a generalised fully coupled hybrid-mixed finite element ... · and fabian brunner who supported me...

A Generalised Fully CoupledHybrid-Mixed Finite Element

Approximation forMulti-Component Two-Phase

Flows in Porous Media

Der Naturwissenschaftlichen Fakultat

der Friedrich–Alexander–Universitat

Erlangen–Nurnberg

zur

Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von

Torsten Muller

aus Neustrelitz

Als Dissertation genehmigt

von der Naturwissenschaftlichen Fakultat

der Friedrich-Alexander-Universitat Erlangen–Nurnberg

Tag der mundlichen Prufung: 25. April 2013

Vorsitzender des Promotionsorgans: Prof. Dr. Johannes Barth

Gutachter/in: Prof. Dr. Peter Knabner

Prof. Dr. Peter Bastian

Acknowledgements

I would like to thank all of those people who helped make this thesis possible.

First of all, I feel indebted to my supervisor Prof. Dr. Peter Knabner who gave

me the opportunity to work on this fascinating subject. I would particularly like

to thank Dr. Estelle Marchand for the support and guidance she showed me

throughout my dissertation writing. This thesis would not have been possible

without her knowledge and prior work on this topic. I could approach her with

any questions at any time. I am also deeply grateful to Dr. Joachim Hoffmann

and Fabian Brunner who supported me in all problems concerning the software

M++. Besides I owe special thankfulness to Marion Muller and Roland Voet for

their time-consuming proof-reading and suggestions for linguistic improvements.

Furthermore, I would like to show my gratitude to Dr. Ellen Rochlitzer. With

her patience and encouragement during the last four years she made a major

contribution to the completion of the project. Last but not least I am obligated

to all my colleagues at the Chair of Applied Mathematics I for the excellent

working atmosphere and the helpfulness I experienced during the last years.

i

Contents

Acknowledgements i

1 Introduction 1

1.1 Latest State of the Art . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Objective of this Work . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Overview of this Work . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Continuous Model 7

2.1 Recent Approaches and Principal Variables . . . . . . . . . . . . . 9

2.2 State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Static Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Other static laws . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.3 Subsumption of all Closure Relationsships . . . . . . . . . 20

2.5 Reformulation with Complementarity Constraints . . . . . . . . . 21

2.6 Summary of the Model . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7 Generalised Formulation . . . . . . . . . . . . . . . . . . . . . . . 26

3 Discretisation 29

3.1 Discretisation in Time . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Spatial Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.2 Diffusion laws . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.3 Hybridisation . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.4 Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.5 Static Condensation . . . . . . . . . . . . . . . . . . . . . 41

3.3 Implementation Aspects . . . . . . . . . . . . . . . . . . . . . . . 44

iii

iv CONTENTS

4 Numerics 47

4.1 Phase Appearance and Disappearance . . . . . . . . . . . . . . . 49

4.1.1 Setup of the Experiment . . . . . . . . . . . . . . . . . . . 50

4.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Convergence Properties . . . . . . . . . . . . . . . . . . . . . . . . 58


4.2.2 Convergence in Space . . . . . . . . . . . . . . . . . . . . . 65

4.2.3 Convergence in Time . . . . . . . . . . . . . . . . . . . . . 69

4.2.4 Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3 Liquid Phase Disappearance . . . . . . . . . . . . . . . . . . . . . 74


4.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4 Two-Phase Flow in Heterogeneous Porous Media . . . . . . . . . 78


4.4.2 Results (A) . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4.3 Results (B) . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.4.4 Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.5 Inhom. Init. Conditions in an Insolated Domain . . . . . . . . . . 86


4.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.6 2D Test Case: Comparison of Models . . . . . . . . . . . . . . . . 92


4.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.6.3 Subsumption . . . . . . . . . . . . . . . . . . . . . . . . . 99

A Derivatives 101

A.1 Derivatives of the Resolution Function . . . . . . . . . . . . . . . 101

A.2 Derivatives of the Flux Continuity Equations . . . . . . . . . . . . 102

Summary 104

Deutscher Titel und Zusammenfassung 106

Bibliography 109

List of Figures

2.1 Phase diagram for a simple equilibrium law . . . . . . . . . . . . . 16

2.2 Phase diagram for example 2 . . . . . . . . . . . . . . . . . . . . . 18

3.1 Triangulation of Ω . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Triangular element with degrees of freedom for the flux basis func-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Overview about considered unknowns . . . . . . . . . . . . . . . . 40

3.4 Scheme of the algorithm for time step n . . . . . . . . . . . . . . 42

4.1 Geometrical setup for numerical experiment 1 . . . . . . . . . . . 51

4.2 Coarse and fine mesh for numerical experiment 1 . . . . . . . . . 51

4.3 Profiles of phase pressures Pℓ, Pg and gas saturation S at Γin . . . 53

4.4 Profiles of component fluxes at Γout . . . . . . . . . . . . . . . . . 53

4.5 Liquid phase pressure Pℓ as function of x at t = 100, 140 (first

row), t = 200, 500 (second row), t = 5000, 6700 (third row) and

t = 8400, 10000 (last row) centuries. . . . . . . . . . . . . . . . . . 54

4.6 Total molar fraction of hydrogen (i.e. X) as function of x at

t = 100, 140 (first row), t = 200, 500 (second row), t = 5000, 6700

(third row) and t = 8400, 10000 (last row) centuries. . . . . . . . . 55

4.7 Gas saturation (i.e. S) as function of x at t = 100, 140 (first

row), t = 200, 500 (second row), t = 5000, 6700 (third row) and

t = 8400, 10000 (last row) centuries. . . . . . . . . . . . . . . . . . 56

4.8 Gas phase pressure Pg as function of x at t = 100, 140 (first row),

t = 200, 500 (second row), t = 5000, 6700 (third row) and t =

8400, 10000 (last row) centuries. . . . . . . . . . . . . . . . . . . . 57

4.9 Capillary pressure Pc as function of x at t = 100, 140 (first row),

t = 200, 500 (second row), t = 5000, 6700 (third row) and t =

8400, 10000 (last row) centuries. . . . . . . . . . . . . . . . . . . . 58

4.10 Geometrical setup for the test of convergence . . . . . . . . . . . . 60

4.11 Computational coarse grid . . . . . . . . . . . . . . . . . . . . . . 60

v

vi LIST OF FIGURES

4.12 Evolution of Pℓ in time . . . . . . . . . . . . . . . . . . . . . . . . 61

4.13 Evolution of Pg in time . . . . . . . . . . . . . . . . . . . . . . . . 62

4.14 Evolution of X in time . . . . . . . . . . . . . . . . . . . . . . . . 63

4.15 Evolution of q(1) in time . . . . . . . . . . . . . . . . . . . . . . . 63

4.16 Evolution of S in time . . . . . . . . . . . . . . . . . . . . . . . . 64

4.17 Uniform refinement of a cell Ωi . . . . . . . . . . . . . . . . . . . 65

4.18 Profiles of Pℓ for varying refinement levels l = 0, . . . , 4 . . . . . . . 67

4.19 Profiles of Pg for varying refinement levels l = 0, . . . , 4 . . . . . . . 67

4.20 Profiles of X for varying refinement levels l = 0, . . . , 4 . . . . . . . 68

4.21 Profiles of S for varying refinement levels l = 0, . . . , 4 . . . . . . . 68

4.22 Profiles of q(1) for varying refinement levels l = 0, . . . , 4 . . . . . . 69

4.23 Profiles of Pℓ at T = Tsim for different time step sizes ∆t . . . . . 71

4.24 Profiles of Pg at T = Tsim for different time step sizes ∆t . . . . . 72

4.25 Profiles of X at T = Tsim for different time step sizes ∆t . . . . . 72

4.26 Saturation profiles at T = Tsim for different time step sizes ∆t . . 73

4.27 Profiles of q(1) at T = Tsim for different time step sizes ∆t . . . . 73


4.29 Evolution of S at x ≈ 50m (left) and x ≈ 140m (right) in time. . . 77

4.30 Evolution of X at x ≈ 50m (left) and x ≈ 140m (right) in time. . 77

4.31 Values of S and X at t = 3005 centuries. . . . . . . . . . . . . . . 78

4.32 Values of S and X at t = 13500 centuries. . . . . . . . . . . . . . 78

4.33 Values of S and X at t = 27500 centuries. . . . . . . . . . . . . . 79


4.35 (A) Evolution of Pℓ in time . . . . . . . . . . . . . . . . . . . . . 82

4.36 (A) Evolution of Pg in time . . . . . . . . . . . . . . . . . . . . . 83

4.37 (A) Evolution of S in time . . . . . . . . . . . . . . . . . . . . . . 83

4.38 (B) Evolution of Pℓ in time . . . . . . . . . . . . . . . . . . . . . . 84

4.39 (B) Evolution of Pg in time . . . . . . . . . . . . . . . . . . . . . 85

4.40 (B) Evolution of S in time . . . . . . . . . . . . . . . . . . . . . . 85


4.42 Profiles of Pℓ, Pg and S at t = 10s. . . . . . . . . . . . . . . . . . . 88

4.43 Profiles of Pℓ, Pg and S at t = 100s. . . . . . . . . . . . . . . . . . 88


4.45 Profiles of Pℓ, Pg and S at t = 1000s. . . . . . . . . . . . . . . . . 89





4.50 Profiles of Pℓ, Pg and S at t = 2 · 105s. . . . . . . . . . . . . . . . 91

LIST OF FIGURES vii

4.51 Profiles of Pℓ, Pg and S at t = 5 · 105s. . . . . . . . . . . . . . . . 91


4.53 Geometrical setup for 2D test case . . . . . . . . . . . . . . . . . 93

4.54 Partially refined mesh. . . . . . . . . . . . . . . . . . . . . . . . . 94

4.55 Evolution of Pℓ in time. . . . . . . . . . . . . . . . . . . . . . . . 95

4.56 Evolution of X in time. . . . . . . . . . . . . . . . . . . . . . . . . 95

4.57 Evolution of S in time. . . . . . . . . . . . . . . . . . . . . . . . . 96

4.58 Difference between standard and alternative model for Pℓ . . . . . 96

4.59 Difference between standard and alternative model for X . . . . . 97

4.60 Difference between standard and alternative model for S . . . . . 97

4.61 Influence of gravity on Pℓ in time . . . . . . . . . . . . . . . . . . 98

4.62 Influence of gravity on X in time . . . . . . . . . . . . . . . . . . 98

4.63 Influence of gravity on S in time . . . . . . . . . . . . . . . . . . . 99

4.64 Difference between standard and alternative model for Pℓ including

gravitational effects. . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.65 Difference between standard and alternative model forX including


4.66 Difference between standard and alternative model for S including


viii LIST OF FIGURES

List of Tables

2.1 State variables for a two-phase fluid . . . . . . . . . . . . . . . . . 11

2.2 Choices of coefficient functions for the standard model . . . . . . 27

2.3 Crossover to the alternative model . . . . . . . . . . . . . . . . . 27

4.1 Used parameters for the Van Genuchten parametrisation . . . . . 48

4.2 Soil and fluid parameters for numerical experiments. . . . . . . . . 49

4.3 Parameters for numerical experiment 1 . . . . . . . . . . . . . . . 50

4.4 Parameters for the study of convergence . . . . . . . . . . . . . . 60

4.5 Evolution of the error ‖u-uh‖L2(Ω) in space for the scalar unknowns

u ∈ P,X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.6 Evolution of the error ‖qα-qαh‖L2(Ω) in space for the flux unknowns

with α ∈ (1), (2) . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.7 Evolution of the error ‖u-uh‖L2(Ω) in time for the scalar unknowns

u ∈ P,X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.8 Evolution of the error ‖qα-qαh‖L2(Ω) in time for the flux unknowns

with α ∈ (1), (2) . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.9 Parameters and constants for numerical experiment 2 . . . . . . . 76

4.10 Simulation parameters for numerical experiment 3 . . . . . . . . . 80

4.11 Varying soil and fluid parameters . . . . . . . . . . . . . . . . . . 80

4.12 Parameters for numerical experiment 4 . . . . . . . . . . . . . . . 86

4.13 Parameters for the comparison of models. . . . . . . . . . . . . . . 92

ix

x LIST OF TABLES

Chapter 1

Introduction

Although multi-phase multi-component flow processes have been discussed for

some decades it attracted wide interest during the last years. Many applications

with respect to multi-phase multi-component flows containing water, oil and gas

have been developed to describe fluid flow in petroleum reservoirs. More recent

topics deal for example with hydrogen flow processes in underground nuclear

waste storage sites. So far radioactive material is often stored in sealed containers.

For reasons of corrosive processes [52] as well as the radiolysis of water [16]

hydrogen may occur on the one hand within the containers and on the other

hand within the storage site. Both processes lead to an increase of the gas phase

pressure within the container respectively storage site. Finally, when the pressure

reaches a critical value, these processes can lead to cracks and thus to a release

of radioactive material.

Another controversial discussed topic in today’s society is the reduction of

atmospheric carbon dioxide (CO2) by the so-called CO2 sequestration, i.e. the

storage of CO2 in the subsurface. Instead of being released to the atmosphere,

in this process CO2 is condensed into the super-critical state and pumped into

subsurface reservoirs which have to fulfil special conditions, e.g. concerning min-

imum depth, minimum temperature and the presence of a so-called cap-rock. A

cap-rock is a formation that is impermeable for CO2 and thus prevents the CO2

from rising back to the surface. This process is also called structural trapping,

followed by several other trapping mechanisms [19].

The two above mentioned applications are challenging with respect to the

physics the models have to include and also with respect to the numerical methods

that are necessary to solve these models by the use of simulation tools. It is very

important to have reliable numerical methods that provide predictions in order

to assess risks for storage sites.

1

2 CHAPTER 1. INTRODUCTION

1.1 Latest State of the Art

The models dealing with multi-phase multi-component flows in porous media are

highly non-linear and the equations are strongly coupled. This leads to a number

of numerical difficulties that have to be overcome. Even the choice of principal

unknowns is not obvious since classical choices become meaningless when the

disappearance of single phases is incorporated into the model. Further details

about that are given in section 2.1.

Different toolboxes (DuMuX [31], MUFTE [36], ROCKFLOW [15], TOUGH2

[47], IPARS-CO2 [20], etc.) have been developed to simulate multi-phase multi-

component flows. Due to the importance of the CO2 sequestration, benchmarks

were set up to compare the different tools including different numerical methods

[23]. The toolboxes differ with respect to the chosen discretisation in space and

time. For time discretisation the implicit Euler method is often used. One ex-

ception is the tool IPARS-CO2 that computes the pressures implicitly whereas

the corresponding concentrations are determined explicitly.

The chosen spatial discretisation is much more variable. For example, inte-

grated finite difference methods are used by TOUGH2. DuMuX and MUFTE

use a box method [6, 32] as spatial discretisation. Finite elements are applied in

ROCKFLOW and mixed finite element methods are utilised in IPARS-CO2.

Nevertheless, the methods and models are very often problem-specific and not

generalised which may become problematic when it shall be used for a wider range

of applications including numerous physical effects. In this work a generalised

approach is presented in which the phase transitions are enabled by reformulating

the given problem into a complementarity formulation. For the incompressible

case (e.g. valid for systems containing water and oil) a proof of the existence

and uniqueness of a solution is already available (e.g. in [2]). With respect

to compressible multi-phase multi-component flows a proof for the case of two

compressible and partially miscible phase flow in porous media [56] is already

available. For other cases this has still to be done.

1.2 Objective of this Work

The goal of this work is the development of a generalised hybrid mixed finite

element approximation for compositional two-phase flows in porous media. The

two phases mostly consist of two components. The presented model formulation

is a generalisation of the one given in [12]. The choice of principal unknowns

varies because the considered model used in this work also allows the liquid

1.3. OVERVIEW OF THIS WORK 3

phase to appear and disappear locally respectively globally. The main problem

is that parts of the model (the parabolic partial differential equations (PDEs),

single variables) degenerate when one phase is missing. This leads to numerical

difficulties that have been overcome by choosing principal variables which allow

to compute all state variables, no matter which phases are present, and by using

a “double” complementarity formulation (see [40], [43]) for the resulting static

system of equations. The convergence properties of the method are determined

numerically. Finally, the scheme is applied to problems of a MoMaS benchmark

[11] comprising academic test cases and to a realistic scenario with respect to gas

(hydrogen) migration processes in nuclear waste repositories.

In consequence of the complexity of the topic and in order to realise the goal of

this work the content was split up into two parts, the formulation and properties of

the model as well as the numerical scheme and numerical results which Dr. Estelle

Marchand and myself were working on. Due to her experiences on this topic and

her prior work Dr. Estelle Marchand mainly focused on the model formulation

(given in chapter 2) which provides the basis for further considerations. Referring

to that I concentrated on the numerical scheme and the examination of numerical

properties including numerous experiments. As a result of this splitting, two

separate papers were written.[49, 48]

1.3 Overview of this Work

The used model for the miscible two-phase two-component flow, which is a gener-

alisation of the one given by Bourgeat et.al.[12], is presented in chapter 2. After

a short introduction of parameters that deal with rock properties, important fea-

tures of the model, as well as necessary assumptions that have been made and

some notations are given. Multi-phase multi-component flow processes have al-

ready been discussed by other researchers [12, 35, 7]. The non-linearities and the

strong coupling of the model equations cause numerical difficulties that are sur-

mounted by different approaches. In section 2.1 the choice of principal unknowns

is discussed and a list of recent approaches is presented. Section 2.2 provides

a list of all state variables including the dimensions. Additionally, all possible

states are defined there.

In section 2.3 the mass conservation equations for each of the possible states

are given, followed by a modification of the model that leads to an alternative

model formulation including different physical properties. In section 4.6 both

model variations are tested to reveal these differences. The static part of the

model that is essential to close the system of equation is shown in section 2.4.


Here, a short description of phase diagrams as well as a summary of all closure

relationships is given.

After introducing all model equations and closure relations, a reformulation

into a double complementarity formulation is presented in section 2.5. This im-

plies the possibility of appearing and disappearing of both phases, which is one of

the major features of this model. Finally, in section 2.6 the problem is summarised

including the system of conservation equations, the system of static equations, a

list of all closure relationships as well as boundary and initial conditions. With

respect to the implementation a general formulation closes this chapter.

In chapter 3 the system of mass conservation equations is rewritten in a mixed

formulation and by choosing adequate ansatz spaces the weak formulation is de-

rived. First, in section 3.1 the time discretisation is completed by applying an

Euler scheme. The Euler schemes allow different choices (implicit resp. explicit)

of time discretisation for different parts of the model. In general, an implicit

Euler scheme is employed on the terms that deal with diffusive processes whereas

the convective terms are treated explicitly. One exception is the treatment of

the diffusion coefficients that can be chosen either implicitly or explicitly. This is

followed by the spatial discretisation (section 3.2) in which the full discrete mixed

variational formulation is derived. A hybridisation technique is applied and new

variables, the so-called Lagrange multipliers, are introduced to the problem as

additional unknowns. Finally, the number of unknowns is reduced by static con-

densation. In a first step the diffusive fluxes are eliminated explicitly on each cell.

These fluxes are plugged into the remaining equations and a resolution function

is set up. By solving this locally with a Newton’s method, a global system of

equations is obtained that is only dependent on the Lagrange multipliers. The de-

tailed explanation of the algorithm is given in section 3.2.5. This chapter is closed

by information about the software tool in which the algorithm is implemented.

Chapter 4 deals with a number of numerical examples that are set up in order

to show the properties of the presented method. Before details of each experiment

are given, physical parameters are briefly introduced and assumptions are given

as well as information about the used parametrisation for the capillary pressure

saturation relation. The examples presented in this work are motivated either by

a MoMaS benchmark [11] or by the aim to show single effects.

One of the test cases from the above mentioned benchmark is presented in

section 4.1. The purpose of this experiment is to show the gas phase appearance

and disappearance. Next to the original experiment (to enable the comparison

of the results with those of other groups), the setup is modified to be able to

compare the different treatments of the diffusion coefficient and the quality of

1.3. OVERVIEW OF THIS WORK 5

the solution with respect to varying time step sizes and mesh refinement levels.

