geir terje eigestad - semantic scholarpreface this thesis constitutes the work that i have done...

Reservoir simulation with imposed fluxcontinuity conditions on heterogeneous and

anisotropic media for general geometries, andthe inclusion of hysteresis in forward modeling

Dr.Scient Thesis, Reservoir Mechanics

Geir Terje Eigestad

Department of Mathematics

University of Bergen

24th April 2003

For my sister Monica

Preface

This thesis constitutes the work that I have done during the time as a Dr. Sci-ent. student at the Department of Mathematics, University of Bergen (UiB). It iswritten in accordance with the criteria stated by the University Board. The workhas been done in collaboration with the research group at Norsk Hydro and theDepartment of Mathematics at University of Bergen.

Magne Espedal (UiB) and Ivar Aavatsmark (UiB/Norsk Hydro ResearchCentre) have been advisors for me, and Norsk Hydro has provided the fundingfor the work.

The thesis’ main research part is the five papers A-E, and fourof these havebeen published in conference proceedings and journals. Themain focus of thesepapers is on issues related to reservoir simulation with improved reliability, and inparticular discretization methods which should be valid for complex geometry.

The governing equations for fluid flow in oil and gas reservoirs can beclassified according to conventional theory of applied mathematics. This doesnot mean that all issues regarding reservoir simulation have been solved. Reliablereservoir simulation is a complicated field, and many issuesare still not known to asatisfactory level. The oil and gas reservoirs are very complex physical media, andthe level of heterogeneity seen in real sandstones can sometimes overwhelm thenovice mathematician. Sandstones may have rapidly varyingstructure, porosityand permeability on a small scale. It can sometimes be hard toconvince oneselfthat it is possible to model enormous amounts of fluids on a large scale when beingfaced with such complexity.

Regardless of these complexities, applied reservoir simulation does studymovement of fluids in complex media. The goal for scientists is to include as muchinformation as possible in the models for both the fluid flow aswell as the physicalmedia themselves. At multi discipline research centres, geologists, geophysicists,physicists and mathematicians work together with this drive. Each group maywork with their own ’small’ problem, but all these pieces should eventually bewoven together. For the most complex reservoirs one might never be able toperform accurate and detailed predictions of the future performance. But beingable to see ’the main trends’ is very useful, and this is really the goal even the

scientist should aim for.The contributions in this thesis are mainly in the mathematical and numer-

ical modelling of fluid flow in oil and gas reservoirs. For a person not directlyinvolved, even the field of numerical modelling may seem likea very narrowissue to focus on, and merely a detail in the big picture of oilexploration andproduction. We, however argue that this is not the case. For future predictionof the performance of an oil reservoir, one has to build a mathematical model.This model should be reliable for a wide range of input, such as rock types, welldata, fluid composition etc. Although there will always be uncertainty involvedwith such parameters, the bottom line for numerical modelling should be that themodels give us reliable output for cases that can be verified.

For most of the work in this thesis, the physical parameters that describe thereservoir are assumed to be given. The issue of reliable parameter determinationis a field of it’s own, and when these are known, the techniquesof numericalmodelling/discretisation might be applied to more complexinput.

Part I of the thesis gives an introduction to reservoir simulation and theissues that I have been working with. Part II is a collection of research papers thathave been written during the PhD period.

Outline of Thesis

The thesis is divided into two main parts. Part I gives an overview and summaryof the theory that lies behind the flow equations and the discretisation principlesused in the work. Part II is a collection of research papers that have been writtenby the candidate (in collaboration with others).

The main objective of this thesis is the discretisation of anelliptic PDE whichdescribes the pressure in a porous medium. The porous mediumwill in generalbe described by permeability tensors which are heterogeneous and anisotropic. Inaddition, the geometry is often complex for practical applications. This requiresdiscretization approaches that are suited for the problemsin mind. The discret-isation approaches used here are based on imposed flux and potential continuity,and will be discussed in detail in Chapter 3 of Part I. These methods are calledMulti Point Flux Approximation Methods, and the acronym MPFA will be usedfor them. Issues related to these methods will be the main issue of this thesis.

The rest of this thesis is organised as follows:

Part I : Chapter 1 gives a brief overview of the physics and mathematicsbehind reservoir simulation. The standard mass balance equations are presented,and we try to explain what reservoir simulation is. Some standard discretizationsmethods are briefly discussed in Chapter 2.

The main focus in Part I is on the MPFA discretisation approach for variousgeometries, and is given in Chapter 3. Some details may have been left out in thepapers of Part II, and the section serves both as a summary of the discretisationmethod(s), as well as a more detailed description than what is found in the papers.

In Chapter 4, extensions to handle time dependent and nonlinear problems arediscussed. Some of the numerical examples presented in PartII deal with twophase flow, and are based on the extension given in this chapter.

Chapter 5 discusses numerical results that have been obtained for the MPFAmethods for elliptic problems, and Chapter 6 deals with issues related to propertiesof the discrete set of (one phase) pressure equations.

Chapter 7 contains summaries of the research papers found inPart II.

Part II : This part contains 5 research papers. The papers mainly deal withMPFA methods, and issues related to the discrete set of equations that areobtained by these discretisation methods. Also, one of the papers (Paper E) dealswith the inclusion of hysteresis for forward simulation of two phase flow. Thetheme in Paper E is therefore also related to discretisationissues.

The papers included in Part II are:

Paper A: Symmetry and M-matrix Issues for the MPFA O-method onan Unstructured Grid . Published inComputational Geosciences, Volume 6,Editors M. G. Edwards, R. D. Lazarov, I. Yotov.

Paper B:MPFA applied to Irregular grids and Faults . Published inDe-velopments in Water SciencesVolume 47, Editors: Hassanizadeh, Schotting,Gray, Pinder.

Paper C:MPFA for Faults with Crossing Layers and Zig-zag Patterns.In Proceedings ofECMOR VIII, 2002.

Paper D:A note on convergence of the MPFA O-method; numerical ex-periments on some 2D and 3D grids. Draft manuscript.

Paper E:Numerical Modeling of Capillary Transition Zones. SPE 64374, inProceedings of the SPE APOGCE, 2000.

Acknowledgements

This work would never have been completed without the help ofmy advisorsMagne Espedal and Ivar Aavatsmark. I owe them a big thank for their guidanceand encouragement throughout this time. I would especiallythank Magne for hispositiveness and ability to make me perform more even in the hardest situations.Ivar needs a big thank for sharing his deep insight in many issues in the fieldof applied mathematics. They also both deserve my gratitudefor their ability tohandle my moods during times of intense stress.

During the time as a PhD student many people have been helpfuland goodto talk to. At Norsk Hydro Research Centre, I would like to thank the followingpeople: Edel Reiso, Hilde Reme, Rune Teigland and Tor Barkve. Formerly em-ployed Johne Alex Larsen has also been good support. The management of NorskHydro needs a big thank for funding the work.

At CSIRO Division of Petroleum Resources, Dr. Chris Dyt and Dr. CedricGriffiths deserve my thanks for letting me stay there for 6 months in 2000/2001.I would also thank Professor Thomas Russell for letting me stay at University ofColorado at Denver for three weeks during spring 2002.

At the Mathematical Institute I have had very good colleagues. I want to thankErlend Øian for many interesting discussions regarding reservoir simulation andimplementation. Jan Martin Nordbotten is also a good sourceof inspiration. HansFredrik Nordhaug has been of tremendous help in all Latex issues, and all generalcomputer questions. His attitude to questions is admirableeven when the answeris trivial. I would also like to thank Torbjørn Aadland and Jarle Haukas for lettingme use parts of their C++ code.

Helge Dahle is a very nice person to have at the department; both academicallyand personally. I want to thank him for good discussions and taking me and otherPhD students hunting and hiking near his cabin.

Runhild Klausen has been a good friend and colleague. I wouldlike to thankher for a good cooperation in the academic field and her friendliness at all times.

My family has been fantastic to me all my life, including my period as PhDstudent. They have always supported me, and it has been good to retire to themwhen the times have been hard.

My loving wife Melanie is the one who made this possible. Without her Iwould never even have started doing a PhD. I owe her all my love.

Geir Terje EigestadBergen, April 2003.

Contents

I Introduction 1

1 Basics 31.1 Darcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Multiphase flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Capillary pressure; inclusion of hysteresis . . . . . . . . .. . . . 71.4 Geology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Discretisation methods; various formulations 112.1 Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Finite element methods . . . . . . . . . . . . . . . . . . . . . . . 122.3 Mixed finite element methods . . . . . . . . . . . . . . . . . . . 14

3 Multi Point Flux Approximation methods 173.1 Transmissibility calculations . . . . . . . . . . . . . . . . . . . .173.2 Unstructured grids; 2D polygonal CVs . . . . . . . . . . . . . . . 253.3 Faults with crossing grid lines . . . . . . . . . . . . . . . . . . . 283.4 Treatment of Dirichlet boundary conditions . . . . . . . . . .. . 34

4 Time dependent problems 394.1 Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Numerical results 415.1 Analytical solutions . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Properties of discrete system 456.1 M-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.2 Notes on violation of maximum principle . . . . . . . . . . . . . 46

7 Summary of the papers 497.1 Summary of Papers A and B . . . . . . . . . . . . . . . . . . . . 497.2 Summary of paper C . . . . . . . . . . . . . . . . . . . . . . . . 52

0

7.3 Summary of paper D . . . . . . . . . . . . . . . . . . . . . . . . 547.4 Summary of paper E . . . . . . . . . . . . . . . . . . . . . . . . 55

8 Further work 57

Bibliography 59

II Published and submitted work 65

A Symmetry and M-matrix issues for the O-method on an UnstructuredGrid 67

B MPFA applied to Irregular Grids and Faults 69

C MPFA for Faults with Crossing Layers and Zig-zag Patterns 71

D A note on convergence of the MPFA O-method; numerical experi-ments on some 2D and 3D grids 73

E Numerical Modeling of Capillary Transition Zones 75

Part I

Introduction

Chapter 1

Basics

Reservoir simulation is the study of how fluids flow or behave in a reservoir. Areservoir is a porous medium where hydrocarbons exist in thepore space, and inthe oil industry the goal is to determine how hydrocarbons and water flow/behavein a reservoir under different conditions.

To do so, one has to derive mathematical and physical models for the pro-cesses that occur in the reservoir. Often the physical processes themselves areof interest, and many physical phenomena (such as ’interface-interaction’ whendifferent phases are present, and complexities associatedwith hysteresis) are notfully understood. This thesis approaches reservoir simulation from a mathemat-ical point of view, and some physical assumptions must be made to complete themathematical approach. The theory that leads to the partialdifferential equations(PDEs) that describe the fluid flow in a porous medium, is standard in many text-books in the area of reservoir simulation, and further theory details may be foundin for example Aziz [39].

For simulation of fluid flow in a hydrocarbon reservoir (eg. toquantify howmuch of a specific fluid flows through a specific area of a reservoir due to pressuredifferences), one has to solve a set of coupled PDEs. This setof equations is oftenthe starting point for applied mathematics, and discretisation of the equations isa field that has received a lot of attention in the past decades. The PDEs arisefrom the principle of mass conservation and a physical description of the velo-city/flux dependency of the pressure (Darcy’s law). Commonly studied are themass conservation equations for each of the fluid phases present in a reservoir. Init’s original form, the mass conservation equation for a fluid with mass densitym(mass per volume) and fluid densityρ on a domainΩ is given by

∫

Ω

∂

∂tmdV +

∫

∂Ω

ρv ·ndS =

∫

Ω

qdV . (1.1)

Herev is the volumetric flow density of the fluid,n is the outward normal

4 Basics

vector of the surface∂Ω, andq is a source term. In other words; Eq. (1.1) saysthat the accumulation of mass of the fluid within a volumeΩ is balanced by thefluxes (across the edges of the volume) and possible sourcesq. Eq. (1.1) is anintegral equation. The integral form of the mass balance equation(s) will be usedthroughout this thesis. A discretisation approach based onthe integral form iscalled a Control Volume (CV) formulation. We will use this form because of theadvantage that local mass conservation is satisfied.

When applied to a porous medium, we need to define the porosityof the me-dium/rock. This is the fraction of the volume of the rock where mobile fluid mayexist, and will be denotedφ. Hence, we may formally define it by

φ =Vp

Vc, (1.2)

whereVp is the effective pore volume available for the fluid, andVc is the totalvolume of the rock. The total pore volume of a rock may be larger thanVp, butthis extra volume is not ’connected’ to the pore space where fluids may flow. Itshould be mentioned that a typical sand stone found in reservoirs from the NorthSea has a porosity of 10-20 percent.

