

Rkin

, (2.25)

where Req ∈ RJeq , Rkin ∈ RJkin , Rex ∈ RJex, Rmob ∈ RJmob , Rsorp ∈ RJsorp ,

Rmin ∈ RJmin .

Using the above notation, the source and sink terms have the representation

f i =

Jkin∑j=1

φs`Skin,ijRkin,j(c`, cs) +

Jeq∑j=1

φs`Seq,ijReq,j .

Here, f i denotes the i-th component of the vector f defined on p. 36. Let us

conclude this chapter with a summary and statement of the full model in concise

notation. It consists of PDEs representing the mass balance of the components in

the gas and liquid phase, ODEs for the mass conservation of the immobile com-


ponents and algebraic equations (AEs) associated with the equilibrium reactions,

∂t(φsgρmol,g) + Lgρmol,g = −φs`R1 (2.26)

∂t(φs`ci`) + L`ci` = φs`

J∑j=1

S`,ijRj , i = 1, . . . , I` , (2.27)

∂t(φs`cis) = φs`

J∑j=1

Ss,ijRj , i = 1, . . . , Is , (2.28)

φeq(c`, cs) = 0 . (2.29)

Here, R1, . . . , RJ denote the components of the vector R, and L` and Lg denote

the transport operators of the liquid and the gas phase, respectively,

Lgc = ∇ · (qgc) , L`c = ∇ · (−ρmol,`D`(s`, q`)∇(ρ−1mol,` c) + q`c) . (2.30)

The function φeq represents the chemical equilibrium conditions and is defined

by

φeq(c`, cs) :=

φmob(c`)

φsorp(c`, cs,nmin)

φmin(c`, cs,min)

∈ RJ∗eq ,

where J∗eq := Jmob + Jsorp + Jmin and φmob,φsorp and φmin denote the equilibrium

conditions associated with the equilibrium reactions governed by the mass action

law,

φmob(c`) = ST`,mob ln(c`)− ln(Kmob) , (2.31)

φsorp(c`, cs,nmin) = ST`,sorp ln(c`) + STs,sorp ln(cs,nmin)− ln(Ksorp) , (2.32)

φmin(c`, cs,min) = minϕmin(c`) , cs,min (2.33)

where ϕmin(c`) := ST`,min ln(c`) − ln(Kmin). Note that the minimum function

operates componentwise in (2.33). The representations (2.31)–(2.33) are based

on ideal activities for all components of the liquid phase. If we assume constant

activity for the component water and if water participates in a reaction, the

definitions (2.31)–(2.33) can be modified by replacing the matrices S`,mob, S`,sorp,

2.4. REACTIVE MULTIPHASE MULTICOMPONENT MODEL 45

Ss,sorp and S`,min with matrices S`,mob, S`,sorp, Ss,sorp and S`,min containing zero

entries in the rows associated with the component water.

In contrast to the single phase reactive transport model considered in [KK05,

KK07, Kra08, Hof10, HKK12], where the Darcy velocity field was given or a priori

computed by solving Richards’ equation, the transport operators in the above

multiphase multicomponent model introduce a bidirectional coupling between

flow and transport through the saturation and pressure variables. In matrix

form, the system reads as follows:

∂t(φsgρmol,g) + Lgρmol,g = φs`Sg,eqReq , (2.34)

∂t(φs`c`) + L`c` = φs`S`,kinRkin(c`, cs,nmin) + φs`S`,eqReq , (2.35)

∂t(φs`cs) = φs`Ss,kinRkin(c`, cs,nmin) + φs`Ss,eqReq , (2.36)

φeq(c`, cs) = 0 . (2.37)

It is closed by the solubility law (2.20), the compressibility laws (2.6)–(2.7), the

capillary pressure law (2.8)–(2.9), the volume balance

sg + s` = 1 (2.38)

and the constraints

ρmol,` =

I∑i=1

ci` , (2.39)

ρmass,` =

I∑i=1

M ici` , (2.40)

where M i [ kgmol

] denotes the molar mass of component i. Note that (2.39) ensures

that the mole fractions of the components of the liquid phase sum up to one,

I∑i=1

χi` =

I∑i=1

ci`ρmol,`

= 1 .

Clearly, the above system is fully coupled and exhibits strong nonlinearities in

the kinetic reaction rates, the transport operators, the chemical equilibrium laws,

and the nonlinear closure relationships.


The efficient and accurate numerical solution of this system is challenging in

several respects. On the one hand, this is due to the strong coupling of flow,

transport, and chemical reactions, including mass transfer across the phases and

the local (dis-)appearance of phases. On the other hand the problem may become

computationally expensive if large-scale and long-term simulations are carried out

and if large reaction systems including many reactants are considered.

In an attempt to reduce the computational complexity, vertical equilibrium mod-

els have been developed based on the assumption that CO2 and brine have seg-

regated due to buoyancy and reached a hydrostatic pressure distribution in the

vertical direction, see, e. g., [GNC09, GNC11]. Such models are computationally

efficient due to the dimensional reduction, but accuracy must be sacrificed for

this purpose.

In this work, the full model is used for simulations. In the next chapter, a model-

preserving transformation of the mathematical equations is carried out, which

allows an elimination of certain variables on the level of the nonlinear solver. This

approach is well-suited for parallel computations. Moreover, by using a global

implicit approach and employing Newton’s method for linearization, the use of

iterative splitting techniques and of first order fixed point methods is avoided,

which may severely limit efficiency due to slow convergence.

Chapter 3

The reduction scheme

In this chapter, we apply the reduction scheme of Krautle, Knabner and Hoff-

mann [KK05, KK07, HKK12] to our multiphase multicomponent system. The

general idea is to take specific linear combinations of equations (which is done

in each phase separately) in order to isolate the equilibrium reaction rates each

in a single transformed equation. Afterwards, provided that one is not inter-

ested in the equilibrium reaction rates, these equations can be dropped. Finally,

the equilibrium equations are used to eliminate certain variables by defining a

resolution function, which further reduces the size of the system. The advan-

tage of the method of Krautle and Knabner compared to other transformation

techniques is that in the transformed equations, all transport operators depend

linearly on transformed concentration variables after solving and substituting the

algebraic equations into the remaining equations. This avoids the generation of

additional nonzero entries in the Jacobian. For a comparison with the methods

of [Fri91, FR92] and [MCAS04], we refer to [KK05] and [KK07], respectively.

In the single phase reactive transport model considered in [KK05, KK07, HKK12],

there are two blocks of equations that decouple from the rest of the system after

employing the reduction scheme: one block of PDEs associated with mobile trans-

formed variables and one block of ODEs associated with immobile transformed

variables. Due to the coupling of flow and transport, this is no longer valid for

our multiphase multicomponent model, implying that all PDEs remain coupled

through the pressure and saturation variables arising in the transport operators.

We will show, however, that at least one block of ODEs can be eliminated from

the global system if the primary variables are chosen appropriately. Moreover, it

47

48 CHAPTER 3. THE REDUCTION SCHEME

should be noted that if a splitting approach is employed – decomposing the mul-

tiphase multicomponent problem into a two-phase flow problem and a reactive

transport problem – all decoupling blocks of the reduction scheme are retained

within the reactive transport problem. This makes our formulation attractive

for iterative splitting approaches, where the reactive transport problem has to be

solved many times.

3.1 Transformation of the system of equations

In the following, we assume that all columns of S are linearly independent. If this

is not the case, there exists a linearly independent set of reactions such that any

other reaction can be expressed as a linear combination of them. Moreover, for the

reduction scheme, we shall require that the columns of (S`,ex S`,mob S`,sorp S`,min)

and the columns of Ss,sorp are linearly independent. When the linear indepen-

dence condition is not satisfied because some of the columns of (S`,mob S`,sorp)

are linearly dependent, the use of the reduction scheme is still possible after

employing preprocessing steps (for example, adding two equilibrium reactions

corresponds to multiplying their equilibrium constants). In this case, a block

Ss,immo representing equilibrium reactions running solely in the solid phase must

possibly be added, cf. [KK07]. Taking into account the partitioning (2.22)–(2.24)

of the stoichiometric matrix, the system (2.34)–(2.37) can be rewritten in concise

notation as

∂t(φsgρmol,g) + Lgρmol,g = φs`Sg

Req

0

, (3.1)

∂t(φs`c`) + L`c` = φs`S`

Req

Rkin(c`, cs,nmin)

, (3.2)

∂t(φs`cs) = φs`Ss

Req

Rkin(c`, cs,nmin)

, (3.3)

φeq(c`, cs) = 0 . (3.4)

The following presentation of the reduction scheme closely follows the one in

3.1. TRANSFORMATION OF THE SYSTEM OF EQUATIONS 49

[Hof10]. In the first step, we define matrices S∗` and S∗g which consist of a

maximal system of linearly independent columns of S` and Ss, respectively. With

the linear independence assumption made above, they have the form

S∗` = (S`,ex S`,mob S`,sorp S`,min S∗`,kin) , (3.5)

S∗s =

Ss,sorp 0 S∗s,kin

0 IJmin0

, (3.6)

where S∗`,kin ∈ R(I`,J∗`,kin) and S∗s,kin ∈ R(Is,J∗s,kin) consist of a subset of the columns

of S`,kin and Ss,kin, respectively. Let A` and As be matrices such that

S` = S∗`A` , Ss = S∗sAs . (3.7)

Taking into account the structure of S` and Ss, we obtain

A` =

IJex 0 0 0 A`,ex

0 IJmob0 0 A`,mob

0 0 IJsorp 0 A`,sorp

0 0 0 IJminA`,min

0 0 0 0 A`,kin

,

As =

0 0 IJsorp 0 As,sorp

0 0 0 IJmin0

0 0 0 0 As,kin

,

with A`,ex ∈ R(Jex,Jkin), A`,mob ∈ R(Jmob,Jkin), A`,sorp ∈ R(Jsorp,Jkin), A`,min ∈R(Jmin,Jkin), A`,kin ∈ R(J∗`,kin,Jkin), As,sorp ∈ R(Jsorp,Jkin) and As,kin ∈ R(J∗s,kin,Jkin).

The next step is to construct matrices S⊥` and S⊥s such that their columns repre-

sent a maximal set of linearly independent vectors perpendicular to all columns

of S∗` and S∗s, respectively. In other words,

S⊥`TS∗` = 0 , S⊥s

TS∗s = 0 . (3.8)

Obviously, S⊥` has I` − Jex − Jmob − Jsorp − Jmin − J∗`,kin columns and S⊥s has


Is − Jsorp − Jmin − J∗s,kin columns. Moreover, we choose matrices B` and Bs

having the same size as S∗` and S∗g such that (B` S⊥` ) and (Bs S

⊥s ) are a basis

of the whole space and Bs has the form∗ 0 ∗

0 IJmin0

. (3.9)

Analogously to S⊥` and S⊥s , matrices B⊥` and B⊥s are constructed such that

B⊥`TB` = 0 , B⊥s

TBs = 0 .

In the reduction scheme proposed in [KK05, KK07], the matricesB` andBs were

chosen as B` = S∗` and Bs = S∗s, which satisfy the above conditions. The more

general form was introduced in [Hof10, HKK12]. Multiplying the block (3.2) with

the matrices

(S⊥`TB⊥` )−1S⊥`

Tand (BT

` S∗`)−1BT

`

and the block (3.3) with the matrices

(S⊥sTB⊥s )−1S⊥s

Tand (BT

s S∗s)−1BT

s

and using (3.7), we obtain the system

∂t(φsgρmol,g) + Lgρmol,g = φs`Sg(Req

0

),

(S⊥`TB⊥` )−1S⊥`

T(∂t(φs`c`) + L`c`) = φs`(S

⊥`

TB⊥` )−1S⊥`

TS∗`A`

(Req

Rkin(c`,cs,nmin)

),

(BT` S∗`)−1BT

` (∂t(φs`c`) + L`c`) = φs`(BT` S∗`)−1BT

` S∗`A`

(Req

Rkin(c`,cs,nmin)

),

(S⊥sTB⊥s )−1S⊥s

T∂t(φs`cs) = φs`(S

⊥s

TB⊥s )−1S⊥s

TS∗sAs

(Req

Rkin(c`,cs,nmin)

),

(BTs S∗s)−1BT

s ∂t(φs`cs) = φs`(BTs S∗s)−1BT

s S∗sAs

(Req

Rkin(c`,cs,nmin)

).

Using the orthogonality relations (3.8) and the fact that the transport operators


are independent of the chemical species, we infer

∂t(φsgρmol,g) + Lgρmol,g = φs`Sg(Req

0

),

∂t(φs`(S⊥`

TB⊥` )−1S⊥`

Tc`) + L`((S⊥`

TB⊥` )−1S⊥`

Tc`) = 0 ,

∂t(φs`(BT` S∗`)−1BT

` c`) + L`((BT` S∗`)−1BT

` c`) = φsÀ`

(Req

Rkin(c`,cs,nmin)

),

∂t(φs`(S⊥s

TB⊥s )−1S⊥s

Tcs) = 0 ,

∂t(φs`(BTs S∗s)−1BT

s cs) = φsÀs

(Req

Rkin(c`,cs,nmin)

).

This motivates the definition of the transformed variables

η` := (S⊥`TB⊥` )−1S⊥`

Tc` , ξ` := (BT

` S∗`)−1BT

` c` ,

ηs := (S⊥sTB⊥s )−1S⊥s

Tcs , ξs := (BT

s S∗s)−1BT

s cs ,

where η` ∈ RI`−Jex−Jmob−Jsorp−Jmin−J∗`,kin , ξ` ∈ RJex+Jmob+Jsorp+Jmin+J∗`,kin , ηs ∈RIs−Jsorp−Jmin−J∗s,kin and ξs ∈ RJsorp+Jmin+J∗s,kin . The transformed variables (η`, ξ`)

and (ηs, ξs) represent the concentrations c` and cs in another basis of RI` and

RIs , respectively. It is easy to verify that the retransformation reads

c` = S∗`ξ` +B⊥` η` ,

cs = S∗sξs +B⊥s ηs .

The variables ξ` and ξs are partitioned in the same way as the columns of S∗ànd S∗s in (3.5) and (3.6), respectively,

ξ` =

ξ`,ex

ξ`,mob

ξ`,sorp

ξ`,min

ξ`,kin

, ξs =

ξs,sorp

ξs,min

ξs,kin

,

where ξ`,ex ∈ RJex , ξ`,mob ∈ RJmob , ξ`,sorp ∈ RJsorp , ξ`,min ∈ RJmin , ξ`,kin ∈ RJ∗`,kin ,

ξs,sorp ∈ RJsorp , ξs,min ∈ RJmin , ξs,kin ∈ RJ∗s,kin . It follows from the block structure


of Bs and S∗s (cf. (3.6) and (3.9)) that the last Jmin rows of B⊥s and S⊥s are zero,

B⊥s =

B⊥s0

, S⊥s =

S⊥s0

.

Consequently, by the definition of ηs, it depends only on cnmin but not on cmin.

The same holds for ξs,sorp, and ξs,kin. With the above partitioning, the retrans-

formation reads

c` = S`,exξ`,ex + S`,mobξ`,mob + S`,sorpξ`,sorp + S`,minξ`,min

+ S∗`,kinξ`,kin +B⊥` η` ,

cs =

cs,nmin

cs,min

=

Ss,sorpξs,sorp + S∗s,kinξs,kin + B⊥s ηs

ξs,min

.

(3.10)

In the next step of the reduction mechanism, the equilibrium reaction rates are

eliminated. For this purpose, the block structure of the matrices A` and As and

the partitioning of the variables ξ` and ξs are used to expand the differential

equations, which yields

∂t(φsgρmol,g) + Lgρmol,g = −φs`Rex ,

∂t(φs`η`) + L`η` = 0 ,

∂t

φs`

ξ`,ex

ξ`,mob

ξ`,sorp

ξ`,min

ξ`,kin

+ L`

ξ`,ex

ξ`,mob

ξ`,sorp

ξ`,min

ξ`,kin

= φs`

IJex 0 0 0 A`,ex

0 IJmob0 0 A`,mob

0 0 IJsorp 0 A`,sorp

0 0 0 IJminA`,min

0 0 0 0 A`,kin

Rex

Rmob

Rsorp

Rmin

Rkin

,

∂t(φs`ηs) = 0 ,

∂t

φs`ξs,sorp

ξs,min

ξs,kin

= φs`

0 0 IJsorp 0 As,sorp

0 0 0 IJmin0

0 0 0 0 As,kin

Rex

Rmob

Rsorp

Rmin

Rkin

.


This leads to the system

∂t(φsgρmol,g) + Lgρmol,g = −φs`Rex , (3.11)

∂t(φs`η`) + L`η` = 0 , (3.12)

∂t(φs`ξ`,ex) + L`ξ`,ex = φs`Rex + φsÀ`,exRkin(c`, cs,nmin) , (3.13)

∂t(φs`ξ`,mob) + L`ξ`,mob = φs`Rmob + φsÀ`,mobRkin(c`, cs,nmin) , (3.14)

∂t(φs`ξ`,sorp) + L`ξ`,sorp = φs`Rsorp + φsÀ`,sorpRkin(c`, cs,nmin) , (3.15)

∂t(φs`ξ`,min) + L`ξ`,min = φs`Rmin + φsÀ`,minRkin(c`, cs,nmin) , (3.16)

∂t(φs`ξ`,kin) + L`ξ`,kin = φsÀ`,kinRkin(c`, cs,nmin) , (3.17)

∂t(φs`ηs) = 0 , (3.18)

∂t(φs`ξs,sorp) = φs`Rsorp + φsÀs,sorpRkin(c`, cs,nmin) , (3.19)

∂t(φs`ξs,min) = φs`Rmin , (3.20)

∂t(φs`ξs,kin) = φsÀs,kinRkin(c`, cs,nmin) . (3.21)

Finally, the equilibrium reaction rates are eliminated. First, the block (3.14) can

be dropped since the reaction rates associated with homogeneous equilibrium

reactions in the liquid phase occur only in these equations. Moreover, by adding

(3.11) and (3.13), subtracting (3.20) from (3.16) and (3.19) from (3.15), the

reaction rates Rex, Rsorp and Rmin can be isolated in the equations (3.11), (3.20)

and (3.19), respectively, which are then omitted. The resulting system reads

∂t(φs`η`) + L`η` = 0 , (3.22)

∂t(φs`ηs) = 0 , (3.23)

∂t(φs`ξ`,ex + φsgρmol,g) + L`ξ`,ex + Lgρmol,g = φsÀ`,exRkin(cnmin) , (3.24)

∂t(φs`ξ`,sorp) + L`ξ`,sorp − ∂t(φs`ξs,sorp) = φsÀsorpRkin(cnmin) , (3.25)

∂t(φs`ξ`,min) + L`ξ`,min − ∂t(φs`ξs,min) = φsÀ`,minRkin(cnmin) , (3.26)

∂t(φs`ξ`,kin) + L`ξ`,kin = φsÀ`,kinRkin(cnmin) , (3.27)

∂t(φs`ξs,kin) = φsÀs,kinRkin(cnmin) , (3.28)

whereAsorp := A`,sorp−As,sorp. It is closed by the equilibrium laws (2.37), the sol-

ubility law (2.20), the constraints (2.38)–(2.40) and the constitutive relationships

introduced in Section 2.2.

When the reduction scheme is applied to flow problems where the Darcy velocity


and the volumetric water content θ = φs` are given or calculated by solving

Richards’ equation, the block (3.22) decouples from the rest of the system. As a

consequence, these equations can be solved independently of the others in each

time step. In our multiphase model, however, all partial differential equations are

coupled through the nonlinear transport operator L` which depends on p` and s`.

Nevertheless, the generation of zero entries on the right hand side of some of the

equations reduces the number of nonzero entries in the global Jacobian. It should

be noted that the resulting formulation is also of interest for solving the global

system using a splitting approach, where the problem is decomposed into a flow

problem and a reactive transport problem. Then, in the reactive transport step,

the pressures and saturations are known, and the blocks (3.22)–(3.23) decouple.

If the splitting is done iteratively and the reactive transport problem has to be

solved many times, this can have a significant impact on the CPU time.

3.2 Choice of primary variables

The next step of the reduction mechanism is the elimination of variables in terms

of a nonlinear resolution function, which resembles the so-called direct substitu-

tion approach. For this purpose, the variables are split into primary and sec-

ondary variables such that the secondary variables can be expressed in terms of

the primary variables with the help of the algebraic equations (including chemical

equilibrium laws, constraints and discretized ODEs). The choice of appropriate

primary variables is essential for the performance of the numerical solver that

is used for numerical simulations, and several issues should be addressed when

making this choice.

In order to keep the size of the global system small, it is desirable to eliminate as

many variables as possible. This helps to reduce the CPU time and is beneficial

for parallelization since the local elimination can be efficiently done in paral-

lel. On the other hand, one should account for the particular structure of the

remaining global problem and the nonlinearities that are involved. In our nu-

merical experiments, the Newton solver required significantly more steps to solve

the nonlinear systems when it contained nonlinearities in the pressure gradients.

This can be avoided by including two phase pressures in the set of primary vari-

ables, which has the additional advantage that most coefficient functions depend

3.2. CHOICE OF PRIMARY VARIABLES 55

only on one primary variable in this case.

Another issue that should be addressed when selecting the primary variables is

the local appearance and disappearance of the gas phase. This causes problems

since in those regions where the pores are filled with liquid only, the equations

(3.22)–(3.28) degenerate to the system

∂t(φη`) + L`η` = 0 , (3.29)

∂t(φηs) = 0 , (3.30)

∂t(φξ`,ex) + L`ξ`,ex = φA`,exRkin(cnmin) , (3.31)

∂t(φξ`,sorp) + L`ξ`,sorp − ∂t(φξs,sorp) = φAsorpRkin(cnmin) , (3.32)

∂t(φξ`,min) + L`ξ`,min − ∂t(φξs,min) = φA`,minRkin(cnmin) , (3.33)

∂t(φξ`,kin) + L`ξ`,kin = φA`,kinRkin(cnmin) , (3.34)

∂t(φξs,kin) = φAs,kinRkin(cnmin) , (3.35)

which does no longer incorporate variables associated with the gas phase. There-

fore, it is not possible to use a standard set of primary variables for compositional

two-phase flow that would include the saturation of one fluid phase and the pres-

sure of the other fluid phase. One possibility to deal with this problem is the

switch of primary variables when a phase appears or disappears. In this work, we

want to avoid such a discontinuous switch. Instead, we chose primary variables

that are suitable to describe both the saturated and the undersaturated phase

state. Such variables are called persistent since they remain valid if one phase is

absent. An overview over different approaches for handling vanishing phases is

given in the next subsection.


3.2.1 Existing approaches for phase (dis-)appearance

Modeling and simulation of compositional multiphase flow including the appear-

ance and disappearance of fluid phases has been an active field of research in

recent years, and various approaches were developed to handle the difficulties

that are associated with it. These methods differ mainly in the choice of primary

variables that are used for the computations and can be divided into two large

classes: (1) use of natural variables and a switching procedure if the phase state

changes, and (2) use of persistent variables. In the following, we give a brief

overview over the existing approaches.

Primary variable switching

A widely used strategy to deal with the appearance or disappearance of a fluid

phase is the switch (or substitution) of primary variables, see, e. g., [FS91, CHB02].

The idea is to update the set of primary variables in the case that a variable has

exceeded its physical bounds. In such an event, at least one primary variable

has to be replaced by another one to ensure that all variables remain physically

meaningful, and the solution procedure can be continued.

For example, if a liquid and a gas phase coexist, one saturation and one pressure

are often used as primary variables, whereas the saturation is replaced by a mass

or mole fraction as soon as the gas phase disappears. This kind of change of the

phase state is detected if the saturation becomes negative.

Clearly, if one phase exists only locally, this substitution process can lead to dif-

ferent sets of primary variables within the domain, which may be altered during

the simulation. Consequently, the residual function is typically discontinuous,

which may cause problems for nonlinear solvers requiring smooth (or at least

semismooth) functions. Moreover, the implementation of variable switching is

more complex than that of methods using persistent primary variables that re-

main valid at all times and in the whole domain.

Persistent primary variables

In most cases, persistent variables are defined by artificially extending the scope

of physical variables to nonphysical phase states, or by using fully nonphysical or


nonstandard variables.

Panfilov et. al. [AP09, PR10, PP14] introduce an extended saturation S which

may take negative values and values greater than one. More precisely, the cases

0 ≤ S ≤ 1 in the two-phase region,

S > 1 in the oversaturated gas,

S < 0 in the undersaturated liquid

are distinguished. They use S and an extended gas pressure pg as primary

variables and require consistence conditions to ensure the physical consistence

between the two-phase and the single-phase model. In the two-phase region, all

extended variables are equal to their physical correspondents.

Marchand et al. [MMK12, MMK13, MK14] consider a two-component model

describing the evolution of a heavy component (water) and a light component

(e. g., hydrogen, CO2) in a two-phase liquid-gas system. They use the total

molar fraction

X =ρmol,gχ

1gsg + ρmol,`χ

1`s`

ρmol,gsg + ρmol,`s`

of the light component and a mean pressure

P = γ(sg)pg + (1− γ(sg))p`

as primary variables, where γ is an appropriate weight function. This set of

variables remains meaningful regardless of the phase state, and the model is

therefore capable of handling the disappearance of either phase. The system of

equations that are used to reconstruct all variables from X and P represents a

nonlinear and nonsmooth complementarity problem. It is solved with the help

of the Semismooth Newton method.

Bourgeat et al. [BJS09, BJS13, BGS13] use liquid pressure p` and the total

mass concentration

ρhtot = s`ρh` + sgρ

hg


of hydrogen as main unknowns to simulate hydrogen migration in an under-

ground nuclear waste repository with the help of a compositional two-phase

two-component model. Here, ρh` and ρhg denote the hydrogen mass concen-

trations in the liquid and the gas phase, respectively. If capillary forces are

included, it is proposed to replace ρhtot by the hydrogen mass concentration ρh` .

Using Henry’s law, the saturations are computed explicitly using the inverse of

the capillary pressure function. As long as the liquid phase is present, these

variables are always physically meaningful. One advantage of the use of p`

and ρh` as primary variables is that they are both continuous across material

heterogeneities.

Neumann et al. [NBI13, dC15] extend the scope of the gas phase pressure and

the capillary pressure and use them as primary unknowns in a two-phase two-

component model. In the absence of the gas phase, the extended gas pressure

pg is defined as the pressure corresponding to the amount of dissolved gas in

the liquid phase according to the solubility function,

χ1` = ψ(pg) , (3.36)

where ψ needs to be a strictly increasing function of gas pressure. If Henry’s law

is used, ψ represents a linear function of pg. Similarly, the extended capillary

pressure pc is defined by

pc = pg − p` . (3.37)

These variables can be interpreted as an algebraic transformation of liquid pres-

sure and the molar fraction of the dissolved gas in the liquid phase by means

of (3.36) and (3.37). An advantage of this approach is that the phase state de-

pends solely on one primary variable since sg can be explicitly computed from

pc using the capillary pressure function. In our formulation, we adopt this idea

by including the extended gas and capillary pressures in the set of primary vari-

ables. This allows us to solve the local equations in several sequential steps, cf.

Section 6.4.3.

Angelini et al. [ACC+11] use the phase pressures pg and p` as unknowns, where

pg represents an extension of the physical gas pressure according to Henry’s law,


i. e., it always satisfies

pg =ρh`

H ·Mh,

where ρh` is the hydrogen mass concentration in the liquid phase, H is Henry’s

constant and Mh is the molar mass of hydrogen. The saturation is obtained

with the help of the inverse of the capillary pressure function by means of

sg(pg, p`) = p−1c (pg − p`) .

Ben Gharbia and Jaffre [BG12, BJ14] propose the use of p` and s` along with

the molar fraction χh` of hydrogen as global unknowns for a two-phase two-

component model. They introduce complementarity constraints to ensure that

s` remains in the physical range between 0 and 1, cf. [JS10]. In the case that

the gas phase exists, it holds that

1− s` > 0 ∧ HMh(p` + pc(s`))− ρh` = 0 ,

whereas in the absence of the gas phase

1− s` = 0 ∧ HMh(p` + pc(s`))− ρh` ≥ 0 .

These constraints are rewritten as a nonlinear complementarity problem by

means of

(1− s`)(HMh(p` + pc(s`))− ρh` ) = 0 , (3.38)

1− s` ≥ 0 , HMh(p` + pc(s`))− ρh` ≥ 0 . (3.39)

In contrast to [MMK13], the equation resulting from (3.38)–(3.39) after employ-

ing a complementarity function is not used as a static equation and resolved

locally, but it is kept as a global equation along with the mass conservation

equations. Consequently, three global unknowns – s`, p` and χh` – are used for

the computations, and the Semismooth Newton method is employed to solve

the global nonlinear problem in each time step.


Lauser et al. [LHHW11] consider a non-isothermal compositional model con-

sisting of M phases and N components. Phase transitions are handled with the

help of M complementarity conditions of the form

1−N∑i=1

χiα ≥ 0 , sα ≥ 0 , sα(1−N∑i=1

χiα) = 0 , α ∈ 1, . . . ,M .

It is shown that these constraints are equivalent to the nonsmooth equations

sα −max

0, Cα

(1−

N∑i=1

χiα

)= 0 , α ∈ 1, . . . ,M , (3.40)

where Cα > 0 denote arbitrary positive constants. These equations are solved

along with one energy conservation equation and N component mass balance

equations. Accordingly, M + N + 1 primary variables are required, and the

authors use p1, χ11, . . . , χ

N1 , s2, . . . , sM and T with T denoting temperature. In

each Newton step, the equations (3.40) are used to eliminate M equations from

the linear system that remains to be solved after linearization. This corresponds

to an elimination on the linear level. In our work, we prefer elimination on the

nonlinear level in terms of a nonlinear resolution function, cf. Chapter 4. In their

numerical results, the authors report that a substantial amount of computation

time can be saved if the complementarity approach is used instead of primary

variable switching.

Most of the methods described above have in common that the secondary vari-

ables must be calculated from the primary variables in each step of the global

nonlinear solver. In the case that saturation is treated as a secondary variable,

the appearance or disappearance of a phase is thus shifted to the local level.

Depending on the retention curve, the static equations may be nonsmooth. For

particular choices of primary variables [NBI13], the secondary variables can even

be expressed explicitly in terms of the primary variables, but the corresponding

functional relationship may be nonsmooth as well. If compositional quantities

are used as primary variables, the process of computing the secondary in terms

of the primary variables is often referred to as flash calculations in the chemical

and engineering literature, cf., e. g., [WM89, SS01].


3.2.2 Persistent primary variables for our model

The numerical solution of (3.22)–(3.28) may lead to ill-conditioned linear systems

when the reaction constants of the equilibrium reactions are large. Hence, the

simulation of realistic problems may not be possible. For this reason, we follow

[Hof10, HKK12] and introduce the additional variables

ξ :=

ξsorp

ξmin

:=

ξ`,sorp − ξs,sorp

ξ`,min − ξs,min

.

It was shown in [Hof10, Section 3.5.1] that with their choice of local and global

variables, the matrix representing the derivatives of the local with respect to the

global transformed variables has entries that are bounded independently of the

concentration values and the reaction constants, which would not be the case

without the additional variables. Consequently, for a model problem with one

equilibrium sorption reaction (and assuming ∆t = 0), they showed that the con-

dition number of the Jacobian is bounded independently of the concentrations

and the reaction constants, whereas the condition number of the Jacobian asso-

ciated with the formulation without ξ is proportional to the (potentially large)

reaction constants.

Besides an improvement of the condition number, the additional variables ξmin

simplify the implementation of the method in a computer code. This is due to the

fact that in the absence of the additional variables, the partitioning into global

and local variables depends on the mineral assemblage, cf. Section 3.4.1. With

the additional variables, however, this is not the case. In fact, we will prove

in Chapter 4 that they provide a unique and persistent partitioning into global

and local unknowns, regardless of the phase state and the mineral assemblage.

This fact and the better conditioning of the system justifies the introduction of

ξ, although the global system is enlarged by incorporating the defining equations

of ξsorp and ξmin as additional equations, cf. (3.51)–(3.52).

With the additional variables, some freedom arises in the choice of the retrans-

formation. We define it by plugging the relations ξ`,sorp = ξsorp + ξs,sorp and


ξ`,min = ξmin + ξs,min into (3.10). Then, the new retransformation reads

c` = S`,exξ`,ex + S`,mobξ`,mob + S`,sorp(ξsorp + ξs,sorp)

+ S`,min(ξmin + ξs,min) + S∗`,kinξ`,kin +B⊥` η` ,

cs =

cs,nmin

cs,min

=

Ss,sorpξs,sorp + S∗s,kinξs,kin + B⊥s ηs

ξs,min

.

(3.41)

Finally, an extended capillary pressure pc, defined by the equation

pc − pg + p` = 0 , (3.42)

is introduced as a global variable, cf. [NBI13]. In the case that pc has not reached

the entry pressure threshold, the gas phase is absent and it holds that

pc ≤ pc(0) ∧ sg = 0 .