In the results section there can be seen that the method produces similar results

although the variations are chosen extremely.

As a consequence of those results the convergence properties of the method are

determined in section 4.2. The objective of this section is to determine whether

the method converges in space and time and if it does to find out at which rate.

Since all convergence results are obtained numerically the proceeding is described

first. The general setup for the experiment is given and simulation results are

shown for the reference solution that is the base for the computation of the

convergence rates. The convergence in space (resp. time) is presented in section

4.2.2 (resp. section 4.2.3). In addition to the development of the corresponding

discretisation error a number of figures are presented to show that the method

converges.

Each of the following examples focuses on a special feature of the model. One

of the major features is the liquid phase appearance and disappearance, set out

in section 4.3. Fluid flow in a heterogeneous porous medium is determined in

the numerical experiments 3 and 4 (sections 4.4 and 4.5). Here, different initial

setups are chosen. The first part of experiment 3 is to show the diffusive flow

of a light component through a porous medium which consists of different rock

types where soil parameters are chosen differently. The gas phase appearance is

still taken into account. In the second part phase transitions are neglected. Soil

parameters and also the parametrisation parameters are set up such that one

part of the rock is almost impermeable for the light component. In experiment

4, within a closed domain, two different saturations of the light component are

assumed. For an infinite time diffusion leads to equilibrium state at which the

phase pressures and saturations are the same in the computational domain.

The last experiment presented is based on an example given in [12]. A source

of hydrogen is defined within a 2D domain. Starting in a purely liquid state,

the gas phase appears at a certain time. This setup is modified in order to

compare the two different model formulations. This is done either neglecting

gravity or taking into account gravitational effects. For both cases, simulation

results received by using the standard model as well as the error between both

models are visualised.

Chapter 2

Continuous Model

The aim is to model compositional two-phase flow in porous media. A porous

medium is a solid (e.g. rock) that contains void spaces called pores. The porosity

φ is a measure of the void spaces within the material. It is defined as the ratio

between the volume of the void space within a representative elementary volume

Vv (see [9], section 1.2.2) and the total volume of it VT

φ =VvVT

and takes values between zero and one [28]. Typically it ranges from zero or near

zero to more than 0.6 depending on the material (e.g. granite: φ < 0.01, silt:

φ ∈ [0.34, 0.61]) [51].

The pore space may contain a large variety of different fluids, e.g. air, water or

minority fluids such as hydrocarbon [28]. Depending on whether these fluids are

miscible or immiscible, they establish one or more phases i with corresponding

phase saturations Si defined as the ratio of the volume Vi of pores filled with

phase i to the overall volume V of the pore space [35]

Si(x, t) =Vi(x, t)

V. (2.1)

The pores themselves are randomly distributed and they are interconnected,

i.e. the void space is connected [9]. The degree of connectivity of the pores

determines the ease with which a fluid can move through the porous medium,

called intrinsic permeability and is denoted by K. This is a parameter character-

ising the porous medium. In comparison to the intrinsic permeability the relative

permeability kri is a dimensionless number with

0 ≤nphases∑

i=1

kri(Si) ≤ 1 (2.2)

7

8 CHAPTER 2. CONTINUOUS MODEL

describing in what way the presence of one phase influences the flow behaviour

of another phase and depends on the phase saturations Si. The product of the

intrinsic permeability and the relative permeability is called effective permeability

[35].

Regarding the planned application, two phases (phase index i with i = ℓ for

the liquid phase and i = g for the gas phase) and two components (component

index α) are considered. Both phases (mostly) contain two components: a light

component (denoted by α = (1)), e.g. carbon dioxide (CO2) or hydrogen (H2),

and a heavy component (denoted by α = (2)), e.g. water (H2O).

Concerning the application of this model to underground gas migration pro-

cesses (e.g. CO2 sequestration, hydrogen migration in underground nuclear waste

storage sites) the model includes the possibility that one phase can disappear lo-

cally as well as globally (e.g. CO2 sequestration: the disappearance of the gas

phase when the mixture of injected CO2 and vaporised water in a storage site

becomes totally dissolved in the surrounding groundwater due to very high pres-

sures). The model also takes into account an exchange of components between

the two phases. For example, the heavy component (herein: water) vaporises

depending on the ambient temperature, ambient pressure and time and thus it

also becomes part of the gas phase. Reversely, CO2 is dissolved in water at a

certain rate depending on the ambient pressure, ambient temperature and time.

Additionally, capillary effects, i.e. effects caused by the difference between the

phase pressures due to the interfacial surface tension between the phases, are

taken into account.

The entire fluid system is assumed to be in thermal equilibrium and the porous

medium is rigid which means that the porosity φ is only a function of space that

is constant in time. Moreover, it is supposed that the entire pore space is either

filled with the liquid phase or the gas phase,∑

i

Si = 1 for i = ℓ, g (2.3)

where Si is the saturation of phase i.

The presented model involves mass conservation equations for each compo-

nent α, generalised Darcy’s law which describes the effects caused by the dif-

ferences in the phase pressures as well as the influence of the gravity, Fick’s law

which represents the effects caused by the differences in the molar fractions, mass

ratios or molar ratios of each component (totally or with respect to each com-

ponent) and phase pressures. One difficulty in modelling is that each part of

the model depends on different state variables. Each mass conservation equation

2.1. RECENT APPROACHES AND PRINCIPAL VARIABLES 9

is written in component densities or component amounts of substance, Darcy’s

law is a volumetric law which deals with saturations, a gravity law is a mass

law that is about phase density and Fick’s law is about molar fractions. Molar

fractions are closely linked to densities but this relationship is no longer linear

since we have several components. However, the main difficulty is to describe the

transition between the two-phase state and the states when one phase is missing

(called one-phase state). In that case the model equations degenerate causing

numerical problems.

2.1 Recent Approaches and Principal Variables

The inclusion of phase appearance and disappearance into the model influences

the choice of principal unknowns. For the system of two mass conservation equa-

tions two principal unknowns are needed. One standard choice would be to use

the liquid pressure Pℓ as one principal unknown [12, 34, 35]. But as soon as

the potential disappearance of each of the phases is considered here this variable

has no meaning when the liquid phase is missing. The saturation is also not a

good choice either since it is constant when one phase is missing and therefore

the number of thermodynamic degrees of freedom becomes 1. [46] The principal

unknowns have to ensure that the number of thermodynamic degrees of freedom

on both sides of both phase transitions remains 2 and that all the other state

variables can be determined by those two unknowns. A good choice of principal

unknowns for the considered applications is a mean pressure P that is a convex

combination of the phase pressures,

P = γ(Sg)Pℓ + (1− γ(Sg))Pg (2.4)

with Pi the phase pressures, Sg (for shortening reasons denoted by S) the satu-

ration of the gas phase and γ(S) a monotone an increasing linear or non-linear

weight function with

γ(S) ∈ [0, 1], γ(0) = 0, γ(1) = 1 , e.g. γ(S) = S,

and the total molar fraction of the light component X = X(1) defined as

X =Ngx

(1)g S +Nℓx

(1)ℓ (1− S)

N(2.5)

withNi the molar density of phase i, x(1)i the molar fraction of the light component

in phase i, N the total molar density and S the saturation of the gas phase.


Definition (2.4) supports that P always remains meaningful for any value of the

saturation S, especially when one phase is missing.

Generally, the mass conservation equations for both components are strongly

coupled and thus inaccuracies in one unknown directly affect all other unknowns.

[14] One traditional approach is to combine the two equations in order to reduce

the coupling, e.g. by the introduction of fictitious variables, for example the

global pressure [21], and to apply a splitting method [7]. The global pressure

formulation presumes that either the gas concentration remains weak or that

both fluids have similar properties. Splitting methods have been developed to

deal with the rising complexity of partial differential equations [38]. Beside lots

of advantages these methods also have some disadvantages depending on the used

technique. Either an additional error, the so-called splitting error, is introduced or

the number of splitting steps (iterative splitting) makes the method become very

expensive with respect to computational resources and time. Nevertheless, these

methods are commonly used as they are a good way to overcome the problems

resulting from complicated equations, although other methods may have better

convergence properties.

An approach for dealing with phase appearance and disappearance is the so-

called method of negative saturations [1]. In order to overcome the numerical

problems the two-phase zone and oversaturated zones, i.e. a single phase gas

or single phase liquid, are described by a uniform system of classical two-phase

equations where the concept of saturation is extended so that the phase saturation

can also take values larger than one or less than zero. Physically this means that

the oversaturated single-phase states are considered as pseudo two-phase states

characterised by a negative saturation of the imaginary phase.

The model presented in this work keeps the coupling. It is an extension of the

model presented in [12]. This is based on the fact that the physical description is

very close to the model in [12], in particular concerning the choice of principal un-

knowns. A wider framework is enabled by the formulation as a complementarity

problem.

2.2 State Variables

All variables which are used to describe the state and the composition of the

fluid completely, in the following called state variables, are summarised in table

2.1 including the corresponding dimensions based on the international system of

units (SI).

2.2. STATE VARIABLES 11

A fraction (1 − φ) of the soil is filled with a rock matrix where φ ∈ (0, 1) is

the porosity. Moreover, the fluid is assumed to be homogeneous. By considering

molar quantities, a volume 1/φ m3 of the soil contains

• S m3 of the gas phase at the pressure Pg and (1−S) m3 of the liquid phase

at the pressure Pℓ,

• N mol overall amount of substance separated into SNg mol in the gas phase

and (1− S)Nℓ mol in the liquid phase,

• NX mol of the light component separated into SNgx(1)g mol in the gas phase

and (1− S)Nℓx(1)ℓ mol in the liquid phase,

• N(1 −X) mol of the heavy component separated into SNgx(2)g mol in the

gas phase and (1− S)Nℓx(2)ℓ mol in the liquid phase.

Pg pressure of gas phase [Pa]

Pℓ pressure of liquid phase [Pa]

Sg = S ∈ [0, 1] saturation of gas phase [-]

x(1)ℓ ≡ xℓ ∈ [0, 1] mol. fraction of the light comp. [-]

in the liquid phase

x(1)g ≡ xg ∈ [0, 1] mol. fraction of the light comp. [-]

in the gas phase

x(2)ℓ ≡ 1− xℓ mol. fraction of heavy comp. [-]

in the liquid phase

x(2)g ≡ 1− xg mol. fraction of heavy comp. [-]

in the gas phase

M (1) (constant) mol. mass of the light comp. [g/mol]

M (2) (constant) mol. mass of the heavy comp. [g/mol]

Ng mol. density of the gas phase [mol/m3]

Nℓ mol. density of the liquid phase [mol/m3]

N = SNg + (1− S)Nℓ total mol. density [mol/m3]

X(1) = X =NgxgS+Nℓxℓ(1−S)

Ntot. mol. fraction of the light comp. [-]

X(2) = 1−X tot. mol. fraction of the heavy comp. [-]

Table 2.1: State variables for a two-phase fluid

Instead of molar quantities mass quantities could be considered. The molar

mass of phase i is defined by

Mi =M (1)x(1)i +M (2)x

(2)i . (2.6)


so that the influence of the gravity depends on the density ρi (i.e. mass density)

of each phase i defined as

ρi =MiNi. (2.7)

By using the equations (2.6) and (2.7) a volume 1/φ m3 of soil contains

• ρ = NXM (1) +N(1−X)M (2) kg of overall mass separated into Sρg kg in

the gas phase and (1− S)ρℓ kg in the liquid phase,

• NXM (1) kg of light component separated into SNgx(1)g M (1) kg in the gas

phase and (1− S)Nℓx(1)ℓ M (1) kg in the liquid phase,

• N(1−X)M (2) kg of heavy component separated into SNgx(2)g M (2) kg in

the gas phase and (1− S)Nℓx(2)ℓ M (2) kg in the liquid phase.

Thereof the corresponding mass fractions can be computed.

The presented model is an equilibrium model [8]. A fluid is called saturated

when there are two phases whose composition is determined by the pressures and

saturations of both, which provides a relation between mass fractions, saturations

and pressures of both phases. The fluid is called oversaturated when either the

total molar fraction (resp. concentration) of the light component is so high or

the gas pressure is so low that only the gas phase is present. Finally, a fluid is

called undersaturated when either the total molar fraction (resp. concentration)

of the light component is so low or the liquid pressure is so high that only the

liquid phase is present. In general, when only one phase is present, the fluid is

called unsaturated.

On the hypothesis that variations of temperature are neglected, the fluid is at

the limit of equilibrium when the state of the fluid changes in one of the following

ways:

• either only one phase is present, but a second phase can appear with an

elementary variation of pressure or composition, or

• both phases are present, but one of them can disappear with an elementary

variation of pressure or composition.

2.3 Conservation Equations

In the case of both phases being present or at the limit of equilibrium, the con-

servation of mass for both components α = (1), (2) is ensured by the following

2.3. CONSERVATION EQUATIONS 13

system of non-linear PDEs:

φ∂t(NXα) +

∑

i=ℓ,g

∇· (Nixαi V i +NiSiW

αi ) = Fα (2.8)

where

V i = − Kkri(Si)

µi(∇Pi −MiNig) (generalised Darcy’s law) (2.9)

W αi = − Dα

i φ∇xαi (Fick’s law) (2.10)

with K the absolute permeability tensor (in m2) of the porous medium, µi the

viscosity of phase i (in Pa·s), kri the relative permeability of phase i in the porous

medium, g the gravity (in m/s2), Dαi the diffusion coefficient of component α in

phase i (in m2/s) and Fα is the source term of component α (in mol.m−3.s−1).

The term V i is the capillary redistribution which represents the effects of the

difference in phase pressures (extended Darcy’s law [25]). The term W αi is the

diffusion term and reveals the effects that are caused by concentration gradients

(Fick’s diffusion law). By multiplying the equation (2.8) by the constant Mα, it

is possible to get the conservation of mass of component α. In that case the right

hand side becomes MαFα (in kg.m−3.s−1).

The conservation equations are the same as used in [12]. Other approaches

(already discussed before) use variations of these conservation equations. For

example, in [39] for example the molar fractions xαℓ are replaced by mass fractions

ραi =xαiM

α

x(1)i M (1) + x

(2)i M (2)

for α = (1), (2) and i = ℓ, g.

In the case of an oversaturated fluid (i = g) or undersaturated fluid (i = ℓ)

equation (2.8) simplifies to

φ∂t(NiXα) +∇· (NiX

αV i +NiWαi ) = Fα for α = (1), (2). (2.11)

On the supposition that the diffusion is proportional to the gradient of the

concentration instead to the gradient of the molar fractions the equations (2.8)

and (2.10) alter to

φ∂t(NXα) +

∑

i=ℓ,g

∇·(

Nixαi V i + SiW

α

i

)

= Fα for α = (1), (2) (2.12)

with Vi as before (equation (2.9)) and

Wα

i = − Dαi φ∇ (Nix

αi ) for i = ℓ, g, α = (1), (2) (2.13)

when both phases are present or at the limit of equilibrium. Having an unsatu-

rated fluid (i = ℓ or i = g) equation (2.12) becomes

φ∂t(NiXα) +∇·

(

NiXαV i + W

α

i

)

= Fα for α = (1), (2). (2.14)


Model Denominations For the following considerations the model correspond-

ing to the system of equations (2.8), (2.11) is termed standard model, whereas

the model associated with equations (2.12), (2.14) is called the alternative

model. In section 4.1 the comparison of the results of the two models will show

that the difference of the two model formulations is not of importance, although

the solution of the alternative model shows an intersting property in terms of the

liquid pressure until the gas phase appears.

Theorem 2.1. From now on we assume Ω to be an open and bounded domain

with Lipschitz measurable boundaries. In addition, assume that the physical laws

and parameters are chosen such that

(H1) Sg = 0 ⇒ Pg = Pℓ ,

(H2) Nℓx(2)ℓ |Sg=0 being a continuously increasing differentiable function of Pℓ and

∂

∂Pℓ

(

Nℓx(2)ℓ

)

> − KkℓµℓDℓφ

Nℓx(2)ℓ ,

(H3) F (2) = 0

is complied. Then, any solution of the alternative model (equations (2.12),(2.14)) with Dirichlet boundary conditions and initial conditions such as

Sg(t = 0, .) ≡ 0

Pℓ(t = 0, .) ≡ const

Pℓ(t = 0, .) satisfies the Dirichlet boundary conditions

fulfills Pℓ(t, .) ≡ Pℓ(t = 0, .) a.e. as long as Sg ≡ 0. This is not valid for the

solutions of the standard model (equations (2.8), (2.11)).Proof: The proof is given in [49].

2.4 Static Equations

By having a closer look at table 2.1 there are eight unknowns that have to be

determined. They are the two-phase pressures Pℓ and Pg, the molar fractions

of the light component in the two phases x(1)ℓ and x

(1)g , the molar densities of

both phases Nℓ and Ng, the total molar fraction of the light component X and

the saturation of the gas phase S. Beside the two mass conservation equations,

with which one can only determine two variables, additional closure relationships

are indispensable. These closure relations are chosen that way that all state

2.4. STATIC EQUATIONS 15

variables of table 2.1 can be computed depending on two of the eight unknowns,

alternatively on the two principal variables.

First of all, one unknown can easily be eliminated by using the local capillary

equilibrium: the capillary pressure Pc. The capillary pressure is defined as the

difference between the phase pressures of two fluids [35] and, in the considered

context, it depends on the saturation of the gas phase

Pg − Pℓ = Pc(S). (2.15)

Applying the definition of the mean pressure P , it is possible to compute each of

the phase pressures: When both phases are present, additional to the given law

P = γ(S)Pg + (1− γ(S))Pℓ ⇒ Pg = P + (1− γ(S))Pc(S),

(definition of mean pressure P ) Pℓ = P − γ(S)Pc(S)

for the capillary equilibrium (2.15) and the definition of the total molar fraction

X (equation (2.5)) four further closure relationships are necessary to compute the

above mentioned eight unknowns of two of them. Nℓ and Ng are seen as explicit

functions and thus the number of additional closure relationships reduces to two.

In the case that only phase i is present, i.e. S = 0 or S = 1, only the three

unknowns Pi, x(1)i = X and Ni = N are relevant due to multiplications by zero

(see table 2.1) and thus only one more closure relationship is necessary. The

representation of equilibrium laws by means of phase diagrams will provide two

additional closure relationships.

2.4.1 Phase Diagrams

In general, a phase diagram is a graph that shows conditions on which phases

that are distinct by some physical condition can occur in equilibrium. Com-

mon components of phase diagrams are lines of equilibrium respectively phase

boundaries that mark conditions under which multiple phases can coexist in equi-

librium. Phase transitions appear along the lines of equilibrium. Phase diagrams

are classified based on the number of involved components in the system [53].

In the context considered here a binary phase diagram is used to get two more

closure relationships. Traditionally, binary phase diagrams describe a mixture of

two components without taking the presence of a porous medium into account

(i.e. without capillary pressure). Here, the phase diagram provides both closure

laws for the two-phase case and criteria directly based on the values of principal

variables to determine which phases are present (see condition (2.19) with jus-

tification). In the following section phase diagrams for molar fractions will be

given.


b

b

liquid

liquid + gas

gasa

c

b

Pℓ Pg

XComposition of Composition ofliquid phase gas phase

Xm(Pℓ)XM(Pg)

pressure ofliquid phase

pressure ofgas phase

Figure 2.1: Phase diagram for a simple equilibrium law

At first a simple equilibrium law is considered.

Example 1. On the assumption that two phases are present whereas the com-

position of each phase and the corresponding phase pressure are those of the one

phase state at the limit of equilibrium, for S ∈ (0, 1) or at the limit of equilibrium

holds

x(1)ℓ = Xm(Pℓ) , x(1)g = XM(Pg). (2.16)

In figure 2.1 the corresponding phase diagram is shown. Two closure relation-

ships that can be seen are

a

a+ b=SNg

Nand c = Pc(S). (2.17)

Usually Xm and XM are linear or non-linear increasing functions of the appropri-

ate phase pressure. This is reasonable since a small pressure Pℓ resp. Pg facilitates

the development of the gas phase. The phase diagram in figure 2.1 is chosen so

that for a fixed X at the bottom only the gas phase is present for low pressures.