The mass density for a representative volume is henceρφ. Applying thistogether with the divergence theorem, we obtain the integral form of the massconservation equation for a single phase fluid within a volume ΩV of a porousmedium

∫

ΩV

(∂(ρφ)∂t

+∇·(ρv)− q) dV = 0. (1.3)

This equation is referred to as the single phase flow equation, or the singlephase pressure equation. The reason for it’s name is that with some further as-sumptions discussed below (Darcy’s law + density function of pressure), this is aintegral PDE where only the pressure needs to be treated as anunknown.

1.1 Darcy’s law

The French engineer Henry Darcy experimented with flow through different typesof sand during the 19th century. In a range of experiments with flow verticallythrough samples of sand, he concluded that the flow through the sands was pro-portional to the pressure difference between the top and bottom pressure. This lawis now known as Darcy’s law. In it’s most primitive form (1D, single phase, grav-ity neglected), the relation says that the volumetric flow density (Darcy velocity)v is proportional to the gradient of the pressurep:

1.2 Multiphase flow 5

v = −k∂p

∂x. (1.4)

The proportionality constantk is called the permeability or conductivity ofthe medium. For flow in higher dimensions, the permeability will be a (spatiallyvarying) tensorK, and when gravity is included, Darcy’s law for single phase flowreads

v = −K (∇p+ρg∇d). (1.5)

This may now be inserted in Eq. (1.3), and pressures will hence enter theequation. If it is assumed that the fluid at all times occupy the pore space, weobtain a parabolic PDE which states how the pressure varies in space and time:

∫

ΩV

(∂(ρφ)∂t

−∇·(ρK (∇p+ρg∇d))− q)dV = 0. (1.6)

If the fluid further is assumed to be incompressible, the firstterm will vanish.When no sources or sinks are present, one hence obtains a purely elliptic pressureequation (time independent)

−

∫

∂Ω

∇·(K∇u) = 0. (1.7)

where we have defined the potentialu of the fluid through the relation

∇u = ∇p+ρg∇d. (1.8)

When suitable boundary conditions are posed, Equation (1.7) has a unique solu-tion.

Whereas reservoir simulation rarely is as simple as to solvetime independentproblems, many discretisation methods that are applied fortime dependent prob-lems, are based on the purely elliptic equation. Time discretisation is briefly dis-cussed in Chapter 4, and is often included by a standard backward Euler schemefor time dependent terms.

Most of the work in this thesis (papers A-D) uses Eq. (1.7) as the basis for thediscretisations to be dealt with.

1.2 Multiphase flow

When several phases or components are present in a porous medium, mass con-servation must be posed for each of the phases or components.In reservoir simu-lation, there are usually 3 phases that may exist; oil, gas and water. In the Black

6 Basics

Oil formulation, hydrocarbons will be grouped, and either be classified as gas oroil. Heavy components will be grouped as oil, and lighter components will begrouped as gas components. Depending on temperature, density and pressure,lighter hydrocarbons may exist in both oil and gas phase. This means that the gascomponents may exist in either oil or gas phase. The water component will onlyexist in the water phase.

For the oil and water component respectively, the mass conservation equationis given by.

∫

ΩV

(∂(ρiSiφ)

∂t−∇· (

kri

µiρiK∇ui))dV =

∫

ΩV

qi, i = o,w. (1.9)

For the gas component, the mass conservation equation is slightly differentbecause the component may exist both in oil and gas phase, andwe refer to for in-stance [40] for details. Eq. (1.9) is a parabolic type of equation, which contains anelliptic term (second term on left hand side). Although eachmass balance equa-tion is parabolic, the total system of mass balance equations has a more hyperboliccharacter. This is for instance discussed in [52]. The new quantitiesSi,kri andµiof Eq. (1.9) will be explained in the following: If the three components/phasesoil, gas and water exist in our system, three mass conservation equations must besolved in the region we investigate. The efficient pore spacewill be assumed tobe filled with fluid at all times. Depending on the processes, the different phaseswill compete to occupy this space. It is then natural to definephase saturations.By this we mean the ratio of the available pore space the different phases occupy.Formally, the saturation of phasei in a representative volumeV is defined as

Si =Vi

V, (1.10)

whereVi is the volume occupied by the specific phase. Hence, the saturations varybetween 0 and 1. Since the phases compete to occupy pore space, the effectivepermeabilities will be reduced, and this is reflected in the occurrence of the entitieskri, of Eq. (1.9). These entities are calledrelative permeabilities, kri, and will bereduction factors of the permeabilityK. The relative permeabilities are functionswhich depend on the saturation of phasei, ie. kri = kri(Si). They take valuesbetween 0 and 1.

For hysteresis models, the relative permeabilities also depend on the directionof the flow processes, as discussed in the next subsection.

The entityµi is called the viscosity of the phasei. This is a weakly nonlinearterm, and for many applications the viscosities of pure fluids will be treated asconstants. For gases, simple models use the linear relationship

µg = bp. (1.11)

1.3 Capillary pressure; inclusion of hysteresis 7

The quantityb will be a gas dependent constant. For a three phase system thethreesaturations will be unknowns of the three equations of the form (1.9). In additionto this, each of the three phase pressures will be unknowns. As densities,ρi, arealso assumed to depend on the pressures, the total number of unknowns in thethree coupled mass conservation equations is 6. To close thesystem of equations,three closure relations are needed.

One closure relation is intuitive if the pore space is filled with fluids at alltimes:

∑

Si = 1,i = w,o,g, (1.12)

where the different subscripts denote water, oil and gas respectively. To get thetwo remaining closure conditions, one needs to know more about the physics ofmultiphase interactions. Moreover, to make the system of mass balance equationssolvable, the physics must be quantified.

1.3 Capillary pressure; inclusion of hysteresis

In black oil models, it is commonly assumed that the phase pressures are re-lated through a quantity called the capillary pressure. When only two phasesare present, the capillary pressure is given by the pressuredifference between thenon-wetting fluid and the wetting fluid:

Pc = pnw −pw. (1.13)

For two phase problems in the oil industry this may be oil and water, or gasand water. If all three phases (water, oil and gas) exist, twocapillary pressures willbe used; the capillary pressure between oil and water and thecapillary pressurebetween gas and oil:

Pcow = po−pw, Pcgo = pg−po. (1.14)

Experimental results show thatPcow is (mainly) a function of the water satura-tion (Sw), and thatPcgo is (mainly) a function of the gas saturation (Sg). The twocapillary pressure relationships then close the system of equations obtained whenmodelling three phase flow. Since the capillary pressure andthe relative permeab-ilities of Eq. (1.9) depend on the solutionsSi, the set of mass balance equationsare nonlinear (with respect to solution variables).

Much emphasis is made in experimental reservoir physics to describe the ca-pillary pressure behaviour. In addition to being dependenton the water saturation,the capillary pressure description also depends on whethera phase is entering or

8 Basics

Figure 1.1: Primary drainage capillary pressure curve withimbibition capillarypressure curves starting out from different points on the primary drainage curve.

exiting the pore space. In physics this is referred to as irreversible processes orhysteresis.

When water (wetting phase) enters the pore space, this is called imbibition,and when oil (non-wetting phase) enters the pore space this is called drainage.

In Paper E in Part II of this thesis, a consistent capillary pressure and relativepermeability hysteresis model is included for forward modelling of two phase flowin a reservoir. The idea is that the fluid flow simulator shoulddynamically updatecapillary pressure and relative permeabilities accordingto what direction the flowhas. For reasons explained in the paper, the reservoir will be initialised with a(single)primary drainagecapillary pressure curve prior to production from thereservoir. Figure 1.1 illustrates possible imbibition curves that may start out fromthe primary drainage curve.

If pure imbibition takes place in a reservoir, the capillarypressure time evolu-tion will be described by imbibition curves. When gravity isincluded in a model,and the system is assumed to be in vertical equilibrium, the capillary pressurewill vary linearly with height, and different grid cells at different height start outwith different saturations. This means that even for pure imbibition throughoutthe reservoir, different capillary pressure descriptionsapply for different parts ofthe reservoir. Such cases have been investigated in Paper E.The model allows fordifferent capillary pressure curves for all grid cells. In addition to (primary andsecondary) imbibition curves, arbitrary secondary drainage curves may apply forthe different cells throughout the simulation. Examples ofsuch curves are visual-ised in Fig. 1.2a. Here we depict examples of capillary pressure scanning curvesfor cells that have their origin at some point of the primary drainage curve. InFig. 1.2b relative permeability scanning curves are visualised, and are explainedas follows: A cell may have an initial water saturation corresponding to the satur-ation at point (1). If imbibition starts for this cell, the relative permeability curve

1.4 Geology 9

a. Capillary pressure. b. Relative permeability

1

2

Figure 1.2: a: Capillary pressure scanning curves. Scanning curves are closedloops, and arbitrary reversals may occur within the loops. b: Relative permeabilityscanning curves.

will follow the lowermost path to point (2). If the directionof the flow is reversedat point (2), the relative permeability curve will follow a different path back to(1). Notice that the loop between (1) and (2) is closed; this is an experimentalobservation, and is included in the scanning curve generation.

Similar explanations may be given for the curves of Fig. 1.2a, and details maybe found in Paper E.

1.4 Geology

Reservoir simulation has become very advanced over the pastdecades. This is dueto at least two main reasons. The first reason is of course the computer revolu-tion, which allows large models to be implemented for computing. The secondreason is that the geology of a North Sea reservoir may often be known quite ac-curately (compared to other porous media). Simulation grids may then be verylarge, and the level of details can be very high. Figure 1.3 shows the simulationgrid for a realistic field example. One of the reasons for the high level of de-tails is the considerable effort which is put into seismic measuring prior to drillingexploration wells. Also, the history may be well known for a reservoir that hasbeen producing hydrocarbons for many years (5-15 years) through logging andmeasured production etc. This again may be used to verify or recalculate geo-physical data. Parameter estimation is an important area init’s own, and produc-tion data is essential to recalculate/calibrate a model with respect to porosity andpermeability. Because the geology may be known at such a detailed level, onemay be required to model flow on grids that include faults and general complex

10 Basics

Figure 1.3: Simulation grid for realistic North Sea field example

Figure 1.4: Extracted part of sandstone found in the Oslo Fault.

geometry. Complex geometry has been a key issue in the work ofthis thesis. Pa-pers B and C discuss MPFA discretisation for simulation grids where faults havebeen included. Faults may be observed and studied in ’analogues’ to North Seareservoirs. One examples of such an analogue may be found in the Oslo Fault.Such analogues may be useful for understanding the flow mechanisms in faultsand fractures. Also, they support the need to model fluid flow on simulation gridswhich contain faults, have large aspect ratios and grid cells that may degenerateto triangles/tetrahedra.

A part of some sandstone found in the Oslo Fault is visualisedin Fig. 1.4,where the scale is ’a few meters’. Even on this small scale thepermeability mayvary extensively.

Chapter 2

Discretisation methods; variousformulations

Discretisation of the governing equations for fluid flow has been the issue of manyresearchers involved in porous media flow. Finite difference methods (FD) havebeen used extensively in commercial reservoir simulation,whereas finite elementmethods (FEM) have a strong support in applied mathematics.

We will here briefly describe the following methods: (Control volume) finitedifferences, finite element- and mixed finite element methods.

The starting point for the discretisation methods to be discussed is the purelyelliptic equation, which is obtained when a possible sourceq has been added inEq. (1.7):

−∇· (K∇u) = q. (2.1)

Suitable boundary conditions will also apply, and for reservoir simulation thesewill usually be no-flow at boundaries, ie.K∇u · n at the boundary∂Ω of somedomainΩ.

2.1 Finite Differences

Finite difference based discretisation (FD) is used extensively in conventionalreservoir simulation. The basic idea is to approximate derivatives by discrete dif-ferences. In 1D the use of finite difference approximations for the elliptic equation(2.1) is justified since the permeabilityK = k then is a scalar entity.

