Mathematical models and finite elements for reservoir simulation: single phase, multiphase, and multicomponent flows through porous media

MATHEMATICAL MODELS AND FINITE ELEMENTS FOR RESERVOIR SIMULATION Single Phase, Multiphase and Multicomponent Flows thro...

Author: Guy Chavent | Jerome Jaffre

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MATHEMATICAL MODELS AND FINITE ELEMENTS FOR RESERVOIR SIMULATION Single Phase, Multiphase and Multicomponent Flows through Porous Media

STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 17

Editors: J . L. LIONS, Paris G. PAPANICOLAOU, New York H. FUJITA, Tokyo H. B. KELLER, Pasadena

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD .TOKYO

MIATHEMIATICAL MODELS A N D FINITE ELEMENTS FOR RESERVOIR SIMULAmON Single Phase, Multiphase and Multicomponent Flows through Porous Media GUY CHAVENT CEREMADE Universitk Paris Dauphine Institut National de Recherche en Informatique etAutomatique Paris, France

JEROME JAFFRE Znstitut National de Recherche en Informatique et Automatique Paris, France

1986 NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD *TOKYO

0

Elsevier Science Publishers B.V., 1986

All rights reserved. N o part of this publication may be reproduced, stored in a retrieval sysiem. or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 70099 4

Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1Y91 1OOOBZ AMSTERDAM THE NETHERLANDS Sole distributors for the U.S.A . and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52VANDERBILT AVENUE NEW YORK, N.Y. 10017 U.S.A. PRINTED IN THE NETHERLANDS

Library of Congress Catalogingin-PublicationData

Chavent, Guy, 1943Mathematical models and f i n i t e elements for reservoir simulation. (Studies in mathematics and its applications ; v. 17) Bibliography: p. 1. Oil reservoir engineering--Mathematical models. 2. Fluid dynamics-Mathematical models. 3. Finite element method. I. Jaffri, J6r6me. 11. Title. 111. Series. TN671.Ch54 1966 622l.1682 86-16830 ISBN 0-444-70099-4 (U.S. )

PRINTED IN THE NETHERLANDS

V

PREFACE During the last twenty years, many numerical simulators for oil reservoirs have been developed. They are now widely used by the oil companies and they include refined physical and thermodynamical effects. However, this research has taken place primarily within the oil companies, and has been thus somewhat inaccessible to the scientific community. The obstacle to communication has not been any kind of confidentiality, but has rather come from the specialized language used in the oil industry, together with an increasingcomplexity of the physical models underlying the numerical simulators. So one aim of this book is to try to remedy t h i s situation and make the models used in res-

ervoir simulation understandable by the non-specialized scientific community. By understandable we mean not only that the models can be recognized by the mathematical reader as being of a known type of equation, but also that the physicaleffectscorresponding to each term are well identified. This goal is achieved by using a synthetic presentation of all models using the new feature of the ‘global pressure’, which enables us to write all models in the form of one pressure equation coupled with one or several saturation or concentration equations.

In order to complete this synthetic presentation of the main models used in reservoir simulation, which are all based on Muskat’s generalization of Darcy’s law, we present in the first chapter the models which are used in other application areas (Richards equation for unsaturated soils, Baiocchi’s dam model, etc ...) and show that they are all in fact special cases of the Muskat model. A second objective of this book is to initiate a rigorous mathematical study of the immiscible flow models. Though the existence theorems presented in this book are far from

covering the most general multiphase problem, we believe that our treatment of incompressible two-phase problems, based on the above-mentioned ‘global pressure’ approach, is the most comprehensive today. It is the hope of the authors that the other models presented in this book, for which no existence theorems are given, such as compressible, three-phase, black-oil, or compositional models, will suggest further research to some of the readers and will be an incentive for further research in this area. The last objective of this book is to present a finite element approximation technique based on the global pressure variational model. The goal is to show how new numerical techniques can be used in reservoir simulation, but it is not to review all methods of discretization (for fmite differences, see the books by Peaceman and by Aziz and Settari). We consider the case of two-phase incompressible flow and the method includes mixed fmite elements for the pressure equation and upstream weighted discontinuous finite elements with slope limiters for the saturation equation.

vi

Preface

The general plan of the book is as follows. In chapter I, we present the basic laws and various models for fluid flow through porous media, and the relation between these models. Chapter I1 deals with slightly compressible monophasic fields. Chapter I11 is devoted to two-phase incompressible displacements; the global pressure is introduced and the model is studied mathematically. Chapter IV generalizes the notion of the global pressure to compressible, three-phase, black-oil and compositional models. Chapter V presents a f d t e element method for two-phase incompressible flow. Chapters I through IV have been written by G. Chavent and chapter V by J. Jaffrk.

Let us state that a large part of the book - chapters I and 11, the second part of chapter 111, chapter W ,and part of chapter V - present original material, which has never been presented elsewhere. Finally, we hope that t h i s book will be of interest for both applied mathematicians, who will find here an introduction to the reservoir simulation area including various ready-tothink-about mathematical and numerical models, and for reservoir engineers involved in numerical simulation; who will find here an alternative approach to their usualview of reservoir modeling. We express many thanks to Brigitte Marchand for her patient typing of the manuscript in a language foreign to her, and to Jean Roberts who equally patiently proofread the manuscript, corrected our shaky english and suggested many improvements. We express also our gratitude to Gary Cohen whose collaboration was very helpful to us. We are also indebted to P. Lemmonier, D. Guerillot and L. Weill from the Institut Franqais du Pktrole, and to G. Barrb, R. Eymard and J.L. Porcheron from the Soci6tC Nationale Elf Aquitaine (Production) for the many sthmlating discussions we have had with them and for the support they expressed for our work.

Vii

CONTENTS

Preface

......................................................

v

CHAPTER I: BASIC LAWS AND MODELS FOR FLOW m POROUS MEDIA. . . . . . .1 I.

GENERALlTIES..........................................

1

I1.

THE GEOMETRY OF THE FIELD

.............................

4

THE BASIC LAWS FOR ONE-AND TWO-PHASE FLOW . . . . . . . . . . . . . .8 The Darcy Law for One-Phase Flow ............................. 8 111.2. The Case of Fully Miscible Flows .............................. 11 111.3. Two-Phase Immiscible Flow ................................. 12 111.3.1. The Muskat Relative Permeabilities Model ........................ 12 111.3.2. The Capillary Pressure Law .................................. 15

111. 111.1.

IV. IV.l. IV.2. IV.3. IV.3.1. IV.3.2. IV.3.3. IV.3.4. IV.4.

THEBASICMODELS ..................................... 17 The Monophasic Model..................................... 17 The Fully Miscible Model ................................... 17 The Two-Phase Immiscible Model .............................. 19 The Relative Permeabilities Capillary Pressure (RPCP) Model . . . . . . . . . . . .19 The Muskat Free Boundary Model ............................. 22 The Richards Approximation................................. 27 The Baiocchi Free Boundafy Model 32 Summary of the Different Models 34

V.

QUALITATIVE BEHAVIOR OF THE SOLUTION IN THE NO-DIFFUSION AND NOCAPILLARY PRESSURE CASE . . . . . . . . . .36 The Miscible or Immiscible Model.............................. 36 Behavior of One-Dimensional Solutions .......................... 39 The Miscible Case ........................................ 40 The Immiscible Case ...................................... 42 Behavior of Two-Dimensional Miscible Solutions .................... 45 . Behavior of Two-Dimensional Immiscible Solutions. . . . . . . . . . . . . . . . . 46

V.l. V.2. V.2.1. V.2.2. V.3. V.4.

............................ ..............................

CHAPTER II: SLIGHTLY COMPRESSIBLE MONOPHASIC FIELDS............ 51

I.

CONSTRUCTION OF THE PRESSURE EQUATION

.................51

viii

Contents

....................

11.

EXISTENCE AND UNIQUENESS THEOREMS

111.

AN ALTERNATIVE MODEL OF MONOPHASIC WELLS .............65 An Exactly Equivalent Representation of Wells by Source or Sink Terms . . . .67 An Approximately Equivalent Representation of Wells by Source or SinkTerms ........................................... 70

III.1. III.2.

56

CHAPTER Ill: INCOMPRESSIBLE TWO-PHASERESERVOJRS. . . . . . . . . . . . . . .89 I.

.

I1 11.1. 11.2. II.3. 11.3.1. 11.3.1.1. 11.3.1.2. 11.3.2. 11.3.2.1. 11.3.2.2. 11.3.2.3.

INTRODUCTION ........................................

89

CONSTRUCTION OF THE STATE EQUATIONS................... 92 The Equation Inside a:The Notion of Global Pressure................93 The Pressure Boundary Conditions ............................ 102 108 The Saturation Boundary Conditions .......................... Saturation Boundary Conditions on the Injection Boundary r- . . . . . . . . . 109 Dirichlet Condition ...................................... 109 Given Water Injection Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110 Saturation Boundary Conditions on the Production Boundary I’+ . . . . . . . 112 Dirichlet Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Unilateral Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Given Water/Oil Production Ratio (WOR) ....................... 116

I11.

SUMMARY OF EQUATIONS OF TWO-PHASE FLOWS FOR INCOMPRESSIBLE FLUIDS AND ROCK ..................... 117 111.1. Characteristics Depending Only on the Reservoir 0 . . . . . . . . . . . . . . . . . 117 117 III.2. Physical Unknowns ....................................... 117 111.3. Characteristics Depending Only on the Fluids ..................... Characteristics Depending Both on Fluids and Rock . . . . . . . . . . . . . . . . . 118 111.4. 119 111.5. Auxiliary Dependent Variables............................... 111.6. Traces on r = as2 of the Dependent Variables..................... 119 111.7. Partitions of the Boundary r of the Porous Medium S2 . . . . . . . . . . . . . . . 120 Functions and Coefficients Depending on Reduced Saturation S Only . . . . . 120 111.8. 121 III.9. Main Dependent Variables.................................. 121 ni.10. Equations for Pressure. Saturation and Flow Vectors

................

IV.

AN ALTERNATIVE MODEL FOR DIPHASIC WELLS . . . . . . . . . . . . . . 125

V.

MATHEMATICAL STUDY OF THE INCOMPRESSIBLE TWO-PHASE FLOWPROBLEMS .................................... 130 Setting of the Problem .................................... 131 135 Variational Formulations .................................. 139 Some Preliminary Lemmas ................................. Resolution in the NonDegenerate Case . . . . . . . . . . . . . . . . . . . . . . . . .147 Resolution in the Degenerate Case ............................ 155

V.l. V.2. V.3. V.4. V.5.

Contents

ix

V.6. V.6.1. V.6.2.

The Case of Decoupled Pressure and Saturation Equations . . . . . . . . . . . . 157 Regularity and Asymptotic Behavior for the Non-Degenerate Case . . . . . . . 159 Regularity and Asymptotic Behavior for the Degenerate Case . . . . . . . . . . 171

.

THE CASE OF FIELDS WITH DIFFERENT ROCK TYPES . . . . . . . . . . .177 The Different Rock Models ................................. 178 The Case of a Field with M Different Rock Types . . . . . . . . . . . . . . . . . .182

V1 VI.1. VI.2.

CHAPTER IV:GENERALIZATION TO COMPRESSIBLE. THREE.PHASE. BLACK OIL OR COMPOSITIONAL MODELS . . . . . . . . . . . . . 189

I. 1.1. 1.1.1. 1.1 2 1.1.3.

.

1.2.

1.2.1. 1.2.2. 1.2.3. 1.2.4. 1.3.

THE TWO-PHASE COMPRESSIBLE MODEL. .................... 189 Equations from the Physics ................................. 190 190 Conservation Laws....................................... Muskat Law (Relative Permeabilities) .......................... 191 191 Capillary Pressure Law .................................... Simplifying Hypotheses ................................... 191 Pressure Dependent Coefficients.............................. 191 Choice of a Rock Model ................................... 192 Limitation of the Pressure Range 194 Summary of the Resulting Equations .......................... 195 The Global Pressure Equation ............................... 196 The Saturation Equation ................................... 198 Summary of the Two-Phase Compressible Model . . . . . . . . . . . . . . . . . . . 199 The Case of Slightly Compressible Rock and Fluids . . . . . . . . . . . . . . . . .202

.............................

1.4.

1.5. 1.6.

I1.

THE THREE-PHASE COMPRESSIBLE MODEL . . . . . . . . . . . . . . . . . . . 203 Equations from the Physics ................................. 204 Conservation Law ....................................... 204 Muskat Law (Relative Permeabilities) 204 Capillary Pressure Law .................................... 205 Simplifying Hypotheses ................................... 205 Pressure Dependent Coefficients.............................. 205 Choice of a Rock Model ................................... 206 The “Total Differential“ (TD) Condition on the Rock Model........... 209 Limitation of the Pressure Range ............................. 213 Summary of the Resulting Equations 214 The Global Pressure Equation 214 216 The Saturation Equation 11.4. 11.4.1. Determination of the Equations .............................. 216 11.4.2. A Hyperbolicity Condition ................................. 218 Construction of Three-Phase Data Satisfying the TD Condition (2.24). . . . . 220 11.5. 220 11.5.1. The Practically Available Data ............................... II.5.2. Continuation of Capillary Pressures 222 11.5.3. Continuation of Relative Permeabilities 222

11.1 11.1.1. 11.12 11.1.3. 11.2. 11.2.1. 11.2.2. 11.2.3. 11.2.4. 11.2.5. II.3. I

.

..........................

.......................... ............................... ...................................

............................ .........................

Contents

X

Numerical Algorithm for the Computation of TD Three-Phase 225 Relative Permeabilities 229 11.5.5. Examples of TD Three-Phase Data ............................ 234 11.5.6. The Hyperbolicity Condition ................................ Summary of the Three-Phase Compressible Model . . . . . . . . . . . . . . . . . .235 11.6. II.5.4.

.

111

III.1. 111.2. III.3. 111.4. 111.5.

111.6. 111.7.

111.8.

IV. IV.1. IV.2. IV.3. IV.3.1. IV.3.2. IV.3.3. IV.3.4. IV.3.5. IV.3.6. IV.3.7. IV.4. IV.4.1. IV.4.2. IV.4.3. IV.5. IV.5.1. IV.5.2. IV.6. IV.7. IV.8. IV.8.1. IV.8.2. IV.8.3. IV.8.4.

..................................

THE BLACK OIL MODEL ................................. Range of Validity Components and Phases ................................... Description of Phases Equili%rium ............................ Description of Phases Characteristics........................... Governing Equations from the Physics Global Pressure and Pressure Equation .......................... Saturations/Dissolution Factors Equations ....................... Summary of the Black Oil Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.......................................

..........................

237 237 238 239 242 243 244 250 253

A COMPOSITIONAL MODEL ............................... 255 Range of Validity 255 Description of the Thermodynamic EquiLiirium ................... 256 Description of Phase(s) Characteristics.......................... 260 In the One-Phase Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 In the Two-Phase Domain (General) . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Compositional Residual Saturations ........................... 262 Compositional Relative Permeabilities .......................... 263 Compositional Capillary Pressure ............................. 267 Hypothesis on Pressure Dependent Data ........................ 269 An Example of Compositional Two-Phase Data .................... 270 Governing Equations from the Physics 274 In the One-Phase Domain .................................. 274 In the Two-Phase Domain 275 Matching of TwoPhase and One-Phase Equations . . . . . . . . . . . . . . . . . .276 Introducing a Global Pressure ............................... 277 Some Preliminaries ...................................... 277 Defmition of the Global Pressure P ............................ 280 The Global Pressure Equation ............................... 284 The Concentration Equations ............................... 290 Regularity of the Equations ................................. 291 Coupling Between Pressure and Concentration Equations ............. 294 Some Preliminaries for the Study of the Regularity . . . . . . . . . . . . . . . . .295 Regularity of the Pressure Equation ........................... 300 Regularity of the Concentration Equation . . . . . . . . . . . . . . . . . . . . . . .304

.......................................

.......................... ..................................

CHAPTER V: A FINITE ELEMENT METHOD FOR INCOMPRESSIBLE TWO-PHASE FLOW ..............................

311

I.

3 11

INTRODUCTION .......................................

xi

Conrenrs

....................................

311 313 314

1.1. 1.2. 1.3.

Introductory Remarks Equations of Incompressible Two-Phase Flow ..................... Discretization ..........................................

I1.

APPROXIMATION OF THE PRESSURE-VELOCITY EQUATIONS .....315 Approximation Spaces .................................... 315 Approximation Equations .................................. 318

11.1. 11.2.

.

111

111.1. III.2. 111.3. N

.

Iv.l. IV.2. N.3. N.4. Iv.5.

.

V

v.l. v.2.

VI. vI.1. vI.2. vI.3. VI 4. vI.5.

RESOLUTION OF THE ALGEBRAIC SYSTEM FOR 322 PRESSURE-VELOCITY ................................. Introduction ........................................... 322 The Mixed-Hybrid Formulation of the Pressure-Velocity Equations ...... 323 The Algebraic System Derived from the MixedHybrid Formulation 326

......

APPROXIMATION OF THE ONE-DIMENSIONAL SATURATION EQUATION: THE CASE WITH NEITHER CAPILLARYPRESSURE NOR GRAVITY ...................................... 330 330 Introduction ........................................... A General Discontinuous Finite Element Scheme ..................332 The Case k = 0: Piecewise Constant Approximation .................335 The Case k = 1 : Piecewise Iinear Approximation. . . . . . . . . . . . . . . . . . . 335 A Slope Limiter ........................................ 338 APPROXIMATION OF THE ONE-DIMENSIONAL SATURATION 345 EQUATION IN THE GENERAL CASE ........................ 345 The Gravity Effects ...................................... The Capillary Pressure Effects . . . . . . . . . . ..................... 347 APPROXIMATION OF THE SATURATION EQUATION IN TWODIMENSIONS.................................... Approximation Spaces .................................... Approximation Equations Integration Formulas A Slope Limiter Some Theoretical Results

.................................. ..................................... ........................................ . .................................. NOTES AND REMARKS .................................. vII. VII.1. The Pressure Equation .................................... VII.2. The Saturation Equation ................................... VII .3. The Coupled System ..................................... REFERENCES...............................................

355 355 356 358 359 362 362 362 363 364

365

This Page Intentionally Left Blank

1

CHAPTER BASIC

LAWS

AND

MODELS

FOR

I FLOW

IN

POROUS

MEDIA

I-GENERALITIES

An oil reservoir is a porous medium, whose pores contain Some hydrocarbon components, usually designated by the generic term ''oil". The porous medium is often heterogeneous, which means that the rock properties may vary from one place to another. The most heterogeneous oil fields are the so-called "fractured oil fields", which consist of a collection of blocks of porous medium separated by a net of fractures. In such a fractured reservoir, rock properties such as permeability may vary from 1 (in the blocks) to 10 000 (in the fractures). Though such reservoirs are governed in essence by the same equations as those with slowly varying properties, they present some additional specific properties, which can be studied by the so-called "homogeneizationTttechnique. Homogeneization is a tool for studying the mean properties of solutions of partial differential equations having periodic coefficients. For a general presentation of homogeneization techniques, we refer to SANCHEZ-PALENCIA, BENSOUSSAN-LIONSPAPANICOLAOU, and to BOURGEAT for an application to a two-phase fractured reservoir. The mathematical models developped in this book will take into account the heterogeneity of the porous medium, (and hence cover the case of fractured reservoirs), but the numerical models of chapter V will be derived primarily for non-fractured reservoirs. One other important characteristic of an oil reservoir is the nature of the fluids filling the porous medium, which influences Strongly the underlying mathematical model. The simplest case is that of a monophasic oil field in which the whole porous medium is filled with a single fluid (usually gas or Oil). Such fields can be found among fields in their very early stage of development,

when

the

gas

or

oil

is

produced

by

simple

natural

decompression. This monophasic stage of the field ends rapidly, when the pressure equilibrium between the oil field and the atmosphere is attained :

Ch. I: Basic Laws and Models for Flow in Porous Media

2

t h e n a t u r a l p r o d u c t i o n o f o i l o r g a s s t o p s , though o n l y a small p e r c e n t a g e of t h e t o t a l amount of o i l o r gas h a s been produced. T h i s f i r s t s t a g e is called

"primary

recovery"

in

the

technical

literature,

and

the

c o r r e s p o n d i n g mathematical model w i l l be s t u d i e d i n c h a p t e r 11. I n o r d e r t o r e c o v e r p a r t of t h e remaining o i l , one could t h i n k of pumping o f f a t t h e w e l l s , c r e a t i n g a p r e s s u r e drop which would draw t h e

O i l

t o t h e t h e s e wells. T h i s would have two draw-backs : f i r s t , t h e p r e s s u r e around t h e wells could f a l l below t h e bubble p r e s s u r e of t h e o i l , s o t h a t t h e wells would produce almost o n l y g a s , and t h e h e a v i e r components would mainly remain t r a p p e d i n t h e f i e l d . Second, d i m i n i s h i n g t h e p r e s s u r e i n t h e f l u i d phase could

l e a d t h e rock t o c o l l a p s e , r e s u l t i n g i n a f i e l d w i t h

lower p e r m e a b i l i t y and hence more d i f f i c u l t t o produce, n o t t o speak of t h e s u b s i d e n c e phenomenon which could be f e l t a t t h e e a r t h ' s s u r f a c e . This

is

why

oil

engineers

use

an

alternative

method

called

"secondary r e c o v e r y " : t h e y d i v i d e t h e a v a i l a b l e wells i n t o two sets : one

set of

i n j e c t i o n wells,

and one s e t of

p r o d u c t i o n wells. The i n j e c t i o n

w e l l s are t h e n used t o i n j e c t an i n e x p e n s i v e f l u i d ( u s u a l l y water) i n t o t h e Porous medium, i n o r d e r t o push t h e o i l toward t h e p r o d u c t i o n wells. During t h i s p r o c e s s , t h e p r e s s u r e i n s i d e t h e f i e l d is m a i n t a i n e d above its i n i t i a l level,

so t h a t t h e two above mentionned draw backs may be more e a s i l y

avoided. For

this

secondary

recovery

process,

two

cases

are

to

be

considered :

- E i t h e r t h e p r e s s u r e can be m a i n t a i n e d always above t h e bubble p r e s s u r e of t h e o i l : t h e flow i n t h e r e s e r v o i r is t h e n of t h e two-phase immiscible t y p e , one phase b e i n g water and t h e o t h e r b e i n g o i l , w i t h no mass t r a n s f e r between t h e phases. T h i s c a s e w i l l be s t u d i e d e x t e n s i v e l y i n t h i s book, both from t h e mathematical p o i n t of view i n c h a p t e r I11 and from t h e numerical p o i n t of view i n c h a p t e r V .

- Or

t h e p r e s s u r e may d r o p ,

p r e s s u r e of

the o i l

phase)

split

may

:

into

then t h e o i l one

liquid

a t some p o i n t s ,

below t h e bubble

( o r more p r e c i s e l y t h e hydrocarbon phase

and

one

gaseous

phase

at

thermodynamical e q u i l i b r i u m . T h i s is t h e s o - c a l l e d " b l a c k - o i l q f r e s e r v o i r , w i t h one water phase, which does n o t exchange mass w i t h t h e o t h e r phases, and two hydrocarbon phases (one l i q u i d phase and one gaseous p h a s e ) , which exchange mass when t h e p r e s s u r e and t e m p e r a t u r e change. The c o r r e s p o n d i n g model w i l l be g i v e n i n s e c t i o n I11 of c h a p t e r I V . The above w a t e r f l o o d i n g t e c h n i q u e makes i t p o s s i b l e t o r e c o v e r a

I. Generalities

3

certain percent (up to 40% in the very good cases) of the oil contained in the field. There are three main reasons for this low figure of recovery : first, there exists regions which are never flooded by the water, and hence whose oil is not going to be produced; second, even in the regions which have been completely flooded by water, a non negligible part of the oil (up to 20 to 30) percent) remains trapped in the pores by the action of the capillary forces

:

the oil saturation never goes below the so-called residual oil

saturation when only displacement techniques such as water flooding are used (see paragraph 111.3.1

of this chapter);

third, when the oil is heavy and viscous, the water is extremely mobile in comparison to the oil. Then, instead of "pushing" the oil towards the production well, the water finds very quickly its own way to the production well, getting the oil to move only very slowly toward the production well (see paragraphs V . 3 then

starts

to

produce water

and V . 4

very

early

in this chapter).This latter and

in

quickly

increasing

proportions, and has to be turned off for economical reasons.

In order to go beyond the above level of recuperation, the oil industry develops now a set of different techniques known under the generic name of "tertiary recovery techniques" or "enhanced recovery techniques". One of the main goals of thoses techniques is to achieve miscibility of the fluids, thus eliminating the residual oil saturation, which was one cause of low recovery with the water flooding technique. This miscibility is sought using temperature increase (in-situ combustion techniques, which also yield a mobility increase of the fluids) or the introduction of other (usually

expensive)

components,

as

certain

polymers,

which

yield

miscibility of oil and water when in the right proportions. Similarly, miscibility of the gas and liquid phases in a black-oil type flow may be restored by addition of a medium weight hydrocarbon component in adequate proportion. So one typical situation for tertiary recovery is the so-called

"partially miscible flow" or "compositional flow", where only the number of chemical components is a priori given. The number of phases, and the composition of each phase in terms of the given components, depend on the thermodynamical

conditions

(temperature,

pressure)

and

the

overall

concentration of each component. Such flows will be described in more

Ch.I: Basic Laws and Models for Flow in Porous Media

4 d e t a i l i n s e c t i o n I V of c h a p t e r I V .

One can n o t i c e t h a t one p r a c t i c a l l y never e n c o u n t e r s i n r e s e r v o i r s i m u l a t i o n p r a c t i c e f u l l y m i s c i b l e f l o w s ( a s would be f o r example t h e c a s e f o r water and s a l t water). However, s u c h models share some s i m i l a r i t y with t h e p a r t i a l l y m i s c i b l e f l o w models, and hence a r e u s e f u l f o r t h e d e s i g n and t h e t e s t of numerical methods. chapter

i n paragraph IV.2,

Moreover, as w i l l be p o i n t e d o u t i n t h i s

the

f u l l y miscible e q u a t i o n s t u r n o u t t o be

( a l m o s t ) a s p e c i a l c a s e of t h e two-phase immiscible e q u a t i o n s , which w i l l be developped i n c h a p t e r 111. 11-

THE

GEOMETRY

OF

-

W e shall consider a f i e l d Q boundary

-

an,

from which the

K s m a l l domains

D,

O i l

,...,DK

in

E

THE

mn, n

=

FIELD 1,2,

or 3 with exterior

is produced through K wells, r e p r e s e n t e d by 0.

One can t h i n k of t h o s e domains

b e i n g s m a l l d i s k s ( f o r a two-dimensional

DK as

" h o r i z o n t a l " f i e l d as shown i n

f i g u r e 1 ) o r small c y l i n d e r s ( f o r a t h r e e d i m e n s i o n a l f i e l d ) . In any c a s e we s h a l l d e n o t e by

aDK

t h e boundary of

DK, J = 1 , 2 , . . . , K .

Figure 1 : The o i l f i e l d Let u s d e n o t e by

g i v e n by

-

(2.1)

Q = Q

t h e porous medium i t s e l f , which is o b v i o u s l y

K

- u

k- 1 As

R

t h e boundary

Dk

aDk

Of t h e domains

Dk

may have non void

I

i n t e r s e c t i o n s w i t h the f i e l d boundary

an,

t h e boundary

30

of t h e porous

5

II. The Geometry of the Field

medium

0

is not exactly

rk

=

aD

r,

=

ai

k

K aiiU U j=1

- aDk

ai n

(boundary of the porous medium in contact with kth well), k=l, 2,.

K (

u

medium),

so that we get a partition of the boundary

K

(2.3)

=

an

=

r u( u r k )

'

k=l

., K ,

("lateral" boundary of the porous

aDk)

k= 1

r

:

J

n ai

(2.2) -

Let us define

aD..

with

I

r of

the porous medium 0 :

,...K,

r, n r J.

=

0,

j

r . n rk

=

0,

j , k = 1,2

J

=

1,2

,...K,

j

f

k.

This appears clearly in the two examples of figures 2 and 3.

Figure 2 : 2-Dnhorizontal~1mode1

Figure 3 : 2-Dnvertica11~model

The effect of the gravity beeing not always negligible, we have now to specify the position of our field with respect to the depth. We define for this purpose

:


6

Z(x)

=

depth of the point x

g

=

gravity acceleration.

E

a,

Examples are shown in figures 4 and 5.

Figure 4 : Z ( x ) for a 1-D model

Figure 5 : Z ( x ) f o r a 2-D "vertical

"

model.

The description of our field is now completed, except in the case of

one o r

two-dimensional models, for which we have to specify the

remaining dimensions. So we introduce a function section" of the field at point x) such that o(x) dx

(2.5) Hence

=

o(x)

("generalized

:

3-D volume of the element dx

=

dx, dx2---dxn-

: *

section of the field at abscissa x for 1-D

domains as in figure 4, *

o(x)

(2.6)

thickness of the field at point (x,y) f o r

2-D "horizontal" models as in figure 2,

=

*

width of the field at point (x,z) for 2-D

"vertical" models as in figure 3,

-

At every point -i

denote by v the normal to gravity acceleration.

s

r

1 for 3-D models.

of the boundary r of the porous medium R , we pointing out of R. We shall denote by g the

II. The Geomehy of the Field

Remark 1 :

One

7

often

uses,

for

local

studies

around

denote by

r

a

well,

Oz. If we

axisymmetric models around a vertical axis

the distance to the axis of symmetry, we

obtain from ( 2 . 5 ) with the notations of figure 6

:

21rrH for 1-D axisymmetric models (2.7)

(J(x) =

2nr

for 2-D "R-Z" axisymmetric models.

1-D M o d e l : x = r

' 2-D " R - Z " M o d e l

Figure 6 : Axisymmetric models

:

x=(r,z)


8

111-

THE

B A S I C

LAWS

FOR

ONE-AND

TWO-PHASE

FLOW

All the laws we are going to consider will be valid at a so-called "first macroscopic levell', i.e. for a volume of porous medium which is infinitely large with respect to the size of the fluid particles and of the pores, but which can be infinitely small with respect to the size of the field itself. At this first macroscopic level, the rock properties may vary continuously with the position x $(x,P)

2 0

P (0 s

$(XI

K(x) > 0

Q :

is the porosity at point x

Q and at pressure

5 1).

is the permeability at point x

a (we suppose for

simplicity that the porous medium is isotropic everywhere,

(3.1)

but anisotropicity can be taken into account by replacing the positive scalar K(x) by a symmetric positive definite matrix K(x)), cR(X) 5 0 is the compressibility of the rock a (c, = ap LOg$(x,P) evaluated at some reference pressure).

-

On the contrary, at the microscopic example takes only the values 0 and 1 ,

level, the porosity for

and the fluid(s)

follow the

Navier-Stokes equation. Some attempts have been made to derive the macroscopic equations from the equations at the microscopic level by some averaging procedure : MATHERON studied in his book the case of monophasic flow and obtained a justification of the experimental Darcy law; MARLES [ 2 ] sought macroscopic equations Of multiphase flow in porous media, and obtained equations which contain, as a special case, the experimental Muskat relative permeabilities model; FITREMAN devoted his study to the general case of multiphase flows, not necessarily restricted to the case of a porous medium. 1 1 1 . 1 - THE DARCY LAW FOR ONE-PHASE FLOW

The flow of a fluid in a saturated porous medium was studied by th century. His model, who was supported DARCy in the middle of the 19 later by other experimental studies and some theoretical considerations,

9

III. The Basic Laws for One- and Tko-Phase Flow

is basically valid when the inertial forces can be neglected for the determination of the motion of the fluid. If this is not the case, a generalization of Darcy's law has to be used ; see for example AMIRAT [21 for a mathematical study of such a case. We come back now to the study of Darcy' s law. Let us first define, at each point of a 3-D porous medium, a +

macroscopic apparent velocity 21 such that the volumetric flow-rate of the +

fluid through any surface + + 21." ds : + +

(3.2)

u - v ds

=

ds

with unit normal vector v is given by

volumetric flow rate (m3/s for example) through a surface ds

normal to unit vector

The macroscopic apparent velocity

+

+ V.

is also called the Darcy

21

velocity, or the seepage velocity. Inside the porous medium, the actual + macroscopic velocity is equal to u / b ( x ) where is the porosity. With this definition, the Darcy law can be written as : + = - -K(x) u

(3.3) where

u

[gradP - pg grad Z(x)

3,

:

K(x) is the permeability of the porous medium at point x to the fluid under consideration. As mentionned in (3.l), K(x) could be taken as a symmetric positive definite matrix, but for simplicity we will in this book always suppose that K(x) is a Positive scalar quantity, which corresponds to the hypothesis that the medium is isotropic. Moreover, K(x) depends on the nature of the fluid saturating the porous medium : if K (X) is for example the permeability to the water, then the permeability to the oil will be K =

k Kw(X)

where

k

(X)

is a given positive constant. The

permeability is homogeneous to a squared length unit of permeability is the Darcy.