The liquid is then called undersaturated. On the other hand, if the entry pressure

has been exceeded, the gas phase exists and the capillary pressure–saturation

relationship yields

sg > 0 ∧ pc = pc(sg) ,

which is referred to as the saturated state. Altogether, both states can be de-

scribed by the conditions

((sg > 0) ∧ (pc(sg)− pc = 0)) ∨ ((sg = 0) ∧ (pc(sg)− pc ≥ 0)) ,

which is equivalent to the complementarity problem

sg(pc(sg)− pc) = 0 ∧ sg ≥ 0 ∧ pc(sg)− pc ≥ 0 .

Using the minimum function as a complementarity function, this complementar-

ity problem translates into the equivalent equation

φcap(sg, pc) := minsg, pc(sg)− pc = 0 , (3.43)


which provides a unique and consistent description of the phase state. Adding

(3.43), the defining equations of ξmin, ξsorp and pc, the algebraic equations asso-

ciated with equilibrium reactions and interphase mass exchange, the compress-

ibility laws and the physical constraints, we end up with the following system of

equations:

∂t(φs`η`) + L`η` = 0 ,

∂t(φs`ηs) = 0 ,

∂t(φ(s`ξ`,ex + sgρmol,g))+L`ξ`,ex +Lgρmol,g = φsÀ`,exRkin(cnmin) ,

∂t(φs`ξsorp) + L`ξ`,sorp = φsÀsorpRkin(cnmin) ,

∂t(φs`ξmin) + L`ξ`,min = φsÀ`,minRkin(cnmin) ,

∂t(φs`ξ`,kin) + L`ξ`,kin = φsÀ`,kinRkin(cnmin) ,

∂t(φs`ξs,kin) = φsÀs,kinRkin(cnmin) ,

(3.44)

(3.45)

(3.46)

(3.47)

(3.48)

(3.49)

(3.50)

Def

.eq

n.

ξsorp − ξ`,sorp + ξs,sorp = 0 ,

ξmin − ξ`,min + ξs,min = 0 ,

pc − pg + p` = 0 ,

(3.51)

(3.52)

(3.53)

Eq.

reac

tion

s

φex(c`, pg) = 0 ,

φmob(c`) = 0 ,

φsorp(c`, cs,nmin) = 0 ,

φmin(c`, cs,min) = 0 ,

(3.54)

(3.55)

(3.56)

(3.57)

Con

stra

ints

φcap(sg, pc) = 0 ,

ρmol,` −I∑i=1

ci` = 0 ,

ρmass,` −I∑i=1

M ici` = 0 ,

ρmol,` − f`(c`) = 0 ,

ρmol,g − fg(pg) = 0 ,

ρmass,g −M1ρmol,g = 0 .

(3.58)

(3.59)

(3.60)

(3.61)

(3.62)

(3.63)

Note that the above system involves I` + Is + Jsorp + Jmin + 8 unknowns, which


corresponds to the number of equations. It is not recommended, however, to use

this global system for numerical computations, which would be costly due to the

large number of unknowns. Instead, it is our goal to select an appropriate set

of primary (or global) variables Xglob, and to use the algebraic equations (3.53)–

(3.63) to express as many variables as possible – the so-called local or secondary

variables, denoted Xloc – in terms of the primary variables. Since some of these

equations are nonlinear, this can be done only implicitly. The implicit function

expressing the secondary in terms of the primary variables is called resolution

function. Clearly, the efficiency of the global method depends strongly on the

number and choice of primary variables and on the computational effort that

is necessary to evaluate the resolution function using some iteration method.

This has to be done whenever the primary variables change, e. g., when they are

updated during the solution process.

In Chapter 4, we will prove the existence of a resolution function Xglob 7→ Xloc

for the choice

Xglob =

pc

pg

η`

ξ`,ex

ξ`,sorp

ξ`,min

ξ`,kin

ξsorp

ξmin

∈ Rnglob , Xloc =

sg

p`

ηs

ξ`,mob

ξs,sorp

ξs,min

ξs,kin

ρmol,`

ρmass,`

ρmol,g

ρmass,g

∈ Rnloc (3.64)

of global and local unknowns, where nglob = I` − Jmob + Jsorp + Jmin + 2 and

nloc = Is + Jmob + 6. With this choice of primary variables, the evaluation

of the resolution functions decomposes into different subproblems that can be

solved sequentially. For example, using pc as a global variable, the current phase

state is obtained independently from the chemical subproblem, which, on the

3.3. THE RESULTING NONLINEAR SYSTEM 65

other hand, provides all transformed variables and concentrations. Since there

are no spatial couplings in the algebraic equations, these local computations can

be efficiently carried out in parallel. The detailed algorithm for evaluating the

resolution function is presented in Section 6.4.3.

3.3 The resulting nonlinear system

Finally, by employing the resolution function, cf. Chapter 4, the system

∂t(φs`η`) + L`η` = 0 , (3.65)

∂t(φs`ξ`,ex + φsgρmol,g) + L`ξ`,ex + Lgρmol,g = φsÀ`,exRkin(Xglob) , (3.66)

∂t(φs`ξsorp) + L`ξ`,sorp = φsÀsorpRkin(Xglob) , (3.67)

∂t(φs`ξmin) + L`ξ`,min = φsÀ`,minRkin(Xglob) , (3.68)

∂t(φs`ξ`,kin) + L`ξ`,kin = φsÀ`,kinRkin(Xglob) (3.69)

ξsorp − ξ`,sorp + ξs,sorp(Xglob) = 0 , (3.70)

ξmin − ξ`,min + ξs,min(Xglob) = 0 , (3.71)

ρmol,` − f`(c`(Xglob)) = 0 , (3.72)

φex(c`(Xglob), pg) = 0 , (3.73)

of nglob equations remains to be solved, where s` = 1 − sg and the transport

operators L` and Lg are defined as in (2.30). To keep the notation tight, the

dependence of s`, sg, ρmol,` and ρmol,g on the global variables was not displayed in

the above equations. Since ξsorp and ξmin were introduced as additional variables,

their defining equations are included in the global system. Moreover, as the

pressures pc and pg are treated as global variables, the algebraic equations (3.72)–

(3.73) are kept in the global system. A variant with fewer global pressure variables

is discussed in Section 3.4.2. The remaining algebraic equations defining the


resolution function read


∂t(φs`ηs) = 0 , (3.75)

φcap(sg, pc) = 0 , (3.76)

φmob(c`) = 0 , (3.77)

φsorp(c`, cs,nmin) = 0 , (3.78)

φmin(c`, cs,min) = 0 , (3.79)

pc − pg + p` = 0 , (3.80)

ρmol,` −I∑i=1

ci` = 0 , (3.81)

ρmass,` −I∑i=1

M ici` = 0 , (3.82)

ρmol,g − fg(pg) = 0 , (3.83)

ρmass,g −M1ρmol,g = 0 . (3.84)

The process of solving this system numerically is described in Section 6.4.3.

3.4 Variants and special cases

3.4.1 No additional transformed variables

The variables ξsorp and ξmin represent additional variables that are introduced to

improve the convergence properties of the numerical solver, cf. [Hof10, Section

3.5.2]. Another advantage of the variables ξmin is the possibility of using persis-

tent primary variables associated with the equilibrium mineral reactions, which

would not be the case without them. To illustrate this, we follow [Hof10, Sec-

tion 3.6.1] and consider, for a moment, the equations (3.22)–(3.28) for the case

that there are no additional variables. Moreover, we assume that there is only

one equilibrium mineral reaction and no other reactions. The resulting equations

3.4. VARIANTS AND SPECIAL CASES 67

(including equilibrium conditions and constraints) read

∂t(φs`η`) + L`η` = 0 , (3.85)

∂t(φs`ξ`,ex + φsgρmol,g)+L`ξ`,ex +Lgρmol,g = 0 , (3.86)

∂t(φs`ξ`,min) + L`ξ`,min = ∂t(φs`ξs,min) , (3.87)

φmin(c`, cs,min) = 0 , (3.88)

φex(c`, pg) = 0 , (3.89)

φcap(sg, pc) = 0 , (3.90)

pc − pg + p` = 0 , (3.91)

ρmol,` − f`(c`) = 0 , (3.92)

ρmol,g − fg(pg) = 0 , (3.93)

ρmol,` −I∑i=1

ci` = 0 , (3.94)

ρmass,` −I∑i=1

M ici` = 0 , (3.95)

ρmass,g −M1ρmol,g = 0 , (3.96)

where cs,min denotes the concentration of the mineral and

φmin(c`, cs,min) = minϕmin(c`), cs,min

represents the equilibrium condition associated with the mineral reaction. The

concentrations c` and cs,min are related to the transformed variables by means of

the retransformation (3.10), which, in this particular case, reduces to

c` = S`,exξ`,ex + S`,minξ`,min +B⊥` η` ,

cs = cs,min = ξs,min .(3.97)

The problem of this formulation is that the algebraic equation (3.88) cannot

be solved for one unknown since the second argument of the minimum function

depends only on cs,min = ξs,min, but the first argument does not. This is in


contrast to the formulation including the additional variables ξmin, where the

retransformation reads

c` = S`,ex ξ`,ex + S`,min(ξmin + ξs,min) ,

cs = cs,min = ξs,min ,(3.98)

and both arguments of the minimum function depend on ξs,min.

To tackle this problem in a numerical code, one can make a case differentiation

with respect to the mineral assemblage. Let the following considerations refer to

a fully discrete form of the system (3.85)–(3.96), where the Semismooth Newton

method is used to solve the nonlinear system of equations in each time step.

Then, one can define the sets

ΩA := x ∈ Ω : ϕmin(c`(x)) > cs,min(x) ,

ΩI := x ∈ Ω : ϕmin(c`(x)) ≤ cs,min(x) ,

indicating which mineral state locally holds for the current Newton iterate:

φmin(c`) =

cs,min in ΩA ,

ϕmin(c`) in ΩI .(3.99)

Thus, the Semismooth Newton method using the minimum function as a comple-

mentarity function may be regarded as an active set strategy, cf. [HIK03]. Note

that the above partitioning refers to the evaluation of the minimum function

based on the current Newton iterate and is updated after each Newton step.

Let us now show how the system (3.85)–(3.96) is used to compute numerical

solutions for all unknowns. For this purpose, we assume that pc, pg, η`, ξ`,ex

and ξ`,min|ΩA represent global variables, i. e., they are known when the residual is

assembled. In the first step, the variables sg and p` can be computed explicitly

from (3.90) and (3.91), respectively. Then, (3.88) and (3.99) imply that

ξs,min = 0 in ΩA , (3.100)

ϕmin(c`) = 0 in ΩI . (3.101)

Thus, we can solve (3.101) for ξ`,min|ΩI . After this step, the variable ξ`,min is known


on the whole domain Ω, and the concentrations c` and cs can be computed on

Ω with the help of the retransformation (3.97). By inserting the saturation, the

pressures and ξ`,min into (3.87), we obtain ξs,min on ΩI by evaluating the left

hand side of the PDE on ΩI . Finally, the PDE (3.87), associated with the global

variable ξ`,min|ΩA , remains to be solved on ΩA as a global equation. Altogether,

the global system reads

∂t(φs`η`) + L`η` = 0 in Ω ,

∂t(φs`ξ`,ex + φsgρmol,g)+L`ξ`,ex +Lgρmol,g = 0 in Ω ,

∂t(φs`ξ`,min) + L`ξ`,min = ∂t(φs`ξs,min) in ΩA ,

ρmol,` − f`(c`) = 0 in Ω ,

φex(c`, pg) = 0 , in Ω ,

and it is solved for the primary variables pc, pg, η`, ξ`,ex in Ω and for ξ`,min in

ΩA. The remaining unknowns are given in terms of the primary unknowns by

the local equations (3.88)–(3.91), (3.93)–(3.96) and the PDE (3.87), restricted to

ΩI .

On the one hand, the calculation of the secondary unknowns requires the evalu-

ation of a differential operator, which is a nonstandard way of using a PDE for

numerical computations. On the other hand, the above strategy implies that the

global system contains PDEs that are valid only on a part of the domain and,

as a consequence, the number of unknowns may vary within the domain. It may

even be altered within one Newton iteration when the set of active and inactive

nodes is updated. This can be avoided if the additional variables are used, cf.

Chapter 4.

3.4.2 No extended capillary pressure variable

The additional variables ξsorp = ξ`,sorp−ξs,sorp and ξmin = ξ`,min−ξs,min represent

heterogeneous reactions between the liquid and the solid phase. Similarly, the

additional pressure variable pc = pg−p` represents the interphase mass exchange,

i. e., a heterogeneous reaction between the gas phase and the liquid phase. To

illustrate the benefit of the use of pc, let us consider a formulation relying solely

on pg and p`, but not on the additional pressure pc. In this case, the capillary


pressure–saturation relationship can be substituted into (3.42), which yields

pg = p` + pc(sg) .

This relation is inserted into (2.20) and equipped with complementarity con-

straints to obtain the modified solubility law

φex(c`, sg) := minsg, φex(c`, p` + pc(sg)) = 0 . (3.102)

It is clear that at least one pressure variable must be included in the set of

primary variables to be able to treat the other pressures as secondary variables.

Since (3.102) depends on p` and sg, the use of p` as a global variable allows the

computation of sg on the local level with the help of this equation. This suggests

the following splitting into local and global variables:

Xglob =

p`

η`

ξ`,ex

ξ`,sorp

ξ`,min

ξ`,kin

ξsorp

ξmin

, Xloc =

sg

pg

ηs

ξ`,mob

ξs,sorp

ξs,min

ξs,kin

ρmol,`

ρmass,`

ρmol,g

ρmass,g

.


The resulting global system reads

∂t(φs`η`) + L`η` = 0 , (3.103)

∂t(φs`ξ`,ex + φsgρmol,g) + L`ξ`,ex + Lgρmol,g = φsÀ`,exRkin(Xglob) , (3.104)

∂t(φs`ξsorp) + L`ξ`,sorp = φsÀsorpRkin(Xglob) , (3.105)

∂t(φs`ξmin) + L`ξ`,min = φsÀ`,minRkin(Xglob) , (3.106)

∂t(φs`ξ`,kin) + L`ξ`,kin = φsÀ`,kinRkin(Xglob) , (3.107)

ξsorp − ξ`,sorp + ξs,sorp(Xglob) = 0 , (3.108)

ξmin − ξ`,min + ξs,min(Xglob) = 0 , (3.109)

ρmol,` − f`(c`(Xglob)) = 0 , (3.110)

and has one unknown less than the global system (3.65)–(3.73). Nevertheless, in

our numerical tests, this formulation required more computation time than the

formulation based on the global system (3.65)–(3.73). One reason for this is the

structure of the local problem, which consists of the equations


∂t(φs`ηs) = 0 , (3.112)

φex(c`, sg) = 0 , (3.113)

φmob(c`) = 0 , (3.114)

φsorp(c`, cs,nmin) = 0 , (3.115)

φmin(c`, cs,min) = 0 , (3.116)

ρmol,g − fg(p` + pc(sg)) = 0 , (3.117)

ρmol,` −I∑i=1

ci` = 0 , (3.118)

ρmass,` −I∑i=1

M ici` = 0 , (3.119)

ρmass,g −M1ρmol,g = 0 . (3.120)


In the local problem, the equations (3.111)–(3.116) consisting of Is + Jex + Jmob

equations are fully coupled by the saturations and concentrations. Consequently,

they have to be solved simultaneously to obtain the saturations and the local

transformed variables. If the logarithms of the concentration variables are used

as unknowns instead of the transformed variables, the system is even larger since

the defining equations of transformed variables must be added to the system, cf.

Section 6.4.3.

On the other hand, if the additional pressure pc is used, the saturation depends

only on pc and can be computed independently from the rest of the system.

Afterwards, the variables ηs are obtained explicitly, and the chemical problem

consisting of (3.77)–(3.79) along with (3.74) can be solved independently, cf. Sec-

tion 6.4.3. Thus, despite of the fact that pc represents an additional variable,

the local problems – having the same size as in the formulation considered here

– can be solved more efficiently if it is included because they decompose into

several smaller subproblems. Another advantage of using pc as a global variable

is that the derivative of sg with respect to the global variables is zero except for

the derivative with respect to pc. Hence, the global Jacobian can be assembled

faster, and it has fewer nonzero entries than the Jacobian that results from the

formulation considered in this section.

3.4.3 Gas and liquid phase pressures as local unknowns

Another possibility to reduce the number of global pressure variables is to use

only the extended capillary pressure pc as global unknown and to treat pg and p`

as local variables. This is accomplished by shifting the equation (3.73) from the

global to the local system. In the resulting formulation, the property that the

local system decomposes into smaller subproblems is preserved. Nevertheless, in

a numerical experiment related to CO2 sequestration, the computations based on

this formulation required significantly more computation time than the compu-

tations based on the formulation (3.65)–(3.73). This is because more steps of the

Newton solver were necessary to solve the nonlinear systems in each time step.

Potential reasons for this behavior are discussed in Section 7.3.1.


3.4.4 Two-phase two-component flow without reactions

Let us finally consider the special case of two-phase two-component flow without

chemical reactions, where I` = 2, Is = 0, Jex = 1 and Jmob = Jsorp = Jmin =

Jkin = 0. This implies

S =

Sg

S`

=

−1

1

0

, B` = S∗` = S` =

1

0

, S⊥` =

0

1

,

and the variable transformation of Section 3.1 yields the transformed variables

η` = (S⊥`TB⊥` )−1S⊥`

Tc` = c2

` ,

ξ` = (BT` S∗`)−1BT

` c` = c1` .

The resulting system of global equations reads (cf. (3.65)–(3.73))

∂t(φs`η1` ) + L`η1

` = 0 , (3.121)

∂t(φs`ξ1`,ex + φsgρmol,g) + L`ξ1

`,ex + Lgρmol,g = 0 , (3.122)

ρmol,` − f`(c`(Xglob)) = 0 , (3.123)

φex(c`(Xglob), pg) = 0 . (3.124)

Here, (3.121) represents conservation of component 1 (water) and (3.122) repre-

sents conservation of component 2 (e. g., hydrogen, CO2). The global and local

variables are given by

Xglob =

pc

pg

η1`

ξ1`,ex

∈ R4 , Xloc =

sg

p`

ρmol,`

ρmass,`

ρmol,g

ρmass,g

∈ R6 ,


and the resolution function Xglob 7→Xloc is defined by the local equations

pc − pg + p` = 0 , φcap(sg, pc) = 0

and the constraints

ρmol,` −I∑i=1

ci` = 0 , ρmass,` −I∑i=1

M ici` = 0 ,

ρmol,g − fg(pg) = 0 , ρmass,g −M1ρmol,g = 0 .

In this special case, the number of global equations can be reduced further by

shifting two of the global variables, e. g., η1` and pg, to the set of local vari-

ables and using (3.123)–(3.124) as local equations instead of global equations.

The resulting global system consists of two partial differential equations for two

global unknowns, which is the standard number of unknowns in two-phase two-

component flow, see, e. g., [NBI13, MMK13, BJS09]. This formulation is used for

the benchmark computations in Section 7.1.

Chapter 4

Resolution function

One of the key issues of the reduction mechanism is the partitioning of unknowns

into primary/global unknowns and secondary/local unknowns and the use of

algebraic equations to eliminate local unknowns from the fully coupled DAE

system in terms of a resolution function Xloc = Xloc(Xglob). Since the chemistry

equations and the constitutive laws are nonlinear, this elimination has to be done

implicitly, i. e., the evaluation of the resolution function requires the solution of

a nonlinear system of equations.

Clearly, the efficiency of the numerical solver depends strongly on the choice

of primary and secondary variables. On the one hand, one has to ensure that

the secondary variables Xloc can be uniquely determined for a given set of global

variablesXglob. This is only the case if there are sufficiently many global variables.

On the other hand, it is beneficial to have as many local variables as possible

since the computational effort that is required for solving the local problems

is significantly lower than the effort required for solving the global nonlinear

problem.

Typically, in large parts of the computational domain, only very few steps of

the nested Newton iteration are necessary to determine the local variables, e. g.,

in regions where chemical reactions proceed slowly or do not take place at all.

Moreover, the solution of the previous time step provides a potentially good initial

guess for these local iterations. Finally, let us mention that the local problems

can be solved efficiently on parallel machines without the need for communication

between the processors.

75

76 CHAPTER 4. RESOLUTION FUNCTION

In this chapter, the existence of a resolution function Xglob 7→ Xloc is studied

for the choice (3.64) of primary and secondary variables. More precisely, we will

prove existence of a local resolution function for the general reaction system and

existence of a global resolution function for the simplified case that there are no

kinetic reactions. This implies that our primary variables are persistent.

4.1 Local resolution function

In this section, the existence of a local resolution function Xglob 7→Xloc is shown

with the help of an implicit function theorem. Since the minimum function is a

nonsmooth function, the classical version of the implicit function theorem cannot

be applied to the local equations (3.74)–(3.84). Instead, a generalized version

for piecewise smooth functions is used. We begin with the statement of some

relevant definitions and results.

Definition 4.1.1. Let U ⊆ Rn be open and let F : U → Rm be locally Lip-

schitz continuous. Denote by ΩF ⊆ U the set of points at which F fails to be

differentiable. Then, the B-subdifferential of F at x ∈ U is defined by

∂BF (x) = M ∈ R(m,n) : M = limxk→x,xk 6∈ΩF

JF (xk) ,

where JF (·) denotes the classical Jacobian of F . Note that by Rademacher’s

Theorem, ΩF has measure zero. Similarly, if U = V ×W ⊆ Rn−p × Rp, the set

∂By F (x,y) = My ∈ R(m,p) : M = (Mx|My) ∈ ∂BF (x,y)

provides a generalization of the partial derivatives of F with respect to y.

Definition 4.1.2. The convex hull

∂clF (x) = conv(∂BF (x))

is called (Clarke’s) generalized Jacobian of F at x.

It is indeed a generalization of the classical Jacobian, since the following theorem

holds (see [Cla83, Proposition 2.2.4]).

4.1. LOCAL RESOLUTION FUNCTION 77

Theorem 4.1.3. Let f : Rn → Rm be continuously differentiable in a neighbor-

hood of x ∈ Rn. Then ∂clF (x) = JF (x).

Lipschitz continuity and Clarke’s generalized Jacobian provide a framework in

which an implicit function theorem holds, cf. [Cla83, p. 256]. This theorem,

however, is not strong enough for our needs, since it requires that all elements of

∂clF (x) are of maximal rank, which is hard to verify for our problem. Instead,

we use the additional regularity provided by our local equations, which are not

only Lipschitz continuous but also piecewise differentiable.

Definition 4.1.4. A continuous function F : U → Rm on an open set U ⊆ Rn is

PCr (r ≥ 1) if for every x ∈ U there exists a finite family Gii∈I of Cr selection

functions Gi : O → Rm, i ∈ I, where O ⊆ U is an open neighborhood of x, such

that F (z) ∈ Gi(z) : i ∈ I for all z ∈ O. A selection function Gi of a PCr

function F is called essentially active at x if x ∈ intz ∈ O : Gi(z) = F (z).The set of all essentially active indices of F at x is denoted by Ie(F,x).

The following lemma shows how the B-subdifferential can be evaluated for piece-

wise smooth functions.

Lemma 4.1.5 [RS97]. If Gii∈I is a family of selection functions for the PC1

function F at x, then

∂BF (x) = JGi(x) : i ∈ Ie(F,x) .

The decisive concept for a generalization of the implicit function theorem to

piecewise differentiable functions is the following notion of coherent orientation.

Definition 4.1.6. Let V ⊆ Rm and W ⊆ Rn be open sets, F : V ×W → Rn

be a PC1 function, and (x0,y0) ∈ V × W . Denote by Λ(x0,y0) the set of all

matrices M ∈ R(n,n) with the property that there exist matrices M1, . . . ,Mn ∈∂By F (x0,y0) such that the p-th row of M coincides with the p-th row of Mp for

all p ∈ 1, . . . , n. The function F is called completely coherently oriented with

respect to y at (x0,y0) if all matrices M ∈ Λ(x0,y0) have the same nonvanishing

determinantal sign.

Using the notion of completely coherent orientation, we are now ready to state

an implicit function theorem for piecewise smooth functions.


Theorem 4.1.7 [RS97] (Implicit function theorem for PCr functions). Let

V ⊆ Rm and W ⊆ Rn be open sets and F : V ×W → Rn be a PCr function,

where r ≥ 1. If F is completely coherently oriented with respect to y at a root

(x0,y0) ∈ V ×W of F , then the equation F (x,y) = 0 determines an implicit

PCr function x 7→ y(x) in a neighborhood of (x0,y0).

Corollary 4.1.8 [RS97]. Let V ⊆ Rm and W ⊆ Rn be open and let F : V ×W →Rn denote the componentwise minimum function, given by

F (x,y) = minh1(x,y),h2(x,y) , h(i)(x,y) =

hi1(x,y)

...

hin(x,y)

, i ∈ 1, 2 ,

where h1,h2 : V ×W → Rn are Cr functions, r ≥ 1. Moreover, let (x0,y0) ∈V ×W be a root of F . If all matrices M = (m1, . . . ,mn) ∈ R(n,n) with mp ∈∇yh1

p(x0,y0),∇yh2

p(x0,y0), for each p ∈ 1, . . . , n, have the same nonzero

determinantal sign, then there exists a PCr function x 7→ y(x) on a neighborhood

of x0 such that (x,y(x)) is the locally unique solution of F (x,y) = 0.

Let us now apply the above results to our local equations and show the existence

of a local resolution function.

Theorem 4.1.9. Let Xglob ∈ Rnglob and Xloc ∈ Rnloc be defined as in (3.64) and

assume that

P = (Xglob,Xloc) ∈ Rnglob × Rnloc : cnmin(Xglob,Xloc) ∈ RInmin+

is nonempty, where the concentrations cnmin = cnmin(Xglob,Xloc) are consid-

ered as a function of the transformed variables by means of the retransformation

(3.41). Moreover, let the kinetic and equilibrium reactions be governed by the

mass action laws (2.14), (2.16) and (2.19), and let pc ∈ C1([0, 1);R) be a strictly

increasing function of sg. Assume further that the ODEs (3.45) and (3.50) are

discretized with the implicit Euler method, where ∆t > 0 denotes the time step

size and the superscript (·)old refers to the value at the previous time step. Then,


if ∆t is sufficiently small and if X0 = (X0glob,X

0loc) ∈ P satisfies

φcap(s0g, p

0c) = mins0

g, pc(s0g)− p0

c = 0 , 0 ≤ s0g < 1 , (4.1)

φeq(c`(X0), cs(X

0)) = 0 , (4.2)

φs0`η

0s − φsold

` ηolds = 0 , (4.3)

φs0`ξ

0s,kin − φsold

` ξolds,kin −∆tφs0

`As,kinRkin(cnmin(X0)) = 0 , (4.4)

there exists a PC1 resolution function Xglob 7→ X loc in a neighborhood of X0,

where s0g, s

0` , p

0c, η

0s and ξ0

s,kin represent entries of X0 according to the partitioning

(3.64).

Proof. First, let us note that equation (3.43) defines an explicit global PC1 reso-

lution function pc 7→ sg by means of

sg(pc) =

0 if pc ≤ pc(0) ,

p−1c (pc) if pc > pc(0) .

Note that since s 7→ pc(s) is strictly increasing, it admits an inverse function

p−1c . In the next step, we show that the chemical equilibrium laws (4.2) and the

discretized ODEs (4.3)–(4.4) define a local PC1 resolution function

Xglob 7→ (ξ`,mob, ξs,sorp, ξs,min,ηs, ξs,kin)

in a neighborhood of X0. For that purpose, we define the function φkin : P →RIs−Jsorp−Jmin associated with the discretized ODEs by

φkin(X) =

φ(s`ηs − sold` η

olds )

φ(s`ξs,kin − sold` ξ

olds,kin)−∆tφs`As,kinRkin(cnmin(X))

.

Moreover, for a subset I ⊆ 1, . . . , Jmin with complement A := 1, . . . , Jmin\I,


let φIeq : P → RJ∗eq be defined by

φIeq(X) =

φmob(c`(X))

φsorp(c`(X), cs,nmin(X))

φImin(c`(X), cs,min(X))

, (4.5)

where φImin represents the selection functions of φmin associated with the indices

j ∈ I,

φImin,j(c`(X), cs,min(X)) =

ϕmin,j(c`(X)) if j ∈ I ,cjs,min if j ∈ A .

Here, ϕmin,j denotes the j-th component of ϕmin, cf. p. 44. Note that φIeq is not

necessarily an essentially active selection function of φeq at X0 because the min-

imum in the j-th component of φmin is not necessarily attained in the first argu-

ment if j ∈ I. If only the derivatives with respect to (ξ`,mob, ξs,sorp, ξs,min,ηs, ξs,kin)

of all essentially active selection functions of φ in X0 were considered, we could

only show that φ is coherently oriented [RS97, Def. 4] in X0, whereas we need it

to be completely coherently oriented to derive the existence of a local resolution

function with the help of Corollary 4.1.8. Let us now show that the determinant

of

M =

∂φIeq∂(ξ`,mob,ξs,sorp,ξs,min)

∂φIeq∂(ηs,ξs,kin)

∂φkin

∂(ξ`,mob,ξs,sorp,ξs,min)∂φkin

∂(ηs,ξs,kin)

=:

M 11 M 12

0 M 22

is positive at X0 ∈ P for an arbitrary choice of the index set I ⊆ 1, . . . , Jmin.After rearranging the mineral reactions and renumbering the variables ξs,min ac-

cordingly (note that this does not change the determinantal sign of M), we may

assume that I = 1, 2, . . . , |I|, A = |I|+ 1, . . . , Jmin and that S`,min provides

a partitioning

S`,min = (S`,min,I S`,min,A) ,

where S`,min,I denotes the submatrix consisting of the first |I| columns of S`,min


and S`,min,A the submatrix consisting of the last |A| columns. Then, M 11 reads

M 11 =

ST`,mobΛ`S`,mob ST`,mobΛ`S`,sorp ST`,mobΛ`S`,min

ST`,sorpΛ`S`,mobST`,sorpΛ`S`,sorp

+STs,sorpΛs,nminSs,sorpST`,sorpΛ`S`,min

ST`,min,IΛ`S`,mob ST`,min,IΛ`S`,sorp ST`,min,IΛ`S`,min

0 0 (0 I |A|)

,

where the block diagonal matrices Λ` and Λs,nmin are given by

Λ` =

1/c1

`

. . .

1/cI``

, Λs,nmin =

1/c1

s,nmin

. . .

1/cIs,nmin

s,nmin

.

Note that the concentrations c1` , . . . , c

I`` and c1

s,nmin, . . . , cIs,nmin

s,nmin are positive for all

X ∈ P . Using the matrices

N 1 =

ST`,mob 0

ST`,sorp STs,sorp

ST`,min,I 0

Λ`

Λs,nmin

S`,mob S`,sorp S`,min,I

0 Ss,sorp 0

,

N 2 =

ST`,mob

ST`,sorp

ST`,min,I

Λ`S`,min,A ,

we can rewrite M 11 as

M 11 =

N 1 N 2

0 I |A|

.

Note that, by assumption, the columns of (S`,mob S`,sorp S`,min,I) and of Ss,sorp

are linearly independent. Consequently, N 1 is symmetric and positive definite

on P , which implies det(M 11) > 0 for all X ∈ P . Moreover, for ∆t = 0, M


admits the representation

M =

M 11 ∗

0 φs`IIs−Jsorp−Jmin

,

which implies det(M ) > 0 for all X ∈ P , provided 0 ≤ sg < 1. By a continuity

argument, we obtain det(M ) > 0 at X0 for ∆t sufficiently small (note that the

bound for ∆t may depend on X0). Since I ⊆ 1, . . . , Jmin was arbitrary, there

exists a PC1 resolution function

Xglob 7→ (ξ`,mob, ξs,sorp, ξs,min,ηs, ξs,kin)

locally around X0 by Corollary 4.1.8. The proof is completed by noting that all

other unknowns are given explicitly in terms of the variables Xglob, ξ`,mob, ξs,sorp,

ξs,min, ηs, ξs,kin. More precisely, the concentrations c` and cs are obtained from

the retransformation (3.41), the liquid pressure p` is provided by (3.80), and the

molar and mass densities are given by (3.81)–(3.84).

4.2 Global resolution function

It is well known that, in general, the existence of a local resolution function

around each point of a domain does not imply the existence of a global resolution

function, i. e., the existence and uniqueness of local variables Xloc for given global

variables Xglob. However, if there are no kinetic reactions, one can prove the

existence and uniqueness of such a global resolution function Xglob 7→Xloc. This

is accomplished by showing that our local equations are equivalent to the KKT

system of a convex minimization problem, for which existence and uniqueness

can be established by standard methods of convex optimization. Similar ideas

were employed in the [Kra08] and [HKK12].

4.2.1 General statements

In this section, we briefly recall some notation and results from the field of opti-

mization. The definitions and theorems below can be found in many entry level

4.2. GLOBAL RESOLUTION FUNCTION 83

textbooks on optimization, e. g., [GK02, BV04]. First, let us compile some facts

about extended functions, i. e., functions defined on R := R ∪ +∞.