By increasing the pressures the liquid phase occurs and the two-phase state is

present (in figure 2.1 denoted by liquid + gas). At a certain point the pressures

are so high that the gas phase will disappear and only the liquid phase will re-

main. This is assumed always to be true although it is invalid under extreme

conditions, which for instance exist at storage sites for CO2 (see [57]).

Because of theorem 2.2 the weight function γ have to be chosen such that for


all P and for S ∈ (0, 1)

SX ′M(Pg(P, S))

∂Pg(P, S)

∂S− (1− S)X ′

m(Pℓ(P, S))∂Pℓ(P, S)

∂S

> Xm(Pℓ(P, S))−XM(Pg(P, S))(2.18)

(with Pℓ(P, S) and Pg(P, S) as before) is satisfied. Because of the monotonicity

of Xm and XM , this condition is satisfied in particular if

∂Pg(P, S)

∂S≥ 0 and

∂Pℓ(P, S)

∂S≤ 0

which is true for

γ′ ≤ (1− γ)P ′c

Pc.

Theorem 2.2. If Xm and XM are increasing functions of Pℓ (resp. Pg) with

Xm < XM and if condition (2.18) is satisfied, two phases are present if and only

if

Xm(P ) < X < XM(P ). (2.19)

Proof: The proof can be found in [49].

Hence there is no need to compute the saturation S in order to know which

phases are present. A consequence of this model (see Figure 2.1) is the lever rule

[46]:a

a + b=SNg

N=

X −Xm(Pℓ)

XM(Pg)−Xm(Pℓ)

that is equivalent toSNg

(1− S)Nℓ

=X −Xm(Pℓ)

XM(Pg)−X.

Note that for the one-phase states

S = 0 ⇔ X −Xm(Pℓ) = 0 and

1− S = 0 ⇔ XM(Pg)−X = 0

is valid.

In the following a more general model than this of equations (2.16) is consid-

ered:

x(1)ℓ = Xm(P, S)

x(1)g = XM(P, S),

i.e. “composition of each phase is then uniquely determined by its phase pressure

and saturation” [12].


liquid

liquid + gas

gas

Pℓ Pg

XComposition of Composition ofliquid phase gas phase

Xm(P, 0)

pressure ofliquid phase

pressure ofgas phaseb b

XM(P, 1) ≡ 1

c

a b

Figure 2.2: Phase diagram for example 2

Theorem 2.3. If Xm(P, 0) < XM(P, 1) and if for all P and for all S ∈ (0, 1)

(1− S)∂Xm(P, S)

∂S+ S

∂XM (P, S)

∂S> Xm(P, S)−XM(P, S) (2.20)

holds, the condition for having two phases present is

Xm(P, 0) < X < XM(P, 1) .

The lever rule readsSNg

(1− S)Nℓ

=X −Xm(P, S)

XM(P, S)−X.

Proof: As for theorem 2.2 the proof can be found in [49].

Example 2. In [12, 39] Henry’s law is used on the assumptions that

• there is no water evaporation (i.e. no exchange of water between the liquid

and the gas phase), i.e. x(1)g = 1, and

• that the liquid phase is incompressible.

A corresponding phase diagram is given in figure 2.2. Just like in figure 2.1

the two closure relations (2.17) hold. In general, Henry’s law [8] is valid for

small concentrations of the light component within the liquid phase. It expresses

a relation between the partial pressure of a component α in a gas phase (i.e.

xαg · Pg) and its concentration in a liquid phase (Nℓxαℓ ) in equilibrium:

x(1)g · Pg = K ·Nℓx(1)ℓ (for a saturated fluid). (2.21)


The constant K is known as Henry’s law constant and depends on the temper-

ature. Regarding the assumption above, the gas phase contains only the light

component and hence it follows

XM(P, S) ≡ 1.

Furthermore it is assumed that the concentration of the heavy component in the

liquid phase (Nℓx(2)ℓ ) is dependent only on Pℓ (incompressible heavy component

in the liquid phase):

Nℓx(2)ℓ =: N

(2)ℓ (Pℓ). (2.22)

Then,

Pg = KN

(2)ℓ (Pℓ)

1−Xm(P, S)Xm(P, S)

i.e.

Xm(P, S) =Pg

Pg +K ·N (2)ℓ (Pℓ)

with Pg = Pℓ + Pc(S).

When using phase diagrams, the two closure relationships Xm and XM are

always given for the two-phase state. Thus two more closure relationships are

needed for that state or rather one more closure relationship is necessary when

only one phase is present.

2.4.2 Other static laws

The missing two closure relationships for the case of two present phases are

compressibility laws which link the volume and the number of moles of both

phases. By using (2.22) and assuming that

N(std)ℓ =

ρ(std)ℓ

M (2)

is constant with ρ(std)ℓ being the mass density of the heavy component at standard

conditions for temperature and pressure, an incompressibility law for the heavy

component is represented by

N(2)ℓ (Pℓ) :=

ρ(std)ℓ

M (2)Bℓ(Pℓ)=

N(std)ℓ

Bℓ(Pℓ). (2.23)

Bℓ is a function depending on the liquid pressure that is 1 at standard conditions.

The compressibility of the light component is given by a combination of the ideal


gas law and Dalton’s law [12] which is valid for low values of pressure Pg:

Ng =PgRT

(2.24)

with R being the universal gas constant and T the temperature. For the one-

phase states just one closure relationship is missing. Consequently, when only the

gas phase is present, equation (2.24) is merely considered and conversely equation

(2.23) is used when only the liquid phase exists.

2.4.3 Subsumption of all Closure Relationsships

The sum of all closure relationships forms the “static part” of the model. In order

to define all closure relationships

• the strictly positive constants

R, T,M (2), ρ(std)ℓ =M (2)N

(std)ℓ ,

• the two non-linear functions dependent on two variables

Xm(P, S) and XM(P, S),

• one function dependent on one variable Bℓ(Pℓ) as well as

• a monotone and increasing linear or non-linear weight function as before

are needed. For P and X given, the closure relationships for each of the three

possible states are:

• X ∈ (Xm(P, 0), XM(P, 1)) (saturated fluid, presence of both phases): the

five variables x(1)ℓ , x

(1)g , Nℓ, Ng, S are determined by solving the non-linear

system of equations below.

x(1)ℓ = Xm(P, S)

x(1)g = XM(P, S)

Ng =P + (1− γ(S))Pc(S)

RT

Nℓ(1− x(1)ℓ ) =

N(std)ℓ

Bℓ(P − γ(S)Pc(S))

S =Nℓ(X − x

(1)ℓ )

Ng(x(1)g −X) +Nℓ(X − x

(1)ℓ )

.

2.5. REFORMULATION WITH COMPLEMENTARITY CONSTRAINTS 21

• X ≤ Xm(P, 0) (undersaturated fluid, presence of liquid phase only): the

three variables x(1)ℓ , Nℓ, S are determined by

S = 0

x(1)ℓ = X

Nℓ =N

(std)ℓ

Bℓ(P )(1−X).

• X ≥ XM(P, 1) (oversaturated fluid, presence of gas phase only): the three

variables x(1)g , Ng, S are determined by

S = 1

x(1)g = X

Ng =P

RT.

For the last two states (unsaturated fluids) no other variables are required since

the mass conservation equations (2.8) are simplified due to multiplications by 0.

2.5 Reformulation with Complementarity Con-

straints

As seen before, the two-phase problem consists of a system of two partial dif-

ferential equations (mass conservation equations presented in section 2.3) and

of a system of static equations (presented in section 2.4) with state variables

listed in table 2.1. First the system of static equations is solved for two princi-

pal variables and then the mass conservation equations are treated as equations

only depending on those two formerly chosen unknowns. Usually, in the static

system of equations the variables Nℓ, Ng are explicitly defined as functions of

X,P, S, x(1)ℓ , x

(1)g and Pℓ, Pg are explicitly defined as functions of P, S.

The aim of this section is to reformulate the static part of the model as a

complementarity problem [43]. The use of complementarity constraints is an

efficient way to deal with a variable definition of the static part of the model.

The variability here is caused by different values of the state variables which

determine the different states of the system (for S = 0, S = 1 and S ∈ (0, 1)).

More generally, the static system of equations is reformulated as a system of

“classical” equations and conditional equations in the form

ab = 0 ∧ a ≥ 0 ∧ b ≥ 0


where a and b are functions of the state variables X , P , S, x(1)ℓ and x

(1)g . The

three possible states given in section 2.4.3 can be rewritten as

(C1) ∨ (C2) ∨ (C3) (2.25)

where

(C1) : S = 0 ∧ X ≤ Xm(P, 0)

∧ x(1)ℓ = X (2.26)

(C2) : S ∈ [0, 1] ∧ Xm(P, S)− x(1)ℓ = 0 (2.27)

∧ x(1)g −XM(P, S) = 0 (2.28)

∧ X =SNgx

(1)g + (1− S)Nℓx

(1)ℓ

SNg + (1− S)Nℓ

(2.29)

∧ Xm(P, S) ≤ X ≤ XM(P, S) (2.30)

(C3) : S = 1 ∧ X ≥ XM(P, 1)

∧ x(1)g = X. (2.31)

In the transition from the one-phase state, in which only the liquid phase is

present, to the two-phase state, both (C1) and (C2) are valid. Continuing with

the transition from the two-phase state to the state in which only the gas phase

exists, both (C2) and (C3) are true. Note that the state variables x(1)g and Ng

(resp. x(1)ℓ and Nℓ) are not defined when the corresponding phase is missing, i.e.

for X ≤ Xm(P, 0) (resp. X ≥ XM(P, 1)). It is desirable that the composition of

the missing phase is defined in such way that the conditional formulation is being

simplified. This can be enabled by defining the composition of the missing phase

such that equation (2.29) of (C2) is true even when one of the cases (C1) or (C3)

are valid. Moreover, the equation (2.30) is a consequence of (2.27), (2.28) and

(2.29). As long as the molar fraction x(1)i and the molar density Ni of the missing

phase i have finite values, equations (2.26) in (C1) and (2.31) in (C3) will be

equivalent to equation (2.29) and the problem (2.25) will be equivalent to

(C1′) ∨ (C2′) ∨ (C3′) (2.32)

SNg(X − x(1)g ) + (1− S)Nℓ(X − x

(1)ℓ )

SNg + (1− S)Nℓ

= 0 (2.33)

with

(C1′) : S = 0 ∧ x(1)ℓ ≤ Xm(P, 0)

(C2′) : 0 ≤ S ≤ 1 ∧ Xm(P, S)− x(1)ℓ = 0

∧ x(1)g −XM(P, S) = 0

(C3′) : S = 1 ∧ x(1)g ≥ XM(P, 1).

2.5. REFORMULATION WITH COMPLEMENTARITY CONSTRAINTS 23

This system of equations consists of a conditional equation (2.32) and a classical

equation (2.33). When equation (2.32) holds,

S(1− S)(Xm(P, S)− x(1)ℓ ) = 0 and

S(1− S)(x(1)g −XM(P, S)) = 0

have to be fulfilled. In order to get a convenient complementarity formulation it

is desirable that

S(1− S)(Xm(P, S)− x(1)ℓ ) = S(1− S)(x(1)g −XM(P, S)) = 0

⇒ S(Xm(P, S)− x(1)ℓ ) = (1− S)(x(1)g −XM(P, S)) = 0.

Obviously, this is true if

[S = 0 ⇒ x(1)g −XM(P, S) = 0

]and

[

1− S = 0 ⇒ Xm(P, S)− x(1)ℓ = 0

]

.

Consequently, the composition of the missing phases are defined as follows:

• in the absence of gas (only the liquid phase is present, S = 0):

x(1)g := XM(P, 0) (2.34)

• in the absence of liquid (only the gas phase is present, S = 1):

x(1)ℓ := Xm(P, 1). (2.35)

Fortunately, the inequalities

x(1)ℓ ≤ Xm(P, 0) and x(1)g ≥ XM(P, 1)

are still true for all S ∈ (0, 1). Then, the final formulation of the complementarity

problem equivalent to (2.25) is

S(Xm(P, S)− x(1)ℓ ) = 0 ∧ S ≥ 0 ∧ Xm(P, S)− x

(1)ℓ ≥ 0 (2.36)

(1− S)(x(1)g −XM(P, S)) = 0

∧ 1− S ≥ 0 ∧ x(1)g −XM(P, S) ≥ 0(2.37)

SNg(X − x(1)g ) + (1− S)Nℓ(X − x

(1)ℓ )

SNg + (1− S)Nℓ

= 0

with unknowns S, x(1)ℓ , x

(1)g .


The idea is now to replace the complementarity constraints (2.36) and (2.37)

by a function

ϕ : R2 → R

that satisfies

ϕ(a, b) = 0 ⇔ a · b = 0 ∧ a ≥ 0 ∧ b ≥ 0. (2.38)

Some common choices for this function ϕ (see [40]) are

• the Fisher-Burmeister-function: ϕ(a, b) =√a2 + b2 − a− b,

• the minimum function: ϕ(a, b) := mina, b,

• ϕ(a, b) = −ab +min2(0, a) +min2(0, b).

An efficient choice is the minimum function. The same approach has already

been used in [37], which deals with reactive transport and mineral dissolution

and precipitation processes in the subsurface. The application of this approach

to the equations (2.36) and (2.37), considering the minimum function, leads to

the following system of equations:

ϕ(

S , Xm(P, S)− x(1)ℓ

)

= 0 (2.39)

ϕ(1− S , x(1)g −XM(P, S)

)= 0 (2.40)

SNg(X − x(1)g ) + (1− S)Nℓ(X − x

(1)ℓ )

SNg + (1− S)Nℓ

= 0. (2.41)

This system of equations is non-smooth due to the minimum function and can

be solved by using a semi-smooth Newton’s method [44]. The advantages of this

method are that

• local convergence of the semismooth Newton’s method is obtained and

• the rate of convergence is still quadratic. [40, 44]

Remark 1. The denominator in equation (2.41) is for normalization purpose.

The final formulation for the static part of the model (2.39) - (2.41) entails the

possibility for the saturation to become zero resp. one with P and X still varying.

Equations (2.39), (2.40) are fulfilled due to the definition of x(1)g in absence of the

gas phase (2.34) and x(1)ℓ in absence of the liquid phase (2.35).

2.6. SUMMARY OF THE MODEL 25

Remark 2. Many numerical examples only take into account the disappear-

ance of the gas phase (see [12, 34]). Although the saturation takes values from

zero to one in the solution of the problem, within the Newton scheme it might

happen that S temporarily becomes larger than one when the disappearance of

the liquid phase is also incorporated into the model. The complementarity for-

mulation for the transition from the two-phase state to the state in which only

the gas phase is present, makes it easy to deal with these “unphysical” values for

the saturation in the Newton algorithm.

2.6 Summary of the Model

Let J be the time interval and let n be the outer unit normal vector to Ω. The

complete two-phase flow model for phases i = ℓ, g and components α = (1), (2)

in Ω× J reads:

• system of mass conservation equations:

φ∂t(NXα) +

∑

i=ℓ,g

∇· (Nixαi V i +NiSiW

αi ) = Fα

V i = − Kkri(Si)

µi(∇Pi −MiNig)

W αi = − Dα

i φ∇xαi

• system of static equations:

0 = ϕ(

S , Xm(P, S)− x(1)ℓ

)

0 = ϕ(1− S , x(1)g −XM(P, S)

)

0 =SNg(X − x

(1)g ) + (1− S)Nℓ(X − x

(1)ℓ )

SNg + (1− S)Nℓ

• specific closure relationships:

Sℓ + Sg = 1

Pg − Pℓ = Pc(S)

Pℓ = P − γ(S)Pc(S)

Ng =P + (1− γ(S))Pc(S)

RT=

PgRT

Nℓ(1− x(1)ℓ ) =

N(std)ℓ

Bℓ(P − γ(S)Pc(S))=

N(std)ℓ

Bℓ(Pℓ)


with initial and boundary conditions

P (x, 0) = P0(x) , X(x, 0) = X0(x) in Ω× 0P (x, t) = PD, X(x, t) = XD on ΓD × J

∑

i=ℓ,g

(Nixαi V i +NiSiW

αi ) · n = qαF on ΓαF × J

2.7 Generalised Formulation

Since different models can be used (see section 2.3) it is reasonable to introduce

a more compact formulation of the system of mass conservation equations. On

the one hand this enables a uniform notation of the discretisation independent

from the choice of the coefficient functions, on the other hand this allows a very

flexible program code in which it is very easy to switch between the different

model formulations. For α = (1), (2), the problem formulation given in section

2.6 is rewritten as

∂tmα(x, y) +∇·

(

−∑

τ

bατ (x, y)∇fατ (x, y) + hα(x, y)g

)

= Fα in Ω× J (2.42)

with boundary and initial conditions

x(·, 0) = x0(·) , y(·, 0) = y0(·) in Ω× 0 (2.43)

x(·, t) = xD, y(·, t) = yD on ΓD × J (2.44)(

−∑

τ

bατ (x, y)∇fατ (x, y) + hα(x, y)g

)

· n = qαF on ΓαF × J. (2.45)

The terms m, f, b and h are coefficient functions. The terms mα represent conser-

vative quantities, bατ represent the diffusion coefficients, fατ embodies potentials,

hα represent the convection coefficients, vector g is the gravity and n is the outer

unit normal vector to Ω. Here, the variables x and y are the principal unknowns

(for example the fields of P and X) and τ corresponds to different kind of dif-

fusion terms. For a model without any simplifications with respect to physical

properties, τ takes values from 1 to 4.

As mentioned before, this formulation allows to switch between the two differ-

ent models by choosing the corresponding coefficient functions. Table 2.2 gives an

overview of the choice of coefficient functions for the two-phase equations (2.42)

in such a way that the standard model (2.8) - (2.11) is obtained.

By changing the choices of b and f for τ = 3, 4 the alternative model cor-

responding to the system of equations (2.12), (2.9), (2.13), (2.14) is obtained.

2.7. GENERALISED FORMULATION 27

mα = φ(SNg + (1− S)Nℓ)Xα

bα1 = NgxαgKkrgµg

fα1 = Pg

bα2 = NℓxαℓKkrℓµℓ

fα2 = Pℓ

bα3 = NgDgSφ fα3 = xαg

bα4 = NℓDℓ(1− S)φ fα4 = xαℓ

hα = Kkrgµg

MgN2gx

αg +

Kkrℓµℓ

MℓN2ℓ x

αℓ

Table 2.2: Choices of coefficient functions for the standard model

Table 2.3 shows a summary of what has to be changed to move from the standard

model to the alternative model. These choices are always valid, even when one

bα3 = NgDgSφ →fα3 = xαg →bα4 = NℓDℓ(1− S)φ →fα4 = xαℓ →

bα3 = DgSφ

fα3 = xαgNg

bα4 = Dℓ(1− S)φ

fα4 = xαℓNℓ

Table 2.3: Crossover to the alternative model

phase is missing due to the definition of the composition of the missing phases

(2.34) and (2.35).

Remark 3. By assuming any simplifications for the model, some of the choices

for the coefficient functions may change to zero. For example, having a situation

in which a phase is equivalent to one component and where the liquid phase is

incompressible, b(1)2 = 0 and b

(2)1 = 0.

Chapter 3

Discretisation

In this section the discretisation of the system (2.42) - (2.45) is presented. The

spatial discretisation is carried out by using the RT0 mixed finite element method.

Time discretisation is realised by a modified Euler scheme presented in section

3.1. Rewriting the equations (2.42) - (2.45) in their mixed form leads to the

following system of equations:

∂tmα(x, y) +∇·qαdiff +∇· (hα(x, y)g) = Fα in Ω× J (3.1)

qαdiff = −∑

τ

bατ (x, y)∇fατ (x, y) in Ω× J (3.2)

with boundary and initial conditions

x(·, t) = xD, y(·, t) = yD on ΓD × J (3.3)

qα · n = qαdiff · n+ hα(x, y)g · n = qαF on ΓαF × J (3.4)

x(·, 0) = x0(·), y(·, 0) = y0(·) in Ω× 0. (3.5)

The discretisation is only presented for one component α since the procedure is

analogous for each of the components and - for reasons of simplification - the

component index α is omitted from now on.

