Modeling and Analysis of Almost Unidirectional Flows in Porous Media Alaa Armiti

157 Pages·2017·3.14 MB·English

Modeling and Analysis of Almost Unidirectional
Flows in Porous Media
Von der Fakult¨at fu¨r Mathematik und Physik der Universit¨at Stuttgart
zur Erlangung der Wu¨rde eines Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigte Abhandlung
Vorgelegt von
Alaa Armiti
aus Nablus (Palestine)
Hauptberichter: Prof. Dr. Christian Rohde
Mitberichter: Prof. Dr. Sorin Pop
Prof. Dr. Dominik G¨oddeke
Tag der mu¨ndlichen Pru¨fung: 11.10.2017
Institut fu¨r Angewandte Analysis und Numerische Simulation, Universit¨at Stuttgart
2017
D93

Acknowledgements
At the end of PhD journey, I would like to thank all people who supported me
to make this thesis possible. First and foremost, I would like to express my deepest
sense of gratitude to my advisor Professor Dr. Christian Rohde for giving me the
opportunity to work in his research group. I thank him for the guidance and great
effort he put into training me in the scientific field. I also deeply appreciate his
support since the days I arrived Germany.
IamthankfulandproudtobeagraduatestudentoftheUniversityofStuttgart.
I would like to express my deepest gratitude to the Deutscher Akademischer Aus-
tauschdienst (DAAD) for the financial support.
I would like to thank the Non-linearities and Upscaling in Porous Media (NU-
PUS) for giving me the opportunity to participate in several events and lectures
that enriched my knowledge in porous media.
I would like to thank Prof. Sorin Pop, Prof. Dominik Go¨ddeke and Prof.
Guido Schneider for reviewing this thesis and taking the time for my exam.
My thanks to all my colleges at the IANS in Stuttgart for the great time and
all kinds of help. Especially, I would like to thank Dr. Jan Giesselmann and Dr.
Iryna Rybak for proofreading the thesis and the helpful discussions. My special
appreciation to my friend Emna Mejri from the IWS in Stuttgart for all kinds of
support and the careful reading of the thesis.
My steadfast husband, Ayser. Your continued love, faith in me, and sharing
my dream made the completion of thesis possible. You helped me to keep focused
at times I thought that it is impossible to continue. My sons, you have inspired
me with your love and patience. Thank you Mohammed and Abdulrahman for
being such great boys.
My mother and father, you dreamed and believed that I will succeed in this
task since I was a young girl. Your love and countless prayers guided me all the
time to make this thesis possible.
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Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction 1
1.1 Modeling Fluid Flows in the Subsurface . . . . . . . . . . . . . . . . 3
1.2 Well-Posedness of Mathematical Models . . . . . . . . . . . . . . . 4
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Model Problems and Mathematical Preliminaries 9
2.1 The Two-Phase Flow Model . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Physical Properties and Relationships . . . . . . . . . . . . . 9
2.1.2 The Two-Phase Flow Model . . . . . . . . . . . . . . . . . . 14
2.1.3 Fractional Flow Formulation . . . . . . . . . . . . . . . . . . 15
2.2 Sobolev Spaces and Compactness Theorems . . . . . . . . . . . . . 16
2.2.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Bochner Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Compactness Theorems . . . . . . . . . . . . . . . . . . . . 18
2.2.4 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
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3 The Vertical Equilibrium Model 21
3.1 Problem Description and Related Work . . . . . . . . . . . . . . . . 22
3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 The Two-Phase Flow Model . . . . . . . . . . . . . . . . . . 24
3.2.2 The Vertically Integrated Model (VI-Model) . . . . . . . . . 25
3.2.3 The Multiscale Model . . . . . . . . . . . . . . . . . . . . . 27
3.3 The Vertical Equilibrium Model (VE-Model) . . . . . . . . . . . . . 29
3.3.1 Dimensionless Two-Phase Flow Model . . . . . . . . . . . . 30
3.3.2 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Models Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.1 Finite-Volume Scheme . . . . . . . . . . . . . . . . . . . . . 35
3.4.2 VE-Model vs. Two-Phase Flow Model . . . . . . . . . . . . 39
3.4.3 VE-Model vs. VI-Model . . . . . . . . . . . . . . . . . . . . 41
3.4.4 VE-Model vs. Multiscale Model . . . . . . . . . . . . . . . . 46
3.5 Reduced Regularity of the VE-Model . . . . . . . . . . . . . . . . . 51
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 The Brinkman Vertical Equilibrium Model 55
4.1 Brinkman Two-Phase Flow Model . . . . . . . . . . . . . . . . . . . 56
4.2 The Brinkman VE-Model . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.1 Dimensionless Model . . . . . . . . . . . . . . . . . . . . . . 59