:

the MKS

is the dynamic viscosity of the fluid. It is homogeneous to a mass/length/time; the MKS unit of viscosity is the Poise. P

is the pressure in the fluid; the MKS unit of pressure is the Pascal.

p

is the density (i.e. mass of unit volume) of the fluid.

g

is the gravity acceleration.

It is important to notice that the volumes used in the definition (3.2)

Of

1;

are evaluated at the pressure P existing in the fluid at the

point of the porous medium where

-t

21

is taken. As this pressure will vary


10

through the porous medium, these volumes cannot be used directly to write a conservation law when the fluid is compressible. So in order to comply with current use in oil reservoir engineering where volumes are prefered to masses

(oil is sold by barrils, not by tons...),

reference density

(3.4)

we will introduce a

:

density

=

'ref

of

the

fluid

under

some

reference

conditions and a volume factor

which will

enable

us to evaluate all volumes at the same reference

conditions. Moreover, as we have seen in paragraph 11, the domain Q in which we are going to write our partial differential equations does not always coincide with the porous medium itself, due to simplifying hypothesis on the shape of the field :Q can be a subset of IR or IRz or IR',

whereas the

porous medium itself is always a susbset of

IR'. So we will define at each point of $2 +

(with n=1,2 or 3 ) a

lRn

E

volumetric flow vector q Such that, for any (n-1)-dimensional surface ds + v

with unit normal vector

E

lRn,

the 3-D

volumetric flow-rate of the

fluid, evaluated at reference conditions, through the surface d s + + by q'v ds : + + q-v ds

(3.6)

is given

3-D volumetric flow rate, evaluated at reference conditions, through an (n-1) dimensional surface ds + n normal to the unit vector v E IR

=

.

For example, f o r 1 - D

models (n=l ) ,

+

q

is a scalar quantity

homogeneous to a 3-D volumetric flow rate (m3/s); for a 3-D model ( n = 3 ) , + then q is homogeneous to a velocity (m/s). From the definitions ( 2 . 5 ) , obtain immediately that + q

(3.7) i.e.

:

(3.8)

'

(3.2) and (3.6) of

:

+

=

=

oB(P) u

-'K ( x ) 'T

[gradP - p(P) g grad Z(x)].

+ o,

21

and

+ 9 we

III. The Basic Laws for One- and Two-PhaseFlow

"s

I t is t h e v e c t o r

t h i s book.

We s h a l l

11

rather t h a n

6

which w i l l be used throughout

as t h e ( v o l u m e t r i c ) flow

i n the sequel r e f e r t o

vector of t h e f l u i d , o f t e n omitting llvolumetricll. 111.2 - THE CASE OF FULLY MISCIBLE FLOWS

We suppose now t h a t t h e f l u i d s a t u r a t i n g t h e porous medium is obtained

by

the

mixing of

two m i s c i b l e components,

s a y 1 and 2.

The

composition of t h i s f l u i d can be d e s c r i b e d by i t s mass c o n c e n t r a t i o n C :

(3.9)

C

=

mass o f component 1 mass of component 1 + mass of component 2

Hence t h e d e n s i t y C,

and

if

we

choose

and v i s c o s i t y

p

some

reference

p

a r e now f u n c t i o n s of P and

conditions

(for

pressure

c o n c e n t r a t i o n ) . we s t i l l can d e f i n e a volume f a c t o r f o r t h e f l u i d

The Darcy law a p p l i e s t h e n t o t h e f l u i d

and

:

:

+

w i t h t h e v o l u m e t r i c flow v e c t o r q Of t h e f l u i d s t i l l d e f i n e d by ( 3 . 6 ) . Here + o f c o u r s e t h e volumes used f o r q a r e f i c t i t i o u s and do n o t correspond t o

any

physical

reality

:

they

are just

some e q u i v a l e n t way of

measuring

masses. One can a l s o d e f i n e , f o r t h e component 1 a l o n e , a " v o l u m e t r i c " +

i n a S i m i l a r way t o t h a t used i n ( 3 . 6 ) f o r t h e d e f i n i t i o n of flow v e c t o r ,$, + q. The f l u x of component 1 h a s two o r i g i n s : first,

component 1

is

f l o w i n g because t h e whole f l u i d is + + t o ,$1,

flowing; t h i s r e s u l t s i n a contribution C q second,

component

1

is

f l o w i n g because t h e molecular and

t u r b u l e n t d i f f u s i o n t e n d t o e q u a l i z e t h e c o n c e n t r a t i o n p r o f i l e through t h e + porous medium : t h i s r e s u l t s i n a c o n t r i b u t i o n D g r a d C t o @1I where D is t h e d i f f u s i o n t e n s o r (nxn s y m e t r i c p o s i t i v e d e f i n i t e m a t r i x ) .


12

Summing up we get

:

The diffusion tensor D is usually of the form (see DOUGLAS [2])

:

,.

where

E

is the rnolecular diffusion (very small) and D is a nxn

definite

positive

,.

eigenvector

of

D

matrix

representing

associated

to

its

the

turbulent

symmetric

diffusion.

largest eigenvalue

The

(longitudinal

-f

dispersion) is parallel to 111.3

111.3.1

q-

- TWO-PHASE IMMISCIBLE FLOW - The Muskat relative permeabilities model When two immiscible fluids share the pore spaces, MUSKAT [ 2 ] has

shown experimentally that the Darcy law is still applicable to each fluid -f separately, at the price of a slight modification : if IJ. is the Darcy J

velocity of the jthfluid (j=1 or 2), defined as in (3.2), then one can write.

(3.1 1 ) where kr.(z,x) J

is the relative permeability of the jth fluid. This adimensional number indicates to what extent the presence, in the pores, of the second fluid prevents the first one from flowing.

-

is the saturation of the fluid number j, defined by S. = Of j These adimensional numbers J volume of fluid 1+2

.

j ’

of course satisfy the relations -use only S=S1

-

FI. . P .

J

J

s1+s2=1. we

are the viscosity and density of jth fluid.

SO

will

III. The Basic Laws for One- and nvo-Phase How

P.

13

is the pressure in the jth fluid (each fluid is supposed to occupy a connected region of the pore net, so that p 1 and P2 are expected to be continuous over n).

3

-/

I

I

II

1 I I

for a simple waterflooding process.

+Q‘

II

I )


14

We will often use the reduced saturation S defined

s = -s, -

(3.12)

-

by

sm

-

M' - m' As for the monophasic case, we will use in this book volumetric

+ flow vectors ,+

i

rather than velocities u. J

4j

(3.13)

:

4. J

=

a B.

=

-oK(x)

=

volumetric flow vector of fluid j

J

kr . ( S , x ) B . J [ gradP. -p.g grad Z(x)] u J. J J

where 8 . is the volume factor of the jth fluid

:

J

P.P.1

B,(P.)

(3.14)

=

'jref

J

As

3 5 .

mentionned

above,

the

notion

of

relative permeabilities

depending only on saturation levels results from experiments, and as such, is only an approximation to reality. In fact, it turns out that this approximation is rather rough : for example, in the water + oil case, the experiment yields different relative permeability curves according to whether the saturation of the wetting fluid (see figure 9 ) increases during the experience ("imbibition") or decreases ("drainage") : some hysteresis phenomenon should be taken into account. A detailed discussion on relative permeabilites can be found in MARLES [ l ] . However,

this

model

is,

despite

its

imperfections, almost

exclusively used for the simulation of oil reservoirs. One reason for that may be that the resulting system of equations is already very difficult to solve, and that the errors due to the numerical approximation process by far exceed those resulting from neglecting the hysteresis of relative permeabilities. The throughout this numerical one.

book

Muskat relative permeability model

will

be

used

for the mathematical study as well as for the

III. The Basic Laws for One- and Two-Phase Flow

111.3.2

-

15

The capillary pressure law Because of the presence of two pressures (one for each phase) one

needs an additional relation in order to get a closed system of equations. This additional relationship is the capillary pressure law, which results froin the curvature of the contact surface between the two fluids. One in turn admits that the curvature depends o n l y , in an extremely rough approximation, on the saturation level of the two fluids (3.15)

P1

Figure 8

:

-

P2

=

:

Pc(E,x,.

A typical shape f o r the water-oil capillary pressure

curve (usually sC = SM)-

The convention in (3.15) is the opposite of that usually used by oil engineers (who would write P2-P1 = Pel. With the choice (3.151, Pc - will always be an increasing function of S, defined over the [S ,&Iinterval, and vanishing for some saturation S, E [?m,$,l.


16

P

Hence the general shape of fixed. The saturation

-

Sc

is known once the value of

for which the capillary pressure vanishes

depends on the wettability of the fluids usually, (3.16)

Pc

=

Sc

=

1

:

0 when the wetting fluid is at its maximum

saturation, i.e. when residual saturation. Hence

sc is

if fluid

the non-wetting

fluid

1 is the wetting fluid, and

is at

Sc

=

0

its

if

fluid 2 is the wetting fluid. In order to know which of the fluids is the wetting one, one has to look at the meniscus separating the two fluids in a capillary tube : the concavity of the meniscus is oriented towards the non wetting fluid see figure 9 ) .

wetting

Figure 9 :

For

water

Determination of the wetting phase

exemple, oil is the non-wetting phase in water-oil displacement,

whereas it becomes the wetting phase in an oil-gas displacement. For a detailed discussion of the concept of capillary pressure one can consult

MARLES [11.

17

IV. The Basic Models

IV

-

THE

BASIC

MODELS

They will be obtained by adding, to the basic laws of section 111, the conservation laws for each phase or component. We will also consider some simplified models which have been proposed in the literature.

IV.1

- THE MONOPHASIC MODEL With the notations of paragraph 111.1, the conservation law for a

single fluid occupying the pore space can be written as

:

(4.1)

which, together with the Darcy Law

(3.8) and initial and boundary conditions, is a standard (mildly nonlinear) parabolic equation. The case of slightly compressible rock and fluid, where the

equation becomes linear, will be studied in detail in

parabolic

chapter TI. IV.2

-

THE FULLY MISCIBLE MODEL Using the notations of paragraph 111.2, we have now to write

two

conservation laws : one for the fluid occupying the pore space which yields

(4.1) and (3.8) with (4.2)

(3.10)

a

p,

B

and

p

[~(x) $(x,P) B(P,C)] =

-o(x) K(x)

depending on P +

div

q’

=

and

C, i.e.:

0

[gradP - p(P,C) B o !.l(P,C)

g grad Z(x) ]

which is called the pressure equation, and one other for one of the components, say component 1 , which yields :


18

(4.3)

i,= -D(;)

(3.lObis)

grad C

+

C ;,

which is called the concentration equation. As the volume factor B(P,C) of the fluid depends mainly on the pressure P and little on the concentration C, the pressure equation is a parabolic equation, whose diffusion coefficient contains a

- factor

which may vary extremely rapidly with C, as the viscosities of the separate components may be very different. The concentration equation then is a diffusion-transport equation, +

with a (usually small) diffusion term

div(D(q) grad C) and a usually -i

preponderant linear transport term Remark 2 :

div (Cq).

In the case of incompressible rock and fluid, if we moreover suppose

that

concentration

the

fluid

(which

is

is

density the

case

independant if

the

two

of

the

separate

components have the same density and if the mixing occurs without volumetric change), the miscible equations reduce to

(4.4)

(4.5)

U(x)

o ( ~ )ac

+

div (-D(;)

grad C

+

C

4')

=

0

(concentration equation). These equations have been extensively studied especially from the point of view of approximation, by DOUGLAS [ 2 ] , [l],

DOUGLAS-EWING-WHEELER

DOUGLAS-ROBERTS, EWING-WHEELER.

In fact, the miscible equations ( 4. 2) ,

(3.10),

(4.3) (3.10bis),

can be seen as a special case of the immiscible equations in their equivalent form presented in chapter IV section 1, up to the diffusion + term, whose coefficient D(q) is a "velocity" dependant tensor in the miscible case whereas it will be a saturation dependant scalar in the

:


19

immiscible case. This miscible-immiscible analogy is presented in detail in CHAVENT [l 1

.

Hence most of the results or techniques presented in this book for immiscible flows can he adapted to the miscible case which we will not investigate further, except in section V of this chapter, where we will point out, using a simplified example, the distinctive features of the miscible and immiscible problems.

IV.3 - THE TWO-PHASE IMMISCIBLE MODEL After presenting the more general

RPCP model, we will review some

other models proposed in the literature for different applications.

IV.3.1

- The Relative Permeabilities Capillary Pressure model With

the

notation

of

paragraph

111.3,

conservation of each of the two immiscible fluids

we

can

write

the

:

j = l ,2

(4.6)

which, together with Muskat's generalization of Darcy's law kr . ( S ,x)B . (P . )

i. -o(x)K(x) J

(3.13)

:

[gradP.-p.ggradZ (X)]

=

J

P. J

J

j = l ,2,

the capillary pressure law : P1 - P2

(3.15)

=

PC(S,X),

and the algebraic relation -

-

s1 s2 =

(4.7)

:

+

1

(S

=

S,)

yield the sought system of equations. The mathematical nature of obvious

:

this

system of equations is not

due to the shape of the relative permeability functions

kr., J

which may vanish for extreme values of the saturation (see figure 7 ) , the


20

equations

-

-

(4.6),

(3.13) may be

identically satisfied at places where - S2 = 1-SM. Hence the actual number of

S1 Sm or at places where equations is not a prioril known. It is however well known (see for example PEACEMAN) that one can combine the two conservation laws (4.6) for j=l and 2 in order to obtain a saturation equation (similar to the concentration

equation

(4.3),

(3.10bis)

of the miscible case), where the capillary

pressure appears as a diffusion term, and a pressure equation, expected to be similar to the pressure equation (4.21, (3.10) of the miscible case, but which looks actually quite different because of the presence of the two pressure unknowns P1 and P2 instead of the single unknown P of the miscible case. Of course, if the capillary pressure is neglected, this obstacle disappears, as one can set (4.8)

p

=

p1

=

P2.

Then if one defines (4.9)

(4.10)

(4.11) the above mentioned combination becomes

:

(4.12) (4.13)

(4.14)

(4.15)

(saturation equation)

IV. The Basic Models Now

21

the

immiscible pressure equation (4.12),

(4.13)

strongly

resembles the miscible pressure equation (4.2), (3.10), and the immiscible saturation equation (4.14), (4.15) the miscible concentration equation. The main difference lies in the expression of the term J1 : in the immiscible +

case, $1 contains

a ( P - p ) g gradZ(x) term which represents the 1 2 differential action of gravity on the two fluids, which tends to bring the heavier one to the bottom of the reservoir and the lighter one to the top; such a term is missing in the miscible case, as this action on a dissolved component is extremely weak and has been neglected in the expression (3.10bis) for

+

ol. On

the other hand, the miscible volumetric flow vector

+

o1

+ contains a diffusion term -D(q) grad C , whereas there is no diffusion term

in the immiscible case, as in equations (4.12) to (4.15) the capillary pressure has been neglected. However, and this is an important similarity, +

+

+

both expressions for o1 ctntain a term proportional to q , namely Cq in 1 + the miscible case, and q in the immiscible case. 2 One of the ideas which will be developped throughout this book is that one can pursue further this similarity between the miscible and the immiscible case even for the case where the capillary pressure is taken into account, by introducing a pressure P intermediate between P1 and P2' which we call the global pressure. This will be done in chapter 111 for the simple two-phase incompressible case (where it will be shown that the full RPCP model ( 4 . 6 ) , (3.13), (3.15), (4.7) is in fact equivalent to the pressure and saturation equation (4.12) through (4.15) with an additional capillary diffusion term in (4.15)). This idea will be applied, in the remaining paragraphs of chapter IV, to more complicated models which are not described in this first chapter, such as three phase, black-oil or compositional models. The advantage of such an approach is that it produces models of identifiable mathematical nature (parabolic equation, diffusion + transport equations, etc ...) which are amenable to rigorous mathematical treatment (as for example in chapter 111 for the two-phase incompressible case)

and

are

well

suited

for

numerical

approximation on

a

sound

mathematical basis (as in chapter V). This approach is somewhat different from that often taken in the oil industry, where the RPCP model in its original form of two COrISerVatiOn laws (4.6), (3.131, (3.15), (4.7) is used for the construction of the numerical

approximation

:

usually

the water

saturation and

the oil


22

pressure are chosen as the main unknowns, the remaining pressure and saturation unknowns are eliminated using (3.15), (4.71, and the resulting system of equations, which has lost any symmetry property, is discretized using a finite difference method.

-

IV.3.2

The Muskat free boundary model

In the thirties, i.e. ten years before he introduced the relative permeability concept, MUSKAT [ l ]

proposed, for the study of water-coning

under a production well, the following free boundary model (see figure 10) : water and oil are supposed incompressible and separated by a (free) boundary

L

;

in domain

Q,,

occuped by water, the monophasic model of

paragraph 4.1 applies : (4.16) with

:

(4.17)

Q2, occuped by oil, one has similarly :

in domain (4.18)

div

1

grad(P2-p2 g Z ) 1

$,

with (4.19)

$ 2K2= ; (Jr 2

= 0

in Q2

IV. The &sic Models

23

production well

It.

Figure 10

:

The Muskat free boundary coning model

Z, one has continuity of the pressure

on the free-boundary

(Muskat neglected the capillary pressure) and of the fluxes (4.20)

P1

(4.21)

- Q ~grad

=

P

2

on

1

(P1-pl g Z )

=

-$I, grad ( P - P gZ) 2 2

The normal speed V v Of the free boundary

_ _

(4.22) vv

where

+ I ,

-

:

, , -

+

Q1 grad ( P - p gZ ) v 0

$(sM-”m’

is a unit normal to

Z is given by

-

Z.

This model is also sometimes ci led the ttpistont’ mo

n

?

oi

engineering literature, because the water displaces the oil as a pistqn would, without allowing for any sharing of the pore space by the two fluids. It ha3 been used by some authors (for example AMIRAT [l].

NGUYEN

TRI HUE) for the design of numerical simulators approximating the free

Ch. I: Basic Laws and Modelsfor Row in Pomus Medh

24

boundary E by a linear broken line which was displaced at each time step using its normal velocity computed from (4.22). In fact, the Muskat free-boundary or llpistonll model can be seen as a special case of the general RP without CP model (4.12) through (4.15) when the fluids are incompressible and the relative permeabilities independant of x. Then it reduces to

(4.23)

1%";

:

(g)+k2(z)) -= 0-aK(x)(kl 1

[gradP-p(S)ggradZ(x)

]

(pressure equation),

(saturation equation). If one defines in all

P (4.25)

-

-

S =

$7

:

p1 in

-

SM in

ill

and

Ql

and

P2

-

Sm

in

Q2,

in

Q2,

the equations (4.16) through ( 4 . 2 1 ) in Muskat's free boundary model simply S satisfy the pressure equation (4.23) (with express that P and K~ K kr1 c.5M ) and K2 = K kr2 (5m 1). Then will be one weak solution of the non linear first-order saturation equation (4.24) if the normal speed of its discontinuity on E is given by the Rankine-Hugoniot relation :

-

-

From the definition (4.24)

s

of

f

and the properties of the relative

permeabilities shown in figure 7, (4.26) reduces to


which

is

25

exactly

the

formula

(4.22)

used

in

d e f i n i t i o n of t h e normal speed of t h e free-boundary However,

the saturation profile

-

Muskat's

model

for

the

1.

S does n o t n e c e s s a r i l y s a t i s f y

t h e e n t r o p y c o n d i t i o n a s s o c i a t e d with t h e t t p h y s i c a l " s o l u t i o n of ( 4 . 2 4 ) . s o may be a "wrong" s o l u t i o n of t h e RP

t h a t t h e Muskat free-boundary model without CP i n c o m p r e s s i b l e model. course,

Of

we

if

modify

the

saturation

equation

(4.24)

by

-t

f(s) by

replacing

(so

that

-

t h e s a t u r a t i o n equation

(4.24),

miscible c o n c e n t r a t i o n e q u a t i o n ( 4 . 3 ) ,

becomes s i m i l a r

(3.10bis) with

-

D(G)

t o the fully =

01, then the

s a t u r a t i o n p r o f i l e S of t h e Muskat model is e x a c t l y t h e unique s o l u t i o n of t h i s modified s a t u r a t i o n e q u a t i o n .

I n o r d e r t o see i n which c a s e s t h e Muskat s a t u r a t i o n p r o f i l e S

s a t i s f i e s t h e e n t r o p y c o n d i t i o n of t h e o r i g i n a l s a t u r a t i o n e q u a t i o n ( 4 . 2 4 ) , + we suppose f i r s t t h a t t h e normal I t o t h e f r e e boundary is o r i e n t e d from domain toward t h e o i l domain, so t h a t S has a decreasing + discontinuity i n the v direction. + V, t h e e n t r o p y c o n d i t i o n , t o be s a t i s f i e d by t h e With t h i s c h o i c e o f t h e water

-

d i s c o n t i n u i t y of

A

S

can be w r i t t e n as :

sufficient condition for

(4.29) t o hold,

and hence

Muskat f r e e boundary s o l u t i o n t o be t h e a c t u a l s o l u t i o n of t h e CP

problem, is:

RP

for the without

Ch.I: Basic Lows and Models for Flow in Porous Media

26

(4.31)

( p -p

1

grad Z ( x ) S 0 on

2)

1.

The last condition simply means that the heavier fluid must always to be located under the lighter one (which is usually the case in the coning problems). In order to interprete the first condition (4.30), let us suppose that the two fluids have 'tcrossrelative permeabilities"

:

and define the mobility ratio of the two fluids =

(4.33)

2= U,

mobility of fluid 1 mobility of fluid 2

'

Then (4.30) becomes

which means that the low mobility fluid has to be the displacing fluid. In the case of general relative permeabilities, we will see in section V of this chapter that the right inequalities in (4.30) will be - satisfied f o r all s E [Sm, Swelge 1 with Swelge + when the mobility + + ratio vanishes (for q'v 2 0 ) or tends to infinity (for

Summarizing

the case where

the

entropy

sM *;

5 0).

condition

(4.29)

is

satisfied, we see that the Muskat free boundary model can be used when a less

mobile

fluid

displacements"),

is

with

displacing the heavier

a

more

fluid

mobile

one

("low mobilitx

remaining always beneath the

1ighter one.

At present, no mathematical results are available concerning the existence or uniqueness of the solution of the Muskat free boundary problem

IV. me Basic Models (4.16)

27

through (4.22)

or of the RP without CP problem (4.23),

(4.24).

However, if one replaces, in the Muskat problem, the pressure continuity requirement (4.20) by a potential continuity requirement (as was done by Muskat himself in his original paper as a first approximation) (4.35)

p, - P1 @(x)

=

:

on 1,

Pc - p 2 @(x)

then the problem can be simplified using a current-function type change variable, and theoretical results obtained (see YOUCEF OUALI).

IV.3.3

-

Of

The Richards approximation Around 1930, RICHARDS [l],

[2] proposed a model for the study of

the evolution of moisture content in unsaturated soils for agricultural purposes. His model was based on the water continuity equation together with an (erroneous) generalization of the Darcy law. Richards model has been extensively studied since 1950, both from the theoretical and numerical point of view, and is still today almost exclusively used for the study of unsaturated soils, where water and air share the pore space of the porous medium. SUpriSingly, the

link

with

the

relative

permeability model

introduced by Muskat in 1949 and used since in petroleum engineering Was not made before the ~ O ' S , maybe b?cause Muskat'y original paper concerned only incompressible fluids, so that a slight generalization was required to handle the case of water and air (see MOREL-SEYTOUX [ll, [21). So we will derive here the Richards model as an approximation to the general RPCP model under the following hypothesis:

-

(4.36)

incompressible water and porous medium, relative permeabilities and capillary pressure independant the air everywhere at atmospheric pressure.

The

most

restrictive hypothesis

here

is

the last one, as

experimentation shows that the air often has difficulties escaping from the soil during the infiltration process, so that the speed of infiltration of water may be lowered by many order of magnitude. Hence there seem to be


28

very few situations where the hypothesis that the air be at a constant pressure in the porous medium is a realistic one. Nevertheless, as we mentioned above, the Richards approximation is widely used in agricultural studies probably because of its (apparent) simplicity. Under the hypothesis (4.36). model (4.6), (4.7) with

(4.37)

reduces to

and the notation (4.10) the RPCP

water

=

fluid

1

(wetting)

air S

=

fluid

-

2

(non-wetting)

=

S1

=

water saturation

IJ

=

p,

=

water viscosity

K(x)

=

absolute permeability to the residual air - at krl(SM) = 11,

water in presence Of saturation (so that

:

(4.40)

p,-P2

(4.41 1

p2

=

=

PC(3

given constant (atmospheric) pressure.

We now transform this system of equations into the Richards equation. Because of the shape of the capillary pressure curve Pc(s) (See - figure 8 with Sc = S M ) , the unsaturated regions of 0 (i.e. where S < SM) correspond necessarily to those where P1 < p2, and hence in the saturated -regions (i.e. where S=sM) one necessarily has P1 2 P2. Therefore we will distinguish two cases

:

i) in the saturated regions, where

-

from (4.38) through ( 4 . 4 1 ) , -div { 2mu

as

krl(SM)

=

1

S

=

and p 2

sMand =

p 1 tP2' we get

constant

[grad (P1-P 2 ) - p l g grad Z(x)] ]

:

=

0


29

which, w i t h t h e d e f i n i t i o n s

1 - p2

(4.42)

reduces t o -div {

(4.43)

gu [grad F - R g grad Z(x)] IJ -

i i ) i n t h e u n s a t u r a t e d r e g i o n s , where S

]

0;

=

< SM

and

P2,

the

and

+

$1 = - u ( x u) K ( x ) [ k r l

(4.44)

(5)P;(s)

grad?-kr,

(5)p 1

P1

O for all F, equation (4.49) has only one singularity at F=O, and yields always a water -saturation S>S, (the "wet case"). If one wants to handle the "dry case",

-

where in some regions the water is at its residual saturation Sm, then c(F)

-

has to vanish for some value F (see figures 1 1 and 1 2 ) , and the equation (4.49)

has one additional singularity at F=F-

Remark 3 :

Hydrologists

use,

permeabilities

instead

and

capillary

of

saturation,

pressure,

some

relative equivalent

quantities such as water content, hydraulic conductivity and suction. We give here the hydraulic terminology for the case of a 3-D (o(x) 5 1 ) homogeneous, porous medium ( @ ( x ) S , K(x)

=

K)

0

=

eR

=

Bs

=

volumetric moisture content @ Sm (R stands for l7retention1') ( S stands f o r "saturated") $I SM

:

-5 =

@

(dimensionless)

(s)

krl

yri(e) (4.50)

$(e)

=

l.l pl

K

pc(s)/plg

=

D(e)

=

FH(e)

=

KH(e)

=

D(a)da

=

suction, or matrix potential (length) =

=

=

Kirchhoff's potential ( length2/time)

c (F ) H

diffusivity (lenth'ltime)

-KLJ F ( S )

?S

e

hydraulic conductivity (velocity)

=

2 (el

e

I

g

H

=

9 c(F)

(dimensionless).

31


Figure 1 1 :

The

1 *

Kirchhoff

s

potential F as a function of the water saturation in the

s

saturated zone. When

5 +sm,F

tends to a finite value

F

or

to infinity depending on the shape of the water relative permeability and pressure curves.

capillary

Y

1)

Figure 12 : The water saturation S and water pressure P 1 as functions of the Kirchhoff potential F.

Ch.I: Basic Laws and Models for How in Porous Media

32

The Richards equation (4.49)

a

(4.51 1

CH(FH) - AFH

+

aKH az

=

then becomes

:

0.

As we mentioned at the beginning of this paragraph, the Richards equations (4.49) has been extensively studied: GILDING-PELETIER and GILDING studied first the case where

-

F > -

L*

(the "dry" case); FASANO-PRIMICERIO

and BERTSCH-LEPELETIER studied existence and positivity of the solution for more general cases. HORNUNG studied the general n-dimensional case with with a monotone lipschitz continuous function

F >-

C(F), together with

unilateral boundary conditions (the above list of references is by no means exhaustive on the subject).

IV.3.4

- The Baiocchi free boundary model At the beginning of the seventies, BAIOCCHI introduced a model for

the free boundary of the water inside a porous dam (see figure 1 3 , and compare with figure 10 for the Muskat coning model). This model is as follows

- Water

is

:

incompressible, and

separated from the air by a

free-surface; *

in the domain

paragraph IV.l applies

0,

occupied by water, the monophasic model of

:

in a

1

with

(4.53)

- in the domain

n2

occupied by air, the air pressure

constant and equal to the atmospheric pressure

(4.54)

2 '

=

'atm

'atm

:

in Q2;

p2

is


33

on the free boundary E ,

one has continuity of the pressure

(Baiocchi neglected the capillary pressure ) (4.55)

P1

=

on 1;

P2

the normal speed (4.56)

_ -

vv -

:

vv

of he free boundary

E

is given Y :

+ JI, grad(P1-plgz 1 - v 0 4l

'EM-EmI

Figure 13

:

The Baiocchi dam problem

If we compare the above Baiocchi's model with Muskat's free boundary model (4.16) though ( 4 . 2 2 ) , we see that Baiocchi model can be seen as the limiting case of Muskat's model when the air viscosity !J 2- and density p 2 go to zero (and hence the mobility ratio M defined in (4.33) goes to zero). Of course, this is only formal, as no mathematical results are available for the Muskat model.

If we now check for conditions (4.30) and ( 4 . 3 1 ) , that the Muskat solution is the entropy solution model, we see that

of

the

RP

which ensure without

CP

34

Ch. I: Basic Laws and Models for Flow in Porous Medh

*

(4.31) is satisfied in dam problems, where water occupies the

bottom of the dam, * (4.30) or equivalently (4.34) is satisfied if cross relative permeabilities are used as we have seen that M

0

such t h a t

(f)

for almost a l l x E 8

v(x) 6 M

Ch. II: Slightly Compressible Monophasic Fields

58 and e q u i p p e d w i t h t h e norm

We w i l l a l s o u s e t h e S o b o l e v s p a c e H’ ( R ) , which c o n s i s t s o f t h o s e f u n c t i o n s of

L 2(R)

whose

derivatives,

i n d e e d f u n c t i o n s of v

L2(0)

H’(R)

E

taken

in

the

sense

of

distributions,

are

:

v E L2 ( R )

and

ax.

E

L ~ ( R ) i,= l , . . . ,

n.

is a H i l b e r t s p a c e when i t is e q u i p p e d

The S o b o l e v s p a c e H’(R)

w i t h t h e f o l l o w i n g s c a l a r p r o d u c t an d norm :

(2. 10)

W e make now t h e f o l l o w i n g h y p o t h e s e s : (2.1 1 )

R

Rn is a c o n n e c t e d o p en s e t w i t h r e g u l a r b o u n d ar y T ,

E

(2.12)

(2.13) (2. 14)

(2.15)

u

0

6

H’(Q)

and s a t i s f i e s t h e b o u n d ar y c o n d i t i o n ( 2 . 2 ) a n d t h e

seco n d p a r t of ( 2 . 4 ) ,

(2.16)

uo

=

0

on

TD, uo

=

i.e.

:

a c o n s t a n t o n Tw.

Let u s d e f i n e t h e f o l l o w i n g f u n c t i o n s p a c e s

(2.17)

59

II. Existence and Uniqueness Theorems

H

(2.18)

L2(0).

=

We e q u i p

and u s e f o r Of

w i t h t h e s c a l a r p r o d u c t a n d norm

V

t h e scalar p r o d u c t a n d norm d e f i n e d i n ( 2 . 8 ) .