Definition 4.2.1. Let f : Rn → R be an extended function. Then f is called

proper if

dom(f) := x ∈ Rn : f(x) < +∞ 6= ∅ .

Definition 4.2.2. Let f : Rn → R be a proper extended function. Then f is

called lower semicontinuous at a point x0 ∈ Rn if

lim infy→x0

f(y) ≥ f(x0) .

Moreover, it is called lower semicontinuous on Rn if it is lower semicontinuous

at every point x ∈ Rn.

Lemma 4.2.3. Let f : Rn → R be a proper extended function. Then the following

statements are equivalent:

(i) f is lower semicontinuous on Rn.

(ii) The level sets L(c) = x ∈ Rn : f(x) ≤ c are closed (respectively empty)

for all c ∈ R.

Since we are going to work with a convex programming problem in the following

section, let us further recall some important definitions and statements from the

field of convex optimization.

Definition 4.2.4. A subset S ⊆ Rn is said to be convex if x,y ∈ S implies

λx+ (1− λ)y ∈ S for all λ ∈ [0, 1].

Definition 4.2.5. Let S ⊆ Rn be a nonempty, convex set. A function f : S → Ris said to be

(i) convex on S if

f(λx+ (1− λ)y) ≤ λf(x) + (1− λ)f(y)

for all x,y ∈ S and all λ ∈ (0, 1).


(ii) strictly convex on S if

f(λx+ (1− λ)y) < λf(x) + (1− λ)f(y)

for all x,y ∈ S, x 6= y, and all λ ∈ (0, 1).

The following lemma provides a sufficient condition for a function to be strictly

convex, given it is at least two times continuously differentiable.

Lemma 4.2.6. Let S ⊆ Rn be a nonempty, convex open set and let f ∈ C2(S).

If the Hessian matrix Hf(x) is positive definite for all x ∈ S, then f is strictly

convex on S.

Lemma 4.2.7. Let S ⊆ Rn be a nonempty, convex set and let f : S → R be

strictly convex. Then the minimization problem

minx∈S

f(x)

has at most one (global) minimum.

The basic idea of our proof of existence of a global resolution function is to

show that some of the local nonlinear equations resulting from the equilibrium

reactions are equivalent to the KKT system of a convex constrained minimization

problem, for which we can show that it admits a unique solution. The following

well-known theorem provides necessary and sufficient optimality conditions for a

constrained convex programming problem.

Theorem 4.2.8 (Kuhn-Tucker Theorem). Let S ⊂ Rn be a convex open set

and let fi : S → R, i ∈ 0, . . . ,m, be convex and differentiable functionals.

Consider the following convex optimization problem:

(P) minx∈Sf0(x) : f1(x) ≤ 0, . . . , fm(x) ≤ 0 .

Let x ∈ S be a feasible point of (P), i. e., fi(x) ≤ 0 for i ∈ 1, . . . ,m.

(a) Assume that there exist nonnegative multipliers (λ1, . . . , λm) 6= (0, . . . , 0)


such that the following relationships hold:

λifi(x) = 0 for i ∈ 1, . . . ,m (complementary slackness) ,

∇f0(x) +m∑i=1

λi∇fi(x) = 0 (Lagrange inclusion) .

Then x is a solution of (P).

(b) Conversely, if x ∈ S is a solution of (P), then there exist nonnegative

multipliers (λ0, . . . , λm) 6= (0, . . . , 0) such that the complementary slackness

condition of part (a) holds as well as

m∑i=0

λi∇fi(x) = 0 .

(c) For λ0 to be nonzero in (b) it is sufficient that there exists a point x ∈ Ssuch that Slater’s constraint qualification holds:

fi(x) < 0 , i ∈ 1, . . . ,m .


4.2.2 Existence of a global resolution function

Let us now use Theorem 4.2.8 to prove that in the absence of kinetic reactions

there exists a global resolution function Xglob 7→Xloc for

Xglob =

pc

pg

η`

ξ`,ex

ξ`,sorp

ξ`,min

ξsorp

ξmin

∈ Rnglob , Xloc =

sg

p`

ηs

ξ`,mob

ξs,sorp

ξs,min

ρmol,`

ρmass,`

ρmol,g

ρmass,g

∈ Rnloc .

More precisely, we show that for given Xglob, the local variables are uniquely

determined by the algebraic equations (3.51)–(3.63) and the equation

φ

∆t(s`ηs − sold

` ηolds ) = 0 , (4.6)

which results from the discretization of (3.45).

Proposition 4.2.9. Let s 7→ pc(s) be a strictly increasing function. Then, there

exists a global resolution function

Xglob 7→ (sg,ηs) .

Proof. The saturation of the gas phase can be computed explicitly with the help

of equation (3.58) by means of

sg(pc) =

0 if pc ≤ pc(0) ,



Once sg and s` = 1− sg are known, ηs is obtained from (4.6), which yields

ηs =1

s`sold` η

olds . (4.7)

Having calculated ηs, all transformed variables but ξ`,mob, ξs,sorp and ξs,min are

known. Using the notations

ξloc =

ξ`,mob

ξs,sorp

ξs,min

, vglob =

S`,exξ`,ex + S`,sorpξsorp + S`,minξmin +B⊥` η`

B⊥s ηs

0

,

the retransformation defines an affine-linear map between the concentrations and

ξloc by means of

c(ξloc) =

c`(ξloc)

cs,nmin(ξloc)

cs,min(ξloc)

= Seqξloc + vglob , (4.8)

where Seq denotes the submatrix

Seq =

S`,mob S`,sorp S`,min

0 Ss,sorp 0

0 0 IJmin

∈ R(I`+Is,J∗eq)

of the stoichiometric matrix S. Note that due to the assumptions we made

on the columns and subblocks of S (cf. p. 48), all columns of Seq are linearly

independent.

The main step in the proof of existence of a global resolution function is to show

that ξloc is uniquely determined by the local nonlinear equations (2.29), which are

considered in terms of ξloc by means of (4.8). This is accomplished by showing the

equivalence of these equations to the KKT system of a constrained minimization

problem based on the following Gibbs functional.


Definition 4.2.10. For c = ( cnmincmin ) ∈ RI`+Is with cnmin ∈ RI`+Is,nmin

0+ and cmin ∈RJmin let the Gibbs functional G be defined by

G(c) =

I`+Is,nmin∑i=1

µ(ci)ci + e1−µi0 +

I`+Is∑i=I`+Is,nmin+1

µ(ci)ci ,

where

µ(ci) =

µi0 − 1 + ln(ci) for i = 1, . . . , I` + Is,nmin ,

µi0 for i = I` + Is,nmin + 1, . . . , I` + Is .

Here, µ0 = (µ10, . . . , µ

I`+Is0 )T ∈ RI`+Is represents a solution of the linear system

ST

eqµ0 = −

ln(Kmob)

ln(Ksorp)

ln(Kmin)

, (4.9)

which exists since all columns of Seq are linearly independent. Let us further

define the set

S = ξloc ∈ RJ∗eq : (c`, cs,nmin, cs,min)(ξloc) ∈ RI`+ × RIs,nmin

+ × RJmin0+ , (4.10)

where the concentrations are considered as functions of ξloc by means of (4.8) and

ηs = ηs(Xglob) depends on Xglob through the resolution function Xglob 7→ (sg,ηs)

from Proposition 4.2.9.

Note that since limc0 ln(c)c = 0, the functional G can be extended to S, cf.

Proposition 4.2.11. For later use, let us also mention that

µ(ci)ci + e1−µi0 ≥ ci (4.11)

for i ∈ 1, . . . , I`+Is,nmin, which follows from the specific choice of the constants

e1−µi0 in the definition of G.

It is our goal to prove that the constrained minimization problem

(MIN) minξloc∈S

G(c(ξloc))


admits a unique solution and that the KKT system associated with it is equiva-

lent to the local equations (2.31)–(2.33). This will imply that these local equa-

tions are uniquely solvable and define an implicit resolution function Xglob 7→(ξ`,mob, ξs,sorp, ξs,min).

Proposition 4.2.11. Let S 6= ∅. Then problem (MIN) is equivalent to the

minimization problem

(MIN1) minξloc∈RJ

∗eq

G(ξloc) ,

where the extended functional G : RJ∗eq → R is defined by

G(ξloc) :=

G(c(ξloc)) on S ,

+∞ otherwise .

Proof.

(i) Let ξloc ∈ S denote a solution of (MIN). Then

G(ξloc) = G(ξloc) ≤ G(ξloc) = G(ξloc) for all ξloc ∈ S .

Since G is continuous on S, we even have G(ξloc) ≤ G(ξloc) for all ξloc ∈ S.

Hence, ξloc is a solution of (MIN1).

(ii) Conversely, let ξloc ∈ RJ∗eq denote a solution of (MIN1). This implies

ξloc ∈ S, and it remains to show that ξloc ∈ S. Let us define the index sets

Inmin := 1, . . . , I` + Is,nmin , Imin := I` + Is,nmin + 1, . . . , I` + Is ,

I0 := i ∈ Inmin : ci(ξloc) = 0 , I1 := Inmin \ I0 ,

and let us assume that ξloc 6∈ S, i. e., I0 6= ∅. Since S is nonempty, we may

select an element ξ∗ ∈ S and define

ξε := εξ∗ + (1− ε)ξloc ,

which belongs to S for any ε ∈ (0, 1) since

c(ξε) = εc(ξ∗) + (1− ε)c(ξloc)


according to (4.8). In the following, we will derive a contradiction to the

assumption that ξloc solves problem (MIN). For this purpose, we define

the functional E : (0, 1)→ R by

E(ε) = G(c(ξε))−G(c(ξloc)) .

Then, E is differentiable, and it holds that

E ′(ε) = ∇cG(c(ξε))T Seq(ξ∗ − ξloc) = ∇cG(c(ξε))T (c(ξ∗)− c(ξloc))

=∑i∈I0

(µi0 + ln(εci(ξ∗))ci(ξ∗) +∑i∈I1

(µi0 + ln(ci(ξε)))(ci(ξ∗)− ci(ξloc))

+∑i∈Imin

µi0(ci(ξ∗)− ci(ξloc)) ,

which implies E ′(ε)→ −∞ as ε→ 0+ as∑i∈I1

(µi0 + ln(ci(ξε)))(ci(ξ∗)− ci(ξloc)) +∑i∈Imin

µi0(ci(ξ∗)− ci(ξloc))

ε→0+−→∑i∈I1

(µi0 + ln(ci(ξloc)))(ci(ξ∗)− ci(ξloc)) +

∑i∈Imin

µi0(ci(ξ∗)− ci(ξloc)) .

Since E is continuous and limε→0+ E(ε) = 0, there exists ε0 > 0 such that

E(ε0) < 0. This implies G(c(ξε0)) < G(c(ξloc)) for ξε0 ∈ S, which is a

contradiction to the assumption that ξloc solves (MIN1).

Proposition 4.2.11 implies that if there exists a solution ξloc of (MIN1), then

the nonmineral concentrations associated with ξloc are strictly positive. Conse-

quently, G will be differentiable at its minimum.

Proposition 4.2.12. G is strictly convex on S.

Proof. Obviously, G(c(·)) is well-defined and C∞ on the open convex set

S ′ := ξloc ∈ RJ∗eq : (c`, cs,nmin)(ξloc) ∈ RI`+ × RIs,nmin

+ .


It follows from (4.8) that ∂c∂ξloc

= Seq, which implies

∇ξlocG(c(ξloc)) = ST

eq∇cG(c(ξloc)) = ST

eqµ0 + ST

eq

ln(c`(ξloc))

ln(cs,nmin(ξloc))

0

= −

ln(Kmob)

ln(Ksorp)

ln(Kmin)

+ ST

eq

ln(c`(ξloc))

ln(cs,nmin(ξloc))

0

.

(4.12)

Hence, the Hessian matrix has the representation

H =∂2G(c(ξloc))

∂2ξloc

= ST

eqΛSeq , (4.13)

where Λ = diag(1/c1(ξloc)), . . . , 1/cI`+Is,nmin(ξloc), 0, . . . , 0). Using the partition-

ings

Seq =

Seq,nmin

Seq,min

, Λ =

Λnmin

0Jmin

with the block matrices

Seq,nmin =

S`,mob S`,sorp S`,min

0 Ss,sorp 0

,

Seq,min =(0 0 IJmin

)and the diagonal matrix

Λnmin = diag(1/c1(ξloc), . . . , 1/cI`+Is,nmin(ξloc)) ,

(4.13) is equivalently rewritten as

H =(ST

eq,nmin, ST

eq,min

)Λnmin

0Jmin

Seq,nmin

Seq,min

= S

T

eq,nminΛnminSeq,nmin .


Since the columns of Seq,nmin are linearly independent and Λnmin is positive

definite for ξloc ∈ S ′, the Hessian matrix is symmetric and positive definite

for ξloc ∈ S. Consequently, G is strictly convex on S ′ and on S ⊆ S ′ by

Lemma 4.2.6.

Proposition 4.2.13. Let S 6= ∅. Then problem (MIN) has a unique solution

ξloc ∈ S.

Proof. Since problem (MIN) is equivalent to problem (MIN1), it suffices to

show that problem (MIN1) admits a solution. Note that by definition, G is

lower semicontinuous. This implies (cf. Lemma 4.2.3) that the level sets

Lα := ξloc ∈ RJ∗eq : G(ξloc) ≤ α , α ∈ R ,

are closed. Moreover, since S is nonempty, there exists some α0 > 0 such that the

level set Lα0 is nonempty. Clearly, by the definition of G, we have Lα0 ⊂ S. Let

us assume for a moment that Lα0 is unbounded. Then, there exists a sequence

(ξnloc)n∈N ⊂ Lα0 , ξnloc =

ξn`,mob

ξns,sorp

ξns,min

with ‖ξnloc‖∞ →∞ as n→∞. Defining

v`,glob := S`,exξ`,ex + S`,sorpξsorp + S`,minξmin +B⊥` η`

and recalling (4.8), it holds that

c`(ξnloc) = (S`,mob S`,sorp S`,min)ξnloc + v`,glob ,

and since the columns of (S`,mob S`,sorp S`,min) are linearly independent, there

exists a constant γ > 0 such that

‖c`(ξnloc)‖∞ ≥ γ‖ξnloc‖∞ − ‖v`,glob‖∞


for all n ∈ N. Moreover, as ‖ξnloc‖∞ →∞, there exists an index N ∈ N such that

γ‖ξnloc‖∞ − ‖v`,glob‖∞ > 0 , (4.14)

mini=1,...,I`+Is,nmin

µi0 − 1 + ln(γ‖ξnloc‖∞ − ‖v`,glob‖∞) > 0 (4.15)

for all n ≥ N . Let us now select an index jn ∈ 1, . . . , I` such that

cjn(ξnloc) ≥ γ‖ξnloc‖∞ − ‖v`,glob‖∞ .

Then, letting

µmin := mini∈Inmin

µi0 , µmax := maxi∈Imin

|µi0| ,

where the index sets Inmin and Imin are defined as in the proof of Proposi-

tion 4.2.11 above, it follows from (4.14), (4.15), and (4.11) (the latter implying

that (µi0 − 1 + ln(ci))ci + e1−µi0 ≥ ci ≥ 0 for i ∈ Inmin on S ) that

G(ξnloc) ≥ (µjn0 − 1 + ln(cjn(ξnloc)))cjn(ξnloc) +

∑i∈Imin

µi0ci(ξnloc)

≥ (µmin − 1 + ln(γ‖ξnloc‖∞ − ‖v`,glob‖∞))(γ‖ξnloc‖∞ − ‖v`,glob‖∞)− Cµmax‖ξnloc‖∞

for all n ≥ N , which implies G(ξnloc) −→ ∞ as n → ∞, since for arbitrary

constants c1, c2, c3, c4 with c2, c3, c4 > 0 it holds that

limx→∞

c1 + ln(c2x− c3)(c2x− c3)− c4x =∞ .

This implies that G(ξnloc) > α0 for n sufficiently large, which is a contradiction

to the assumption (ξnloc)n∈N ⊂ Lα0 . Hence, Lα0 must be bounded. Since it is also

closed, it is compact and G attains a global minimum ξloc ∈ S, which represents

a solution of (MIN1). Finally, it follows from Proposition 4.2.11 that ξloc ∈ Sand that ξloc solves problem (MIN). The uniqueness of ξloc follows from the

strict convexity of G on S, cf. Lemma 4.2.7 and Proposition 4.2.12.


The proof of Proposition 4.2.13 appears somewhat technical. This is because, in

general, we have no a priori information on the signs of the entries of µ0, which is

defined as a solution of the linear system (4.9). It should be noted, however, that

the proof can be simplified under the additional assumption that there exists a

vector s⊥ with only strictly positive entries which is orthogonal to all columns

of the stoichiometric matrix S. Then, there exists a solution µ0 of (4.9) having

only strictly positive entries, and the estimate

G(c(ξloc)) ≥ C

I`+Is∑i=1

ci(ξloc) = C‖c(ξloc)‖1

holds for all ξloc ∈ S, which immediately implies the boundedness of the level

set Lα0 . The assumption that such a vector s⊥ exists is called the assumption

of the conservation of the number of atoms and is satisfied for many reactive

networks. It is often used in the literature, e. g., in [Kra08] to prove existence

of a multispecies ODE batch problem with kinetic reactions governed by the law

of mass action, or in [Hof10] to prove existence of solutions for a multispecies

transport problem involving kinetic mineral reactions.

Let us now use the above results to prove existence of a global resolution function

Xglob 7→Xloc in the absence of kinetic reactions.

Theorem 4.2.14. Assume that there are no kinetic reactions and let the global

variables Xglob = (pc, pg,η`, ξ`,ex, ξ`,sorp, ξ`,min, ξsorp, ξmin) be given such that

S = ξloc ∈ RJ∗eq : (c`, cs,nmin, cs,min)(ξloc) ∈ RI`+ × RIs,nmin

+ × RJmin0+

is nonempty, where the concentrations are given in terms of ξloc by means of

(4.8) with ηs = ηs(Xglob) provided by (4.7). Then, all local variables are uniquely

determined by the local equations (3.75)–(3.84).

Proof. It was show in Proposition 4.2.9 that there exists a global resolution func-

tion Xglob 7→ (sg,ηs), defined by the equations (3.75)–(3.76). Let us now employ

Theorem 4.2.8 to prove that the variables ξ`,mob, ξs,sorp and ξs,min are uniquely

determined by the local equations (3.77)–(3.79). For this purpose, note that

problem (MIN) is equivalent to the minimization problem

(MIN2) minξloc∈S′

G(c(ξloc)) : ξs,min ≥ 0 ,


with S ′ ⊃ S defined as in the proof of Proposition 4.2.12. Let ξloc =

(ξ`,mob

ξs,sorpξs,min

)∈ S

denote the unique solution of (MIN2) according to Proposition 4.2.13. Then,

Slater’s constraint qualification is satisfied for the element ξloc +(

00ε

)with ε =

(ε, . . . , ε)T ∈ RJmin+ , which is feasible for (MIN2) for ε sufficiently small. Next, we

combine Theorem 4.2.8 b-c), (4.12) and (4.9) to deduce existence of nonnegative

multipliers λ ∈ RJmin0+ such that the KKT system

0 = ST

eqµ0 + Seq

ln(c`(ξloc))

ln(cs,nmin(ξloc))

0

+

0

0

−λ

= −

ln(Kmob)

ln(Ksorp)

ln(Kmin)

+ ST

eq

ln(c`(ξloc))

ln(cs,nmin(ξloc))

0

+

0

0

−λ

=

φmob(c`(ξloc))

φsorp((c`, cs,nmin)(ξloc))

φmin(c`(ξloc))

+

0

0

−λ

,

λj ξjs,min = 0 , j = 1, . . . , Jmin

is satisfied, which is equivalent to the local equations (3.77)–(3.79). Hence, they

are solvable for ξ`,mob, ξs,sorp and ξs,min. The uniqueness of the solution follows

from Theorem 4.2.8 a) and Proposition 4.2.13.

Having calculated ξ`,mob, ξs,sorp and ξs,min, all transformed variables are known

and the concentrations c` and cs can be reconstructed using the retransformation

(3.41). The densities are then given explicitly by (3.81)–(3.84), and the proof is

completed by noting that the liquid pressure is given explicitly in terms of pc and

pg by (3.80).


4.3 The Semismooth Newton method

When complementarity constraints are involved, the resulting problem is typi-

cally nonsmooth since the minimum function or the Fisher-Burmeister function

are not C1 functions. This means that the assumptions that are usually posed on

the classical Newton method to converge locally quadratic (e. g., “C1 with Lip-

schitz continuous derivative”, or similar, cf. [Deu11]) are not met. However, it is

well known that the assumptions can be weakened to the assumption of strong

semismoothness. In the following, we give the definition of the Semismooth New-

ton method and state a convergence theorem from [dLFK00].

Definition 4.3.1. A locally Lipschitz continuous function f : Rn → Rn is said

to be semismooth at x ∈ Rn if f is directionally differentiable at x and

Md− f ′(x;d) = o(‖d‖)

for any d → 0 and any M ∈ ∂clf(x + d), where f ′(x;d) denotes the direc-

tional derivative of f at x in the direction d. Analogously, f is called strongly

semismooth at x if

Md− f ′(x;d) = O(‖d‖2) .

f is called (strongly) semismooth if f is (strongly) semismooth at any point

x ∈ Rn.

Definition 4.3.2. Let f : Rn → Rn be locally Lipschitz continuous.

(i) The Semismooth Newton method is defined such as the classical Newton

method, but with the classical Jacobian replaced by an arbitrary element of

the B-subdifferential.

(ii) A solution x of f(x) = 0 is called BD-regular if all elements M ∈ ∂Bf(x)

are nonsingular.

Theorem 4.3.3. Let f : Rn → Rn be semismooth (strongly semismooth) and let

x be a BD-regular solution of f(x) = 0. Then, for any initial value sufficiently

close to x, the Semismooth Newton method generates a sequence which converges

superlinearly (quadratically, respectively) to x.

4.3. THE SEMISMOOTH NEWTON METHOD 97

Let us now demonstrate how the Semismooth Newton method is applied to solve

the local equations (3.74), (3.77)–(3.79) for the local transformed variables

ξloc =

ξ`,mob

ξs,sorp

ξs,min

ξs,kin

for a given set of global variables Xglob. This represents the main step to evaluate

the resolution function. Note that in this section, the definition of ξloc involves

the variables ξs,kin, whereas in Section 4.2.2, kinetic reactions were not included.

Using the Semismooth Newton method, we would like to generate a sequence

ξklock∈N converging to the solution of the root-finding problem

φ(ξloc) = 0 ,

where φ is defined by

φ(ξloc) =

φeq(c`(ξloc), cs(ξloc))


olds,kin)−∆tφs`As,kinRkin(cnmin(ξloc))

and represents the residual function associated with the local equations (3.74) and

(3.77)–(3.79). Here, the concentrations c = c(ξloc) are represented in terms of the

unknown local variables by an affine linear map induced by the retransformation

for given global variables, cf. (4.8) for the case that there are no kinetic reactions.

In practice, the iteration is stopped as soon as a certain stopping criterion is

satisfied, e. g.,

‖φ(ξkloc)‖ < tol ,

for a given tolerance tol. For the k-th iterate ξkloc, we define the index set

Ik := j ∈ 1, . . . , Jmin : ϕmin,j(c`(ξkloc)) ≤ cj,ks,min

that specifies in which argument the minimum of the minimum function asso-

ciated with the equilibrium mineral condition (2.33) is attained. Here, ϕmin,j


denotes the j-th component of ϕmin, cf. p. 44. Then, the function

φIk

(ξglob) =

φIk

eq (ξglob)


olds,kin)−∆tφs`As,kinRkin(cnmin(ξloc))

(4.16)

with φIk

eq defined as in (4.5) represents an essentially active selection function of

φ in the point ξkloc, cf. Definition 4.1.4. According to Lemma 4.1.5, the matrix

Jk :=∂φI

k

eq (ξloc)

∂ξloc

represents an element of the B-subdifferential of φ at ξkglob. Hence, we may define

the k-th step of the Semismooth Newton method by

Jk∆ξkloc = −φ(ξkloc) ,

ξk+1loc = ξkloc + ∆ξkloc .

Note that it follows from the proof of Theorem 4.1.9, where we showed that the

determinantal sign of Jk is positive, that the Newton step is well-defined for ∆t

sufficiently small. Having solved the linear system for ∆ξkloc and updated the

current iterate, the index set Ik+1 is computed, and the next Newton step can

be started.

The index set Ik indicates which minerals are present in the current Newton step

and which are not. Using the minimum function as a complementarity function,

one of the following two cases holds:

φIk

min,j(c`(ξkloc), cs,min(ξkloc)) =

ϕmin,j(c`(ξkloc)) if j ∈ Ik ,

cjs,min otherwise .

In the first case, the j-th mineral is present, whereas in the second case, it is fully

dissolved. Thus, the Semismooth Newton method using the minimum function

may be regarded as an active set strategy, cf. [HIK03].

Chapter 5

Discretization

In this chapter, we present the discretization of the system system (3.65)–(3.73)

based on the implicit Euler method in time and a LFEM-FV scheme in space,

resulting in a nonlinear system of discrete equations for each time step. These

nonlinear problems are then solved by some linearization method, e. g., the Semis-

mooth Newton method.

In the LFEM-FV scheme presented below, all terms of the partial differential

equations but the advective ones are treated by standard Galerkin linear finite

elements (LFEM), whereas the advective term is approximated by finite volumes

(FV). If mass lumping is employed to all terms without spatial derivatives and the

Donald diagram is used to construct the dual grid of a conforming triangulation

of a 2D domain, it can be shown that this scheme is equivalent to the finite

volume method described in [KA03, Sec. 6.2]. This is because the diffusive terms

of the linear finite element scheme coincide with the diffusive terms of the finite

volume scheme, cf. [KA03, Lemma 6.11]. Due to this equivalence, the LFEM-FV

scheme is locally conservative, a property that is highly valued in computational

fluid dynamics because such methods reflect the physical principle behind the

mass balance equations and prevent the artificial generation or loss of mass due to

numerical errors. Locally mass conservative methods of mixed type are addressed

in the second part of this work.

The discretization of the system (3.65)–(3.73) is illustrated for a model problem

resulting for the special case of two-phase two-component flow, cf. Section 3.4.4.

For convenience, let us recall the global system that is obtained from the reduction

99

100 CHAPTER 5. DISCRETIZATION

scheme, which consists of the equations

∂t(φs`η`) + L`η` = 0 , (5.1)

∂t(φs`ξ`,ex + φsgρmol,g) + L`ξ`,ex + Lgρmol,g = 0 , (5.2)

ρmol,` − f`(c`(Xglob)) = 0 , (5.3)

φex(c`(Xglob), pg) = 0 . (5.4)

Here, the concentrations c` = c`(Xglob) are related to the transformed variables

via the resolution function Xglob 7→ Xloc and the retransformation (3.41). The

global and local variables associated with (5.1)–(5.4) read

Xglob =

pc

pg

η`

ξ`,ex

, Xloc =

sg

p`

ρmol,`

ρmass,`

ρmol,g

ρmass,g

.

For the ease of presentation, let us assume that Ω is a polygonally bounded

domain in R2 and that Th represents a conforming triangulation of Ω, i. e., a

decomposition of Ω into closed triangles K ∈ Th with no hanging nodes. The

same ideas, however, can be transferred to bilinear elements on rectangles and to

the case of three space dimensions with minor modifications. In the numerical

solver developed as a part of this work, linear elements on triangles/tetrahedra

and bilinear elements on rectangles/hexahedra were implemented.

The system (5.1)–(5.4) is completed by prescribing initial and boundary condi-

tions.

5.1. INITIAL AND BOUNDARY CONDITIONS 101

5.1 Initial and boundary conditions

Since we would like to admit Dirichlet and flux boundary conditions, let ∂Ω

consist of two disjoint parts ΓD and ΓF such that

∂Ω = ΓD∪ΓF .

It is assumed that each boundary face of Th entirely belongs either to ΓD or to ΓF .

Moreover, let us assume that the above partitioning refers to all global variables,

i. e., Dirichlet values are prescribed for all variables on ΓD.

Initial conditions

For the global variables, we provide the following initial conditions:

pc = p0c on Ω× 0 ,

pg = p0g on Ω× 0 ,

η` = η0` on Ω× 0 ,

ξ`,ex = ξ0`,ex on Ω× 0 ,

where p0c , p

0g, η

0` and ξ0

`,ex represent given functions depending on space.

Dirichlet boundary conditions

On ΓD, Dirichlet values are prescribed for the global variables. They are repre-

sented by given functions pDc , pDg , ηD` and ξD`,ex, i. e.,

pc = pDc on ΓD × (0, Tend) ,

pg = pDg on ΓD × (0, Tend) ,

η` = ηD` on ΓD × (0, Tend) ,

ξ`,ex = ξD`,ex on ΓD × (0, Tend) .

Typically, the initial and boundary conditions are specified for nontransformed

and physical variables in the literature. In this case, it may be necessary to

evaluate the resolution function and the variable transformation to compute the

values of p0c , p

0g, η

0` , ξ

0`,ex, pDc , pDg , ηD` and ξD`,ex from the given physical variables.


Flux boundary conditions

To simulate scenarios involving the injection of chemical substances, inhomoge-

neous flux boundary conditions are admitted on ΓF :(−D` ρmol,`∇

(1

ρmol,`

ξ`,ex

)+ q`ξ`,ex + qg ρmol,g

)· n∂Ω = u1

F on ΓF ,

(−D` ρmol,`∇

(1

ρmol,`

η`

)+ q`η`

)· n∂Ω = u2

F on ΓF ,

where u1F , u

2F denote the injection rates and n∂Ω is the outer unit normal vector

of ∂Ω.

5.2 Discretization with linear finite elements

In this section, the discretization of (5.1)–(5.4) based on the linear finite ele-

ment method is presented. This method belongs to the class of Lagrange finite

elements, where the degrees of freedom are given as function values at certain

points of the triangles, called nodes. More precisely, let a1, . . . ,aM denote the

vertices of the triangles T ∈ Th, and let them be numerated in such a way that

a1, . . . ,aM1 ∈ Ω ∪ ΓF ,

aM1+1, . . . ,aM ∈ ΓD .

Let further ϕ1, . . . , ϕM denote a set of piecewise linear functions satisfying

ϕi|T ∈ P1(T ) , i ∈ 1, . . . ,M, T ∈ Th ,

ϕi(aj) = δij , i, j ∈ 1, . . . ,M

with δij denoting the usual Kronecker-Delta. The function spaces associated with

the LFEM scheme read

Vh = spanϕ1, . . . , ϕM = v ∈ C(Ω) : v|T ∈ P1(T ) for all T ∈ Th ,

Vh,0 = spanϕ1, . . . , ϕM1 = v ∈ Vh : v|ΓD = 0 .

5.2. DISCRETIZATION WITH LINEAR FINITE ELEMENTS 103

For the time discretization, let the time elapsed at the n-th time step be denoted

by tn, 0 ≤ n ≤ N , and let ∆tn = tn−tn−1 denote the corresponding time step size.

Using an Euler-implicit time stepping scheme, all time derivatives are replaced

by backward difference quotients,

un − un−1

∆tn,

where un denotes the approxmation of a generic variable u at time t = tn. The

superscript (·)n for a coefficient function means that it is evaluated at time t = tn.

The fully discrete problem associated with (5.1)–(5.4) reads as follows.

Problem 1. For all tn, n ∈ 1, . . . , N, find Xnglob,h = (pnc,h, p

ng,h, η

n`,h, ξ

n`,ex,h) ∈

Vh × Vh × Vh × Vh such that

1

∆tn

∫Ω

(φsn`,hηn`,h − φsn−1

`,h ηn−1`,h )ϕh dx−

∫Ω

qn`,hηn`,h∇ϕh dx

+

∫Ω

Dn`,hρ

nmol,`,h∇

(1

ρnmol,`,h

ηn`,h

)∇ϕh dx = −

∫ΓF

u1F ϕh dσ ,

(5.5)1

∆tn

∫Ω

φ(sn`,hξn`,ex,h + sng,hρ

nmol,g,h − sn−1

`,h ξn−1`,ex,h − sn−1

g,h ρn−1mol,g,h)ϕh dx

−∫

Ω

qn`,hξn`,ex,h∇ϕh dx−

∫Ω

qng,hρnmol,g,h∇ϕh dx

+

∫Ω

Dn`,hρ

nmol,`,h∇

(1

ρnmol,`,h

ξn`,ex,h

)∇ϕh dx = −

∫ΓF

u2F ϕh dσ

(5.6)for all ϕh ∈ Vh,0,

ρnmol,`,h(aj)− f`(c`(Xnglob,h(aj))) = 0 , j = 1, . . . ,M , (5.7)

φex(c`(Xnglob,h(aj)), p

ng,h(aj)) = 0 , j = 1, . . . ,M , (5.8)

pnc,h(aj) = pDc (aj, tn) , j = M1 + 1, . . . ,M , (5.9)

png,h(aj) = pDg (aj, tn) , j = M1 + 1, . . . ,M , (5.10)

ηn`,h(aj) = ηD` (aj, tn) , j = M1 + 1, . . . ,M , (5.11)

ξn`,ex,h(aj) = ξD`,ex(aj, tn) , j = M1 + 1, . . . ,M , (5.12)

and X0glob,h(aj) = (p0

c , p0g, η

0` , ξ

0`,ex)(aj) for j = 1, . . . ,M .