For 1D, the second order problemu′′ = f may be discretised by applying theforward discretisation for the first derivativeu′(x) = (u(x+ h) − u(x))/h, whichleads to a similar treatment ofu′′(x). For multi dimensional problems with non-orthogonal grids and general tensor permeability description, such differences are

12 Discretisation methods; various formulations

not directly applicable to Eq. (2.1), as the differentiation involves the permeabilityas well as the pressure. The 1D version of Eq. (2.1) is

∂

∂x(k(x)

∂u(x)∂x

) = 0, (2.2)

so that differentiation of the permeability must also be ’included’. DenotingF =

k ∂u∂x

, the discrete version of (2.2) takes the form (F (x+ h) −F (x))/h. Withoutgoing into detail, there are possible grids to use for finite difference discretisations:point- or cell distributed grids. Cell centred grids (non-uniform) are preferableto use because they yield local mass conservation, and have consistent fluxes.This is enforced through harmonic averages of permeabilities obtained at mediadiscontinuity points. It should be noted that the FD discretisation on cell centredgrid in 1D is second order convergent [34], even though the discretisation schemeis inconsistent.

For 2D orthogonal grids where the principal axes of the permeability tensorsare aligned with the coordinate axes, the 1D principles may be utilised to ob-tain discretisation schemes on cell centred grids. Such grids will be termedK-orthogonal grids. Convergence on cell centred grids is proven in [34] and [26].

For general non-orthogonal grids in 2D, and generally varying permeabilitytensors, direct finite difference discretisation does not lead to convergent schemes.The numerical solutions convergence to the wrong solution,and this may in somecases be interpreted as convergence to solutions for continuous problems wherethe permeability tensors are not symmetric. The MPFA discretisation methodsto be discussed in Chapter 3 will be able to handle full tensorrepresentation ofthe permeabilities and non-orthogonal grids. Our numerical results (eq. PaperD) indicate/show convergence for examples with skew grids and full permeabilitytensor description.

2.2 Finite element methods

We will here briefly discuss the variational formulation of the problem (2.1), andthe use of finite element methods to discretise the equation.

The equation−∇ · (K∇u) = q can be rewritten in order to obtain a weak for-mulation of the equation: take the original equation, multiply by a suitabletestfunctionof some function spaceV [47], and integrate over a domainΩ:

−

∫

Ω

(∇· (K∇u)vdσ =

∫

Ω

qvdσ. (2.3)

Eq. (2.3) may be rewritten by using the rules for partial integration:∫

Ω

K∇u ·∇vdσ−

∫

∂Ω

K∇u ·nvdS =

∫

Ω

qvdσ. (2.4)

2.2 Finite element methods 13

If the boundary conditions areK∇u ·n = 0 at∂Ω, the second term of the left handside vanishes, and the variational equation reads

∫

Ω

K∇u ·∇vdσ =

∫

Ω

qvdσ, ∀v ∈ V . (2.5)

This type of boundary condition is called anatural boundary condition, whereasit is possible to define boundary conditions for the functions v ∈ V by v = 0 on∂Ω. The latter boundary condition is called anessentialboundary condition. It iscommon to use the notation (·,·) for inner products in some inner product space,see for example [47]. Letting (·,·) here denote the inner product in the innerproduct spaceL2(Ω), we may rewrite Eq. (2.5) as

(K∇u,∇v) = (q,v), ∀v ∈H1(Ω), (2.6)

The solution of Eq. (2.3) will be denoted aweak solutionof the original equation(2.1). One hence wants to solve this equation foru. The equivalence of the weaksolution and the original solution for smoothK is proven in various textbookson finite element analysis, see for example [47] and [50]. Theequivalence ofthe weak solution and the solution of the integral equation−

∫

∇ · (K∇u) =∫

q

is also valid for discontinuousK. Although not discussing discontinuousK, theequivalence of the solutions is discussed in [48] for solutions with regularity lessthanH2.

The term (K∇u,∇v) of Eq. (2.6) is abilinear form, see [47]. SinceK is sym-metric, this bilinear form will besymmetric. The bilinear form will bebounded(orequivalentlycontinuous) provided thatK is bounded (K ∈ L∞(Ω)). Further, thebilinear form will becoercivebecause of the positive definiteness ofK. Existenceand uniqueness of a solutionu of Eq. (2.6) is then proven by the Lax-Milgramtheorem [47].

To solve a discrete problem of the form (2.6), one may choose afinite-dimensional subspaceVh ⊂ H1. The discrete problem then reads: finduh ∈ Vhsuch that

(K∇uh,∇v) = (q,v), ∀v ∈ Vh. (2.7)

This method is called the (Continuous) Galerkin Finite Element Method. Underthe same conditions as above (Vh closed subspace ofH1, (K∇uh,∇v) bounded,symmetric bilinear form that is coercive onVh), there exists a uniqueuh that solves(2.7). The Galerkin Finite Element Method is an example of a method where thediscrete solution is sought in a subspaceVh ⊂ V , whereasnonconforming methodsdo not seek a solution in a subspace ofV . Discontinuous Galerkin Methods [37]are also methods where jumps in the pressure are allowed.

14 Discretisation methods; various formulations

2.3 Mixed finite element methods

Fluxes are not calculated directly by the Galerkin Finite Element Method. A postprocessing needs to be done in order to find the fluxes from the discrete pressures,and it is known [52] that there is a loss of accuracy when fluxesare found fromapproximate pressures. For discontinuous coefficient problems, the fluxes foundfrom the Galerkin FEM are not continuous. This was one of the reasons that mixedfinite element methods (and other locally conservative methods) were developedfor simulation of fluid flow in porous media.

The starting idea behind the mixed finite element method is todo a splittingof the second order elliptic pressure equation into two equations, where the twoequations express the physics of the pressure,u, and the velocity,v, respectively.One then expresses Eq. (2.1) as a system of two first order equations

v = −K∇u, (2.8)

and

∇·v = q. (2.9)

The two equations (2.8) and (2.9) will both be used for weak formulations.To obtain a weak formulation for the two equations, one multiplies by suitable

test functions, and integrates over some domainΩ. For the first equation we obtain

∫

Ω

K−1v ·wdσ = −

∫

Ω

∇u ·wdσ = −

∫

Ω

∇·wudσ+

∫

∂Ω

w ·nuds

The boundary conditionu = 0 on∂Ω is chosen (natural boundary condition here),and the last term on the right hand side then cancels.

Using the same inner product notation as in Sec. 2.2, the weakformulation ofthe mixed system reads: Find (v,u) ∈H (div)×L2 such that

(K−1v,w)+ (u,∇·w) = 0, ∀w ∈H (div),

(∇· v,z) = (q,z), ∀z ∈ L2. (2.10)

The discrete problem reads: Find (vh,uh) ∈ V h×Ph ⊂H (div)×L2 such that

(K−1vh,w)+ (uh,∇·w) = 0, ∀w ∈ V h,

(∇·vh,z) = (q,z), ∀z ∈ Ph. (2.11)

2.3 Mixed finite element methods 15

The discrete spacePh is the space of piecewise constants (which is a subspaceof L), and the discrete spaceV h is in many applications the lowest order RaviartThomas space [27], [28]

The power of the mixed methods lies in the way they are able to handle mediadiscontinuities. The methods may be shown to be locally massconservative whenthe media have discontinuities. One of the drawbacks of the method is that thediscrete system of equations is a saddle point problem, which is more costly tosolve than symmetric positive definite systems. In addition, more unknowns needto be solved for.

The mixed finite element framework has also been extended to account for acontrol volume formulation by Russell [29], [30] and coworkers. The ExpandedMixed Finite Element Method was developed by Wheeler, Yotovand coworkers[35], [36]. The relationship between this method and the MPFA O-method wasrecently found by Klausen [32].

The next chapter deals with the MPFA discretisation methods, and are al-ternative discretisation methods to the methods discussedin this chapter.

Chapter 3

Multi Point Flux Approximationmethods

The main work in this thesis has been on various Multi Point Flux Approximationmethods, and discretisation issues related to these. It is important for a discretisa-tion method to handle complex geometry in order for it to be applied for realisticcases. In addition, the discretisation methods should be applicable in physicalspace if this is required. The MPFA methods have this advantage.

MPFA methods have also been the issue in work by Edwards and coworkers[17], [18] and [19].

We will here give a thorough description of the methods for the geometriesthat have been considered in the papers: 2D quadrilaterals,2D polygonal con-trol volumes, 3D quadrilaterals with crossing grid lines and handling of generalDirichlet boundary conditions.

3.1 Transmissibility calculations

The basic idea behind the MPFA methods is to define discrete fluxes for edgesof control volumes by a reasonable restriction of the medium. The ideas can becarried out for several types of geometries, and can be illustrated in 2D by thegeneral quadrilateral grid visualised in Fig. 3.1.

In practical reservoir simulation, the geometry may very well have similartrends as in this figure. Fluxes should therefore be calculated for each cell edge,where the orientation of the different edges may vary highlythroughout the grid.Furthermore, the permeability tensorK will be heterogeneous and anisotropic,and the principal directions are likely to vary from grid cell to grid cell. How-ever,K is assumed to be symmetric and positive definite. A ’good’ discretisationmethod should take this physical picture into account, and produce reliable fluxes

18 Multi Point Flux Approximation methods

Figure 3.1: Possible local gridding situation; grid cells need not be orthogonal.

a. Polygonals b. Triangles

Figure 3.2: Unstructured, conforming 2D grids.

for all cell edges that are encountered.The MPFA approach differs from methods such as FEM (continuous Galerkin

method) [47], and Finite Differences [39], [40] by the fact that the MPFA meth-ods discretise the flux itself, whereas the methods mentioned above discretise thepressure. But the MPFA methods also yield pressure equations to solve instead offlux equations. Mixed finite element methods consider both pressures and fluxesas unknowns, as discussed in Sec. 2.3.

We will first try to give a unified treatment of the ideas that lie behind theMPFA discretisation method for variousconforminggrids. By conforming gridswe mean grids where corners of grid cells meet other corners.The grid shown inFig. 3.1 is an example of such grids. Some other examples are shown in Fig. 3.2.

As mentioned above, the basic ideas behind the MPFA control-volume formu-lation is to determine fluxes across all edges of all grid cells in our grid. Hence,all grid cells are control volumes for which we pose mass conservation. By thisapproach local mass conservation will be satisfied. Further, we want to expressthe edge fluxes by some local formula/molecule. In doing so, we define acornerof degree nat corners wheren grid cells meet. This definition goes for both tri-angular, quadrilateral and polygonal control volumes, seeFig. 3.3. The MPFAapproach has also been discussed for non-conforming grids in [6], [7] and [12].

3.1 Transmissibility calculations 19

i

Figure 3.3: Left: Interaction region and associated cornerfor 2D quadrilateralcontrol volumes, degree of corner is 4. Right: Interaction region for polygonalcontrol volumes, degree of corner is 3.

Thedegreeof the corner is defined to be the number of grid cells that meetinthe corner. Around each corner aninteraction regionis defined, and the interactionregions make up a dual grid to the original control volume grid. The controlvolume grid and associated dual grid for a general conforming 2D quadrilateralgrid is visualised in Fig. 3.4

Figure 3.4: Control volume grid (fully drawn lines) and associated interactionregions (dotted lines).

Here, the interaction regions (indicated by dotted lines) may in general have8 edges, and a particular interaction region is visualised on the left hand side ofFig. 3.3. For some special polygonal control volumes the interaction region maybe a triangle as in the rhs. of Fig. 3.3. The reason why the interaction region has8 edges for the quadrilateral case is explained by the algorithm for constructingthe interaction regions in Fig. 3.3: From each node we draw (dotted) lines tomidpoints of cell edges, and an octagon arises around each corner. The midpointsof the edges are denoteddividing pointsor continuity points. The dividing pointsneed not be midpoints of the edges, but for the 2D discussion in this thesis, the


midpoints of edges are chosen as dividing points/continuity points.For the interaction depicted on the left hand side of Fig. 3.3, four grid cells

meet at the corner associated with the interaction region. There will be four celledges that meet in the corner, and parts of these cell edges will be defined to belongto the interaction region. These parts will be definedsub-interfacesor half-edges,and it is these we want to calculate fluxes for within the interaction region. Awhole cell edge for both quadrilaterals and general polygonals will be covered bytwo neighbouring interaction regions. The quadrilateral case is depicted in Fig.3.5.

E

III

Figure 3.5: Whole cell edge is covered by two neighbouring interaction regions.Total flux obtained by summing fluxes for two half edges. Interaction regions forquadrilateral grid cells depicted.