H

course,

d e f i n e s a norm o n

(2.19)

V,

which is e q u i v a l e n t t o

norm ( 2 . 1 0 ) , o n l y i f t h e P o i n c a r 6 i n e q u a l i t y h o l d s , i.e.

rD

t h e r-measure of

(2.20)

is non z e r o and

if

is bounded,

R

which we s h a l l s u p p o s e from now on. If

Remark 2 :

(2.20)

does n o t h o l d , a l l t h a t f o l l o w s r e m a i n s t r u e by -it which unknown f u n c t i o n u -* 0. = u e

making a c h a n g e of

makes a p p e a r i n t h e l e f t hand s i d e of (2.1)

a hU

term, so

t h a t one can equip V w i t h t h e complete s t a n d a r d product i n

H'. As

O

V is a d e n s e s u b s e t o f

H,

we c a n embed

H'

in

V'

with

d e n s e embedding, s o t h a t we h a v e , V

(2.21

c

and

H

HI

dense

c

V'

.

dense

If now we decide t o i d e n t i f y H w i t h its d u a l H' u s i n g t h e s c a l a r

product (2.8). (2.22)

( 2 . 2 1 ) becomes : V c H c V'

, each i n c l u s i o n being dense

and we h a v e :

(2.23)

Y

f

C

H c V ' ,

Y

v e V : < f , V > V'

v

=(f,V)

Ch.II: Slightly CompressibleMonophasic Fields

60

,

s o we s h a l l u s e t h e same n o t a t i o n (

f o r both t h e s c a l a r product i n H

)

( a s d e f i n e d i n ( 2 . 8 ) ) and t h e d u a l i t y between V' and V. We now g i v e t h e

-

THEOREW 1

:

:

Under

assumptions

(2.1 1 )

(2.24)

u E L-(o,T;v)

(2.25)

d" dt

(2.26)

-div

LZ(O,T;H)

f

($(XI

=

L2(Q

grad u)

E

x

(2.1 6 ) , t h e e q u a t i o n s

through

(2.1) t h r o u g h (2.5) have a u n i q u e s o l u t i o n

u

satisfying :

]O,T[)

L2(0,T;H)

L2(Q

=

]O,T[).

x

Before g i v i n g t h e proof o f t h i s theorem, we remark t h a t i t g i v e s a l'strongT1, s o l u t i o n of

the

(2.1)

,... ( 2 . 5 ) )

-

(2.12) H-l / 2

(2.1)

E

holds

in

L2(nx]O,T[)

*.

,(2.5

s i n c e each

L2(Qx]0,T[) u s i n g ( 2 . 2 5 ) ,

-

(2.2)

holds in

-

(2.3)

holds

as the regularity

),

in

of

t h e two l e f t

hand

(2.12) and ( 2 . 2 6 ) ;

L2(0,T;H1/2(r,))

from ( 2 . 2 4 ) ;

L2(0,T;H-'/*(rN))

s i n c e we have from

(2.24),

t h a t t h e v e c t o r f i e l d JI g r a d u f L 2 ( 0 , T ; H ( d i v ; Q ) ) , s o t h a t

and (2.26)

component

on

the

boundary

of

Sa

has

a

trace

in

L'(0.T;

(r));

1,

.

(2.1 ) ,

:

s i d e terms are i n

its normal

equations

g i v e s a s e n s e t o e a c h t e r m of t h e s t a t e e q u a t i o n s

r e s u l t s (2.24),.. . , ( 2 . 2 6 )

(2.4)

holds

in

L2(0,T),

as

soon

as

'drW

0, so

=

that

H1'2(r);(')

'D ~~~

(1)

The h y p o t h e s l s

arw = b,

wtllch means t h a t

rW

i s a closed c u r v e ( f o r a

2D problem), is s a t i s f i e d o f t e n i n p r a c t i c e - see f i g u r e 2 o f c h a p t e r I .

I1 Existence and Uniqueness Theorems

-

61

h o l d s a t least

(2.5)

i n H.

as

u

and

au e L’(0,T;H) at

from

(2.24) a n d (2.25). We s h a l l u s e t h e a u x i l i a r y unknown

Proof of THEORM 1 :

and u s e t w i c e t h e Lax-Milgram t h e o r e m : o n c e t o d e t e r m i n e u s u p p o s i n g

W

is known, a n o n c e t o d e t e r m i n e W. Step 1

:

Suppose we h a v e been a b l e t o f i n d W

E

and t r y

L’(nx]O.T])

to

d e t e r m i n e u. Obviously

is g i v e n a t a n y t i m e t b y a s t a t i o n a r y e l l i p t i c

u(t)

problem :

(2.29)

u(t)

=

0

on

rD,

u(t)

=

unknown c o n s t a n t o n

rW’

(2.31 1

By m u l t i p l y i n g

(2.28)

by a t e s t f u n c t i o n v

V

E

v a r i a t i o n a l f o r m u l a t i o n of problem ( 2 . 2 8 ) - ( 2 . 3 1 )

So i f we d e f i n e L ( t )

( (Au,v)

=

n

E

V‘

and

A E g(V,Vi)

$ ( XgIr a d u g r a d v dx

=

we g e t t h e w e l l known

:

as

((u,v))

$’

Y u, v

E

V,


62

w e first see t h a t

:

(2.34)

L

H'(0,T;V' )

(2.35)

A

E

(from (2.14)),

is a n isomorphism from

V

onto

V'

(from (2.12) a n d t h e

Lax-Milgram t h e o r e m ) ,

r

and t h a t t h e v a r i a t i o n a l e q u a t i o n ( 2 . 3 2 ) may b e r e w r i t t e n a s

(2.36)

Au(t)

=

W(t)

+

L ( t ) a.e

Thus e q u a t i o n (2.32)

t

E

]O,T[,

W

C

L*(nx]O,T[)

which

obviously

in

E

h a s a unique s o l u t i o n u ( t ) for almost every satisfies

(2.24)

and

(2.26)

V'

0

f

soon

)

as

.

w i t h t h e scalar p r o d u c t ( (

,

t r a n s p o r t e d from t h e scalar p r o d u c t and norm ( 2 . 1 9 ) on

u s i n g t h e isomorphism A d e f i n e d i n (2.33)

since u

as

n L m ( O , T , V ' 1.

Let u s now e q u i p t h e s p a c e

11 11

1

JO,T[.

Step 2 : E x i s t e n c e and u n i q u e n e s s o f W E L ~ ( Q X ] O , T [ ) n L"(0,T;V'

and norm

:

V from ( 2 . 1 5 ) ;

:

( 2 . 1 6 ) , and L E H'(0,T;V' ) n Z ( C 0 , T l ; V ' ) ,

and c o n s i d e r t h e v a r i a t i o n a l e v o l u t i o n equation

))*

V

II. Existence and Uniqueness Theorems

63

f i n d W E L'(0,T;H)

I

such t h a t

v

SL

for a.e.

E H,

t

E

lO,TC,

wo.

W(0) =

(2.41)

1

Equation

(2.40),

is

(2.41)

a

standard

variational

evolution

e q u a t i o n , b u t w i t h a f f s h i f t f f i n t h e s p a c e s V , H a n d V' :

- t h e r o l e of V

is p l a y e d h e r e by H ,

- t h e r o l e of H

i s p l a y e d by V' ,

-

is p l a y e d by V

t h e r o l e of H'

=

A-l

V' ,

- t h e r o l e o f V' is p l a y e d by H , s o t h a t t h e d i a g r a m c o r r e s p o n d i n g t o (2.22)

[ (2.42)

I

H

V A-'~

c

~

V

C

is h e r e

H

each i n c l u s i o n b e i n g dense

and t h e s t a n d a r d r e s u l t s f o r p a r a b o l i c e q u a t i o n s a p p l y t o (2.40),

W e need t o check t h a t

:

t h e b i l i n e a r form on H x H : ( u , v ) which is o b v i o u s from h y p o t h e s i s (2.12), *

the right

h a n d s i d e of

which is a l s o o b v i o u s from (2.121, Hense solution W

we

(2.41 ).

know

satisfying :

that

(2.40) (2.13),

equations

+

1 R

uv

dx is H - e l l i p t i c ,

is i n L ' ( 0 , T ; H ' )

=

L'(0,T;H)

,

(2.14). (2.40),

(2.41)

have

a

unique

Ch.II: Slightly Compressible Monophasic Fields

64

Step 3

:

Let

and d e f i n e

be t h e unique s o l u t i o n o f (2.401,

W

u

by

Then

(2.36).

u

found i n s t e p 2,

(2.41)

is t h e s o u g h t s o l u t i o n o f e q u a t i o n s

(2.1 1, (2.5).

Using (2.37)

we rewrite (2.40)

Using (2.36)

and (2.28)

as

we g e t :

(2.45)

Y v

du

and hence we g e t

E

i.e.

L'(Qx]O,T[)

We c a n now m u l t i p l y (2.46)

r e s u l t from (2.29),

...,(2.3l),

E

H , a.e.

on l O , T [ .

(2.25).

by @ ( x ) t o g e t (2.1).

and (2.5)

Then (2.21,

r e s u l t s from (2.39)

o u r f u n c t i o n u s a t i s f i e s a l l e q u a t i o n s ( 2 . 1 ) t h r o u g h (2.5)

(2.3)

and (2.36).

So

and h a s moreover

t h e r e g u l a r i t y announced i n Theorem 1.

Step 4 : Uniqueness Let

u

through (2.26). from (2.46)

b e a s o l u t i o n o f (2.1) Define W

t o (2.40),

t h e n , from s t e p 1 , t h a t Remark 3 :

=

t h r o u g h (2.5)

which shows ( s t e p 2 ) t h a t W is u n i q u e l y d e f i n e d , and

u

is unique.

Suppose t h a t t h e boundary

rK

is r e g u l a r w i t h

t h a t t h e r e e x i s t s a neighbourhood

regular.

s a t i s f y i n g (2.24)

- d i v ( $ g r a d u ) . I n s t e p 3, we c a n go backward now

Then

the

L 2 ( o J T ; H 2 ( (52Pk-Dk))

as

qk of

potential follows

u

Dk

= 71

immedi;itely

ar K

=

0. and

on which $ is belmgs

to

from

the

65

III. An Alternative Model of Monophasic Wells

theorems

regularity

(2.28),...,(2.31) (2.28).

...,(2.31)

.

for

More to

elliptic

equations

generally

infer

it

is

regularity

applied

very

results

for

u

r e g u l a r i t y r e s u l t s for e l l i p t i c e q u a t i o n s . Remark 4 :

to

easy using

from 0

The a b o v e p r o o f r e m a i n s v a l i d i f t h e f u n c t i o n

0

depends

b o t h o n x and o n t a n d s a t i s f i e s o n l y :

The u s u a l v a r i a t i o n a l f o r m u l a t i o n o f e q u a t i o n s (2.1) t h r o u g h

Remark 5 :

(2.5) is :

I

I

(2.48)

Q u v +

In

I)

gradu gradv

=

I f v I gV +

n Y v

u(0)

=

+ GV

rN a.e.

6 V,

on 10,TC.

uo.

To s o l v e i t i n t h e u s u a l way, we h a v e t o e q u i p H

a scalar p r o d u c t (

(u,v),

(2.49)

,

)@

=

L2(n) w i t h

a d a p t e d t o t h e f i r s t term of (2.48)

Q(x) u ( x ) v ( x ) dx

=

n

s o t h a t t h e i d e n t i f i c a t i o n of H w i t h its d u a l H' become more complicated.

Moreover,

the

proof

does

h y p o t h e s i s (2.42), t o t h e case where

@

n o t go over,

under

is time d e p e n d a n t . For

these r e a s o n s we do n o t u s e t h e u s u a l v a r i a t i o n a l f o r m u l a t i o n (2.48)

and p r e f e r t h e s t r o n g e r

f o r m u l a t i o n (2.361, (2.40),

(2.41 1.

111

-

AN

ALTERNATIVE

0

MODEL

OF

MONOPHASIC

WELLS

I n t h e above model, a l l t h e wells i n t h e f i e l d have been modelled by small " h o l e s t * i n t h e p o r o u s medium, o n t h e boundary o f which p r e s s u r e

Ch. II: Slightly Compressible Monophasic FieMs

66

and flow c o n d i t i o n s a r e imposed. T h i s r e p r e s e n t a t i o n i s o f c o u r s e v e r y c l o s e t o t h e p h y s i c , b u t is n o t used by o i l e n g i n e e r s , who p r e f e r t o n e g l e c t t h e hole

in

n due t o t h e well ( i t s d i a m e t e r is g e n e r a l l y very small i n

comparison t o t h e mesh s i z e s used f o r numerical c o m p u t a t i o n s ) and t o model t h e well

by a s o u r c e o r

equation

(2.1).

This

s i n k term a p p e a r i n g

i n t h e right-hand

side of

l a s t model is p o s s i b l e o n l y f o r wells where f l o w

c o n d i t i o n s a r e imposed, b u t t h i s is n o t a g r e a t r e s t r i c t i o n i n p r a c t i c e as such wells a r e t h e m a j o r i t y .

W e s h a l l c o n s i d e r h e r e o n l y t h e c a s e o f wells whose b o r e h o l e Dk h a s a s e t d i a m e t e r v e r y s m a l l i n comparison w i t h t h e f i e l d s i z e , s o t h a t t h e y can b e s e e n as a p o i n t ( a s i n t h e 2-D h o r i z o n t a l model o f f i g u r e 2 of c h a p t e r I ) . The c a s e o f wells whose " l i m i t shape" is a l i n e ( a s i n t h e 2-D v e r t i c a l model o f Fig.

3 of c h a p t e r I I ) , s h a l l n o t b e c o n s i d e r e d h e r e , b u t

could be handled by similar t e c h n i c s . For s i m p l i c i t y , we s h a l l c o n s i d e r t h e c a s e of f i e l d s produced o n l y by a s i n g l e well ( K = l subscript K=l

i n t h e n o t a t i o n s o f s e c t i o n I ) , s o we s h a l l omit t h e

and s h a l l write D i n s t e a d of D1

f o r t h e well b o r e h o l e ;

of

c o u r s e t h e r e s u l t s s t i l l h o l d f o r more t h a n one well. The r e l a t i o n s between t h e porous body 0 , t h e f i e l d domain

n

and

t h e i r b o u n d a r i e s a r e t h e n , u s i n g t h e n o t a t i o n (1.18) :

We s h a l l u s e t h e n o t a t i o n :

W e

(2.1),.. .,(2.5)

want

to

replace

the

partial

differential

d e f i n e d on il by one d e f i n e d on t h e whole of

i,

equation

namely :


61

(3.5)

-u

(3.1) where 4 , q , (2.1)

-

= u

1, F

,...,( 2 . 5 )

on

at t

= 0,

i.n , have t o b e chosen s o t h a t

and

and

n"

of ( 3 . 3 )

,...,( 3 . 6 )

F i g u r e 1 : The domains

Q

and

ii

t h e s o l u t i o n s u of

a r e as c l o s e as p o s s i b l e on

il.

when the wells are modelled

r e s p e c t i v e l y by boundary c o n d i t i o n s a n d by s o u r c e terms.

111.1

- AN EXACTLY EQUIVALENT REPRESENTATION OF WELLS BY SOURCE OR S I N K

TERMS The idea h e r e is t o e x t e n d t o

t h e c o e f f i c i e n t s and s o l u t i o n of

( 2 . 1 ) t h r o u g h (2.5) w i t h s u f f i c i e n t r e g u l a r i t y t o e n s u r e t h a t t h e e q u a t i o n s t i l l h o l d s i n a l l of

n"

when a n a d hoc r i g h t hand s i d e F is added.

Ch. It Slightly Compressible Monophasic Fields

68

H y p o t h e s i s a n d n o t a t i o n of Theorem 1 and Remark 3

-THRORm 2 :

Let

(%

b e t h e n e i g h b o u r h o o d of

d e f i n e d i n Remark 3.

D

T , Go of

Then t h e r e e x i s t s e x t e n s i o n s 9, 1, and a f u n c t i o n F s a t i s f y i n g (3.7)

o s m

(3.8)

o s

(3.9)

T(x)

=

(3.10)

-uo

0

and

~ ( x s) ZM a.e.

x

, F

[O,T]

...,(3.6)

ii, I c

on

to

iiL

%I(%),

L~(C~),

e

, Go

rD

on

(3.31,

that

T

o a.e. on D,

SuppF c D such

f , u,

:

E

(3.11)

$, J,,

s i ( x ) 5 M a.e. on ii,

m/2 5

=

-

H’(6)

z

€

L2(G

admits

a uniqu

solution

U,

+

r e s t r i c t i o n t o R is t h e u n i q u e s o l u t i o n u of (2.1),...,(2.5). Moreover, F is r e l a t e d t o

j

(3.12)

F ( x , t ) dx -

D

Proof :

j

0

au

and

G

dx

=

G(t)

u

by a.e.

on lO,T[.

D

The p r o o f is v e r y s i m p l e a n d r e l i e s o n e x t e n s i o n t h e o r e m s f o r

S o b o l e v s p a c e s ( c f . LIONS [ 21 ). Let 0 (3.9)

and

ij

( t h e e x i s t e n c e of

be a n y e x t e n s i o n s of @ a n d J, s a t i s f y i n g (3.8).

1 r e s u l t s from

t h e r e g u l a r i t y h y p o t h e s i s of Remark

3 and from e x t e n s i o n t h e o r e m s ) . Let

Let (2.1)

T be u

c

t h e e x t e n s i o n of f d e f i n e d by (3.9).

H’(Q) n L‘(0.T;

t h r o u g h (2.5)

H2(q-D)

be

the

(unique)

s o l u t i o n of

g i v e n by theorem 1 a n d Remark 3. The e x t e n s i o n t h e o r e m s

a p p l i e d t o H1(Q) and H 2 ( q L D ) y i e l d t h e e x i s t e n c e of a n e x t e n s i o n satisfying

:

6


69

which, t o g e t h e r w i t h (3.81, g i v e s

From (3.13),

(3.14)

we g e t ( b y t h e same techniques as i n Theorem

1) :

i

H'(n))

@'(COT];

E

which is obviously a n e x t e n s i o n of uo s a t i s f y i n g (3.10), and

for (x,t) C D

0

(3.15)

F(x,t)

x

]O,T[,

=

5(x) which s a t i s f i e s (3.11)

a;

-[div I ( x ) g r a d a )

from (3.13),

for (x,t)

6

Dx]O,T[,

(3.14).

W e check now t h a t u s a t i s f i e s t h e equations (3.3) through (3.6).

Equation (3.3) is s a t i s f i e d , a.e i n

at

]O,T[,

by c o n s t r u c t i o n o f

at

is a n e x t e n s i o n of u ) and i n (i) (3.15) of F ) . As t h e i n t e r i o r and e x t e r i o r t r a c e s on rw of i, &? ,and 5 a r e e q u a l ( c f . (3.13) and

separately in

(9)

(because

(where it reduces t o a n i d e n t i t y by t h e - d e f i n i t i o n av

(3.8)).

we

at(R u D)

in

get =

that

equation

(6) and

Equations (3.4)

(3.3)

through (3.6)

extension of u and from t h e d e f i n i t i o n There remains t o prove (3.12)

a Green formula y i e l d s (:

to

Dj

satisfied

a.e.

in

]O,T[

f o l l o w from t h e f a c t t h a t

ii

is a n

0' :

i n t e g r a t i n g (3.3) over D and using rw e x t e r i o r t o R, hence i n t e r i o r

is t h e normal t o

:

which y i e l d s (3.12)

rW*

is

L2(6 1.

hence i n

u s i n g (2.4) ,.id

the fact th a t

=

$

and

av

=

av on


I0 111.2

-

OR

AN APPROXIMATELY EQUIVALENT REPRESENTATION OF WELLS BY SOURCE

SINK TERMS

It

of

is

continuation

5,

i$,

course

impossible

in

practice

to

calculate

the

O0 and the equivalent distributed F when existence is

asserted by Theorem 2. Hence engineers commonly use an -Approximate

representation of wells by source or sink terms

-

5, C I ~ , T are any continuations of 0, $, uo, f to instance using constant functions on D) satisfying :

4,

i

(f

(3.17) rn s

$(XI

2 M

,

m 2 i$(x) 2 M ,

and the equivalent distributed source F(x,t) is chosen constant on D as to satisfy approximately (3.12) :

Of course, the solution

of (3.3)

,...,(3.6)

differs from the solution u of (2.1 ),. . . , ( 2 . 5 ) . difference becomes small as the diameter

E

with (3.17),

(3.18)

But we shall see that this

of the well borehole D tends

toward zero. In order to emphasize this dependance on notation

:

i 1

DE, r E , Q E

u -€

and u

instead of D , rw,O,

instead of

and

u,

E,

we shall use the

71


and we s h a l l see t h a t b o t h t h e same

limiting

GE

and uE converge, when

E

t e n d s t o z e r o , toward

c o r r e s p o n d i n g t o a p o i n t s o u r c e with

s o l u t i o n u,

an

i n j e c t i o n r a t e C ( t ) and l o c a t e d a t t h e % e n t e r " of t h e b o r e h o l e , which f o r s i m p l i c i t y we t a k e a s t h e o r i g i n . I n o r d e r t o be able t o d e f i n e t h i s point source s o l u t i o n u, we have t o suppose f i r s t t h a t 0 , $, uo are d e f i n e d on t h e whole domain

6.

Let

1-

fl

(3.20)

be

a

connex bounded open s e t o f lRn,

r e g u l a r boundary

r

=

rD

r N , such

u

t h a t meas

n

=

rD >

2 , 3 with a 0,

(3.21 1

We d e f i n e t h e p o i n t s o u r c e s o l u t i o n

u

E

L2(a

)

a s t h e ultra-weak

(1)

s o l u t i o n of : e(x)

(3.22)

2 at

u = o

on ED

au $ x = g

on E

u = u

on

0

where

6

div($(x)grad u)

i

=

f+ C(t)6

in

a

N at t

=

0,

is t h e Dirac f u n c t i o n a t t h e o r i g i n .

( 1 ) T h i s u l t r a weak s o l u t i o n , o b t a i n e d d e f i n e d i n t h e proof o f theorem 3.

by

transposltion,

is p r e c i s e l y

01. II: Slightly Compressible Monophasic Fields

72 We suppose t h e n t h a t t h e well b o r e h o l e D

Y E

>

satisfies :

0 , DE i s a c l o s e d s u b s e t o f lRn w i t h a r e g u l a r boundary

rE, s u c h

that

(3.23) c B(O,E)

D

and f o r any

E

>

0

= ball

we d e f i n e

OE,$E,uoE,fE

of radius

centered a t the origin 0.

E

:

a s the r e s t r i c t i o n s t o

-

QE

4 E , @ E , 0 0 E , f E as c o n t i n u a t i o n s t o R E of a c c o r d i n g t o (3.171,

(3.24)

of B,$,uo,f @E,$E,~OE,fE

F E a s i n (3.18). We d e f i n e t h e n t h e boundary s o u r c e s o l u t i o n u E E L2(QE) ultra-weak s o l u t i o n " ) ( 2 ) of ( 2 . 1 ) t o ( 2 . 5 ) w i t h O E , $ € , u o E , f

distributed source solution (3.3)

,...,(3.6) 3

-THEOREW

with

mE, qE,

a s the and t h e

L2(G) is t h e ultra-weak s o l u t i o n ( f i ( 2 ) of

Gee,

T and FE. Then we have t h e

(Convergence o f boundary s o u r c e and d i s t r i b u t e d s o u r c e

:

-

s o l u t i o n s towards t h e p o i n t s o u r c e s o l u t i o n ) Under t h e h y p o t h e s i s o f and w i t h t h e n o t a t i o n ( 3 . 1 9 ) , . . . , ( 3 . 2 4 ) , t h e boundary s o u r c e s o l u t i o n

(3.25)

u

u

( c o n t i n u e d by z e r o on D )

If moreover t h e c o n t i n u a t i o n s QE,

satisfy

satisfies

5,.

-*

QoE

:

u s t r o n g l y i n L2(G

).

i n (3.18) are chosen s o as t o

:

(3.26)

II QE-@ll,

+

0.

f o o t n o t e o f l a s t page. s o l u t i o n c o i n c i d e s w i t h t h e s t r o n g s o l u t i o n g i v e n by theorem 1 i f uo, g and G a r e t a k e n s o as t o s a t i s f y (2.14) t h r o u g h ( 2 . 1 6 ) . (1) Cf.

(2) T h i s ultra-weak

73

III. An Alternative Model of Monophasic Wells -1 1 6

(3.27)

E

II qE-*IL

(3.28)

IaoE-uoI L2(i

(3.29)

IFE-fILZ(;)

0,

+

0.

+

+

0,

then, the distributed source solution UE satisfies UE

(3.30)

Remark 6

:

+

u strongly in L2(a) when

E

+

:

0.

DAMLAMIAN-LI TA TSIEN have proven under a stronger hypothesis on D

(DE star-shaped with respect to the origin and having

the ratio of the largest to smallest radius bounded above and below independantly of (3.31)

GE

(3.32)

GE+G

+

E)

that

G strongly in L2(OT) => u (continued by 0 ) weakly in L2(OT) u

:

+

u strongly in L2(b),

+

u weakly in L2(a ).

=>

(continued by 0)

A slight modification of the proof of Theorem 3 yields easily (3.31),

but

not

(3.32),

which would

requires a "strong

convergence" version of Lemma 4. Both techniques however rely on an elliptic lemma (Lemma 2 in the proof) due to LI TA TSIEN-CHEN SHU XING.

Remark 7

:

Conditions (3.26),...,(3.29)

are obviously satisfied if one

makes the following reasonnable choices : InfEss O(s) 5 @,(x) 2 SupEss @(x) ser, scrE

(3.33)

InfEss scrE

$(XI

uOE(x)

=

5 $,(x)

5 SupEss

$(XI

Y x E DE, Y x

E

DE,

srr C

Y x

E

DE

,

C

ndependant of

E,

c independant of

E.


14

Proof o f t h e f i r s t p a r t of Theorem 3: solution

u

convergence o f t h e boundary s o u r c e

t o t h e p o i n t s o u r c e s o l u t i o n u.

W e f i r s t r e c a l l t h e d e f i n i t i o n s of t h e p o i n t s o u r c e s o l u t i o n u and

of t h e boundary s o u r c e s o l u t i o n

u

.

The p o i n t s o u r c e s o l u t i o n u

E

L2(G) is d e f i n e d by

:

T

(3.34)

Y w where v i s r e l a t e d t o w by

L2(G),

:

(3.35)

\

V(T)

=

o

on

i.

The mapping w+v d e f i n e d by ( 3 . 3 5 ) is an isomorphism from

L2(6)

o n t o X ( c f . Theorem 1 f o r i n s t a n c e ) , where :

As t h e c o e f f i c i e n t s 8 , $ a r e r e g u l a r on

is less t h an o r e q u a l t o 3 ,

5

and t h e s p a c e dimension

there e x i s t s a neighbourhood

9

of 0

such

X,

which

that

s o t h a t t h e r i g h t hand s i d e of proves, solution

by t r a n s p o s i t i o n u

6

L z ( Q 1.

of

(3.34)

i s co n t i n u o u s l i n e a r on

t h e mapping w+v,

that

(3.3‘1)

h as a unique


IS

The boundary source solution uE E L2(QE) is defined by

v where v

is related to wE by

:

w ~ ~ L ~ ( Q ~ )

:

w

in

QE,

on Z D' 0

(3.39)

on EN, a constant a.e. on ]O,T[,

The mapping onto XE, where

(3.41)

V E = {V

E

w +v E

E

defined by (3.39) is an isomorphism from E

H1(RE) Ivl

=

L2(8)

an unknown constant }.

rD

The right-hand side of (3.38) is a continuous linear mapping on X E . Hence (3.38) has a unique solution

uE6L2(QE).

We define now some notation for the continuations to functions defined on Sl (resp. Q E ) :

(3.42)

i

6

(resp.8) of

for any v E L2(RE), we denote by E L2(6 the zero continuation of v to DE ; for any v E V E , we denote by 0 E V the continuation of v to D by its constant value on rE.


16

This n o t a t i o n is compatible with the Let t h e n

-

n o t a t i o n introduced i n (3.24).

:

i E=

(3.43)

{v

E

H'(n")

IvI

=

, v l D E = unknown c o n s t a n t ]

0

rD

and

i=

(3.44)

u

tE= {v

E HI(;)

(vIr

=

We g i v e f i r s t two t e c h n i c a l lemmas Lemma 1

:

(3.45)

0,

3

E>O,

D

E>O

v

ID

=

constant}.

E

:

Under h y p o t h e s i s ( 3 . 2 0 ) ( 3 . 2 3 ) w e h a v e

,.

V

is d e n s e i n

V.

w

Proof : is d e n s e i n

As

q'(6)

$'(c)

n V is dense i n V ,

n V.

Moreover, as

it is s u f f i c i e n t t o p r o v e t h a t

is enough t o p r o v e t h e d e n s i t y f o r t h e s p e c i a l case where t h e D

of c e n t e r 0

We h a v e t h e n

and r a d i u s

E.

i

DE c B(O,E) ( h y p o t h e s i s ( 3 . 2 3 ) ) , i t are d i s k s

III. An Alternotive Model OfMonophasic Wells

I7

As t h e f i r s t r i g h t hand s i d e

term t e n d s t o z e r o , t h e convergence

of wE towards v i n V s h a l l b e proved i f we check t h a t t h e l a s t term of

(3.47) t e n d s towards z e r o w i t h

E.

W e do t h i s now i n t h e c a s e n = 2 ( b u t t h e

same proof works for o t h e r s p a c e d i m e n s i o n s ) . Let u s d e n o t e by r a d i a l c o o r d i n a t e s and p u t C

-- D2€

But by c o n s t r u c t i o n of w

where

T

-

DE.

(r,e) the

Then

we have, f o r

E

2 r 5 2~ :

is t h e u n i t v e c t o r o r t h o g o n a l t o r. Hence

which e n d s t h e proof of lemma

We

admit

TSIEN-CHEN SHU XING.

then

the

following

elliptic

lemma,

due

to

LI

TA

ch. 11: Slightly CompressibleMonophasic Fields

78

,Leuma

Under h y p o t h e s i s (3.20).

2 :

(3.49)

V P,

where

p,

strongly in

P

+

gradpE]

and where

p

6

L‘(6)

-div (3.51

[J,

=

L’(Z

)

0 in

r N u rD

on

p E = o

(3.50)

(3.23), we have

is t h e unique u l t r a weak s o l u t i o n of

L2(nE)

E

(3.21),

=

aii,

is t h e unique u l t r a weak s o l u t i o n

gradp]

=

6

in

Of

:

5,

)

p = o

I

on

r N U r D

=aii.

W e use now lemmas 1 and 2 t o s t u d y t h e p r o p e r t i e s of t h e mapping

w

-*

E

v

d e f i n e d by (3.39). E

Lemma 3 -(3.19),

:

(A

...,(3.24).

the solution v

(3.52)

priori

e s t i m a t i o n f o r vc)

:

Hypotheses and n o t a t i o n

There e x i s t s a c o n s t a n t c independant of

of (3.39) s a t i s f i e s

:

11 V ~ I I L m ( O T ;H1(QE)) S C l WE I L 2 ( Q E ) ’

E

such t h a t

79


Proof : M u l t i p l y i n g t h e f i r s t e q u a t i o n o f (3.39) by (3.54)) and i n t e g r a t i n g over

RE

x

p,/O,

(pE

d ef i n ed i n

y i e l d s ( a s i n t h e proof o f Theorem

]t,T[

1) :

where

11

11 ,*

is t h e norm on

V',

d u a l t o t h e norm

(((I

on

(notation

VE

*€

3.56) we g e t a l s o

:

which p r o v es (3.52). We u s e

now lemma 2 i n o r d e r t o prove (3.55).

d ef i n ed a s i n lemma 2, and l e t

x

E

@(i

) be

Let

pc~L2(RE)

a regular truncating function

such t h a t : . O l x l l Mu l t i p ly in g

p,

t h e Green formula y i e l d s

,

x

by

pEx,

1

on a neighbourhood of 0.

i n t e g r a t i n g o v er

RE

and u si n g t w i ce

:

Developping t h e l a s t term, u s i n g t h e f i r s t eq u at i o n of (3.50) using once Gr ee n ' s formula y i e l d s

:

and

Ch.II: Srightly Compressible Monophasic Fields

80

From (3.58) we get, using the fact that

] ~ ~ l ~ ~ is( bounded ~ ~ )

are bounded on

(lemma 2), that x and its first and second derivatives (and hence on QE independantly of E), and that $ € is

the restriction to

61E

independantly of

E

of

$J

€

W'."(fi)

But using the Poincar6 inequality on

:

fi yields

The sought estimation (3.55)

:

results then from (3.59),

(3.54) and (3.52). 'Lemma 4

(3.19)

:

(3.601, 0

(passing to the limit in (3.39))

,...,(3.24).

Let wcLz(6

)

and wE

€

:

Hypothesis and notation'

L2(QE),

E

> 0, be given such

that (3.61 1

x

weakly in Lz(6 1.