By multiplying equation (3.1) with a sufficiently regular test function w and

equation (3.2) with a sufficiently regular test function v and by integrating these

two equations over Ω one yields:∫

Ω

∂tm(x, y)w dx+

∫

Ω

∇·qdiffw dx+

∫

Ω

∇· (h(x, y)g)w dx =

∫

Ω

Fw dx

(3.6)∫

Ω

qdiff · v dx = −∫

Ω

(∑

τ

bτ (x, y)∇fτ (x, y) · v)

dx (3.7)

29

30 CHAPTER 3. DISCRETISATION

3.1 Discretisation in Time

In a first step the discretisation in time is achieved with an Euler scheme that is

optionally

• implicit for the diffusion terms fτ and explicit for the diffusion coefficients

bτ and explicit for the convection terms h

• implicit for the diffusion terms fτ and the diffusion coefficients bτ and ex-

plicit for the convection terms h.

The different treatment of the diffusion coefficients bτ is the result of single ap-

plications in which either the implicit or the explicit treatment lead to numerical

difficulties whereby one of the Newton schemes failed. A similar idea is used in

[33]. In the numerical experiment 1 given in section 4.1 a comparison of both the

implicit and the explicit treatment is presented. Hereafter, only the discretisa-

tion for the first of the two different treatments of the diffusion coefficients will

be given.

The time interval J = (0, T ) is devided into N subintervals (tn−1, tn). In the

implicit Euler scheme the time derivative is replaced by the backward difference

quotientm(xn, yn)−m(xn−1, yn−1)

∆t

where ∆t := tn − tn−1 is the time step size. For the sake of simplicity only the

unknowns at the old point of time tn−1 get an index. Any quantity without an

index for the time step is evaluated at the current time point tn. The application

of the first of the above mentioned Euler schemes (implicit diffusion terms, explicit

diffusion coefficients and convection terms) to the equations (3.6) - (3.7) leads to

1

∆t

(∫

Ω

m(x, y)w dx−∫

Ω

m(xn−1, yn−1)w dx

)

+

∫

Ω

∇·qdiffw dx =

∫

Ω

Fw dx−∫

Ω

∇·(h(xn−1, yn−1)g

)w dx.

(3.8)

∫

Ω

qdiff · v dx = −∫

Ω

(∑

τ

bτ (xn−1, yn−1)∇fτ (x, y) · v

)

dx (3.9)

for all n ∈ 1, . . . , N and v, w sufficiently regular.

Due to their explicit treatment the convective terms act as additional source

terms on the right hand side of the equation (3.8).

3.2. SPATIAL DISCRETISATION 31

3.2 Spatial Discretisation

Let Th be a triangulation of the computational domain Ω into subdomains Ωi ∈ Thwith i = 1, . . . , NC (NC is the number of subdomains). The set of all edges is

denoted by E and let ED resp. EF be the set of edges at the Dirichlet resp. flux

boundary. The boundary edges El ∈ E with l = 1, . . . , NB are adjacent to one

subdomain Ωi1(l) whereas the interior edges Ek ∈ E with k = NB + 1, . . . , NE

are adjacent to the subdomains Ωi1(k) and Ωi2(k). The normal vectors ni to ∂Ωiare directed towards the exterior of subdomain Ωi. The normal vector nk to

Ek = Ωi1(k) ∩ Ωi2(k) ⊂ Ω is directed from subdomain Ωi1(k) to Ωi2(k) (arbitrary

definition of i1(k), i2(k) for k > NB).

∂Ω

Ωi1(k)Ωi2(k)

Ωi1(l)

nl

nk

Figure 3.1: Triangulation of Ω

For the spatial discretisation with the RT0 mixed finite element method on

triangles the two function spaces

Vh = q ∈ H(div,Ω)| q|Ωi∈ RT0(Ωi) ∀Ωi ∈ Th

Wh = w ∈ L2(Ω)| w|Ωi∈ P0(Ωi) ∀Ωi ∈ Th

are defined with RT0(Ωi) = (P0(Ωi))2 ⊕ ((x1, x2)P0(Ωi)) for i = 1, . . . , NC and

Pl(Ωi) is the space of polynomials of maximum degree l ∈ N on Ωi with the

notation ⊕ indicating a direct sum and (x1, x2)Pl(Ωi) = (x1Pl(Ωi), x2Pl(Ωi))[22, 30]. These spaces were the first mixed finite element spaces introduced by

P.-A. Raviart and J.-M. Thomas in 1977 [54]. To characterise a function v in Vhthe degrees of freedom are chosen as the values of the flux across the edges of Th,i.e. ∫

Ek

v · nk .


Ωi

Figure 3.2: Triangular element with degrees of freedom for the flux basis functions

By the transition from the continuous function spaces to the discrete ones,

the discrete variational formulation reads

If Vh and Wh were regular enough one should find(qdiff,h, (xh, yh)

)∈ (Vh × (Wh ×Wh)) such that

1

∆t

(∫

Ω

m(xh, yh)wh dx−∫

Ω

m(xn−1h , yn−1

h )wh dx

)

+

∫

Ω

∇·qdiff,hwh dx =

∫

Ω

Fwh dx−∫

Ω

∇·(h∗(xn−1

h , yn−1h )g

)wh dx

(3.10)

∫

Ω

qdiff,h · vh dx+

∫

Ω

(∑

τ

bτ (xh, yh)∇fτ (xh, yh) · vh)

dx = 0 (3.11)

for all n ∈ 1, . . . , N and vh ∈ Vh and wh ∈ Wh.

Details about the terms h∗ are given in remark 5. Actually it is not correct to

write “∇fτ (xh, yh)” and it can be seen in section 3.2.2 how to deal with this. In

the following parts some more details for the single parts of the discretised model

equations will be given.

3.2.1 Conservation laws

By the transition from the continuous function spaces to the discrete ones, for

equation (3.8) one gets the mass conservation equation for any subdomain Ωidue to the fact that the test function wh ∈ Wh is constant on subdomain Ωiand vanishes anywhere else. Additionally, the application of Gauss’s divergence


theorem to the integral containing the convective part results in

1

∆t

∫

Ωi

m(xh, yh) dx+

∫

∂Ωi

qdiff,h · nexti dσ = +

∫

Ωi

F dx

1

∆t

∫

Ωi

m(xn−1h , yn−1

h ) dx−∫

∂Ωi

h∗(xn−1h , yn−1

h )g · nexti dσ ,

(3.12)

in which nexti is the field of normal vectors to ∂Ωi directed towards the exterior

of cell Ωi.

Remark 4. There is a well known equivalence between lowest-order mixed

finite elements and finite volumes [62, 60]. Here, both methods lead to equation

(3.12). Using the RT0 mixed finite element method with a test function wh ∈ Wh

which is constant on subdomain Ωi and vanishes anywhere else results in an

equation that only contains integrals over subdomain Ωi. This is equivalent

to a finite volume discretisation. In the finite volume method the domain Ω is

subdivided into subdomains Ωi, also called control volumes. Integrating equation

(3.1) over each control volume Ωi and applying Gauss’s divergence theorem ends

up in the same equation (3.12). More details about finite volumes can be found

in [42].

Finally, for all Ωi and for all n ∈ 1, . . . , N one gets

|Ωi|∆t

m(Xi,h, Yi,h) +BlocQ[i]diff,h =

|Ωi|∆t

m(Xn−1i,h , Y n−1

i,h )−∑

k,Ek⊂∂Ωi

h∗(Xn−1i,h , Y n−1

i,h )ǫikgk + Fi.(3.13)

The scalar values Xi,h, Yi,h are degrees of freedom corresponding to values of x, y

on subdomain Ωi at the current time step tn, Bloc is a row vector containing 1

or −1 (depending on the definition of i1(k)), the vector Q[i]diff,h that is of size 3

(resp. 4) for triangular (resp. quadrilateral) subdomains contains the diffusive

fluxes of the considered component through each edge of Ωi and

ǫik = nexti · nk for Ek ⊂ ∂Ωi

gk =

∫

Ek

g · nk dσ ∀k = 1, . . . , NE.

Fi =

∫

Ωi

F dx.


Remark 5 (Convection terms h∗α(x, y) for α = (1), (2)). There is a number

of possibilities to deal with the boundary integrals over the coefficient functions

hα(xn−1, yn−1). One choice is to apply a Godunov scheme [21, 10]. But this is not

obvious when two-dimensional variable fields are considered, even in the linear

case (see for example [45]). Here, as an alternative, an upwind scheme is applied

that reads as follows: the coefficients hα(xn−1, yn−1), which except for the factor

gk embody the numerical convective fluxes, can be written as

(h∗(1)(xn−1, yn−1)

h∗(2)(xn−1, yn−1)

)

= H(xL, xR, yL, yR) =

(H(1)(xL, xR, yL, yR)

H(2)(xL, xR, yL, yR)

)

with

• if gk ≥ 0:

xL = Xn−1i1(k)

, xR = Xn−1i2(k)

and yL = Y n−1i1(k)

, yR = Y n−1i2(k)

• if gk < 0:

xL = Xn−1i2(k)

, xR = Xn−1i1(k)

and yL = Y n−1i2(k)

, yR = Y n−1i1(k)

and H(1), H(2) as can be seen below remark 7.

Remark 6. hα is, upto the factor ‖g‖, the projection of the convective flux

along the axis g. The order of the arguments of H (i.e. the definition of L(eft),

R(ight)) is defined by following a chosen direction. Here it is the direction of the

axis g, and it is independent from the sign of hα, i.e. the local direction of the

physical flux. However, this choice should be based on properties of hα (for the

opposite choice one would consider the functions −hα). For example, convexity

properties should be satisfied.

Remark 7. In the case when gk = 0 the definition of L and R is of no impor-

tance.

The upwind scheme should only be applied when the matrix

A(xL, xR, yL, yR) :=

(h

′(1)

h′(2)

)

is a diagonal matrix with hα monotone. For component α = (1) H reads


• if m(1)(xL, yL) ≤ m(1)(xR, yR):

H(1)(xL, xR, yL, yR) = min(h(1)(xL, yL), h

(1)(xR, yR))

• if m(1)(xL, yL) > m(1)(xR, yR):

H(1)(xL, xR, yL, yR) = max(h(1)(xL, yL), h

(1)(xR, yR)).

It is well known that, in order to keep the convergence, the time step size ∆t

becomes restricted when explicit time schemes are used for numerical solutions.

By ignoring the restriction the method will produce incorrect result in which the

solution may oscillates and blows up. That restriction means that the time step

size has to be less than a certain time. This context is known as the Courant-

Friedrichs-Lewy (CFL) condition that is named after Richard Courant, Kurt

Friedrichs and Hans Lewy. [24] Nevertheless, this condition is only a very weak

constraint here since the problems to be solved are not advection dominated,

even for the case of the presence of phase transitions. Therefore, a formulation

for the constraint on ∆t is omitted in this work.

Remark 8 (Alternatives to the Upwinding). There are a number of alter-

natives of dealing with the advective part of the model. One commonly used

approach (e.g. in [14, 13]) is to consider the local flux variable as

q = −∑

τ

bτ (x, y)∇fτ (x, y) + h(x, y)g

which is called the “strong divergence form” [26]. One difference between these

two formulations concerns the required smoothness assumptions (about q resp.

qdiff ). The change of the decomposition of the flux variable leads to an alterna-

tive discretisation in which the advective terms could be treated implicitly with

only little expense. For advection dominated problems these terms have to be

stabilized, for example by an upwind formula [13]. Beside the fact that for the

two possible choices of the definition of the fluxes the weak formulations become

different, so do the convergence properties (weak divergence form is better than

the strong one)[26, 29].

A second alternative approach presented by T. Arbogast and M. Wheeler uses

the same decomposition of the flux variable as in (3.1),(3.2) in which the con-

vective/advective parts of the flux remain in the mass conservation equation. In

[3] a characteristics-mixed finite element method for advection dominated prob-

lems is presented in which a characteristic approximation is used to handle the


convection/advection in time. In general, the goal of the method of character-

istics is to change coordinates from (x, t) to a new coordinate system in which

the PDE becomes an ordinary differential equation (ODE) along certain curves,

called characteristic curves or in short characteristics. Once the solution of the

ODE is found it can be solved along the characteristics and then it can be trans-

formed into a solution for the original PDE. But these methods are very hard to

implement and without any modifications, of which one possibility is presented

in [3], there is no local conservation of mass. The mass conservation is ensured

only globally over all of Ω.

3.2.2 Diffusion laws

Continuing with the diffusion laws (3.11), one defines for xh, yh and qdiff,h regular

enough

Ta(i) :=

∫

Ωi

qdiff,h · vi,h dx

Tτ (i) :=

∫

Ωi

bτ (xn−1h , yn−1

h )∇fτ (xh, yh) · vi,h dx

for any basis function vi,h ∈ Vh. Integration by parts results in a decomposition

of Tτ (i) into the boundary terms Tbd and interior terms Tint

Tτ (i) =

∫

∂Ωi


h )fτ (xh, yh)vi,h · nexti dσ

︸︷︷︸

Tbd(i)

−∫

Ωi

fτ (x, y)∇·(bτ (x

n−1h , yn−1

h )vi,h)dx

︸︷︷︸

Tint(i)

.

(3.14)

Remark 9. The local boundary term reads

Tbd(i) =∑

k,Ek⊂∂Ωi

∫

Ek


h )fτ (xh, yh)vi,h · nexti dσ .

Without hybridisation these terms vanish when Ek is an edge of two adjacent

cells Ωi1(k) and Ωi2(k) (Ek = ∂Ωi1(k) ∩ ∂Ωi2(k))∫

Ek


h )fτ (xh, yh)vi,h · ni1(k) dσ

+

∫

Ek


h )fτ (xh, yh)vi,h · ni2(k) dσ = 0


and therefore only terms at the global boundary remain when all equations are

summed up

Nc∑

i=1

Tbd(i) =∑

τ

∑

k,Ek⊂∂Ω

∫

Ek


h )fτ (xh, yh)vi,h · nk dσ

=∑

τ

∫

ΓD

bτ (xD, yD)fτ (xD, yD)vh · nk dσ.

(3.15)

As soon as hybridisation is applied this is no longer valid. Moreover, the basis

functions vi,h are chosen such that vi,h ·nk = 0 with the exception of the Dirichlet

boundary. Thus one gets

Nc∑

i=1

Tbd(i) =∑

τ

∑

k,Ek⊂ΓD

∫

Ek

bτ (xD, yD)fτ (xD, yD)vi,h · nk dσ.

If Xi,h, Yi,h represent a constant approximation of x, y over Ωi and by applying

again Gauss’s divergence theorem, the local interior terms Tint(i) of (3.14) can

be approximated by

Tint(i) = fτ (Xi,h, Yi,h)

∫

∂Ωi


h )vi,h · nexti dσ

= fτ (Xi,h, Yi,h)∑

k|Ek⊂∂Ωi

∫

Ek


h )vh · nexti dσ.

Remark 10. Note, that within Tint(i) and Tbd(i) integrals over the coefficient

functions b and f over the boundary of each cell Ωi have to be determined.

But these functions b are defined on each cell Ωi and not on ∂Ωi. One possible

answer to deal with this could be the computation of a mean value of b of the

two adjacent cells to Ek. Since hybridisation is used the coefficient functions are

evaluated with the help of the additional unknowns which are introduced by the

hybridisation. These additional unknowns are the Lagrange multipliers that are

allocated to the global edges.

For all i = 1, · · · , Nc let A[i] be the local mass matrix defined by

A[i]k,l =

∫

Ωi

vi,k · vi,l dx

in which vi,k is a basis function associated to cell Ωi and edge Ek. Finally,


∀i ∈ 1, . . . , Nc, Ek ⊂ ∂Ωi the equations for the diffusion laws reads[

A[i]Q[i]diff,h

]

k+∑

τ

bτ (λn−1k , µn−1

k )fτ (λk, µk)

−∑

τ

fτ (Xi,h, Yi,h)bτ (λn−1k , µn−1

k ) = 0. (3.16)

with the Lagrange multipliers λk (resp. µk), k = 1, . . . , NE, associated to x (resp.

y).

3.2.3 Hybridisation

Finally the hybridisation is applied to equations (3.10)-(3.11). By hybridisation

the discrete function space Vh is replaced by a new function space such that

the continuity of the normal components of the fluxes over interior edges is not

ensured anymore. Let

Vh = qα ∈ (L2(Ω))2| qα|Ωi∈ RT0(Ωi) ∀Ωi ∈ Th

be this new function space. The continuity of the normal components of the fluxes

over interior edges is ensured by additional equation and consequently additional

unknowns, the so-called Lagrange multipliers. By introducing the function spaces

Λh = λ ∈ L2(E)| λ|E ∈ P0(E) ∀E ∈ E

Λg,h = λ ∈ Λh|∫

E

(λ− g) = 0 ∀E ∈ ED

the right hand side of equation (3.15) can be reformulated as follows:

−∫

ΓD

(∑

τ

bτ (xD, yD)fτ (xD, yD)vh · nk

)

dσ =

−∫

ΓD

(∑

τ

bτ (xD, yD)fτ (xD, yD)vh · nk

)

dσ

−∑

Ωi∈Th

∑

Ek⊂∂Ωi

Ek 6⊂ΓD

∫

Ek

(∑

τ


k )fτ (λk, µk)vi,h · nexti

)

dσ

for vh ∈ Vh and (λk, µk) ∈ ΛgD,h. In the additional terms any values λk and µk on

Ek ∈ ∂Ωi, not necessarily the Langrange multipliers, can be chosen. For vh ∈ Vhthese additional terms are zero since for the interior edges

ni1(k) + ni2(k) = 0 on Ek


hold with Ek being an edge of two adjacent cells Ωi1(k) and Ωi2(k) with Ek =

∂Ωi1(k)∩∂Ωi2(k). The terms on ΓF are zero due to the choice of the test functions

vh ∈ Vh. When replacing the space Vh by the extended function space Vh and

assuming that λk and µk are the Lagrange multipliers on edge Ek, equation (3.11)

becomes∫

Ω

qdiff,h · vh dx−∫

Ω

(∑

τ

fτ (xh, yh)∇·(bτ (xn−1h , yn−1

h )vh)

)

dx =

−∑

Ωi∈Th

∑

Ek∈∂Ωi

∫

Ek

(∑

τ



)

dσ

with (λk, µk) ∈ ΛgD,h. Because the number of unknowns was increased, additional

equations are required to determine all degrees of freedom. These additional

equations are the flux continuity conditions. When using the flux boundary

conditions (3.4) for the chosen definition of the flux variable qdiff these equations

are∑

Ωi∈Th

∫

∂Ωi

ηqdiff,h · ni dσ =

∫

ΓD

ηqdiff,h · nk dσ

︸︷︷︸

=0

+

∫

ΓF

η(qF − h∗(xn−1

h , yn−1h )g · nk

)dσ

for η ∈ Λ0,h. In detail, when Ek is an interior edge of two adjacent subdomains

Ωi1(k) and Ωi2(k) the requirement of the flux continuity reads[

Q[i1(k)]diff,h

]

k=[

Q[i2(k)]diff,h

]

k.

The full discrete hybrid-mixed variational formulation of problem (3.10) -

(3.11) reads

Find (qdiff,h, (xh, yh), (λh, µh)) ∈ Vh × (Wh ×Wh)× (ΛgD,h × ΛgD,h) such that

1

∆t

(∫

Ωi

m(xh, yh)wh dx−∫

Ωi

m(xn−1h , yn−1

h )wh dx

)

+

∫

Ωi

∇·qdiff,hwh dx

=

∫

Ωi

Fwh dx−∫

Ωi

∇·(h∗(xn−1

h , yn−1h )g

)wh dx ,

(3.17)

∫

Ωi

qdiff,h · vi,h dx−∫

Ωi

(∑

τ

fτ (xh, yh)∇·(bτ (xn−1h , yn−1

h )vi,h)

)

dx =

−∑

Ek∈∂Ωi

∫

Ek

(∑

τ



)

dσ ,

(3.18)


∑

Ωi∈Th

∫

∂Ωi

ηqdiff,h · ni dσ =

∫

ΓF

η(qF − h∗(xn−1

h , yn−1h )g · nk

)dσ (3.19)

for all n ∈ 1, . . . , N and vi,h ∈ Vh, wh ∈ Wh, η ∈ Λ0,h.