If the degree of the corner associated with an interaction region isn, we saythatn grid cells interact in the interaction region. The number ofinteracting cellsdetermine how theflux-moleculefor a sub-interface will be. For each sub-interfacethe flux will be a weighted sum of the potentials at the nodes ofthe cells thatinteract:

fi =

n∑

j=1

tijuj. (3.1)

These weights will be denotedtransmissibilities, and the discretisation prob-lem for the MPFA control volume method is to determine these.When the gridis quadrilateral,n = 4 in Eq. (3.1), and for the 2D polygonals studied in Paper A,n = 3. We may express then sub-interface fluxes of an interaction region by thegeneral matrix notation

f = T n×nu, (3.2)

wheref andu aren×1 vectors, andT n×n is ann×n matrix. The flux across thewhole cell edgeE of Fig. 3.5 is found by summing two half edge fluxes of two


neighbouring interaction regions. In total this yields a flux that is a weighted sumof the 6 (node) potential values of the cells that touch upon the whole cell edge.Similarly, the flux for a whole cell edge of the polygonal control volumes (withtriangular interaction regions) discussed in Paper A, is given by a weighted sumof 4 potential values.

Looking at the continuous case, fluxes across cell edges are given by

f = −

∫

S

K∇u ·n dS. (3.3)

Assume that the flux across sub-interfacei of the left interaction region of Fig3.3 is to be considered. If the flux is to be approximated by using the informationto the left of the sub-interface, the gradient of the potential for this cell is needed.By the assumption that the potential is linear for the part ofthe grid cell containedin the interaction region,∇u is hence constant, and the (discrete) flux across thesub-interface will also be constant (see Eq. (3.4) below).

Likewise, the flux may be approximated by the information to the right of thesub-interface, and the two discrete flux expressions shouldthen be equal. Thediscrete flux across the cell edge can hence be expressed as

fi = −nTi K i±∇ui±, (3.4)

where the indexi± denotes if the flux is approximated by the information of eitherthe cell in the positive or anti-positive direction of the edge (positive directiondefined by positive direction of normal vector). If, for instance, the potential wasassumed to be bilinear, fluxes would not be constant across cell edges.

Formally, the flux continuity condition for each sub-interfacei of an interac-tion region reads

fi,i− = fi,i+, (3.5)

The gradient of the potentials in each of the sub-cells contained in an interac-tion region will now be derived in order to use explicit expressions for Eq. (3.4).

Since the potentialU is linear on each sub-cellj of the control volume, theequation that describes a potential plane applies:

Uj(x) = ∇Uj(x−x0)+U0, (3.6)

wherex is the position vector,x0 is the node of grid cellj, andU0 is the potentialvalue at the node.

The potentials at the continuity points may be used as the basis for determ-ining the gradient of the potential in each sub-cell of an interaction region. Thisinformation may be inserted in Eq. (3.6) to obtain


x x

νν

0 12

1

−

x−

2

Figure 3.6: Variational triangle associated with part of quadrilateral controlvolume contained in interaction region;x0 is node of grid cell,x1 and x2 arecorresponding dividing points/continuity points

∇U · (xk −x0) = uk −u0, k = 1,2. (3.7)

This system may be written as

X∇U =

[

u1−u0

u2−u0

]

, (3.8)

whereX is the matrix

X =

[

(x1− x0)T

(x2− x0)T

]

. (3.9)

The inverse ofX must hence be found. Inward normal vectors of the vari-ational triangle are introduced:

ν1 =R(x2−x0),ν2 = −R(x1−x0), (3.10)

where

R =

[

0 1−1 0

]

. (3.11)

The vectors and points used here are illustrated in Fig. 3.6,where the gradientof the potential is to be determined for a sub-cell of a quadrilateral grid cell. Thematrix R has the property that for vectorsa and b, aRb is equal to the thirdcomponent of the cross product betweena andb. Moreover, the determinant ofX is

T = det(X) = (x1−x0)TR(x2−x0). (3.12)


which is equal to twice the area spanned by the pointsx0, x1 and x2 (in a righthanded system seen fromx0). To find the inverse, observe that by Eq. (3.10)matrix multiplication yields

[

(x1− x0)T

(x2− x0)T

]

·[

(ν1,ν2)]

=

[

(x1− x0)T ν1 (x1− x0)Tν2

(x2− x0)T ν1 (x2− x0)Tν2

]

.

=

[

2F 00 2F

]

. (3.13)

Hence,

X−1=

12F

[

ν1,ν2]

. (3.14)

And finally, the gradient of the potential reads

∇U =X−1[

u1−u0

u2−u0

]

. (3.15)

Explicit fluxes across sub-interfaces are now obtained by inserting the expres-sion for the gradients in (3.4).Having found an expression for the gradient of the potentialof each sub-cell ofan interaction region, it is seen that the discretisation principles may be appliedto a whole range of grids. In the case of conforming quadrilaterals in 2D, fourflux conditions of the form (3.5) are posed. In [8] scalar coefficientsωijk areintroduced, and they are defined by

ωijk =nTi Kjνjk

Tj, (3.16)

whereni is the positive normal vector of sub-interfacei, Kj is the permeability ofcell j, andνjk is local outward normal vectork of cell j. The entityTj is equalto twice the area of the triangle for which the two normal vectorsνjk,k = 1,2 arelegs of.

The flux continuity conditions (3.5) lead to a local system ofequations thatwill define the transmissibilities of Eq. (3.1). Each sub-cell uses two continuitypoints (in addition to the node) to determine∇u. Inserting the explicit expressionsfor the gradients yields discrete fluxes that depend on the potentials at continuitypoints, and are hence not on the form (3.1). However, the potentials at the con-tinuity points can be eliminated so that discrete fluxes can be given that are onlydependent on the potentials at the nodes of the control volumes.


This is illustrated by rewriting the system of flux continuity equations (3.5) inmatrix form, where the information about the potentials at the continuity pointsand the potentials at the nodes has been ’split’

Av = Bu. (3.17)

The vectorv = [u1, u2, u3, u4] then contains the potentials at all continuitypoints, andu = [u1,u2,u3,u4] contains the information of the potentials at thenodes. The 2 matricesA andB are 4×4, and their elements consist of the scalarcoefficients of (3.16). Details may be found in [8]. To complete the derivation,the left hand side of (3.5) is rewritten in terms of potentials at continuity pointsand nodes:

f = Cv−Du. (3.18)

The matricesC andD also have elements that consist of coefficients (3.16).If A is non-singular,v can now be eliminated from (3.17);v =A−1Bu. Insert-

ing this in (3.18), the flux expressions of the desired form are obtained

f = Tu = CA−1Bu−Du. (3.19)

We now briefly comment on the solvability of the system, and hence the de-termination of the transmissibility matrixT of Eq. (3.19). This discussion is alsovalid for the 2D unstructured grids discussed in the next section.

In an interaction region wheren grid cells interact, 3n degrees of freedom areoriginally available to determine then piecewise linear potential planes (as func-tions of the position vectorx). Since each potential plane must honour the nodes(of the sub-cell in question),n degrees of freedom will be locked. This meansthat 2n degrees of freedom are left for flux and potential continuity. There arensub-interfaces contained in the interaction region, and from Eq. (3.5) there arenflux continuity equations. Now, there aren degrees of freedom left for potentialcontinuity. To derive explicit expressions for the gradients of the potentials, thecontinuity points were used as a basis, see Eq. (3.8). The vector v of Eq. (3.17)contains then unique potential values at the continuity points. This is how the lastn degrees of freedom have been used; namely that the potentials are continuous atthe dividing points.

Full potential continuity can in general not be achieved by the assumption ofpiecewise linear potentials. This is the price that has to bepaid to get (piecewise)constant fluxes across sub-interfaces.

3.2 Unstructured grids; 2D polygonal CVs 25

3.2 Unstructured grids; 2D polygonal CVs

A discussion of the transmissibility calculation for the sub-interfaces of triangularinteraction regions will now be given. In Paper A, the MPFA O-method is dis-cussed for polygonal control volumes, and it is the transmissibility calculationsthat apply for this geometry that will be discussed.

E

Figure 3.7: Left: Polygonal control volume grid with associated triangular interac-tion regions. Right: Whole cell edge,E, covered by two neighbouring triangularinteraction regions.

The geometry may be illustrated by Fig. 3.7, and the notationused for thisgeometry will be the same as in [3], [4] and in Paper A.

The interaction regions of the polygonal control volumes discussed in PaperA, will be triangles. Such an interaction region was visualised in Fig. 3.3. Forthis case, fluxes across sub-interfaces will also be given bythe general formula(3.1). Now 3 grid cells will interact, so thatn = 3 in Eq. (3.1). Further, 3 fluxcontinuity equations of the form (3.5) must hold. Similar coefficients as (3.16)were introduced in [3], but the notation is slightly different. More specifically, thescalar coefficients for the case of polygonal control volumes are given by

ω±ik=

nTi Kkν±k

2Fk

. (3.20)

The quantities used in the above coefficients are explained by Fig. 3.8.Note thatparts of the triangle edges are used in the definition of the scalar

coefficients (3.20). In Paper A, the continuity points/dividing points of the tri-angle edges are midpoints of the triangular edges, such thatthe outward normalvectorsν+i andν−

p(i) are equal. However, as the method was derived for arbitrarycontinuity points along the triangle edges, we have kept theoriginal notation.

The transmissibilities for the sub-interfaces of this triangular interaction re-gion are derived using exactly the same principles as for quadrilateral controlvolumes. Flux continuity is posed across the sub-interfaces, and potential continu-ity is posed at dividing points. As for the quadrilateral case, explicit expressions


ν−1

ν+

1 ν−2

ν+2

ν−3

ν+

3

n1

n2

n3

Figure 3.8: Triangular interaction region with quantitiesused in scalar coeffi-cients.

are obtained for the gradients of the potentials for each sub-cell in the interactionregion. For each sub-cell, the potentials at the node and thedividing points areused as a basis for expressing the gradient. This is illustrated by Fig. 3.9, whichshows the corresponding picture to Fig. 3.6, where inward normal vectors of theinteraction region were used for expressing the gradients of the potentials.

ν

ν

xx

k

m(k)

kk −

x−

m(k)

Figure 3.9: Variational triangle

Using these points and vectors, one obtains the general expression for thegradient of the potential of sub-cellk,k = 1, ..,3 in a triangular interaction region:

∇Uk = −1

2F(ν+

k(um(k) −uk)+ν

−k

(uk −uk)), (3.21)

wherem(k) is a backward integer function defined by

m(k) =

k−1, for k > 1,

3, for k = 1.(3.22)

The flux continuity conditions are with this notation statedby

3.2 Unstructured grids; 2D polygonal CVs 27

fi,i = fi,p(i), i = 1, ..,3. (3.23)

The indexingp(i) denotes the forward integer function

p(i) =

i+1, for i < 3,

3, for i = 3.(3.24)

When inserting the scalar coefficients in Eq. (3.23) and the flux expressionfi = −nTi Kj∇Uj for sub-interfacei, one obtains a similar system of equations asEq. (3.17). In the original paper [3], the system of flux continuity equations waswritten as

Av+Bu = Cv+Du. (3.25)

As in the discussion of the transmissibility calculations for the case of quadri-lateral grids,v denotes pressure values at the continuity points, andu denotes thepressures at the nodes of the cells that interact.

The above system of equations is written differently from the system (3.17)by the fact that the information there was split, so that matrices acting onv andu respectively were defined. This is purely an issue of definition. The matrixdifference (A−C) of Eq. (3.25) serves as the matrixA of system (3.17), and thematrix difference (D−B) of (3.25) serves as the matrixB of system (3.17).

The matricesA,B,C andD of the system of equations (3.25) are defined,and can be found on p.6 of Paper A. Their elements consist of scalar coefficients(3.20). As in the discussion of the transmissibility calculations for the quadri-lateral case, the potentials at the continuity points may beeliminated. From Eq.(3.17) one obtains

v = (A−C)−1(D−B)u. (3.26)

Together with, for example, the left hand side expression for the flux of (3.17),the transmissibilities are found:

f =A(A−C)−1(D−B)u+Bu. (3.27)

Hence, the transmissibility matrix is

T =A(A−C)−1(D−B)+B. (3.28)

Having found the transmissibilities, the coefficient matrix for the total systemof discrete pressure equations may be found by an assembly procedure. This isalso discussed in Paper A.


3.3 Faults with crossing grid lines

3D structured grids are the most commonly used grids for fieldreservoir simula-tion. This is mostly because of the frontier commercial simulatorEclipse [57],where structured, orthogonal, conforming quadrilateralswere used in the first re-leased versions. Although the grids are structured, one is faced with several chal-lenges if for instance faults need to be taken into account. Grid cells need nolonger be conforming, and the discretisation of fluxes across cell edges must bere-investigated.