Here, the secondary variables are defined in terms of the primary variables by

pn`,h(x) =M∑j=1

p`(Xnglob,h(aj))ϕj(x) ,

snα,h(x) =M∑j=1

sα(Xnglob,h(aj))ϕj(x) , α ∈ `, g ,

ρnmol/mass,α(x) =M∑j=1

ρmol/mass,α(Xnglob,h(aj))ϕj(x) , α ∈ `, g ,

where (p`, sα, ρmol/mass,α)(·) denotes the resolution function. The discrete diffusive

and advective fluxes are given by

qnα,h(x) = −K(x)krα(snα,h(x))

µα(∇pnα,h(x)− ρnmass,α,h(x) g) , (5.13)

Dn`,h(x) = αT |qn`,h(x)|I + (αL − αT )

qn`,h(x)⊗ qn`,h(x)

|qn`,h(x)| + φ43 sn`,h(x)

103 Ddiff,`I .

(5.14)

Finally, by expanding the global unknowns in terms of the basis functions of Vh,

pnc,h =M∑j=1

pnc,jϕj , png,h =M∑j=1

png,jϕj , ηn`,h =M∑j=1

ηn`,jϕj , ξn`,ex,h =M∑j=1

ξn`,ex,jϕj ,

and using the basis functions ϕj, j ∈ 1, . . . ,M1 of Vh,0 as test functions

in (5.5)–(5.6), a nonlinear system of equations for the unknowns Xnglob,j :=

(pnc,j, png,j, η

n`,j, ξ

n`,ex,j), j ∈ 1, . . . ,M1, is obtained in each time step.

Integrals of the form ∫Ω

ϕi(x)ϕj(x) dx ,

and integrals resulting from sources and sinks are discretized with the help of

mass lumping, using a quadrature of the form

I(f) =M∑i=1

ωif(ai) .

5.3. FV STABILIZATION 105

b

b

b

b

b

b

b

b

b

b

b

aj

Ωj

Figure 5.1: Control volume Ωj associated with aj

5.3 Finite volume stabilization for advection–

dominated problems

When advective transport dominates diffusion, the scheme (5.5)–(5.6) produces

numerical solutions containing unphysical oscillations. To tackle this problem, a

stabilization technique proposed in [Hof10] (see also [BFK14]) was implemented,

where the advective terms in (5.5)–(5.6) are discretized by finite volumes. In

the following, this stabilization is presented for the case that the underlying

discretization is the linear finite element method on triangles, cf. Section 5.2.

It should be noted, however, that it is also applicable for bilinear elements on

rectangles and for linear/bilinear elements on tetrahedral/hexahedral meshes in

three space dimensions, respectively. Given a conforming triangulation Th of Ω, a

dual grid is defined by connecting all edge midpoints of all triangles T ∈ Th with

the barycenter of the associated triangle. Thus, a control volume Ωj is generated

around each node aj, cf. Figure 5.1. The family of control volumes obtained by

this construction is called Donald diagram.


The FV stabilization is based on the approximations∫Ω

∇ · (qn`,hηn`,h)ϕj dx ≈∫

Ωj

∇ · (qn`,hηn`,h) dx =

∫∂Ωj

(qn`,hηn`,h) · n∂Ωj dσ ,∫

Ω

∇ · (qng,hρnmol,g,h)ϕj dx ≈∫

Ωj

∇ · (qng,hρnmol,g,h) dx =

∫∂Ωj

(qng,hρnmol,g,h) · n∂Ωj dσ ,

where n∂Ωj denotes the outer unit normal vector of ∂Ωj. It consists in treating

the boundary integrals on the right hand side with an upwind-weighted finite

volume method. The integral∫∂Ωj

(qn`,hηn`,h) · n∂Ωj dσ =

∑T∈Th

∫∂Ωj∩T

(qn`,hηn`,h) · n∂Ωj dσ

is approximated by (see Figure 5.2 for the used notation)∫∂Ωj∩T

(qn`,hηn`,h) · n∂Ωj dσ ≈ qn`,h((a0 + ac)/2) · n[a0,ac](r01η

n`,h(a0)

+ (1− r01)ηn`,h(a1)) + qn`,h((ac + a2)/2) · n[ac,a2](r02ηn`,h(a0) + (1− r02)ηn`,h(a2)) ,

where the normals n[a0,ac] and n[ac,a2] have the length |a0 − ac| and |ac − a2|,respectively. The integral∫

∂Ωj

(qng,hρnmol,g,h) · n∂Ωj dσ =

∑T∈Th

∫∂Ωj∩T

(qng,hρnmol,g,h) · n∂Ωj dσ

is treated analogously. For the weights r01 and r02, there exist different ap-

proaches. For example, the choice r01 = r02 = 0.5 corresponds to a central

scheme, which is not suitable for advection–dominated problems. A full upwind

scheme is obtained for

r01 =

1 if qn`,h((a0 + ac)/2) · n[a0,ac] ≥ 0 ,

0 otherwise ,

r02 =

1 if qn`,h((ac + a2)/2) · n[ac,a2] ≥ 0 ,

0 otherwise .

In contrast to the upwind stabilization used in [Hof10], our upwind scheme is

5.3. FV STABILIZATION 107

b

b

b

b

b

b

Ωj ∩ T

a2

a0

a1

a0

ac

a1

a2

n[ac,a2]

n[a0,ac]

Figure 5.2: Intersection of a control volume Ωj with a triangle T

fully nonlinear since the velocities qn`,h and qng,h depend nonlinearly on the phase

pressures, saturations and mass densities, which themselves are nonlinear func-

tions of the primary variables pnc,h, png,h, η

n`,h and ξn`,ex,h by means of the resolution

function and (5.13).

For simple model problems involving a linear transport operator and a scalar

diffusion coefficient which is piecewise constant, it can be shown that the full

upwind scheme is inverse monotone on grids that consist of nonobtuse triangles,

cf. [KA03, Theorem 6.19]. This means that monotonicity is preserved and that

overshoots and undershoots are prevented. In our numerical tests the full upwind

scheme worked well for the multiphase multicomponent model involving nonlinear

transport.

Since the the full upwind scheme introduces a significant amount of artificial

diffusion to the problem, resulting in a smearing of sharp fronts in the numerical

solution, it is recommended to use full upwinding only in those parts of the

domain where the PDEs are advection dominated. This can is accomplished

by using partial upwinding or exponential upwinding, where the upwind weights

depend on the local grid Peclet number, cf. [KA03].


5.4 Implicit elimination

After the discretization in time and space, a nonlinear system of equations of the

form

Rn(Xnglob,h,Xloc(X

nglob,h)) = 0

remains to be solved in each time step, where Rn represents the residual function

associated with the discrete global problem and Xnglob,h denotes the vector of

global unknowns at time t = tn. As we solve this system using the (Semismooth)

Newton method, the derivative ∂Xloc

∂Xglobof the local variables with respect to the

global variables must be evaluated in each iteration step. If the resolution function

exists in an open neighborhood U of Xnglob,h, it satisfies

φloc(Xglob,Xloc(Xglob)) = 0 for all Xglob ∈ U ,

where φloc represents all local equations defining the resolution function. Then,

the implicit function theorem implies

∂φloc

∂Xglob

+∂φloc

∂Xloc

∂Xloc

∂Xglob

= 0 on U , (5.15)

which represents a linear system of equations for ∂Xloc

∂Xglob. Thus, despite of the

fact that an explicit representation of the resolution function is not available, its

derivative can be computed by solving (5.15). Note that in general, the resolution

function is only piecewise smooth, cf. Theorem 4.1.9. If it is not differentiable at

some point, an element of the B-subdifferential can be determined by replacing

φloc in (5.15) by one of its essentially active selection functions in that point, cf.

Lemma 4.1.5.

Let us now apply this approach to our problem. First, we make the approximation

DXglobξs,kin ≈ 0 . (5.16)

This assumption is justified since the variable ξs,kin corresponds to kinetic reac-

tions which proceed slowly compared to equilibrium reactions. Moreover, as the

local equations do not depend on ξ`,sorp and ξ`,min, cf. (3.74)–(3.84) and (3.41),

it holds that Dξ`,sorpXloc = 0 and Dξ`,minXloc = 0. Moreover, p` depends only on

5.4. IMPLICIT ELIMINATION 109

the global variables pg and pc by means of p` = pg− pc, which implies ∂p`∂pg

= 1 and∂p`∂pc

= −1. The remaining derivatives can be computed in five sequential steps:

(1) The variable sg depends only on the global variable pc, and we obtain the

explicit representation

∂sg∂pc

=

0 if pc ≤ pc(0) ,(∂pc∂sg

)−1

if pc > pc(0)

for the derivative of sg with respect to pc.

(2) The variables ηs depend only on pc by means of

ηs =1

s`(pc)s0`η

0s ,

cf. (3.75), where s0` and η0

s denote the initial values of s` and ηs, respec-

tively. This implies

∂ηs∂pc

=1

s2`

∂sg∂pc

s0`η

0s .

(3) The derivatives

∂(ξ`,mob, ξs,sorp, ξs,min)

∂(η`, ξ`,ex, ξ`,kin, ξsorp, ξmin)

can be computed by solving the linear systems

∂(φmob,φsorp,φmin)

∂(ξ`,mob, ξs,sorp, ξs,min)· ∂(ξ`,mob, ξs,sorp, ξs,min)

∂(η`, ξ`,ex, ξ`,kin, ξsorp, ξmin)

=∂(φmob,φsorp,φmin)

∂(η`, ξ`,ex, ξ`,kin, ξsorp, ξmin).

(5.17)

After possibly rearranging the equilibrium mineral reactions and the entries

of ξs,min with respect to the current mineral assemblage, specified by the

index sets A and I representing the active and inactive mineral reactions,

we obtain


∂(φ

mob,φ

sorp,φ

min

)

∂(ξ`,

mob,ξ

s,so

rp,ξ

s,m

in)

=

=

ST `,

mobΛ`S

`,m

ob

ST `,

mobΛ`S

s,so

rpST `,

mobΛ`S

`,m

in,I

ST `,

mobΛ`S

`,m

in,A

ST `,

sorp

Λ`S

`,m

ob

ST `,sorp

Λ`S`,sorp

+ST s,sorp

Λs,n

minSs,sorp

ST `,

sorp

Λ`S

`,m

in,I

ST `,

sorp

Λ`S

`,m

in,A

ST `,

min,I

Λ`S

`,m

ob

ST `,

min,I

Λ`S

s,so

rpST `,

min,I

Λ`S

`,m

in,IST `,

min,I

Λ`S

`,m

in,A

00

0I|A|

,

∂(φ

mob,φ

sorp,φ

min

)

∂(η

`,ξ`,

ex,ξ

`,kin,ξ

sorp,ξ

min

)=

=

ST `,

mobΛ`B⊥ `

ST `,

mobΛ`S

`,ex

ST `,

mobΛ`S∗ `,k

inST `,

mobΛ`S

`,so

rpST `,

mobΛ`S

`,m

in

ST `,

sorp

Λ`B⊥ `

ST `,

sorp

Λ`S

`,ex

ST `,

sorp

Λ`S∗ `,k

inST `,

sorp

Λ`S

`,so

rpST `,

sorp

Λ`S

`,m

in

ST `,

min,I

Λ`B⊥ `ST `,

min,I

Λ`S

`,ex

ST `,

min,I

Λ`S∗ `,k

inST `,

min,I

Λ`S

`,so

rpST `,

min,I

Λ`S

`,m

in

00

00

0

.

5.4. IMPLICIT ELIMINATION 111

Using the matrices

B =

S`,mob S`,sorp S`,min,I

0 Ss,sorp 0

, Λnmin =

Λ`

Λs,nmin

,

where Λ` and Λs,nmin are defined as in Section 4.1, the system matrix of

these linear systems can be rewritten as

∂(φmob,φsorp,φmin)

∂(ξ`,mob, ξs,sorp, ξs,min)=

BTΛnminB ∗

0 I |A|

.

Since the columns of B are linearly independent, the matrix BTΛnminB

is symmetric and positive definite. This implies that the above matrix is

regular and that the linear systems (5.17) are uniquely solvable. Note that

since the system matrix is the same for all right hand sides, the solutions of

these systems are obtained by computing an LU factorization of the matrix

and using backward/forward elimination for each right hand side.

(4) As soon as the systems (5.17) have been solved, the derivatives∂ξ`,mob

∂Xglob,

∂ξs,sorp∂Xglob

and∂ξs,min

∂Xglobcan be used to compute the derivatives of the concentra-

tions with respect to the global variables by means of

∂c`∂Xglob

= S`,glob +(S`,mob S`,sorp S`,min

)

∂ξ`,mob

∂Xglob

∂ξs,sorp∂Xglob

∂ξs,min

∂Xglob

,

∂cs,nmin

∂Xglob

= B⊥s

∂ηs∂Xglob

+ Ss,sorp

∂ξs,sorp

∂Xglob

,

∂cs,min

∂Xglob

=∂ξs,min

∂Xglob

,


which is a consequence of the retransformation

c` = S`,globXglob + S`,locXloc ,

cs,nmin = Ss,locXloc ,

cs,min = ξs,min ,

cf. (3.41), and the approximation (5.16). Here, the block matrices S`,glob,S`,loc

and Ss,loc are defined by

S`,glob =(

0 0 B⊥` S`,ex 0 0 S∗`,kin S`,sorp S`,min

),

S`,loc =(

0 0 0 S`,mob S`,sorp S`,min 0 0 0 0 0),

Ss,loc =(

0 0 B⊥s 0 Ss,sorp 0 S∗s,kin 0 0 0 0

)and exhibit the same block structure as the global and local variables,

respectively, cf. (3.64).

(5) Finally, we obtain from (2.39), (2.40), (2.6) and the above considerations

that the derivatives of the molar and mass densities with respect to the

global variables have the representations

∂ρmol,g

∂Xglob

=∂fg∂pg

∂pg∂Xglob

,∂ρmass,g

∂Xglob

= M1 · ∂ρmol,g

∂Xglob

,

∂ρmass,`

∂Xglob

=

I∑i=1

M i ∂ci`∂Xglob

,∂ρmol,`

∂Xglob

=

I∑i=1

∂ci`∂Xglob

.

Chapter 6

The numerical framework

6.1 The M++ toolbox

The algorithms developed as a part of this thesis were implemented in M++, a

finite element toolbox for parallel computations based on C++ using the MPI

standard for parallelization, cf. [Wie, Wie05, Wie10]. It relies on data structures

for parallel computing which are directly linked to geometric quantities of an

underlying mesh, and which are adapted to the requirements of a general finite

element realization. More precisely, all geometric objects (cells, faces, edges)

are referenced by their midpoints, and all algebraic data structures (vectors and

matrices) are tied to the nodal points of the finite elements. Together, they build

Distributed Point Objects (DPO), where the parallel distribution of the grid is

made transparent by processor lists assigned to the points. All objects are stored

in hash tables (where the keys are points) so that pointers can be completely

avoided. The general concept is described in the paper [Wie10].

M++ provides basic functionality for solving finite element problems, e. g., iterative

linear solvers (GMRES, BiCGStab, QMRCGStab) and preconditioners (SSOR,

ILU). Moreover, different types of finite elements (linear, mixed, Crouzeix-Raviart

etc.) and output formats for visualization (vtk, gp) are supported.

Our code is written for problems in 2D and 3D on unstructured simplicial or

quadrilateral/hexahedral meshes. The coarse grid is specified in a text file con-

taining information on the grid topology and remains fixed throughout the com-

putation. By not incorporating grid adaptivity, M++ represents a compromise

113

114 CHAPTER 6. THE NUMERICAL FRAMEWORK

between flexibility, compactness of the code, and the requirements for an optimal

performance.

The user specifies the problem in a script file including information about the

finite element ansatz spaces that should be used, the linear and nonlinear solvers,

stopping criteria, discretization parameters etc. In our implementation, the num-

ber of chemical species and chemical reactions may be given in the script file,

and the transformation of the nonlinear system using the reduction scheme is

performed automatically. Consequently, an arbitrary number of chemical species

and reactions can be handled. The code comprises all techniques that are de-

scribed in the following section including cutting strategies, starting value search

for the local Newton iterations, and special numerical treatment of the local

systems.

6.2 Adaptive time stepping

Since for explicit time stepping schemes, very strict time step constraints may

apply as the result of a CFL condition, we use the backward Euler method for time

discretization, which provides good stability properties (B-stability, L-stability)

and enables the use of large time steps. As a consequence, a nonlinear system

of equations has to be solved in each time step using some fixed point method.

To ensure efficiency, we use a damped Newton’s method for linearization, which

converges quadratically if the residual function is sufficiently smooth and if the

initial guess of the iteration is located close enough to the solution. Typically,

this is the case if the time step size is sufficiently small. For very large time steps,

however, the starting value may not lie in the region of quadratic convergence,

and the Newton solver may not converge at all or need many steps until the

stopping criterion is satisfied. For this reason, we update the time step size after

each time step depending on the number of Newton iterations that were necessary

to solve the nonlinear problem of the previous time step. More precisely, after

each successful time step, the current time step size is multiplied with the factor

β =Number of global Newton steps

OptSteps,

where OptSteps is a user-defined value for the optimal number of Newton steps.

With this strategy, the number of Newton steps tries to level off at this value.

6.3. THE GLOBAL NEWTON SOLVER 115

β =Number of global Newton steps

OptSteps

∆t *= β

∆t > ∆tmaxTRUE FALSE

∆t = ∆tmax ∅

∆t < ∆tminTRUE FALSE

∆t = ∆tmin ∅

t+∆t > TendTRUE FALSE

∆t = Tend − t ∅

Figure 6.1: Update of the time step size

Additionally, a maximum and a minimum step size can be specified by the user.

The detailed instructions for updating the time step size after a successful time

step are given in Figure 6.1.

For some applications, the number of Newton steps can be significantly reduced

if a linear extrapolation of the values of the previous two time steps is used as

an initial guess instead of the value of the previous time step. The resulting time

stepping algorithm is illustrated in Figure 6.2.

6.3 The global Newton solver

In the implementation done for this work, a damped Newton’s method is used

to linearize the nonlinear discrete equations resulting from the global and the

local problems. More precisely, Armijo’s rule is is used as a line search strategy

when updating the Newton step. Note that by the definition of the Semismooth

Newton method, the Jacobian of any essentially active selection function can


Initialize: Xglob = X0glob ; Xold

glob = Xglob ; t = 0 ; told = 0 ; tnew = ∆t ; tsteps = 0 ;

while t < Tend

Extrapolation of previous time steps

TRUE FALSE

∆Xglob = ∆tt−told

(Xoldglob −Xglob)

Xnewglob -= ∆Xglob

∅

Solve nonlinear problem at time tnew with the Semismooth Newton method

using Xnewglob as iteration variable

Newton iteration converged

TRUE FALSE

Update ∆t depending on number

of Newton steps, cf. Figure 6.1

tsteps += 1

Xoldglob = Xglob

Xglob = Xnewglob

told = t

t = tnew

tnew += ∆t

Reduce time step size for restart

of time step:

∆t *= 0.5

tnew -= ∆t

Xnewglob = Xglob

tsteps > max steps ∆t < ∆tmin

TRUE FALSE TRUE FALSE

Error: max. number

of time steps reached

BREAK

∅Error: ∆t < ∆tmin

BREAK ∅

Figure 6.2: The adaptive time stepping algorithm

6.3. THE GLOBAL NEWTON SOLVER 117

be taken if the residual function is not differentiable for some Newton iterate.

The detailed algorithm for the global Newton solver is summarized in Figure 6.3.

The update of the global Newton step requires some special treatment, which is

described in the following section.

Solve local problems to determine Xloc = Xloc(Xglob)

Calculate defect R = R(Xglob,Xloc)

d = ‖R‖

while stopping criterion not satisfied

Assemble Jacobian J

Calculate Newton update by solving the linear system

J∆Xglob = −R

Update Xglob and Xloc using the Newton update ∆Xglob


j = 0

while j < maxLineSearchSteps

d1 = ‖R‖

d1 < d

TRUE FALSE

BREAK

∆Xglob *= 0.5

Update Xglob and Xloc using theupdate −∆Xglob


j += 1

d = d1

Figure 6.3: The damped Newton algorithm


6.4 Special numerical treatment

6.4.1 Updating the global Newton step

In the k-th step of the global Newton iteration, the local problems have to be

solved in each grid point to determine the local variables Xkloc = Xloc(X

kglob) as a

function of the global variables. This requires a suitable starting value Xk,0loc for

the local Newton iteration. It was reported in [Hof10] that after solving the η

problem, i. e., the linear subproblem that decouples from the rest of the system

after employing the reduction scheme, it may happen that the new η variables

together with the other transformed variables correspond to negative concentra-

tions. This is not admitted, however, since the logarithms of the concentrations

are used as unknowns for solving the local problems, cf. Section 6.4.3. Therefore,

a feasible starting value has to be searched for the local Newton iterations. In

our formulation, there is no decoupling of η equations, but it may still happen

that, after adding the Newton update ∆Xk−1glob , the new global variables

Xkglob = Xk−1

glob + ∆Xk−1glob

together with the local variables Xk−1loc = Xk−1,end

loc of the previous Newton step

(which represent a natural choice for the starting value Xk,0loc of the local Newton

iteration) correspond to negative concentrations according to the retransforma-

tion. One possible solution of this problem is to restart the global Newton step

and add only a part of the computed Newton update,

Xkglob = Xk−1

glob + τ∆Xk−1glob , τ ∈ (0, 1) .

If this is done globally, i. e., for all nodes, this corresponds to a damping of the

global Newton method. It is preferred, however, to modify the variables locally,

i. e., only in those nodes where the starting value is unfeasible. In our numerical

experiments, the number of these nodes was less than 1% of all nodes. Since we

have Xk−1,endloc = Xloc(X

k−1glob ), a feasible starting value will be recovered by this

strategy if τ is sufficiently small.

This approach has the drawback that the current iterate of the global Newton

iteration is modified, which may affect the speed of convergence of the method.

Another possible strategy is to modify only the local transformed concentration

6.4. SPECIAL NUMERICAL TREATMENT 119

variables to obtain a feasible starting value. To keep the magnitude of the mod-

ification as small as possible, the following optimization problem can be solved.

min ‖∆ξ`,mob‖+ ‖∆ξs,sorp‖+ ‖∆ξs,min‖

s. t.

c` + S`,mob∆ξ`,mob + S`,sorp∆ξs,sorp + S`,min∆ξs,min ≥ Ec`,max1I`

cs,nmin + Ss,sorp∆ξs,sorp ≥ Ecs,max1Is,nmin

cs,min + ∆ξs,min ≥ 0

where 1n = (1, . . . , 1)t ∈ Rn, c`,max = maxi=1,...,I` ci`, cs,max = maxi=1,...,Is,nmin

cisand 0 < E 1. Defining the positive and negative parts of the updates,

∆ξ+`,mob := max0,∆ξ`,mob , ∆ξ−`,mob := max0,−∆ξ`,mob ,

∆ξ+s,sorp := max0,∆ξs,sorp , ∆ξ−s,sorp := max0,−∆ξs,sorp ,

∆ξ+s,min := max0,∆ξs,min , ∆ξ−s,min := max0,−∆ξs,min ,

introducing slack variables ξslack ≥ 0 and letting

∆ξloc =

∆ξ+`,mob

∆ξ−`,mob

∆ξ+s,sorp

∆ξ−s,sorp

∆ξ+s,min

∆ξ−s,min

,Sc =

−S`,mob S`,mob −S`,sorp S`,sorp −S`,min S`,min

0 0 −Ss,sorp Ss,sorp 0 0

0 0 0 0 −IJminIJmin

,

this system can be transformed into a linear program in standard form:

min (1, . . . , 1) ·∆ξloc

s. t.

Sc ∆ξloc + ξslack =

c` − Ec`,max1I`

cs,nmin − Ecs,max1Is,nmin

cs,min

∆ξloc ≥ 0, ξslack ≥ 0 .


The variables c`, cs,nmin and cs,min are those (possibly negative) concentrations

that correspond to the current values of Xglob and Xloc according to the retrans-

formation. Since all entries in the vector (1, . . . , 1) are nonnegative, the simplex

tableau associated with the above linear program is dual feasible. Therefore, the

standard two-phase simplex method can be avoided and we are ready to start

with the dual simplex method.

Under the assumption that the local problem is solvable, its solution represents

a feasible starting value for the local problems. Then, if E is sufficiently small,

the above minimization problem admits a solution. On the other hand, if the

simplex method fails to converge, the set S defined in (4.10) is empty, and the

local problems are not solvable. In this case, the global Newton step has to be

modified. To overcome this problem, one could modify both the local and the

global variables with an update obtained from solving the extended minimization

problem

minεloc(‖∆ξ`,mob‖+ ‖∆ξs,sorp‖+ ‖∆ξs,min‖) + εglob(‖∆ξ`,ex‖+ ‖∆ξsorp‖

+ ‖∆ξmin‖+ ‖∆ξ`,kin‖+ ‖∆η`‖+ ‖∆ηs‖)

s. t.

c` + S`,ex∆ξ`,ex + S`,mob∆ξ`,mob + S`,sorp(∆ξsorp + ∆ξs,sorp)

+ S`,min(∆ξmin + ∆ξs,min) + S∗`,kin∆ξ`,kin +B⊥` ∆η` ≥ Ec`,max1I`

cs,nmin + Ss,sorp∆ξs,sorp + B⊥s ∆ηs ≥ Ecs,max1Is,nmin

cs,min + ∆ξs,min ≥ 0 ,

where we choose εglob εloc to ensure that the local variables are modified in

the first place.

Since these optimization problems are large and computationally expensive to

solve, we follow a different strategy that additionally tackles another problem that

may occur when the local problems are solved: even if a feasible starting value

is available or found during starting value search, it may happen that the local

Newton iteration fails to converge. One potential reason for this is that the global

Newton solver has reached an unphysical state where there exists no nonnegative

solution for the concentrations. In this case, it is inevitable to modify the global

Newton iterate, e. g., by taking only a part of the global Newton update. More

precisely, we employ a backtracking line search along the Newton update using


Armijo’s rule. Since the previous time step was successful, the local problems

will be solvable and the starting value search will converge if a sufficiently small

part of the full Newton update is used. Moreover, it should be noted that the

problem arises only in very few grid points. Hence, the convergence of the global

Newton method is hardly affected by this strategy. The resulting algorithm for

updating the global Newton step is summarized in Figure 6.4.

Xk+1glob = Xk

glob +∆Xkglob ; Xk+1,0

loc = Xk,endloc ; A = 0 ;

while A 6= 1

(cℓ, cs,nmin)(Xk+1glob ,X

k+1,0loc ) > 0

TRUE FALSE

Solve local problem with

starting value Xk+1,0loc

starting value search → ∆Xk+1loc

successful

NO YES

Xk+1,0loc += ∆Xk+1

loc

solve local problem with

starting value Xk+1,0loc

successful

YES NO

∆Xkglob*= 0.5

Xk+1glob

-= ∆Xkglob

Xk+1,0loc

-= ∆Xk+1loc

A = 1

converged

YES NO

A = 1 ∆Xkglob*= 0.5

Xk+1glob -= ∆Xk

glob

Figure 6.4: Algorithm for employing the global Newton update using startingvalue search

Instead of taking the valueXk,endloc as the initial value for solving the local problems

in the k + 1-th Newton step, an extrapolation based on a linearization of the

resolution function can be used:

Xk+1,0loc = Xk

loc +∂Xloc(X

kglob,X

kloc)

∂Xglob

·∆Xkglob .

Then, fewer steps are typically necessary to reach the stopping criterion of the


local Newton iteration. On the other hand, our numerical experiments showed

that the number of grid points where starting value search is necessary increases

if this extrapolation is used. In our code, the use of the extrapolation is optional

and can be enabled/disabled in the script file.

6.4.2 Gas phase appearance and disappearance

−1.0 · 106

−0.80 · 106

−0.60 · 106

−0.40 · 106

−0.20 · 106

0.00 · 106

0.20 · 106

−1.0 · 1060 2 · 105 4 · 105 6 · 105 8 · 105 1060

time [years]

p c[Pa]

−1.0 · 106

−0.80 · 106

−0.60 · 106

−0.40 · 106

−1.0 · 1060 2 · 105 4 · 1050

Figure 6.5: Evolution of pc at Γin

One benefit of using pc as a global unknown is the fact that the saturations sg and

s` depend solely on that variable. This does not only simplify the evaluation of

the resolution function, but can be further exploited to improve the convergence

of the global Newton solver by employing a suitable cutting strategy when the

gas phase appears. The appearance of the gas phase in a grid point may cause

problems for the global Newton solver. Figure 6.5 shows the evolution of pc at

the inflow for the numerical example considered in Section 7.2 (hard test case).

Before the gas phase starts to appear, the variable pc increases almost linearly.

Once the gas phase has appeared, however, it remains approximately constant.

Therefore, in the vicinity of the the phase transition point, the Newton solver

may dramatically overestimate the update for pc, resulting in an unphysical jump

of the gas phase saturation from 0 to a value close to 1. As a consequence, very

small time steps may be required to reproduce the phase transition correctly


by the numerical solver. The cutting strategy proposed in [dC15] detects the

appearance of the gas phase by testing in each grid point and in every Newton

step if the entry pressure threshold has been succeeded (for the first time) by the

variable pc. If this is the case, it is manually set to a value closely above the entry

pressure. The positive impact of this cutting strategy on the convergence of the

global Newton solver was demonstrated in [dC15].

6.4.3 The local problems

The resolution function Xglob 7→ Xloc is evaluated numerically by solving the

algebraic equations (3.74)–(3.84) for the local unknowns. With our choice of

primary variables, this problem decomposes into smaller subproblems that can

be solved in several sequential steps.

(1) First, the variable sg is calculated from the equation (3.76). If the retention

curve sg 7→ pc(sg) admits an explicit inverse p−1c (this applies for the van

Genuchten–Mualem and the Brooks–Corey relations), we obtain

sg(pc) =

0 if pc ≤ pc(0) ,


(2) As soon as the saturation s` = 1− sg is known, equation (3.75) can be used

to determine ηs explicitly by means of

ηs =1

s`(pc)s`(0)ηs(0) .

Note that this is always possible since our model excludes the case that the

liquid phase completely vanishes at any point of the domain.


(3) In the next step, the nonlinear system

φmob(c`) = 0 , (6.1)

φsorp(c`, cs,nmin) = 0 , (6.2)

φmin(c`, cmin) = 0 , (6.3)


olds,kin)

∆t= φs`As,kinRkin(c`, cs,nmin) (6.4)

is solved for the unknowns ξ`,mob, ξs,sorp, ξs,min and ξs,kin using Newton’s

method (see below). Afterwards, the concentrations c`, cs,nmin and cs,min

can be computed with the help of the retransformation (3.41).

(4) Finally, the defining equation of pc provides the liquid pressure,

p` = pg − pc ,

and the molar and mass densities are reconstructed from (3.81)–(3.84).

When the nonlinear system (6.1)–(6.4) is solved in terms of the transformed

variables, the linear systems arising in each Newton step may be ill-conditioned,

e. g., when the concentration values vary over a wide range of magnitudes. For this

reason, the logarithms of the variables should be used as unknowns. However, this

is not immediately possible since the equilibrium conditions contain the logarithm

of a sum, e. g.,

0 = Kmob + ST`,mob ln(c`) = Kmob + ST`,mob ln(S`,exξ`,ex + . . .+B⊥` η` ) .

To tackle this problem, the local problems are solved in terms of the concentra-

tions c` and cs instead of the transformed variables, cf. [Hof10]. If we replace the

transformed variables ξ`,mob, ξs,sorp and ξs,min by the concentrations, the number

of unknowns is increased by I`+Is−(Jmob +Jsorp +Jmin). Hence, additional equa-

tions are needed to close the system. Therefore, the defining equations for the

global variables η`, ηs, ξ`,ex, ξ`,kin, ξs,kin, ξsorp and ξmin are added to (6.1)–(6.4).