3.2.4 Resume

As a result of the hybridisation the number of unknowns has increased. Figure 3.3

gives an overview which unknowns are considered for Raviart-Thomas elements

of the lowest order. For any subdomain Ωi for i = 1, . . . , Nc one obtains

b

bb

×

Xi, Yi

[

Q[i]diff

]

l

λk, µk

Figure 3.3: Overview about considered unknowns

• two scalar unknowns Xi, Yi for the the two fields x and y (constant per

subdomain) and

• one flux unknown per global edge

when no hybridisation is applied. By hybridising the problem one gets

• one vector Q[i]diff (for each component α) of size 3 (resp. 4 when a 3-

dimensional domain is considered) containing the diffusive fluxes over each

boundary edge of the subdomain Ωi[

Q[i]diff

]

l=[Qdiff

]

j

for l = 1, . . . , 3 and j = 1, . . . , 2NE −NB and

• the additional unknowns λk, µk for k = 1, . . . , NE that are allocated on each

global edge.


3.2.5 Static Condensation

The problem (3.17) - (3.19) can be simplified by the so-called static condensation.

A detailed description of this method is presented in [55]. In general, at each

time step the procedure contains two steps and finally leads to a global system

of equations which only depends on the Lagrange multipliers

(λn, µn) 7→ (λn+1, µn+1)

instead of all unknowns listed before. In a first step the diffusive fluxes can be

eliminated explicitly. In the second step the fluxes are introduced into the mass

conservation equations and the unknowns Xi, Yi can be eliminated implicitely

by solving local equations on each Ωi. Finally, the determined fluxes and values

for the scalar unknowns are introduced into the flux continuity equations and

the global system of equations depending only on the Lagrange multipliers is

obtained. This is solved by a global Newton’s method and the obtained values

for the Lagrange multipliers are used to recompute the new iterates for Xi, Yi,

which are again the initial data for the global Newton iteration to be solved. This

is repeated until a specified precision ǫ is reached. In figure 3.4 this procedure is

summarised.

Elimination of the Diffusive Fluxes

With

H [i] =(A[i])−1 ∀i = 1, . . . , Nc

and k chosen such that Ek ⊂ ∂Ωi, the diffusion equations (3.16) can be rewritten

and an explicit expression of the diffusive fluxes is obtained

Q[i]diff,h =

∑

Ek′⊂∂Ωi

H [i]kk′

∑

τ

[fτ (Xi,h, Yi,h)− fτ (λk′, µk′)] bτ(λn−1k′ , µn−1

k′

).

This can be introduced into the mass conservation equations (3.13) which be-

comes

|Ωi|∆t

m(Xi,h, Yi,h) +∑

Ek⊂∂Ωi

ai,k∑

τ

[fτ (Xi,h, Yi,h)− fτ (λk, µk)] bτ (λn−1k , µn−1

k )

=|Ωi|∆t

m(Xn−1i,h , Y n−1

i,h )−∑

k,Ek⊂∂Ωi

h∗(Xn−1i,h , Y n−1

i,h )ǫikgk + Fi.

(3.20)

with

ai,k =∑

Ek′⊂∂Ωi

H[i]k′k.


Initialisation of stepn, l = 0

Xn0,0i := X

n−1,lend

i

λn0,0k := λ

n−1,lend

k

Yn0,0i := Y

n−1,lend

i

µn0,0k := µ

n−1,lend

k

principal variables:

Lagrange multipliers:

secondary variables:Sn0,0i := S

n−1,lend

i , xn0,0ℓi

:= xn−1,lend

ℓi, x

n0,0gi := x

n−1,lendgi

local problems,m = 0

ComputeXn,l+1i , Y n,l+1

i by a local Newton iteration on eachΩiFi(λ

n,l, µn,l, Xnm,li , Y nm,l

i ) = 0 (3.22)

local Newton step(

Xnm,li , Y

nm,li

)

7→(

Xnm+1,li , Y

nm+1,li

)

(

Xnm+1,l

i , Ynm+1,l

i

) (

Snm,li , x

nm,lℓi

, xnm,lgi

)

m := m+ 1

solve compl. problem(2.39) - (2.41)(

Snm,li , x

nm,lℓi

, xnm,lgi

)

7→(

Snm+1,li , x

nm+1,lℓi

, xnm+1,lgi

)

by a semi-smooth Newton’s method

local ǫ fulfilled

local ǫnot fulfilled

(

Xn,l+1i , Y n,l+1

i

)

,(

Sn,l+1i , xn,l+1

ℓi, xn,l+1

gi

)

l := l + 1

global problem

computeλn,l+1k , µn,l+1

k ∀Ek ∈ E by a global Newton iterationglobalǫnot fulfilled

Xn,lend

i , Y n,lend

i , λn,lend

k , µn,lend

k

Sn,lend

i , xn,lend

ℓi, xn,lend

gi

globalǫ fullfilled

Figure 3.4: Scheme of the algorithm for time step n

Based on this equation the two scalar unknowns Xi, Yi are eliminated in the

second step of the static condensation. The introduction of the diffusive fluxes


into the flux continuity equations results in∑

i∈i1(k),i2(k)

∑

Ek′⊂∂Ωi

H[i]kk′

∑

τ

[fτ (Xi,h, Yi,h)− fτ (λk′, µk′)] bτ (λn−1k′ , µn−1

k′ ) = 0

(3.21)

∀k ∈ Nb + 1, . . . , NE . These equations will form the global system of equations

dependent only on the Lagrange multipliers after the scalar unknowns have been

determined.

Implicit Elimination of Scalar Unknowns

For a given set of Lagrange multipliers the corresponding values Xi, Yi to x, y on

any subdomain Ωi can be computed by considering a local implicit function

(Xi,h, Yi,h) = ψi(λ, µ) ⇔ Fi(λ, µ,Xi,h, Yi,h) = 0

where F is given by the mass conservation equation (3.20). If Fi has a root

(λ0, µ0, X0, Y0) with(

∂X,Y Fi

)

λ0,µ0,X0,Y0

∈ Isom(R2,R)

then ψ will be well defined and C1 on a neighborhoodN of (λ0, µ0) and for Λ ∈ N

d ψΛ = −[(

∂X,Y F)

Λ,ψ(Λ)

]−1

(

∂λ,µF)

Λ,ψ(Λ).

The function

Fi =

(

F(1)i

F(2)i

)

, F αi : (RNE × R

NE)× (R× R) → R

is defined by the mass conservation equations (3.20)

F αi (λ,µ,Xi,h, Yi,h) =

|Ωi|∆t

(mα(Xi,h, Yi,h)−mα(Xn−1

i,h , Y n−1i,h )

)

+∑

Ek⊂∂Ωi

ai,k∑

τ

[ fατ (Xi,h, Yi,h)− fατ (λk, µk)] bατ

(λn−1k , µn−1

k

)(3.22)

+∑

Ek⊂∂Ωi

hα∗

(Xn−1i,h , Y n−1

i,h )ǫikgk − Fαi ∀i = 1, . . . , Nc

for α = (1), (2). This equation is solved locally on each subdomain Ωi by a local

Newton’s method. Thus the Jacobian matrix of F αi is needed. If the Jacobian

matrix of

(m(1)

m(2)

)

with respect to (x, y) is diagonal dominant and if ∆t is

small enough,(

∂x,yFi

)

(λ,µ,x,y)is diagonal dominant and therefore invertible. The

needed derivatives of F αi are given in appendix A.1.


Flux Continuity Equations

Finally, the global function to be inverted by the iterative solver at each time

step is determined by the flux continuity equations (3.21): ∀k ∈ Nb + 1, . . . , NE ,

for α = (1), (2) and for d = 1, 2 let

i := id(k) , x := ψxi (λ, µ) , y := ψyi (λ, µ)

and

Gαd,k(λ, µ) :=

∑

Ek′⊂∂Ωi

H[i]kk′

∑

τ

[fατ (x, y)− fατ (λk′, µk′)] bατ (λ

n−1k′ , µn−1

k′ ).

Then, for α = (1), (2) the flux continuity equations read:

Gαk (λ, µ) := Gα

1,k(λ, µ) + Gα2,k(λ, µ) = 0. (3.23)

The Jacobian matrix of G - essential for the global Newton’s method - is given

in appendix A.2.

3.3 Implementation Aspects

The algorithm was implemented using a finite elements library for parallel compu-

tations in the context of (time-dependent non-linear) PDEs called M++ (Multigrid,

Meshes and More)[61]. It is an object oriented code written in C++ that uses MPI

(Message Passing Interface) for parallelization.

The code enables computations in 2D as well as 3D. Computations in 1D

are not supported. It uses finite elements on unstructured grids. These grids

are specified by the user in a modified geo-file. These grids are usually given as

coarse grids that for example contain information about the allocation and type

of boundary edges and special structures within the grid (e.g. holes). These grids

can be refined uniformly.

M++ consists on the one hand of a kernel that provides any components that

are independent from the problem to be solved (uniform mesh refinement, frame-

work for assembling, implicit Euler method, iterative solvers, linear solvers and

preconditioners, output generation) and on the other hand of problem specific

components. All problem specific parameters can be modified by the user via a

data file, i.e the ansatz space, any discretisation parameters, the type of linear

solver and preconditioner to be used, the stopping criteria for the non-linear and

linear solver, and the output format. To visualise the data M++ creates text files

in different formats to be opened with gnuplot, OpenDX or ParaView.

3.3. IMPLEMENTATION ASPECTS 45

Remark 11 (1D computations). Some of the examples in chapter 4 involve

one-dimensional computations. This problem is overcome by using a 2-dimensional

computation domain with uniform boundary and initial conditions in one spatial

direction.

In figure 3.4 the implemented algorithm is shown. After the initialisation of all

variables (Lagrange multipliers, principal unknowns, state variables) an implicit

Euler scheme is applied. Within each time step a damped Newton’s method

[4, 27] is used to solve the non-linear system of equations. Note that there are

Newton iterations on different “levels” of the algorithm. First of all one global

Newton’s method is applied to solve the global system of equations depending on

the Lagrange multipliers. Within this global Newton iteration a local Newton’s

method is used to solve a local system of equations for the principal variables

on each cell Ωi. For each iteration within the local Newton method the state

variables have to be updated by the semi-smooth Newton’s method for the static

system of equations.

Chapter 4

Numerics

In this chapter a number of numerical results is presented. The objective is on

the one hand to show convergence properties of the method and on the other

hand to present numerous examples which show important features. While the

first examples given in sections 4.1 to 4.5 are academic experiments in 1D to show

important properties of the model (phase appearance and disappearance, liquid

pressure behaviour, convergence properties), the last one deals with a realistic and

thus more complex scenario in 2D. This experiment will be also used to compare

the two different model formulations (given in section 2.3). The experiments of

sections 4.1, 4.4 and 4.5 are very similar to the problems 1 to 3 of the MoMaS

benchmark [11].

Nevertheless, the following assumptions about the physics are valid for all

presented numerical experiments:

• The liquid phase is incompressible in the sense of x(2)ℓ Nℓ ≡ ρ

(std)ℓ

M (2) .

• The gas phase behaves like a perfect gas: Pg =Ng

RT.

• A phase diagram as given in example 2 (figure 2.1) is used.

• To model the relative permeabilities and capillary pressures the Van Genuchten

parametrisation is applied.

Van Genuchten parametrisation To describe the relative permeability sat-

uration relations two models are commonly used: the one by Brooks-Corey and

the model by Van Genuchten in conjunction with the approach of Mualem [35].

The parameters needed to apply the Van Genuchten parametrisation are given

in table 4.1. Note that Sgr + Sℓr < 1 holds. The parameters ǫvg and γvg are form

47

48 CHAPTER 4. NUMERICS

Sgr, Sℓr ∈ [0, 1] residual saturation for both phases [-]

n ∈ [1,+∞[, m = 1− 1n

Van Genuchten parameters [-]

ǫvg ∈]0, 1[, γvg ∈]0, 1[ Van Genuchten form parameters [-]

P 0c = 1

α> 0 α > 0 Van Genuchten parameter [Pa]

Table 4.1: Used parameters for the Van Genuchten parametrisation

parameters that describe the connectivity of the pores [50]. With the effective

saturations

sℓ =Sℓ − Sℓr

1− Sgr − Sℓr

sg =Sg − Sgr

1− Sℓr − Sgr

the capillary pressure can be formulated as a function of the relative saturation

sℓ

Pc(S) = P 0c

(

s− 1

m

ℓ − 1) 1

n

S(Pc) =1− Sgr − Sℓr(

Pnc

(P 0c )

n + 1)m + Sℓr

and the corresponding relative permeability saturation relations reads as

krℓ(S) = sǫvgℓ

(

1−(

1− s1m

ℓ

)m)2

krg(S) = sγvgg

(

1− (1− sg)1m

)2m

.

More detailed information can be found for example in [7, 59].

Remark 12 (Regularisation). The function Pc(S) is only defined for S ∈ [0, 1)

and shows unbounded slopes for S close to 0 and 1. Since it might happen that

S becomes larger than 1 within the Newton iterations it is necessary to slightly

modify and to extend the definition of Pc(S) for S ∈ R. This is meaningful since

the iterative schemes would terminate due to infinite values of Pc. It is already

assured that S ∈ [Slg, 1− Slr] by the use of the complementarity formulation.

Let 0 < ǫ << 1. For

• S ∈ [Sgr, 1− Sℓr]:

S := Sgr + (1− ǫ)(S − Sgr) +ǫ

2(1− Sgr − Sℓr).

S ∈ [Sgr +ǫ

2(1− Sgr − Sℓr), 1− Sℓr −

ǫ

2(1− Sgr − Sℓr)],

Pc(S) = Pc(S).

4.1. PHASE APPEARANCE AND DISAPPEARANCE 49

• S < Sgr:

Pc(S) = Pc(Sgr) + P ′c(Sgr)(S − Sgr).

• S > 1− Slr:

Pc(S) = Pc(1− Sℓr) + P ′c(1− Sℓr)(S − 1 + Sℓr).

The choices of values for the soil and fluid parameters for all simulations

based on the MoMaS benchmark [11] are stated in table 4.2. When necessary,

different values for physical parameters are given for each subdomain within the

description of each experiment when a heterogeneous porous medium is assumed.

Fluid parameters105

RT39.6938 bar.m3/mol for T = 303K fixed

Henry 0.765 mol/bar/m3 (7.65.10−6 mol/Pa/m3)

ρ(std)l 1000 kg/m3

M (2) 0.01 kg/mol

Soil Parameters

φ 0.15

K 5.10−20m2

D 9.4670814 m/century (i.e. 3.10−9 m/s)

µℓ 3.161.10−18 bar.century (i.e. 1.0e-8 bar.s)

µg 2.845.10−20 bar.century (i.e. 9.0e-11 bar.s)

Van Genuchten Parameters

P 0c 20 bar

n 1.49

ǫ 0.5

γ 0.59

Sgr 0

Sℓr 0.4 [0.01]

Table 4.2: Soil and fluid parameters for numerical experiments.

4.1 Phase Appearance and Disappearance

This first numerical experiment is motivated by the already mentioned MoMaS

benchmark started in 2010 [11]. It deals with the appearance and disappearance


of a gas phase (hydrogen) for a compressible and partially miscible two-phase

flow in a homogeneous porous medium. Beside the original questioning within

the benchmark the setup is also used

• to test convergence in space and time by comparing results from simulations

with different meshes and different time step sizes and

• to compare the results which are received by using the implicit and explicit

treatment of the diffusion coefficient.

A detailed test that the scheme converges in space and time is presented in section

4.2.

4.1.1 Setup of the Experiment

The experiment starts and ends in a purely liquid state. Due to an injection

of hydrogen into the domain Ω, the concentration of the light component in-

creases until a threshold is reached where the gas phase appears. The injection is

stopped at a certain time Tinj. From that moment on, the concentration of the

hydrogen decreases until the gas phase disappears. The simulation parameters

(additional to the ones already known from table 4.2) are given in table 4.3. The

computational domain is rectangular and the presented experiment is quasi -1D.

This means that the initial conditions and boundary conditions are chosen as

homogeneous in y-direction, and the flow direction is always along the x-axis.

The principal unknowns are Pℓ (can be achieved by choosing γ(S) ≡ 1) and X .

All computations are done with the standard model, and gravitational effects

are neglected. Therefore the convective terms qαconv for α = (1), (2) vanish. The

temperature is fixed to T = 500K.

Lx 200m (i.e. [0m, 200m])

Ly 20m (i.e. [−10m, 10m])

Qh −2.785 · 10−1 mol/century/m2 (i.e. −5.57 · 10−6 kg/year/m2)

P outℓ 10 bar (i.e. 106 Pa)

Tinj 5 · 103 centuries (i.e. 5 · 105 years)

Tsim 104 centuries (i.e. 106 years)

Table 4.3: Parameters for numerical experiment 1

Geometry The geometrical setup is shown in figure 4.1. In order to examine

the convergence in space, all computations are done with two structured meshes.


Ω

Lx

Ly

Γimp

Γimp

Γin Γout

Figure 4.1: Geometrical setup for numerical experiment 1

First, a triangular mesh that contains triangles of high quality (almost the same

size, no sharp angles) is used, afterwards all computations are repeated with a

very coarse grid with triangular elements of low quality (i.e. sharp angles). Both

meshes are shown in figure 4.2.

Figure 4.2: Coarse and fine mesh for numerical experiment 1

To examine the convergence in time two different initial time step sizes ∆t =

0.2 centuries and ∆t = 4.0 centuries are used. In both cases the time step size

has to be reduced in some of the simulations. This became necessary close to the

phase transition depending on the chosen mesh, time step size and treatment of

the diffusion coefficient.

Boundary and Initial Conditions At the left boundary Γin a constant inflow

of hydrogen is assumed while there is no inflow of water:

q(1)diff · next =

Qh for 0 ≤ t ≤ Tinj

0 else

q(2)diff · next = 0.

The inflow is limited to a certain time Tinj (see table 4.3). At the right boundary

Γout Dirichlet boundary conditions are chosen as

Pℓ = P outℓ X = 10−5.

The upper and lower boundaries Γimp are impervious, i.e.

q(1)diff · next = 0



The source terms Fα for α = (1), (2) are fixed to zero for this experiment. Ini-

tial conditions for P and X are chosen such that the experiment starts in an

undersaturated state where almost no hydrogen is present in the domain Ω:

P 0ℓ = P out

ℓ X0 = 10−5.

4.1.2 Results

Altogether eight simulations are run, one for each combination of the three test

parameters (mesh, time step size, treatment of the diffusion coefficient).

In general, the injection of hydrogen at Γin causes an increase of the concentra-

tion of hydrogen. At about t = 140 centuries the threshold of the concentration

of hydrogen is reached where the phase transition appears at the inflow bound-

ary. Through diffusion the hydrogen is transported towards the domain to the

right boundary and the allocation of the phase transition also moves to the right.

This process continues until the injection of hydrogen is stopped at t = 5000

centuries. After that date the concentration decreases until it drops below the

threshold and the gas phase disappears. The results are presented in figures 4.5 to

4.9 where each figure shows one state variable as a function of the space variable

x at the dates t = 100, 140 (first row), t = 200, 500 (second row), t = 5000, 6700

(third row) and t = 8400, 10000 (fourth row) centuries. The legend for each plot

contains short notations for each of the three test parameters, which are:

• expl/impl: explicit resp. implicit treatment of the diffusion coefficients,

• fine/coarse: used mesh (see figure 4.2),

• small/large: initial ∆t = 0.2 resp. ∆t = 4 centuries.

When the implicit treatment of the diffusion coefficient is used the Newton’s

scheme fails to converge even when ∆t is reduced. Thus, there is not the same

number of curves in each figure.