Paper C of Part II extends the MPFA methods to handle cases where lateralgrid lines are allowed to intersect. Examples of such grids are encountered whenusing (Geophysical) models for real fields. In particular, the North Sea reservoirVisunduses a reservoir description where such faults are found. The simulatorEclipse allows reservoir simulation for this grid type, butthe flux discretisationuses a standard two-point flux expression, see [6] and [57] for details.

MPFA Transmissibility calculations to handle faults with crossing lateral gridlines are presented in Paper C, and the discretisation uses the same principles asfor previously investigated grid types. Note, however, that some additional issuesneed to be taken care of because of the under-determined character the system ofequations that determines the transmissibilities has.

Before discussing these special 3D faults, we briefly discuss standard faults in2D and 3D. The extension of the MPFA method to handle faults was first done for2D quadrilateral grids in [6]. Here the concept of non-conforming grids was used,and interaction regions were defined around such corners. Asseen in Fig. 3.10,three grid cells meet in a corner.

Fault line

Figure 3.10: Corners when faults are encountered in 2D

A natural interaction region to use here is the triangular interaction regionwhich was introduced in Sec. 3.2, or a similar interaction region where three gridcells interact. It has been reported that this interaction region has an apparent dis-advantage when one of the grid cells of the interaction region is inactive [6], whereit was shown that there will be no flow across one of the three sub-interfaces for

3.3 Faults with crossing grid lines 29

certain situations. Instead, in [6] it was suggested to use an extended interaction

.

.

.

.

.

.

.

.

.

.

12

34

5

Figure 3.11: Interaction region associated with 2D faultedcorners. Five grid cellsinteract

among grid cells. The suggested interaction region will contain information from5 grid cells instead of 3. Hence, the fluxes across each of the sub-interfacesi areexpressed as

fi =

5∑

j=1

tijuj. (3.29)

This is visualised in Fig. 3.11. The MPFA method was extendedto faultedcorners in 3D in [7]. Similar ideas as for the 2D faults were used to derive trans-missibilities for the extended interaction regions, and details may be found in [7].Five different situations may occur, and the flux molecules will contain informa-tion from 8 to 11 grid cells. One possible situation is illustrated in Fig. 3.12, where7 grid cells meet in a corner. For this case the flux molecules contain informationfrom 9 grid cells. These types of corners will be termedregular corners, and 5-8

Figure 3.12: Ordinary faulted corners in 3D

grid cells will touch upon such corners. However, in [7] the case where grid linesintersect was not discussed. This situation was discussed in Paper C. Lateral gridlines may for some simulation grids cross, and the flux acrossgeneral polygonalflux interfaces must be discretised. The point for which lateral grid lines intersect,will be termed anirregular corner, and an interaction region will be associated


with it. How these new corners arise is depicted in Fig. 3.13.For this corner, 4grid cells will touch upon it, and these 4 grid cells will be the interacting grid cellsin the associated interaction region. The sub-interfaces for which the transmiss-ibilities need to be calculated are visualised in Fig. 3.14.Here, 6 sub-interfaceswill be associated with the interaction region. To understand this, consider Fig.3.13. The local numbering of the four grid cells is as follows: Local grid cell1 is the front cell, grid cell 2 is the back cell. Above cell 1 there will be a gridcell (not drawn), and this cell has local number 3. Above cell2, grid cell 4 willbe located (also not drawn). A cell interface is a a sharing ofa common surfacebetween two neighbouring grid cells. From Fig. 3.13 it is seen that there will be6 cell interfaces touching upon the irregular corner. Denoting these sharings bya tuple (a1,a2) (wherea1 anda2 correspond to the local numbering of the gridcells), the 6 sharings are given by (1,3), (2,4), (1,2), (1,4), (3,2) and (3,4). The6 sub-interfaces will be parts of these 6 cell interfaces, and this is discussed below.

For each of the 6 sub-interfaces we want to find fluxes that are linear combin-ations of these 4 interacting grid cells:

fi =

4∑

j=1

tijuj, i = 1, ...,6. (3.30)

1

2

3

4

Figure 3.13: Crossing grid lines arising from faults

As for the 2D case,∇U is needed in order to derive discrete fluxes and trans-missibilities for the sub-interfaces belonging to this newtype of interaction region.This is done by using the same methodology as in [8]. For three-dimensionalcases (ie. both this situation and the situation with ordinary faulted corners),∇Ufor each sub-cell of an interaction region, can be defined implicitly by the valuesof the potentials at three corresponding potential continuity points:


2

3

4

5

6

1

Figure 3.14: Sub-interfaces associated with irregular corner

X∇U =

u1−u0

u2−u0

u3−u0

. (3.31)

The derivation of the inverse ofX follows exactly the same ideas as for the2 dimensional case. We introduce variational tetrahedra spanned by the cellcentrex0 (node) and three local continuity pointsx1,x2,x3 that are located onthe three sub-interfaces seen from the sub-cell in question. These points are loc-ally numbered cyclically. A variational tetrahedron used for the faulted corners in3D discussed in [7] is visualised in the left hand side of Fig.3.15. The continuitypoints are three of the corners of the tetrahedron, and are located on 3 separatesurfaces of the cell in question. Due to the special geometrywhen lateral gridlines are allowed to intersect, the variational tetrahedrawill be somewhat differ-ent from the regular case. This is depicted in the right hand side of Fig. 3.15. Twoof the tetrahedron corners will be located on the same cell interface (clarified byFig. 3.17).

4 of the 6 sub-interfaces are parts of side surfaces of the grid cells that interact,whereas the two remaining sub-interfaces are parts of the top/bottom surfaces ofthe cells of the interaction region. This is visualised in Fig. 3.14. The 4 sub-interfaces that lie in the surface generated by the side surfaces of the grid cellsthat interact are numbered 3-6 in the figure. In Paper C it was assumed that theside surfaces of the cells are planar, and the surfaces 3-6 will hence also be planar.

To better see how these four sub-interfaces arise, it may be useful to visualisethe situation orthogonally on the intersection, as in Fig. 3.16.

Different situations may occur, and the whole cell interfaces that touch uponthe corner may have from 3 to 6 corners. Each of these corners will correspondto an interaction region. For a complete discretisation of the fluxes across thewhole cell interfaces, the cell interface will be shared among the 3-6 interactionregions which the corners of the whole flux interfaces correspond to. The sharingof flux-interfaces among the interaction regions may easilybe defined by dividinga whole cell interface withn corners inton segments/sub-interfaces. This is done


x

x

x

x x

x

1

2

3

1

3

2

Figure 3.15: Variational tetrahedra for a) regular case b) crossing grid lines.

Front

Back

Figure 3.16: Intersection of lateral grid lines seen orthogonally on the intersection

by finding the geometric centre of the cell interface (x′ = 1/n∑

xi), and drawingstraight lines to the midpoint of all edges. All sub-interfaces will then have 4edges, and the approach is easy to implement.

34

5

6

Figure 3.17: Sub-interfaces belonging to side surfaces of grid cells

Fig. 3.17 explains how sub-interfaces 3-6 of Fig. 3.14 arise. At each sub-interface there will be one point for which the potentials ofneighbouring grid cellsare continuous, ie. a continuity point. These points are marked with dots, andtheir numbering corresponds to the numbering of sub-interfaces. This explains


why two of the corners of the variational tetrahedra used here are located on thesame side surface (of the cell in question).

For all 6 sub-interfaces, flux continuity will be posed, and (together with po-tential continuity at 6 points) this leads to a local system of equations to determinethe transmissibilities of the interfaces. This system is written out in Paper C, andthe redundancy of the system is discussed. The redundancy implies that moreconditions are needed in order to determine the transmissibilities. The determina-tion of transmissibilities for the sub-interfaces was partially answered by using acriterium for no-flow and a criterium for uniform flow in PaperC. Details may befound in the paper.

Linear flow assumes a homogeneous medium, and it is not clear whether thetransmissibility calculations can be generalised to arbitrary heterogeneous mediawithout additional information.


Figure 3.18: Gridding of region of interest; inner grid cells are sketched by con-tinuous lines. Grid cells surrounding the area of interest are indicated by dottedlines.

3.4 Treatment of Dirichlet boundary conditions

The first published papers on MPFA methods [1]-[8] handled boundaries withhomogeneous Neumann conditions. The implementation of such conditions werestraight forward since the transmissibility coefficients could be calculated by thesame ideas as for the fully active interaction region. Details may be found in [3].

Paper D investigates convergence of the O-method on some quadrilateral gridsin 2D. Such a study requires more general conditions than homogeneous Neu-mann conditions. All our analytical examples required using fixed potentials atsome boundary.

We now discuss various ways of how to extend the MPFA O-methodfor 2Dquadrilateral grids to account for general Dirichlet conditions.

The simplest way to include fixed potentials at boundaries isto add an en-circling layer of artificial grid cells around the medium which we study. If theanalytical pressure solution is known, we may fix the potentials at these outercells, and solve the discrete pressure equations for all theinner grid cells (at thenodes). At each refinement level one solves the pressure equation on the samephysical medium, as illustrated by Fig. 3.18.

This approach does not incorporate the boundary conditions(Dirichlet andNeumann) for the ’active area’ exactly, but rather approximately. This can beseen in Figure 3.19, where an extended interaction region isshown. The pointsx3a andx4a are nodes of the artificial grid cells, and at these points we use exactpressures. The transmissibility calculations then yield approximate fluxesf1d andf2d and corresponding discrete pressures at the continuity pointsx2d andx4d.

As the approach above requires that the analytical solutionis known outsidethe area of interest, a better approach of handling general Dirichlet boundary con-ditions is to actually incorporate fixed potentials/pressures at the actual boundariesof the medium we study. Utilising the ideas from the originaltransmissibility de-

3.4 Treatment of Dirichlet boundary conditions 35

x x

xx

f fx x

1 2

xx1d 2d

− −4d 2d

3a4a

Figure 3.19: Incorporation of approximated boundary conditions

x x

x

x

x

1 2

2D1D

F

Figure 3.20: Treatment of Dirichlet boundary conditions

rivations, this can be done by studying an interaction region which covers a partof the boundary.

Consider the extracted part of a grid visualised in Fig. 3.20. The nodes of thetwo grid cells arex1 andx2 respectively. The pointsxD1 andxD2 are points onthe boundary, and the potentials at these point will be specified by the Dirichletboundary conditions. To derive the transmissibilities forthe sub-interface goingfrom xF to the boundary, we pose continuity of the flux at the sub-interface andcontinuity of potential at the dividing pointxF . When the potential is assumedto be piecewise linear in each of the sub-cells for the cells that interact in theinteraction region, the local system is closed with respectto degrees of freedom.

Hence, we may derive the transmissibilities through the equation

−K1∇U1 ·n = −K2∇U2 ·n. (3.32)

The gradients can be calculated by using variational triangles defined on eachsub-cell and the notation used in Sec. 3.1. Hence,

∇Ui =1Ti

∑

νik(uik −ui), (3.33)

for the each of the cellsi = 1,2. The vectorsνik are as before the inward normalvectors of the edges (xij−xi) of the variational triangles (j,k cyclic,k local num-bering of dividing points of triangle).Ti is equal to twice the area of variational


trianglei. The potentials ¯uik are the potentials at the pointsxik (ie. either potentialat boundary or at continuity pointxF ).Having expressed the gradients, the flux continuity equation (3.33) now reads

−1T1

K1(ν11(u1D −u1)+ν12(uF −u1))n =

−1T2

K2(ν21(uF −u2)+ν22(u2D −u2))n. (3.34)

This may be written as

av = bu+k1u′, (3.35)

wherea (scalar),b andk1 (1x2 vectors) depend on geometry and permeability.Further,v = uF is the unknown potential at the dividing/continuity pointxF , u =

[u1,u2] represents the vector of node potentials, andu′ = [u1D,u2D] is the vectorinvolving the pressuresu1D andu2D specified by the boundary conditions.