Indeed, the number of these defining equations is

I` − (Jex + Jmob + Jsorp + Jmin + J∗`,kin) + Is − (Jsorp + Jmin + J∗s,kin)

+ Jex + Jsorp + Jmin + J∗`,kin + J∗s,kin

= I` + Is − Jmob − Jsorp − Jmin ,

which corresponds to the number of additional unknowns. Then, in the local

problems, the variables c`, cs and ξs,kin are determined by solving the extended

system

φmob(c`) = 0 ,

φsorp(c`, cs,nmin) = 0 ,

φmin(c`, cs,min) = 0 ,

η` = (S⊥`TB⊥` )−1S⊥`

Tc`

ηs = (S⊥sTB⊥s )−1S⊥s

Tcs

ξ`,ex =((BT

` S∗`)−1BT

` c`)i=1,...,Jex

ξ`,kin =((BT

` S∗`)−1BT

` c`)i=Jeq+1,...,Jeq+J∗`,kin

ξsorp =((BT

` S∗`)−1BT

` c`)i=Jex+Jmob+1,...,Jex+Jmob+Jsorp

−((BT

s S∗s)−1BT

s cs)i=1,...,Jsorp

ξmin =((BT

` S∗`)−1BT

` c`)i=Jex+Jmob+Jsorp+1,...,Jeq

− cs,min

ξs,kin =((BT

s S∗s)−1BT

s cs)i=Jsorp+Jmin+1,...,Jsorp+Jmin+J∗s,kin


olds,kin)

∆t= φs`As,kinRkin(c`, cs,nmin) .

Now, the logarithms of the concentrations can be used as unknowns in the equi-

librium laws. Defining l` := ln(c`) and ls,nmin := ln(cnmin), the system translates


into

φmob(l`) = 0 , (6.5)

φsorp(l`, ls,nmin) = 0 , (6.6)

φmin(l`, cs,min) = 0 , (6.7)

η` = (S⊥`TB⊥` )−1S⊥`

Texp(l`) (6.8)

ηs = (S⊥sTB⊥s )−1S⊥s

T(

exp(ls,nmin)cs,min

)(6.9)

ξ`,ex =((BT

` S∗`)−1BT

` exp(l`))i=1,...,Jex

(6.10)

ξ`,kin =((BT

` S∗`)−1BT

` exp(l`))i=Jeq+1,...,Jeq+J∗`,kin

(6.11)

ξsorp =((BT

` S∗`)−1BT

` exp(l`))i=Jex+Jmob+1,...,Jex+Jmob+Jsorp

−(

(BTs S∗s)−1BT

s

(exp(ls,nmin)cs,min

))i=1,...,Jsorp

(6.12)

ξmin =((BT

` S∗`)−1BT

` exp(l`))i=Jex+Jmob+Jsorp+1,...,Jeq

− cs,min

(6.13)

ξs,kin =(

(BTs S∗s)−1BT

s

(exp(ls,nmin)cs,min

))i=Jsorp+Jmin+1,...,Jsorp+Jmin+J∗s,kin

(6.14)


olds,kin)

∆t= φs`As,kinRkin(exp(l`), exp(ls,nmin)) (6.15)

Note that due to the structure of S∗s and Bs, the variables ηs, ξs,kin and ξsorp do

not depend on cs,min, cf. (3.6) and (3.9). Consequently, the variable cs,min appears

only in the blocks (6.7) and (6.13). It was shown in [Hof10] that this fact can be

exploited to further reduce the system by eliminating the concentrations cs,min

and reconstructing them according to the explicit representation of (6.13) (in the

case that the mineral is present) or by setting cks,min = 0 (if it is not present).

Accordingly, either the k-th equation of block (6.13) or the k-th equation of block

(6.7) can be omitted. This is possible as long as each mineral concentration occurs

only once in the system (6.5)–(6.15). If kinetic mineral reactions are included

into the model, however, the mineral concentrations additionally occur in the


For each mineral reaction k

ϕmin,k(lℓ) > cmin,k

TRUE FALSE

A[k] = 0 A[k] = 1

cks,min = 0 ∅

Assemble local defect (6.5)–(6.15) without (6.7)

For each mineral reaction k

A[k]

0 1

def[Jmob + Jsorp + k] = ϕmin,k(lℓ)def[Jmob + Jsorp + k] = 0

Figure 6.6: Detailed algorithm for assembling the local defect

terms resulting from the discretization of their time derivative, cf. Section 6.4.4.

Therefore, the above elimination is no longer possible. Since the code written for

this thesis includes the case of kinetic mineral reactions, the full system (6.5)–

(6.15) is solved.

When the defect of the system (6.5)–(6.15) is calculated, it must be checked for

each mineral reaction whether the corresponding mineral is present in the next

iteration step or not. If the minimum in (2.33) is attained in the first argument,

the mineral will be present, whereas if it is attained in the second argument, it

will not be present. In the code written as part of this work, the result is stored

in a vector A, where A[k] = 1 corresponds to the case that the k-th mineral is

present and A[k] = 0 means that it is absent. The algorithm for calculating the

local defect is summarized in Figure 6.6. After solving the system (6.5)–(6.15),

the transformed variables ξ`,mob and ξs,sorp are calculated with the help of their

definitions

ξ`,mob =((BT

` S∗`)−1BT

` c`)i=Jex+1,...,Jex+Jmob

, ξs,sorp =((BT

s S∗s)−1BT

s cs)i=1,...,Jsorp

,


and the transformed variable ξs,min is equal to cs,min. It should be noted, however,

that ξ`,mob is not needed to assemble the defect associated with the global system

(3.65)–(3.73).

When the system (6.5)–(6.15) is solved using Newton’s method, it may happen

that the Jacobian becomes numerically singular. This is the case if a global

variable is defined as a linear combination of concentrations where all coefficients

have the same sign, and if the value of this variable is close to zero. This is because

it holds that ∂ exp(l`)∂l`

= exp(l`) = c`, i. e., the line in the Jacobian corresponding

to the defining equation of this global variable has entries that are close to zero.

Following [Hof10], we do not solve the linear system J∆x = d to compute the

Newton update ∆x, but the minimization problem (assume for a moment that

J ∈ R(n,n),d ∈ Rn)

Minimize ‖∆x‖22 on

L(d) := z ∈ Rn : ‖Jz − d‖22 ≤ ‖Jy − d‖2

2 for all y ∈ Rn .

This problem admits a unique solution which can be computed with the help

of a QR factorization for rank-deficient matrices. If column pivoting is applied,

the rows of the Jacobian that are numerically singular are detected, and it is

ensured that an error is introduced only in those equations associated with these

rows, i. e., all other equations are solved exactly. The order of magnitude of the

total error that is created by solving the minimization problem instead of the

linear system lies in the range of the machine precision. The detailed algorithm

is described in [Hof10].

6.4.4 Kinetic mineral reactions

Kinetic reactions involving minerals require some special treatment since pure

solids are typically assumed to have constant activity. Consequently, if the mass

action law is used to model the rate function of a kinetic mineral reaction, the

reaction rate does not depend on the concentration of the associated mineral.

This is in contrast to the kinetic reactions between nonminerals considered in

Section 2.3.1. Consider, for example, the reaction

A+ 2B ←→M ,


where A and B represent two components of the liquid phase, and M is a mineral.

The reaction rate according to the mass action law is given by

R(cA` , cB` ) = kfc

A` (cB` )2 − kb , (6.16)

where cA` and cB` denote the concentrations of A and B in the liquid phase.

Assuming that there is no gas phase, the equations describing the evolution of

the concentrations of A, B and M read

∂t(φcA` ) + L`cA` = −φR(cA` , c

B` ) ,

∂t(φcB` ) + L`cB` = −2φR(cA` , c

B` ) ,

∂t(φcMs ) = φR(cA` , c

B` ) .

Obviously, the rate law (6.16) cannot be valid in the case that

cMs = 0 ∧ kfcA` (cB` )2 − kb < 0 ,

since a negative rate R would imply a negative mineral concentration cMs . A well-

known strategy to comply with the nonnegativity of the mineral concentration

is the use of a set valued rate function, cf. [vDK97, vDKS98, KvDH95, vDP04].

For the above example, this approach reads

∂t(φcMs ) = φ(kfc

A` (cB` )2 − wkb) , (6.17)

w ∈ H(cMs ) , (6.18)

where H denotes the set-valued Heaviside “function”,

H(u) =

1 if u > 0 ,

[0, 1] if u = 0 ,

0 if u < 0 .


In [Hof10], a complementarity approach was proposed to treat kinetic mineral

reactions, which reads

cMs (∂t(φcMs )− φ(kfc

A` (cB` )2 − kb)) = 0 , (6.19)

cMs ≥ 0 , ∂t(φcMs )− φ(kfc

A` (cB` )2 − kb) ≥ 0 . (6.20)

It can be shown that the weak formulations of (6.17)–(6.18) and of (6.19)–(6.20)

are equivalent, cf. [Hof10, Theorem 5.2]. While the first one is well suited for the-

oretical considerations, the complementarity approach is preferred for numerical

computations since it admits an equivalent reformulation as an equation using

the minimum function,

mincMs , ∂t(φcMs )− φ(kfcA` (cB` )2 − kb) = 0 .

The complementarity approach ensures the nonnegativity of the mineral concen-

tration, and it does not require a regularization parameter, which is typically

used to discretize problems involving set-valued functions, see, e. g., [DPvDC08].

In the following, it is shown how the complementarity approach can be incor-

porated into our model. If cjs,min represents the j-th mineral and if this mineral

participates in a kinetic mineral reaction, the equation describing the evolution

of cjs,min reads

φmin,j(c`, cjs,min) := mincjs,min, ∂t(φs`c

js,min)− φs`Rmin,j(c`) = 0 , (6.21)

where Rmin,j denotes the reaction rate depending on the concentrations of the

components of the liquid phase,

Rmin,j(c`) = kf,j

I∏i=1sij<0

ai(c)−sij − kb,j

I∏i=1sij>0

ai(c)+sij .

In its fully discrete form, (6.21) yields a nonlinear algebraic equation that can be

treated in the same way as the equilibrium conditions resulting from equilibrium

reactions. More precisely, if the j-th mineral reaction is kinetic, (6.21) it is incor-

porated as the j-th equation in the block (3.79) of algebraic equations associated

with equilibrium minerals, which is in contrast to the kinetic reactions without

minerals, the rates of which appear on the right hand side of (3.66)–(3.69).

Chapter 7

Numerical results

7.1 The MoMaS benchmark on multiphase flow

In the first numerical example, we consider a benchmark problem related to the

migration of hydrogen in geological formations, which are discussed as potential

storage sites for radioactive waste. The storage concept is based on the presence of

multiple barriers that ensure storage security. On the one hand, there is the deep

rock environment which is required to be stable, low permeable and unaffected by

environmental change for a long time period (hundreds of thousands of years). On

the other hand, there are engineered barriers like the container of the radioactive

material, or a buffer or backfill material filling the space between the container

and the rock. One potential risk of this storage option is the generation of

highly explosive hydrogen gas due to corrosion, and a pressure buildup. As a

consequence, the engineered barriers or even the host rock may be perturbed,

and the waste material may be released.

The MoMaS benchmark on multiphase flow [MoM10, BGS13] was proposed in

order to improve the simulation of hydrogen migration in deep geological nuclear

waste storage sites. We recompute here the first test case which addresses the

problem of gas phase appearance and disappearance during a long-term injec-

tion process of hydrogen into a homogeneous and low-permeable porous medium.

The computational domain Ω is a rectangle of length 200m and width 20m, cf.

Figure 7.1. The domain is initially fully saturated with water, and hydrogen is

injected through the left part of the domain for a time period of Tinj = 5 · 105

131

132 CHAPTER 7. NUMERICAL RESULTS

Γin ΓD

Γimp

Γimp

Ω

Figure 7.1: Computational domain for the MoMaS benchmark problem

years. Since gravity is neglected, the problem is a pseudo 1D problem.

7.1.1 Parameters and setup

The hydrogen migration is driven by a two-phase two-component model, where

the first component is water and the second component is pure hydrogen. The

liquid phase may contain water and hydrogen, whereas the gas phase consists

only of hydrogen. We converted the benchmark description to match the formu-

lation in this work. In particular, molar concentrations are used instead of mass

concentrations.

Governing equations

The governing equations of the benchmark are

∂t(φs`c2`) + L`c2

` = 0 , (7.1)

∂t(φs`c1` + φsgρmol,g) + L`c1

` + Lgρmol,g = 0 , (7.2)

where c1` represents the molar concentration of hydrogen and c2

` is the molar

concentration of water in the liquid phase. The variable transformation for this

system was presented in Section 3.4.4 and yields the transformed variables η1` = c2

`

and ξ1`,ex = c1

` . If we use pc and ξ1`,ex as primary unknowns, our formulation

corresponds well to one of the formulations proposed in [NBI13] for the special

case of two-phase two-component flow.

7.1. THE MOMAS BENCHMARK ON MULTIPHASE FLOW 133

Initial and boundary conditions

Initially, the domain is liquid-saturated, and there is no hydrogen dissolved in

the liquid phase. Accordingly, the initial conditions read

p`(·, 0) = p0` , c1

`(·, 0) = 0 .

Note that the initial and boundary conditions must be converted to the specific

choice of primary variables used by our code. In this work, the extended pressure

pc and the transformed variable ξ1`,ex = c1

` were used as primary unknowns, and

were initialized by the values pc(·, 0) = −p0` and ξ1

`,ex(·, 0) = 0.

Denoting the hydrogen total flux and the water total flux by

Q1 = c1`q` + ρmol,gqg + j1

` , Q2 = c2`q` + j2

` ,

respectively, the following boundary conditions are imposed on ∂Ω = Γin∪ΓD∪Γimp:

1. No-flux on Γimp:

Q1 · n = 0 , Q2 · n = 0 ,

where n denotes the outer unit normal vector of Γimp.

2. Hydrogen injection on Γin:

Q2 · n = 0 , and Q1 · n =

q1in if t ≤ Tinj ,

0 otherwise .

3. Pure liquid water and a fixed pressure on ΓD:

p` = p`,out , c1` = 0 .

Closure relationships

Let us now choose the solubility function φex and phase compressibility laws f`

and fg consistently with the benchmark description. The hydrogen interphase


mass exchange follows Henry’s law, which is represented by the choice

φex(c`, pg) := Hpg − c1` .

For the compressibility laws, we set

f`(c`) := c1` + c2,std

` , (7.3)

fg(pg) :=pgRT

, (7.4)

where c2,std` denotes the molar density of water at standard conditions, T is the

(constant) temperature and R the universal gas constant. Note that (7.3) and

the constraint ρmol,` = c1` + c2

` are consistent with the condition c2` = c2,std

` from

the benchmark description, i. e., the concentration of water in the liquid phase

remains constant. The diffusion–dispersion tensor reads

Diα = φs`D

idiff,αI .

Finally, the assumption

j1` + j2

` = 0

is used to express the water diffusive flux in terms of the hydrogen diffusive flux.

Hence, only the diffusion coefficient D1diff,` must be specified. Finally, capillary

pressure and relative permeabilities are given by the van Genuchten–Mualem

model, cf. (2.9)–(2.11). The parameters of this test case are listed in Table 7.1.

Note that the molar mass of water was incorrect in the benchmark description.

The physically correct value is 1.8 · 10−2 kg mol−1.

The computations were carried out on a rectangular grid consisting of 160×16 =

2560 squares. Figure 7.2 shows the evolution of the gas phase saturation and

the phase pressures at the left boundary over time. Initially, the domain is

fully water-saturated. During the first injection period, all injected hydrogen

dissolves into the liquid phase and is transported by advection and diffusion.

Meanwhile, the liquid pressure remains almost constant. At t ≈ 13000 years,

the gas phase starts to appear and the gas phase saturation and phase pressures

start to increase. Between 105 years and 5 · 105 years, hydrogen is still injected

and the gas phase saturation keeps growing, while the phase pressures start to


variable symbol value SI unit

porosity φ 0.15 [-]

absolute permeability K 5 · 10−20 m2

temperature T 303 K

Van Genuchten parameter s`,res 0.4 [-]

Van Genuchten parameter sg,res 0.0 [-]

Van Genuchten parameter n 1.49 [-]

Van Genuchten parameter α 5 · 10−7 Pa

Henry constant (T = 303K) H 7.65 · 10−6 mol m−3 Pa−1

molar mass of hydrogen M1 2 · 10−3 kg mol−1

molar mass of water M2 10−2 kg mol−1

initial value p0` 106 Pa

diffusion coefficient D1diff,` 3 · 10−9 m2 s−1

hydrogen injection rate q1in 1.77 · 10−13 kg m−2 s−1

viscosity of the liq. phase µ` 10−3 Pa.s

viscosity of the gas phase µg 9 · 10−6 Pa.s

water standard density c2,std` 105 mol m−3

Table 7.1: Parameters of the MoMaS benchmark problem (test case 1)

decrease after reaching their maximum value at t ≈ 91000 years and t ≈ 156000

years. After the end of the gas injection, the gas is progressively distributed

due to advective and diffusive transport, and the gas phase starts to disappear.

This imposes a flux of water from the right to the left in order to quickly replace

the volume occupied by the gas phase, cf. [ACC+11]. Consequently, the liquid

pressure drops below the value that is imposed on the right boundary, before it

grows once more to reach its initial value.

Our computational results are in good agreement with the results reported by

other groups, cf. [BGS13]. It should be noted, however, that different formula-

tions may lead to different outcomes for the gas pressure variables in those parts

of the domain where the gas phase is not present. This is because different exten-

sions/definitions of pg are used in monophasic region. In our case, the extended

gas pressure pg was defined to satisfy the solubility law (in this particular case:

Henry’s law) regardless of the phase state.

The good correspondence to the results of other groups shows that our discretiza-


0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.0000 2 · 105 4 · 105 6 · 105 8 · 105 1060

time [years]

saturation

[-]

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.0000 2 · 105 4 · 1050

sg

0.70 · 106

0.80 · 106

0.90 · 106

1.00 · 106

1.10 · 106

1.20 · 106

1.30 · 106

1.40 · 106

1.50 · 106

0.70 · 1060 2 · 105 4 · 105 6 · 105 8 · 105 1060

time [years]

pressure

[Pa]

0.70 · 106

0.80 · 106

0.90 · 106

1.00 · 106

1.10 · 106

0.70 · 1060 2 · 105 4 · 1050

pgpl

Figure 7.2: Evolution of gas phase saturation and pressures at Γin

tion and the code written in the framework of this thesis provide accurate results.

In particular, our formulation is capable of handling gas phase appearance and

disappearance. Although the two-phase two-component model studied in this

benchmark represents only a special case of the full reactive multiphase mul-

ticomponent model considered in this work, it contains all relevant transport

mechanisms of the full model.

7.1.2 Comparison of GIA and SIA

Since multiphase flow and multicomponent reactive transport models involve

large nonlinear and fully coupled systems of partial differential equations, ordi-

nary differential equations and algebraic equations, many codes utilize a splitting

approach to solve them numerically [HLL07, WF12, HSN12, XSS+12, AKV15].

Typically, the global problem is decomposed into a flow problem and a reactive

transport problem, which are solved sequentially. More precisely, in a first step,

pressures and saturations are determined as the solution of a two-phase two-

component flow problem for two dominant components (e. g., water and CO2),

neglecting the other components and their chemical reactions. In a second step,

the calculated pressures and the resulting Darcy velocities are used to update the

concentrations by solving a reactive transport problem. This approach is referred

to as sequential non-iterative approach (SNIA) or operator splitting. In doing so,

one can benefit from the availability of customized solvers for specific subprob-

lems, which can be easily coupled or combined. For example, the flow module


PFLOW and the transport module PTRAN were merged into the multiphase

multicomponent reactive transport code PFLOTRAN, cf. [HLL07].

One drawback of SNIA is a splitting error, which may become large when the

physics are tightly coupled. Moreover, by neglecting chemical reactions of the

dominant components during the solution of the flow problem, an additional

modeling error is introduced. In the case of CO2 injection, this assumption may

not be justified since CO2 itself is reactive.

To eliminate the splitting error, the sequential iterative approach (SIA) can be

used, where the subproblems are solved alternately until they both converge

within prescribed tolerances. This may require many iterations in each time

step to meet the convergence criteria, in particular if fast kinetics and strong

chemical interactions are involved. When the iteration has converged, the solution

represents a solution of the global nonlinear problem.

While in the 1980s, Yeh and Tripathi [YT89] conclude that SIA is preferable,

the availability of increasing computational resources has attracted growing in-

terest in the global implicit approach for the simulation of reactive transport

problems during the past two decades [dDE10, AK10, KK05, KK07, Hof10]. In

the MoMaS benchmark on reactive transport, global methods proved to be very

competitive to other methods, and the code of Hoffmann et. al. showed equiv-

alent or lower CPU times than required by all other codes, cf. [CHK+10]. Very

recently, the global implicit approach has also been applied to coupled flow and

reactive transport problems, cf. [FDT12, SVG+13].

In this work, the global implicit approach is used, which means that all primary

variables are computed simultaneously during each time step by employing New-

ton’s method to the global system of nonlinear equations which results after the

model-preserving variable transformation has been employed. To demonstrate

that the resulting method is efficient and competitive to SIA, we recompute the

MoMaS benchmark problem from the previous section using SIA. For this pur-

pose, a sequential iterative approach is defined for the MoMaS benchmark by

decoupling the equations (7.1) and (7.2). More precisely, the k-th iteration step

associated with time step n (note that the index n and the discretization param-

eter h are omitted for a tighter notation) consists of two subproblems:


Subproblem 1

In the first step, the pressure variable pkc is determined by solving the equation

∂t(φs`c2`) + L`c2

` = 0 ,

where the secondary variables s`, c2` and p` (needed in the transport operator)

are expressed in terms of pkc with the help of the resolution function by means of

c2`(p

kc ) = c2,std

` ,

s`(pkc ) =

0 if pc < pc(0) ,

p−1c (pkc ) otherwise ,

p`(pkc ) =

c1,k−1`

H− pkc .

Subproblem 2

In the second step, pkc is known and c1,k` is determined by solving

∂t(φs`c1,k` + φsgρmol,g) + L`c1,k

` + Lgρmol,g = 0

with the secondary unknowns s`, sg, pg, ρmol,g and p` defined as

s` = s`(pkc ) , sg = 1− s` , pg(c

1,k` ) =

c1,k`

H,

ρmol,g(c1,k` ) =

c1,k`

HRT, p`(c

1,k` , pkc ) =

c1,k`

H− pkc .

The iteration is continued until the residuals corresponding to both equations

are within a given tolerance εabs for the pair (pkc , c1,k` ). Both subproblems are

treated fully implicitly, and the resulting nonlinear problems are linearized using

Newton’s method. In each time step, the time step size is adapted depending

on the total number N of Newton steps that were carried out in the previous

time step (summed over all iteration steps). More precisely, it is multiplied by

the factor N30

as long as the minimum/maximum time step size is not reached.

Moreover, the time step size is halved and the time step is repeated if the iteration

has not converged after 60 iteration steps. The detailed algorithm that is used

for the sequential iterative approach is illustrated in Figure 7.3. Since each of the


subproblems consists only of one equation and is solved for one variable only, the

memory demand is clearly lower and the linear systems that have to be solved

are smaller than the linear systems resulting from the use of the global implicit

approach. Hence, the efficiency will be dominated by the number of iteration

steps that are necessary until both subproblems converge.

To compare the efficiency of SIA and GIA, the MoMaS benchmark is recomputed

using the same mesh as in the previous section and requiring an absolute tolerance

εabs to be reached by both methods. For GIA, pc and c1` are used as primary

variables. Moreover, the control parameter OptSteps=4 (cf. Section 6.2) is used

in the adaptive time stepping algorithm. The results of the performance test for

different values of εabs are displayed in Table 7.2, which shows the number of time

steps and the total number of Newton steps that were required by GIA and SIA

for different values of εabs. Obviously, the fixed point iteration of the sequential

approach converges slowly, resulting in rather poor convergence of SIA for strict

stopping criteria. This is due to a dramatic reduction of time step size by the

adaptive time stepper. On the other hand, for GIA, the global Newton method

converges quadratically and the time step size remains within a reasonable range

of values. The evolution of time step size for both approaches for εabs = 10−10

is illustrated in Figure 7.4. In this case, 15973 Newton steps were carried out by

the sequential approach for solving the subproblems 1 and 2, whereas only 409

Newton steps were necessary for the global implicit one. Although each Newton

step is computationally less expensive using SIA, the computation time was more

than ten times larger than the computation time of GIA. For more complex

problems involving chemical interactions, the difference in the performance of

the two methods is expected be even more significant. We conclude that the

global implicit approach represents an efficient and competitive way of solving

coupled flow and reactive transport problems, in particular if a high accuracy of

the nonlinear solver is required.


k = 0 , d1 = 99 , d2 = 99 , N = 0

while (d1 > εabs or d2 > εabs) and k < 60

k += 1

Solve Subproblem 1 for pk

N1 = Newton iterations required for subproblem 1

Solve Subproblem 2 for c1,kℓ

N2 = Newton iterations required for subproblem 2

N=N1 +N2

d1 = ‖Residual of Subproblem 1 for (pk, c1,kℓ )‖d2 = ‖Residual of Subproblem 2 for (pk, c1,kℓ )‖

d1 < εabs and d2 < εabs

TRUE FALSE

∆t *= N30 ∆t *= 0.5

Proceed to next time step Repeat time step

Figure 7.3: Schematic of the sequential iterative approach for the MoMaS bench-mark problem

0

1 · 1042 · 1043 · 1044 · 1045 · 1046 · 1047 · 1048 · 1049 · 10410 · 104

00 2 · 105 4 · 105 6 · 105 8 · 105 1060

time [years]

∆t[years]

GIASIA

0

1 · 1042 · 1043 · 1044 · 1045 · 1046 · 1047 · 1048 · 1049 · 10410 · 104

00 2 · 1050

b b bbbb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b b b b b b b bb b b

bb b

b b

b

b

b

b

b

b b

b b b b b b b b b bb b b b

b b b b b b b b

b

b

b

b

b

b b bbb

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b b b b b

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b b b b b b b b

b b b b b b b b b

b

b

b

b b

b

Figure 7.4: Evolution of the time step size for SIA and GIA for εabs = 10−10

7.2. INFLUENCE OF DIFFERENT RETENTION CURVES 141

GIA SIA

εabs time st. Newton st. CPU time time st. Newton st. CPU time

10−5 54 206 4.3 min 46 1215 6.4 min

10−6 56 211 4.4 min 75 2060 10.9 min

10−7 67 260 5.9 min 118 3397 17.9 min

10−8 74 299 6.2 min 190 5550 29.5 min

10−9 89 351 7.1 min 324 9591 51.7 min

10−10 101 409 8.4 min 536 15973 86.6 min

Table 7.2: Performance of GIA and SIA for the MoMaS benchmark

7.2 Influence of different retention curves

The performance of the global numerical solver is strongly influenced by the soil

parameters of the problem. In the previous section, hydrogen injection into a very

dense and low-permeable material was simulated, which requires different soil pa-

rameters than CO2 injection into deep saline aquifers consisting of sandstone. To

investigate further the effect of different retention curves on the convergence of

the Newton solver, a simple modification of the MoMaS benchmark considered in

the previous section was proposed in [dC15]. The modified benchmark problem

involves two test cases using two different values of the van Genuchten param-

eter α in the definition of the capillary pressure–saturation curve: the value

α1 = 5 · 10−7 Pa was adopted from the MoMaS benchmark and corresponds to a

very low-permeable rock material, and α2 = 5 · 10−4 Pa represents a sandstone

aquifer. The resulting retention curves are shown in Figure 7.5.

Governing equations

The system is governed by the conservation equations (7.1)–(7.2), where diffusion

of component 1 (hydrogen) in the liquid phase is now expressed in terms of the

gradient of its molar concentration,

j1` = −D1

`∇c1` .


0.0 · 108

0.5 · 108

1.0 · 108

1.5 · 108

2.0 · 108

0.0 · 1080.0 0.2 0.4 0.6 0.8 1.00.0

effective liquid phase saturation Sℓ,e [-]

capillary

pressure

p c[Pa]

0.0 · 108

0.5 · 108

1.0 · 108

0.0 · 1080.0 0.2 0.40.0

α1 = 5 · 10−7

α2 = 5 · 10−4

Figure 7.5: Retention curves of the modified MoMaS benchmark problem

The diffusion-dispersion tensor of hydrogen in the liquid phase follows an ap-

proach of Jin and Jury [JJ96] and is given by

D1` = φ

43 s2`D

1diff,`I .

For the diffusive flux of water in the liquid phase, the relation j2` = −j1

` is used.


The computational domain is initially liquid-saturated with no hydrogen being

dissolved in the liquid phase. The initial conditions read

p`(·, 0) = p0` , c1

`(·, 0) = 0 .

As in the MoMaS benchmark, hydrogen is injected through the left part of the

computational domain at a constant rate q1in at Γin. At Tinj = 5 · 105 years, the

injection is stopped and the simulation is continued until the final time Tend =

106 years. Accordingly, the the following boundary conditions are imposed on

∂Ω = Γin∪ΓD∪Γimp:



Q1 · n = 0 , Q2 · n = 0 .

2. Hydrogen injection through Γin:

Q2 · n = 0 , Q1 · n =

q1in if t ≤ Tinj ,

0 otherwise .

3. Pure liquid water and a fixed liquid pressure on ΓD:

p` = p`,out , c1` = 0 .

The computational domain Ω = (0, 200) of this benchmark problem is one dimen-

sional, but since our code was written for 2D and 3D simulations, we consider it

as a pseudo 1D problem and run the computations on a thin stripe consisting of

N rectangles, cf. Figure 7.6. On the upper and lower boundary, no flux conditions

are imposed, and on the right boundary, Dirichlet conditions are imposed for c1`

and p`.

(0, 0) (200, 0)

(200, 1)(0, 1)

T1 T2 T2 TN. . .Γin ΓD

Γimp

Γimp

Figure 7.6: Computational mesh for the modified MoMaS benchmark problem

Closure relationships

For the hydrogen interphase mass exchange, Henry’s law is employed, i. e.,

φex(c`) := Hpg − c1` ,


and the compressibility laws are given by the functions

f`(c`) := c2,std` ,

fg(pg) :=pgRT

,

where the constant c2,std` denotes the molar density of water at standard condi-

tions. As a consequence, the liquid phase has constant molar concentration (but

varying mass density) in this example. All parameters associated with this test

problem are listed in Table 7.3. Since the simulation turned out to be much more

challenging using the parameter α2, we refer to this case as the “hard” test case,

whereas the use of the parameter α1 is referred to as the “easy” test case.


porosity φ 0.15 [-]

absolute permeability K 5 · 10−20 m2

temperature T 303 K

Van Genuchten parameter s`,res 0.4 [-]

Van Genuchten parameter sg,res 0.0 [-]

Van Genuchten parameter n 1.49 [-]

Van Genuchten parameter α1 5 · 10−7 Pa

Van Genuchten parameter α2 5 · 10−4 Pa

Henry constant (T = 303K) H 7.65 · 10−6 mol m−3 Pa−1

molar mass of hydrogen M1 2.0 · 10−3 kg mol−1

molar mass of water M2 1.8 · 10−2 kg mol−1

initial value p0` 106 Pa

diffusion coefficient D2diff,` 3 · 10−9 m2 s−1

diffusion coefficient D1diff,g 0 m2 s−1

hydrogen injection rate q1in 1.77 · 10−13 kg m−2 s−1

viscosity of the liq. phase µ` 10−3 Pa.s

viscosity of the gas phase µg 9 · 10−6 Pa.s

standard water density c2,std` 55555.56 mol m−3

Table 7.3: Parameters of the modified benchmark problem


Grid convergence

A grid convergence study is carried out for both test cases. For this purpose, the

numerical solution of the problem is calculated on different levels of refinement,

where the number of rectangles in x direction is doubled in each refinement step,

resulting in a grid consisting of N = 100 · 2i cells on level i. As in the previous

section, pc and c1` are used as primary variables. Since an analytical solution of

the problem is not known, the solution on level i = 10 (N = 102400) is used as

a reference solution. It is illustrated in Figures 7.7, where the evolution of the

gas phase saturation and the evolution of the pressures at the inflow boundary

are shown. On each level of refinement, the experimental order of convergence is

determined by

EOCi+1 =1

log(2)| log

(eiei+1

)| ,

where ei denotes the L2 error between the solution on level i and the reference

solution. The relative L2 error ereli is defined as the ratio between ei and the L2

norm of the reference solution. All computations are run with a constant time

step size ∆t = 0.5 years, which is chosen sufficiently small to ensure that the

time discretization error is negligible compared to the space discretization error.

The results of the convergence study are listed in Tables 7.4 and 7.5. For the

easy test case, first order convergence is observed for the variables sg, p` and c1` ,

which is optimal for the upwind LFEM–FV scheme. For the hard test case, the

convergence rate is slightly lower for the variable sg. This is in accordance with

the results reported in [dC15].