Additional to the profiles at single dates the development over the entire

period of time gives a good overview of what happens. In the first two graphs

in figure 4.3 the behaviour of the pressures of both phases Pℓ, Pg (left) and the

saturation S (right) at the inflow boundary is given.

Before the gas phase appears, the phase pressures stay (almost) constant and

the saturation is constant zero. After the phase transition, the phase pressures

and the saturation increase rapidly. After reaching a maximum value both pres-

sures begin to decrease due to a smaller ∇Pℓ and due to the fact that there is


6

7

8

9

10

11

12

13

14

15

16

0 2000 4000 6000 8000 10000

pres

sure

s (b

ar)

time (centuries)

PlPg

0

0.5

1

1.5

2

2.5

0 2000 4000 6000 8000 10000

satu

ratio

n (%

)

time (centuries)

S

Figure 4.3: Profiles of phase pressures Pℓ, Pg and gas saturation S at Γin

no water injection. The saturation still increases until the inflow of hydrogen is

stopped at t = 5000 centuries. At this point the saturation decreases. Both phase

pressures drop down to values below the initial pressures, caused by capillary ef-

fects. At a certain point (about t = 6700 centuries) the gas phase disappears

and thus the saturation is constant zero. With ongoing time the values of the

phase pressures move back to their initial values. The corresponding fluxes of

both components at the outflow boundary Γout are shown in figure 4.4.

-1

0

1

2

3

4

5

6

0 2000 4000 6000 8000 10000

hydr

ogen

flux

(m

ol/m

^2/c

entu

ry)

time (centuries)

hydrogen flux

-600

-500

-400

-300

-200

-100

0

100

200

300

400

0 2000 4000 6000 8000 10000

wat

er fl

ux (

mol

/m^2

/cen

tury

)

time (centuries)

water flux

Figure 4.4: Profiles of component fluxes at Γout

Beside the fact that the computations using the implicit treatment of the

diffusion coefficient fail at certain points in time the curves are quite similar. In

general, all curves show the same behaviour of the state variables, except for the

anomalies shown at t = 140 centuries. There is a gap between the results received

by using the fine and the coarse mesh at this time point. This is reasonable since

the state variables are linked to the barycenter of the elements. So the fine mesh

“notices” the phase appearance earlier than the coarse mesh.


0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10

10.001

10.002

10.003

10.004

10.005

10.006

expl. fine, smallimpl. fine, smallexpl. fine, largeimpl. fine, large

expl. coarse, smallimpl. coarse, smallexpl. coarse, largeimpl. coarse, large

0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10

10.1

10.2

10.3

10.4

10.5

10.6



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10

10.2

10.4

10.6

10.8

11

11.2



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10 10.2 10.4 10.6 10.8

11 11.2 11.4 11.6 11.8

12


expl. coarse, smallexpl. coarse, large

0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10 10.02 10.04 10.06 10.08

10.1 10.12 10.14 10.16

expl. fine, smallexpl. fine, largeimpl. fine, large


0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

8 8.2 8.4 8.6 8.8

9 9.2 9.4 9.6 9.8 10



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10

10.0005

10.001

10.0015

10.002

10.0025

expl. fine, smallexpl. fine, large


0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10 10.0001 10.0002 10.0003 10.0004 10.0005 10.0006 10.0007 10.0008 10.0009

10.001



Figure 4.5: Liquid phase pressure Pℓ as function of x at t = 100, 140 (first row),

t = 200, 500 (second row), t = 5000, 6700 (third row) and t = 8400, 10000 (last

row) centuries.


0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0 1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 8e-05



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0 1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 8e-05 9e-05



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 8e-05 9e-05

0.0001 0.00011



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0 2e-05 4e-05 6e-05 8e-05

0.0001 0.00012 0.00014 0.00016



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0 2e-05 4e-05 6e-05 8e-05

0.0001 0.00012 0.00014 0.00016 0.00018

0.0002 0.00022



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 8e-05 9e-05



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

1e-05

1.5e-05

2e-05

2.5e-05

3e-05

3.5e-05

4e-05



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

1e-05

1.2e-05

1.4e-05

1.6e-05

1.8e-05

2e-05

2.2e-05



Figure 4.6: Total molar fraction of hydrogen (i.e. X) as function of x at

t = 100, 140 (first row), t = 200, 500 (second row), t = 5000, 6700 (third row) and

t = 8400, 10000 (last row) centuries.


0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

-1

-0.5

0

0.5

1



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0 0.0002 0.0004 0.0006 0.0008

0.001 0.0012 0.0014 0.0016 0.0018



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0 0.0005

0.001 0.0015

0.002 0.0025

0.003 0.0035

0.004 0.0045



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0

0.002

0.004

0.006

0.008

0.01

0.012



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0

0.005

0.01

0.015

0.02

0.025



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0

0.001

0.002

0.003

0.004

0.005

0.006



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

-1

-0.5

0

0.5

1



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

-1

-0.5

0

0.5

1



Figure 4.7: Gas saturation (i.e. S) as function of x at t = 100, 140 (first row),


row) centuries.


0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10

10.001

10.002

10.003

10.004

10.005

10.006



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10

10.2

10.4

10.6

10.8

11

11.2



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10

10.5

11

11.5

12

12.5



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10 10.5

11 11.5

12 12.5

13 13.5

14 14.5



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10

11

12

13

14

15

16



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

8.6 8.8

9 9.2 9.4 9.6 9.8 10



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10

10.0005

10.001

10.0015

10.002

10.0025



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

10 10.0001 10.0002 10.0003 10.0004 10.0005 10.0006 10.0007 10.0008 10.0009 10.001



Figure 4.8: Gas phase pressure Pg as function of x at t = 100, 140 (first row),


row) centuries.


0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

1.91e-14 1.915e-14

1.92e-14 1.925e-14

1.93e-14 1.935e-14

1.94e-14 1.945e-14

1.95e-14 1.955e-14



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0 0.2 0.4 0.6 0.8

1 1.2 1.4



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0

0.5

1

1.5

2

2.5

3



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

5



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

1.91e-14 1.915e-14

1.92e-14 1.925e-14

1.93e-14 1.935e-14

1.94e-14 1.945e-14

1.95e-14 1.955e-14



0 20 40 60 80 100 120 140 160 180 200-10-8-6-4-2 0 2 4 6 8 10

1.91e-14 1.915e-14

1.92e-14 1.925e-14

1.93e-14 1.935e-14

1.94e-14 1.945e-14

1.95e-14 1.955e-14



Figure 4.9: Capillary pressure Pc as function of x at t = 100, 140 (first row),


row) centuries.

4.2 Convergence Properties

As mentioned before this section is meant to validate the convergence of the

scheme in space and time. The numerical experiment is based on an experiment

4.2. CONVERGENCE PROPERTIES 59

given in [12]. While the parameters of this experiment are kept the same, the

geometry of the mesh is modified for reasons of simplicity and to save computa-

tional time. In general, a rectangular domain Ω is considered in which a constant

source of hydrogen Ωs is allocated at the center of Ω. Both the convergence in

space and in time are studied individually.

In a first part, the convergence in space is determined by fixing the time step

size ∆t for a fixed simulation time Tsim and by varying the refinement level of

the mesh. Here, the time step size has to be chosen so that the influence of the

discretisation error caused by the time step size can be neglected. The solution

of each simulation is compared to a reference solution and the evolution of the

discretisation error is determined.

Second, for the convergence in time, the mesh refinement level and simulation

time Tsim are fixed and the time step size ∆t is modified. Here, the refinement

level is chosen such that the discretisation error coming from the spatial discreti-

sation can be neglected. Again the solution of each simulation is compared to a

reference solution and the evolution of the discretisation error is determined.


Initially, the two principal unknowns P = Pℓ and X are chosen such that the

experiment starts in a purely liquid state. There is a source of hydrogen within

the computational domain Ω limited to a subdomain Ωs that is allocated to the

center Ωs of Ω. For t ∈ [0, Tsim] hydrogen is introduced at a constant rate Qh.

This leads to the appearance of the gas phase at about t = 0.8 centuries. Since the

phase transitions are the most challenging part within the numerical simulations,

Tsim is chosen such that these effects are taken into account for the estimation of

the convergence of the presented scheme.

The effects of gravity are neglected, the temperature is fixed to T = 303K,

and for all simulations the standard model with the explicit treatment of the

diffusion coefficients is used.

Geometry The geometrical setup, with parameters Lx, Ly, Ls given in table

4.4, is shown in 4.10. This domain is a subdomain of the experiment in 4.6. The

allocation of the source at the center of the domain will cause a spreading of the

light component to the left and right boundary. The computational grid that is

used for all simulations in this section can be seen in figure 4.11


Lx 200m (i.e. [0m, 200m])

Ly 20m (i.e. [−10m, 10m])

Ls 20m (i.e. [90m, 110m])

Qh 1.25 mol/m2/century (i.e. 8 · 10−13 kg/m2/s)

PDℓ 10 bar (i.e. 106 Pa)


Table 4.4: Parameters for the study of convergence

Ω

Lx

Ly

Γimp

Γimp

ΓD ΓDΩs

Ls

Figure 4.10: Geometrical setup for the test of convergence

Figure 4.11: Computational coarse grid

Boundary and Initial Conditions At the boundary Γ = ΓD Dirichlet bound-

ary conditions for the principal unknowns are assumed:

P = PDℓ on ΓD

X = 10−5 on ΓD.

The upper and lower boundaries Γimp are again impervious, i.e.



The source terms Fα for α = (1), (2) are fixed to

F (1) =

Qh on Ωs

0 on Ω \ ΩsF (2) = 0


for this experiment. Initial conditions for Pℓ and X are chosen appropriate to

the boundary conditions:

P 0ℓ = PD

ℓ X0 = 10−5.

Simulation Results For both the convergence in space and time reference

solutions are computed. All figures within this section give an overview of one

state variable at the cross section of Ω at [0m, 200m] × 0m. Figures 4.12 to

4.15 show the simulation results for ∆t = 7.8125e− 4 centuries with the coarse

grid refined 5 times.

Figure 4.12: Evolution of Pℓ in time

As already known the phase pressures stay almost constant until the gas phase

appears. And exactly this circumstance can be seen when having a closer look at

the center of the x-axis in figures 4.12 and 4.13. After the phase transition both

the liquid and the gas phase pressure increase rapidly.

In figure 4.14 the evolution of the total molar fraction of the light component

in time is presented. At the beginning of the experiment the light component


Figure 4.13: Evolution of Pg in time

is almost limited to Ωs. With ongoing time the light component diffuses to the

left and the right boundary. The rate at which this diffusive transport process

takes place increases with time depending on the gradient of X (see figure 4.15

for the evolution of the (diffusive) flux of the light component). Moreover, as a

consequence of the initial pure liquid state the saturation is constant zero until at

about t = 0.8 centuries the gas phase appears. Directly after the first gas phase

appearance the saturation adopts values of about 10−7 to 10−6 and increases with

an increasing value of X . In summary it can be stated that the results match

the effects which have been expected from theoretical considerations (see section

2.3). In comparison to the first numerical experiment 4.1 it can be said that the

properties of the model and the observed numerical difficulties fit together.

In the following sections 4.2.2 and 4.2.3 one can see that this solution is

already very close to the corresponding reference solution which will be used for

the estimation of the numerical rate of convergence.


Figure 4.14: Evolution of X in time

Figure 4.15: Evolution of q(1) in time


Figure 4.16: Evolution of S in time

Convergence Rate Estimation Let hl be a positive decreasing sequence.

The index l denotes the level of the mesh refinement. The order of numerical

convergence αN in h of the discretisation scheme satisfies

αN ≥ ln

(‖u− uhl−1‖L2(Ω)

‖u− uhl‖L2(Ω)

)/

ln

(hl−1

hl

)

(4.1)

where u represents the exact solution of the considered unknown and uhl the so-

lution on the mesh Ωhl. In order to determine the rate of convergence a numerical

experiment is set up in which (ideally) the exact solution is known. In general,

the procedure is to compute a corresponding right hand side to any given solu-

tion of the problem. This has not been done in this work since the given model

equations are highly non-linear and coupled such that it is extremely complicated

to compute a right hand side for a given solution for the two principal unknowns.

Another - not necessarily a less important - reason is that such an artificial prob-

lem is far away from the applications in mind. This could lead to results in which

a much better rate of convergence is achieved than for the application of the

method to any realistic scenario.

In 4.1 a reference solution uref can also be used as an alternative to an exact

solution u. This reference solution is achieved by the same simulation setup with

a very small h and time step size ∆t. By means of this approach it is possible


to use a simplified test scenario that is based on a realistic problem, and thus

it gives a rate of convergence of the method that can (almost) be expected for

realistic scenarios. In order to be up to do this one has to ensure that the given

problem has a unique solution. Otherwise the algorithm could converge, but to a

wrong solution. Here, a reference solution is used denoted by the index ref . Any

results shown here are based on that reference solution.

4.2.2 Convergence in Space

For the study of the spatial convergence simulations with a fixed setup are re-

peated for different refinement levels of the coarse computational mesh shown in

figure 4.11 with a fixed time step size ∆t = 1.5625e-3 centuries. The time step

size is chosen such that the time discretisation error is negligibly small. By one

uniform refinement step one cell Ωi is separated into 4 cells as can be seen in

figure 4.17.

Ωi at leveli = 0

Ωi at leveli = 1

Ωi at leveli = 2

Figure 4.17: Uniform refinement of a cell Ωi

In the tables 4.5 and 4.6 the evolution of the error ‖u-uh‖L2(Ω) for the scalar

unknowns P,X as well as the flux unknowns q(1), q(2) are given. With respect

Level l #cells ‖P -Ph‖L2(Ω) αN ‖X-Xh‖L2(Ω) αN0 242 2.5316 − 1.8922e-4 −1 968 1.5557 0.7024 1.1052e-4 0.7758

2 3872 0.8124 0.9373 5.5844e-5 0.9848

3 15488 0.3836 1.0826 2.6934e-5 1.0519

4 61952 0.1798 1.0932 1.3062e-5 1.0441

5 247808 0.0843 1.0928 6.3250e-6 1.0462

ref 3717120 − − − −

Table 4.5: Evolution of the error ‖u-uh‖L2(Ω) in space for the scalar unknowns

u ∈ P,X


Level l #cells ‖q(1)-q(1)h ‖L2(Ω) αN ‖q(2)-q

(2)h ‖L2(Ω) αN

0 242 5.6311 − 156.7498 −1 968 3.3267 0.7593 89.6133 0.8067

2 3872 1.731 0.9425 44.6102 1.0063

3 15488 0.8513 1.0239 21.3223 1.065

4 61952 0.4136 1.0414 10.253 1.0563

5 247808 0.2011 1.0403 4.9211 1.059

ref 3717120 − − − −

Table 4.6: Evolution of the error ‖qα-qαh‖L2(Ω) in space for the flux unknowns

with α ∈ (1), (2)

to the used elements RT0, one expects convergence of order 1 for both the scalar

and the flux unknowns which also can be observed in the experiment [5, 26, 29].

Note, that the magnitudes of ‖·‖L2(Ω) varyy greatly for the single unknowns. This

is caused by different scales of the physical dimensions (e.g. pressures: about 10

to 20 bar; total molar fraction: about 10−8 to 1). By scaling of ‖ · ‖L2(Ω) with

the norm of the reference solution for each unknown (e.g. ‖P -Ph‖L2(Ω)/‖P‖L2(Ω))

one gets a comparable magnitude for all observed unknown.

In figures 4.18 to 4.22 the profiles of the state variables Pℓ, Pg, X, S, q(1) are

shown with respect to the reference solution. In the legend the index corre-

sponds to the refinement level l. Although there are also simulations done with

refinement level 5 it is not meaningful to plot the profiles for these one. The

simulation results received with the mesh on refinement level 4 already match

the reference solution very well. To see any difference to the reference solution

one has to zoom into the figures as demonstrated in each of the figures 4.18 to

4.22. The most interesting areas are the center of the domain as well as areas

with kinks/discontinuities and areas close to the bottom of the curves. For this

reason image details are given for these areas only.

By having a look at the figures 4.18 to 4.20 it can be seen that a coarse

mesh causes an over-estimation of the phase pressures as well as the total molar

fractionX . This leads to a higher diffusion (see figure 4.22) of the light component

towards the domain, which finally influences the area in which gaseous hydrogen

is present. This is reasonable since the faster the values for P = Pℓ and X

reach the “threshold”, the faster the gas phase appears. This can be seen in

the comparison of the solution of S in figure 4.21. As a consequence, it can be

observed in the numerical experiments that the date at which the phase transition

appears first differ slightly.


Figure 4.18: Profiles of Pℓ for varying refinement levels l = 0, . . . , 4

Figure 4.19: Profiles of Pg for varying refinement levels l = 0, . . . , 4


Figure 4.20: Profiles of X for varying refinement levels l = 0, . . . , 4

Figure 4.21: Profiles of S for varying refinement levels l = 0, . . . , 4


Figure 4.22: Profiles of q(1) for varying refinement levels l = 0, . . . , 4

With an increasing refinement level these effects disappear and the solution

for each of the state variables converges towards the reference solution.

4.2.3 Convergence in Time

After the spatial convergence is shown numerically, in this section the same sim-

ulation setup is used to determine the convergence properties in time. For that

purpose the mesh refinement level is fixed to l = 4 and the time step size ∆t

varies. In this case the refinement level is chosen such that the spatial discretisa-

tion error is so small that it can be neglected. The time step size ∆t is computed

by following

∆t = 2−i∆tinit with i = 0, . . . , 5 (4.2)

with ∆tinit = 0.25 centuries. In addition to the resulting six simulations, a

reference solution is computed for the same mesh refinement level with ∆t = 2e−5

centuries which is about 40 times smaller than the finest time step size. All

results are compared to that reference solution. Note, that this affects the rate

of numerical convergence.

In the tables 4.7 and 4.8 the evolution of the error for the scalar unknowns

P,X and the flux unknowns qα for α = (1), (2) are presented. As described in

the section about the spatial convergence, image details are given for the areas

around the center of the computational domain, the areas in which the solutions


show kinks and the areas at the bottom of the curves. It can be recognised that

for large time steps the solution shows considerable variations with respect to

the reference solution. Repeatedly it can be observed that the method converges

for all determined state variables at an almost constant rate, even for the first

refinements of the time step size. As before the order of magnitude of ‖ · ‖L2(Ω)

for each of the unknowns varies. By applying a scaling as in section 4.2.2 the

order of magnitude becomes comparable for each considered unknown.

i ∆t ‖P -Ph‖L2(Ω) αN ‖X-Xh‖L2(Ω) αN0 2.5000e-2 1.2441 − 3.7830e-6 −1 1.2500e-2 0.6753 0.88 2.0243e-6 0.90

2 6.2500e-3 0.3428 0.98 1.0427e-6 0.96

3 3.1250e-3 0.1696 1.02 5.1027e-7 1.03

4 1.5625e-3 0.0839 1.01 2.5109e-7 1.02

5 7.8125e-4 0.0416 1.01 1.2350e-7 1.02

ref 2.0000e-5 − − − −

Table 4.7: Evolution of the error ‖u-uh‖L2(Ω) in time for the scalar unknowns

u ∈ P,X

i ∆t ‖q(1)-q(1)h ‖L2(Ω) αN ‖q(2)-q

(2)h ‖L2(Ω) αN

0 2.5000e-2 1.2665e-1 − 4.2814e+1 −1 1.2500e-2 6.5458e-2 0.95 2.1848e+1 0.97

2 6.2500e-3 3.2130e-2 1.03 1.0809e+1 1.01

3 3.1250e-3 1.5811e-2 1.02 5.3467e+0 1.02

4 1.5625e-3 7.8124e-3 1.02 2.6263e+0 1.02

5 7.8125e-4 3.8519e-3 1.02 1.1802e+0 1.01

ref 2.0000e-5 − − − −

Table 4.8: Evolution of the error ‖qα-qαh‖L2(Ω) in time for the flux unknowns with

α ∈ (1), (2)

This behaviour of convergence can also be seen in figures 4.23 to 4.27. Anal-

ogously to the visualisation of the solution of the state variables Pℓ, Pg, X, S and

q(1) in the previous section, the results of the simulation using the finest ∆t are

not presented in the corresponding figures since even at a very large zoom al-

most no difference to the reference solution can be seen. In the legend the index

displays the value of i that determines the time step size following equation (4.2).