The entitiesa,b andk1 of Eq. (3.35) can be given explicitly:

a = −1T1

K1ν12n+1T2

K2ν21n

b = [−1T1

K1(ν11+ν12)n,1T2

K2(ν21+ν22)n]

k1 = [1T1

K1ν11n,−1T2

K2ν22n] (3.36)

To express the transmissibilities for the sub-interface, we may write the left handside of (3.34) as

f = cv−du+k2u1D, (3.37)

wherec (scalar),d (1x2 vector) andk2 (scalar) depend on geometry and permeab-ility. These can also be given explicitly:

c =1T1

K1ν11n

d = [−1T1

K1(ν11+ν12)n,0]

k2 = −1T1

K1ν11n (3.38)

The unknown ¯uF may now be eliminated from Eq. (3.35):

3.4 Treatment of Dirichlet boundary conditions 37

x

x

x

e

ee1D 2D1D

1

2D

1 Fx

x

2

Figure 3.21: Boundary interaction region, used to include Dirichlet boundary con-ditions.

v = a−1(bu+k1u′), (3.39)

and inserting this in Eq. (3.37) yields

f = (ca−1b−d)u+ ca−1k1u′+k2u1D. (3.40)

Hence, the flux is expressed as

f = Tu+T ′u′. (3.41)

The transmissibilities fromT will enter the system matrixA for the discreteset of (pressure) equationsMU =R, and the transmissibilities fromT ′ will enterthe right hand sideR of the system. To construct the full system matrix, a loopingover all interaction regions will be done, and for inner interaction regions wecalculate transmissibilities in the standard way, discussed in Sec. 3.1.

The above method to take Dirichlet boundary conditions intoaccount seemto be a reasonable way to handle the problem. However, forK-orthogonal grids,some information seems to be lost from the boundary because of the orthogonalityof the grid. Moreover, the elements ofT ′ of Eq. 3.41 will be zero for the casedepicted in Fig. 3.22. It is possible to move the continuity point and the boundarypointsx1D andx2D in such a way that the same points as in Edwards’ [17] MPFAmethod are obtained. When this is done, the elements ofT ′ in (3.41) will ingeneral be nonzero.

The third way to deal with Dirichlet boundary conditions forthe MPFAmethod is to somehow include fluxes on the boundary as well.

The information about the pressure on the boundary will as before be includedfor the transmissibility calculations, as illustrated in figure 3.21. The potentialvalues at the boundary are used in the expressions for the gradient of potentials


K

Ky

x

Figure 3.22:K-orthogonal grid; ’Dirichlet’ transmissibilities will bezero

for each cell. From this, the discrete fluxes for the sub-interfaces are

f11 = −K1∇U1 ·ne1 (3.42)

f12 = −K2∇U2 ·ne1 (3.43)

f1D = −K1∇U1 ·ne1D (3.44)

f2D = −K2∇U2 ·ne2D (3.45)

The gradient,∇U i, i = 1,2 will use the potential values at the pointsxi, xDi andxF . The flux from cell 1 into the flux interfacee1 , calculated by the informationfrom cell 1, is denotedf11, and equals the flux calculated for interfacee1 into cell2, (information from cell 2),f12. The potential at the dividing pointxF may hencebe eliminated, from Eq. (3.42) and Eq. (3.43). If we next use the potential at thedividing pointxF in (3.43)-(3.45), we are left with the fluxf = (f12,f1D,f2D)T asa function of the potentials of the active cells 1 and 2,u= (u1,u2)T , and potentialsat the boundary,u′ = (u(xD1),u(xD2))T .

The fluxes can now be expressed as:

f = Tu+ T u′

The transmissibilities ofT will be incorporated in the the discrete version of(2.1), while T u′ incorporates the boundary conditions on the right hand sideofthe corresponding discrete system.

Chapter 4

Time dependent problems

Reservoir simulation models are used to predict future performance of reservoirs.For multiphase problems the flow is hardly ever stationary, so in order to makeforward simulations (in time), the time dependent terms of the governing PDEsmust also be discretised.

Several options are possible, but it is common to use backward Euler timediscretisation to obtain fully implicit discrete equations. This means that all un-knowns are coupled at a given time level, and the equations for the unknowns needto be solved simultaneously. A semi discrete version of the mass balance equationfor phaseα then reads

mnα −mn−1

α +∆tn∑

j

fnαj = qnα. (4.1)

The entities above are:mα accumulated mass of phaseα, n andn−1 time levels,fαj phase fluxes through surface numberj of a control volume, andqα a possiblesource term. Nonlinearities and how these affect the numerical solution strategyare discussed next.

4.1 Nonlinearities

The discretisation of multiphase flow equations/problems yields sets of non-linearequations to solve (with respect to solution variables). The non-linearities arisebecause of the appearance of relative mobilities (see Chapter 1) of the phase fluxes∫

kriµiK∇ui ·n, and the inclusion of capillary pressure. Relative mobilities will be

handled by standard upstream evaluation across edges. A possible modificationof this has not been an issue for the MPFA O-method yet. This means that thethe transmissibilities derived for single phase flow will bescaled by relative mo-bilities according to the flow direction. By an implicit solution strategy, relative

40 Time dependent problems

permeabilities will be updated at each iteration level.At each time level, indicated by Eq. (4.1), nonlinear algebraic equations must

be solved. Our in-house research simulator solves these algebraic equations byusing a Newton discretisation approach, see for instance [39], [40].

Based on the time stepping, an outer iteration is defined, andthe goal is tosolve the equation based on Eq. (4.1), which determines the change in solutionvariables from one time step to the next time step

r(x) = m(x)−mn−∆t∑

k

fk(x)−∆tq(x) = 0. (4.2)

Herex denotes the vector of the (primary) solution variables in all grid cells.For three-phase flow these may be the three quantitiespo, Sw andSg, and theremaining 3 solution variables (So,pw,pg) are found by the previously discussedclosure relations.

The solution by the Newton discretisation procedure is defined by

(∂r

∂x)ν−1[x(ν) −x(ν−1)] = −r(ν−1), (4.3)

where the superscriptsν and ν − 1 denote iteration levels. Eq. (4.3) is solveduntil x(ν) −x(ν−1) has reached some desired limit (ie. specified convergence cri-teria). The data structure of the research simulator allowsfor general geometry,and this will be reflected in the Jacobi matrix (∂r

∂x). Moreover, there will be more

couplings in the Jacobi matrix when a control volume interacts with more controlvolumes than when two point flux schemes are used for flux discretisation. Thisgeneralisation is accounted for in, for example, the handling of simulation gridswith crossing lateral grid lines in Paper D. Details of the Newton strategy may befound in [39].

Chapter 5

Numerical results

Paper D investigates convergence numerically for the MPFA O-method. In the lit-erature, there are numerous reference examples to be found for the purely ellipticproblem (1.7) with suitable boundary conditions. Linear pressure solutions (inspace) are perhaps the easiest reference solutions to investigate. Although sucha reference solution is very simple, it is a good first test to examine the discretesolution obtained by a discretisation method when the grid is challenging.

The discrete flux operator is examined for various non-orthogonal grids inboth 2D and 3D in Paper D. It is shown that the discrete fluxes (from the MPFAO-method) across sub-interfaces of interaction regions are exact when applied tolinear potentials. Intuitively we should expect that the discrete pressure solutionobtained from the corresponding systemAu = b also should be exact since thediscrete set of pressure equations is constructed by assembly of transmissibilities.

The pressure solution on a randomly perturbated grid is investigated for theSupport Operator Method (SOM) for a homogeneous medium in [44], where theanalytical flow is uniform in a given direction. Their discrete pressure solution isexact for that case.

Fig 5.1 shows a random 8×8 grid, and the underlying medium is homogen-eous. The discrete pressure solution is calculated by the MPFA O-method, wherethe analytical pressure solution is linear (driven by specified (Dirichlet) pressureson the left and right hand side of the medium). The right hand plot of Fig. 5.1reveals that the pointwise pressures are linear and exact for this case.

A similar test can be carried out for a discontinuous coefficient problem, wherethe analytical pressure solution is piecewise linear. The medium is divided intotwo parts, and the permeability ratio is 1/10 across the medium discontinuity. Thediscontinuity line is here given byrx+ sy = 0, wherer = tan(π/3)/(1+ tan(π/3))ands = 1/(1+ tan(π/3)). This is visualised in Fig. 5.2. The permeabilities arekx = ky = 10 for rx+sy < 0 andkx = ky = 1 for rx+sy > 0. The true solution isgiven by

42 Numerical results

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

−0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x−coordinate

Pre

ssur

e

a. Random grid;. b. Pressure vs.x-coord

Figure 5.1: Numerical pressure solution; uniform flow inx-direction.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

0

0.5

−1

0

1

2

3

4

5

a. Random grid. b. Discrete pressure solution

Figure 5.2: Numerical pressure solution; discontinuous coefficient problem.Piecewise linear reference solution.

u(x,y) =

rx+ sy, for rx+ sy < 0,

10(rx+ sy), for rx+ sy > 0.(5.1)

The numerical results are shown in Fig. 5.2, where the pressures at the grid nodesare employed. The discrete pressure solution on our 16× 16 grid is piecewiselinear and exact also for this case.

More sophisticated cases are tested in Paper D. Especially interesting testcases are cases where the permeability of the medium has sharp jumps, and theanalytical solution exhibits singularities in the velocity field. Such reference solu-tions are discussed in [46].

5.1 Analytical solutions 43

−0.5

0

0.5

−0.5

0

0.5−10

−5

0

5

10

Figure 5.3: Plot of analytical solution on [−0.5,0.5]× [−0.5,0.5]. Solution exhib-its singularity at the origin.

5.1 Analytical solutions

The problems dealt with in Paper D are special cases where analytical solutionsof the elliptic equation (1.7) are known. The cases are discussed because theyregard important physics which discretisation methods should handle. In reservoirsimulation, non-orthogonal grids are used extensively, and the permeabilities varyfrom grid cell to grid cell. Four grid cells will meet at each corner for a conforminggrid. If the grid cells have different permeability, the pressure may be less regularnear these corners than in areas where the medium is homogeneous.

In particular, cases where the analytical solutions exhibit singularities, shouldbe discussed. Such a solution is depicted in Fig. 5.3 on a square domain. Usingpolar coordinates, this is a solution of the form

ui(r,θ) = rβ (acos(βθ)+bsin(βθ)), (5.2)

where the indexi denotes regions with different permeability. The parameter βdefines the regularity of the solution, and will depend on heterogeneity and theangles of the grid cells that meet at a corner. These functions are known to notbelong to the function spaceH2(Ω), but rather the fractional spaceH1+ν (Ω),whereν < β, see [46]. The solutions are derived based on separation of variables,and details may be found in Strang and Fix [46].

Chapter 6

Properties of discrete system

There has recently been work done to investigate the relationships between MFE-types of methods and MPFA methods, [30] and [32]. Furthermore, these relationsare used to analyse convergence of the MPFA O-method on a general quadrilateralgrid (with continuously varying heterogeneity) theoretically [32].

As the MPFA method does not directly build on a variational formulation ofthe governing (elliptic) equation, the approach in Paper A was not to investig-ate convergence but rather to investigate the coefficient matrix that arises whendiscretising Eq. 2.1. This matrix is also frequently referred to as theDiscreteoperator(corresponding to the continuous operator).

Various works in the literature, [41], [42] etc., indicate that convergence of adiscretisation method is related to three important issues; conservation of mass,consistency of fluxes (see discussion in [4]) and stability.MPFA-methods arelocally mass conservative, so this is not an issue to be discussed. The concept ofconsistent flux means that the discrete fluxes are ’reasonably’ approximated. Weclaim that this is the case for fluxes obtained by the MPFA approach, since theseallow for general geometry, and general tensor representation of permeabilities.Exceptions are cases where the solutions are far fromH2-regular, as indicated bythe last numerical example of Paper D.

Stability issues are among the reasons for studying the discrete operator. Cer-tain important properties of the continuous elliptic operator should be inherited forthe discrete operator. The operator∇· (K∇ ) is a self adjoint operator, and thediscrete operator should then ideally be symmetric [43]. This was one of the issuesdiscussed in Paper A (see discussion of Paper A below). Stability of a method isassured if the discrete system satisfies the coercivity condition [8], [51]:

uTAu ≥ µuTDu, (6.1)

whereA denotes the coefficient matrix of the system, andu denotes the discretesolution vector. For the coercivity condition to be satisfied, A should be positive

46 Properties of discrete system

definite [49].A special class of matrices that are positive definite, are symmetric M-matrices:

6.1 M-matrices

The following definition is taken from [51].

Definition: A matrixA is called an M-matrix if the following is satisfied:

a) The diagonal elementsaii are positive whereas the off-diagonal termsaij,i 6= j are non-positive.

b)A is non-singular andA−1 ≥ 0.

Such matrices are appealing when looking at the continuous problem−∇ · (K∇u) = q. For positive sourcesq, the solutionu should also be pos-itive, and this is assured if the coefficient matrix that arises from the discretisationof −∇ · (K∇ ) is an M-matrix. Moreover, the positiveness of the solutionisassured if the coefficient matrix has a non-negative inverse. Such matrices arecalledmonotone(operators), see also [8] for a discussion.