Performance

In addition to the grid convergence study, this benchmark problem was proposed

to enable a comparison of the performance of different numerical solvers. For

that purpose, we recompute both test cases using the adaptive time stepping

strategy described in Section 6.2 and the control parameters of Table 7.6. In


0.000

0.005

0.010

0.015

0.020

0.0000 2 · 105 4 · 105 6 · 105 8 · 105 1060

time [years]

saturation

[-]

0.000

0.005

0.010

0.015

0.020

0.0000 2 · 105 4 · 1050

sg

0.70 · 106

0.80 · 106

0.90 · 106

1.00 · 106

1.10 · 106

1.20 · 106

1.30 · 106

1.40 · 106

1.50 · 106

0.70 · 1060 2 · 105 4 · 105 6 · 105 8 · 105 1060

time [years]

pressure

[Pa]

0.70 · 106

0.80 · 106

0.90 · 106

1.00 · 106

1.10 · 106

0.70 · 1060 2 · 105 4 · 1050

pgpℓ

0.00

0.02

0.04

0.06

0.08

0.10

0.000 2 · 105 4 · 105 6 · 105 8 · 105 1060

time [years]

saturation

[-]

0.00

0.02

0.04

0.000 2 · 105 4 · 1050

sg

0.70 · 106

0.80 · 106

0.90 · 106

1.00 · 106

1.10 · 106

1.20 · 106

1.30 · 106

1.40 · 106

1.50 · 106

0.70 · 1060 2 · 105 4 · 105 6 · 105 8 · 105 1060

time [years]

pressure

[Pa]

0.70 · 106

0.80 · 106

0.90 · 106

1.00 · 106

1.10 · 106

0.70 · 1060 2 · 105 4 · 1050

pgpℓ

Figure 7.7: Evolution of gas phase saturation and pressures at Γin for the easytest case (top) and the hard test case (bottom)

sg p` c1`

i N ereli EOCi erel

i EOCi ereli EOCi

0 100 1.78e-02 3.26e-04 9.02e-03

1 200 9.21e-03 0.95 1.92e-04 0.77 4.66e-03 0.95

2 400 4.67e-03 0.98 9.18e-05 1.06 2.36e-03 0.98

3 800 2.35e-03 0.99 4.74e-05 0.95 1.18e-03 0.99

4 1600 1.16e-03 1.01 2.31e-05 1.04 5.97e-04 0.99

5 3200 5.70e-04 1.03 1.13e-05 1.03 2.98e-04 1.00

6 6400 2.80e-04 1.03 5.61e-06 1.01 1.41e-04 1.08

7 12800 1.31e-04 1.10 2.62e-06 1.10 6.58e-05 1.10

Table 7.4: Relative L2-errors and experimental orders of convergence (EOC) forthe easy test case (α1 = 5 · 10−7) at t = 105 years.


sg p` c1`

i N ereli EOCi erel

i EOCi ereli EOCi

0 100 7.72e-2 1.68e-03 2.99e-02

1 200 4.88e-2 0.66 1.13e-03 0.57 1.71e-02 0.81

2 400 3.16e-2 0.63 5.82e-04 0.95 9.64e-03 0.82

3 800 2.11e-2 0.59 3.33e-04 0.81 5.33e-03 0.85

4 1600 1.38e-2 0.60 1.94e-04 0.78 2.89e-03 0.88

5 3200 8.97e-3 0.63 1.12e-04 0.80 1.52e-03 0.93

6 6400 5.50e-3 0.70 5.37e-05 1.06 7.71e-04 0.98

7 12800 3.15e-3 0.80 2.64e-05 1.03 3.72e-04 1.05

Table 7.5: Relative L2-errors and experimental orders of convergence (EOC) forthe hard test case (α2 = 5 · 10−4) at t = 105 years.

Tables 7.7 and 7.8, we list the number of time steps and Newton iterations (suc-

cessful and failed) that were necessary to compute the numerical solution on

different levels of refinement. For these computations, the linear extrapolation

of the previous time steps described in Section 6.2 was disabled and the value of

the previous time step was used as an initial guess for the global Newton method

at each time step.

Clearly, a significant higher number of Newton steps is required for the hard

test case than for the easy test case. More precisely, the number of Newton

steps almost doubles in each refinement step for the hard test case, whereas it

remains approximately constant for the easy test case. This is in accordance

with the results reported by R. de Cuveland and P. Bastian from the University

of Heidelberg [dC15], cf. Tables 7.9 and 7.10 . They use capillary pressure and

gas pressure as primary unknowns and discretize the equations with an Euler-

implicit cell-centered finite volume scheme. The nonlinear systems of equations

are solved with an inexact Newton method using the same stopping criterion as

in this work.

Although we solved a 2D problem with approximately twice as many unknowns as

for the corresponding 1D problem, our solver needed fewer time steps and Newton

iterations on all refinement levels. One of the reasons for this might be the fact

that we use the analytical Jacobian in our Newton method. Thus, no numerical

differentiation is required, which may lead to inaccurate approximations of the


control parameter value

εrel 10−6

εabs 0.0

max. line search steps 3

max. Newton iterations 30

∆tmin 1 year

∆tmax 106 years

OptSteps 10

Table 7.6: Control parameters for the performance test

Jacobian. Moreover, our time stepping algorithm and line search strategy is

different from the ones used in [dC15], which may have a strong impact on the

global performance. The choice of primary variables, however, seems to have

only little influence on the performance for this test problem. In further numerical

tests, we obtained the identical number of time steps and Newton iterations using

p` instead if pc or using pg instead of c1` = ξ1

`,ex.

We conclude this section by noting that our solver provides an accurate and

efficient numerical scheme which proved to be competitive to other fully implicit

solvers for the two-phase two-component model considered in this and in the

previous section.

N unknowns time steps (failed) Newton steps

100 404 21 (2) 153

200 804 20 (1) 144

400 1604 18 (3) 123

800 3204 29 (9) 246

1600 64042 31 (12) 253

3200 12804 32 (13) 263

Table 7.7: Benchmark performance results for the easy test case (α1 = 5 · 10−4)


N unknowns time steps (failed) Newton steps

100 404 93 (11) 994

200 804 167 (25) 1756

400 1604 273 (46) 3060

800 3204 454 (64) 4932

1600 6404 883 (166) 9659

3200 12804 1490 (380) 16626

Table 7.8: Benchmark performance results for the hard test case (α2 = 5 · 10−4)

N time steps (failed) Newton steps (failed)

100 48 (7) 309 (70)

200 62 (11) 422 (110)

400 76 (15) 511 (150)

800 89 (18) 630 (180)

160 92 (19) 643 (190)

3200 89 (18) 629 (180)

Table 7.9: Computational results reported by Uni Heidelberg (R. de Cuvelandand P. Bastian) for the easy test case (α1 = 5 · 10−7)

N time steps (failed) Newton steps (failed)

100 213 (55) 1335 (550)

200 403 (107) 2641 (1070)

400 781 (206) 5535 (2060)

800 1477 (390) 10701 (3990)

1600 2827 (746) 20751 (7440)

3200 5488 (1441) 38942 (13699)

Table 7.10: Computational results reported by Uni Heidelberg (R. de Cuvelandand P. Bastian) for the hard test case (α2 = 5 · 10−4)


7.3 CO2 sequestration

In this section, we consider the injection of CO2 into a deep aquifer covered by

an impermeable layer of rock for the purpose of permanent geological seques-

tration. This is considered as a promising approach to reduce the emission of

greenhouse gases into the atmosphere. The storage concept is based on different

trapping mechanisms that prevent the injected CO2 from being released into the

atmosphere.

Typically, CO2 is injected in its supercritical state where it adopts properties

midway between a gas and a liquid. More precisely, it has the low viscosity of

a gas and the high density of a liquid. Consequently, the ideal gas law used for

hydrogen injection is not applicable for carbon sequestration. Instead, we employ

the EOS of Duan [DMW92] to compute the density of CO2, cf. Figure 7.8.

When CO2 dissolves into the liquid phase, it causes density variations of the liquid

phase that may influence the flow regime. According to experimental data, the

density of aqueous solutions of CO2 can be as much as 2-3% higher than that of

pure water, which triggers downward migration due to gravitational forces. We

use the approach of Garcia [Gar01] to define the function f` that represents the

molar liquid phase density.

Regarding the solubility of CO2 in water, Henry’s law, which was used in the

previous numerical examples, is invalid at the high pressure conditions prevailing

at a deep underground CO2 storage site. We use the EOS of Spycher and Pruess

[SP05] to define the function φex representing the interphase mass exchange of

CO2. Figure 7.9 shows the solubility curves of CO2 in water for different tem-

peratures, neglecting salinity and the presence of other components in the liq-

uid phase. For temperatures below the critical temperature (Tcrit = 304.15K),

the curves exhibit a non-differentiable kink at the transition point between the

gaseous and liquid phase state. For the viscosity of the gas phase, an approach

of Fenghour et al. [FWV98] is used, cf. Figure 7.10.

Since the coefficient functions described above are given only implicitly, their

evaluation may be computationally expensive. Therefore, lookup tables are ini-

tially generated and cubic splines are used to evaluate the coefficient functions

and their derivatives during the simulation.

7.3. CO2 SEQUESTRATION 151

0

200

400

600

800

1000

1200

00 1 · 107 2 · 107 3 · 107 4 · 107 5 · 107 6 · 1070

pg [Pa]

ρmass,g[kg/m

3]

0

200

400

600

800

1000

1200

00 1 · 107 2 · 107 3 · 1070

283.15 K293.15 K303.15 K313.15 K323.15 K

Figure 7.8: Mass density of CO2 as a function of pressure

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.0000 1 · 107 2 · 107 3 · 107 4 · 107 5 · 107 6 · 1070

pg [Pa]

CO

2molefraction

χ1 ℓ[m

ol/m

ol]

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.0000 1 · 107 2 · 107 3 · 1070

283.15 K293.15 K303.15 K313.15 K323.15 K

Figure 7.9: Solubility of CO2 as a function of pressure


0.0 · 10−4

0.2 · 10−4

0.4 · 10−4

0.6 · 10−4

0.8 · 10−4

1.0 · 10−4

1.2 · 10−4

1.4 · 10−4

1.6 · 10−4

1.8 · 10−4

0.0 · 10−4

0 1 · 107 2 · 107 3 · 107 4 · 107 5 · 107 6 · 1070

pg [Pa]

µg[Pa·s

]

0.0 · 10−4

0.2 · 10−4

0.4 · 10−4

0.6 · 10−4

0.8 · 10−4

1.0 · 10−4

1.2 · 10−4

1.4 · 10−4

1.6 · 10−4

1.8 · 10−4

0.0 · 10−4

0 1 · 107 2 · 107 3 · 107 4 · 107 5 · 107 6 · 1070

283.15 K293.15 K303.15 K313.15 K323.15 K

Figure 7.10: Viscosity of CO2 as a function of pressure

7.3.1 CO2 injection without chemical reactions

In the next test case we simulate the injection of CO2 into an initially fully

water saturated domain based on a two-phase two-component model, cf. [NBI13].

This is numerically more challenging than the previous examples related to the

injection of hydrogen because all fluid properties and coefficient functions are

nonlinear, cf. Figures 7.8 to 7.10, while in the MoMaS benchmark, linear relations

(Henry’s law and the ideal gas law) are used to model the solubility of hydrogen in

the liquid phase and the compressibility of hydrogen gas, respectively. Moreover,

the soil parameters used in this example lead to a retention curve that corresponds

to the one of hard test case of Section 7.2. More precisely, the relations of

Brooks and Corey are used for capillary pressure and relative permeabilites. The

parameters are listed in Table 7.11. With the help of this numerical example we

are able to demonstrate that the incorporation of pc and pg into the set of primary

variables (cf. (3.64)) may help to significantly reduce the total computation time

since the nonlinear solver converges more rapidly than for other choices of primary

variables, e. g., pc and c1` . Consequently, fewer time steps are required by the

adaptive time stepper.

The computational domain is located 800m below the surface, providing super-


Γin

ΓimpΓD

Γimp

Γimp

Ω

10m

90m

600m

Figure 7.11: Computational domain for the CO2 injection scenario

critical conditions for the CO2 that is injected through the lower left part of the

domain at a constant rate q1in, cf. Figure 7.11.

Governing equations

The system is governed by the conservation equations (7.1)–(7.2), where compo-

nent 1 represents CO2 and component 2 represents water. Diffusion is expressed

in terms of the gradients of the molar fractions, cf. (2.4). Dispersion is not con-

sidered in this example, and molecular diffusion follows an approach of Millington

and Quirk [MQ61], resulting in the representation

Dα = φ43 s

103α Ddiff,αI (7.5)

for the diffusion-dispersion tensor.


At the initial time, the domain is fully liquid-saturated with no CO2 being dis-

solved in the liquid phase. The initial conditions read

p`(·, 0) = p0` , c1

`(·, 0) = 0 ,

where

p0`(x, y) = 105 + (900− y)ρstd

mass,` · 9.81 Pa


represents hydrostatic pressure conditions in the domain. Here, ρstdmass,` denotes

the mass density of pure water.

The following boundary conditions are imposed on ∂Ω = Γin∪ΓD∪Γimp:


Q1 · n = 0 , Q2 · n = 0 .

2. CO2 injection through Γin:

Q2 · n = 0 and Q1 · n = q1in .

3. Pure liquid water and hydrostatic pressure conditions on ΓD:

p` = p0` , c1

` = 0 .


porosity φ 0.2 [-]

absolute permeability K 10−12 m2

temperature T 313.15 K

Brooks–Corey parameter s`,res 0.0 [-]

Brooks–Corey parameter sg,res 0.0 [-]

Brooks–Corey parameter λ 2.0 [-]

Brooks–Corey parameter pentry 1000 Pa

molar mass of CO2 M1 4.4 · 10−2 kg mol−1


pure water mass density (T = 313.15K) ρstdmass,` 992 kg m−3

diffusion coefficient Ddiff,` 2 · 10−9 m2 s−1

CO2 injection rate q1in 4 · 10−2 kg m−2 s−1

viscosity of the liq. phase (T = 313.15K) µ` 6.526 · 10−4 Pa.s

Table 7.11: Parameters for the CO2 injection scenario


Primary variables pc and pg Primary variables pc and c1`

cells time steps CPU time [s] time steps CPU time [s]

150 · 41 748 23 671 33

150 · 42 752 111 1600 270

150 · 43 2169 1559 4645 3469

150 · 44 6354 23323 13606 56695

Table 7.12: Comparison of two different choices of primary variables for the CO2

injection scenario

Comparison of different choices of primary variables

The numerical results are illustrated in Figure 7.12. Each of the graphics shows

the gas phase saturation at a specific point in time, calculated on a grid consisting

of 38400 rectangles. The computations were done in parallel on 20 processors.

Shortly after the injection of CO2 commences, a gas phase starts to appear and,

due to gravitational forces, propagates upwards. At the same time, the CO2

spreads in horizontal direction and finally accumulates below the upper boundary,

which is impermeable. To study the influence of the selected primary variables,

the computations were done for two different choices of primary variables, using

the same control parameters for the nonlinear and linear solvers. The number of

total time steps (successful and unsuccessful) and the computation time required

to reach the final simulation time Tend = 65 days are listed in Table 7.12. Clearly,

the use of two pressures pc and pg leads to a significant lower number of time steps

and CPU time than the use of pc along with the concentration c1` . This is because

the nonlinear solver converges more rapidly for the first choice, resulting in larger

time steps due to the adaptive time step control. There are two possible reasons

for this behavior: first, the coefficient functions presented above are functions

of pg and depend directly on a primary variable if the first formulation is used.

Secondly, the use of pc and pg implies that the gradient terms in the transport

operators depend linearly on the primary variables,

q` = −K kr`µ`

(∇(pg − pc︸︷︷︸=p`

)− ρmass,` g) , qg = −K krgµg

(∇pg − ρmass,g g) .


Figure 7.12: Gas phase saturation during the CO2 injection process after 7, 20and 65 days

If the same test is carried out for the MoMaS benchmark, the same number of

time steps is required for both formulations since Henry’s law implies c1` = Hpg,

i. e., both formulations translate into each other by linear rescaling of the primary

variables.

The above results motivate the incorporation of pc and pg into our global set of

primary variables for the general multiphase multicomponent model, cf. (3.64).

Although pg could be treated as a local variable by removing (3.73) from the set

of global equations, the formulation with both pressures was more efficient due

to the better convergence of the nonlinear solver for that case, despite of the fact

that the resulting linear systems are larger.


7.3.2 CO2 injection with chemical reactions

During the past years, the injection of CO2 into the subsurface with the goal

of reducing greenhouse gas emissions into the atmosphere has attracted wide

interest. A desirable effect in the geologic sequestration of CO2 would be if the

carbon precipitates forming minerals, which will effectively bind it to the rock

and thus permit a long-term storage. The desired mechanism is modeled by the

following set of generic chemical reactions (cf. [Kra08]):

CO(g)2

R1←→ CO(`)2

CO(`)2 + H2O

R2←→ HCO−3 + H+

Calcite + H+ R3←→ Ca2+ + HCO−3

Min A + 3H+ R4←→ Me3+ + SiO2

Min B + 2H+ R5←→ Me3+ + HCO−3

The reaction system consists of three minerals (Min A, Min B and calcite) and

seven aqueous reacting species, including water. The first reaction represents the

interphase mass exchange of CO2 between the gas and the liquid phase. The

second and the third reaction allow a transition of liquid CO2 into HCO−3 and

calcite. They both affect and are affected by the pH, i. e., the concentration of

H+. If a silicate (Min A) is present at the initial state, it dissolves at high H+

concentrations, releasing metal ions by reaction R4. Finally, the metal ions can

initiate the precipitation of Min B, trapping the carbon as a carbonate mineral.

Defining the concentration vectors

c` = (c1` , . . . , c

7`)T := (c

CO(`)2

` , cH2O` , c

HCO−3` , cH+

` , cCa2+

` , cMe3+

` , cSiO2` )T ,

cs = (c1s, c

2s, c

3s)T := (ccalcite

s , cMin As , cMin B

s )T ,


the stoichiometric matrix associated with this order of the chemical species reads

S =

Sg,eq

S`,eq

Ss,eq

=

−1 0 0

S`,ex S`,mob S`,min

0 0 IJmin

=

−1 0 0 0 0

1 −1 0 0 0

0 −1 0 0 0

0 1 −1 0 −1

0 1 1 3 2

0 0 −1 0 0

0 0 0 −1 −1

0 0 0 −1 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

.

The columns of this matrix and of its submatrices satisfy the linear indepen-

dence assumptions of Section 3.1, such that the reduction scheme can be applied.

Defining

S∗` =

1 −1 −1 0 0

0 −1 0 0 0

0 1 −1 0 −1

0 1 0 3 2

0 0 −1 0 0

0 0 0 −1 −1

0 0 0 −1 0

, S⊥` =

1

39

0 0

15 9

9 8

6 1

−3 −7

3 −6

15 9

,


we obtain the transformed variables

η` = (η1` , η

2` )T = (S⊥`

TS⊥` )−1S⊥`

Tc` =

0 1 0 1 1 2 1

0 0 1 −1 −2 −3 0

c1`

c2`

...

c7`

,

and

ξ` = (ξ`,ex, ξ`,mob, ξ1`,min, ξ

2`,min, ξ

3`,min)T

= (S∗`TS∗`)

−1S∗`Tc` =

1

39

39 −24 9 6 −3 3 15

0 −24 9 6 −3 3 15

0 −3 −7 4 −28 15 −3

0 15 9 6 −3 3 −24

0 −12 −15 3 18 −18 27

c1`

c2`

...

c6`

c7`

.

For this example, the system (3.65)–(3.73) is solved numerically with the trans-

formed variables as defined above. The diffusive fluxes are given by (2.4) using

the Bear–Scheidegger diffusion–dispersion tensor, cf. (2.5). For relative perme-

abilities and capillary pressure, we use the Brooks–Corey parametrizations. All

parameters are listed in Table 7.13. For the mass density, the solubility and the

viscosity of CO2 we use the same coefficient functions as in the previous section,

cf. Figures 7.8–7.10. All reactions are assumed at equilibrium with R1 being

governed by the solubility law and R2–R5 by the equilibrium mass action law.

The solubility products of these reactions are

cHCO−3` cH+

`

cCO

(`)2

`

= 10−3 ,cH+

`

cHCO−3` cCa2+

`

= 1 if ccalcites > 0 ,

(cH+

` )3

cMe3+

` cSiO2`

= 10−3 if cMin As > 0 ,

(cH+

` )2

cHCO−3` cMe3+

`

= 0.8 if cMin Bs > 0 ,


where the (constant) activity of H2O is already incorporated. The computational

domain is shown in Figure 7.13. It is located 800 m below the surface, providing

supercritical conditions for the CO2 that is injected through a square with side

length 10m located at the lower part of the left boundary.

Figure 7.13: Computational domain for the mineral trapping scenario


The initial conditions must be consistent with the equilibrium conditions. For

the liquid pressure, we set

p0`(·, 0) = p0

` ,

where

p0`(x, y, z) = 105 + (900− z)ρstd

mass,` · 9.81 Pa

represents hydrostatic pressure conditions within the domain. The initial con-

centrations are constant within the computational domain,

ci`(·, 0) = ci`,0 , i ∈ 1, . . . , 7 , cis(·, 0) = cis,0 , i ∈ 1, 2, 3 ,

where the values c1`,0, . . . , c

7`,0 and c1

s,0, c2s,0, c

3s,0 must satisfy (2.7), i. e., they must

be consistent with the liquid phase compressibility law. Their values are listed

in Table 7.13. Denoting by Q1 = c1`q` + ρmol,gqg + j1

` the total flux of CO2 and

by Qi = ci`q` + ji` the total flux of component i, i ∈ 2, . . . , 7, the following

boundary conditions are imposed on ∂Ω = Γin∪ΓD∪Γimp:


1. No-flux conditions on ∂Ω \ (ΓD ∪ Γin) for all components:

Qi` · n = 0 , i ∈ 1, . . . , 7 .

2. CO2 injection through Γin:

Q1 · n = q1in , Qi · n = 0 , i ∈ 2, . . . , 7 .

3. The Dirichlet values prescribed at ΓD for the pressures and the concentra-

tions coincide with the initial values:

p` = p0` , ci` = ci`,0 , i ∈ 1, . . . , 7 .

The boundary and initial values for all other variables are induced by the variable

transformation and the resolution function.

Numerical results

The numerical results obtained on a grid consisting of 40960 hexahedra are shown

in Figures 7.14–7.16. Shortly after the injection of CO2 has started, a gas phase

appears, which continuously moves upwards due to gravity and accumulates be-

low the impermeable upper boundary. The CO2 that has dissolved into the liquid

phase causes the pH to decrease, inducing the dissolution of calcite and Min A.

Thereby, metal ions and bicarbonate are released, which initiates the precipita-

tion of Min B.

The propagation of the concentration fronts, in particular the appearance of the

gas phase and the precipitation and dissolution of minerals are well reproduced

by our solver. The computations were carried out using a maximum time step

size ∆tmax = 5000s and the control parameter OptSteps=6 for the time step

adaption. With these parameters, 2742 time steps (including 667 unsuccessful

ones) were carried out to simulate a time span of 85 days with an average number

of 4.32 Newton steps per time step.



porosity φ 0.2 [-]

absolute permeability K 10−12 m2

temperature T 313.15 K

Brooks–Corey parameter s`,res 0.0 [-]

Brooks–Corey parameter sg,res 0.0 [-]

Brooks–Corey parameter λ 2.0 [-]

Brooks–Corey parameter pentry 1000 Pa

molar mass of CO2 M1 4.4 · 10−2 kg mol−1


molar mass of bicarbonate M3 6.1 · 10−2 kg mol−1

molar mass of hydrogen M4 10−3 kg mol−1

molar mass of calcium M5 4 · 10−2 kg mol−1

molar mass of metal ions M6 1.5 · 10−2 kg mol−1

molar mass of silicon dioxide M7 6 · 10−2 kg mol−1

initial concentration of CO2 c1`,0 10−2 kg m−3 mol−1

initial concentration of water c2`,0 55333.33 kg m−3 mol−1

initial concentration of bicarbonate c3`,0 10−2 kg m−3 mol−1

initial concentration of hydrogen c4`,0 10−3 kg m−3 mol−1

initial concentration of calcium c5`,0 10−1 kg m−3 mol−1

initial concentration of metal ions c6`,0 10−4 kg m−3 mol−1

initial concentration of silicon dioxide c7`,0 10−2 kg m−3 mol−1

initial concentration of calcite c1s,0 0.1 kg m−3 mol−1

initial concentration of Min A c2s,0 0.2 kg m−3 mol−1

initial concentration of Min B c3s,0 0.0 kg m−3 mol−1

pure water mass density (T = 313.15K) ρstdmass,` 992 kg m−3

diffusion coefficient D` 2 · 10−9 m2 s−1

longitudinal dispersion coefficient αL 0.1 m2 s−1

transversal dispersion coefficient αT 0.01 m2 s−1

CO2 injection rate q1in 2 · 10−2 kg m−2 s−1

viscosity of the liq. phase (T = 313.15K) µ` 6.526 · 10−4 Pa.s

Table 7.13: Parameters for the mineral trapping scenario


Figure 7.14: Results after 7 days

Part II

Analysis of robust mixed hybrid

finite element discretizations for

advection–diffusion problems

166

Chapter 1

Introduction

In the second part of this work, we study mixed hybrid finite element approxi-

mations of the linear parabolic advection–diffusion problem

∂tc−∇ · (D∇c−Qc) = f in QT , (1.1)

c = c0 on Ω× 0 , (1.2)

c = 0 on ∂Ω× J , (1.3)

where Ω is a bounded domain in R2 with sufficiently smooth boundary, 0 <

Tend < ∞ denotes the final time, J = (0, Tend) and QT = Ω × J . This problem

may be regarded as a model problem for a wide range of applications arising,

e. g., in hydrology, civil engineering, petroleum engineering, economics and many

other fields, and the need for accurate, efficient and reliable schemes to solve

them numerically is well recognized. Despite of the simplicity of the problem and

extensive research over the last decades, the design of such methods remains a

challenging task, in particular when advection is strongly dominant. In this case,

numerical instabilities typically occur when standard numerical methods are used

since sharp layers in the solution cannot be resolved properly.

1.1 Current state of research

Generally, the interest of mixed methods lies in the fact that they are locally

mass conservative and provide accurate approximations of a scalar and a flux

167


unknown. Moreover, the flux approximations are continuous across interelement

boundaries.

There is a rich literature on mixed finite element methods and their conver-

gence. For a general overview on mixed finite element methods and their ap-

plications, we refer the reader to the books [BBF13, BBD+08, BF91], the re-

view articles [RT91, YAD10], and the references therein. Approximations of

advection–diffusion problems using the Raviart–Thomas element were studied

in [DR82, DR85], upwind-mixed methods are treated in [Daw98, DA99] and in

[RSH+11, Voh07]. Stabilization techniques based on using quadrature formulas

for the mass matrix are employed in [MSS01, SS97].

For approximations of a diffusion equation using the BDM1 element, we refer to

[BDM85]. The phenomenon of suboptimal convergence of the total flux variable

in the presence of an additional advection term is reported in [Dem02], where a

general second order advection–diffusion equation is analyzed.

1.2 Objective of this work

In this part, we study Euler-implicit mixed hybrid finite element methods to ap-

proximate the advection–diffusion problem (1.1)–(1.3) using the RT0 element and

the BDM1 element. While the discretization of the advective term relies on the

cellwise constant approximations of the scalar unknown if standard mixed meth-

ods are employed, cf. [DR82, DR85, Dem02], the schemes considered here use

the Lagrange multipliers arising in the hybrid problem formulation to discretize

the advective fluxes. This is motivated by the fact that the Lagrange multipliers

represent approximations of the scalar unknown on the edges of the triangula-

tion. They can even be used to define a higher order reconstruction in the space

of linear Crouzeix-Raviart elements; see, e. g., [AB85, BDM85].

For the RT0 element, we derive optimal order convergence estimates for a new

class of Euler-implicit methods designed to improve the robustness of the stan-

dard scheme for problems involving high Peclet numbers, including an upwind-

mixed hybrid scheme which is suitable for strongly advection–dominant problems,

cf. [RSH+11]. The definition of this new class involves a reconstruction of the

advective fluxes based on a general weight function, which may depend on the

Lagrange multipliers and the cellwise constant approximations of the scalar un-

1.3. OVERVIEW OVER THIS WORK 169

known. Moreover, the weight function is purely local, i. e., on each element its

definition involves only quantities defined on that element. As a consequence,

the flux unknowns and the scalar unknowns can be eliminated locally, result-

ing in a linear system for the Lagrange multipliers only. This local elimination

process is called static condensation, cf. Section 3.2.2. On the other hand, the

standard upwind-mixed scheme of Dawson [Daw98] relies on non-local upwind

weights requiring information from neighbor cells, such that static condensation

is not applicable. Therefore, our scheme can be implemented more efficiently

than the standard scheme, while the accuracy obtained by both methods in a

numerical experiment was almost identical, cf. Section 3.5.

If the same ideas are employed to discretizations based on theBDM1 element, it is

not only possible to improve the robustness with respect to advection–dominance:

using the Lagrange multipliers in the discretization of the advective term, the

order of convergence for the total flux variable is increased, and optimal second

order convergence in L2(Ω) is restored, which is not provided by the standard

BDM1 scheme if an advection term is present. The results presented in the

following have been published in the articles [BBKR13, BRK14], and the preprint

[BFK15].

1.3 Overview over this work

The second part of this thesis is structured as follows. In the next section, we

introduce some basic notation and state assumptions on the partial differential

equation and the computational mesh. In Chapter 2, a continuous mixed varia-

tional formulation associated with (1.1)–(1.3) is defined, and sufficient conditions

for existence and uniqueness are given. In Chapter 3, we study approximations

of (1.1)–(1.3) based on the RT0 element. More precisely, a new class of methods

is introduced, and optimal order convergence estimates are derived for the flux

and the scalar variable. At the end of Chapter 3, we discuss different schemes

belonging to this class, and we illustrate the theoretical results by numerical ex-

periments. Finally, in Chapter 4, we study approximations of (1.1)–(1.3) based

on the BDM1 element. The numerical results confirm that our modified scheme

approximates the total flux variable with optimal second order accuracy in L2(Ω),

whereas the standard scheme provides only suboptimal first order.


1.4 Notations and assumptions

Throughout the following, the common notations of functional analysis are used.

In particular, let (·, ·) denote the inner product in L2(Ω) or (L2(Ω))2, respectively,

and ‖ · ‖0 the corresponding norm. Further, 〈·, ·〉 indicates the inner product in

L2(∂Ω), and ‖ · ‖k stands for the norm in Hk(Ω) = W k,2(Ω). Similarly, if K ⊂ Ω

is measurable, let (·, ·)K denote the inner product in L2(K), ‖ · ‖k,K the norm

in Hk(K), and 〈·, ·〉∂K the inner product in L2(∂K). As usual, H(div,Ω) is

defined as the space of functions in (L2(Ω))2 having the (weak) divergence in

L2(Ω). For the time discretization, let the time elapsed at the n-th time step

be denoted by tn := n · τ , 0 ≤ n ≤ N , where N is an integer and τ := TendN

denotes the (constant) time step size. The time derivative of a function c will be

approximated by the backward difference quotient ∂cn := 1τ(cn−cn−1), where the

superscript n indicates the evaluation of a function at the discrete time t = tn.

Assumptions on the partial differential equation

Throughout this work, we solve the equation (1.1) along with the initial and

boundary conditions (1.2)–(1.3). The following assumptions are made on the

coefficient functions.

(A1) The coefficient matrix D = (Dij)ij is symmetric and Dij ∈ C(QT ) for

i, j ∈ 1, 2. Furthermore, the uniform ellipticity condition

C1|ξ|2 ≤2∑

i,j=1

Dij(x, t)ξiξj ≤ C2|ξ|2

holds for all ξ = (ξ1, ξ2)T ∈ R2, (x, t) ∈ QT , where C1, C2 > 0 are fixed

constants.

(A2) The velocity field Q and the source term f satisfy Q ∈ C(J ; (W 1,∞(Ω))2)

and f ∈ C(J ;L2(Ω)), respectively.

(A3) The initial value has the regularity c0 ∈ H10 (Ω).

In addition to the regularity obtained by Theorem 2.0.1, we shall assume that

the solution (q, c) of the mixed variational problem (2.5)–(2.6) satisfies

1.4. NOTATIONS AND ASSUMPTIONS 171

(A4) (q, c) ∈ C(J ; (H1(Ω))2)×H1(J ;H1(Ω)) ∩H2(J ;L2(Ω)).

Assumptions on the grid

Let Thh>0 denote a family of triangular decompositions of Ω such that

(M1) Ω =⋃K∈Th K and K1 ∩ K2 = ∅ if K1 6= K2, where the elements K ∈ Th

are closed triangles.

(M2) If K1 ∩K2 6= ∅ and K1 6= K2, then K1 ∩K2 is either a vertex or a full edge

of each.

(M3) If K ⊂ Ω, then K has straight edges only.

(M4) If K is a boundary triangle, the boundary edge can be curved.

(M5) hK := diam(K), h := maxK∈Th

hK .

(M6) Thh>0 is shape-regular, i. e., there exists a constant σmax > 0 such that

hK ≤ σmax ρK for all K ∈ Th, where

ρK := supdiam(S) : S is a disc in R2 and S ⊂ K

is the diameter of the inscribed circle of K.

The collection of all edges of an element K is denoted by E(K), whereas Eh =

EIh ∪ EDh represents the set of all edges of Th consisting of the disjoint subsets EIhand EDh of interior and boundary edges, respectively. By L2(E) for some E ∈ Eh,we denote the L2 space with respect to the surface measure on E. The notation

L2(Eh) refers to the L2 space with respect to the surface measure on the union of

all edges.