'W

Let v and v (3.39). (3.62)

Then

be the corresponding solutions of (3.35) and

:

VE

+

weak y in L~(o,T;~~(ii)),

v

V

(3.64)

VE + Ov weakly in L~(G), d v VE] z[OE ~d[ ( b v ] weakly in L2(a),

(3.65)

uE (defined in (3.54)+-Uiv($gradv)

(3.66)

vElrE

(3.63)

(bE

+

+

~ ( 0 , s )weakly in L2(0,T).

weakly in L2(a 1,

81

III. An Alternative Model ofMonophasic Wells

Proof :

Let u s n o t e f i r s t t h a t

u E is d i f f e r e n t from -div(QgradVE)

(this

l a t t e r d o e s n o t n e c e s s a r i l y b e l o n g t o L 2 ( a ) ! ) , s o t h a t i t is n o t p o s s i b l e t o p a s s t o t h e l i m i t d i r e c t l y i n (3.39). But

multiplicating

*

Y

E

VE, i n t e g r a t i n g o v e r

Sl

E'

fi

a l l the i n t e g r a l s over

the

f i r s t equation of

(3.39)

by a f u n c t i o n

u s i n g a Green's f o r m u l a and t h e n o n l y e x t e n d i n g

shows t h a t V E s a t i s f i e s t h e f o l l o w i n g v a r i a t i o n a l

formulation :

We g e t t h e n from lemma 3 t h e f o l l o w i n g bounds : I

where

x by

E

c

is

a

constant

independant

L 2 ( 0 , T ; H ' ( i ) ) and a subsequence o f

vE,

Hence

E.

VE,

there

exists

which we s h a l l s t i l l d e n o t e

such t h a t :

(3.69)

VE

+

OE

VE

weakly i n L 2 ( 0 , T ;

x

x weakly i n L , ( a ) ( a s BE+@ s t r o n g l y i n d

and hence

H'(ii)), V

+

dt

(3.70)

of

*"

:

V

OE VE(T)

+

O x ( T ) weakly i n L 2 ( i i ) .

L2(6)),

Ch.I..: Slightly Compressible Monophasic Fields

82

Using (3.69) and (3.70) we can now p a s s t o t h e l i m i t i n (3.671, which shows t h a t

x

satisfies :

(3.71)

But w e know from lemma 1 t h a t holds i n f a c t f o r every

in

Y

V

s t a n d a r d p a r a b o l i c e q u a t i o n (3.71). etc...

V is dense i n :

x

converge i n (3.69) and ( 3 . 7 0 ) ; moreover a s t h e s o l u t i o n

Hence (3.69),

(3.72) prove (3.62) through (3.641, and from t h e f i r s t e q u a t i o n of

(3.64)

f vJrE(t)Y(t)dt

(3.73)

0

=

/

I

x,r,

x

Y

V,

-

/

We know from lemma 2

(3.62)

that

2

@€ VEJ

v

of

and (3.65) r e s u l t s from

(3.39).

I n o r d e r t o prove

which y i e l d s (3.58). :

V

/

Y p, J, gradV,gradX

-

6

d together with

Y

Q -

V

:

(3.66), we m u l t i p l y , as i n lemma 3, )I, by p,x, Multiplying (3.58) b y a f u n c t i o n Y E L2(0,T), we g e t

p

div(J, gradx).

,+

p

s t r o n g l y i n L2(d 1,

which

and (3.65) e n a b l e s u s t o p a s s t o t h e l i m i t i n t h e

r i g h t hand s i d e of (3.73)

:

T

(3.74)

s o t h a t (3.71)

Hence t h e whole sequences V,,

(3.35) is obviously t h e s o l u t i o n of (3.71) we g e t

(3.61),

V,

is t h e unique s o l u t i o n of t h e

lim

/

v E I r E ( t ) Y ( t ) d t = - ( d i v ( J , gradv)pxY-2

/

E+O

0

d

d

-

IY 6

p

v d i v (JI gradx).

YP

J,gradv gradx


83

Using i n (3.74) t h e i d e n t i t y

yields : lim

1

E+O

0

T

T

vElrE(t) Y ( t ) d t

I v(0,t)

=

Y(t) dt

0

which proves (3.66).

0

We prove now t h e weak convergence of be given, and take, i n (3.38), w

(3.38) can b e r e w r i t t e n as

Y- -E

toward u. Let w

e q u a l t o t h e r e s t r i c t i o n of

e

Lz(a

w t o RE. Then

:

V

uo aE

+

VE(0).

I

R w and

By c o n s t r u c t i o n , of lemma 4 ( i n f a c t

8

+

{w,,

E

w strongly in

>

01

s a t i s f y t h e h y p o t h e s i s (3.61)

L z ( d ) by t h e Lebesgue convergence

theorem), s o t h a t w e can p a s s t o t h e l i m i t i n t h e r i g h t hand s i d e of (3.77) V

using t h e r e s u l t s o f lemma 4, and t h e f a c t t h a t f E -t f s t r o n g l y i n (3.78)

l i m Iu, w E+O

a

=

f0 C ( t )

which, t o g e t h e r w i t h (3.341,

v(0.t) d t

+/

f v

B means t h a t

uE

+

g v

+

+

I- uo

L2(a ) :

0 v(0)

R u weakly i n L z ( 6 ) .

We prove now t h e s t r o n g convergence of u- toward u. c

W e i n t r o d u c e as u s u a l t h e q u a n t i t y :

and prove t h a t i t t e n d s t o z e r o w i t h As

u

and

{uE,

E

>

0

E.

]

s a t i s f y now t h e h y p o t h e s i s (3.61) of

lemma 4 , we can go t o t h e l i m i t i n t h e r i g h t hand s i d e of (3.77) w

=

u

E'

and hence go t o t h e l i m i t i n (3.79)

:

with

Ch. II: Sliphtly CompressibleMonophasic Fiekis

84

(3.80) +

T a k i n g i n (3.34)

w=u

1a

uo 0 v ( 0 )

-

u2.

6

shows t h a t t h e r i g h t hand s i d e of (3.80) is

e q u a l t o z e r o a n d h e n c e c o m p l e t e s t h e p r o o f o f t h e f i r s t p a r t (3.25) o f t h e theorem. 0

Proof

of

theorem 3

t h e s e c o n d p a r t of

source solution

:

c o n v e r g e n c e of t h e d i s t r i b u t e d

a- t o t h e p o i n t s o u r c e s o l u t i o n u. c

source

Now UE is a n a p p r o x i m a t i o n of u n o t only because t h e point 6 is a p p r o x i m a t e d by a d i s t r i b u t e d s o u r c e F E , b u t a l s o b e c a u s e

the c o e f f i c i e n t s

O,Q

and t h e d a t a

(which are less r e g u l a r t h a n

0 , Q)

the distributed source solution

f,g,uo and

UE E

fE’

L2(a)

a r e a p p r o x i m a t e d by gEP

UOE

0

E

’

@E’

:

is d e f i n e d by

(3.82)

The

proof

is

very

similar

to

that

of

the f i r s t p a r t of

the

theorem: h e n c e we s h a l l omit t h e t e c h n i c a l d b t a i l s e x c e p t when t h e p r o o f is different.


Under h y p o t h e s i s (3.2O), ( 3 . 2 1 ) , (3.23) we have

-Lema 5 :

(3.83) where

85

p,

+

is t h e unique ultra-weak s o l u t i o n of

L2(ij)

p, E

~ ~ ( 0 )

strongly in

p

:

I

1 i f x E D,

- d i v ( $ gradp,)

=

me1as^ x, i n

8 , where x , ( x )

=

0 i f xLD,

E

(3.84) p,

=

and where

o p

aP

L on =r rD , @ N av

on

L 2 ( i j ) is t h e unique ultra-weak s o l u t i o n of

€

- d i v ($grad p )

in

6

=

:

-

R

(3.85)

rD , $ 2 = o o n r N .

on

p = o

The proof of t h i s lemma is s i m i l a r , b u t s i m p l e r ,

t o t h a t o f lemma

2 and s h a l l be o m i t t e d here.

6

,Lemma

w

c

:

Hypothesis

L 2 ( a ) and [w,

L2(G ) , and l e t

(3.17),

L2(8),

€

v

E

and

and ( 3 . 8 2 ) . Then we have

>

(3.20),

(3.211,

(3.23).

0} be given such t h a t

w,

+

v

(3.87)

- +

(3.88)

-div(G,

+

v

d;

:

(3.89)

F,

a

i n LZ(OT; V) weakly

dv

dt

dt

v

i n L 2 ( a ) weakly

gradGE)

+

) 0

then-

be t h e c o r r e s p o n d i n g s o l u t i o n s o f (3.35)

I

(3.86)

Let

w weakly in

+

-div($ gradv)

C ( t ) v(O,t) dt.

in

L2(6)

weakly


86

Proof

proof is similar t o t h a t of lemma 4

: The

:

(3.86),. ..,(3.88)

are

e a s i l y obtained by p a s s i n g t o t h e l i m i t i n t h e v a r i a t i o n a l formulation of

-(3.82), $-

and (3.89)

is o b t a i n e d using lemma 5 and (3.88)

5,

with

= 4

$,

7

-Le-

:

Hypothesis (3.20), (3.21), ( 3 . 2 3 ) , (3.28).

(3.90) Then

ProoP

and 0

[Ie

-t

u

:

$.

s t r o n g l y i n L2(6).

I t is very similar t o t h a t of t h e f i r s t p a r t of t h e theorem (one

:

uses

lemma 6

(3.90)

).

t o p a s s t o t h e l i m i t i n (3.81

),

v

where

because of 0

8 :

'Lemma

I, =

and

= 4

QE

(3.29) and

Hypothesis

and

(3.27). Then (3.91)

F,(;,-V)

-*

notation

of

lemma

6,

plus

(3.261,-

:

0.

4 i

Proof : Let -V,

AV =

One checks t h a t

64 =

6

-4,

6$

=

-$,

6W =

W -W.

Av s a t i s f i e s - d i v [ $ grad Av]

a t + div[GJI gradV,] avE

= 64

+ 6w

in@?G), on AV(T)

Let t h e n

p,

=

o

on

ED

i.

be d e f i n e d as :

,

aAv JI av

=

0

on

EN,

III. An AIternative Model OfMonophasic Wells

[ 0 at a% (3.93)

pE

=

-

!J,(O)

d i v [ $ gradp,]

=

= FE

,

on ED

0

87

0,

which, from lemma 7 , s a t i s i f e s (3.94)

p,

strongly

+ p

where p is t h e s o l u t i o n o f ( 3 . 9 3 ) w i t h satisfies,

f o r some neighbourhood

r e g u l a r i t y of 0 , $ )

-

where

v

C ( t ) 6 r e p l a c i n g F,.

of

the o r i g i n 0 ,

Moreover,

PE

(theorem 1 plus

:

s a t i s f i e s ( f r o m lemma 6 ) :

(3.97)

As

n=2

For

E

3

or

we g e t from t h e S o b o l e v imbedding theorem :

small enough t h a t D

and (3.98)

:

C

9,

we g e t from ( 3 . 9 6 ) u s i n g ( 3 . 9 5 ) ,

(3.97)


88

But,

s o t h a t (3.99) becomes

11 FE Avl

2

:

c I16811m

a

which t e n d s t o z e r o u s i n g (3.26), Bw

+

c IISJIII,

(measDE)

a (3.27),

( 3 . 9 1 ) and t h e weak convergence o f

towards zero. T h is ends t h e proof o f lemma 8. 0

The proof of t h e convergence o f

cIE

+

u

is t h en done, as i n t h e

proof o f t h e f i r s t p a r t of t h e theorem, by p a s s i n g t o t h e l i m i t i n (3.81) u s i n g lemma 6 and 8. T h i s ends t h e proof o f theorem 3.

89

CHAPTER

INCOMPRESSIBLE

I -

111

TWO-PHASE

RESERVOIRS

INTRODUCTION

We shall consider in this chapter the flow of two incompressible immiscible fluids through a porous medium. Though this problem is of great practical importance, as it corresponds to the simplest case of secondary oil recovery technique, where the resident oil is displaced by injected water,

it

has received only recently attention from the mathematical

community, despite the huge amount of oil engineering technical literature on the subject. The reason for this may be that even this "simple" (for oil engineers) model has such a complicated and non-standard structure that the usual mathematical tools cannot be applied in an evident manner. The key to the reduction of these two-phase equations to the more familiar system elliptic

of one parabolic saturation equation coupled with an

pressure

equation

is

a

mathematical

transformation of

the

equations, which replaces the two pressure unknowns (one per phase) by only one

pressure

unknown,

called

the

global, or

the

reduced,

or

the

intermediate pressure. This transformation was discovered independently by CHAVENT [ l ] , authors

in 1975 (q9globalpressure") and by two several some Russian

("reduced

ANTONCEV-MONAHOV).

pressure"

see

the

references

1

though

3

of

A detailed explanation is given in section I1 below,

together with a careful discussion of the choice of boundary conditions. The resulting system of equations is summarized in section 111.

As we already mentioned it, rather few mathematical results have been available for this problem. In order to organize logically these results,

we

emphasize

first

the

importance of

the

notion of water

breakthrough time, which corresponds to the time at which the injected water first starts being produced with the oil at a given production well.

Ch.III: Incompressible Two-Phase Reservoirs

90

This time is economically important, as the water oil ratio (WOR) increases very quickly after the water breakthrough time, so that the production well has to be turned off. Depending on the boundary conditions which are used for the model, the breakthrough phenomenon may or may not be well represented. One other mathematical

difficulty

associated

with

this

problem

is

that, under

standard conditions, the parabolic saturation equation is degenerate, (and practically

very

close

to

a

first order hyperbolic equation).

This

degeneracy may or may not be taken into account. Among the papers which do not take properly into account the breakthrough phenomenon, we find CHAVENT [l],

1975 (existence theorem for a

degenerate parabolic equation with simple Dirichlet and Neumann conditions, coupled with a family of elliptic pressure equations), KRUZKOV-SUKORJANSKI,

1977 (existence of classical solutions for the non degenerate problem with Dirichlet and Neumann boundary conditions), ANTONCEV-MONAHOV, 1978 (existence of weak solutions for the degenerate problem with Dirichlet and Neumann boundary conditions, plus some regularity and stability results for simplified problems), ALT-DIBENEDETTO, 1983 (existence of a weak solution for

the

degenerate

problem

with

two

unilateral

overflow

boundary

conditions, these conditions do not allow simultaneous production of oil and water) KROENER-LUCKHAUS, 1984 (existence of solutions for the partially degenerate problem with Dirichlet and Neumann boundary conditions, this author works with the original set of equations, not the transformed one). Concerning the models that take properly into account the water breakthrough phenomenon, an adequate unilateral boundary condition was formulated in CHAVENT [lbis], without an existence theorem (the main part of the paper was devoted to mathematical problems related to the estimation of the non-linearities appearing in the saturation equation). Existence theorems for the resulting degenerate variational inequality were given in CHAVENT [ Z ] for one-dimensional problems (where the pressure and saturation equations decouple), and will be given in section V of this chapter for the general multidimensional case. Regularity results, a description of the asymptotical behaviour, a precise definition and some properties of the water breakthrough time can be found in the work of GAGNEUX [l]-[4]

in the

case where the saturation and pressure equations decouple; part of these results are recalled in this chapter (still in section V).

So we b e l i e v e t h a t t h e material i n s e c t i o n V below is t o d a y ' s most

comprehensive

mathematical

treatment

of

the

two-phase

equations,

a s it

takes i n t o a c c o u n t t h e l a r g e s t number o f r e l e v a n t p h y s i c a l p r o p e r t i e s .

The reader may h a v e n o t i c e d t h a t none of t h e p a p e r s was c i t e d f o r a u n i q u e n e s s theorem. T h i s is b e c a u s e o f t h e c o u p l i n g between t h e p r e s s u r e and

saturation

r e g u l a r i t y of still have

equations,

which

the solution

t o get

such

a

theorem,

it

difficult

t h e uniqueness.

to

obtain

enough

T h i s problem is open

case, t h o u g h KRUZKOV-SUKORJANSKI claim t h e y

t h e non-degenerate

for

makes

because

they

suppose

i n the h y p o t h e s i s

that

the

s o l u t i o n is i n d e e d r e g u l a r .

Tn o r d e r t o be d e f i n i t e , we make p r e c i s e now t h e way i n which t h e reservoir described

i n c h a p t e r I w i l l b e p r o d u c s d t h r o u g h o u t Lhis c h a p t e r

(cf. Figure 1 ) : ( 1 .l )

r II

The l a t e r a l boundary

is s u p p o s e d c l o s e d ;

-

water is i n j e c t e d t h r o u g h t h e wells 1 , 2 , . . . k ,

-k

(1.2)

:

=

U k= 1

rk

and we d e n o t e by

t h e i n j e c t i o f ? boundary.

...K, and

o i l is prodilned til-ouyh t h e r e m a i n i n g wells k+l , K

(1.3)

we d e n o t e by

rs

=

U

-

T k : t h e p r o d u c t i o n boundary.

k= k+ 1

Figure 1 : Secondary recovery of an o i l reservoir

Ch. III: Incompressible nYo-Phase Reservoirs

92

For the sake of simplicity, we will consider in the sequel the case of one

injection well and one production well

11-

CONSTRUCTION

We

first

give

OF

the

(z=l,K = Z ) .

THE

STATE

characteristics

EQUATIONS

of

the

two

fluids

(see

paragraphs 111.3 and IV.3 of chapter I) They (2.1 1

are

both

chapter

supposed where

11,

incompressible (in opposition to

compressibility

phenomenon). Hence B = B =1 1 2-

was

the

driving

.

They are immiscible, so two distinct phases are present in the pores of the porous medium. Hence we have pressures (2.2)

P1

and

P2

corresponding

two

respectively

distinct to

the

pressure in the wetting phase (water) and tp the pressure in the non-wetting phase (oil).

pj

> 0 is the mass of a unit volume of the

jth fluid,

j=l ,2,

(2.3)

>

0

is the viscosity of the jth fluid, j=1,2.

Let us define at each point x

6 Ci

:

!+J

@ . = flow vector of the jth fluid, j=1,2

(cf. ( 3 . 1 2 ) in chapter I for a precise definition).

s(x)

=

saturation in fluid 1 at x

=

vol of fluid 1 around x, vol of fluid 1 + 2

(2.5) 1 - s(x)

=

saturation in fluid 2 at x.

So we have now three dependant variables

:

93

I1 Construction of the State Equations

(2.6)

11.1

-

pressure of the wetting phase (water),

P1

=

P2

=

pressure of the non wetting phase (oil),

?.

=

wetting phase (water) saturation.

THE EQUATIONS INSIDE

Inside 52,

Sl :

THE NOTION OF GLOBAL PRESSURE

the Darcy law applies separately to each of the two

fluids (cf. Chapter I 3111.3.1) with a reduction

k . rJ

where the relative permeabilities

k , of permeability : rJ

depend on

S as shown in figure 7

of chapter I, and also generally on x .

-

A s can be seen in figure 7 of chapter I, the saturation S

remains always,

as long as only displacement phenomena are considered, which is the case in this chapter, between

5,

and

:

is the water residual saturation : for S 6 Sm' there is m' so little water that it is "trapped" by the capillary forces in the pores

.

of the porous medias and can no longer be displaced.

-

1-s,

oil

is the

residual saturation and

is

interpreted

similarly to 'ma

A s the rock and the fluids are supposed incompressible, the only

accumulation term is due to a change in saturation so the conservation laws

for each of the two fluids are (compare with (1.46) of chapter I)

(2.9)

O(X)

@(x)

(2.10)

a ( x ) $(x)

+

xa

i,

div = 0, V xcn, (water conservation law)

-

V t c 10,TC,

-t

div $2 = 0, fF x (oil conservation law)

(13)

+

:

c 52,

V

t E l0,TC.

Ch.III: Incompressible Two-PhaseReservoirs

94

At this point we have, in n, only two equations (2.9), the three unknown

P 1 P2

s.

and

(2.10) for

The missing equation is given by the

capillary pressure law (see (3.15) of chapter I) (2.11 1 Pl - P2 = PC(S,X), depends on

where P

-

as shown i n figure 8 of chapter I.

S

Two things are to be noted concerning these capillary pressure curves :

- They always have a positive derivative

:

(2.12)

-

S [,

- The capillary pressure vanishes for only one value

-

interval :

S ],

- P 1 - P = P =o s = s . 2 c Usually Sc = SM (resp. Sc = Sm) when

(2.13)

(resp.

-

Sc of the

-

non-wetting

< 5e < m

S

phase)

saturation.

However

-

is the wetting phase

S

it

can

happen

that

the reduced

water

zM.so we shall distinguish in the sequel between -Sc and -sM.

In order to simplify notation, we shall now use saturation S instead of the actual saturation S :

so that equations (2.7), (2.9),

as at

div

i,

...,(2.11)

(2.15)

@(x)

(2.16)

Q(x) ~ ( 1 - S +) div $2

(2.17)

P1

-

P*

(2.18)

$.J

=

-@(x) k.(S,x) grad [ P . - p .

+

a

=

:

0, +

=

become

=

0,

PC(S,X),

J

J

J

g 21,

j

=

1,2,


95

where we have defined (caution than in chapter 11)

:

Q and J, represent different quantities

:

kr.(Sm+S(zM-zm),x) (2.21)

k.(S,x) J

=

=

u.

mobility of the

jthfluidj=1,2,

J

P (s,x)

(2.22)

=

P~(S,+S(Z~-S~), x)

=

capillary pressure.

In all the sequel we shall make the following assumptions on the spatial dependance of

k. and

P

J

:

The mobilities, as functions of the reduced saturation, are independant of x

(2.23)

k.(S,x) and

=

:

k.(S),

+XER,

J

:

the (2.24)

capillary

pressure,

as

saturation, is independent of P (S,X)

=

a x

fuhction up

of

the

reduced

to a scaling factor

:

PCM(X) PC(S),

where (2.25)

(2.26)

I

I

P

(x) 2 0 is the maximum of the absolute value of the

capillary pressure at the point x, CM pc(S) is a dimensionless function such that -1 2 pc(S) 51 and pc(Sc) = 0. With the choice (2.25) we see in figure 8 of chapter I that p is always an increasing function of S .

The hypotheses

(2.23),

(2.24)

are

very

usual, as

they

are

sufficient to provide a good model of the important phenomena f o r the oil engineers

(including

heterogeneous media).

for

example

counter-flows

by

imbibition

in


96

However, equations (2.151, (2.16) are not suited for a proper mathematical study

:

for example

disappears as

k (1)

2

in

the zones of

where S = 1 ,

equation (2.16)

0; similarly equation (2.15) disappears in the zones

=

where S EO.

So we are now going to transforme equations (2.15) through (2.18), under hypotheses (2.23), (2.241, into a more tractable form. We shall carry out this transformation in 3 steps

Step 1 (2.27)

:

Sum equation (2.15), (2.16) to get

:

:

+ + div ($1+$2) = 0.

Use then the algebraic identity

and (2.15) to get

(2.29)

Q

:

as

- div { $

k t k2 grad

1: (P - P gz 1- ( P 2 - ~ 2 g z) 1 )

4

kl+k2 +

div {

-

+

+

Using (2.17) in (2.29) and adding (2.28) to (2.31) gives

:

(2.30)

In order to simplify notation we define

:

S

(2.32)

a(S)

=

\

a(s) ds

increasing (viscosity-’)

b kl (S)

(2.33)

bO(S)

= kl

(S)+k2(S)

“fractional flow1’,increasing (dimensionless)


97

(2.34)

(2.35)

Pc (x)

(2.36)

=

-p

m g Z(x)

=

gravity potential (same dimension as a pressure).

r

(2.37)

I

(2.38)

@

as +

at

[PCM(x) grad a(S)+bl(S) grad PCM(X)

b2 ( S ) grad PG(x)I]

+

div ($1

[$(XI

div

+

+

G2)

=

+

+

+

div {bO(S)(01+02)l

We see that in equations ( 2 . 3 7 ) , S

= 0,

(2.38)

the only dependant + + $1+$2,

and the water + oil flow Vector

which itself depends on S, P1 and

P2

through the definition ( 2 . 1 8 ) .

The notion of global pressure (CHAVENT

:

+

0.

variables are the saturation

Step 2

-

(2.27) with the above notation

We rewrite now equations (2.301,

[11).

We show now that it is possible to introduce a new unknown called the global pressure, which is a point function of

so that the water+oil flow

+

+

$1+$2

S,

P1

P

and P 2 ,

can be expressed in terms of S, P and grad

-P only. Thus the number of unknowns reduces to two. We define first some notation

-

corresponding to Sc Y(S) =

is the reduced saturation

defined in ( 2 . 1 3 1 , so pc(Sc) S

(2.391

(Sc

J

1 dPc (bo(S)- 7 ) d~ ( S ) d s

=

0) :

(dimensionless),

sC

(dimensionless). One has obviously (2.41)

:

‘f(S)+Y1(S) = (bo(S)

- 71 ) pc(S).

Ch.III: Incompressible nYo-Phase Reservoirs

98

A o(=/oSa (T) dT

1 Figure 2

Figure 3

:

:

S b

: 0

1

The functions a(S) and a(S)

The shapes of the functions ,,b

b,

and

b2

s

99

I1 Constructionof the State Equations

pressure P by : P(x,t)

(2.42)

=

7 1 CP,(x,t)

+

P2(x,t)l

One checks then easily, as

pc(Sc)

+

=

Y(S(X,t))

0 and

PCM(X).

O

01 = p r o d u c t i o n bolundary.

V

R

F i g u r e 6 : The p a r t i t i o n of Remark 7 :

r

into

r-

For a f i e l d w i t h c l o s e d o u t e r b o u n d a r y , made up of t h e o u t e r boundary boundary well

re

and

boundary

imbibiLion

or

r+

rs.

r

r+

and

r-

is g e n e r a l l y

and t h e i n j e c t i o n w e l l

.?

is g e n e r a l l y made up of t h e p r o d u c t i o n Xowever,

laboratory

in

some e x p e r i e n c e

displacements,

the

such

notion

as of

i n j e c t i o n and p r o d u c t i o n boundary may n o t be c l e a r l y d e f i n e d ,

so we u s e ( 2 . 7 6 ) One

key

for,

i,+le

dilicil

choice

:?.?not, g i v e r i s e t o a n y a m b i g u i t y .

of

physically

admissible

saturation

boundary c o n d i t i o n s is t o remember t h a t t h e y have t o s a t i s f y t h e r u l e of p r e s s u r e c o n t i n u i t y f o r e a c h o f t h e f ' l ? l i d ? ?'Loding d w o s s t h e boundary

:

II. Construction of the State Equations

109

- if only one fluid is flowing across the boundary, its continuity is ensured by the pressure boundary condition, and the boundary saturation is unimportant. - if two fluids are simultaneously flowing across the boundary, their two pressures unique pressure

P1

Pe

and

P2

r

on

have to be both equal to the

so that using the property (2.13) of the

existing outside of the porous medium Sl,

on I", i.e. P -P = O 1 2 capillary pressure curve, this implies

necessarily

s=SC-

(usually S = 1

if

is the

S

wetting fluid saturation). So a necessary condition for our saturation boundary conditions to be physically admissible is that they satisfy

11.3.1

-

Saturation boundary conditions on the injection boundary

One can use for

-

11.3.1.1

where

:

r- two types of boundary conditions

r-

:

Dirichlet condition :

Se is a given boundary saturation.

Remark 8

The boundary condition S=S

C'

which satisfies (2.77), can be

used, in the usual case where

S

=1,

for the modelling of

water injection. However, when the capillary diffusion term div

f

in

(2.54)

is

non-zero,

this

boundary

condition

generally leads to a production of oil through the water injection

boundary;

though

this

oil

production

can

be

observed under certain experimental circumstances, it is not present under the usual field conditions. So condition (2.78) has to be used for the modelling of water injection at high injection rates only, for which the parasitic oil production occurs only during the very short period when the porous

Ch.HI: Incompressible f i d h a s e Reservoirs

110

medium is not yet saturaLed with water in t h e vinicity of the injection boundary I#--( cf. fig.") 0

- oil

-

4-'p2

no oil flow

- -water -

,water

9'

'1

+

+ oil +water

'0

st

,oil+water

'0

oil +water

!

1

0

r-

spacell

spacer,

AT INITIAL

DURING

Figure 7

A

SHORT PERIOD

TIME :

space

LATER

Approximate modelling of water injection by a S=1

Dirichlet boundary condition in case of high injection rates Remark 9 :

Conditions ( 2 . 7 8 ) with

Se<Sc is mathematically correct, but

does not necessarily satisfy (2.77). However the condition S=O

can be used for,the modelling of oil injection high

injection rates, as the p a r a s i t i c water production (during when (2.77) is violated) occurs only during the short period where the porous medium is not yet saturated in oil in the vicinity of 11.3.1.2

-

r-.

Given water injection rate :

0

111

II. Construction of the State Equations where

:

Q.(s,t)

jth fluid production rate density at point

=

s and

(2.80) is negative in case of injection).

Remark 10

:

On the part of

fluid), -

+ + qo-v

T- where

< 0 (global injection of

one can model injection of

pure water or Oil by

taking the following values for Q , in (2.79)

+

(2.81)

Q1

=

(2.82)

Q

=

1

+

qo-w < 0 0

Q2

0

=

:

water injection,

oil injection.

A water injection rate

Q1

different from (2.81) or (2.82)

would generally lead to simultaneous flow of water and oil

Remark 1 1

:

and hence violate (2.77) as we could not ensure that

S

would remain equal to 1.

0

On the part of

r-

where

+ + qo*v

=

0

(globally closed

boundary) one can model closure conditions by taking Q 1=0, which automatically ensures Q2=0. Remark 12

:

In the case where the overall water injection rate Ql,(t) known through only a part

rw

of

is

r-, (2.79) can be replaced

by

(2.83) IS

=

unknown constant on

rW.

The second condition of (2.83) is mathematically and practically convenient, but has, to my knowledge, no physical justification.

0

112

Ch. III: Incompressible llvo-f'hase Reservoirs

-

11.3.2.

One c a n u s e f o r

-

11.3.2.1

where

r+

Saturation boundary conditions on the production boundary

r+

three t y p e s of boundary c o n d i t i o n s :

Dirichlet conditions :

Se

is a g i v e n boundary s a t u r a t i o n .

Remark 13 :

For

a

displacement

s a t u r a t i o n is S

=

can

0

without

of

used

be

oil

over

0

violating

by

water,

the f i e l d ,

where

condition

the

initial

(2.84)

with

b e f o r e t h e w a t e r b r e a k t h r o u g h time,

(2.77).

However

condition

this

is

not

a d m i s s i b l e a f t e r t h e b r e a k t h r o u g h time, when o i l and w a t e r flow simultaneously. 0

Remark 14 :

The D i r i c h l e t c o n d i t i o n ( 2 . 8 4 ) w i t h :

always s a t i s f i e s (2.77) However,

generally leads, term,

to

and t h u s is p h y s i c a l l y a d m i s s i b l e .

b e f o r e t h e water b r e a k t h r o u g h time t h i s c o n d i t i o n i n t h e p r e s e n c e of

intrusion

p r o d u c t i o n boundary experimentally

of

water

r+ ( c f . f i g .

observed.

in

a capillary diffusion field

the

through

the

8 ) , which is g e n e r a l l y n o t

However,

condition

this

s i m p l e and l e a d s , t o g e t h e r w i t h S=S

on

r-

is

very

( c f . Remark 8 ) t o

s i m p l e homogeneous D i r i c h l e t boundary c o n d i t i o n s f o r t h e o i l saturation

Remark 15 :

1-S

i n t h e u s u a l c a s e where Sc

We have s e e n t h a t t h e water

S=O

=

1.

is a good c o n d i t i o n on

b r e a k t h r o u g h time (remark 1 3 ) and t h a t

0

r+

before S=S

is

s a t i s f a c t o r y a f t e r t h e b r e a k t h r o u g h time (remark 1 4 ) . Thus a more r e a l i s t i c c o n d i t i o n would be t o t a k e t o 1 a t t h e b r e a k t h r o u g h time. Dirichlet

But

this

Se jumping from 0

is n o t a s i m p l e

c o n d i t i o n , as t h e b r e a k t h r o u g h time is n o t known

II. Construction of the State Equations

113

oil

oil%

2 ‘

4

water c1‘-

no water flow

oi i+water -3L

4

oil +water

L

-

&water

n

1

D

Oh

initial time Figure 8

:

r+r-

space

stationary curve

space

transitory period

The effect of an

production boundary

S-Sc

T+

b e f o r e h a i d . 31,

r+

later

Dirichlet boundary condition on the

f o r a sufficciently long 1-D field.

W H Iirtvc? t n

u s e t h e u n i l a t e r a l c o n d i t i o n which

we describe now.