The M-matrix property is examined in Paper A. However, this property maybe too restrictive for a method in order to say something about the monotonicity.It is well known that a matrix may have a non-negative inversewithout being anM-matrix. This means that the matrices one really is lookingfor, are matricesthat are symmetric, positive definite, and have non-negative inverses. As far as weknow, it is not known how such matrices can be classified basedon their elements,but work has been done recently to analyse the monotonicity of the O-method.

6.2 Notes on violation of maximum principle

A specific case has been investigated in [15] where it was of interest to find outwhether a non-negative inverse still could be obtained eventhough the conditionsfor the matrix to be an M-matrix were violized. The O-method was examined on aparallelogram grid (2D) for homogeneous media. The coefficient matrix will thenbe described by the legs of the grid cells (and the corresponding inner angles ofthe grid cells). The elements of the coefficient matrix are given by the cell stencilfor the central cell (cell 1) of Fig. 6.1. These coefficients are denotedm1.....m9,and correspond to the local numbering of grid cells in Fig. 6.1. For the O-methodon a parallelogram grid on a homogeneous medium, these coefficients are given

6.2 Notes on violation of maximum principle 47

by [8]

1 2

345

6

7 8 9

Figure 6.1: Local numbering of grid cells

a = a1TKa1, b = a2

TKa2, c = a1TKa2, (6.2)

and

m2 = −a+c2

d, m3 = −

c

2−

c2

2d, m4 = −b+

c2

d, m5 =

c

2−

c2

2d, (6.3)

andmi+4 = mi, i = 2, ...,5. The variabled is given by 2ab/(a+ b), andm1 =

−∑

i=29mi. The permeabilityK is positive definite such thata > 0,b > 0, andc2 < ab. It is shown that two criteria are sufficient for the coefficient matrixA tohave a non-negative inverse. These criteria are

(1) m2 ·m4−m3 ·m1 ≥ 0, (2)m2 ·m4−m5 ·m1 ≥ 0, (6.4)

which are the same as|c|(2− c2

ab) ≤ d. Since these criteria only are sufficient, the

inverse will be non-negative even if (6.4) is not satisfied. In [15] it is stated that forgrids with approximately unity aspect ratios, grid skewness of about 45 degreesmay be allowed. For stronger aspect ratios and/or anisotropy ratios, the limitationson grid skewness are more severe. The reader may notice that grid skewness of45 degrees has been tested in Paper D for various reference solutions.

When the inverse is investigated numerically on ann× n grid, it is found thatangles down toπ/6 yield positive inverses for largen (n approximately 20) foraspect ratios in the interval (1/2,2).

This result may be interpreted as an unfortunate way of choosing neighbouringinformation for the fluxes across cell edges when the grid cells are very skewed.When the aspect ratio of the grid is close to 1, it is possible to construct a schemewhere the bandwidth is reduced. This should intuitively lead to a more diagonallydominant matrix than the 9-point scheme does. Such a scheme can be illustratedby Fig. 6.2.

48 Properties of discrete system

23

4

1

Figure 6.2: Possible 7-point cell molecule. Combination ofdifferent types ofinteraction regions

This molecule can be obtained by combining two types of interaction regions.Interaction region 2 and 4 are triangular interaction regions which are special casesof the triangles investigated in Paper A. Interaction region 1 and 3 are standard in-teraction regions associated with (conforming) corners for 2D quadrilaterals. Thisscheme has not been implemented by us, but for the homogeneous case the schememay easily be generated by the use of explicit expressions for transmissibilities ofsub-interfaces of the two different types of interaction regions.

Chapter 7

Summary of the papers

Summaries of the papers which Part II of this thesis consistsof, will now be given.Some issues may also have been left out of the papers themselves, and some ofthese are discussed here. Details are also to be found in the previous sections.

7.1 Summary of Papers A and B

This Subsection will give an overview of Paper A and Paper B:

Paper A: Symmetry and M-matrix Issues for the O-method on an Unstruc-tured GridAuthors: G.T. Eigestad, I. Aavatsmark, M. Espedal

Paper B:MPFA applied to Irregular grids and Faults.Authors: G.T. Eigestad, I. Aavatsmark, E. Reiso, H. Reme, R.Teigland.

The theme is similar for both papers. The MPFA O-method is discussedfor irregular 2D grids in Paper A, and an extension to accountfor certain griddingsituations which arise when faults occur in 2D, is discussedin Paper B.

Paper A discusses the O-method for polygonal control volumes. These willbe control volumes for which the interaction regions are triangles. The MPFA O-method for such cases is also referred to as theTriangle methodin the literature.

It is of interest to study the coefficient matrix that arises when the ellipticterm −∇(K∇u) is discretised. Because the underlying continuous operator isself-adjoint, this should ideally be inherited for a corresponding discrete operator(coefficient matrix). This means that the matrix should be symmetric and positivedefinite (sinceK is positive definite). There exist alternative discretisation meth-ods (Support Operator Methods, [43]), which actually use the preservation of such

50 Summary of the papers

properties for the construction of discrete schemes. Secondly, the physics behindthe purely elliptic problem (2.1) implies that positive sourcesq should lead topositive pressures (for homogeneous Dirichlet boundary conditions). Translatedto the discrete problem, this means that the coefficient matrix should be monotone,see Chapter 6 and [8].

The interaction regions are as mentioned above triangular,and three grid cellsinteract in such an interaction region. This was visualisedin Fig. 3.8. The deriva-tion of the transmissibilities are discussed in Sec. 3.2.

The first case studied in the paper is the case with homogeneous permeability(for the whole medium). For this case, the coefficient matrixis shown to be sym-metric as long as the dividing points (see Sec. 3.2) are midpoints of triangle edges.Further, the dual triangular grid should satisfy the Delauney condition in order forthe coefficient matrix to be an M-matrix. This means that opposite angles of tri-angles which share an edge, should satisfy the conditionθ1+θ2 < π.For general permeabilities, symmetry of the discrete operator is not satisfied dir-ectly for the MPFA O-method applied to polygonal control volumes. This wasfurther examined in the paper, and the question raised was whether it is possibleto obtain a symmetric, discrete operator. To answer this question,symmetry equa-tions were defined locally for the interaction regions. The dividing points werefixed (midpoints of triangle edges), and the equations were solved with respectto the location of the triangle points inside the interaction regions. The controlvolumes change with this procedure, but the goal of the experiment is to find geo-metrical situations for which symmetry of the discrete operator is satisfied. Solu-tions of the symmetry equations may be sought for arbitrary heterogeneity andanisotropy ratios inside the interaction regions. The algorithm for finding trianglepoints that yield symmetric coefficient matrices, may easily be implemented fornumerical purposes. We were not able to obtain solutions forall cases we tested(anisotropic cases), but we were able to extract important information for layeredmedia (two cells of an interaction region have the same permeability). For thesecases it is possible to construct gridsa priori that honour the media discontinuities,and at the same time yield coefficient matrices that are symmetric. For isotropiccases it was found that the triangle point should be located on a straight line whichconnects two dividing points, and we refer to Paper A for details. If the anisotropyratios are different, but the principal axes of the permeability tensors are alignedwith the discontinuity line, the triangle point should alsobe located on this straightline.

This result was found using the symmetry algorithm numerically, and is laterproven analytically in the paper.

The next question raised in the paper is how the dual triangular should bein order for the local contributions from the interaction regions to the coefficientmatrix to have ’the correct’ signs. This is referred to aslocal M-matrix property

7.1 Summary of Papers A and B 51

in the paper. It is found that this local M-matrix property yields a constraint onangles of the dual triangular grid, heterogeneity and anisotropy combined.

The local M-matrix property does not fully answer the question whether thetotal coefficient matrix is an M-matrix, as a neighbouring interaction region maycompensate for unwanted signs/contributions from one interaction region. TheM-matrix property derived for homogeneous media in Paper A is an example ofthis. Such an analysis may be carried out in the same manner asin Paper A toanswer this for specific cases.Paper B is an overview paper, and the symmetry results found in Paper A areused in a discussion of practical gridding. Paper B also discusses the fullyheterogeneous case on a right angled triangle. Symmetry maybe obtained forthis case when the media are isotropic. Further, faults in 2Dare illustrated, andare seen to be generalisations of the interactions of a triangular interaction region.

Main results of the papers:

The main results of the papers may be summarised:

• Operator fully analysed on homogeneous media

• A general symmetry algorithm was investigated

• Layered, isotropic media can always be gridded to obtain symmetry andlocal M-matrix property for the triangle method.

• Layered media with varying anisotropy ratios can be handled for specialcases

• A 90 degree triangle has been found to have ’good properties’ regardingsymmetry for cases where all three grid cells have differentpermeability

• Practical gridding based on symmetry results has been sketched

• An overview of gridding issues regarding 2D faults is given


7.2 Summary of paper C

This subsection is a summary of Paper C:

MPFA for faults with crossing grid lines and zig-zag patterns.Authors: G.T. Eigestad, I. Aavatsmark, E. Reiso, H. Reme.

This paper discusses a possible extension of the MPFA methodfor faultsto cases where the lateral grid lines of control volumes/grid cells are allowed tointersect. At the point where the grid lines intersect we define anirregular corner,and we associate an interaction region to it. Such cases are occasionally seen forgrids that are used for reservoir simulation of North Sea fields. The geometry forsuch faults was illustrated in Sec. 3.3. The idea behind the work was to be ableto simulate fluid flow on grids with general faults in the corner point geometrysetting.

A simulation grid may then contain both faulted corners as discussed in [7],in addition to this new type of corners. In the paper we have shown how to cal-culate transmissibilities for the sub-interfaces in the interaction region associatedwith the irregular corner. To do so, we used the same framework as for ordinarycorners [7] in addition to using the principle of exact discrete fluxes for linearflow. Linear flow assumes a homogeneous medium, and it is not clear whether thetransmissibility calculations can be generalised to arbitrary heterogeneous mediawithout additional information.

A structured 3D (quadrilateral) grid where all corners are matching has a reas-onable simple data-structure. Fluxes across cell surfaceswill in general be 18-point flux molecules, and this is only modified when inactive grid cells exist orwhen the grid cell is on the boundary.

When faults are allowed, the flux molecules may be much more general, as agrid cell may have many more and different neighbours that for the matching grid.This has an impact on the Jacobian matrix which is needed whena fully implicitproblem is solved.

The data structure gets more complicated when crossing gridlines are allowed.Surrounding all corners, suitable interaction regions aredefined. This means thatdifferent parts of a cell surface is shared between several interaction regions, andtransmissibilities will be calculated for sub-interfacesof the different interactionregions.

The paper contains only one example where we have simulated two phaseflow on a simulation grid which contains irregular corners. The grid is, however,a rather extreme example as nearly all cells have crossing lateral grid lines. Thisis not likely to occur for a realistic grid.

The paper further investigates planar faults vs. zig-zag faults in 3D by a

7.2 Summary of paper C 53

selection of numerical examples.

Notes on data-structure, crossing grid lines

In the work of implementing this new type of geometry, a substantial amountof time was spent on generalising the data structure in the in-house reservoirsimulator. To sum up, the following data structure needs to be defined in order tocompute transmissibilities, calculate phase fluxes and define the Jacobian matrixproperly:

• All irregular corners must be found

• Cells that touch upon corners must be known

• Whole flux-interfaces must be divided and shared among different interac-tion regions

• Sub-interfaces in different types of interaction regionsmust be defined

• Corners that touch upon whole flux-interfaces must be known

• All corners (both regular and irregular) must know their neighbour corners

• All flux-interfaces must know their neighbouring flux-interfaces

The main results of Paper C can be summarised as:

Main results of the paper:

• New geometry handled by the MPFA methods

• Data structure made more general; simulation grid may contain generalfaults

• Transmissibilities derived for interaction region associated with cornerarising when lateral grid lines cross

• Additional physical constraints discussed in the derivation of transmissibil-ities

• Zig-zag vs. planar faults in 3D investigated numerically


7.3 Summary of paper D

This subsection gives a summary of Paper D:

A note on the convergence of the MPFA O-method; numerical experimentson some 2D and 3D gridsAuthors: G.T. Eigestad, R. Klausen.

Paper D investigates convergence of the MPFA O-method on general quad-rilaterals in 2D and some special 3D grids.