Chapter 2

Continuous mixed variational

formulation

Mixed finite element methods for problem (1.1)–(1.3) are based on a mixed re-

formulation of the problem. In this reformulation, the total flux q is introduced

as an additional explicit variable, and (1.1)–(1.3) translates into the system

∂tc = −∇ · q + f in QT , (2.1)

q = −D∇c+Qc in QT , (2.2)

c = c0 on Ω× 0 , (2.3)

c = 0 on ∂Ω× J . (2.4)

A mixed finite element method consists in approximating the quantities q and c

simultaneously in appropriate function spaces by discretizing the system (2.1)–

(2.4). The continuous mixed variational problem associated with (2.1)–(2.4) reads

as follows.

Problem 2. Find (q, c) ∈ L2(J ;H(div,Ω))×H1(J ;L2(Ω)) with c|t=0 = c0 such

that for almost every t ∈ J

(D−1q(·, t),v)− (∇ · v, c(·, t))− (D−1Qc(·, t),v) = 0 , (2.5)

(∂tc(·, t), w) + (∇ · q(·, t), w) = (f, w) (2.6)

for all (v, w) ∈ H(div,Ω)× L2(Ω).

172

173

Note that here and in the following, the arguments of the coefficient functions

are omitted in order to simplify the notation.

Theorem 2.0.1. Let (A1)–(A3) be satisfied. Then, there exists a unique so-

lution (q, c) ∈ L2(J ;H(div,Ω)) × H1(J ;L2(Ω)) ∩ L2(J ;H2(Ω)) of Problem 2.

Moreover, it holds that

q = −D∇c+Qc . (2.7)

Proof. Let us first prove the existence of a solution. Under the above assumptions,

equation (1.1) is equivalent to the advection–diffusion problem

∂tc−∇ · (D∇c) +Q · ∇c+ (∇ ·Q)c = f , (2.8)

supplemented with the initial and boundary conditions (1.2) and (1.3), respec-

tively. The existence of a unique solution of (2.8) along with (1.2)–(1.3) in the

space H1(J ;L2(Ω))∩L2(J ;H2(Ω)) is provided by [LSU68, Theorem IV.9.1]. Note

that since c is a weak solution of (1.1), the pair (q, c) := (−D∇c+Qc, c) satisfies

(∂tc, ψ)− (f, ψ) = (q,∇ψ)

for all ψ ∈ H1(Ω) and a. e. t ∈ J . Consequently, q ∈ L2(J ;H(div,Ω)) and

∂tc + ∇ · q = f in L2(Ω) for a. e. t ∈ J . This implies (2.6), whereas (2.5)

follows from Green’s formula. To show uniqueness, assume that Problem 2 has

two solutions (q1, c1) and (q2, c2) in L2(J ;H(div,Ω)) × H1(J ;L2(Ω)). Defining

c := c1 − c2 and q := q1 − q2, the error equations

(D−1q(·, t),v)− (∇ · v, c(·, t))− (D−1Qc(·, t),v) = 0 , (2.9)

(∂tc(·, t), w) + (∇ · q(·, t), w) = 0 (2.10)

hold for all (v, w) ∈ H(div,Ω) × L2(Ω) and a. e. t ∈ J . Next, by adding the

equations (2.9)–(2.10) and testing them with v := q(·, t) and w := c(·, t), we

infer

(D−1q(·, t), q(·, t)) + (∂tc(·, t), c(·, t))− (D−1Qc(·, t), q(·, t)) = 0 .

Note that since c(·, 0) = 0, integrating the last equation from 0 to some arbitrary

t′ < Tend and using the ellipticity and boundedness of D−1 and Q, respectively,

174 CHAPTER 2. CONTINUOUS MIXED VARIATIONAL FORMULATION

yields∫ t′

0

‖q(·, t)‖20 dt+ ‖c(·, t′)‖2

0 ≤C

4δ

∫ t′

0

‖c(·, t)‖20 dt+ Cδ

∫ t′

0

‖q(·, t)‖20 dt

for any δ > 0. Finally, pushing back the last term for δ sufficiently small and

applying the Gronwall lemma, we obtain ‖c(·, t′)‖0 = 0. Since t′ < Tend was

arbitrary, this implies the uniqueness of the scalar variable. The uniqueness of

flux variable follows immediately from∫ t′

0‖q(·, t)‖2

0 dt = 0 for all t′ < Tend.

Chapter 3

Mixed hybrid finite element

discretizations based on the RT0

element

3.1 Approximation spaces and projection oper-

ators

In this chapter, we study approximations of the advection–diffusion problem

(1.1)–(1.3) based on the Raviart–Thomas mixed finite element of lowest order.

For this purpose, we define the approximation spaces

V h = RT0(Ω, Th) = v ∈ H(div,Ω) : v|K ∈ RT0(K) for all K ∈ Th ,

Wh = w ∈ L2(Ω) : w|K ∈ P0(K) for all K ∈ Th ,

where Pk(K) denotes the restriction of polynomials of total degree not greater

than k to K, and

RT0(K) = v ∈ (L2(K))2 : v(x) = a+ bx, a ∈ R2, b ∈ R

is the local Raviart–Thomas space of lowest order. Mixed finite element schemes

may be equivalently rewritten in a hybrid formulation, where the continuity con-

straints of the flux variable across element interfaces are relaxed and imposed

175

176 CHAPTER 3. MHFE SCHEMES BASED ON THE RT0 ELEMENT

by requiring additional variational equations and introducing Lagrange multipli-

ers. The approximation spaces associated with the mixed hybrid formulation are

defined as

V h = v ∈ (L2(Ω))2 : v|K ∈ RT0(K) for all K ∈ Th ,

Λh = λ ∈ L2(Eh) : λ|E ∈ P0(E) for all E ∈ EIh , λ|E = 0 for all E ∈ EDh .

In our convergence analysis, we make use of the usual projection operator Πh×Ph :

(H1(Ω))2×L2(Ω)→ V h×Wh [RT77], which can be extended to domains having

a curved boundary, cf. [DR85]. The following properties hold for the projectors:

(i) Ph is the L2(Ω)-orthogonal projection onto Wh;

(ii) for any v ∈ (H1(Ω))2 and any w ∈ L2(Ω),

(∇ · (v − Πhv), wh) = 0 for all wh ∈ Wh , (3.1a)

(∇ · vh, w − Phw) = 0 for all vh ∈ V h ; (3.1b)

(iii) the following approximation properties hold:

‖v − Πhv‖0 ≤ C‖v‖1 h for all v ∈ (H1(Ω))2 , (3.2a)

‖w − Phw‖0 ≤ C‖w‖1 h for all w ∈ H1(Ω) . (3.2b)

Here and in the following, C denotes a generic constant which is independent of

the unknowns and the discretization parameters. The projection Πh|K is uniquely

determined by means of moments over the edges,

〈(Πh|Kq − q) · nE, 1〉E = 0 for all E ∈ E(K) ,

where nE is the outer unit normal vector of the edge E ⊂ ∂K. In the same man-

ner, basis functions vKEK∈Th,E∈E(K) of V h can be constructed with supp(vKE) ⊆K and

〈vKE · nE′ , 1〉E′ = δEE′ for all E,E ′ ∈ E(K) .

For further use, let us mention that the basis functions are uniformly bounded in

3.2. A NEW CLASS OF EULER-IMPLICIT MHFE SCHEMES 177

the L2-norm,

‖vKE‖0 ≤ C for all K ∈ Th , E ∈ E(K) . (3.3)

If K is a straight-sided triangle, an explicit representation of the basis functions

associated with K ∈ Th is given by

vKE(x) =x− xE

2|K| χK(x) , E ∈ E(K) ,

where xE denotes the corner point of K facing the edge E, |K| is the measure

of the element K, and χK the characteristic function of K. Then, (3.3) is an

immediate consequence of the shape-regularity (M6). For the spaces Λh and

Wh, basis functions are given by characteristic functions µEE∈EIh of the interior

edges and χKK∈Th of the triangles, respectively.

3.2 A new class of Euler-implicit mixed hybrid

finite element schemes of lowest order

3.2.1 The mixed finite element schemes

In this section, we introduce new class of mixed hybrid finite element schemes.

First, let us recall the classical mixed approximation method associated with

(2.1)–(2.4), cf. [DR85] for the corresponding elliptic problem.

Problem 3. Let n ∈ 1, . . . N and cn−1h ∈ Wh be given. Find (qnh, c

nh) ∈ V h×Wh

such that

((Dn)−1qnh,vh)− (∇ · vh, cnh)− ((Dn)−1Qncnh,vh) = 0 , (3.4)

(∂cnh, wh) + (∇ · qnh, wh) = (fn, wh) (3.5)

for all (vh, wh) ∈ V h ×Wh.

It is well known that the linear algebraic system resulting when basis functions

of V h and Wh are employed in (3.4)–(3.5) is typically indefinite. Hence, com-

mon iterative solvers requiring a system matrix that is symmetric and positive

definite may fail to converge. To overcome this problem, a hybridization process


can be employed, where the space V h is replaced by the augmented space V h,

i. e., the continuity of the normal fluxes across interelement boundaries is relaxed,

cf. [BF91]. The continuity constraints are then imposed by requiring additional

equations and introducing Lagrange multipliers from the space Λh. The resulting

hybrid formulation is equivalent to the original formulation. Moreover, by apply-

ing a local elimination procedure, one can derive a linear system in terms of the

Lagrange multipliers only, cf. Sec. 3.2.2. Hence, the resulting system has fewer

unknowns than that of the non-hybrid method. The mixed hybrid formulation

associated with (3.4)–(3.5) reads as follows.


nh, λ

nh) ∈

V h ×Wh × Λh such that

((Dn)−1qnh,vh)−∑K∈Th

(∇ · vh, cnh)K − ((Dn)−1Qncnh,vh)

= −∑K∈Th〈λnh,vh · n∂K〉∂K ,

(3.6)

(∂cnh, wh) +∑K∈Th

(∇ · qnh, wh)K = (fn, wh) , (3.7)

∑K∈Th〈µh, qnh · n∂K〉∂K = 0 (3.8)

for all (vh, wh, µh) ∈ V h ×Wh × Λh.

Here, n∂K denotes the outer unit normal vector of ∂K. The Lagrange multipliers

may be regarded as an approximation of the scalar unknown on the element

interfaces. In fact, they carry some extra information about the exact solution

which can be used to reconstruct a higher order nonconforming approximation in

L2(Ω). We refer to [AB85], where this was shown for an elliptic model problem.

The use of the Lagrange multipliers in the discretization of the advective term is

one of the key ideas in the definition of the new class of methods. Moreover, the

exact velocity field is replaced by the approximation

Qnh := ΠhQ

n =∑K∈Th

∑E∈E(K)

QnKEvKE ∈ V h ,


which satisfies (cf. [Dur88])

‖Qn −Qnh‖L∞(Ω) ≤ Ch‖Qn‖W 1,∞(Ω) . (3.9)

Note that in many applications, an approximation of the velocity field Qn is

computed in a preprocessing step by solving Richards’ equation. If the RT0

element is used for this calculation, the coefficients QnKE are directly known.

In the schemes studied here, the advective terms are discretized using the re-

construction operator B : Λh × Wh → V h, which—given a piecewise constant

approximation λnh =∑

E∈EIhλnEµE ∈ Λh of cn on the edges of the triangulation

and the cellwise constant approximation cnh =∑

K∈Th cnKχK—reconstructs an ap-

proximation for the advective flux in the space V h. More precisely, the operator

B is defined to act on Λh ×Wh as

B(λnh, cnh) =

∑K∈Th

∑E∈E(K)

αnKE(cnK , λnE)Qn

KEvKE ,

where the weights αnKE are linear functionals on R×R and satisfy the condition

|αnKE(cnK , λnE)− cnK | ≤ C|λnE − cnK | . (3.10)

In contrast to the standard scheme, where the approximation of the advective

flux relies on the cellwise constant approximations cnK only, the discretization of

the advective fluxes in the mixed methods based on the reconstruction operator Bmay additionally depend on the Lagrange multipliers. Different schemes that are

recovered for specific choices of the weights αnKE are discussed in Section 3.4. The

class of mixed methods based on the reconstruction operator B and the weight

functions αnKE reads as follows.


nh, λ

nh) ∈

V h ×Wh × Λh with the representations

qnh =∑K∈Th

∑E∈E(K)

qnKEvKE , cnh =∑K∈Th

cnKχK , λnh =∑E∈EIh

λnEµE (3.11)


such that


(∇ · vh, cnh)K − ((Dn)−1B(λnh, cnh),vh)

= −∑K∈Th〈λnh,vh · n∂K〉∂K ,

(3.12)


(∇ · qnh, wh)K = (fn, wh) , (3.13)

∑K∈Th〈µh, qnh · n∂K〉∂K = 0 (3.14)


Note that an extension of the method to three space dimensions using Raviart–

Thomas elements of lowest order on tetrahedral grids is immediate and requires

only minor changes in the proof of convergence below, cf. Theorem 3.3.4.

For later use, let us finally state the standard upwind-mixed method, cf. [Daw98,

Voh07], which approximates the diffusive flux q = −D∇c.Problem 6. Let n ∈ 1, . . . , N and cn−1

h be given. Find (qnh, cnh) ∈ V h ×Wh

satisfying

((Dn)−1qnh,vh)− (∇ · vh, cnh) = 0 ,

(∂cnh, wh) + (∇ · qnh, wh) +∑K∈Th

∑E∈E(K)

QnKEσ

nKEwK = (fn, wh)

for all (vh, wh) ∈ V h ×Wh, where wK = wh|K and

σnKE =

cnK if QnKE ≥ 0 ,

cnK′ otherwise

if E is a common edge of K and K ′, and

σnKE =

cnK if QnKE ≥ 0 ,

0 otherwise

if E is a boundary edge.


3.2.2 Static condensation

Let us assume for a moment that there exists a unique solution (qnh, cnh, λ

nh) ∈ V h×

Wh × Λh of Problem 5 having the basis representations (3.11). Employing these

representations in the equations for the fluxes (3.12), using the basis functions as

test functions in (3.12)–(3.14) and the relations

(∇ · vKE, χK)K = 1 , 〈µE,vKE′ · n∂K〉∂K = δEE′ for K ∈ Th , E,E ′ ∈ E(K) ,

the following system of linear equations for the fluxes is obtained from (3.12):∑E∈E(K)

qnKE((Dn)−1vKE,vKE′)−∑

E∈E(K)

QnKEα

nKE(cnK , λ

nE)((Dn)−1vKE,vKE′)

= cnK − λE′ for all K ∈ Th , E ′ ∈ E(K) .

DefiningBnKE′E := ((Dn)−1vKE,vKE′),B

nK := Bn

KEE′E,E′∈E(K),C := (1, 1, 1)T ,

qnK = qnKEE∈E(K), λnK = λnEE∈E(K), these equations can be rewritten in matrix

form as

BnKq

nK = CcnK +Bn

K

QnKE1

αnKE1(cnK , λ

nE1

)

QnKE2

αnKE2(cnK , λ

nE2

)

QnKE3

αnKE3(cnK , λ

nE3

)

− λnK , K ∈ Th . (3.15)

It is easy to show that the matricesBnK are positive definite. Hence, the equations

(3.15) can be solved explicitly for the flux unknowns on each element K ∈ Th. In

the next step, the fluxes are inserted into the mass conservation equations (3.13)

to obtain a representation cnK = cnK(λnK) of the scalar unknown in terms of the

Lagrange multipliers. Finally, by substituting the flux and scalar unknowns into

the equations (3.14), a linear system for the Lagrange multipliers remains to be

solved in each time step. Once it has been solved, the other variables can be

reconstructed efficiently on each element in a postprocessing step. Note that this

static condensation process can be applied only if the upwind weights depend on

variables defined on a single element K.


3.3 Error analysis of the fully discrete problem

In this section, we present the error analysis of the mixed hybrid finite element

scheme defined in Problem 5. First, let us demonstrate existence and uniqueness

for the scheme (3.12)–(3.14).

Theorem 3.3.1. Assume (A1)-(A2). Then, for every n ∈ 1, . . . , N, Prob-

lem 5 admits a unique solution.

Proof. Since the problem is linear, it suffices to show uniqueness. Thus, let

cn−1h be given, and assume that there exist two solutions (qnh,1, c

nh,1, λ

nh,1) and

(qnh,2, cnh,2, λ

nh,2) ∈ V h × Wh × Λh of (3.12)–(3.14), where cnh,i and λnh,i have the

basis representations

cnh,i =∑K∈Th

cnK,iχK , λnh,i =∑E∈EIh

λnE,iµE , i = 1, 2 ,

respectively. Then, by subtracting, the error equations

((Dn)−1(qnh,1 − qnh,2),vh)−∑K∈Th

(∇ · vh, cnh,1−cnh,2)K +∑K∈Th〈λnh,1−λnh,2,vh · n∂K〉∂K

−∑K∈Th

∑E∈E(K)

QnKEα

nKE(cnK,1 − cnK,2, λnE,1 − λnE,2)((Dn)−1vKE,vh) = 0 ,

(3.16)

(cnh,1 − cnh,2, wh) + τ∑K∈Th

(∇ · (qnh,1 − qnh,2), wh)K = 0 , (3.17)

∑K∈Th〈µh, (qnh,1 − qnh,2) · n∂K〉∂K = 0 (3.18)

hold for all (vh, wh, µh) ∈ V h×Wh×Λh. Next, we proceed similarly as in [AB85]

to obtain an estimate for the errors ‖λnh,1 − λnh,2‖0,E on the edges. For K ∈ Thand E ∈ E(K) let τE denote the unique element of V h with supp(τE) ⊆ K and

τE · nE′ =

λnh,1 − λnh,2 if E = E ′ ,

0 otherwise

3.3. ERROR ANALYSIS OF THE FULLY DISCRETE PROBLEM 183

for all edges E ′ ∈ E(K). Then, it follows from a simple scaling argument that

hK‖τE‖1,K + ‖τE‖0,K ≤ Ch1/2K ‖λnh,1 − λnh,2‖0,E , (3.19)

and we obtain

‖λnh,1 − λnh,2‖0,E ≤ C(h

1/2K ‖qnh,1 − qnh,2‖0,K + h

−1/2K ‖cnh,1 − cnh,2‖0,K

+ h1/2K

∑E′∈E(K)

|E ′|(|cnK,1 − cnK,2|+ |λnE′,1 − λnE′,2|))

≤ C(h

1/2K ‖qnh,1 − qnh,2‖0,K + h

−1/2K ‖cnh,1 − cnh,2‖0,K

+ hK∑

E′∈E(K)

‖λnh,1 − λnh,2‖0,E′)

by using τE as a test function in (3.16) and employing (3.26) and (3.19), respec-

tively. Summing this estimate over the edges E ∈ E(K) and pushing back the

last term for h sufficiently small yields

‖λnh,1 − λnh,2‖0,E ≤ C(h

1/2K ‖qnh,1 − qnh,2‖0,K + h

−1/2K ‖cnh,1 − cnh,2‖0,K

). (3.20)

Next, we take vh := τ(qnh,1 − qnh,2), wh := cnh,1 − cnh,2 and µh := τ(λnh,1 − λnh,2) as

test functions in (3.16)–(3.18) and add the resulting equations to find

τ((Dn)−1(qnh,1 − qnh,2), qnh,1 − qnh,2) + ‖cnh,1 − cnh,2‖20

− τ∑K∈Th

∑E∈E(K)

QnKEα

nKE(cnK,1 − cnK,2, λnE,1 − λnE,2)((Dn)−1vKE, q

nh,1 − qnh,2) = 0 .

Further, using (3.20), (3.3), and (A1)-(A2), we obtain

‖cnh,1 − cnh,2‖20 + τ‖qnh,1 − qnh,2‖2

0 ≤ Cτ∑K∈Th

∑E∈E(K)

|E||λnE,1 − λnE,2|‖qnh,1 − qnh,2‖0,K

+ Cτ∑K∈Th

∑E∈E(K)

|E||cnK,1 − cnK,2|‖qnh,1 − qnh,2‖0,K


≤ Cτ∑K∈Th

h1/2K

∑E∈E(K)

‖λnh,1 − λnh,2‖0,E‖qnh,1 − qnh,2‖0,K

+ Cτ∑K∈Th

‖cnh,1 − cnh,2‖0,K‖qnh,1 − qnh,2‖0,K

≤ Cτ∑K∈Th

hK‖qnh,1 − qnh,2‖20,K + Cτ

∑K∈Th

‖cnh,1 − cnh,2‖0,K‖qnh,1 − qnh,2‖0,K

≤ Cδ‖cnh,1 − cnh,2‖20 +

Cτ 2

4δ‖qnh,1 − qnh,2‖2

0 + Cτh‖qnh,1 − qnh,2‖20

so that, by taking δ, τ and h sufficiently small, cnh,1 − cnh,2 and qnh,1 − qnh,2 vanish.

From (3.20), we infer that λnh,1 − λnh,2 vanishes.

Let us now turn to the error analysis of the fully discrete mixed hybrid scheme

(3.12)–(3.14). One of the main steps in the proof of convergence is the derivation

of a priori estimates for the differences |λnE−cnK | which occur in the error equations

resulting from (2.5)–(2.6) and (3.12)–(3.14). This is done in the following lemma

by employing similar techniques as in the a posteriori error analysis in [AB85],

where the Lagrange multipliers were used to obtain a higher order approximation

of the scalar unknown by postprocessing.

Lemma 3.3.2. Let (A1)-(A3) hold and assume that the solution (q, c) of Prob-

lem 2 satisfies (A4). Further, let (qnh, cnh, λ

nh) ∈ V h ×Wh × Λh be the solution

of Problem 5. Then, for h sufficiently small, there exists a constant C > 0,

independent of n and h, such that∑K∈Th

∑E∈E(K)

|E|2(λnE − cnK)2 ≤ C(‖cn − cnh‖2

0 + h2‖qn − qnh‖20

+ h2‖Qnhcnh −Qncn‖2

0 + h2‖cn‖21

).

(3.21)

Proof. In the first step, we show that

‖λnh −Q0hcn‖0,E ≤C

(h

1/2K ‖qn − qnh‖0,K + h

1/2K

∑E′∈E(K)

|E ′||λnE′ − cnK |

+ h1/2K ‖Qn

hcnh −Qncn‖0,K + h

−1/2K ‖cn − cnh‖0,K

),

(3.22)

where Q0h denotes the projection operator defined locally by L2(E)-orthogonal


projection onto P0(E) for some E ∈ Eh, i. e.,

〈Q0hcn − cn, 1〉E = 0 for all E ∈ Eh .

For some fixed K ∈ Th and E ∈ E(K), define τE as the unique element of V h

such that supp(τE) ⊆ K and

τE · nE′ =

λnh −Q0hcn if E = E ′,

0 otherwise

for all E ′ ∈ E(K). A simple scaling argument shows that

hK‖τE‖1,K + ‖τE‖0,K ≤ Ch1/2K ‖λnh −Q0

hcn‖0,E . (3.23)

Next, we use τE as a test function in (3.12) to obtain

((Dn)−1qnh, τE)K − (∇ · τE, cnh)K −∑

E′∈E(K)

QnKE′α

nKE′((D

n)−1vKE′ , τE)K

= −〈λnh, λnh −Q0hcn〉E .

(3.24)

Moreover, (2.7) and Green’s formula imply

((Dn)−1qn, τE)K − (∇ · τE, cn)K − ((Dn)−1Qncn, τE)K = −〈cn, λnh −Q0hcn〉E .

(3.25)

Subtracting (3.24) from (3.25), using the definition of Q0h, the estimate

|QnKE| = |〈Qn · nE, 1〉E| ≤ C|E| , (3.26)


the property (3.3), the condition (3.10), and the assumption (A1), we get

‖λnh −Q0hcn‖2

0,E = 〈λnh −Q0hcn, λnh − cn〉E

= ((Dn)−1(qn − qnh), τE)K − (∇ · τE, cn − cnh)K

−∑

E′∈E(K)

QnKE′(c

nK − αnKE′)((Dn)−1vKE′ , τE)K + ((Dn)−1(Qn

hcnh −Qncn), τE)K

≤ C(‖qn − qnh‖0,K‖τE‖0,K + ‖cn − cnh‖0,K‖τE‖1,K

+∑

E′∈E(K)

|E ′||λnE′ − cnK |‖τE‖0,K + ‖Qnhcnh −Qncn‖0,K‖τE‖0,K

).

Then, (3.22) follows from combining the last estimate with (3.23). Next, we use

the Lagrange multipliers to define a nonconforming approximation cnh of cn in the

space PCR1 (Th) of linear Crouzeix-Raviart elements by means of

Q0hcnh = λnh . (3.27)

Note that cnh is uniquely determined according to [AB85, Lemma 2.1], see also the

remark on boundary elements having a curved edge after Lemma 4.2 in [BDM85].

Similarly, we define cnh ∈ PCR1 (Th) to be the nonconforming projection of cn by

means of

Q0h(c

n − cnh) = 0 . (3.28)

Then, by standard arguments, it follows that

‖cnh − cn‖0 ≤ Ch‖cn‖1 . (3.29)

Moreover, from (3.27) and (3.28), we have

Q0h(c

nh − cnh) = λnE −Q0

hcn ,

which implies

‖cnh − cnh‖0,K ≤ Ch1/2K

∑E∈E(K)

‖λnh −Q0hcn‖0,E


by [AB85, Lemma 2.1]. Combining this with (3.22), we obtain

‖cnh − cnh‖0,K ≤ C(hK‖qn − qnh‖0,K + ‖cn − cnh‖0,K + hK‖Qn

hcnh −Qncn‖0,K

+ hK∑

E∈E(K)

|E||λnE − cnK |)

so that, by summing over all elements,

‖cnh − cnh‖20 ≤ C

(h2‖qn − qnh‖2

0 + ‖cn − cnh‖20 + h2‖Qn

hcnh −Qncn‖2

0

+ h2∑K∈Th

∑E∈E(K)

|E|2(λnE − cnK)2).

(3.30)

Hence, from (3.29) and (3.30), we get

‖cnh − cnh‖20 ≤ 3‖cnh − cnh‖2 + 3‖cnh − cn‖2

0 + 3‖cn − cnh‖20

≤ C(h2‖qn − qnh‖2

0 + ‖cn − cnh‖20 + h2‖Qn

hcnh −Qncn‖2

0

+ h2∑K∈Th

∑E∈E(K)

|E|2(λnE − cnK)2 + h2‖cn‖21

).

(3.31)

Now, we use that for any piecewise linear polynomial ph ∈⊕

K∈Th P1(K), the

estimate

‖ph‖L1(∂K) ≤ C‖ph‖0,K

holds, cf. [BF91, p. 112]. Then, for any K ∈ Th and E ∈ E(K),

|E|(λnE − cnK) = 〈cnh − cnh, 1〉E ≤ C‖cnh − cnh‖L1(∂K) ≤ C‖cnh − cnh‖0,K .

Finally, combining the last estimate with (3.31) yields∑K∈Th

∑E∈E(K)

|E|2(λnE − cnK)2 ≤ C‖cnh − cnh‖20 ≤ C

(h2‖qn − qnh‖2

0 + ‖cn − cnh‖20

+ h2‖Qnhcnh −Qncn‖2

0 + h2∑K∈Th

∑E∈E(K)

|E|2(λnE − cnK)2 + h2‖cn‖21

),

and (3.21) follows by pushing back the last term on the right hand side for

sufficiently small h.


For later use, we state the following discrete integration by parts formula.

Lemma 3.3.3. Let (an)n∈N0 and (bn)n∈N0 be real sequences and let m ∈ N. Then,

it holds that

m∑n=1

(an − an−1)bn = ambm − a0b0 −m∑n=1

an−1(bn − bn−1) .

Theorem 3.3.4. Let the assumptions (A1)–(A3) hold and denote by (q, c) ∈L2(J ;H(div,Ω))×H1(J ;L2(Ω)) the unique solution of Problem 2. Moreover, let

τ and h be sufficiently small such that for every n ∈ 1, . . . , N there exists a

unique solution (qnh, cnh, λ

nh) ∈ V h × Wh × Λh of Problem 5. Then, if (A4) is

satisfied and c0h = Phc0, there exists a constant C > 0, independent of τ and h,

such that

maxm∈1,...,N

‖c(·, tm)− cmh ‖20 + τ

N∑n=1

‖q(·, tn)− qnh‖20 ≤ C(τ 2 + h2) . (3.32)

Proof. We take t = tn in (2.5)–(2.6), subtract (3.12)–(3.13) from the resulting

equations and use the properties (3.1a)–(3.1b) to obtain the error equations

((Dn)−1(qn − qnh),vh)−∑K∈Th

(∇ · vh, Phcn − cnh)K − ((Dn)−1Qncn,vh)

+∑K∈Th

∑E∈E(K)

QnKEα

nKE((Dn)−1vKE,vh) =

∑K∈Th〈λnh,vh · n∂K〉∂K ,

(3.33)

(∂tcn − ∂cnh, wh) +

∑K∈Th

(∇ · (Πhqn − qnh), wh)K = 0 , (3.34)

∑K∈Th〈(Πhq

n − qnh) · n∂K , µh〉∂K = 0 (3.35)

for all (vh, wh, µh) ∈ V h × Wh × Λh. Note that our regularity assumptions

allow the evaluation of c and q and all coefficient functions at t = tn. Taking

vh := Πhqn−qnh, wh := Phc

n−cnh and µh := −λnh as test functions in (3.33)–(3.35)


and adding the resulting equations yields

((Dn)−1(qn − qnh),Πhqn − qnh)− ((Dn)−1Qncn,Πhq

n − qnh)

+ (∂tcn − ∂cnh, Phcn − cnh) +

∑K∈Th

∑E∈E(K)

QnKEα

nKE((Dn)−1vKE,Πhq

n − qnh) = 0 .

The last identity can be rewritten equivalently as

((Dn)−1(qn − qnh),Πhqn − qnh) + ((Dn)−1(Qn

hcnh −Qncn),Πhq

n − qnh)

+∑K∈Th

∑E∈E(K)

QnKE(αnKE − cnK)((Dn)−1vKE,Πhq

n − qnh)

+ (∂tcn − ∂cnh, Phcn − cnh) = 0 .

(3.36)

Further, for any m ∈ N with 1 ≤ m ≤ N , by summing (3.36) from n = 1, . . . ,m,

multiplying by 2τ and expanding, we obtain

T1 + . . .+ T8 := 2τm∑n=1

((Dn)−1(qn − qnh), qn − qnh)

+ 2τm∑n=1

((Dn)−1(qn − qnh),Πhqn − qn)

+ 2τm∑n=1

((Dn)−1(Qnhcnh −Qncn),Πhq

n − qnh)

+ 2τm∑n=1

(∂tcn − ∂cn, cn − cnh)

+ 2τm∑n=1

(∂tcn − ∂cn, Phcn − cn)

+ 2τm∑n=1

(∂(cn − cnh), cn − cnh)

+ 2τm∑n=1

(∂(cn − cnh), Phcn − cn)

+ 2τm∑n=1

∑K∈Th

∑E∈E(K)

QnKE(αnKE − cnK)((Dn)−1vKE,Πhq

n − qnh)

= 0 .


We now proceed to estimate separately each of the terms T1, . . . T8. First, the

terms T1 and T6 are bounded from below. Note that sinceD is uniformly positive

definite, D−1 is also uniformly positive definite. Thus, we have

T1 = 2τm∑n=1

((Dn)−1(qn − qnh), qn − qnh) ≥ Cτm∑n=1

‖qn − qnh‖20 . (3.37)

By the identity 2a(a− b) = a2 − b2 + (a− b)2, we find

T6 = 2m∑n=1

((cn − cnh)− (cn−1 − cn−1h ), cn − cnh)

=m∑n=1

‖cn − cnh‖20 −

m∑n=1

‖cn−1 − cn−1h ‖2

0 +m∑n=1

‖cn − cnh − cn−1 + cn−1h ‖2

0

≥ ‖cm − cmh ‖20 − ‖c0 − c0

h‖20 .

(3.38)

All other terms are passed to the right hand side and bounded from above. Using

a weighted Young’s inequality, we get for any δ2 > 0

|T2| ≤ Cδ2τm∑n=1

‖qn − qnh‖20 +

Cτ

δ2

m∑n=1

‖Πhqn − qn‖2

0 . (3.39)

The term T3 is rewritten as

T3 = 2τm∑n=1

((Dn)−1(Qnhcnh −Qncn), qn − qnh)

+ 2τm∑n=1

((Dn)−1(Qnhcnh −Qncn,Πhq

n − qn) =: T31 + T32 .

Then, for h sufficiently small, it follows from (3.9) and the regularity of Q that

‖Qnhcnh −Qncn‖0 ≤ ‖Qn

h(cn − cnh)‖0 + ‖(Qn −Qnh)cn‖0

≤ ‖Qnh‖L∞(Ω)‖cn − cnh‖0 + ‖Qn −Qn

h‖L∞(Ω)‖cn‖0

≤ C(‖cn − cnh‖0 + h‖cn‖0) .