For the first time steps a large difference between the solution of the consid-

ered state variable and the corresponding reference solution is observed. With a


decreasing ∆t these variations become smaller until the solution almost matches

the reference solution. Contrary to the behaviour of the solutions for an in-

creasing refinement level a too large time step size causes an under-estimation

of the state variables. The decrease of the time step sizes results in two major

observations:

1) The time discretisation error declines rapidly for the first refinement steps.

Afterwards it decreases at an almost constant rate for an evermore reducing

time step size. For i ≥ 3 the global error between the solution for P,X

and the flux unknowns is already so small that it cannot be seen in the

visualisation without zooming far into each graph.

2) For large time step sizes (i ∈ 0, 1) the “first” phase transition is quite

complicated to compute. The consequence is that the time step size has to

be refined when the phase transition appears for the first time. Afterwards

∆t can be increased towards the prior value without any problems. These

numerical difficulties for the first gas phase appearance vanish for smaller

time step sizes.

Figure 4.23: Profiles of Pℓ at T = Tsim for different time step sizes ∆t

In contrast to the effect of an over-estimation of the principal unknowns, the

under-estimation does not have such a strong impact on the date of the first gas

phase appearance. In the previous section we could see that large deviations of


Figure 4.24: Profiles of Pg at T = Tsim for different time step sizes ∆t

Figure 4.25: Profiles of X at T = Tsim for different time step sizes ∆t


Figure 4.26: Saturation profiles at T = Tsim for different time step sizes ∆t

Figure 4.27: Profiles of q(1) at T = Tsim for different time step sizes ∆t


P and X to the corresponding reference solution led to a significant change of

the solution for S. These deviations exist in a lower maximum saturation as well

as in spatial expansion for areas with gaseous hydrogen. This effect cannot be

noticed here. Major differences to the reference solution can only be observed at

Ωs. Here, the under-estimation of the principal unknowns result in an obvious

deviation of the saturation S to the reference solution. This effect vanishes with a

decreased ∆t and the solution for each of the considered state variable converges

towards the corresponding reference solution.

4.2.4 Resume

In the previous two sections the convergence properties in space and time have

been determined. The setup of this experiment is taken from a numerical example

presented in [12]. For simplification reasons only a subdomain of the one in

experiment 2 in [12] is considered whereas all physical parameters are maintained

the same. Nevertheless, similar observations for the convergence in space and time

are obtained for real 2D simulations, but they take much more computational

time so that this was not applicable for the study of the convergence properties.

It could be seen that the method converges in space and time with order

αN ≈ 1 for all determined scalar state variables Pℓ, Pg, X and S as well as the

flux unknowns qα with α ∈ (1), (2). These results are based on a reference

solution and do not represent a proof of the convergence. Nevertheless one gets a

good impression at which order the presented scheme may potentially converge.

A proof of the existence and the uniqueness of a solution as well as error estimates

of the solution of the given problem are not obvious.

4.3 Liquid Phase Disappearance

One of the most important differences to other models as for example presented

in [12] is the additional support of the liquid phase disappearance. By using

a double complementarity formulation as explained in section 2.5 it is ensured

that both phases can disappear (with an appropriate phase diagram). Here, an

example is given in which both the gas phase appearance and the disappearance

of the liquid phase are presented.

In order to realise the liquid phase disappearance it is important to choose a

phase diagram that enables this effect. The one used for the prior experiments

(as given in figure 2.2 of section 2) assumes that XM ≡ 1 and thus it is impossible

for the liquid phase to vanish. Moreover, the modelling of the capillary pressure

4.3. LIQUID PHASE DISAPPEARANCE 75

has to be modified for this experiment. The Van Genuchten parametrisation is

not valid for values of S = 1, i.e. Sℓ = 0, and therefore it cannot be used here.

Any parametrisation in which Pc(S) is defined for S ∈ [0, 1] has to be used when

both phase transitions shall be considered.


Within a rectangular domain with Dirichlet boundary conditions on the left hand

side and impervious boundaries elsewhere, initial conditions are chosen such that

the experiment starts in the purely liquid state. This choice of boundary condi-

tions leads to an increase of the total molar fraction of the light component and

finally to the first phase transition in which the gas phase appears in parts of Ω.

Proceeding the experiment, the value of X , Pℓ and S will continue to increase

until the second threshold is reached at which the liquid phase will vanish. To

allow both phase transitions, the linear capillary pressure - saturation relation

Pc(S) = P 0c · S

with a constant P 0c is used. The remaining closure relationships are given by

krg(S) =S

krℓ(S) =1− S

Xm(P, S) =max

0, Xcritm

PgPg + Pcrit

XM(P, S) =min

XcritM , Xcrit

M

√PgPcrit

γ(S) =

S(2− S), for S ∈ [0, 1]

0, if S ≤ 0,

1, if S ≥ 1

(needed to define the complementarity problem)

Nℓ =min

N stdℓ

Bℓ(1− xcritℓ ),

N stdℓ

Bℓ(1− x(1)ℓ )

Bℓ =min1, 1

1 + s · Pℓ

Ng =A · Pg

with constants P 0c , Pcrit, X

critm , Xcrit

M , xcritℓ , N stdℓ , s, A,Dℓ, Dg and K as listed in ta-

ble 4.9.


Lx 200m (i.e. [0m, 200m])

Ly 20m (i.e. [−10m, 10m])

Tinj 2 · 104 centuries (i.e. 2 · 106 years)

Tsim 3.4815 · 104 centuries

P 0c 20 bar

Pcrit 100P 0c

Xcritm 0.8

XcritM 0.85

xcritℓ 0.9995

N stdℓ 8.5 · 104 mol·m−3

s 10−4 bar−1

A 39.7 mol·bar−1

Dℓ = Dg 0.47 m2/century

K 10−20 m2

Table 4.9: Parameters and constants for numerical experiment 2

Geometry For this experiment the fine mesh given in section 4.1.1 is used.

The setup including the varied type of boundary conditions is shown in figure

4.28 with parameters Lx, Ly given in table 4.9.

Ω

Lx

Ly

Γimp

Γimp

ΓD Γimp


Boundary and Initial Conditions At the left boundary ΓD Dirichlet bound-

ary conditions for P and X are assumed as follows

P =100 bar

X =

0.5, for 0 ≤ t ≤ Tinj

10−4, for Tinj < t ≤ Tsim.

The boundaries Γimp are assumed to be impervious

u(1)diff · next = u

(2)diff · next = 0.

4.3. LIQUID PHASE DISAPPEARANCE 77

For this experiment the source terms Fα for α = (1), (2) are fixed to zero. Initial

conditions for Pℓ and X are chosen uniform with

P 0ℓ = 100 bar and X0 = 10−4.

4.3.2 Results

Since this experiment is meant to demonstrate the gas phase appearance and

liquid phase disappearance, the evolution of the gas saturation S and X in time

is shown. Owing to the setup of the experiment the phase transitions first appear

at the left boundary of Ω. This leads to the presence of different states within Ω.

The simulation results are shown in figures 4.29 to 4.33.

Figure 4.29: Evolution of S at x ≈ 50m (left) and x ≈ 140m (right) in time.

Figure 4.30: Evolution of X at x ≈ 50m (left) and x ≈ 140m (right) in time.

On the left hand side of figure 4.29 it can be seen that within the first centuries

only the gas phase is present. By increasing the total molar fraction of the light

component (as to be seen in figure 4.30) the first phase transition happens at

the chosen location (x ≈ 50m). An further increase of X finally leads to the

disappearance of the liquid phase.


On the right hand side (x ≈ 140m) the first phase appearance can be seen at a

later date due to the larger distance from the left boundary. The saturation also

keeps increasing, until the boundary conditions at ΓD are changed at t = Tinj .

At this point the value of X decreases and so the saturation does. The threshold

for the second phase transition is not reached in this part of Ω.

Figure 4.31: Values of S and X at t = 3005 centuries.


4.4 Two-Phase Flow in Heterogeneous Porous

Media

With respect to the movement of fluids in the subsurface the presence of a ho-

mogeneous porous medium is not very realistic. On the contrary in general the

subsurface is highly heterogeneous. Thus this experiment focuses on transport

processes of hydrogen and water in a computational domain Ω with different

soil/rock properties. That is why the experiment is split up into two parts:

4.4. TWO-PHASE FLOW IN HETEROGENEOUS POROUS MEDIA 79


(A) In a first numerical experiment only the porosity φ and the absolute per-

meability tensor K are chosen differently in 2 subdomains within Ω. For

both parts of Ω the same parametrisation for Pc is assumed, i.e. the Van

Genuchten parameters are the same for the two subdomains.

(B) In addition to the porosity and the permeability tensor the parametrisation

for Pc is altered. To realise this simulation setup, the principal variables

have to be chosen differently to (A). An explanation for this will be given

in section 4.4.3.

A change of the soil/rock parameters as well as changes of the used parametrisa-

tion for the capillary pressure have strong influence on the fluid flow. This is to

be seen in this experiment. Here, the parameters only vary in a way such that

a fluid flow through the corresponding subdomain is still possible. In many ap-

plications (e.g. CO2 sequestration) a so-called “cap-rock” which is impermeable

for the light component is assumed. This will be the main focus of the second

experiment of this section.


The setup of experiment A equals that one of the first experiment in section 4.1

except for a modified mesh that contains two subdomains with different values

for φ and K. Hydrogen is injected into the domain for T ∈ (0, Tsim]. This leads

to an increase of X and thus the gas phase appears after a certain time. Due

to a lower value of φ and K in subdomain Ω2 (see figure 4.34) the movement of

the hydrogen is reduced in this part of Ω which affects the gas phase appearance.

Since less hydrogen is transported into Ω2 the threshold for the phase transition

is reached later and thus the saturation profile will show a kink at the subdomain

boundary.


In addition to φ and K the function of Pc is chosen discontinuously for both

subdomains in experiment B. By choosing Pℓ and X as principal unknowns Pgwill become discontinuous for different definitions of Pc. This is a contradiction

to the model that ensures the continuity of P (and therefore Pℓ and Pg) and X .

By neglecting phase transitions one can choose the phase pressures as principal

variables. Initial conditions are chosen such that the gas phase is already present

in Ω with different values of S in both subdomains. Due to the chosen parameters

only a small amount of hydrogen enters Ω2 and thus the phase transition takes

much more time in this subdomain.

The parameters for the two experiments can be seen in table 4.10. In table

4.11 all soil parameters and Van Genuchten parameters that differ in the two

subdomains are listed. Remind that the Van Genuchten parameters are only

relevant for part B of this experiment. For the simulations of part A the named

first parameters are used. All computations are done with the standard model.

Lx 200m (i.e. [0m, 200m])

Ly 20m (i.e. [−10m, 10m])

L1 20m (i.e. [0m, 20m])

Qh −2.785 · 10−1 mol/century/m2 (i.e. −5.57 · 10−6 kg/year/m2)


P outg 15 bar (i.e. 1.5 106 Pa)

Tsim 104 [103]centuries (i.e. 106 [105] years)

Table 4.10: Simulation parameters for numerical experiment 3

Soil Parameters at Ω1[Ω2]

φ 0.30 [0.15]

K 10−18 5 · 10−20m2

Van Genuchten Parameters at Ω1[Ω2]

Pc0 20 bar [150 bar]

n 1.54 [1.49]

Sgr 0 [0]

Sℓr 0.01 [0.4]

Table 4.11: Varying soil and fluid parameters

The diffusion coefficients are treated explicitly, gravitational effects are neglected

and the temperature is again fixed to T = 303K.


Geometry The geometry of the mesh is shown in figure 4.34. The geometry

including all types of boundaries is similar to the geometry in 4.1 apart from the

fact that Ω is separated into two subdomains Ω1 and Ω2 in order to be able to

simulate two different types of soil.

Ω1

Lx

Ly

Γimp

Γimp

Γin ΓDΩ2

L1


Boundary and Initial Conditions At the left boundary Γin a constant inflow

of hydrogen is assumed while there is no inflow of water:

q(1)diff · next =Q

h


At the right boundary ΓD Dirichlet boundary conditions are set to

(A) Pℓ = P outℓ X = 10−8

(B) Pℓ = P outℓ Pg = P out

g

The upper and lower boundaries Γimp are impervious, i.e.



Like in the first experiment the source terms Fα for α = (1), (2) are fixed to zero.

Initial conditions for P andX (part A) are chosen such that the experiment starts

in an undersaturated state where almost no hydrogen is present in the domain

Ω:

P 0ℓ = P out

ℓ X0 = 10−8.

In the second part of this experiment Pℓ and Pg are the principal unknowns with

initial conditions

P 0ℓ = P out

ℓ Pg = P outg

with consequence S > 0 on Ω.


4.4.2 Results (A)

In figures 4.35 to 4.37 the temporal evolution of Pℓ, Pg and S at the cross-section

[0m, 200m] × 0m of Ω are shown. As long as the hydrogen is remains only in

Ω1 the solution looks the same as in the first numerical experiment presented in

section 4.1. Due to slight porosity φ and absolute permeability K hydrogen only

diffuses at a lower rate from Ω1 to Ω2. This has two important consequences.

(i) Since the total amount of hydrogen is smaller in Ω2 the gas phase takes

more time to appear. As a consequence the profiles of the phase pressures

as well as the profile of S show kinks at x = 20m.

(ii) The fluid flow in Ω2 is reduced in comparison to the flow rates in Ω1. This

causes an accumulation of hydrogen in Ω1 and hence a higher saturation in

time than for the homogeneous case.

Figure 4.35: (A) Evolution of Pℓ in time

It can clearly be seen that because of the soil properties in Ω2 the inflow of

hydrogen is only reduced but not inhibited. The lower permeability and poros-

ity affect the transport processes in the form of a retardation of the hydrogen

migration process. By changing also the used parametrisation for the capillary

pressure saturation relation as done in experiment (B) of this section the retar-

dation becomes so high that Ω2 becomes a barrier.


Figure 4.36: (A) Evolution of Pg in time

Figure 4.37: (A) Evolution of S in time

4.4.3 Results (B)

For this second part of the experiment the parameters for the capillary pressure

saturation relation are switched. Since phase transitions are neglected for this

simulation the initial conditions for the principal unknowns Pℓ and Pg ensure


that both phases are already present at Ω. The profile of the saturation S clearly

demonstrates that the soil in Ω2 now acts as a kind of “cap-rock”, i.e. an almost

impermeable rock. Therefore the hydrogen accumulates in Ω1 which causes an

rapid increase of the gas phase pressure. Due to the high gas phase pressure small

amounts of hydrogen can pass the interface between the two subdomains. There

exists a discontinuity of the saturation between both subdomains, i.e. a jump of

S at the interface.

Figure 4.38: (B) Evolution of Pℓ in time

4.4.4 Resume

By comparing the simulation results of both experiments (A) and (B) it can

clearly be seen that a change of soil properties as well as changes of parameters

of the used capillary pressure saturation relation has a strong influence on the

fluid flow. The submitted model does not support all of the effects. In part (A)

a retardation of the hydrogen flow is presented. For that simulation the model

including the standard choice of principal unknowns could be used without any

complications. As long as the capillary pressure remains continuous the model

facilitates phase transitions. By assuming a discontinuous capillary pressure func-

tion the model as presented does not enable phase transitions any more. In part

(B) phase transitions were neglected and the principal unknowns were chosen

that way that both remain continuous. The handling of such discontinuities is -


Figure 4.39: (B) Evolution of Pg in time

Figure 4.40: (B) Evolution of S in time

independent from the chosen discretisation - one major difficulty that is discussed

by many scientists, e.g. in [18, 17]. One approach of dealing with that is also

presented in [48].


4.5 Inhomogeneous Initial Conditions in an

Insolated Domain

This experiment focuses on a flow that is caused by concentration gradients in

consequence of different initial conditions in two adjacent parts of the compu-

tational domain Ω. It is also part of the benchmark [11]. Starting in a non-

equilibrium state the phase pressures and saturations are observed in a closed

domain as time goes by. For an infinite time the system reaches the equilibrium

in which the phase pressures and saturations are homogeneous in Ω.

Here, the beginning of this process is examined. It is a challenging problem

because of very high concentration gradients and thus high flow rates within the

first time steps of the experiment. With the passing of time changes of the phase

pressures and saturation can only be noticed by using larger time scales. This is

reasonable since the flow rates decrease caused by an increasing mixing of both

components in both subdomains.


As in the examples described before, this is a quasi-1D experiment with principal

unknowns Pℓ and X . Gravitational effects are neglected and thus only diffusive

transport takes place. The outer boundaries are impermeable, i.e. there is no flux

of hydrogen and water out of the domain. The temperature is fixed to T = 303K.

The initial conditions are chosen such that in both subdomains different initial

saturations of hydrogen are present. Within each subdomain the initial conditions

are chosen as homogeneous. Variable time step sizes are used starting off with

Lx 1.0m (i.e. [0m, 1.0m])

Ly 0.1m (i.e. [−0.05m, 0.05m])

L1 0.5m

Pℓ,Ω1, Pℓ,Ω2 10 bar (i.e. 106 Pa)

XΩ1 2.678034 · 10−4 (i.e. Pg,Ω1 = 1.5 · 106 Pa)

XΩ2 1.34355 · 10−3 (i.e. Pg,Ω2 = 2.5 · 106 Pa)

used Model standard

Tsim 106 seconds

Table 4.12: Parameters for numerical experiment 4

small time steps. Then the time steps sizes are increased as follows:

• ∆t = 10s for t ∈ [0, 100] seconds,

4.5. INHOM. INIT. CONDITIONS IN AN INSOLATED DOMAIN 87

• ∆t = 100s for t ∈ [100, 104] seconds,

• ∆t = 250s for t ∈ [104, 5 · 104] seconds and

• ∆t = 500s for t ∈ [5 · 104, Tsim] seconds.

One major problem of that simulation was to choose the stopping criteria for the

Newton’s methods correctly. On the one hand the problem is very difficult to

solve for the first time steps. So the stopping criteria could not be chosen too

strictly. On the other hand the problem can be solved more easily at later time

steps. When the system is “close” to the equilibrium, the initial defect in the

Newton’s methods becomes really small and hence - for badly chosen stopping

criteria - no more Newton iterations are done.

Geometry The computational domain Ω consists of the two subdomains Ω1

and Ω2. They are not necessarily of the same size. The geometrical setup is

shown in figure 4.41 with parameters Lx, Ly, L1 given in table 4.12.

Ω1

Lx

Ly

Γimp

Γimp

Γimp ΓimpΩ2

L1


Boundary and Initial Conditions The boundary Γ is assumed to be imper-

vious

u(1)diff · next = u

(2)diff · next = 0.

For this experiment the source terms Fα for α = (1), (2) are fixed to zero. Hence,

the mass of both components is constant in Ω for t ∈ [0, Tsim].

Initial conditions for Pℓ and X are chosen such that the experiment starts in

a non-equilibrium state:

P 0ℓ,Ω1

= Pℓ,Ω1, X0Ω1

= XΩ1 on Ω1 and

P 0ℓ,Ω2

= Pℓ,Ω2, X0Ω2

= XΩ2 on Ω2.


4.5.2 Results

Figures 4.42 to 4.52 give an overview of what is happening in the course of time.

At the beginning the large concentration gradient causes high flow rates that

lead to a rapid exchange of both components between the two subdomains. The

more mass moves from Ω1 to Ω2, the lower the gradient of the phase pressures

and saturations is, and so the flow rates decrease. Although the liquid pressure

was initially chosen as homogeneous, there is a drop at the interface for the first

time steps on both subdomains. This is caused by the exchange of mass of both

components. The increase of the saturation at the interface in Ω1 corresponds to

an increase of the liquid pressure, whereas the decrease of the saturation at the

interface in subdomain Ω2 is related to a decrease of Pℓ.

Figure 4.42: Profiles of Pℓ, Pg and S at t = 10s.