0

- Unilateral boundary condition

11.3.2.2

On t h e p r o d u c t i o n boundsry w e t t i n g phase

S

Y+, a s long a s t h e s a t u r a t i o n of t h e is s t r i c t l y less t h a n Sc, we see from (2.77) t h a t

e i t h e r water o r o i l , but n o t b o t h , can flow o u t of forces,

the w e t t i n g phase

(irater,

i n c a s e of

i n s i d e t h e porous media as l o n g a s t h e s a t u r a t i o n less t h a n

(2.85)

Sc,

s o t h a t we g e t

s < sc

+

+

$, .LI

>

=

:

0

on

R.

Due t o c a p i l l a r y

w a t e r + o i l flow) is r e t a i n e d

r+.

S

remains s t r i c t l y

Ch. III: Incompressible W o P h a s e Reservoirs

114

S becomes equal to Sc, then water is allowed to get out of

When

the field : this is the breakthrough time. We shall suppose that no water

r+,

is available outside the porous medium on only be directed outward from +

(2.86)

$l'v

+

so that the water flow can

:

r+.

on

2 0

Thus conditions (2.85) (2.86) give the unilateral production boundary condition :

on

(2.87) (Sc-S)

t (s) BT

(2.88)

+

+

$,*V

=

=

r+,

f

t

6

10,T[,

0,

first time when S becomes equal to S at s c

r+.

J

We have drawn for the case of boundary condition (2.871, typical variations of S with time at s

E

r+

(figure 9) and with space for given

times t (figure 10). Remark 16 :

In the usual case where

S

=1, C-

one has simultaneously on

r+

after the breakthrough time : (2.89) which implies, as condition boundary

a(l)=O,

introduces a

r+

that

boundary

as au

=

~

3

.

So the unilateral

layer on

the production

after the breakthrough time (cf. figure 1 0 ) . 0

I1 Constructionof the State Equations

115

s4

Figure 9 : Evolution of the water saturation on the production boundary

r+

with the unilateral boundary condition (2.87).

+-

4 -

P1.v >o

P1.v = 0

space phase @ of figure 9

r+


r+


Figure 10 : Profiles of the water saturation inside the porous medium D near the production boundary

r+

at different times, when

the unilateral boundary condition (2.87) is used.

r+

116

11-3.2.3

Ch. III: Incompressible n v d h a s e Reservoirs

- Given water/oil production ratio (MR). A

r+

widely used condition for

is obtained by neglecting the

boundary layer introduced by the unilateral condition, and by requiring

r+ be proportional to their respective

that the water and oil flow through mobilities on

l‘+

:

+ - +

$l.v - = + + @2*v

(2.90)

kl(S)

a~ t

E

lO,TC,

k2(S)

or equivalently : + +

r-v +

(2.91)

1

+

+

b.(S) qj.v

=

J

0 on

r+

fF t c 10,TC.

j = l ,2

This boundary condition does not satisfy ( 2 . 7 7 ) . +

Since the water+oil flow field

qo

becomes very large near the

well as the diameter of this latter is very small, one checks easily from (2.60),

that near the well (but outside of the boundary layer) one

(2.61)

always has, whatever the saturation boundary condition is, that + kl( S )

$1

(2.92)

i,

z

:

k2(S)

which shows that condition ( 2 . 9 0 ) or ( 2 . 9 1 )

and the unilateral condition

become equivalent for small well diameters. Remark 17

:

It can be a l s o convenient to use a slight variant of ( 2 . 9 0 ) or ( 2 . 9 1 ) . Let

(2.93)

can take

:

1 (:-;

+

1

‘w

j=1,2

S

=

rw

b.(S) 3

c

r+

be one production well boundary. One

+ - +

qjw)

unknown constant on

=

0

‘W

t t c l0,TC V t

E

10,TC.

We shall use this condition to find the equivalent point source model for the saturation equation.

0

III. &Summary of Equations

111-

117

OF

SUMMARY

EQUATIONS

INCOMPRESSIBLE

111.1

-

=

rock porosity,

(3.2)

K(x)

=

rock permeability,

(3.3)

o(x)

=

section of field,

(3.4)

Z(x)

=

depth.

P

(3.6)

P

111.3

1

ROCK

Q

.. .at point x

2

=

pressure inside the wetting phase,

=

pressure inside the non-wetting phase,

=

wetting-phase saturation.

S

-

FLOWS

Q

PHYSICAL U"0WWS

(3.5)

(3.7)

AND

CHARACTERISTICS DEPENDING ONLY ON THE RESERVOIR

@(x)

-

THO-PHASE

FLUIDS

(3.1)

111.2

OF

CHARACTERISTICS DEPENDING ONLY ON THE FLUIDS

of fluid j , (3.9)

(3.11)

p. =

j = l,2

viscosity

The two fluids are incompressible, i.e.

p1 =

cste,

p2 =

CSte

FOR

118

Ch. III: Incompressible nvliuoPhase Reserwoirs

111.4 CHARACTERISTICS DEPENDING BOTH ON FLUIDS AND ROCK

(3.12)

(3.13)

-

Sm(x)

-

SM(x)

=

wetting phase residual saturation at point x

=

1-non-wetting phase residual saturation at point x E R ,

ZM(x)]

k .(s,x) : [sm(x),

x

R

[O,l]

-f

E

R,

relative permeability

=

(3.14) j=1,2,

of fluid j,

(3.15)

P1

-

P

2

=

P (2.x) : [sm(x), sM(x)] c

x

R

+

R

=

capillary pressure.

Hypothesis : If one introduces : S(x,t) - s,(x) (3.16) S(x,t) = then we suppose that

kr.(Sm(x) J

reduced saturation of the wetting phase,

:

-

k.(S)

=

- Sm(X)

SM(X)

=

+

S(FM(x) - sm(x)), x) =

u J.

(3.17)

mobility

is a function of the reduced saturation S only and that there exist functions PCM(x) and pc(S) such that :

where : PCM(x) 2 0

(3.19)

=

maximum of the absolute value of the capillary pressure at

(3.20)

I

x

E

R,

reduced capillary function (dimensionless, increasing), with -1

S pc(S) 5 1,

pc(Sc)

=

0 (where usually

Sc=l).

III. Summary ofEquations

119

PCM (x)

(3.21)

maximum capillary pressure at x c i2 (defined in

=

(3.19)), (3.22)

PG (x)

(3.23)

Q(X)

=

(ZM(XI

(3.24)

$(x)

=

o(x) K(x),

(3.25)

;,(x)

=

-$(x) grad PCM(x) (governs the effects of capillary

(3.26)

G2(x)

=

-$(x) grad PG (x) (governs the effects of gravity).

-p

=

m @(x)

=

gravity potential at x

- Sm(X)l

6 0,

o(x) $(X),

pressure heterogeneity),

111.5

-

AUXILIARY DEPENDANT VARIABLES + =

flow vector of the wetting phase,

=

flow vector of the non-wetting phase,

r

=

part of

+ qo

=

+ + $ l + O2

$

(3.27)

1

+ $2

(3.28)

+

(3.29) (3.30) 111.6

-

+

-t

$

=

1

and of -$2 due to capillary diffusion,

global flow vector.

TRACES ON r=an OF THE DEPENDANT VARIABLES They may be known o r unknown depending on the type of the boundary

condition used.

(3.31)

Pe

=

trace of pressure, (index e stands for "exterior") trace of saturation,

(3.32)

Se

(3.33)

Q1

(3.34)

Q2

=

trace of " * v

(3.35)

Q

=

trace of qo * =~ global production rate density

=

trace of

=

+

$1

+

+

-

=~

=

+

wetting-phase production rate density through r , non-wetting phase production rate density through r , through

Caution

'

Q1

9

Q2

r.

and Q are negative in case of injection into R .

Ch. III: Incompressible ThwPhase Reservoirs

120

r

111.7-PARTITIONS OF THE BOUNDARY

(3.36)

r

=

(3.37)

rD

=

OF THE POROUS MEDIUM Q

r D u rN u r W

where (cf. fig. 1 1 ) :

r where the pressure is specified (D stands

part of

f o r Dirichlet),

rN = part of r where the global flow is specified (N

(3.38)

stands for Neumann),

rw = part of r through which the overall global flow is

(3.39)

specified (W stands for well),

r- u r+ with r-

(3.40)

r

(3.41)

r-

=

[

s

6

rl

Q

=

+ + q o - L 5 0 }=global injection boundary,

(3.42)

r+

=

{

s

6

rl

Q

=

+ + qo-v

111.8

=

n

>

r+

0 where (cf. Fig. 1 2 )

0 }

=

:

global production boundary.

- FUNCTIONS AND COEFFICIENTS DEPENDING ON -

REDUCED SATURATION S

ONLY

(3.43) (3.44)

k.(S)

mobility of jth fluid (defined in (3.17)

=

J

p (S)

j=1,2,

reduced capillary function defined in ( 3 . 1 8 ) , ( 3 . 2 0 ) ,

=

(3.45)

a(S)

=

(3.46)

a(S)

=

k l k2 dPc kl k2 dS ’

positive ,

+

S

a(t) dt,

increasing,

0

(3.47) (3.48)

bo(S) bl(S)

=

=

kl ,

increasing (fractional flow),

k t + k2 k l k2

-PCW

I

k\+k2 k1

(3.49)

bZ(S)

=

k2

k1 + k2

(P,

- P2) Pm

’

III. Summary of Equations

121

reduced saturation f o r which pc(Sc)

(3.50)

S

(3.53)

Y2(S)

=

=

kl p1

+ k 2 p2

x

-

d(S)

(3.54)

(3.56)

’

(kl + k2).

=

111.9 - MAIN DEPENDANT VARIABLES

(3.55)

(cf. 3.20)),

0

1 Pm

k l + k2

=

:

S

=

reduced saturation (defined in (3.16)),

P

=

-21

(P +P 1

2

) +

PCM Y(S)

=

global pressure.

111.10 - EQUATIONS FOR PRESSURE, SATURATION AND PLOW VECTORS

Equations governing the global pressure P f o r every t

- Inside

E

C0,Tl

k l :

+

(3.57)

div qo

(3.58)

Go

0,

=

2 =

-$d(S) gradP

+

d(S)

1

Yj(S)

Gj.

j=l

-

On the boundary

(3.59) (3.60)

P

=

+

r

(we use the partition

P on r e

+

=

TD u TN u

rw

:

(Dirichlet),

D

qo*v = Q on

r

rN

(Neumann),

r’ (well-type) P

=

P

+

unknown constant on

rw.

:

Ch. III: Incompressible nvvo4hase Reservoirs

122

Equations governing the saturation S inside Q

(3.63)

=

OxlO.T]

r’ =

:

-$PcM grad a ( S ) .

-

On the boundary

-

on the global injection boundary

(3.64) or (3.65)

-

S

=

r

S

=

r,)

:

:

(Dirichlet),

S

+ + @,-v

r = rr- one can take

(we use the partition

=

(Neumann);

Q1

on the global production boundary

(3.66)

:

r+ one can take

:

(Dirichlet),

S

or +

+

+

+

(unilateral), (3.67) S 5 Sc, @1. V , ( S c - S ) q , * L = 0 or + + + + (3.68) r-v + 1 b. qv: = 0 (WOR equal to mobility ratio). j=1,2

- At the initial time t=O (3.69)

S

=

So(x)

:

on a.

III. Summary of Equations

123

Figure 12 : Example of saturation boundary conditions compatible with the pressure conditions of fig. 1 1 when Pe is constant over

rD-

Ch. III: Incompressible nyO-Phose Reservoirs

124

Equations for separate phase pressures and flows (3.70) (3.71) (3.72) (3.73)

P1

=

P

-

[Y(S)

-

1

PC(S) 1 PCM’

:

IV. An Alternative Model for Diphasic Wells

IV - A N

125

ALTERNATIVE MODEL F O R D I P H A S I C YELLS

As we have done i n c h a p t e r I1 f o r t h e case of monophasic wells, we

s h a l l t r y here t o r e p l a c e t h e boundary c o n d i t i o n s used i n s e c t i o n s I1 and

I11 f o r t h e model f o r d i p h a s i c wells by p o i n t s o u r c e s a p p e a r i n g i n t h e r i g h t hand s i d e of t h e e q u a t i o n s ( d i s t r i b u t e d s o u r c e s can t h e n be o b t a i n e d by approximating t h e D i r a c f u n c t i o n s ) . The d i p h a s i c e q u a t i o n s being much more i n t r i c a t e t h a n t h e monophasic o n e , we s h a l l proceed f o r m a l l y o n l y .

We c o n s i d e r t h e t y p i c a l s i t u a t i o n of a c l o s e d f i e l d i n j e c t i o n well DE- and one p r o d u c t i o n well D

QE

w i t h one

( c f . f i g u r e 1 3 ) , and suppose

E+

that :

where a , b are t h e ' l c e n t e r s l l of t h e i n j e c t i o n and p r o d u c t i o n wells.

We production

take well

as p r e s s u r e the

boundary c o n d i t i o n s a t t h e

well-type

condition

(3.61),

where

( w a t e r + o i l ) i n j e c t i o n and p r o d u c t i o n r a t e , and where on

r

e

Q T ( t )is t h e e q IS t h e t r a c e

of a given regular function. EL

boundary

We t a k e t h e n as s a t u r a t i o n boundary c o n d i t i o n on t h e p r o d u c t i o n TE+

aDE+

=

the

r a t i o " c o n d i t i o n (2.90).

variant

(2.93)

of t h e l'WOR e q u a l t o m o b i l i t y

which becomes e q u i v a l e n t ( a t l e a s t f o r m a l l y ) t o

t h e u n i l a t e r a l c o n d i t i o n when

E+O.

On t h e i n j e c t i o n boundary

The e q u a t i o n s g o v e r n i n g t h e p r e s s u r e the saturation SE

-

( d e f i n e d up t o a c o n s t a n t ) and

are t h e n : 0 with

=

qoE

PE

rE-=aDE- we

QIT (condition 2.83)).

t a k e as g i v e n t h e o v e r a l l water i n j e c t i o n r a t e

(4.2)

P =P

i n j e c t i o n and

=

0

;

OE

on

r.

=

- @ d E gradPt

+

dE

/ J=1

+t

6

]O,T[,

+ YjEqjE,

i n RE,

126

Ch. 111:Incompressible n v d h a s e Reservoirs

i

J

(4.3)

sly

-Q ,T ,

=

*

$,€

=

c o n s t a n t on

I

+.+ (r.v

+

2 + + 1 b. 9 . j=1 J C J E

o ,

=

SI

Let

so

=

ii

=

on R

c o n s t a n t on L

E+

be t h e r e s e r v o i r ; we u s e t h e same kind of

DE+

t e c h n i q u e a s i n c h a p t e r 11, s e c t i o n 111, but f o r m a l l y o n l y .

We go f i r s t t o t h e l i m i t i n t h e p r e s s u r e e q u a t i o n ( 4 . 2 ) : The p r e s s u r e P

where

v

satisfies, f o r every

depends on

w

t

E

]O,T[

:

by :

- d i v ($d

gradvE)

=

w,

av $dE

r E J.

av

>=

o ,

a\,

=

vElr

,

=

constant,

EJ

8VE *dE

’

at t=O.

-

D

R~

= E+

E+

s

zE-,

E-

I. E-

0 , on

r.

One can go t o t h e l i m i t f o r m a l l y i n (4.7) i f

j =

+,-,

IV. An Alternotive Model for Diphasic Wells

Figure 13

:

127

The field used to determine equivalent point sources

which is expected if

v.+

and the functions whose

Pet+

are the traces are

regular enough. We get then

(4.10)

-1 Pw

=

Q,(t)

v(a

w

E

LZ(Sz)

s.t

n

Y

-R

w

=

0.where v depends on w by

:

Equations (4.10) and (4.11) inem that P is the (ultra weak) Solution

Of :

Ch. III: Incompressible nYoPhase Reservoirs

128

+ divqO = Q , ( t ) (4.12)

Go

=

+

+

6(x-a) - Q,(t)

-+d gradP

+

d

2

1

6(x-b)

in

i,

on

r,

+ Yj qj,

j=l (lo'" = 0

which is very similar t o t h e monophasic r e s u l t .

We go now t o t h e l i m i t saturation

where v

in

s a t u r a t i o n equation (4.3).

the

satisfies :

SE

depends on w by :

av -a 2 at

-

'CM

d i v ( $ PCM a E gradv avE

= W

=

in

QEp

on 1,

(4.14)

\

vE(T)

=

0.

Going f o r m a l l y t o t h e l i m i t i n ( 4 . 1 3 ) , ( 4 . 1 4 )

where

v

depends on -0

(4.16)

J,

w

2at

by : d i v ( + PCM a g r a d v )

P c M a av -=O

v(T)

av

=

0.

yields :

=

w

-

i n Q, on X,

The

129

N.An Alternative Model for Diphasic Wells This means t h a t

S

is f o r m a l l y t h e

( u l t r a - w e a k ) s o l u t i o n of

:

(4.17)

s

=

so

on

ii

We check now t h a t ,

at

t=O.

d e s p i t e the d e l t a - f u n c t i o n s appearing i n the

r i g h t hand s i d e of t h e s a t u r a t i o n e q u a t i o n (4.17).

its s o l u t i o n

S

always

satisfies 0 I S(x,t) I 1

(4.18)

a . e . on

G,

a.e. on

ii

a s soon a s (4.19)

I

0

s SO(X) I 1

0 I Q I T ( tS) Q T ( t )

The f u n c t i o n s a and b .

.I'

for

S E [0,1],

aF

t

E

10,TC.

j=O,l,2, which a r e p h y s i c a l l y d e f i n e d only

have t o be c o n t i n u e d o u t s i d e of t h e i n t e r v a l [0,11, as we

d o n ' t know a p r i o r i t h a t ( 4 . 1 8 ) h o l d s . So we choose

:

f o r r, C [0,13 f o r r, M u l t i p l y i n g t h e f i r s t e q u a t i o n of ( 4 . 1 7 ) by Green f o r m u l a , we g e t

x=(S-l )

+

e

[0,11.

and u s i n g a

Ch.III: Inrompressible Tiuo-Phase Reservoirs

130

where B

:

lR

+

R is defined as one primitive

of

bo

and using the pressure equation (4.12), we obtain

or

(4.24)

which yields

x

0 i.e.

S 5

1

using the hypothesis (4.19) and (4.20).

x

=

-(S

Similarly, multiplying by

yields

:

which yields S 20. Remark 18 :

0

The formulation (4.12), DOUGLAS-EWING-WHEELER

(4.17) is used by some authors, cf.

[l],

EWING-WHEELER for approximation

studies. We shall prefer in the following the boundary-source formulation of sections I1 and I11 which is more general and well suited to our mathematical and numerical tools. 0

V

-

MATHEMATICAL

STUDY OF THE INCOMPRESSIBLE

TWO-PHASE FLOW PROBLEMS The aim of this section is to obtain some rigorous mathematical results on the existence of solutions to the two-phase incompressible flow model summarized i n section 111.

131

V. Mathematical Study

V.l

-

SETTING OF THE PROBLEM

Let,

Rn be a bounded, convex domain, with regular boundary r re u rll u rs (referred to respectively as

R E

partitioned into (5.1)

entry, lateral and output boundaries), and with normal unit v

pointing outwards from Q,

T >O

]O,T[,

be

the given

time

interval of interest,

be the space time domain, and

Q=nx]O,T[

(5.2)

Z = r ~ l o , T [ : (resp. Ze,

I t * Zs).

We

consider

equations

in

Qx]O,T[

the

:

*)

(5.3)

following system of partial differential

div

in Q,

0

q =

(5.4) (5.5) (5.6) (5.7)

+ +

q.v

=

0

on XL,

q.:

=

A(P-Pe)

on zs,

(5.8)

(5.9) (5.10)

on 1 e'

(5.11

on

(5.12)

on zs,

(5.13)

where (5.14)

U(X,O)

=

uo(x)

equations"

zL,

on n at t=O,

Ch. III: Incompressible n v d h a s e Reservoirs

132

and q,b stand for

outside the summation sign Z.

qo, bo

We make the following assumptions on the coefficients in (5.3) through (5.14) (5.15)

(5.16) (5.17)

Q

6

L2(Ze)

Q 2 0 a.e. on Ze,

(5.18)

h

E

L"(ZS)

h 2 m2

d,u,a,b,b.,Y., J

IR (5.19)

R

+

on 1S'

are continuous bounded functions of

such that

d(c) t m,

a(c)

bo(c)

=

b.(c)

= 0 , .tF

J

j=1,2

J

a.e.

b(c)

E

a'(c)

=

t 0,

Y 5

f

R, u ( 0 )

=

0;

Y 5 E R; b(

:

+ F € E .

0

the set

{v e L2(Q)I

=

a(5) 2

:

lemma

2

11 gll.1;.

M,

and Y(S,)

}

2 M

is relatively compact in

r'' operator).

follows simply

from the

compactness of

injection of W defined in (5.28) into L2(0,T;H'-E(i2)) f o r any

E

the

> 0. 0

Lemma 3

I

:

(compactness lemma for the degenerate case)

-

Suppose now that a satisfies hypothesis (5.26) and

3e

(5.44)

l0,ll

E

s.t.

c(a)

=

5- 5

Sup

{ v

=

11 B(V)IIQ

6

5

L ~ ( Q ) ~ o2 v(x,t) S 1 2

+-.

[I /a(?) d-rl'

Then for every M>O the set

sM

B

weakly in (%

,

weakly in L2(0,T; H'(R)), weakly in [L2(Q)ln, weakly in [L2(Q)ln. Using (5.67) i) and ii) and the Lebesgue convergence theorem we obtain that

:

146

Ch.III: Incompressible Two-Phase Reservoirs

- Passing t o the limit

:

Using the weak convergences of (5.67) and the strong convergences of (5.68). we can pass to the limit in (5.30) through (5.31) and (5.65)

,,,

+

+

+

+

( a l l written with u P qp, $211), which shows that = P, = q and + + + p' $ 2 = 4'. As P , q, $ 2 are uniquely defined by (5.301, (5.31) and (5.65), +

w

and as we have seen in (5.68) i) that = B(u) which is also uniquely + + defined, we get the convergence of the whole sequences B(uk ) , Pk, qk, $2k in (5.67) iii) and iv), which proves (5.571, (5.59) through (5.61). We prove now (5.62). From (5.65) and (5.30) with w=B(uk) we get :

T

GZk

(5.70)

grad u

Q

k

1I O

:

lim inf

2

Q

From (5.57) we get

J, lgrad B(uk)Iz

=

I Glgrad

QB(uk) -

From (5.59) we see that with (5.69) gives

PkIZ

O

1

1 h(Pk-Pe)

B(uk)

rs

1 bj(uk)

Gj.grad uk.

j=l Q

re

]grad B(u)Iz.

B(uk)Iz 2

Q

Q

k+m

+

>-

PI z

weakly in L2(Z), which together

:

The last term of (5.70) can be rewritten

:

b.(c) where B . ( c ) J

=

l;z

a

is continuous and bounded (cf. (5.53)).

(c)


147

Similarly as for

(5.68) one can prove that

B.(uk) J

+

B.(u) J

strongly in L2(Q), which together with (5.57) shows that :

2

2

1

lim

B.(uk) :.-grad B(uk)

k + m j=1

Q

. I

J

I Bj(u)

1

=

Q

j-1

;:grad J

B(u)

which ends the proof of (5.62) and of lemma 4. Remark

23

:

0

on the functions bj, j=1,2 is not

The hypothesis (5.53)

constraining from a practical point of view

:

- for a non-degenerate problem it is always satisfied

for a degenerate problem coming from two-phase flow, we get from the definitions of a, b, and b2 in terms of the *

mobilities k. and the capillary pressure J

p,

. *

(5.72)

As p,

is usually a bounded function with positive, bounded

below derivative (cf. figure 8 of Chapter I), the hypothesis (5.53) is practically always satisfied. 0

V.4

-

RESOLUTION IN THE NON DEGENERATE CASE

We

suppose throughout this

paragraph

that hypothesis

(5.42)

holds :

3 q>O

(5.42)

s.t. a(5) 2

q

>

0 a.e. on

R

and we want to show the existence of a solution of problem general hypotheses (5.28).

We

use

essentially that problem. Let

(5.79)

(5.1)

(5.2),

(5.6) through (5.18),

( g under ) (5.23)

the

through

for this a penalization technique. The proof follows given

in CHAVENT

[2]

for

a simpler one-dimensional

Ch. III: Incompressible nve-Phase Reservoirs

148

be g i v e n , and d e f i n e t h e p e n a l i z e d problem

( g c :)

(5.85 (5.86

I t s s o l u t i o n is g i v e n by t h e f o l l o w i n g theorem. .THEOREM 1 : Suppose t h a t

t h r o u g h (5.28)

(gE) admits

(5.87)

and ( 5 . 3 7 ) ,

(5.1), (5.42)

a t least a s o l u t i o n

(5.2),

(5.6)

and (5.79)

through (5.11),

(5.23).

h o l d . Then t h e problem

u,P s a t i s f y i n g

:


149

Proof : - Existence : let u

W be given, and define P,

6

4

by (5.29)

through (5.31) unchanged, and u by :

(5.88)

The family of elliptic equations (5.29) through (5.31) admits (for +

a given u ) a unique solution P , q. Then the non linear parabolic equation (5.88) admits ( u and theorem 1.2 p. 1 6 2 1 ,

being now given) a unique solution u (cf. LIONS [ 2 ] , as it is driven by an operator which is the sum of a

linear elliptic operator and of a penalization operator, this latter being monotone, bounded, and semi-continuous. Using standard bounds, we get

:

So we *ave defined a mapping u+u

from S,

into itself. As SM is

convex and weakly compact in W, we get from the Kakutani theorem the existence of a fixed point of this mapping (i.e. the existence of u ) as

soon as the mapping

u+p

is continuous on

SM

for the weak topology of

W, which can be proved with the same techniques as in lemma 4. - Majorations on

:

(5.87) i) is obtained in a standard way by

i n (5.81) and using the general hypotheses (5.16) through

taking

w=u

(5.20).

Taking then

E-

u

v=-u. in (5.84), using (5.81) through (5.82) with

w=B(v) and integrating between 0 and T we obtain :

which gives, using (5.87) i) and lemma 1

:

150

Ch. III: Incompressible lk-Phase Reservoirs

and (5.87)

ii) follows then immediately.

Taking then v=u in (5.84) and still using (5.81) through (5.82) (with W=B(UE)) we get, with the notation (5.53) :

(5.89)

Using lemma 1 we note that integrating from 0 to t we obtain

(5.90)

i

B ( u + ) 5 1 and IB(-u;)I

2

u-

and

:

2

+

+

1 2m j=1 1 IIgjII: II GjII;L2(Q),n

II Allcn

which yields (5.87)

I(pc-Pe)+lL*(ls) IUJL iii) and

2

(

xs)

iv) using (5.87) i)

One gets then from (5.84)

and

ii).

:

which, with (5.87) i) and ii) and the general hypotheses (5.18) through (5.20), yields the sought result. This ends the proof of Theorem 1. 0 We come back now to problem

(@)

and give

V. Mathematical Study -THEOREM

2

through

:

151

Suppose t h a t ( 5 . 1 ) ,

(5.28).

and

(5.42)

(5.2),

( 5 . 6 ) t h r o u g h (5.111,

Then problem ( @ ) (5.29)

hold.

(5.35), admits a t l e a s t a s o l u t i o n (u,P) s a t i s f y i n g (5.92)

a.e.

0 5 u(x,t) 2 1

on

We s h a l l o b t a i n

:

solutions

(YE,

of

PE)

q

such t h a t

:

as t h e l i m i t of a subsequence of t h e

(u,P) the

through

:

Q,

and t h e r e e x i s t v a r i o u s c o n s t a n t s c independant of

Proof

(5.23+

penalized

problem

using

(PE),

standard

t e c h n i q u e s . We g i v e t h e proof f o r t h e s a k e of completeness. From (5.42) and ( 5 . 8 7 ) i v ) such t h a t

uE e SM

existence of through

a subsequence

(5.62),

uk

uk

->

u

E

v) we s e e t h a t t h e r e e x i s t s M > O From lemma 2 and 4 , we g e t t h e

and of @2 k ( r e s p . P ,

S

5k,

[where P k ,

( r e s p . u ) ] , and s a t i s f y i n g a l s o (5.94)

and

defined i n (5.43).

+M

% q, 9

+€

2

s a t i s f y i n g (5.54)

1 c o r r e s p o n d t o uk

:

weakly i n W.

We check f i r s t t h a t

(u,P)

is a s o l u t i o n of

(8 ) :

- ( 5 . 2 9 ) t h r o u g h (5.31) and (5.35) are s a t i s f i e d by (5.59) through (5.61 1.

uk

+

u

-

- From ( 5 . 5 6 ) and t h e Lebesgue theorem, we g e t t h e convergence of i n L 2 ( Z ) ( f o r a subsequence a t l e a s t ) , which t o g e t h e r w i t h (5.87)

ii) gives

-

u =O

on

Ts, s o t h a t (5.32) is s a t i s f i e d .

-Replacing v by v-uk, v e K

i n ( 5 . 8 4 ) (hence v-11 =o) y i e l d s

(5.95)

IY v

e K,

a . e . on 1 0 , ~ ~ .

:

Ch.III: Incompressible Tbc-Phase Reservoirs

152

Take v=v(t), v c

a

in (5.95)

and integrate from 0 to T :

(5.96)

Using the weak lower semi-continuity of v

-f

-21

Iv(T)Ii on W we see that:

du

(5.97)

k+which together with (5.58), limit in (5.96).

Thus du u-v$@

(z,

(5.61).

(5.62)

makes it possible to pass to the

6 i2

grad (u-v) I 0,

VVE&

I

which is (5.33). - The initial condition (5.3'1) follows from (5.851, and from the continuity of the linear mapping u+u(O) from W into H. So (u,P) is a solution to problem

obviously from (5.87). principle

(5.92)

(5.93)

result

using a maximum

:

- Taking

get, as for (5.89)

v = u ~ v with V = -u- in (5.33)

(v

E

K as v(

-

v

=

(u-l)+ in (5.33),

which is permissi 1

:

=

+ -

(ulr - I ) + e

=

o

=o) we

'curs

:

Then v=O, and u>O a.e. on Q. Taking v = u ~ P with as

(8). The bounds

There remains to prove

(5.9'1)

as u

=

Ire

o

153


we g e t

:

f I bj

j = l 51

(P+1)

GJ.

g r a d P S 0.

The l a s t term v a n i s h e s , u s i n g ( 5 . 1 9 ) and t h e f a c t t h a t P+1 2 1 . Using t h e n

(5.30) w i t h w

=

B ( P + l ) we o b t a i n :

4 $ I'+'(t)Ii +

m

11 B(9(t))l12 I +

X(P-Pe)

B(9+1)

rS

-

f,

Q B(9+1)

0.

=

'e

Noting t h a t B is monotone i n c r e a s i n g and ( t a k e wEl i n ( 5 . 3 0 ) ) t h a t :

we s e e t h a t (5.100)

:

+ 2 I@(t)Ii 11 +

m

B(9(t))l12s

J

x(P-P,)-

[B(9+1)-B(1)1

TS

where t h e r i g h t hand s i d e term v a n i s h e s a s B ( 5 ) is c o n s t a n t f o r 5>1 ( c f . lemma 1 ) . Hence P 3 , i . e .

us1

a.e.

on Q. T h i s completes t h e proof of

theorem 2 . Remark 24

0

:

Suppose t h a t we r e l a x t h e h y p o t h e s e s

i n ( 5 . 1 9 ) . Then we have t o make t h e f o l l o w i n g changes i n t h e p r o o f s of theorems 2 and 3 :

-

To o b t a i n t h e energy bounds on u - i n ( 5 . 8 9 ) through (5.901,

we have t o suppose t h a t :

-either B ( c ) is bounded f o r < > O , which is a c h i e v e d i f (5.102)

0 S 1 - b(

1

Clt III: Incompressible TwoPhase Reservoirs

154

(and then

B(5) 5 B ( 1 )

-) 1

+

5-1

-or (5.1 0 3 )

one knows a p r i o r i t h a t P 2 P

-

rs.

on

To o b t a i n t h a t us1 by t h e maximum p r i n c i p l e i n (5.100) we

have t o suppose t h a t ( 5 . 1 0 3 ) h o l d s , so t h a t t h e l a s t term can be dropped o u t of ( 5 . 1 0 0 ) .

In c o n c l u s i o n , theorems 2 and 3 h o l d -with (5.101) replaced by (5.102)

:

[but then t h e conclusion

u61 of theorem 3 must be r e p l a c e d by t h e weaker one

where

c(q)

-without

->

+

.