(3.40)

Moreover, from the uniform boundedness of D−1 and a weighted Young’s in-


equality, we obtain that for any δ3 > 0

|T31| ≤Cτ

δ3

m∑n=1

‖Qnhcnh −Qncn‖2

0 + Cδ3τm∑n=1

‖qn − qnh‖20

≤ Cτ

δ3

m∑n=1

‖cn − cnh‖20 +

Ch2τ

δ3

m∑n=1

‖cn‖20 + Cδ3τ

m∑n=1

‖qn − qnh‖20 .

(3.41)

Similarly, we derive the estimate

|T32| ≤ Cτ

m∑n=1

‖cn − cnh‖20 + Ch2τ

m∑n=1

‖cn‖20 + Cτ

m∑n=1

‖Πhqn − qn‖2

0 . (3.42)

Next, we use Bochner’s inequality, the Cauchy-Schwarz inequality and the regu-

larity of c to find

|T4| ≤1

τ

m∑n=1

‖τ∂tcn − cn + cn−1‖20 + τ

m∑n=1

‖cn − cnh‖20

=1

τ

m∑n=1

∥∥∥∥∫ tn

tn−1

∫ tn

s

∂ttc(η) dη ds

∥∥∥∥2

0

+ τm∑n=1

‖cn − cnh‖20

≤ τ 2

m∑n=1

∫ tn

tn−1

‖∂ttc(η)‖20 dη + τ

m∑n=1

‖cn − cnh‖20

≤ τ 2‖∂ttc‖2L2(J ;L2(Ω)) + τ

m∑n=1

‖cn − cnh‖20

≤ Cτ 2 + τm∑n=1

‖cn − cnh‖20 .

(3.43)

Analogously,

|T5| ≤ τ 2‖∂ttc‖2L2(J ;L2(Ω)) + τ

m∑n=1

‖Phcn − cn‖20

≤ Cτ 2 + τ

m∑n=1

‖Phcn − cn‖20 .

(3.44)


For T7, we employ Lemma 3.3.3 and the definition c0h = Phc0 to get

|T7| ≤ δ7‖cm − cmh ‖20 +

1

δ7

‖Phcm − cm‖20 + (2 + τ)‖Phc0 − c0‖2

0

+ τ

m−1∑n=1

‖cn − cnh‖20 +

1

τ

m∑n=1

‖(Ph − I)(cn − cn−1)‖20 ,

(3.45)

where I denotes the identity operator. Denoting the last term on the right hand

side of (3.45) by T71, we obtain from Bochner’s inequality, the Cauchy-Schwarz

inequality and the regularity of c that

T71 ≤Ch2

τ

m∑n=1

‖cn − cn−1‖21 =

Ch2

τ

m∑n=1

∥∥∥∥∫ tn

tn−1

∂tc(η) dη

∥∥∥∥2

1

≤ Ch2

∫ tm

0

‖∂tc(η)‖21 dη ≤ Ch2‖∂tc‖2

L2(J ;H1(Ω))

≤ Ch2 .

Consequently,

|T7| ≤ δ7‖cm − cmh ‖20 +

1

δ7

‖Phcm − cm‖20 + (2 + τ)‖Phc0 − c0‖2

0

+ τm∑n=1

‖cn − cnh‖20 + Ch2 .

(3.46)

For the remaining term T8, we combine (3.26) and (3.3) to find

|T8| ≤Cτ

δ8

m∑n=1

∑K∈Th

∑E∈E(K)

|E|2(λnE − cnK)2 + Cδ8τm∑n=1

‖Πhqn − qn‖2

0

+ Cδ8τm∑n=1

‖qn − qnh‖20 ,

3.4. CHOICE OF THE WEIGHT FUNCTION 193

and it follows from Lemma 3.3.2 and (3.40) that

|T8| ≤C

δ8

(h2τ

m∑n=1

‖cn‖21 + τ

m∑n=1

‖cn − cnh‖20 + h2τ

m∑n=1

‖qn − qnh‖20

+ h2τ

m∑n=1

‖cn − cnh‖20 + h4τ

m∑n=1

‖cn‖20

)+ Cτδ8

m∑n=1

‖Πhqn − qn‖2

0

+ Cτδ8

m∑n=1

‖qn − qnh‖20 .

(3.47)

Finally, by collecting the estimates (3.37)–(3.47), choosing δ2, δ3, δ7, δ8, τ and h

sufficiently small, and pushing back terms to the left hand side, we obtain

‖cm − cmh ‖20 + τ

m∑n=1

‖qn − qnh‖20 ≤ Cτ 2 + Ch2 + Cτ

m∑n=1

‖Πhqn − qn‖2

0

+ Cτm∑n=1

‖Phcn − cn‖20 + C‖Phc0 − c0‖2

0 + C‖Phcm − cm‖20

+ Ch2τm∑n=1

‖cn‖20 + Cτ

m∑n=1

‖cn − cnh‖20 + Ch2τ

m∑n=1

‖cn‖21 ,

and (3.32) follows from the properties of the projectors, the regularity of q and

c, and by applying the discrete Gronwall lemma.

Note that since we applied the discrete Gronwall lemma, the constant in the error

estimate (3.32) is potentially large. This is in accordance with the convergence

analysis of the upwind-mixed methods in [Daw91] and [Daw93]. In [DA99], the

use of the Gronwall lemma was avoided, but only suboptimal convergence was

obtained for the semidiscrete problem.

3.4 Choice of the weight function

The proof of convergence given in the previous section applies to all schemes that

are obtained if the weight functions αnKE satisfy (3.10). This condition holds for


the upwind-mixed method resulting from the choice

αn,upKE (cnK , λ

nE) =

cnK if QnKE ≥ 0 ,

2λnE − cnK otherwise ,

which was studied numerically in the paper [RSH+11] in the context of reactive

transport simulation. This upwind weighting formula is motivated by the fact

that the Lagrange multipliers represent an approximation of the scalar unknown

on the edges. Hence, the expression 2λnE − cnK may be regarded as an extrapo-

lation of the concentration on the adjacent cell. The advantage of the resulting

upwind-mixed hybrid method is that it is fully local, i. e., it does not incorporate

information of adjacent cells. Consequently, static condensation can be employed

to eliminate the flux and the scalar unknowns from the system, cf. Section 3.2.2.

This implies that the method can be implemented more efficiently than the stan-

dard upwind-mixed method stated in Problem 6, where the definition of the

upwind weights explicitly involves concentration values of adjacent grid cells. In

the next section, the upwind-mixed hybrid method resulting from the choice αn,upKE

is compared with the standard upwind-mixed scheme in terms of accuracy and

efficiency.

It is well-known that full upwind methods may introduce a significant amount of

numerical diffusion, which leads to an artificial smearing of the numerical solution.

We refer to the numerical experiments in [RSH+11], where the amount of artificial

diffusion of the upwind-mixed hybrid method was quantified and compared with

other discretization schemes. One possibility to reduce the amount of artificial

diffusion is to use the partial upwind scheme obtained for the choice

αn,partKE (cnK , λ

nE) =

cnK if QnKE ≥ 0 ,

(1− νE)cnK + νE(2λnE − cnK) otherwise ,

which obviously satisfies condition (3.10). Here, νE ∈ [0, 1] denotes a coefficient

describing the amount of upstream weighting. It may depend on the local grid

Peclet number to ensure that upwinding is only employed in those parts of the

domain where the flow is advection–dominant. Obviously, for νE = 1, the full

upwind scheme is recovered.

3.4. CHOICE OF THE WEIGHT FUNCTION 195

Qn

cnK1b cnK2

bλnEb

E

Figure 3.1: Scalar unknowns and Lagrange multiplier associated with the com-mon edge of two adjacent triangles K1 and K2

Further admissible choices for the weight function are given by

αn,lagKE (cnK , λ

nE) = λnE ,

which was tested in [RSH+11] and proved to be robust for moderately advection–

dominant problems, and

αn,stdKE (cnK , λnE) = cnK ,

which represents the standard scheme (3.6)–(3.8) with the only difference that

Qn was approximated by Qnh. Another full upwind scheme is recovered for the

choice

αn,up,2KE (cnK , λ

nE) =

cnK if QnKE ≥ 0 ,

λnE otherwise .

This scheme was tested numerically in the paper [BFK14] and performed favor-

ably compared to an upwind-weighted finite volume scheme in terms of robust-

ness, monotonicity properties, and the amount of artificial numerical diffusion

introduced by both methods.


3.5 Numerical results

In this section, we present a numerical example for the upwind-mixed hybrid

method obtained for the weight αn,upKE defined in the previous section. More

precisely, we provide computational evidence that our error estimates are sharp,

and we compare the upwind-mixed hybrid method with the standard upwind-

mixed method of Dawson [Daw98]. Further numerical examples from the context

of contaminant biodegradation with advection–dominated flow are included in

[Bru10]. Moreover, in [RSH+11], the numerical diffusion of the upwind-mixed

hybrid method was quantified and compared with other discretization methods.

In our computations, we solve problem (1.1)–(1.3) on the unit square Ω = (0, 1)2

and the time interval J = (0, 1). The (scalar) diffusion coefficient and the velocity

field are given by D = 1 and Q = (0,−1)T , respectively. The source term f is

chosen so that c(x, y, t) = x(1− x)y(1− y)e−t represents the analytical solution

of the problem, and c0 := c(·, 0) is imposed as the initial value. A simple uniform

triangular mesh with h =√

2 is used as a coarse grid, and τ = 0.5 is chosen as

the initial time step size. In each level of refinement, h and τ are halved, and the

errors

Eτ,h =

‖c(·, tN)− cNh ‖2

0 + τN∑n=1

‖q(·, tn)− qnh‖20

1/2

are computed. For the upwind scheme of Dawson, we use qnh = qnh + Qcnh ∈RT0(Ω, Th) as an approximation of the total flux, where qh denotes the flux vari-

able of the scheme, which approximates the diffusive flux q = −D∇c. Both

methods were implemented in the software package M++ [Wie, Wie05]. To han-

dle the saddle point problems arising from the standard upwind-mixed scheme,

the linear systems are solved with a direct solver based on the SuperLU library

[Li05]. The results of the computations are summarized in Tables 3.1 and 3.2.

They clearly show optimal first order convergence in h and τ for our upwind-

mixed hybrid scheme. Hence, the error bound of Theorem 3.3.4 is sharp. More-

over, the magnitude of the errors almost coincides for both methods, while the

computation time was 50% lower on the finest mesh for the hybrid scheme.

On the one hand, this is due to a lower number of global unknowns resulting

from hybridization and employing static condensation, cf. Sec. 3.2.2. On the

3.5. NUMERICAL RESULTS 197

Ref. level # unknowns Eτ,h reduction CPU time [s]

1 16 3.685e-2 0.01

2 56 2.297e-2 1.60 0.03

3 208 1.236e-2 1.85 0.09

4 800 6.345e-3 1.95 0.47

5 3136 3.205e-3 1.98 3.88

6 12416 1.610e-3 1.99 33.11

7 49408 8.067e-4 2.00 305.90

Table 3.1: Numerical results for the upwind-mixed hybrid method

Ref. level # unknowns Eτ,h reduction CPU time [s]

1 24 3.507e-2 0.01

2 88 2.264e-2 1.54 0.03

3 226 1.234e-2 1.83 0.10

4 1312 6.349e-3 1.94 0.66

5 5184 3.208e-3 1.98 5.79

6 20608 1.611e-3 1.99 57.52

7 82176 8.068e-4 2.00 709.00

Table 3.2: Numerical results for the upwind-mixed method [Daw98]

other hand, the linear system associated with our method is sparser than that

corresponding to the non-hybrid method. Hence, the linear solver is faster and

less memory is required during the computation.

Chapter 4

Mixed hybrid finite element

discretizations based on the

BDM1 element

In the previous chapter, a new class of mixed hybrid finite element methods

based on the RT0 element was studied, where the advective fluxes are discretized

using the Lagrange multipliers introduced during hybridization. Depending on

the particular choice of the weight functions, different schemes are recovered from

this general class, e. g., the standard scheme or an upwind-mixed hybrid scheme

designed for strongly advection–dominated problems.

In this chapter, similar ideas are applied to approximations based on the BDM1

element. More precisely, we will show that suboptimal convergence for the flux

variable, which is known to arise when the standard BDM1 scheme is employed to

advection–diffusion problems, can be overcome by using the Lagrange multipliers

for the discretization of the advective fluxes. This observation was made and

confirmed numerically in the paper [BRBK12]. The error analysis is provided in

the article [BFK15].

198

4.1. SUBOPTIMAL CONVERGENCE OF THE STANDARD SCHEME 199

4.1 Suboptimal convergence of the standard non-

hybrid scheme

Using the Raviart–Thomas spaces for the discretization of (1.1)–(1.3), the flux

and the scalar unknown are approximated with the same order of convergence.

For applications where the main interest lies in the flux variable, the BDM

spaces were introduced [BDM85], which are able to approximate the flux to one

order higher than the scalar variable. Indeed, if the BDM1 mixed finite element

method is used to discretize a pure second order elliptic diffusion equation, the

flux is approximated with second order accuracy in the L2 norm. Error estimates

in L2(Ω) and L∞(Ω) for general second order elliptic problems with nonvanish-

ing advection using the BDM family of methods were derived in [Dem02] and

[Dem04], respectively. More precisely, it was shown that the total flux variable

consisting of diffusive and advective transport is approximated with first order

accuracy in L2(Ω), which is suboptimal since the same order of convergence is

obtained by the computationally less expensive RT0 element. This phenomenon

of suboptimal convergence may occur whenever the mixed finite element spaces

employed use polynomials of higher order for the approximation of the flux vari-

able than for the approximation of the scalar variable. The order of convergence

is then limited by the first order accuracy of the approximation for the conserved

quantity itself.

In the following, we will present a modified mixed hybrid BDM1 scheme which

restores optimal second order convergence of the flux variable in the presence of

an advective transport term. The modification consists in replacing the cellwise

constant approximation of the scalar unknown in the definition of the discrete

advective flux by a reconstruction based on the interelement Lagrange multipliers.

This is motivated by the well-known fact that the Lagrange multipliers carry some

higher order information about the scalar unknown.

200 CHAPTER 4. MHFE SCHEMES BASED ON THE BDM1 ELEMENT

b

b b

b

b b

K

Figure 4.1: Degrees of freedom of the local BDM1 space

4.2 The modified hybrid scheme

Mixed approximation spaces and projections

In the following, we assume that the domain Ω ⊂ R2 is polygonally bounded

and that Thh>0 denotes a family of shape-regular triangulations of Ω where

all triangles have straight edges only, cf. (M1)–(M3), (M5)–(M6). For the

discretization with the BDM1 mixed finite element method, let

V h = BDM1(Ω, Th) = v ∈ H(div,Ω) : v|K ∈ BDM1(K) for all K ∈ Th ,

Wh = w ∈ L2(Ω) : w|K ∈ P0(K) for all K ∈ Th ,

where BDM1(K) = (P1(K))2. The degrees of freedom for BDM1(K) are illus-

trated in Figure 4.1.

The standard BDM1 mixed finite element method for approximating the total

flux q = −D∇c+Qc reads as follows, cf. [Dem02].


nh) ∈ V h×Wh

such that

((Dn)−1qnh,vh)− (∇ · vh, cnh)− ((Dn)−1Qncnh,vh) = 0 , (4.1)

(∂cnh, wh) + (∇ · qnh, wh) = (fn, wh) (4.2)

for all (vh, wh) ∈ V h ×Wh.

In the following, we want to consider the hybrid problem formulation associated

with (4.1)–(4.2). For the BDM1 element, the Lagrange multipliers are linear

functions on each edge and may be discontinuous at the vertices. The approxi-

4.2. THE MODIFIED HYBRID SCHEME 201

mation spaces associated with the hybrid formulation are defined as

V h = v ∈ (L2(Ω))2 : v|K ∈ BDM1(K) for all K ∈ Th ,

Λh = λ ∈ L2(Eh) : λ|E ∈ P1(E) for all E ∈ EIh , λ|E = 0 for all E ∈ EDh .

The hybrid formulation associated with (4.1)–(4.2) reads as follows.


nh, λ

nh) ∈



(∇ · vh, cnh)K((Dn)−1Qncnh,vh)

= −∑K∈Th〈λnh,vh · n〉∂K , (4.3)


(∇ · qnh, wh)K = (fn, wh) , (4.4)

∑K∈Th〈µh, qnh · n〉∂K = 0 (4.5)


To obtain an algebraic formulation of the scheme (4.3)–(4.5), basis functions of

the involved function spaces must be specified. The space Λh consists of functions

which are defined on the edges of the triangulation and which are linear on each

edge. A basis of this space is given by functions vanishing everywhere except for

one interior edge and taking the value 0 at one endpoint and the value 1 at the

other endpoint of this edge. More precisely, for an interior edge E ∈ EIh with

vertices x1E and x2

E, we denote by µiE the function in Λh which takes the value 1

in xiE and the value 0 at the other endpoint of E and vanishes on all other edges.

For V h we define the basis functions to be the vector fields that vanish on every

triangle except for one triangle K, on which the normal flux is prescribed to be

zero at every edge except for one edge E ∈ E(K); on this edge, we require the

(outward) normal flux to be 2/|F | at the point 13(2x1

E + x2E) and to be 0 at the

point 13(x1

E + 2x2E). These basis functions are obtained by transforming the basis

functions on the unit reference triangle K (cf. Table 4.1) to an arbitrary triangle

K ∈ Th using Piola’s transformation. Note that the basis functions satisfy the


following properties:

(∇ · viKE, χK) = 1 ∀K ∈ Th, F ∈ E(K), i = 1, 2 , (4.6a)

〈viKE′ · n∂K , 1〉E = δEE′ ∀K ∈ Th, E,E ′ ∈ E(K), i = 1, 2 , (4.6b)

〈µiEvjKE′ · n∂K , 1〉E = δEE′δij ∀K ∈ Th, E,E ′ ∈ E(K), i, j = 1, 2 . (4.6c)

Here, n∂K denotes the outer unit normal vector of ∂K. The basis functions of

Wh are given by characteristic functions χK for each K ∈ Th. With the help of

the basis functions, we expand the unknowns in terms of the basis functions,

qnh =∑K∈Th

∑E∈E(K)

2∑i=1

qnKEiviKE ,

λnh =∑E∈EIh

2∑i=1

λnEiµiE ,

cnh =∑K∈Th

cnKχK .

Expanding the velocity field Qn as

Qn = Π1hQ

n +Qnr =

∑K∈Th

∑E∈E(K)

2∑i=1

QnKEivKEi +Qn

r =: Qnh +Qn

r ,

where Π1h denotes the usual BDM1 projection operator [BDM85], and using the

abbreviations

BnK,EiE′j

:= ((Dn)−1vKEi ,vKE′j) , F nK := (fn, χK) ,

we can approximate the standard mixed hybrid finite element scheme by the

4.2. THE MODIFIED HYBRID SCHEME 203

linear system (the approximation consisting in using Qnh instead of Qn)

|K|cnK − cn−1

K

τ+

∑E∈E(K)

2∑i=1

qnKEi = F nK ∀K ∈ Th , (4.7)

∑E∈E(K)

2∑i=1

BnK,EiE′j

qnKEi − cnK −∑

E∈E(K)

2∑i=1

QnKEi

cnKBnKEiE′j

= −λnE′j

∀K ∈ Th , E ′ ∈ E(K) , j = 1, 2 ,

(4.8)∑K∈Th,E∈E(K)

qnKEi = 0 ∀E ∈ EIh , i = 1, 2 . (4.9)

Obviously, the discretization of the advective term in (4.8) relies on the cellwise

constant approximation cnh ∈ Wh of the scalar unknown cn. As in the previous

chapter, we define a modified scheme by replacing the cellwise constant value

cnK with a reconstruction based on the Lagrange multiplier λnh (evaluated in the

points dividing an edge into three segments of equal length). The resulting linear

system reads

|K|cnK − cn−1

K

τ+

∑E∈E(K)

2∑i=1

qnKEi = F nK ∀K ∈ Th , (4.10)

∑E∈E(K)

2∑i=1

BnK,EiE′j

qnKEi − cnK −∑

E∈E(K)

2∑i=1

QnKEi

λnE1+ λnE2

+ λnEi3

BnKEiE′j

= −λnE′j

∀K ∈ Th , E ′ ∈ E(K) , j = 1, 2 ,

(4.11)∑K∈Th,E∈E(K)

qnKEi = 0 ∀E ∈ EIh , i = 1, 2 . (4.12)

In order to find a representation of the modified scheme on the finite element

level, we define an operator B : Λh → V h, which reconstructs an approximation

for the advective flux in the space V h. It is defined to act on Λh as

B(λnh) :=∑K∈Th

∑E∈E(K)

2∑i=1

λnEi + λnE1+ λnE2

3QnKEivKEi ,


v1KE1

(x, y) = (−2x, 6x+ 4y − 4)T v2KE1

(x, y) = (4x,−6x− 2y + 2)T

v1KE2

(x, y) = (4x,−2y)T v2KE2

(x, y) = (−2x, 4y)T

v1KE3

(x, y) = (−2x− 6y + 2, 4y)T v2KE3

(x, y) = (4x+ 6y − 4,−2y)T

Table 4.1: BDM1 basis functions on the reference triangle K

i. e., the normal component B(λnh) · n is prescribed at the two points of an edge

dividing the edge into three segments of equal size to match the product of the

normal component of Qnh and the Lagrange multiplier λnh at these points. This

property of the reconstruction operator is crucial for its higher order approxima-

tion property. With the help of the operator B, the modified linear system (4.10)–

(4.12) can be rewritten as the following mixed hybrid finite element method.


nh, λ

nh) ∈



(∇ · vh, cnh)K − ((Dn)−1B(λnh),vh)

= −∑K∈Th〈λnh,vh · n〉∂K , (4.13)


(∇ · qnh, wh)K = (fn, wh) , (4.14)

∑K∈Th〈µh, qnh · n〉∂K = 0 (4.15)


Note that in three space dimensions, a similar modification of the standard scheme

is possible by defining an appropriate reconstruction operator. The theoretical

results continue to hold for this case, cf. [BFK15].

4.3. OPTIMAL CONVERGENCE OF THE MODIFIED SCHEME 205

4.3 Optimal second-order convergence of the mod-

ified hybrid scheme

The convergence of the modified scheme is provided by the following theorem,

which was shown in [BFK15].

Theorem 4.3.1. Let D ∈ L∞(Ω × [0, T ];R2×2) be uniformly elliptic with el-

lipticity constant λ > 0 and upper bound Λ > 0 and let f ∈ L2(Ω × [0, T ]),

Q ∈ L∞([0, T ];W 2,∞(Ω)). Consider a weak solution u ∈ L2([0, T ];H10 (Ω)) to the

problem (1.1)–(1.3) and let q be defined by (2.2). For every n ∈ 1, . . . , N, let

(qnh, cnh, λ

nh) ∈ V h ×Wh × Λh denote a solution of the numerical scheme (4.13)–

(4.15). Then, there exists some hmax = hmax(Ω, σmax, λ,Λ, ||Q||L∞([0,T ],W 1,∞(Ω)))

and some maximal time step size τmax = τmax(Ω, σmax, λ,Λ, ||Q||L∞(Ω×[0,T ])) such

that the following holds: Provided that the smallness conditions τ < τmax and

h < hmax are satisfied, the a priori error estimate

maxm∈1,...,N

||P 0hc(·, tm)− cmh ||20 +

N∑n=1

τ ||q(·, tn)− qnh||20

≤ C(||P 0

hc0 − c0h||20 + ||∂ttc||2L∞(Ω×[0,T ])τ

2 + ||q||2L∞([0,T ];H2(Ω))h4

+ ||c||2L∞([0,T ];H2(Ω))h4)

holds, where C = C(Tend, σmax, λ,Λ,Ω, ||Q||L∞([0,T ];W 2,∞(Ω))) is independent of the

discretization parameters.

The main step in proving this theorem is the derivation of error estimates for the

Lagrange multipliers by employing techniques from the a posteriori error analysis

presented in [BDM85]. Indeed, it can be shown that the Lagrange multiplier λnhrepresents an approximation of cn on the edges. These estimates are then used

to establish approximation properties of the reconstruction operator B. Given

the Lagrange multiplier λnh (defined only on the edges), the reconstruction B(λnh)

provides an approximation of the advective flux Qncn in the space V h. The

precise definition of B implies that the normal component of B(λnh) coincides

with the normal component of Π1hQ

n · λnh in the two points dividing an edge into

three segments of equal length. The approximation properties of the Lagrange

multipliers and the reconstruction operator B entail the above theorem.


4.4 Numerical results

In this section, we illustrate our theoretical results and compare the classical and

the modified BDM1 scheme with the help of a computational experiment. Our

numerical results show that the error bounds stated above are sharp. Further

numerical tests, in which the modified BDM1 scheme was applied to nonlinear

reactive transport problems, are presented in [BRBK12, BBKR13].

To be more precise, problem (1.1)–(1.3) is solved on the unit square Ω = (0, 1)2 ⊂R2 on the time interval J = (0, 1). For the (scalar) diffusion coefficient and the

velocity field, we choose D = 1 and Q = (0,−1)T , respectively. Moreover,

homogeneous Dirichlet conditions are imposed on the boundary ∂Ω. The source

term f is prescribed so that the analytical solution of the problem is given by

u(x, y, t) = x(1− x)y(1− y)e−t .

In each refinement step, the mesh (which initially consists of two triangles) is

uniformly refined, and the errors

E(1)h,τ = τ 1/2

(N∑n=1

‖q(·, tn)− qnh‖20

)1/2

,

E(2)h,τ = max

1≤n≤N‖c(·, tn)− cnh‖0 ,

E(3)h,τ = max

1≤n≤N‖P 0

hc(·, tn)− cnh‖0

are computed. For all computations, a fixed time step size τ = 0.001 is used,

which is sufficiently small to ensure that the time discretization error is neg-

ligible compared to the space discretization error. The experimental orders of

convergence are determined by means of

EOC(i)h,τ = log2

(E

(i)2h,τ

E(i)h,τ

), i ∈ 1, 2, 3 .

The results of the computations are listed in Tables 4.2 and 4.3. As expected,

we obtain optimal second order convergence for the flux variable if the advective

fluxes are approximated using the Lagrange multipliers, whereas only suboptimal

first order convergence is observed when the classical scheme is used. Moreover,

4.4. NUMERICAL RESULTS 207

the scalar unknowns are approximated with first order accuracy for both schemes,

the magnitude of the errors being of almost equal size. For the approximation of

the projection P 0hc of the scalar variable into the space of piecewise constants by

our numerical solution cnh, we observe the usual (second order) superconvergence

result both in the case of the classical scheme and in the case of the modified

scheme.

triangles E(1)h,τ EOC E

(2)h,τ EOC E

(3)h,τ EOC

2 · 40 4.36e-02 2.91e-02 1.80e-02

2 · 41 2.01e-02 1.12 1.60e-02 0.86 6.37e-03 1.50

2 · 42 5.94e-03 1.76 8.57e-03 0.90 1.88e-03 1.76

2 · 43 1.56e-03 1.93 4.36e-03 0.97 4.94e-04 1.93

2 · 44 3.96e-04 1.98 2.19e-03 0.99 1.24e-04 1.99

2 · 45 9.88e-05 2.00 1.10e-03 1.00 3.07e-05 2.02

2 · 46 2.39e-05 2.04 5.48e-04 1.00 7.18e-06 2.10

Table 4.2: L2-errors and experimental orders of convergence (EOC) for the mod-ified BDM1 scheme

triangles E(1)h,τ EOC E

(2)h,τ EOC E

(3)h,τ EOC

2 · 40 4.13e-02 2.91e-02 1.79e-02

2 · 41 2.05e-02 1.01 1.60e-02 0.86 6.34e-03 1.50

2 · 42 6.91e-03 1.57 8.57e-03 0.90 1.87e-03 1.76

2 · 43 2.51e-03 1.46 4.36e-03 0.97 4.91e-04 1.93

2 · 44 1.09e-03 1.21 2.19e-03 0.99 1.23e-04 1.99

2 · 45 5.19e-04 1.07 1.10e-03 1.00 3.05e-05 2.02

2 · 46 2.56e-04 1.02 5.48e-04 1.00 7.12e-06 2.10

Table 4.3: L2-errors and experimental orders of convergence (EOC) for the clas-sical BDM1 scheme

Conclusion

In the first part of this work, we developed a numerical framework for efficiently

simulating partially miscible multiphase multicomponent flow in porous media,

modeled by a system of coupled and strongly nonlinear partial differential equa-

tions, ordinary differential equations, and algebraic equations.

For this purpose, the problem was transformed with the help of a reduction

scheme, which preserves the model and allows the unknown equilibrium reac-

tion rates to be eliminated from the system. More precisely, a variant of the

reduction scheme including additional variables was used, which were shown to

be mandatory to obtain a well-conditioned global problem, cf. [HKK10, Hof10].

In contrast to the work mentioned above, we need an additional transformed

variable representing the interphase mass exchange between the liquid and the

gas phase, and there are no partial differential equations decoupling from the

rest of the system after using the reduction scheme. This is due to the nonlinear

coupling of flow and transport in our multiphase model, which implies that each

partial differential equation depends on the liquid pressure variable. If, however,

the primary variables are chosen appropriately, it remains possible to reduce the

size of the global problem by shifting at least one block of ODEs to the local

level.

In the transformed system, the algebraic equations resulting from the chemical

equilibria are eliminated with the help of a nonlinear, implicitly defined resolution

function. Hereby, the equilibrium conditions consisting of equations and inequal-

ity constraints are formulated as a complementarity problem and rewritten as

algebraic equations using the minimum function. The existence of a resolution

function is established by using the connection of the nonlinear system of alge-

braic equations to a constrained minimization problem based on a modified Gibbs

functional, and numerical strategies to evaluate it numerically are presented. This

208

CONCLUSION 209

requires a formulation based on the logarithms of the concentrations. Moreover,

we present a method to ensure that the local nonlinear solver does not produce

nonphysical values while solving the chemical subproblem. As a nonlinear solver,

the Semismooth Newton method is used, which converges locally quadratically.

The use of the resolution function corresponds to an elimination on the level of

the nonlinear solver, which can be efficiently carried out on multiple processors.

The time discretization of the system obtained after the variable transformation

and the use of the resolution function is done using the implicit Euler method,

and the spatial discretization is based on a linear finite element method, enhanced

by an upwind finite volume stabilization for advection–dominated problems.

Using a global implicit approach, the nonlinear systems that remain to be solved

in each time step are treated with Newton’s method. With the help of the

resolution function, we are ready to use a persistent set of primary variables that

are valid in either phase state and for an arbitrary mineral assemblage. Thus,

the discontinuous switch of variables during the simulation as a result of the

appearance or disappearance of the gas phase or the dissolution and precipitation

of minerals can be avoided.

Regarding the definition of primary variables, our numerical tests indicate that

the use of two extended pressure variables leads to better convergence properties

of the nonlinear solver than the use of only one pressure variable in a smaller

global system. This is because the nonlinear coefficient functions depend directly

on one primary variable if two phase pressures are used as primary variables.

Moreover, there are fewer nonlinearities in the transport operators which involve

the gradients of both phase pressures.

Different numerical tests related to hydrogen migration in deep geological repos-

itories of radioactive waste and to CO2 sequestration show that our solver, which

was designed for 2D and 3D computations and implemented in a parallel finite

element toolbox, provides accurate numerical results, and that it is capable of

handling the strongly nonlinear coupling of flow, transport, chemical reactions,

and mass transfer across the phases. If a strict stopping criterion is used for the

nonlinear solver, the global implicit approach required significantly less compu-

tation time than the sequential iterative approach.

210 CONCLUSION

The second part of this work deals with the analysis of robust mixed hybrid finite

element methods of lowest order for advection–diffusion problems. When advec-

tion strongly dominates diffusion, the standard mixed method [DR85] typically

fails to resolve steep gradients in the analytical solution and produces approx-

imations that are polluted by spurious oscillations. One of many methods to

overcome this is to employ upwinding to the method. In the approach studied

here, this is accomplished by extending the classical scheme with the help of the

Lagrange multipliers that arise from the hybrid problem formulation. It relies

on the fact that the Lagrange multipliers represent approximations of the scalar

unknown on the interelement boundaries and uses them in the approximation of

the advective fluxes. Using techniques from the a posteriori error analysis, we are

able to establish optimal order convergence for a new class of methods including

the standard method, a partial upwind method, and a full upwind method. The

advantage of the specific choice of the upwind weights is the fact that the method

remains local such that static condensation can still be employed, whereas the

standard upwind-mixed method requires information from neighbor cells such

that static condensation is no longer applicable.

Concerning approximations with the BDM1 mixed finite element, we show that

the use of Lagrange multipliers in the discretization of the advective term pro-

vides optimal second order convergence for the total flux variable consisting of

advective and diffusive flux in the L2 norm, whereas the standard mixed method

is known to be of suboptimal first order accuracy only. Thus, the modification,

which was intended to enhance the robustness of the standard method against

advection–dominance for moderately advection dominated problems, leads to a

higher accuracy of the method without increasing the computational costs.

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