4.2.2 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.1 Brinkman VE-model vs. Brinkman Two-Phase Flow Model . 66
4.3.2 Brinkman VE-model vs. VE-model . . . . . . . . . . . . . . 74
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 Well-posedness of the Brinkman Vertical Equilibrium Model 81
5.1 Preliminaries and Assumptions . . . . . . . . . . . . . . . . . . . . 83
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5.2 A priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4 Boundedness of Weak Solutions in L∞(Ω ) . . . . . . . . . . . . . . 99
T
5.5 Uniqueness of Weak Solutions . . . . . . . . . . . . . . . . . . . . . 103
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6 Gravity-Driven Water Infiltration into Unsaturated Zone 109
6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.1.1 Richards’ model . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.1.2 Fourth-Order Model of Cueto-Felgueroso and Juanes . . . . 111
6.2 The Fourth-Order Model . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3 Preliminaries and Assumptions . . . . . . . . . . . . . . . . . . . . 114
6.4 Well-posedness of the Transformed Fourth-Order Model . . . . . . . 116
6.4.1 A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . 119
6.4.2 Convergence Results . . . . . . . . . . . . . . . . . . . . . . 122
6.4.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.6 Well-posedness of the Fourth-Order Model . . . . . . . . . . . . . . 133
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7 Summary and Outlook 137
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Bibliography 141
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Abstract
For many environmental and industrial applications, such as aquifers purification
and extraction of fossil fuels from reservoirs, it is crucial to study fluid flows in
porousmediaandthephenomenonofsaturationovershoots. Typically, mathemat-
ical models are used to describe and predict such flows. These models are often
coupled systems of nonlinear differential equations. As a result, exact solutions are
hard to obtain and only numerical simulations are feasible where two issues must
be tackled. First, it is necessary to prove the well-posedness of these models as a
validating criterion of the simulations. Second, such simulations have high com-
putational complexity depending on the number of unknowns, the effective forces,
and media’s volume. In this thesis, we contribute to three aspects of modeling
transport processes in the subsurface: first, reducing models’ complexity based
on natural properties, second, describing more physical phenomena like saturation
overshoots, and third, proving models’ well-posedness.
In saturated media, such as aquifers and reservoirs, fluid flows are modeled
using the two-phase flow model, which is a coupled system of two unknowns.
Applying this model in such large media might lead to very high computational
complexity. However, transport processes in saturated media are in general almost
horizontal,suchastheflowofextractedoil. Thisnaturalpropertyhasbeenutilized
to reduce model’s complexity. Of specific interest is the approach proposed by
Yortsos (1995), which reduces the number of unknowns. In this thesis, we perform
several numerical examples to analyze the validity and numerical efficiency of this
approach. We also extend it to describe almost horizontal flows in media with high
porosity as well as macroscopically heterogeneous media, where fluids velocity is
often modeled using Brinkman’s equations instead of Darcy’s law. Our extended
approachleadstoathird-orderpseudo-parabolicdifferentialequationofsaturation
only. We provide various numerical tests to show the computational efficiency of
the extended model. Additionally, these tests show the ability of the model to
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describethephenomenonofsaturationovershoots. Finally, weinvestigatethewell-
posedness of the extended model, where we prove the existence of weak solutions
and the uniqueness for a special case of it.
In unsaturated soils, the vertical infiltration processes are modeled using
Richards’ equation, which is a second-order parabolic equation of saturation. This
equation is unable to describe saturation overshoots, as it satisfies the maximum
principle. Therefore, different higher-order extensions have been suggested. We
propose a nonlinear fourth-order extension of Richards’ equation. We investi-
gate the well-posedness of this model by applying Kirchhoff’s transformation
to linearize the higher-order terms. Then, we prove the well-posedness of the
transformed equation. Finally, we use the inverse of Kirchhoff’s transformation
to prove the well-posedness of the original equation.
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