(5.101)

when

11

* 0.

1

if one knows that

P

on

2 P

One s e e s

in

(5.90)

and

(5.100)

that

Is, f o r

.

i n s t a n c e b y t h e maximum p r i n c i p l e of remark 21

Remark 25 :

:

0

acts as a

(P-Pe)-l TS

s o u r c e term f o r exactly,

as

t h e s a t u r a t i o n e q u a t i o n , which c o r r e s p o n d s

formally

h(P-Pe)

+ + =

q*,

on

rs,

to

the

i n t e r p r e t a t i o n of t h e u n i l a t e r a l boundary c o n d i t i o n g i v e n i n remark 19. Remark 26 :

0

The uniqueness of t h e s o l u t i o n of t h e non-degenerate problem h a s n o t y e t been proved. The d i f f i c u l t y f o r t h i s a r i s e s from

the

coupling

between

the

pressure

and

saturation

e q u a t i o n s . As soon as t h e c o u p l i n g f a i l s , t h e n u n i q u e n e s s can be o b t a i n e d by s t a n d a r d methods ( c f . SV.6 below). 0

K Mathematical Study Remark 27 :

155

Asymptotical behaviour of u(t) in the special case where

G2

=

(no gravity and

0

no

G1

=

capillary heterogeneity effects)

and where Pe = constant on rs. As we already noticed in remark 21, one gets in this case using a maximum principle P 2 P

on

rS, so

that the corresponding source term (cf.

remark 25) vanishes. Passing to the limit in (5.891, we obtain

:

which using the Poinearre inequality gives

Hence

:

which shows that

u(t)+O

in L2(Q), i.e.

one can recover all

the mobile o i l of the field by injecting water for a long enough time. This property may fail as soon as gravity or capillary heterogeneity effects are present. V.5

-

RESOLUTION I N THE DEGENERATE CASE We suppose throughout this paragraph that the following hypotheses

hold

:

(5.44)

(5.53)


156

,THEOREM

3

Suppose t h a t ( 5 . 1 ) t h r o u g h ( 5 . 2 1 ,

:

(5.23)

( 5 . 2 8 ) , ( 5 . 3 7 ) , (5.44) and ( 5 . 5 3 ) h o l d , and t h a t moreover

u0

(5.105)

K.

E

Then t h e problem (5.40),

through

:

(5.35),

( P I

admits a t

),

(5.29) t h r o u g h ( 5 . 3 1 ) , ( 5 . 3 8 ) through

least

a solution

(u,P),

which moreover

satisfies : (5.106)

u

/

E

L"(0,T;H).

( 5 . 2 9 ) t h r o u g h (5.31) unchanged,

i) ii)

~ ( u c ~ ~) n % ,

iii)

[s, du

v-u ( t ) ) , + 17

q

(5.1 07 )

iv)

g r a d (v-u ( t ) )2 0

I

R ;2n

+

u (0)

I 5 grad u

n

17

=

17

V v

02ri

-f

17

6

K , a.e.

on 1 0 , T [ ,

uo,

2

ri

v)

grad(v-u ) ( t ) )

=

grad u ( u ) rl

+

B' ( u q )

-

1

bj(uq) ;j*

j=l

w i t h t h e bounds ( j u s t t r a n s l a t e d F r o m ( 5 . 9 2 ) t h r o u g h ( 5 . 9 3 ) ) :

-

157


where the c are constants independant of n. From (5.108) i), iv), v) we see that there exists an M>O such that u € SM defined in (5.45). From lemma 3 and 4 we get the existence of a rl subsequence uk satisfying (5.54) through (5.62). We check now that the couple (u,P) is a solution of @I ) :

v

E

Wn

(5.29) through (5.31) result from (5.59) through (5.60), (5.107) ii), (5.56) and (5.57) -> (5.58) --> zduf q l ' , (5.108) i) and (5.55) -> (5.40), (5.107) v) and (5.61) ->(5.35).

B(u)

,

There remains to prove (5.39) : take in (5.107) iii) v=v(t) with and v(0) = u and integrate from 0 to T :

&

0

t 2

nk

1 J, grad uk grad v

0

which gives

:

2 E L2(Q), one can pass to the limit i n (5.109) by using (5.58), dt (5.61) and (5.108) iv), which yields (5.39). This completes the Proof of

If

theorem 4 .

V.6

-

0

THE CASE OF DECOUPLED PRESSURE AND SATURATION EQUATIONS

The coupling between the pressure and saturation equation makes the mathematical study of the whole system very difficult. For example, it has not yet been possible to prove the uniqueness of the solution of the coupled system of equations, even i n the case of a non degenerate saturation equation with simpler boundary conditions. Similarly, the

Ch.III: Incompressible nYo-Phase Reservoirs

158

demonstration of the existence of a strong solution of the coupled system relieves on the L"-regularity of the transport field not achieved when

+

Go,

which is usually

qo is given as the solution of the elliptic pressure

equation with coefficient d(u). In order to give further results in these two directions, we suppose throughout this paragraph that :

i

the

transport

(5.291,

field

...,(5.31)

is

+ qo

given by

the

pressure

equation

independant of u and of time t and

satisfies

(5.111)

which is the most restrictive assumption, and that

:

(5.112)

Gj

(5.113)

b.

Remark 28 :

Hypothesis (5.111) is satisfied in two cases

J

f

f

CL"(n)ln

j=1,2 j=O,l , 2 .

W''"(IR)

- one-dimensional problems

(q

0

:

is then constant in space)

with constant given global injection rate Q(t)

5

Q on

r

(but

one dimensional problems with Dirichlet pressure conditions on I' and I' do not satisfy (5.111)).

- multidimensional

problems

with

neither +

gravity

nor

+

capillary pressure heterogeneity (hence q1=q2=O) and such that d(u)

1 (cross-mobility curves).

The following results are due to G. GAGNEUX [ll, [2].

0

We shall

follow his proof and hence give only the main steps of the demonstrations.

159


V.6.1

-

Regularity and asymptotic behaviour for the non degenerate case

4

-THEOREM

:

(Regularity of the non degenerate case with compatible

-

initial data). We assume the hypotheses of theorem 2, (5.111) through (5.113) and that

:

(5.114)

uo

K.

6

Then the solution u

of (5.32) through (5.35) given by theorem

2 is unique and satisfies moreover

u

(5.115)

t

L"(0,T; V),

du

(5.116) (5.117)

div

L2(Q),

r' =

-div ({grad

and for almost every t, t-.i

xd

(5.118)

:

a

t

cx(u))

€

C0,Tl and

L2(Q),

uo,

Ci0 E

K

:

lu(x,t) - G(x, t--r)(+ I 0

where u (resp. il) is the solution corresponding to uo (resp. f i g ) ; this implies (5.118bis) Proof

d

lu(t) - Ci(t-T)IL,(a) 2 0.

:

(5.119)

du (E(t), v-a(u(t)), Let then s c ,

c

>

+

f a

i2 grad

(v-a(u(t))

0 be the following approximation of the (sign)+

function for 5 I 0 s

p

=

v E K, a.e. on l0,TC.

2 0 f

for 0 2 5 I for

I 5

E

Ch.III: Incompressible no-Phase Reservoirs

160

and l e t u and G be two s o l u t i o n s of (5.119) c o r r e s p o n d i n g t o t h e i n i t i a l c o n d i t i o n s uo and Go. For

(TI

u 0

cn

Proof

:

The proof of (5.118) in theorems 4 and 5 requires only that

where

s (w)

s,(u(u(t))-a(ii(t))

=

E

V

is positive a.e. on n.

Under hypothesis (5.151) these inequalities are satisfied for the following two choices of u and & : i)

u(t)

=

solution of (5.32),

ii) u(t) E u fF t, 0

Choosing i ) [resp. ii)] (5.157)

..., (5.35),

&(t)

G(t) solution of (5.32),

uo fF t ,

...,(5.35).

we shall show that

u(x,t) 2 u (x) [resp. u(x,t) 2 u (x)l 0

0

a.e. on ~ x l 0 , ~ ~ C .


167

From now on we c o n s i d e r o n l y t h e f i r s t case i n ( 5 . 1 5 7 ) ( t h e second b e i n g t r e a t e d i n t h e same way). Taking t h e n i n (5.118) G=u w i t h O < T < t < T y i e l d s

which, as

U(T)

(5.1 58 ) which

5 u

0

proves

a.e. on R u s i n g ( 5 . 1 5 7 ) , shows t h a t : 2

U(X,t)

a . e . on R , f o r 0 2

U(X,t-T)

(5.152),

decreasing function Hence

:

and

shows

t--u(x,t)

that,

a.e.

for

has a l i m i t ,

x

E

2 t,

R,

the

positive

which we d e n o t e by

uw(x).

:

0

s

u_(x) i 1

a . e . on R ,

(5.159 ) u(t)

+

Let then f (5.160)

f(t)

i n L P ( Q ) f o r every p > t .

U

0,

a . e . on lo,+-.[

f' ( t ) 2 f'

f o r a.e. t , r h 0

Sup

Ess f'(t)

2 0 =

0.

t>O

Hence, n o t i n g t h a t f ' ( t ) = -

I- ddut ( t ) IL ' ( 0 )

, one g e t s

-,

and

f' ( t ) = f ' _ ( t ) = f ' t ) ( t + L )

u we g e t

2 0,

t , T

when t

which p r o v e s (5.153) and shows t h a t fl(t)

=

:

f T 2 0,

Ch. III: Incompressible 7boPhase Reservoirs

168

Taking then v

=

0 in (5.119) yields :

Hence :

2m . J=1

which, using (5.163) shows that

stays in a bounded set of V when

a(u(t))

t-,. Hence there exists a subsequence a(u(t'))

Using

then

(5.159),

(5.163),

such that

(5.165)

and

the

weak

lower

semicontinuity of the norm, one can pass to the limit in (5.119) (for v given) when uo

t-,

which shows that

E

K

u~,,necessarily satisfies (5.155).

In order to prove (5.156) one just remarks that (5.118) holds for uc, solution of (5.155), with act) E urn

satisfying (5.20) and for any

+ t, which

shows that

t u_(x)

io u

(XI

a.e. on

(5.166)

:

Cl

which ends the proof of theorem 6. Remark 29

:

u(x,t) 2 U_(X) a.e. on a

x

lo,.,[

0

In the special case where q1 = q, = 0 ( n o gravitational or capillary heterogeneity effects) and where the given exterior

V. Morhemoticol Study

169

pressure P that PLP m

2

rs,

is constant on

we have seen (cf. remark 21 )

on Ts; we get then from (5.164)

11 a(u(t))l12 s

E(t)

+

:

o

which proves that necessarily ucd= 0. Remark 30 :

The asymptotical behaviour of

0

u(t)

in the case where it

does not evolve monotonically (i.e. when the initial data does not

satisfy

however,

noticing

(5.151)) that

u0 open problem. One has

is an (5.166) holds

hypothesis that 0 5 u (x) 2 1 0

under

the

sole

:

(5.1 67 )

Remark 31

An

example of multiple steady-state solutions u-. Consider a

vertical

1-D porous slab

Q

= ]O,II[,

with insulated lateral

boundary, and with 7Tinjection"boundary and "production" boundary Ts

=

fe

that

:

q1

0 (no capillary heterogeneity effects)

=

{ O } at the top

=

1 1 ) at the bottom. We suppose

(5.168)

Then

u

represents the oil

saturation in one imbibition

experiment ( q 3 ) in a vertical sample maintained in contact with water at the top ( u ( 0 )

=

0), with insulated bottom end

(as the unilateral condition resumes to

+

+

$2'1)

=

0 when u>O),

Ch. III: Incompressible f i o q h a s e Reservoirs

170

and with oil and water mobilities kl and k2 such that

51

J u(1-u) and a capillary pressure curve kl+k2 p (u) = arccos (1-2u).

-=

The steady-state equation (5.155) becomes now : $2

=

u(0)

(5.169 )

constant on lO,n[ =

0

,

u(n) 2 0

$,(n)

,

0

2

- The class of initial data u0 ax]O,-[

u(n)$2(II) E

K

=

0.

such that

au

2 0

a.e. on

contains only the stationary states : uo = ,:u 0 2 a a . urn is defined in figure 15 (if (5.151) holds in

2 II, where

the bracket case, then necessarily necessarily

@20

(il)

$20(II)

2

0 ; hence

0 and uo is one of the stationary

=

solutions). One checks in this example that the equilibrium profile

0

um

is exactly that of the capillary pressure. This property, which is always true, under condition (5.168), in 1-D samples with both ends insulated, is used as a physical definition of the capillary pressure law. Remark 32

:

0

If we replace, in the last remark, the function b2(U) by

(which corresponds to the (unbounded) capillary pressure law

e),

then the steady-state equation (5.155) has p,(u) = Log o n l y one solution urn E 0. This comes from the fact the "equilibrium profile''

x= Log

boundary condition u(0)

=

0.

1 -u

does not satisfy the 0


V.6.2

-

171

Regularity and asymptotic behaviour for the degenerate case We turn now to the degenerate case: in order to handle this case,

we had replaced in the case of the coupled system of equations, the

8)by the weaker (8 1, which required that

saturation equations (5.32) through (5.35) of problem ( formulation (5.38)

u,,

E

through (5.40)

of problem

K in order to get a solution (theorem 3 ) . We shall now treat this case in another way, and, still following

CACNEUX [ l ] , [2], we introduce the following variational formulation :

Problem (@)

6%

,

(5.174)

B(u)E

(5.175)

($ (t), v-a(u(t))),

:

find u such that

$Eq'p

a Y. v

(5.176)

~ ( 0 =) u 0 ,

(5.177)

0 2 u(x,t) 2 1

(v-a(u(t.1,) L 0

;,(t).&ad

+

E

K

a.e. on lO,TC,

a.e. on Q.

we snail be able to show the existence of a solution without the -compatibility condition from [O,T]

u

0

E

K. Since (5.174) implies that u is continuous

into H equipped with the weak topology, the equations (5.175),

(5.176) make sense.

One a'

=

checks

a L 6 > 0, the

easily problems

that,

in

the

non

degenerate case where

( 3 )and (2") are

equivalent. In the

degenerate case, the inequality (5.175) can be formally shown to satisfy the saturation equations ( 5 . 8 ) through (5.13) with the boundary conditions on has a trace on El.

Ee

and

E

a(u)

instead of u in

(which is satisfying because a ( u )

172

Ch. III: Incompressible nuo-Phase Reservoirs

1

(

The corresponding \ solution converges \

a

b

a

Asymptotical behaviour?

n X F i g u r e 15 : Exaaples of initial data

yielding monotonic and uo non-monotonic evolution of the saturation profile.

U


-THEOREM

1 73

7 : ( r e g u l a r i t y f o r t h e d e g e n e r a t e c a s e w i t h non compatible

-

i n i t i a l data). We make t h e h y p o t h e s e s of theorem 3 b u t w i t h (5.105) r e p l a c e d by o < u 0( x ) s l

(5.20)

Then t h e problem one

"entropy"

(PI)(5.174)

solution

u

v i s c o s i t y ) s a t i s f y i n g moreover

$

(5.178)

fi

(5.179)

Proof

:

Let

B(u)

a.e. i n R.

E

t h r o u g h (5.177) a d m i t s a t l e a s t

(defined

by

addition

of

a

vanishing

:

L2(Q),

a(u) e L w ( O , T ; V).

u

n

be

the

s o l u t i o n of

theorem 5

a ( s ) = a ( C ) + q . We know from theorems 2 and 5 t h a t u

( c f . ( 5 . 1 0 8 ) , and (5.148) ( 5 . 1 4 9 ) )

n

corresponding t o

is bounded as f o l l o w s

:

where t h e c o n s t a n t s C are independant o f n. Using t h e same compactness argument a s i n t h e proof of theorem 3 , one c a n p a s s t o t h e l i m i t when u

n

n+O,

and hence show t h a t a subsequence of

converges toward one s o l u t i o n of problem

(5.179 )

.

which s a t i s f i e s ( 5 . 1 7 8 ) , 0

Ch. III: Incompressible TboPhase Reservoirs

174

Remark 33 :

The c h a r a c t e r i z a t i o n of t h e e n t r o p y s o l u t i o n s of t h e proof

results

in

, and

@It)

of t h e i r u n i q u e n e s s , have n o t y e t been done. For that

direction

diffusion-convection

in

the

case of

the

degenerate

e q u a t i o n s ( i n s t e a d o f i n e q u a l i t i e s ) one

can see VOLPERT-HUDJAEV and B R E N I E R . 0

-

THEOREM 8 :

(Asymptotic behaviour i n t h e d e g e n e r a t e case)-

Let t h e h y p o t h e s e s of theorem 7 h o l d , and suppose t h a t

s a t i s f i e s moreover (5.151) and (5.181)

n

j, g r a d uo g r a d v t

0

aF

vfV

(which f o r m a l l y means t h a t d i v ( $ g r a d u,)

on

rk

and

rs).

u(t)

-t

S 0

on R and t h a t

Then t h e "entropy" s o l u t i o n u of (

urn

0

s . t . v t 0 a.e. on $2

strong

j

where uw i s , among t h e s o l u t i o n s of :

t h e o n l y one which s a t i s f i e s (5.156).

i n L'(Q)

au

5 2 0

gTt) defined

theorem 7 ( b y a d d i t i o n of a v a n i s h i n g v i s c o s i t y ) s a t i s f i e s

(5.184)

u

:

:

f o r every, p L 1,

in

175


Proof : -

u

Let

be t h e s o l u t i o n of t h e non d e g e n e r a t e problem i n t r o d u c e d i n

n

t h e proof o f theorem 7 . One knows t h e n t h a t a ( u ) is bounded i n L " ( 1 6 , T C ; V ) n d a ( u ) is bounded i n L 2 ( ] 6 , T , [ ; H ) when 6+0. Then a ( u ) and a ( u )

and t h a t

n

n

are c o n t i n u o u s from

i n t o V (equipped w i t h t h e weak t o p o l o g y ) , and

[O,m]

and u are c o n t i n u o u s from ] O , f > [

We check f i r s t t h a t , f o r a subsequence, s t i l l denoted by u f o l l o w i n g convergence p r o p e r t i e s

u

u

n

i n t o H (equipped w i t h t h e weak t o p o l o g y ) .

n'

one h a s t h e

:

t>O

a ( u (t))

+

a ( u ( t ) ) weakly i n V , s t r o n g l y i n H and a.e. on Q,

un(x,t)

+

u(x,t)

n

U

p

-f

a.e.

u(t)

in

on

Q , and hence

Y

LP(Q)

...when

r~

-f

p 2 1

0.

I n o r d e r t o prove t h i s , we i n t r o d u c e t h e c a n o n i c a l isomorphism A from V o n t o V' D(A)

=

{ v

E

,

associated with t h e s c a l a r product ( (

VIA v

[U

E

6>0, V

)),

and i t s domain

}. Then we g e t from ( 5 . 2 7 ) and (5.178) t h a t

H c V

t , t Ob 6 ,

U v

E

:

D(A) C V ,

and t h e same p r o p e r t y h o l d s f o r a(u ) .

n

Let then { a E the Dirac function

E

L'(IR),

t

>

0

]

be an a p p r o x i m a t i o n sequence of

:

6 ( t ) 2 0,

6E(t)

=

i f It1 2

0

L,

1

W

Then Hence, as

bE(t-tO)

Av

L'(6,T;

E

n

L

>

6 S t0-E

and v

E

to+(1 s E ( t - t 0 ) ( ( a ( u ~ ( t ) ) , v )-)

0

5

[ n + a ( r ) ] d r ) , s o t h a t theorem

0

Hence we have

q>O.

=

:

u ( x , t ) 2 u (x,t+T)

0,

n

n

for a.e

x

E

a.

Using (5.187) one can p a s s t o t h e l i m i t i n (5.190) when

+ t,

(5.191)

so that,

for

u(x,t) 2 u ( x , ~ + T )

T>O

almost

every

x

f

R,

u(x,t)

->

U_(X)

f o r a.e. x

E

q+O:

a.

t + u ( x , t ) is a p o s i t i v e d e c r e a s i n g

f u n c t i o n : l e t u s d e n o t e by uc,(x) its l i m i t (5.192)

n>O,

:

a.e. on a.

t 4 m

T h e f u n c t i o n f d e f i n e d by (5.160) is c o n t i n u o u s on 1 0 ,

c o n t i n u o u s from [O,-[ [O,@[

+

w [ ,

a s u is

i n t o H equipped w i t h t h e weak topology. Let t h e n f :

IR be d e f i n e d as

n

:

VI. The Case of Fields with Different Rock Types

1I7

Yt50. Then we get from (5.187)

:

Y t>O

(5.194)

fn(t)

We see from (5.192)

+

f(t)

when n+O.

and (5.194)

that f satisfies (5.161),

and the

rest of the proof is the same as in theorem 6, of course with (5.155) replaced by (5.186). Remark 34 :

0

Theorem 8 covers the practical case of a field initially saturated with oil, when u (x) E 1 on il

:

0

in this case, the

saturation tends to a stationary profile

uo,

which is

identically zero in the special case where no gravity or capillary heterogeneity effects are present and where the imposed exterior pressure is constant on

r

. 0

V I

-

THE

CASE

OF

F I E L D S WITH

DIFFERENT

ROCK

TYPES

Up to now, we have always supposed that the shapes of the non linearities a, b.

J'

d,

the hypotheses (3.17 )

'I.

J

were the same all over il. This was the result of

through (3.20 )

relative permeabilities and

capillary

concerning the dependance of the pressure

laws

upon the spatial

variable x. In petroleum engineering terms, this would be rephrased by saying that we have considered a field containing a single rock type. This notion of V o c k

type?' thus appears as an hypothesis simplifying the spatial

dependance of the relative permeabilities and capillary pressure curves

:

inside a given rock type, (argiles, or sandstone or...), the porosity $, the permeability K and the maximum capillary pressure PCM may vary from one place to the other, but, at a given point x and for a given (actual) saturation

5, the

relative permeabilities and the capillary pressure are

perfectly determined once one has been given the residual saturations

-

S (x) and m

l-sM(x) at that point.

Ch. III: Incompressible Tivo-Phase Reservoirs

178

VI.1

-

THE DIFFERENT ROCK MODELS

Define : d. J

1 V.

=

j=l,2 =

krl kr2 d=+ -

(6.1)

krl dl

=

=

krl dl

p2

p1

v

mobility of jth fluid,

3

=

+

kr d 2 2

global mobility,

=

fractional flow.

Then choosing values for k r l , kr2 and p choosing values for d,w and p,. terms of d, v and p

.

clearly amounts to

Hence the rock-models will be specified in

In all rock models for incompressible fluids, we take as given the following functions of the reduced saturation S (6.2)

S

->

S

->

:

pc(S) satisfying ( 3 . 2 0 1 , w(S) satisfying

(6.3) w(0) = 0 ,

v(l)

=

1,

L

an increasing function of S.

As for the choice of the global mobility function d , we shall distinguish two cases Case 1 :

:

Rock model of t h e f i r s t kind : we take as given the following

function of the reduced saturation S

S

->

d(S) satisfying

(6.4) d(0) 2 d2,

d(1) 2 d l , d(S) > 0 .

The relative permeabilities generated by this model are (6.5) These

relative

permeabilities

depend

only

on

the

reduced

saturation S , which is the assumption made in sections I1 and 111. All

equations in these paragraphs have been established using a rock model

of the first kind.

VI. The Case of Fields with Different Rock Types

179

This rock model can be used for two phase flows without exchanges between phases, where

-

sM remain away from 0

Sm and

and 1.

However, when exchanges between phases take place (see chapter IV, 8111 and IV). The actual saturation

s

may take values outside of the

and sm(x) may approach to 0 and sM(x) may approach

interval [zm(x,) sM(x)],

1 as one tends toward the critical point. So one will need to calculate the

relative permeabilities (but not the capillary pressure) for values

s of

the

But,

actual saturation lying outside the interval [sm(x), ZM(x)].

continuing the relative permeabilities given by (6.5) outside the interval (x), S (x)] would lead to discontinuous relative permeabilities, as kr 1 m(x,SM ) = - I 1, whereas the physics indicates that krl(x,l) = 1 (cf. So rock models of the first kind are not valid in situations figure::61 [S

El(1)

where the residual saturations ?m(x)sl-sM(x)

approach zero.

Rock model of the second kind : we take as given the following function of the actual saturation :

Case 2 :

l

(6.6)

-

S -> d(0)

d(S)

=

d2,

satisfying d(1)

=

d l , d(S)

>

0

and we suppose that

(6.7)

the function

1 1

given in (6.3) is continued by 0 for S 2 0 and

by 1 for S t 1. Now we can generate relative permeabilities over the entire range of actual saturations

by setting

(6.8) where S is the reduced saturation corresponding to

5

at the point x (given

by (6.16). The formula (6.8) yields continuous relative permeabilities when

-

Sm

0 and

-

Sm + 1 , as shown in figure 17, and hence this rock model has to be chosen when exchanges between phases take place. +

Of course, the shapes of the relative permeability curves, when expressed as functions of the reduced saturation S, will slightly change from one point x to the other.

Ch.III: Incompressible Tbo-Phase Reservoirs

180

I

Figure 16 : The discontinuous limiting relative permeabilities (dashed lines) obtained with a rock model of the first kind when the residual saturations tend towards zero.

0 Figure 17

The continuous limiting relative permeabilities (dashed

line) obtained with a rock model of the second kind when the residual saturations tend towards zero.

181

VI. The Case of Fields with Different Rock Q p e s

Remark 35 :

For the practical determination of functions, it is enough to know

the d(5)

and v ( S )

:

for one point xo, i.e. for one rock sample,

-

one set of relative permeability curves z+kr.( x o , S ) , j = l,2

(6.9)

J

over the whole interval of (non reduced) saturation and the residual saturations sm(xo) and l-zM(xo),

(6.10)

the viscosities 11, Then

d(2)

and

and

\,(S)

u2 of the two fluids. are

determined by

(6.1) without

ambiguity.

0

The two-phase equations developped in sections I1 and I11 for the (implicit) case of a rock model of the first kind, remain valid for a rock model of the second kind, with the following modifications d(S) has to be replaced by d(x,S)

=

:

d(S),

(with an evident abuse of notation), (6.11)

b ( S ) is equal to > ( S ) , 0 b ( S ) , b (S) become bl(x,S), b 2 ( X , S ) . 1

2

and (6.12)

has to be replaced by

The theoretical results developped in section V remain valid with a rock model of the second kind, as the supplementary dependance of d, b l , b2 on x does not change the proofs.

Ch.III: Incompressible no-Phase Reservoirs

182

VI.2

-

THE CASE OF A FIELD WITH M DIFFERENT ROCK TYPES

Let us now consider the case of a field R , which contains M

.

different rock types. Let Om, m=l ..M be the spatial domain occupied by each type of rock, let convention that

rmk=

m

rmll be

the boundary between R and R', with the Q 0 if Rm and R do not meet, or meet only on a line

or a point for n=3 or meet only on a point for n=2, and attach a superscript m to each quantity related to Rm. According to the notion of rock-type, not only the shape of relative permeabilities and of capillary pressure curves, but also the maximum capillary pressure PCM may differ in each Rm.

It is hence

necessary to allow for discontinuities of the maximum capillary pressure P (x) across the boundaries rmQ; so P (x) will consist of M regular CM CM m functions PCM(x), defined over Q m' and which do not necessarily meet continuously at the internal boundaries TmQ. We look now for the equation in R , appropriate when for each rock-type, a rock-model of the first kind is used a)

:

we can proceed as we have done in

Inside each of the Q",

m and Pm the saturation and global

sections I1 and 111. We denote by S m pressure in R , which satisfy :

The

equations +m

capillary flow r

,

governing,

in

the global pressure

Qm

the

saturation Sm,

+m

Pm and the (water+oil) flow q

are (cf. (3.571, (3.58), (3.62), (3.63) and (3.72)) : +m Y x e om, (6.15) div qo = 0

and,

(6.18)

the

v

x e Rm'

183

VI. The Gme of Fields with Different Rock Types

Figure 18 : An example of a field with four different types of rock

R 2 and we have

(here il

+m r

(6.19)

We

13)

boundaries

=

m

r 13

0 and

=

m

r2,,

=

0 with our convention).

m

t x

grad a ( S )

-$PcM

€

am.

have then to choose continuity conditions at the

rmll between

different types of rock.

For convenience, we shall denote the jump of any quantity 0 across

by

mR

[ e

1;

em - e p.

=

On any T m R , Il=1 conditions

...M ,

. m=l.

..L, we have to satisfy the following

:

Continuity of pressure : each of the pressure to be continuous (6.20)

[

Pj

1;

=

P

1

and P 2 has

:

0.

- Conservation of masses has to be continuous

:

:

the flux of each of the two fluids

Ch. III: Incompressible Tluo-PhaseReservoirs

184 +

+ m

(6.21)

t q o * v l I 1 = 0,

(6.22)

[

where

;,.;I;

=

0,

is any normal t o TmL. 'I)

Now we want to obtain from (6.15) to (6.22) equations

valid over all P. We f i r s t i n t r o d u c e f o r t h a t purpose f u n c t i o n s d e f i n e d

........as

(6.24)

+ d i v qo

(6.25)

0

as at

=

+

+

x

E

a, t t >

t x

€

a, t t > 0

f

0

d i v $1

=

0

soon as x

0,

2

(6.26) !

$,(x,t)

=

r'(x,t)

+

1

j=o

b.(x,S(x,t)) G.(x,t) J J

f X E

a, t > O .

E

nm

VI. The Case of Fields with Different Rock Q p e s

185

(6.27) (6.28)

So we first get rid of those gradients by using a variationnal

formulation of (6.16), (6.19). Let the test function

?

be any (regular)

mapping from Om into 1". Multiplying (6.16) and (6.19) scalarly by integrating over Om and using a Green's formula we obtain :

?,

(6.30)

where

+

i m is the exterior normal to the boundary m of Qm. We combine now (6.291, (6.30) in the following ways : +m to * For every regular z:Q -+ R n , we set s = restriction of

m'

and sum up equations (6.29) and (6.30) for all m , which yields

:

9

But the above equations alone are not equivalent to (6.291,

(6.30). We have to use also other combinations.

Ch. III: Incompressible nYo Phase Reservoirs

186

For every m , II

..M, m < I I , for every regular

= 1,2.

g

: Q+lRy

with

zm

support in the interior of u 5, , set gm = the restriction of s' to Qm, = the restriction of s' to QII, and take the difference between ( 6 . 2 9 )

;'

(resp. ( 6 . 3 0 ) ) on ilm and on O R . This yields

1

s'.:

(pm+pR)

Pm div

=

rmL

s'

:

-

P R div . +s

%

nm

(6.33)

Y m,L=1 aF

1

s'

: Q

...M, m < R , +

lRn, supp

2.;

(am(Sm)+aL(SR))

:is in the interior of -Qm crm(Sm)div

=

s'

m'

rml

-

4

r - s -j -

(6.34)

Qm "CM aF

m,R=l...M,

aF ;:Q+IRn,

+I

-

1

s'

I I R

(S )

QII,

div

5

%+

+

r - s -

Q R "CM

m

d(l,O,P)

2 d2(P)

3

or

(2.17) d(0,O.P)

2 d2(P)

d(O,l,P)

5 d3(P)

(rock model of the first kind) 0 2 b.(S

3

1

,s 3 ,P)

1 1 3 s1 = o =>

(2.18)

s l + s3

=

d2(P)

=

d3(P)

(rock model of the second kind)

,S3,P)

2 1 -

‘J

=1=>

= o

1

-

1

v 1 + v 3 =

-

->

s3 = o

d(O,O,P) d(O,l,P)

2 1

,s ,P)+V3(S1

0 5 w (S

,s

d(zl ,P) > 0 3 d(l,O,P) = dl(P)

0

= o

‘!3

0

Shape of capillary pressure curves We make the hypothesis that the capillary pressure curves, when expressed in terms of the reduced saturations, are independant of the space variable x

:

P, - P2

=

PI:

P3 - P2

=

P?2 (S1, S 3 L

(S,,S3)

(2.20) \

and that

4F

P 32(S1,0) = 0, f (2.22)

s1

5 0,

s1

E

ap:’

s3

t 0, Sl+S3 s 1 ;

C0,ll

- ( S ,s ) 2 0 , - ( S ,s ) t 0 , as, 1 3 as3 1 3

D. The Three-Phase Compressible Model

209

These hypotheses reduce, in the common case where as a function of

S1

only and

P22 as a function of

Pb2

is taken

only, to the

S3

usual hypotheses on the shape of diphasic capillary pressure curves for fluids

1-2-3

of decreasing wettability. Notice that hypothenes (2.21),

(2.22) imply that: P 12( S ,s ) 5 0, Pc 32(S,,S3) 2 0, tF s l , s3. c 1 3

(2.23)

We allowed, in the two-phase case, a spatial dependance of the capillary PCM(x).

pressure by

introducing a spatially varying scaling factor

We suppress this dependance in our three-phase model, in view of

the forthcomming "total differential" condition on the rock model which would otherwise become too complicated. Moreover it is common in numerical field simulations to use only one capillary pressure curve inside a given rock type, so this is not a drastic restriction.