The question of whether the O-method converges was raised byreferees in thereviewing process of Paper A. We tried to answer this question by some numericalexperiments in this paper. In parallel, Klausen [33] has analysed this throughrelationships found between Mixed Finite Element methods and MPFA methods.The results show 1st order convergence for problems with smooth coefficientsK. Numerical results for MPFA methods have been published by Edwards [18]and Jeanninet al. [38]. Edwards’ method, however, uses different dividing pointsthan what is used for the MPFA O-method (see Sec. 3.1). In addition to this, thenumerical examples in the above references do not have the same complexitiesas for instance the examples found in Riviereet al [37]. We therefore perform abroader spectrum of tests for the MPFA O-method in Paper D. These problemsranged from the simple case where the flow is (piecewise) uniform, to caseswhere the derivatives of the analytical pressure solutionsmay become infinitelylarge. These cases are likely to occur in real reservoir simulation because sharpheterogeneity contrasts (with corners) must be handled. The convergence isobserved for the examples that have been tested, and the results are summarisedbelow.


• Convergence of the O-method on quadr. 2D grids investigated numerically

• Flux operator investigated analytically on quadrilateral 2D and 3D grids,and found to be exact for linear problems.

• 2nd order convergence observed for theH2-regular problems we tested.

• h2α-order convergence of pressure indicated for problems tested with ana-lytical solutions belonging toH1+α(Ω) for α > 0.39. Lower convergencefor case withα = 0.127.

• Flux is roughlyhα-order convergent for all examples except for case withα = 0.127.

7.4 Summary of paper E 55

7.4 Summary of paper E

The concluding paper has the title:

Numerical Modeling of Capillary Transition Zones.Authors: G.T. Eigestad, J.A. Larsen.

In this paper, a consistent hysteresis model was included ina reservoirsimulator for forward simulation of two phase flow. Hysteresis does apply forboth capillary pressure and for relative permeabilities. The hysteresis modelsare taken from work done by researchers at Stavanger College[53], [54], andthey suggested a hysteresis model to describe capillary pressure (Pc) and relativepermeability (kr) for reservoirs with varying degree of wettability. The entitiesPc and kr will be described by general curves/functions dependent onwatersaturations,Sw, as well the history of reservoir locally (the history of eachindividual grid cell in a discrete model).

Regardless of the generality of the curves, they will usually be described byonly a few parameters. For instance, the capillary pressurefor a primary drainagecurve will be described by the expression

Pcd(S) =cwd

( (S−Swr )(1−Swr )

)awd(7.1)

The parameters used here are:cwd is the threshold pressure, S is the watersaturation,Swr is the irreducible water saturation. The parameterawd is the in-verse of thepore-size distribution index, and is a parameter related to the specificsandstone. Further details are given in the paper and it’s references. The expres-sion (7.1) is the foundation for the generation of general capillary pressure curvesfor media with varying wettability. The wettability of a sandstone says somethingabout the preferred ’glueing/clinging’ of water to the sandstone compared to thesame effect for oil. Sandstones that are classified as mixed-wet, will contain bothpores that are water-wet and oil-wet. From the discussions in [53] and [56], thisimplies capillary pressure curves with both positive and negative capillary pres-sure. For details we refer to [53] and [56] and references herein.

The capillary pressure curves used in Paper D will be combinations of twoterms of the form (7.1), where one of the terms will be the capillary pressuredescription in a completely water-wet system, and the second term will be thecapillary pressure description in a completely oil-wet system. The ’bounding’secondary imbibition and drainage curves will be on the form

Pc(S) =cw

( (S−Swr )(1−Swr )

)aw+

co

( (1−S−Sor)(1−Sor)

)ao(7.2)


Thea’s andc’s are constants, and there is one set each for imbibition curvesand drainage curves respectively. Irreducible water saturation is denotedSor. Thea’s are all positive numbers,cw is a positive number, whereasco is a negativenumber. The curves will crossPc = 0 for some saturation value between the ir-reducible water saturation and the irreducible oil saturation. Furthermore, thecapillary pressure curves will have asymptotes atSwr andSor.

A hysteresis loop logic is introduced in [53], and this allows for arbitraryimbibition and drainage capillary pressure curves to be defined. Furthermore,similar logics will apply for relative permeability, so that our numerical modelwill include a complete description of both capillary pressure and relativepermeability. In paper D, all examples initialise the reservoirs by primarydrainage curves given by Eq. (7.1). The initial saturation distribution in thereservoir is hence a function of height, and the gradual change in saturationdefines the verticaltransition zonebetween oil and water. Water flooding ismodelled by using a water injector at the bottom of the reservoir, and an oilproducer at the bottom of the reservoir. After a specific amount of fluids has beeninjected/produced, the wells will be shut off, and the redistribution of the fluids ismodelled. Different rate regimes are investigated, and theredistribution is highlydependent on the state at the time of well shut off. A comparison was also madewith a hysteresis option available in the commercial simulator Eclipse. The mainresults of the paper are summarised below.


• Inclusion of consistent capillary pressure and relative permeability modelsin forward reservoir simulator

• Numerical handling of solution dependent curves with non-smooth derivat-ives in implicit solver

• Investigation of examples where processes change directions throughoutsimulations

• Comparison of new hysteresis model and hysteresis model available in com-mercial simulator.

Chapter 8

Further work

The work of Paper A dealt with symmetry and M-matrix issues for the MPFAO-method on a polygonal grid. This only partially answers the question about themonotonicity of the method. As mentioned in Chapter 6, analysis has recentlybeen done for the O-method on 2D structured grids in regards to monotonicityissues. These ideas can possibly be extended to expand the analysis of the O-method on polygonal grids. As a starting point one should consider polygonalgrids with set patterns, so that there is a certain structureof the elements of thecoefficient matrix. A natural case to analyse is layered media.

The work with faults in 3D is by no means finished. The paper which dealswith faults for quadrilateral grids with crossing grid lines (Paper C) was initiallystarted because of specific field needs. However, we had certain problems withthe first version of the code that accounts for these types of faults. At the time ofwriting this thesis, we have not been able to successfully simulate flow for realfield cases. This is one of the issues that we will work on at a later stage. Wethen hope to be able to do field studies of a field with corresponding reservoir/gridmodel depicted in Figure 8.1.

In addition to this, there are unresolved issues with the transmissibility calcu-lations. Because of the special character the local system of continuity equationsto determine transmissibilities has, additional principles had to be used to determ-ine the transmissibilities. As the discussion in the paper reveals, one degree offreedom was left undetermined even after the principle of exact fluxes for linearflow was used. The underlying local system is still subject tomore analysis, andmore information may be used to bind the last degree of freedom.

The work of Paper D dealt with numerical convergence results. So far, onlymatching quadrilateral grids have been tested, and most of the examples deal with2D examples. Since the MPFA methods have been developed for general faults,this may certainly be used to perform tests with local grid refinement. Resultsshould then be compared with those of Discontinuous Galerkin Methods and En-

58 Further work

1 473 945 1418 1890 2362 2834 3306 3779 4251 4723 5195 5667 6140 6612 7084

Cells

FloViz 2001A_1

Figure 8.1: Extracted part of real field case with exaggerated faults.

hanced Mixed Finite Element Methods.Streamline issues are currently being considered for the MPFA O-method on

irregular grids [16]. Uniform flow results for the Control Volume Mixed FiniteElement Method have been presented in [31]. This method gives a continuousrepresentation of velocities within grid cells (interpolation). Because of complexgeometry, uniform flow will not be simulated correctly for all shape function [31],but the problem is overcome with different choices of shape functions. For theMPFA O-method, interpolation issues associated with bilinear cell surfaces havenot been considered yet. This is a field where we hope to do moreanalysis andnumerical tests in the future.

Bibliography

[1] I. Aavatsmark, T. Barkve, Ø. Bøe, T. Mannseth:Discretization on non-orthogonal, curvilinear grids for multi-phase flow, Proc. 4th European Con-ference on the Mathematics of Oil Recovery, vol. D, Røros, Norway (1994),17pp.

[2] I.Aavatsmark, T.Barkve, T.Mannseth, Ø. Bøe:Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media, J.Comput. Phys.127(1996), 2-14.

[3] I.Aavatsmark, T.Barkve, Ø.Bøe and T.Mannseth:Discretization on Unstruc-tured Grids for Inhomogeneous, Anisotropic Media. Part I: Derivation of themethods, SIAM J. Sci. Comput.19 (1998), pp. 1700-1716.

[4] I.Aavatsmark, T.Barkve, Ø.Bøe and T.Mannseth:Discretization on Unstruc-tured Grids for Inhomogeneous, Anisotropic Media. Part II:Discussion andnumerical results, SIAM J. Sci. Comput.19 (1998), pp. 1717-1736.

[5] I.Aavatsmark, T.Barkve, Ø.Bøe and T.Mannseth:A Class of DiscretizationMethods for Structured and Unstructured Grids in Anisotropic, Inhomogen-eous Media, Proc. 5th European Conference on the Mathematics of Oil Re-covery, Leoben, Austria, 1996.

[6] I. Aavatsmark, E. Reiso and R. Teigland:Control-volume discretizationmethod for quadrilateral grids with faults and local refinement, Computa-tional Geosciences5, 2001, pp. 1-23.

[7] I. Aavatsmark, E. Reiso, H. Reme, R. Teigland:MPFA for Faults and LocalRefinement in 3D Quadrilateral Grids With Application to Field Simulations,SPE 66356, Proceedings of SPE Reservoir Simulation Symposium, Hous-ton, Texas, 2001.

[8] I. Aavatsmark: An introduction to Multipoint Flux Approximations forQuadrilateral Grids, Computational Geosciences6, 2002, pp. 405-432.

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[10] G.T. Eigestad, I. Aavatsmark, E. Reiso, H. Reme, R. Teigland:MPFA meth-ods applied to irregular grids and faults, Developments in water science47;Editors Hassanizadeh, Schotting, Gray, Pinder.

[11] G.T. Eigestad, I. Aavatsmark, E. Reiso, H. Reme:MPFA for faults withcrossing grid lines and zig-zag patternsECMOR VIII, September 3-6.,2002.

[12] G.T. Eigestad, R. Klausen:A note on the convergence of the MPFA O-method; numerical experiments on some 2D and 3D gridsDraft manuscript.

[13] G.T. Eigestad, J. A. Larsen:Numerical modeling of capillary transitionzones, SPE 64376 APOGCE 2000, 16.-18. October 2002.

[14] G. T. Eigestad:On the Symmetry of a Discrete Operator; Discretizing anElliptic term by the O-method, Cand. Scient Thesis, Department of AppliedMathematics, University of Bergen, Norway.

[15] J. M. Norbotten, I. Aavatsmark:Monotonicity Conditions for ControlVolume Methods on Uniform Parallellogram Grids in Homogeneous Media.Submitted.

[16] H. Hægland:Streamline Tracing on Irregular Grids, Ongoing Master Thesisat Department of Mathematics, University of Bergen.

[17] M.G. Edwards and C.F. Rogers:A flux continuous scheme for the fulltensor pressure equation, Proceedings of the 4th European Conference onthe Mathematics of Oil Recovery, Norway, June 1994.

[18] M.G. Edwards and C.F. Rogers:Finite volume discretization with im-posed flux continuity for the general tensor pressure equationComputationalGeosciences2, 1999.

[19] A.F. Ware, K. Parrott, C. Rogers:A Finite Volume Discretization for PorousMedia Flows Governed by Non-Diagonal Permeability TensorsProceedingsof CFD95, Banff, Canada, 1995.

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Part II

Published and submitted work

Paper A

Symmetry and M-matrix issues forthe O-method on an UnstructuredGrid ∗

A person who is looking for something is not travelling very fast

In Stuart Little by E. B. White

∗ Published in Computational Geosciences, vol. 6, 2002, pp. 381–404.

Paper B

MPFA applied to Irregular Grids andFaults ∗

One day I shall reach the summit, and see how small the other moun-tains are

Du Fu, Chinese Philosopher

∗ Published in Developments in Water Science47, Elsevier, 2002, pp. 413-420.

Paper C

MPFA for Faults with CrossingLayers and Zig-zag Patterns

I was programmed for etiquette, not destruction

3CPO, in Star Wars Episode II

Paper D

A note on convergence of the MPFAO-method; numerical experimentson some 2D and 3D grids

An idea that remains an idea is not a good idea

Unknown

Paper E

Numerical Modeling of CapillaryTransition Zones

Wer reitet so spat durch Nacht und Wind? Es ist der Vater mit seinemKind

Goethe

geir terje eigestad - semantic scholarpreface this thesis constitutes the work that i have done...

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