Figure 4.50: Profiles of Pℓ, Pg and S at t = 2 · 105s.

Figure 4.51: Profiles of Pℓ, Pg and S at t = 5 · 105s.



For this experiment one can conclude that, starting in a non-equilibrium state,

the system is close to the equilibrium after 106 seconds.

4.6 2D Test Case: Comparison of Models

This experiment is very similar to the one presented in section 4.2. Whereas the

simulations in section 4.2 are modified to study the convergence of the method,

here the original numerical experiment presented in [12] is realised. This experi-

ment is also used to compare the two possible choices of physics (“standard” and

“alternative” in section 2.3). It has already been demonstrated that the method

converges in space and time for the choice of the “standard” physics. Based on

this, a comparison of both physics is presented and the difference between the two

will be illustrated in this section. Moreover, the setup of the original experiment

is extended by taking also gravitational effects into account.


Initially, the two principal unknowns P = Pℓ and X are chosen such that the

experiment starts in a purely liquid state. There is a source of hydrogen within the

computational domain Ω limited to a subdomain Ωs that is allocated to the center

of Ω. For t ∈ (0, Tsim] hydrogen is introduced at a constant rate Qh. This leads

to the appearance of the gas phase at about t = 0.80 centuries. The temperature

is fixed to T = 303K. The final time Tsim is set that way that the system is

close to the stationary state at the end of the computation. As mentioned before,

Lx 200m (i.e. [0m, 200m])

Ly 200m (i.e. [−100m, 100m])

Ls 20m

Qh 1.25 mol/m2/century (i.e. 8 · 10−13 kg/m2/s)



Table 4.13: Parameters for the comparison of models.

this experiment contains two different parts. At first gravitational effects are

neglected (A) and afterwards the computations are repeated where g is related

to the earth’s gravity (B).

Geometry For the simulations an unstructured and partially refined mesh is

used. Due to the setup of the experiment it is obvious that major changes take

4.6. 2D TEST CASE: COMPARISON OF MODELS 93

place close to the center of Ω whereas close to the boundary Γout only small

changes of the state variables occur. In line with this, it is reasonable to use a

mesh that is refined encircling the subdomain Ωs. The geometric properties of

the domain Ω are shown in figure 4.53. The left hand side shows the overall mesh,

on the right the section close to Ωs can be seen. The parameters Lx, Ly, Ls in

figure 4.53 are given in table 4.13.

Ω

Ωs Γout

Γout

Γout

Γout

Lx

Ly Ls

Figure 4.53: Geometrical setup for 2D test case

Boundary and Initial Conditions At the boundary Γ = Γout Dirichlet

boundary conditions for the principal unknowns are assumed:

P = P outℓ on Γ

X = 10−5 on Γ.

The source terms Fα for α = (1), (2) are fixed to

F (1) =

Qh on Ωs

0 on Ω \ ΩsF (2) = 0.


Figure 4.54: Partially refined mesh.

for this experiment. Initial conditions for Pℓ and X are chosen appropriately to

the boundary conditions:

P 0ℓ = P out

ℓ X0 = 10−5.

4.6.2 Results

In sum, four simulations have been run, one for each of the physics including

respectively excluding gravitational effects. Simulation results are given in sets

of 6 pictures for each of the state variables Pℓ, X and S. First, the results obtained

by the standard physics are presented at t = 0.6, t = 2, t = 20 centuries (first

row) and t = 35, t = 70, t = 200 centuries (second row).

2D test case without gravitational effects

Figures 4.55 to 4.57 give the simulation results for Pℓ, X and S using the standard

model without taking gravity into account. The figures depict the well known

behaviour: the gas saturation stays constant zero whereas X increases due to

the constant source of hydrogen and Pℓ slightly increases. Gradually, hydrogen

spreads circularly across Ω, the gas phase appears and also spreads until the

stationary state is reached.

In figures 4.58 to 4.60 the difference between the solutions obtained with both

the standard and the alternative model is shown. The difference is taken point-

wise and visualised with ParaView. [58, 41] By comparing the figures for Pℓ at

t = 0.6 one notices that Pℓ is constant for the simulation using the alternative


Figure 4.55: Evolution of Pℓ in time.

Figure 4.56: Evolution of X in time.

model before the phase transition appears. In general, the difference between the

solutions using the two different physics stays within the given specified accuracy

of the Newton solvers (global, local and static) and of the linear solver.


Figure 4.57: Evolution of S in time.

Figure 4.58: Difference between standard and alternative model for Pℓ

2D test case including gravitational effects

After all kinds of effects, without taking gravitational effects into account, have

been studied in the previous computations the same setup is repeated with g

related to the earth’s gravity directed downwards. According to the direction of

gravity the hydrogen is transported by convection towards the bottom boundary,

whereas diffusion leads to a spread in any direction. This can be seen in figures


Figure 4.59: Difference between standard and alternative model for X

Figure 4.60: Difference between standard and alternative model for S

4.61 to 4.63. Note that the gas phase appearance is supported by a lower liquid

phase pressure northern of Ωs. Moreover, because of differences in the phase

densities there is an upward movement of the gas phase.

Figures 4.64 to 4.66 show the differences between the solutions for the cor-

responding state variables. It can be noticed that there is a little effect of the

mesh on the solution. Due to the geometry of the mesh there are a few triangular


Figure 4.61: Influence of gravity on Pℓ in time

Figure 4.62: Influence of gravity on X in time

elements of lower quality within the mesh. These elements are allocated close to

the boundary of Ωs in the north-east and south-west. Among the simulation

results using the standard model this cannot be noticed, but by taking the differ-

ence between the solutions it can clearly be seen. Nevertheless, the error remains

within the given accuracy for the iterative and linear solvers.


Figure 4.63: Influence of gravity on S in time

Figure 4.64: Difference between standard and alternative model for Pℓ including

gravitational effects.

4.6.3 Subsumption

By means of this experiment it was possible to show that the two different model

formulations produce similar simulation results for the given setup. The specific

physical properties of the alternative model presented in section 2.3 can be seen

as expected. Moreover, it was possible to recompute the results that are given in


Figure 4.65: Difference between standard and alternative model for X including


Figure 4.66: Difference between standard and alternative model for S including


[12]. Independently from the existence of gravitational effects both models only

differ within the accuracy of iterative and linear solvers.

Appendix A

Derivatives

A.1 Derivatives of the Resolution Function

Derivation of equation (3.22) with explicit treatment of the diffusion

coefficients

(

∂x,yFi

)

(λ,µ,x,y)=

|Ωi|∆t

(∂m(1)(x,y)

∂x

∂m(1)(x,y)∂y

∂m(2)(x,y)∂x

∂m(2)(x,y)∂y

)

+∑

Ek⊂∂Ωi

∑

τ

ai,k

∂f(1)τ (x,y)∂x

· b(1)τ (λn−1k , µn−1

k ) ∂f(1)τ (x,y)∂y

· b(1)τ (λn−1k , µn−1

k )∂f

(2)τ (x,y)∂x

· b(2)τ (λn−1k , µn−1

k ) ∂f(2)τ (x,y)∂y

· b(2)τ (λn−1k , µn−1

k )

.

For α = (1), (2)

∂F αi

∂λk= −ai,k

∑

τ

[∂fατ (λk, µk)

∂λkbατ (λk, µk)

]

∂F αi

∂µk= −ai,k

∑

τ

[∂fατ (λ

0k, µ

0k)

∂µkbατ (λ

0k, µ

0k)

]

Derivation of (3.22) with implicit treatment of the diffusion coefficients

(

∂x,yFi

)

(λ,µ,x,y)=

|Ωi|∆t

(∂m(1)(x,y)

∂x

∂m(1)(x,y)∂y

∂m(2)(x,y)∂x

∂m(2)(x,y)∂y

)

+∑

Ek⊂∂Ωi

∑

τ

ai,k

∂f(1)τ (x,y)∂x

b(1)τ (λk, µk)

∂f(1)τ (x,y)∂y

b(1)τ (λk, µk)

∂f(2)τ (x,y)∂x

b(2)τ (λk, µk)

∂f(2)τ (x,y)∂y

b(2)τ (λk, µk)

.

101

102 APPENDIX A. DERIVATIVES

For α = (1), (2)

∂F αi

∂λk= ai,k

∑

τ

[

(fατ (x, y)− fατ (λk, µk))∂bατ (λk, µk)

∂λk−∂f

ατ (λk, µk)

∂λkbατ (λk, µk)

]

∂F αi

∂µk= ai,k

∑

τ

[

(fατ (x, y)− fατ (λk, µk))∂bατ (λk, µk)

∂µk−∂f

ατ (λk, µk)

∂µkbατ (λk, µk)

]

A.2 Derivatives of the Flux Continuity Equa-

tions

Derivation of (3.23) with explicit treatment of the diffusion coefficients

For α = (1), (2), d = 1, 2, ∀k ∈ [Nb, Nb +Ni] and El ⊂ ∂Ωid(k) with


∂Gαd,k

∂λl=∑

Ek′⊂Ωi

H[i]kk′

∑

τ

[∂fατ (x, y)

∂x

∂ψxi (λ, µ)

∂λl

+∂fατ (x, y)

∂y

∂ψyi (λ, µ)

∂λl

]

bατ (λn−1k′ , µn−1

k′ )

−H[i]kl

∑

τ

[∂fατ (λl, µl)

∂λlbατ (λ

n−1l , µn−1

l )

]

;

∂Gαd,k

∂µl=∑

Ek′⊂Ωi

H[i]kk′

∑

τ

[∂fατ (x, y)

∂x

∂ψxi (λ, µ)

∂µl

+∂fατ (x, y)

∂y

∂ψyi (λ, µ)

∂µl

]

bατ (λn−1k′ , µn−1

k′ )

−H[i]kl

∑

τ

[∂fατ (λl, µl)

∂µlbατ (λ

n−1l , µn−1

l )

]

.

Derivation of (3.23) with implicit treatment of the diffusion coefficients

For α = (1), (2), d = 1, 2, ∀k ∈ [Nb, Nb +Ni] and El ⊂ ∂Ωid(k) with


A.2. DERIVATIVES OF THE FLUX CONTINUITY EQUATIONS 103

∂Gαd,k

∂λl=∑

Ek′⊂Ωi

H[i]kk′

∑

τ

[∂fατ (x, y)

∂x

∂ψxi (λ, µ)

∂λl

+∂fατ (x, y)

∂y

∂ψyi (λ, µ)

∂λl

]

bατ (λk′, µk′)

+H[i]kl

∑

τ

[

(fατ (x, y)− fατ (λl, µl))∂bατ (λl, µl)

∂λl

−∂fατ (λl, µl)

∂λlbατ (λl, µl)

]

;

∂Gαd,k

∂µl=∑

Ek′⊂Ωi

H[i]kk′

∑

τ

[∂fατ (x, y)

∂x

∂ψxi (λ, µ)

∂µl

+∂fατ (x, y)

∂y

∂ψyi (λ, µ)

∂µl

]

bατ (λk′, µk′)

+H[i]kl

∑

τ

[

(fατ (x, y)− fατ (λl, µl))∂bατ (λl, µl)

∂µl

−∂fατ (λl, µl)

∂µlbατ (λl, µl)

]

.

104 APPENDIX A. DERIVATIVES

Summary

The scope of this thesis is to develop a generalised hybrid mixed finite element ap-

proximation for compositional two-phase flows in porous media. Both phases are

assumed to be miscible and they generally consists of two components. The model

includes a couple of effects such as the appearance and disappearance of both the

gas and the liquid phase, exchanges of components between the two phases as

well as capillary effects. In general, the modelling of multi-phase multi-component

flows is very challenging since the resulting equations are highly non-linear and

strongly coupled which lead to numerical difficulties. This becomes even worse

when phase transitions are taken into account which lead to a degeneration of

the model equations.

With respect to the number of phases a system of two mass conservations is

set up. The choice of principal variables differs in the already existing approaches.

So, a short discussion about possible principal variables is included. The disap-

pearance of the liquid phase disables the possibility to choose the liquid pressure

(which is a common choice in other approaches) as one principal unknown. By

the definition of a mean pressure both phase transitions are enabled and the prin-

cipal variables are well defined for any values of the gas saturation. Additional

to the principal unknowns a number of state variables have to be determined.

In this work all state variables are defined in terms of molar quantities, and a

transformation into mass quantities is given.

Due to the number of unknown variables closure relationships are necessary

to close the system of equations. They form the static part of the model. Within

the static system of equations two of the unknowns, namely the phase densities,

can be expressed explicitly from the other ones such that it can be reduced to a

system of three equations. This system is rewritten into a system of equations

containing one classical equation and two conditional ones by using the approach

of complementarity constraints. This formulation ensures that the saturations

always remain physical. Finally, in order to be able to implement the method

into the toolbox M++, it is rewritten in a general formulation. This becomes ad-

vantageous when modifications of the model are assumed as done in the numeric

105

106 SUMMARY

part of this thesis. By re-defining a single function one gets an implementation

of an alternative model.

Starting off with that generalised formulation two possible choices for time

discretisation can be applied to obtain the semi-discrete variational formulation.

Raviart-Thomas elements of lowest order are assumed for the spatial discreti-

sation. The application of adequate test functions from corresponding function

spaces leads to the full discrete formulation. By applying a hybridisation tech-

nique new unknowns, the Lagrange multipliers, are introduced and static conden-

sation finally leads to a global system of equations only depending on those new

unknowns. This system of equations is solved by a Newton’s method. Within

each iteration all principal variables resp. state variables are determined by local

Newton’s methods resp. semi-smooth Newton’s methods for the system of static

equations.

This is implemented in M++ and applied to numerical challenging scenarios.

The experiments are either based on tasks within an international benchmark

or on examples given in publications that also deal with multi-component multi-

phase flows including phase transitions. All examples are chosen such that im-

portant features of the method can be seen, e.g. phase transitions and phase

pressure behaviours. It turned out that the method is able to deal with both

phase transitions and that, due to the complementarity formulation, the model

remains physical at the end of each time step. All presented choices for the

model formulations and treatments of the diffusion coefficients are compared and

it turned out that for the suggested examples the differences are always within

the accuracy of the non-linear and linear solvers.

Apart from the benchmark examples one experiment is set up to determine

the rate of convergence in space and time. The development of the error of

solutions, that are based either on an increased refinement level of the mesh

or on a decreased time step size, is shown by means of a reference solution. It

became obvious that the optimal order is not reached, which was not unexpected.

Nevertheless it is shown that convergence of order αN > 1 is obtained for the

given problem.

Dt. Titel und Zusammenfassung

Eine Verallgemeinerte Voll-Gekoppelte Approximation mit

Gemischten Hybriden Finiten Elementen fur Mehr-

komponenten-Zweiphasenstromungen in Porosen Medien

Das Hauptziel der vorliegenden Arbeit ist es, eine verallgemeinerte Approxi-

mation fur Zweiphasenstromungen, bestehend aus mehreren Komponenten, in

porosen Medien mithilfe gemischter hybrider finiter Elemente herzuleiten. Dabei

wird angenommen, dass beide Phasen mischbar sind und großtenteils je zwei

Komponenten beinhalten. Dabei soll das verwendete Modell das Entstehen und

Verschwinden von Phasen, den Austausch von Komponenten zwischen den zwei

Phasen sowie kapillare Effekte beinhalten. Grundsatzlich ist die Modellierung

von Mehrphasenstromungen eine große Herausforderung. Die zugehorigen Mod-

ellgleichungen sind sehr stark miteinander gekoppelt und hochgradig nichtlinear,

was zu numerischen Schwierigkeiten fuhrt. Dies wird durch die Einbeziehung von

Phasen- ubergangen noch komplexer da im Falle von verschwindenden Phasen

einige der Gleichungen degenerieren.

Die Gleichungen fur die Massenerhaltung bilden ein Gleichungssystem beste-

hend aus 2 Gleichungen, entsprechend der Anzahl der Phasen. Die Wahl der

Hauptunbekannten in verschiedenen, bereits existierenden, Modellansatzen wird

erortert. Da in dem hier betrachteten Ansatz auch die flussige Phase verschwinden

konnen soll, entfallt eine Standardwahl fur eine der Hauptunbekannten, namlich

der Druck der flussigen Phase. Stattdessen wird ein Mitteldruck definiert, der

auch im Falle verschwindender Phasen wohldefiniert bleibt. Daruber hinaus

gibt es eine Reihe von sekundaren Unbekannten, die zusatzlich bestimmt werden

mussen. Grundsatzlich werden hier molare Großen beschrieben, ein Umrechnung

zu Massen ist erlautert.

Aufgrund der Anzahl von Unbekannten werden weitere Gleichungen benotigt.

Diese zusatzlichen Gleichungen bilden als Gleichungssystem den statische Teil des

Modells. Von den sekundaren Unbekannten konnen zwei explizit in Abhangigkeit

107

108 DEUTSCHER TITEL UND ZUSAMMENFASSUNG

der anderen berechnet werden, wodurch sich die Große des Gleichungssystems auf

3 verringert. Dieses System wird unter Verwendung von Komplementaritats-

bedingungen umgeschrieben. Damit wird unter anderem auch sichergestellt,

dass die Sattigungen physikalisch sinnvoll bleiben. In einer verallgemeinerten

Formulierung wurde das Modell dann in M++ implementiert. Diese allgemeine

Form ist sinnvoll, da mit sehr wenig Aufwand Modellvariationen erzeugt werden

konnen, die auch im Rahmen dieser Arbeit untersucht werden sollen.

Als Zeitdiskretisierung werden zwei verschiedene Euler-Verfahren vorgestellt,

durch deren Anwendung die semi-diskreten Formulierung resultiert. Fur die

raumliche Diskretisierung werden Raviart-Thomas-Elemente niedrigster Ordnung

verwendet. Durch die Wahl geeigneter Testfunktionen aus entsprechenden Funk-

tionenraumen wird die voll-diskrete Formulierung hergeleitet. Durch Hybridi-

sierung werden neue Unbekannte, die sogenannten Lagrange Multiplikatoren,

eingefuhrt. Statische Kondensation fuhrt letztendlich auf das zu losende globale

Gleichungssystem, welches nur noch von den Lagrange Multiplikatoren abhangt.

Innerhalb jeder globalen Newton-Iteration werden lokale Probleme gelost, bei

denen die Hauptunbekannten sowie die sekundaren Unbekannten berechnet wer-

den. Dies erfolgt ebenfalls uber Newton-Verfahren (lokale Newton-Iteration und

“semismooth” Newton-Verfahren).

Das vorgestellte Verfahren ist in M++ implementiert worden und wurde auf nu-

merisch herausfordernde Experimente angewandt. Diese Experimente sind zum

einen durch einen internationalen Benchmark vorgegeben, zum anderen stammen

die Setups aus Publikationen, die sich ebenfalls mit dieser Thematik befassen.

Grundsatzlich wurden die Experimente so ausgewahlt, dass die entscheidenden

Effekte beobachtet werden konnten, wie beispielsweise die Phasenubergange und

das Verhalten einzelner Phasendrucke. Es ist ersichtlich, dass das vorgestellte

Verfahren in der Lage ist, beide Phasenubergange zu realisieren. Des weiteren

ist aus dem Vergleich der verschiedenen Modellwahl beziehungsweise Behand-

lung der Diffusionskoeffizienten ersichtlich, dass sich die Unterschiede zwischen

den Losungen immer innerhalb der vorgegebenen Genauigkeit der verwendeten

Losungsalgorithmen befinden.

Zusatzlich wurde anhand eines numerischen Experiments die Konvergenzord-

nung des Verfahrens untersucht. Dazu wurden Simulationen mit unterschiedlicher

Gitterweite beziehungsweise Zeitschrittweiten mit einer berechneten Referenz-

losung verglichen und der globale L2-Fehler berechnet. Wie zu erwarten war, kon-

nte die theoretisch mogliche Konvergenzordnung nicht erreicht werden. Dennoch

zeigte sich, dass das Verfahren fur das gestellte Problem mit Ordnung αN > 1

sowohl im Raum als auch in der Zeit konvergiert.

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a generalised fully coupled hybrid-mixed finite element ... · and fabian brunner who supported me...

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