-

11.2.3

The "total differential" (TD) condition on the rock model

We suppose that the following condition holds : There exists a function ( S 1' s3' P)

PC(Sl,S3'P)'

+

called the global capillary function, such that for all saturation and pressure distributions S1(x,t), S (X,t)

3

and

P(x,t), one has

:

(2.24) grad P (S ,S , P ) c 1 3

For

convenience,

we

shall

12

=

v l ( S ,S ,P)grad Pc (sl,s3) +

+

'9

+

2 ( s S ,PI grad P. ap 1' 3

1 3

3 ap

(S

refer

S ,P)grad P:2(S,,S3)

1' 3

to

this

condition

differential" (TD) condition on the rock model

(i.e.

+

as

the

"total

on three phase

relative permeabilities and capillary pressure curves). We seek now a necessary and sufficient condition for (2.24) to hold. We want to have

210

Ch.IV: Compressible, Three-Phase,Black Oil or Compositional Models

aPb2

gradP

=

(v,

ap?'

as, +

v

apb'

as,)

wadsl

+

(V

1

as3 +

ap;'

v

3

as3)

gradS

3

C + -ap ap grad P

for any saturation and pressure distributions S, (x,t),S3(x,t), P(x,t), which holds iff apb'

ap c as, (51 , s ~ , P ) =

v1(~1,~3'~)

apc (s1,s3,p)= as3

v

3F (S,,S3,P)

6

- (sl as1

ap;'

'S3)+V3(S,

'S3'P)

as, ('1 ap;'

C0,ll

( s , , s 3 ) + v 3 ( s 1 .S3,P)

x

C0,ll

x

as3 ('1

9'3)s

m.

\

A necessary and sufficient condition for the existence of a function P ( s s ,p) satisfy ng ( 2 . 2 5 ) is (consider p as a parameter) : c 1' 3

(2.26)

\

a2p

which, developping and noticing that

b2

a2

pi2

aslas3 as as and =

3 1

2 32

a'p:'

turns out to be equivalent to -as,as3 - - as3as, ,

(2.27)

avl ap l 2 av a ~ 3av,~ aPcl 2 av a ~ 3 ~ -c+3c=--+3c as3 as, as3 as, as, as3 as, as 3 '

or equivalently,

II. The Three-Phase Compressible Model

21 1

Equation ( 2 . 2 8 ) gives a simple differential condition to be satisfied by the three-phase relative permeabilities and the capilllary pressure curves for the TD condition ( 2 . 2 4 ) to hold. The construction of such three-phase curves from the available two-phase data will be examined in paragraph 11.5.

I

Once ( 2 . 2 8 ) holds, the global capillary function P (sl , s 3 , p )

is given by

In

the

:

usual

case, where

the fractional flows satisfy the

conditions ( 2 . 1 8 ) of remark 1 , we have w 3 ( S 1 , 0 , P ) f 0, and the definition ( 2 . 2 9 ) of P c

simplyfies somewhat.

Ch. IV: Compressible, Three-Phase,Black Oil o r Compositional Models

212

0 Figure 1 : Illustration of the path used i n (2.29) for the calculation

of the global capillary pressure Pc(S,,S3). From (2.29), we get immediately, using the hypotheses (2.21) and

(2.22) on the shape of the capillary pressure curves and the fact that all fractional flows

take values in interval

[O,l

3,

that

(2.30) +

d s 3 2 P22(S1 ,S3,P).

Hence the global capillary function Pc(Sl,S3,P) satisfies

213


11.2.4

-

Limitation on the pressure range Given a rock model satisfying the TD condition we suppose that

P3 inside the field 1 satisfying CPmin' 'max

the pressures P 1 , P2 given interval

3 (2.32)

8 E [O,ll

such that

:

e
0.

From (2.32) we see that equation (2.48)

has always a unique solution

in the interval [Pmin, Pmax], and from (2.31) we see that

P

P

actually

satisfies (2.7). So equation (2.47) becomes

which, together with (2.42), yields the sought pressure equation where the global flow vector

q

is expressed in term of the global pressure

gradient only. If we had not made the total differential hypothesis (2.24) on

the shape of the thee-phase relative permeabilities and capillary

pressure curves, then the right hand side of (2.49) would necessarily contain

gradS,

coupling between stronger.

terms (see for example CHAVENT [ 4 1 ) , and the 3 the saturation and the pressure equations would be

and

gradS

216

11.4

Ch. IV: Compressible, Three-Phase,Bhck Oil or CornposittonalModela

-

THE SATURATION EQUATION

We

11.4.1

-

now to the determination of two equations for t h e

turn

saturations S1

and

S3.

Determination of the equations it is well known, equations (2.38) and (2.39) with

As

+

be rewritten so as to express the water flow field global flow

+

q

@1

j = l , can

in terms of the

and of the gradients of the two capillary pressures. T h i s

can be done as follows. From the identity

and expression

vl+v2+v3

=

1,

Noticing that

(2.52)

+

and

we get, since by definition

@

3' * +2+ and since @ + @ + @ = q, that : 1 2 3 +

$,

(2.51)

G,,

(2.39) of

=

1v 2 [: grad(P1-P2)-(p1-P2) g gradZ 1 I: grad(P1-P3)-(p1-P3) g gradZ I -oKd v 1 v3

-oKd

P1-P 2

G, Let u s

=

+

-

v

PA2 and P1-P

+

u1

pA2 - Pz2, we obtain

=

-oKd [ v ( 1 - v )gradPb2- U, v3 gradP:2 1 1 1 -oKd [ vl(l-~',)(p 1- p 2)-V 1 V 3(P 3-P 2) 1 !3 @'adz+

denote by

differences of density

Apl

and

-

u @ B1 S1

]

+

div

GI

=

0,

+ q.

the following weighted

Ap3

3

(1-v ) ( P -P

3

3

2

)

Then we get for the water saturation equation the following form

a { at

"1

:

- 0

(2.54)

G-

:

II. The Three-Phase CompressibleModel

I,

(2.55)

=

217

-oKd[vl(l-vl) gradPl2-v c 1v 3 gradP:2-Ap1

g gradZ]

vlG.

+

In the above equation, and in the corresponding equation for S

$,

3’

the

capillary

diffusion term

involves the gradient of the two 32 , with different coefficients. capillary pressure functions Pb2 and Pc Hence the computation of this diffusion term by the finite element technique described in Chapter V would require the resolution of four 12, one for linear systems (one for approximating vl(l-vl) gradPC

v1v3 gradPb2, one for

L

3

(1-0

3

)gradP3’ C’ and one for

Vlv3gradP:‘).

We will see now that, thanks to the TD hypothesis made on the shape of the three-phase relative permeabilities and capillary pressure curves, we can express these diffusion terms with the gradient of only two capillary

pressure

type

functions,

and

hence

divide

by

two

the

computational cost of this diffusion term. Noticing that the TD condition (2.24) makes it possible to write i n terms of VPc and VP, we may write equation

v gradPb2 + v gradP3’ 1 3 c (2.55) as follows : (2.56)

il

=

-oKd

[V

1

grad(PL2-Pc)

+

v1

apc ap

-1 gradP -Ap

g grad Z]+ v1

Using the expression (2.49) for the global flow vector can in turn express gradP

(2.57)

o1

=

+

in terms of

q and

-oKd [vlgrad(Pb2-Pc)-(Ap -.v1 +(1 -

ap

-

+

G.

q,

we

VZ, which finally gives :

(1-

ap)-1 p)

g gradZ]

v1 q,

which is equivalent to (2.55). Remark 2 :

In the Buckley-Leverett case where the dependance of the fractional flow curves v . on the pressure can be neglected, then to

:

Pc

J

depends only on

S1

and

S

3’

and (2.57)

reduces

Ch. IV: Compressible, Three-Phase, Black Oil or CompositionalModels

218

i1=

(2.58)

-aKd [ v , grad(Pb2 - Pc) - np

1

g grad2 ]

,

v1

+

0

Equations

(2.54)

-

corresponding to S 3

-

11.4.2

and

and

z3,

and

(2.57),

the

similar

equations

are the sought saturation equations.

An hyperbolicity condition When we neglect the capillary and gravity effects, the saturation

equations reduce to

a { at

(2.58-1

0 B. 5. }

0

J

+

J

P

If we take the pressure given,

is

(2.58-1)

a

G

div { v . J

}

=

0

j=1,2.

and the global flow field

-+

q

as

system of two conservation laws. Then we can

linearise this system around any a-priori given saturation profiles, and 6 S. of the reduced saturation from theses J

take as unknown the deviation profiles :

-

-

ass.

-

(1-SR,-SR2-SR3) o$B.J 2 at

(2.58-2)

av.

1

+

R=l,3

a

+I

as9.

;-grad 6SL +

j=1 ,2,

CI16S,+d=O

L=1

nd d are known functions of sp ce and time. A

where c

n.

condition for the original system (2.58-1)

ne essary

to be well posed is that its

linearized version (2.58-2) is well posed too. We

derive now a necessary condition for this using an 121. Let u s denote, at a given point

argument of BRENIER

the spatial coordinate along the direction of

q'(x,,

(xo, to), by y

to). The linearized

system becomes

-

(2.58-3)

-

-

-

ass.

(1-SR,-SR2-SR3)a@B j

+

I e=1,3

all. ass, A -+

as,

aY

.....

=

0

j=1,2,

which shows that the linearized saturation system is essentially one dimensional in space.

-

-

The hyperbolicity condition for (2.58-3)

-

[ ( (l-SR,-SR2-SR3)a$B. 7-l

J

av.J

as, , YI

i.e. that

is then that the

j , L , = 1 , 3 1 matrix has real eigenvalues,


219

av 1 (2.58-4)

av 3 as3

has real eigenvalues, where B

n = 3 .

(2.58-5)

B1

This is the case as soon as

a

(2.58-6)

Q(TI)

4n

=

\;

-1

23

as3 as1

But, as the value of

B

3

av,

(n --

+

q

as,

av as3

3 ) 2 2 0.

may vary from place to place since B 1

and

depend on P, we will require that

In order to violate (2.58-7) one has to suppose that the fractional flows v,

and

v3

satisfy

:

av as3 av, as3 av, (-) as,

av as, av 3 a v l y 3< --a s , as, as3 av 3 2 a'Jl (-) as3 - ( 2 as3 as,

- - 1 . 3 < 0

2

av av 2- 3 ) 2 as as 1

3

which, after some simple manipulations, resumes to

avl av3 < as3 as, Hence,

a

av

3

:

a\,

3). as, a s 3

Min (0,

necessary

condition for

the

linearized system of

saturation equations to be hyperbolic for any value of the volume factors

-

B. and any saturation profile S. is that : J

J -

220

Ch.I K Compressible, Three-Phase,Black Oil or Compositional Models

everywhere on the ternary diagram.

This condition should be satisfied by any three-phase data set if one wants the small capillary pressure problems to have a chance to be well posed for any volume factors. 11.5

- CONSTRUCTION OF THREE-PHASE DATA SATISFYING THE TD CONDITION (2.24)

The aim of this paragraph

is to show how one can actually

construct three-phase relative permeability and capillary pressure curves satisfying the total differential condition (2.24), by giving a step by step simple numerical procedure for that purpose. Of course the condition (2.24) alone does not uniquely determine the sought three phase data, and the choices left up to the user will be precisely described. Finally, some numerical examples of three-phase data satisfing the TD condition will be compared to the classical three-phase construction,(STONE-DIETRICH-BOUNDOR), which does not satisfy the condition. 11.5.1

- The practically available data The only data usually available to the reservoir engineer are two

sets of two-phase data, water oil and gas oil. So we will take as given (see figure 2) krt2(S,)

=

:

water relative permeabilities

(2.60)

in the water -oi1 system,

22 1


(2.61)

I

kr

(S )

=

gas

32 3 kr ( S )

=

oil

=

capillary pressure curve

3

23

P:~ \ 32


in the gas-oil system.

Figure 2 : Typical shape of the two sets of water-oil and gas-oil data and their location on the ternary diagram.

The problem is then to continue 12

into

p C (s1,S3)

into

P ~ ~ ,s3) ~ ( s ~

kr12 (S1)

into

krl(S1 ,Sj)

kr21( S 1 ) and kr23(S3)

into

kr2(S1 ,S3)

kr32(S3)

into

kr3(S1 , S 3 )

in such a way that

Ch.IV: Compressible. Three-Phase,Black Oil or Compositional Models

222

= o

=> ->

0 5 kr (S

S

krl(l,O)

5

kr2(0,0)

s1+s3= S

3

1

S ) 1' 3 0 2 kr2(S1.S3) 1

0 2 kr (S

3

11.5.2

S ) 1' 3

kr 2 kr

3

=

0

=

0

2 kr3(0,1)

-Continuation of capillary pressures One can choose any convenient continuation of the capillary

pressures. In order to avoid any unnecessary complexity, we will use from now on the following classical definition of the three-phase capillary pressures :

(2.63)

which will lead to an easy computation of the relative permeabilities. 11.5.3

- Continuation of relative permeabilities As

we have seen in paragraph 11.2.2, the knowledge of the three


kr,, kr2, kr is equivalent to the knowledge 3 and v3 of the global mobility d and of the two fractional flows v 1 (and then v2 is equal to 1-v -v ) , where d and v . are defined in

1 3

J

(2.11 ). From data ( 2 . 6 0 ) , (2.61) we define first two sets of two-phase global mobilities and fractional flows (rock model of the first kind)

:

12. The Three-Phase Compressible Model

223

(2.64)

(2.65)

Now, determining the three relative permeabilities krl, kr2, kr

3

all over the ternary diagram amounts to determining - a continuation d

of

d12 and

- a continuation

of

v

Of

'32

- a continuation

J

1

'3

dg2

(global mobility) (water fractional flow)

12

(oil fractional flow).

But, as one searches for functions satisfying the TD condition, it is not possible to choose independantly d, v1 and v3, and we can only choose independantly v1 w

3

+

v3

(i.e. l-u2) and

d

;

the values of

v1 and

will then be determined by the TD condition.

So we suppose chosen

:

. a three-phase oil fractional flow the condition S +S 1 3

(2.66)

(2.67)

1

=>

b1

and

d(Sl,S3,P). in such a way that

b3

that the TD condition (2.28) holds, and that : q0,s

3 ,PI

=

0

,

.' (S

3 1

(Sl,S3,P)satisfying

"2 = 0,

- a three-phase global mobility

We are now left with determining l-v2,

=

b2

,O,P) = 0 .

v1+W3 =

224

Ch. I K Compressible, Three-Phase,Black Oil or Compositional Models

Using the simple continuations (2.63) of the capillary pressure, the TD equivalent condition (2.28) reduces to

:

(2.68)

5

(where form

is a function to be determined), or in equivalent integral

:

S3 dPz2 v1 (S1,s ,PI

=

(S1 ,O,P)

dl

j - (s)

+

3

0

R(S1,s,P) ds

dS3

(2.69)

6 is known (remember (S ,P) which are 3 3 32 3 Hence we have to choose the function 5 in

Equations (2.69) determine uniquely v that

vl(S1

,O,P)

=

v

(S

,P)

12 1 given by (2.64) and (2.65)). such a way that (2.70)

w1(s1,s3,P)+

L

3

and

(S

1'

and v

1 -

once

v ( 0 , s ,P)

S ,P)

3

and that (2.67) holds, which amounts, as

=

=

v

l-v2(S1, S P) 3' \ J ~

(given)

(O,O,P) = v (O,O,P), to :

3

Then the equations (2.69) through (2.71) determine uniquely the function

6 (and hence v and v ) . This will be more clearly seen in a 1 3 discretized version of these equations, which will moreover give a simple algorithm for the computation of the three phase data. In conclusion, we see that, once three phase capillary pressures have been chosen, the three phase relative permeabilities satisfying the TD condition are uniquely determined by the choice of a mobility function d

and an oil fractional flow function

v2'

22s

I1 The Three-Phase Compressible Model

-

11.5.4

Nuuerical algorithm for the computation of TD three-phase relative permeabilities Let u s cover t h e t e r n a r y diagram by a uniform t r i a n g u l a r mesh -

w i t h s i d e s p a r a l l e l t o t h e s i d e s of t h e t e r n a r y diagram. Let AS

1 NPS

=

be

t h e s i z e o f t h e mesh (see f i g u r e 3 ) .

Let u s suppose t h a t t h e t h r e e - p h a s e f u n c t i o n s Pz2

,

B

12 v l , v 2 , v3, P c ,

are c o n t i n u o u s p i e c e w i s e l i n e a r f u n c t i o n s over t h e t e r n a r y

diagram; t h e y a r e hence p e r f e c t l y known once t h e i r v a l u e s

etc...

a t t h e nodes ( i , j ) , i , j

=

l...NPS,

i + j 5 NPS,

ij v1

,

ij

v2

,

of t h e mesh a r e

known. The TD e q u i v a l e n t c o n d i t i o n ( 2 . 6 9 ) g i v e s :

where I

From ( 2 . 7 0 ) ’ a n d (2.72) we g e t :

which is a l i n e a r e q u a t i o n f o r a r e known a t ( i ,j - 1 ) ,

E

a t ( i , j ) , (see f i g u r e 3 ) .

and

6 v3

ij

,

which one can s o l v e once

a r e known a t ( i - 1 ,j ) ,

and

5 \i2

and v1 is known

226

Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models

points w h e r e 0 = 0 and w h e r e \),and V3are know

NPS

+AS+

Figure 3 : The mesh on the ternary diagram for the computation of three-phase fractional flows satisfying the T-D condition

II. The Three-PhaseCompressible Model

227

We describe now step-by-step the construction of a set of three-phase relative permeabilities satisfying the TD condition

:

Step 1 : Initialization:

. Using i,O

v2

di*O ,

and

the available water-oil data, assign values to i

and

0,l

=

. set visa 3

=

... NPS.

o ,

= 0,

i

=

O,I

v

i ,O 1 ’

’

.O,j 2

... NPS.

. Using the available gas-oil data, assign values to do*’, j 0,l ... NPS. . set 0 , go*j 0, j O , I ... NPS.

0,j

v3

=

=

“79’

=

=

Step 2 : Choice of the three-phase oil fractional flow :

. Choose any curves

”O v2

and

convenient continuation of the oil fractional flow to the three-phase domain.

u;”

. assign the corresponding values to

Step 3

:

v3*J

Step 4

condition

. Solve (2.74) for for i 1 ... NPS, j

=

=

:

,

...NPS, j=1 ...NPS,i+j 6NPS. i=l

Determination of three-phase water and gas fractional flows

satisfying the TD

i ‘

ij J~

:

6

ij and use (2.72) for calculating v i ” and

... NPS, i+j 6 NPS.

1

Choice of the three-phase global mobility d For a choice

dij

i=l

...

NPS-1, j

=

1

...NPS-i,

of the

three-phase global mobility d, the three-phase relative permeabilities are given by : ij

(2.75)

kriJ

L

But, if

=

‘Lij dL diJ

,

for

L

=

1 (water), 2 (oil) and 3 (gas).

has been chosen without care, the relative

permeabilities computed from (2.75) may range out of the [O,ll interval, or be decreasing in a direction where one should expect them to be

increasing. So we will proceed in three steps

:


228

Step 4.1

Choose any convenient continuation ;1 of the global mobility,

:

and denote by mobility

d

diJ

its values on the mesh of the ternary diagram. This

will be used as a target f o r

the determination of the

three-phase mobility d in our TD system. Step 4.2

:

Determination of

d

on the water-gas side of the ternary

diagram. The values of di’NPS-i will be determined so that they are close to zi,NPS-i , and so that the relative permeability functions (2.75) and 3, have second derivatives not too large, range between 0

krl, &=l

and 1 and are increasing in the expected directions. This can be done by i solving the following constrained optimization problem, where d stands for di,NPS-l etc.. :

.

Find

d’, d

NPS-1

1

2

,....dNPS-l

{ w1 I di -d- i1 2

which minimize

1

w

+

I2kr&-kr&-’-kr&+l

I 1,

2

L=l,3

i= 1

under the constraints (2.76)

,

2 1

0 1 v

dL

:

aF i=l

1

di+di+l

0 2-

(

’,’,)

+

F,

-> \):+I+

i+l- i

b3

3 Remember that

Step 4.3

:

do

and

L3)

+

Determination of

requiring

that

it

is

,;

di+l-di

: 3 kr’ decreasing). 3

are known from step 1. d

inside the ternary diagram.

Similarly, we will determine by

krl increasing),

2* d

(i.e. i -> dNPS

3

1 7

(i.e. i

di+di+l 2d

&=1 , 3

[0,11),

,,li+’+.,~ d i + ~-d i

i+l- i jl

2dl

0 5-

...NPS-1, fF

(i.e. kr i and kri

close

to

diJ, i=l

...NPS-2, j=1...NPS-i-l,

diJ , and

that

the

relative

permeabilities (2.75) have a Laplacian not too large, take values between 0

and 1, and are increasing in the expected directions.

I1 The Three-PhaseCompressible Model

229

We will denote by T T diagram, and by d , wL etc... T

the set of triangles covering the ternary the value, at the barycenter of a triangle

, of the piecewise linear functions d,

E

denote by

$1

barycenter

of

v

a.

etc. Moreover we will

a unit vector pointing in the direction going from the triangle

T

to

the

vertex

that the krL

S

=

1. These directions

;:

of

the

ternary

diagram

will be used to assure Q +T relative permeability is increasing in the V L direction.

corresponding to

The optimization problem is then as follows :

.. Find dlJ i=l ..NPS-2, j=1.. .NPS-i-1

.

I

which minimize

:

NPS-2 NPS-i-1

{ wll d -d

i=l

1

+

\

w

1

i-1 ,j 16kr&j - krL

Q=l

j=1

- kri,j-l- kri+l,j-l Q 9, (2.77)

3

ij -ij 2

1

under the constraints

-

i+1,j - kri,j+l - kri-l,j+l 2 krQ 2 Q 1 1

:

One can remark that the optimization problems (2.76) and (2.77) have quadratic criterions and linear constraints, so they can be solved by any algorithm for quadratic programming. 11.5.5

- Example of TD three-phase data In order to demonstrate the existence of TD three phase data,

we have computed one set of such data starting from the water-oil and gas-oil two-phase data taken from STONE'S paper on three phase relative permeabilities. This will allow for comparison with one usual way of calculating

three-phase

relative

permeabilities.

As

Stone's

method

involves neither capillary pressure, viscosities nor volume factors,

Ch.IV: Compressible. Three-Phase,Black Oil or Compositionnl Models

230

their data have been completed by reasonable capillary pressure curves, viscosities and volume factors. The resulting set of two-phase data and fluid

Water Viscosities Volume

CcPl :

.5

:

1.

factors

Oil

Gas

1.43 .0128

1.

1.

Figure 4 : The two-phase data and fluid characteristics f o r the determination of a TD three-phase set of capillary pressures and relative permeabilities. The

three phase

capillary pressures are

defined

in

the usual way

S is taken to be equal to 1’ 3 12 the water-oil two-phase capillary pressure P c,2(S,) , and similarly Pz2 ( S , , S ) is taken to be equal to Pcz:(S3). 3 indicated in paragraph 11.5.2, namely Pb2(S

Then we have computed a set of TD-three-phase relative permeabilities using the step by step algorithm indicated in paragraph 11.5.4 : Step 1 : initialization using the data of figure 4


23 1

water .9

oil

water

nas

oil [identical)

I

oil

water

A

.5-

.1, water

oil

Satisfying t h e

TD

water

oil

A

.5

\

.1-

water

0il

Stone

Condition

Figure 5 :One set of three phase water, oil and gas fractional flows v,

, v2,

v3 satisfying the

TD condition (left), and the fractional flows

computed from STONE' s three phase relative permeabilities (right).

23 2


A

5-

water

5,

1

-

A

lwa er

Satisfying t h e

Stone

TD Condition Figure 6 :One three phase global mobility function

d

satisfghing the TD

condition (left), and the corresponding function computed from STONE'S three-phase relative permeabilities (right).

Figure 7 :The water and gaz relative permeabilities in the two-phase water-gas system obtained at the end of step 11.2 of the algorithm described in 311.5.4.


233

water .5-

I

water

oil

-

.7water

I

oil

~

oil

9

I

water

water

0il

.6gas

gas

oil

water

Satisfying the

TD

Stone

Condition

Figure 8 : One set of TD three-phase relative permeabilites (left) as obtained as the end step 4.3 of the algorithm of 911.5.4,

and the STONE'S

three-phase relative permeabilities (right).

234

Ch. IV: Compressible. Three-Phase,Black Oil or Compositional Models

Step 2 : the three-phase oil fractional flow v2

is chosen to

be equal to the one obtained by STONE's method. Step 3 v2

the TD three-phase fractional flows

:

identical to

STONE's)

obtained

at

the

v , , v 2 , v3, (with

end of this step are

illustrated in figure 5, together with the fractional flows obtained using STONE's method. Step 4

:

the global mobility

d

is chosen, both on the

water-gas side (step 4.2) and inside the ternary diagram (step 4.31,

So

that the resulting relative permeability functions will be the most regular ones satisfying the constraints (ie we have taken

w,

=

0 in

(2.76) and (2.77) and have skipped the step (4.1)). The resulting global mobility

d

is shown in figure 6 , together with the global mobility

computed from STONE's relative permeabilities. Finally, the water and gas relative permeabilities in the water-gas system computed at, the end of step 4.2 are shown in figure 7, whereas figure 8 shows the TD-three-phase relative permeabilites compared to STONE' s . 11.5.6

- The hyperbolicity condition We have checked the hyperbolicity condition (2.58-8) on the

three-phase fractional flow obt-jined !using either STONE's method or the TD-algorithm, both represented on

figure 5. It turns out that this

condition is satisfied at every mesh node for the TD fractional flows, but is not satisfied in the lower right corner of the ternary diagramm for STONE's fractional flows, avl -

ds

in this area one has

av

K 3>

>

0

,

avl
o,

O

as1

but, also

as3

a\,

as1

:


235

The fact that

as'

and

aS3

av,

' and as1

have opposite signs (which,

3

3%

have the same sign, 3 makes STONE'S fractional flow violate the hyperbolicity condition) cannot together with the fact that

occur for the TD fractional flows, as we see from the TD condition (2.68) that

av & 3

are always

aL

as3

and

have necessarily the same sign, as

Pz2

and P

12

1

increasing functioris nf t!irir. arguments (see figure 2).

11.6 - SUMMARY FOR THE THREE-PHASE COMPRESSIBLE MODEL

Notations

We quote here only the notations specific to the

:

three-phase case. All others can be found in paragraph 1.5 (summary for the two-phase compressible model). 1

index for water

,

P1

2

index for o i l

,

3

index for gas

,

P

12( S ,S ) c 1 3

P -P2 1

=

=

pressure in water phase

P2

=

pressure in oil phase

P

=

pressure in gas phase,

3

water-oil capillary pressure, cf (2.21)

Pi:

=

maximum absolute value of P k 2 (S1,S) 3

Pzi

=

maximum absolute value of P 2 2 (S1,S3)

P ( S ,S , P )

c

P

=

1

3

P

+ P

2

c

=

global capillary function, defined by (2.29)

=

global pressure (governs the oil+water+gas and P can be defined only when the

flow), ( P

TD condition (2.24) or (2.28) is satisfied). 1,2,3, reduced saturations, cf. (2.9)

S.,

j

d

global mobility, d

J

=

=

=

kr

1

d

1

+

kr2 d2

+

kr3 d3


236

p

=

w e i g h t e d d e n s i t y of f l u i d s , p

1'1 -

=

vlpl

+

v2p2 + v3p3

w e i g h t e d d i f f e r e n c e s of d e n s i t i e s of f l u i d s , c f ( 2 . 5 3 )

AP q

+

global flow vector, q

=

+ =

C$l

+

+ C$2

+

+

C$3.

Pressure equation in il

a q'

{ a

[B1sl

C$

+

B2S2 +

B3s3]

+

div

"q

=

0

ap =

-aKd

{

(1-

2 ) ap

g r a d P - pg g r a d Z

}.

Saturation equation in il

;iai: { a $ B I S-1 } + d i v G

= O

1

- 1 g r a d ( P 1c2 - P c )-(Ap - v l

i,

- oKd

Xa I

o $ B 3 S- 3 1 + d i v ;

=

{ \ l1

apc

apc -1

ap ( 1 - ap)

P ) g gradZ

= o 3

i

=

-oKd

{ v3 g r a d ( P z 2 - P c

3

I

Necessary condition for hyperbolicity av,

av

- as3 as3

1

2

Min

{

av,

0,

av

as, A as

}

:

everywhere on t h e t e r n a r y diagram.

}

III. The Black Oil Model

231

.

I11

The

black-oil

THE BLACK

model

the

is

OIL MODEL

simplest

case

of

compositional

models : it c o n c e r n s t h e f l o w , through t h e porous medium, of one heavy hydrocarbon component "gas")

and

of

(the"oil"),

water.

one l i g h t hydrocarbon component

on

Depending

the

pressure

and

(the

temperature

c o n d i t i o n s , t h e l i g h t component c a n e v e n t u a l l y be c o m p l e t e l y d i s s o l v e d i n t h e heavy

one

( t h e n one h a s a

single

l i q u i d hydrocarbon p h a s e ) , and

c o n v e r s e l y i t c o u l d happen t h a t t h e heavy component v a p o r i z e s c o m p l e t e l y (one

would

then

intermediate gaseous),

have

conditions,

each

of

which

a

single

one

has

hydrocarbon two

contains

gaseous

hydrocarbon the

two

phase).

phases

components

For

( l i q u i d and in

variable

proportions. The there

is

main

difference with

no l o n g e r

coincidence

t h e p r e v i o u s models

between

t h e phases

is hence t h a t

and t h e chemical

components ( w i t h t h e e x c e p t i o n of w a t e r , which w e w i l l suppose t o e x a c t l y c o i n c i d e w i t h t h e aqueous p h a s e ) . In

contrast

to

compositionnal

models

with

three

or

more

components, where t h e mass c o n c e n t r a t i o n s of t h e components a r e u s u a l l y t a k e n as main unknowns (see s e c t i o n I V ) , t h e b l a c k - o i l models a r e u s u a l l y s o l v e d i n terms o f t h e same unknowns as t h e two-phase problem, namely t h e phase s a t u r a t i o n s , t o which new unknowns, t h e d i s s o l u t i o n f a c t o r s Rs and

r s , are added. F o l l o w i n g t h i s l i n e , we a r e going t o write t h e b l a c k - o i l model e q u a t i o n i n a form s i m i l a r t o t h a t of

the previous compressible

model, i n c l u d i n g t h e u s e of a g l o b a l p r e s s u r e f o r t h e s i m p l i f i c a t i o n of t h e p r e s s u r e e q u a t i o n . For more d e t a i l s about b l a c k - o i l

models one can

s e e AZIZ-SETTARI and C I L I G O T - T R A V A I N . 111.1

- RANGE OF VALIDITY We s t u d y an i s o t h e r m a l model ( t h e t e m p e r a t u r e is c o n s t a n t a l o n g

s p a c e and t i m e ) i n a p r e s s u r e range below t h e c r i t i c a l p r e s s u r e of t h e two components ( s e e paragraph 1 1 1 . 3 ) .


238

Though

the

case

where

an

aqueous

phase

is p r e s e n t

in

the

r e s e r v o i r c o u l d be h a n d l e d , by mixing up t h e t e c h n i q u e s of t h i s c h a p t e r and of t h e p r e v i o u s o n e , we a r e going t o develop o u r b l a c k o i l model i n t h e c a s e where no water is p r e s e n t , i n o r d e r t o f o c u s on t h e d i f f i c u l t i e s i n h e r e n t t o t h e hydrocachwl

!)il-wq

eqtiilihrium.

For t h e s a k e of s i m p l i c i t y , we s h a l l a l s o n e g l e c t g r a v i t y and c a p i l l a r y p r e s s u r e h e t e r o g e n e i t y , but s u c h terms c o u l d b e a c c o u n t e d f o r very e a s i l y .

-

111.2

COMPONENTS AND PHASES

One phases

has

to

carefully

distinguish

between

components

and

:

I

gaseous

("gas phase" )

liquid

( " o i l phase")

2 phases

(

( " g a s component" )

("0 i1

However we w i l l u s e t h e index

g

component" )

t o r e f e r t o t h e gaseous phase as w e l l

a s t o t h e l i g h t component, and t h e index o t o r e f e r t h e l i q u i d phase as

well as t o t h e heavy component. I n o r d e r t o d e s c r i b e t h e composition of t h e l i q u i d and g a s e o u s phases,

we

shall

simulation area, and r

(3.1)

S'

use,

following

widespread

use

in

the

reservoir

t h e two d i m e n s i o n l e s s numbers ( d i s s o l u t i o n f a c t o r s ) R

defined i n f i g u r e 9 R

a

=

:

r a t i o , a t reference pressure of t h e volume of t h e g a s component t o t h e t h e o i l component c o n t a i n e d i n a g i v e n volume of t h e l i q u i d phase t a k e n a t reservoir pressure P ,

111. The Black Oil Model

r

(3.2)

If

239

r a t i o , a t r e f e r e n c e p r e s s u r e P r e f , of t h e volume of t h e o i l component t o t h e volume of t h e g a s component c o n t a i n e d i n a g i v e n volume of t h e gaseous phase taken a t reservoir pressure P.

=

one n e g l e c t s t h e change i n volume of t h e l i q u i d phase a t

r e s e r v o i r p r e s s u r e P caused by d i s s o l u t i o n of t h e l i g h t component, t h e of f i g u r e 9 is g i v e n by

volume f a c t o r B

Po(P)

B

0

=

B (P) = 0

) =

o i l component volume f a c t o r .

ref

Under a s i m i l a r h y p o t h e s i s for tilo gaseous p h a s e , one h a s

We can now s p e c i f y t h e main unknowns which w i l l be used i n t h e

d e s c r i p t i o n of t h e b l a c k - o i l model :

-

-

S g, So

=

s a t u r a t i o n of t h e gaseous and l i q u i d p h a s e s ,

P g , Po

=

gaseous phase and l i q u i d phase p r e s s u r e s ,

+ + $g, $o

=

v o l u m e t r i c flow v e c t o r of g a s and o i l components evaluated a t reference pressure

'ref'

One h a s of c o u r s e ,

-

s

+

so

=

1,

g a n d , a s we s h a l l e x p l a i n i n paragraph 1 1 . 4 ,

the saturation

range i n t h e whole i n t e r v a l [ O , l ] .

111.3

S

8

will

- DESCRIPTION OF PHASES EQUIILIBRIUM C o n s i d e r , under r e f e r e n c e p r e s s u r e c o n d i t i o n , a u n i t volume of

o i l component and a volume in a

container pre?surized

V

of g a s component. Then b r i n g them t o g e t h e r a t ttie p r e s s u r e P .

What happens t h e n w i l l

depend on t h e i n i t i a l p r o p o r t i o n s of t h e o i l and g a s components and of the pressure

P , a s shown i n f i g u r e 1 0 .

Ch. IF Compressible, Three-Phase,Black Oil or CompositionalModels

240

i ) If V 2 R(P),

the two components will stay in a single is by definition V

liquid phase, whose dissolution factor Rs with figure 9) and hence satisfies

(compare

:

RS 2 R(P). i i ) If R(P) 5 V 6 l/r(P),

the two components will split into two

distinct liquid and gaseous phases, whose compositions depend only on the pressure, and are given by

:

R

=

R(P)

for the liquid phase,

r

=

r(P)

for the gaseous phase.

Of course, the relative ammount of liquid and gaseous phase thus obtained

will depend on

V.

iii) If l/r(P) 6 V.

the two components will remain in a single

gaseous phase, whose dissolution factor

1

is by definition equal to -

rs

V

(compare With figure 9) and :?ence satisfies :

Take a gaseous phase with dissolution factor add progressively oil component to that phase. Then when it becomes equal to

r(P),

r

r

< r(P), and

increases, and

small drops of liquid phase ("dew")

appear, giving its name to the dew point. Similarly, Rs

take

a

liquid

phase

with

dissolution factor

< R(P), and add progressibly gas component to that phase. Then

increases, until it becomes equal to

R(P).

Rs

At that time, small bubbles

of the gaseous phase appear in the liquid phase, giving its name to the bubble point. P >-

The bubble point function function P pressure

P

->

R(P)

and the dew point

r(P) are increasing functions of pressure. Hence when

increases, R(P)

and

'

ro

tend towards each other, ie the

composition of the liquid and gaseous phase tend to be equal (see figure 9). The value P pressure :

when

crit P=Pcrit

for which the

R(P)

liquid

'

=is called the critical r(P) and gaseous phase become

indistinguishable. When (neither

the two component remain in a single "fluid" P > Perit, liquid nor gaseous) phase, independantly of the relative


24 1 gas

component

gas component component

Reference Pressure (P

Reservoir Pressure (P) l i q u i d phase

gaseous phase

Figure 9 : D e f i n i t i o n of the d i s s o l u t i o n f a c t o r s (boxes represent volumes)

gas

Rs

and rs

component

01I

component

Reference P r e s s u r e (P ) ref

Reservoir Pressure (P)

d

LlOUlD

PHASE

(bubble Point) LIQUID+GASEOUS (dew point) GASEOUS

R(P)

PHASES

l/i(p)

PHASE

Figure 10 : The binary diagram for a two component system (boxes represent volumes)

V

242


proportions of the oil and gas components.

As we mentionned at the end of paragraph 111.1, we restrict our P O .

and lower and

of t h e r e s e r v o i r p r e s s u r e , one h a s

Pmax

( s ) ds,

g

aF

As u s u a l ,

2 ] d~

:

as soon as ( w i t h t h e d i f f e r e n t i a l n o t a t i o n )

f o r 6P

Now, as t h e

S

=

Max

I P c ( S g ) l , and

P

f

[Pmin,

Pmaxl.

s g €CO,ll r e d u c e d s a t u r a t i o n may t a k e v a l u e s o u t s i d e of

g t h e i n t e r v a l [ O , l ] , we are f a c e d w i t h t h e problem of f i n d i n g some r e g u l a r c o n t i n u a t i o n f o r Y.

One checks e a s i l y t h a t :

a


241

Y

which shows that, though

and Pc

cannot be themselves continued out

with continuous derivatives

of the interval L O ,1]

(

dP C d~ (0)

g often are close to infinity, see figure 1 1 1, one still can

(3.23)

continue

-21

continue -

-21 P c (S g )

!

P (S ) c g

+ '((S

g

dP C (1) dS g

:

1 - P (0) 2 c

+

Y(0,P)

1 ,P) by - 7 Pc(0)

+

.i(O,P) for S 2 1 , g

r(Sg,P) by

+

and

for S 2 0 , g

into continuous functions having continuous derivatives. But, from the ichpi?.:.ar'y pressure law (3.11) we get that (3.24)

1 Po+ - P (S )+Y(S ,P) 2 c g g

=

-21 (Po+Pg)+Y(Sg,P)

=

so we can unambiguously define the global pressure

(3.25) P

=

i

p0

+

1 P (S ) 2 c g

1 P (S 2 c g

P g

Y ( S ,PI,

+

+

g

y(sg,P)

P by

:

f o r --

< s

2 I,

o

s

*

:s

SG(X,C,P) has

necessarily a discontinuity at the critical point (for example, one can

*

take for SG in the fluid domain an helice centered at the characteristic point and taking values 0 on the fluid-liquid boundary and value 1 on the fluid-gas boundary).

Hence, in

critical

is

point,

SG

a

very

the

two-phase domain, close to the

quickly

varying

function

of

the

concentrations (see figure 1 8 ) . Hence numerical determination of SG may be difficult, and one objective of the forthcoming transformation is to

*

"eliminate" the unknown SG from the problem formulation, thus obtaining equations with more regular coefficients. IV.5.2 - Definition of the global pressure P

We must define fluid.

P

by reference to the pressure in a flowing

I K A CompositionalModeI

28 1

1

2 Figure 18 : Perspective view of the Punctions Ss (discontinuous at critical point) and R*.

Ch.IV: Compressible, Three-Phase,Bhck Oil or Compositional Models

282

-

In

p r e s s u r e is

the

TI =

(4.59)

f l u i d domain

(see f i g u r e 1 3 ) t h e unique a v a i l a b l e

( s e e paragraph I V . 3 . 1 ) ) . Hence we s e t

PF

:

P = P F = T

- In

strict

the

two-phase

domain

and

the

enlarged

liquid

domain, t h e l i q u i d p r e s s u r e PL makes s e n s e everywhere, is c o n t i n u o u s on t h e bubble p o i n t l i n e ( c f . ( 4 . 4 3 ) ) and meets w i t h PF on t h e f l u i d - l i q u i d boundary ( c f . ( 4 . 3 1 ) ) . Hence, a t a p o i n t x whose c o n c e n t r a t i o n C l i e s i n t h e s t r i c t two-phase

domain o r i n t h e e n l a r g e d l i q u i d domain, and where

the l i q u i d p r e s s s u r e

*

*

where (SG, R )

iS

PL, we d e f i n e t h e g l o b a l p r e s s u r e P by

has been d e f i n e d i n

:

( 4 . 5 8 ) . Using (4.571, we s e e t h a t

e q u a t i o n ( 4 . 6 0 ) d e f i n e s u n i q u e l y P from t h e i m p l i c i t f u n c t i o n theorem. Similarly, g a s domain, i t is

in the strict-two

phase domain and t h e e n l a r g e d

which makes s e n s e everywhere, meets w i t h

PG

t h e r l u i d - g a s boundary, and we d e f i n e t h e g l o b a l p r e s s u r e

P

One f i r s t checks t h a t d e f i n i t i o n s (4.59) t h r o u g h ( 4 . 6 1 ) g l o b a l p r e s s u r e match

-

on

PF

on

by

of t h e

:

t h e s t r i c t two-phase

( 4 . 6 9 ) o f YL and YG one h a s

YL-YG

=

domain,

a s from t h e d e f i n i t i o n

PC

- on t h e f l u i d - l i q u i d boundary, a s t h e r e

PF

- on t h e f l u i d - g a s boundary, as t h e r e

=

PF

PL and PCM 0 PG and PCM E 0 =

s o t h a t t h e g l o b a l p r e s s u r e 2 is iurimbigi1ously d e f i n e d by (4.59) through

(4.61 ) . We check now t h a t t h e g l o b a l p r e s s u r e P regular pressure

f u n c t i o n of

pG

-

space

than

the

is l i k e l y t o be a more

liquid pressure

PL o r t h e g a s

IV. A Composition01Model

1

283

PRESSURES

SPACE

‘G

-1

IGAS SATURATION

GAS

SPACE

Figure 19 : Typical pressure profiles accros a front for a capillary pressure curve p as in figure 17.


284

We

have

illustrated

corresponding pressure

for

that

purpose

in

figure

19

the

profiles across a liquid-gas front, for the

saturation profile along a tie-line shown at the bottom of the figure and the capillary pressure curve p

of figure 17. The full line part of the

PG profile has been drawn using the fact that the liquid pressure profile PL has necessarily a continuous derivative across

sG

-

=

SGR in order to

ensure liquid-phase flow rate continuity, and using the shape of the capillary pressure curve p

of figure 17, which has infinite derivative

for S =O. A similar argument has been used to draw the full line part of G the PL profile. The dotted lines represent the physically meaningless "gas pressure" in the liquid domain and "liquid pressure" in the gaseous domain which could be arbitrarily defined using the extreme values of the capillary pressure. Due to the shape of the relative permeability curve near

SG =

and

0

SG =

1,

the

pG and

discontinuous derivatives, whereas the

a YC -

derivative, as

P

(O,R,P)

profiles

PL

have

strongly

profile always has a regular =

0

(cf. (4.56)

and figure

3% 17).

Hence

numerically

approximating P

is

probably easier than

approximating PL or PG! This has to be tempered by the fact that, in many reservoir simulations, the size of the capillary pressure is small compared to the pressure drop through the field, so that the high gradient zones in P c

or PL would probably not be seen in a pressure

profile drawn at the scale of the whole reservoir. Nevertheless, as it will turn out that the pressure equation for P is simpler (which is of course connected to the regularity of P ) , we think that the global pressure unknown should be used whenever it is available. IV.6 - THE GLOBAL PRESSURE EQUATION

We want to obtain, from the different equations described in paragraphe IV.4, an equation similar to the one-phase pressure equation (4.34), (4.35), but valid throughout the ternary diagram. First, we define in the two-phase domain a global mass flow +

vector q (4.62)

by +

q

+ =

)G

+ +

$L9

IV. A CompositionalModel

285

iG and iLa r e

where

g i v e n by (4.381,

(4.39).

Of c o u r s e , t h i s

c o n t i n u o u s l y w i t h t h e one phase mass f l o w v e c t o r

G,

We t r y now t o e x p r e s s thu?, e l i m i n a t i n g

-

In

gradPL,

d e f i n i t i o n s (4.34) of

-

meets

d e f i n e d i n (4.34) !.

i n a l l cases, i n terms of g r a d P o n l y ,

eG and g r a d P F

fluid

the

G

we

domain,

and ( 4 . 5 9 ) of P

I n t h e s t r i c t two-phase

:

get

immediately,

the

fron

:

domain, both phases are f l o w i n g ;

hence t h e two d e f i n i t i o n s ( 4 . 6 0 ) and (4.61) of P a r e v a l i d . As i n s i d e t h e

*

two phase domain we have SG (4.60), (4.61) y i e l d s gradP

=

gradP

=

SG and R

=

:

*

=

taking the gradient in

R,

ayL a.r gradR+PCM5 ay asG gradSG+PCM 5 gradP

gradP + Y g r a d PCM+PCM L

L

(4.64) grad? + Y w a d PCM+PCM G G"

M u l t i p l y i n g t h e f i r s t e q u a t i o n by

aYG

J.

a-i ay gradSG+PCM5 gradR+PCM gradP.

-1- w G ,

t h e second one by vG, and

summing we g e t , o b s e r v i n g from (4.54) t h a t t h e g r a d SG term v a n i s h e s (l-PcM

(4.65)

g)

grad P

=

v

G

gradPG+vLgradP L

+ P CM

a,

+

( v GYG+ vL YL )gradPCM

grad R.

+ From (4.65) and t h e d e f i n i t i o n s (4.38) and (4.39) of $G and for

+

q

+ =

+

$G+@L t h e e x p r e s s i o n

:

-+ $L,

we get

:

which has t h e same form t h a n t h e equat,ion (4.63) i n t h e f l u i d domain. The

c o e f f i c i e n t of g r a d PCM is t h e f u n c t i o n Y, d e f i n e d i n ( 4 . 5 3 ) .

Ch.IV: Compressible, Three-Phase,Bhck Oil or Compositionalhfodels

286

-

I n t h e enlarged liquid

domain,

only d e f i n i t i o n (4.60)is

v a l i d f o r t h e g l o b a l p r e s s u r e P . D i f f e r e n t i a t i n g i t y i e l d s immediately,

*

as SG

=

0 (Cf. ( 4 . 5 8 ) ) :

*

(b.67)

(l-PCM

ay

*

5

gradP

gradPL

=

*

*

where PCM s t a n d s f o r PCM(x,R , P ) , Y," i) and +

q

=

+

YL gradPCM + PCM

* *

for

I ~ ( S ~ , ,RP ) e t c

ay

*

5

gradR*,

...

i n t h e two-phase p a r t of t h e e n l a r g e d l i q u i d domain, one h a s wG = 0 + (4.67) and t h e d e f i n i t i o n (4.39) of $L y i e l d f o r + + $ G + $, t h e e x p r e s s i o n

iG = 0. Then

4

(4.68)

as ( 4 . 6 6 ) b u t w i t h R* i n s t e a d of R and $ i n s t e a d

of S G .

i n t h e l i q u i d domain, ( 4 . 6 7 ) and t h e d e f i n i t i o n ( 4 . 3 4 ) of t h e mass

ii)

-f

flow v e c t o r q y i e l d , as (4.69)

=

for

*

* )

"q

?I

=

P

.

L '

-K(x)d(C.P) { [l-PCM(x,R

2.i

Y

,P)

aP

Y

*

(SG,R ,P)]gradP-p(C,P)ggradZ

I n t h e e n l a r g e d g a s domain one o b t a i n s , i n a similar way,

t h e e x p r e s s i o n s ( 4 . 6 8 ) ( i n t h e two-phase domain) and (4.69)

w i t h YG

i n s t e a d of YL ( i n t h e gas domain). + Summing up t h e d i f f e r e n t e x p r e s s i o n s o b t a i n e d f o r q , we g e t t h e

sought p r e s s u r e e q u a t i o n

whose

coefficient

are

:

defined

in

table

1.

They

are

all

f u n c t i o n s of C over t h e t e r n a r y diagram, w i t h t h e e x c e p t i o n of discontinous a t the c r i t i c a l point.

continuous

?

which is


0

TABLE 1 Definition of the coefficif;nt#for the compositional equations (4.70). (4.71). (4.75). SG.R are definied in (4.58). CiG and ciL by (4.44). (4.45). and PG and pL by (4.48).

287

288


P

I 1

3

J

I

3

Figure 20

:

The p and ^p functions of concentration for a given pressure

(data of sIV.3.7)

IV. A Compositional Model

289

I

1

1

3

Figure 21

:

The d , ?

and

0

functions of concentration. for a given

pressure (data of sIV.3.7)

Ch. IV: Compressible, Three-Phase, Black Oil or Compositional Models

290 Remark 8 :

We have d i s p l a y e d i n f i g u r e s 20 and 21 t h e g r a p h s o f some of

the

c o e f f i c i e n t s of

compositional

the

pressure

IV.3.7.

d a t a of

equation,

for

the

is a l s o

The vcM f u n c t i o n

d i s p l a y e d i n f i g u r e 23. IV.7

-

0

THE CONCENTRATION EQUATIONS Here

also,

want

we

to

replace

(4.33) ( o n e phase

equations

domain) and (4.37) through (4.40) (two-phase domain) by e q u a t i o n s v a l i d i n a l l c a s e s . We w i l l c o n s i d e r

*

I n t h e one-phase domain, e q u a t i o n (4.33) remains unchanged :

a

% { @ ( x , P ) p(C,P)

(4.72)

(4.37)

a s g i v e n by t h e p r e s s u r e e q u a t i o n .

Ci}

+

}

div (Ci

=

0,

i

=

1,2,3.

I n t h e two-phase domain, w e s t a r t from e q u a t i o n (4.37)

a at

{ @ ( x , P ) [ p G CiG

sG+pL CiL sL] ]

+

d i v {C,,

GG

+

CiL

:

i,1

=

0,

i = l ,2,3.

Using

the

continuation

of

to

p(C,P)

the

two-phase

domain

d e f i n e d i n t a b l e 1 and t h e r e l a t i o n ( 4 . 1 ) we g e t :

(4.73)

P G CiG

From t h e i d e n t i t y (v,

plus

(4.38)

through

sc =

+

PL CiL SL

=

p(C,P)

ci.

l-vG)

(4.40),

t h e c o n t i n u a t i o n of

d ( x , C , P ) t o t h e two

phase domain g i v e n i n t a b l e 1 , and (4.73) we can rewrite (4.37) a s

:


29 1

Comparing (4.72) and (4.74), one sees that they both can be written as

a {

(4.75)

,. o(x,P) p(C,P) Ci]

+

div

Ci(x,C,P)

-K(x) d(x,C,P) ai(x,C,P) [gradnc(x,C,P)-6p(C,P) ggrad7.1 } i

=

=

0

1,2,3

whose coefficients are defined in table 1 . One may notice that the third term of equation (4.75) corr’esponds to exchanges between phases (caused by

capillarity

or

gravity);

concentration Ci of the ith

such

exchanges

influence the overall

component only if this component has a

different concentration in the liquid and gaseous phases, hence the C. -CiL factor in the a. coefficient (see table 1 ) . iG

Remark 9 :

One can see in figures 22, 23 some of the non-linear coefficients

of

the

concentration

equation

(4.75),

computed using the compositional data of paragraph IV.3.7. IV.8 - REGULARITY OF THE EQUATIONS

We discuss now the pressure and concentration equations (4.70), (4.71) and (4.75), especially from the point of view of the regularity of the coefficients, and hence of the solution; we will look particularly c1osel.y at what happens at the critical point. As noticed after (4.58),

the (reduced) gas saturation SG is a

very quickly varying function of the concentrations in the two-phase

*

domain close to the critical point (at the limit, its continuation SG is discontinuous at the critical point, as was seen in figure 18). Hence the practical determination of SG when C is close to the critical point is difficult and yields often innaccurate results. But, we shall see that Y

the coefficients depending on SG in the above equations either tend to be

*

independant of SG or tend

critical point. This shows

tMi:

towards zero when one approaches the the

coefficients of the equation can be

computed with reasonable accuracy even close to the critical point.

292


c 3

a2

I

1

2 1

2 1

Figure 22 : The

Ci and ai functions of concentration

(compositional data of 3IV.3.7)

I


293

F i g u r e 23 : The

x c , xcn

and

6p

functions of concentration

(compositional data of

3IV.3.7).

Ch.IV: Compressible, Three-Phase,Black Oil or CompositionalModels

294

Iv.8.1

-

Coupling between pressure and concentration equations in the strict two phase case, the pressure shows up in the

As

concentration equation (4.75) only in the coefficients and through the -f

global flow vector For

q.

the

pressure

equations

(4.70),

(4.71)

however, the

situation is somewhat different from the strict two-phase case

:

the

concentration gradient appears explicitely i n the expression (4.71) of the global flow vector terms. Of course,

q'

because of the grad aCM(x,C,P) and grad, R*(C)

depends on grad C only through grad R * (see table 1 ) ,

i.e. the derivative of

C

in the direction "toward the critical point"

+

only has an effect on q . One checks easily that trying to merge this grad C term in the grad P term by an ad hoe modification of the definition of the global pressure P requires that the compositional fractional flow v G ( S G . R ) and capillary pressure PG-PL

= P ( S , R ) to satisfy, as in paragraph IV.2 for C G the three phase model, the following total differential condition :

(4.76)

(we have dropped the dependance on x and P, as they appear as parameters for this type of calculation). But, condition (4.76) is very strong. It can be satisfied only if Pc is a function of

F

: [Ol]

-f

vG

(or conversely),

i.e.

if

there

exists

a

function

JR such that :

(4.77)

PG - PL

=

P (SG,R)

=

F ( v (S R ) ) G

G'

Such a function F could be easily determined, as P functions of S G , from the data f o r the 1-3

and vG are increasing

component system. However,

using (4.77) for the definition of the compositional capillary pressure instead of (4.28), ( 4 . 2 9 ) ,

would not be satisfying from a thermodynamical

point of view, as the capillary pressure defined by (4.77), does not tend to zero when R

+

0 (remember that

vc(SG,O)

=

Sc).


295

So we have to give over the hope of integrating the grad

C term

;.

in the grad P one in the expression (4.41) of

IV.8.2 - Some preliminaries f o r the study of the regularity

In order to focus on the main difficulties, we will make, in this

paragraph

simplifications

and

in

the

two next ones, the following technical

:

- The medium is homogeneous, i.e. we drop the dependance on x in all coefficients and non linearities. - The compositional two-phase data has been constructed from the 1-3 component two-phase data as indicated in remarks 5, 6, and 7. - The tie lines satisfy Hand's rule as described in remark 4. These assumptions are not essential and are made only for simplifying the study of the regularity of the coefficients as functions of

the concentration C = ( C ,C ) 1

3

over the whole ternary diagram. As

difficulties will arise principally at the critical point, we shall replace the ( C , , C figure 24.

3

)

variables by the (X,Y) variables as indicated on

Notice that this change of variable depends on the pressure

level P. We moreover require that the critical point is a "regular point" on the bubble/dew point line in the sen3e that

:

- The tangents to the bubble point line and the dew point line meet continuously at t'ne critical point (this hypothesis was implicitly made in the description of paragraph IV.2). - The

critical

point

is

not

an

inflexion

point

of

the

bubbleldew points line. As

usal, these assumptions are not practically restrictive, as

nobody has ever measured the curvature o r the bubbleldew points line! Under

these hypothesis, we

can

conveniently represent the

bubble/dew points line in the neighbourhood of the critical point by

:

Ch.IV: Compressible, Three-Phase, Black Oil or Compositional Models

296

when r is t h e c u r v a t u r e r a d i u s of t h e bubbleldew p o i n t s l i n e a t c r i t i c a l point. A t a g i v e n p r e s s u r e P , t h e r e is o b v i o u s l y a one-to-one mapping

from t h e ( X , Y ) v a r i a b l e s t o t h e ( X , R ) v a r i a b l e d e f i n e d by (X,Y)

(4.79)

->

(X,R

=

Y ( 1 - -) x b

a

- ')

When i t is p o s i t i v e , such R is e x a c t l y t h e parameter> a s s o c i a t e d i n ( 4 . 2 ) t o the tie-line

passing through t h e (X,Y)

point.

Of c o u r s e , when t h e

p r e s s u r e l e v e l P is changing, t h e bubble/dew p o i n t s l i n e w i l l change, So t h a t a , b and r are a c t u a l l y f u n c t i o n of t h e p r e s s u r e P.

W e w i l l suppose

that P

->

a ( P ) , b(P), c(P)

=

rb

(4.79bis) a r e bounded w i t h boiunded d e r i v a t i v e s which is a p h y s i c a l l y r e a s o n a b l e assumption. Finally,

we

express

will

c o n c e n t r a t i o n and p r e s s u r e (C

C

lL3'

S

G

3

b u t a l s o on P ) .

We end t h i s paragraph by e x p l i c i t l y c a l c u l a t i n g

as f u n c t i o n s of X , R

phase c o n c e n t r a t i o n s (X,,YG) p a r a b o l a and a s t r a i g h t l i n e

I

hence

:

xG-xL

=

SG

(notice

( a n d hence

f o r a given

R > O , t h e gaseous and l i q u i d

and ( X ,Y ) , g i v e n by t h e i n t e r s e c t i o n of a L

L

:

,

{

-*

on

and P .

W e first calculate,

(4.80)

depending

P ) using the variables ( X , R , P )

t h a t X and R depend n o t o n l y on C 1 , C

*

quantities

all

ZC(P)

w,

N.A CompositionaIModei

291

As for a given P, C. -GiL

iG

YG-YL, we obtain (4.80bis)

is a linear combination of X -X and G L

:

C. -CiL 1g

+

0 at least as fast as

&?

when

R+O+.

Then, for any (X,R) point inside the two phase region, and distinct from characteristic point, i.e satisfying

(4.81)

R(a(p)-X) 2

the gas saturation

5,

, 2c(P) x2

R > O

is given by :

whose partial derivatives are

:

I

In order to evaluate how quickly the derivatives of 3G may blow up to infinity when one approaches the critical point, we restrict ourselves to some neighbourhood of the critical point, and hence will only consider now those (X,R,P) satisfying (4.81) and (4.85)

O < R < l ,

-a$XSa

298

Ch. IV: Compressible, Three-Phase.Black Oil or Compositional Models

f o r w h i c h we g e t from ( 4 . 8 3 )

:

I

*

W e can now c a l c u l a t e t h e f u n c t i o n s SG ( X , R , P ) d e f i n e d i n (4.581, b u t

*

and R (X,R,P)

e x p r e s s e d i n terms of ( X , R , P ) i n s t e a d o f

(C,P);

one f i r s t h a s o b v i o u s l y :

*

(4.86)

R (X,R,P)

=

R

i n d e p . o f X a n d P.

=

D i f R S O Then u s i n g t h e d e f i n i t i o n ( 4 . 1 4 ) of t h e r e d u c e d s a t u r a t i o n :

where

S G ( X , R , P ) is t h e f u n c t i o n s d e f i n e d i n ( Q . 8 2 ) . From ( 4 . 1 4 ) we g e t ,

u s i n g (4.11)

:

*

G are t h e n g i v e n by

The p a r t i a l d e r i v a t i v e s o f S

:


299

Y

X

Figure 24 : The

XY

variables used instead of the C variables for the

study of regularity of c o e f f i c i e n t s across the c r i t i c a l point.


300

*

-

asG

- = _

ax

asG

asG

._ ax

3

(4.88)

as:

-

asG

-

= - . - + -

ap

"G

asG

asG

ap

ap

'

which, using (4.85),(4.87bis) and hypothesis ( 4 . 1 2 1 , shows that

* ap

constant

(X,R,P) 6

(4.88bis)

7' constant ,

5

R

constant

S

m

.

IV.8.3 - Regularity of the pressure equation We

will

investigate

here

whether,

for

given

regular

concentration distributions in 0, continuity and boundedness of the +

global flow q will imply a regular global pressure profile (by "regular" we understand here that the functions have bounded space derivatives). The practical interest of such a regularity result is that, in all +

situations where q is continuous and bounded, the global pressure will be easy

to

approximate

by

a

finite

difference

or

finite

element

approximation, as its derivatives will be always finite, and possibly discontinuous when one passes through the critical point or the critical tie-line (this situation is similar to that of elliptic or parabolic equations with discontinous diffusion coefficients).

In order to show this regularity result, we first study the regularity of the coefficients of the pressure equation (4.70), (4.71).

(so that the C = ( C l , C 3 , P ) concentration and pressure variables are replaced by the Using the hypothesis and notation of paragraph IV.8.2

(X,R,P)

as :

variables of figure 241, we can rewrite this pressure equation

301


a { q' (4.89)

=

$(PI p(X,R,P) ]

div

+

{GI

=

0

-Kd(X,R,P { [x(X,R,P)-?(X,R,P)

aTCM

(R,P)] grad P

-fi(X,R,P) g grad 2

aT -[?(X,R,P) S ( R , P ) - n(X,R,P)] grad R'].

One checks easily, from their definitions in table 1, that p, d,

5

are

continuous functions of X a n d R over the whole ternary diagram, with bounded continuous partial derivatives everywhere, except possibly on the SG = 0 and SG = 1 lines for p, d, 5 and on the SG = 0 and SG = 1 lines for

5.

Moreover, though p. d and

5

are defined via

?iGo r

SG

1 ), they are easy to calculate near the critical point, where

(cf. Table

sc and SC -

may vary very abruptly, as they tend to become independant of SG or SG when

R+O.

For

x

we get, using the remarks 5 through 7 *

*

We see that, despite the fact that

SG

:

is a discontinuous function of

at the critical point (see (4.5811, the x function is continous as

(X,R)

the integral is multiplied by

the product of the two interpolation

and e,,, each of them vanishing when functions e PC Similarly, we get for 7 :

R

+

0, R

> 0.

which is discontinuous at the critical point. But, on the other hand we have (4.92)

:

%

:A

dP

(R;P) =

- (P) dP

B

PC

(R')


302

From (4.92), (4.93) and (4.30 quarto) we see that

an

Y

continuous and

A

,"," bounded

?

aT

-$

is

over the whole ternary diagram, and that

both functions vanish over the fluid domain. The last coefficient to be studied is then

which, using (4.25), and (4.30 quinto), is also a bounded function on the ternary diagram, which vanishes over the fluid domain.

Remark 10

If we strengthen the hypothesis (4.30 quarto), replacing

%

5 constant by

d0

(R)

(4.95)

-f

0 when

R

-f

0,

aT

then both

CM aR

and

n

become continuous over the

ternary diagram.

0

i: L ' l e

prrssure equation is regular in

the sense given at the beginning of the paragraph. Looking at the second equation of ( 4 . 8 9 ) , we see that, given a regular concentration profile (X,R) (i.e. such that grad X and grad R are bounded over i2) I

(4.96)

d

:

is continuous over 2,

-

aTCM X - Y ap

is continuous over R ,

6

is continuous over 2 ,

g gradZ

- an CM (Y - aR

n ) gradR+

is bounded over R.

Thus the possible discontinuities of this last term have to be compensated by ad hoc discontinuities in grad P in order to yield a + anCM continuous bounded flow vector q. Moreover, the coefficient x - Y +

-ap

stays away from zero, so that the boundedness of q implies that of grad P.


303

1.

%i

2

\

\

\ coefficient

of

grad P

t

2

coefficient Figure 25

:

of

grad R

The coefficients OP grad P (top) and grad R (bottom)

in the pressure equation as functions of the concentration


304

So we have "proved" that the pressure equation is regular (in the sense given at the beginning of this paragraph) as soon as the interpolation functions

BRS,

ed etc.., used for the construction of the (4.251,

compositional two-phase data satisfy ( 4 . 1 2 ) ,

and (4.30 quarto).

The regularity properties of coefficients are summarized in tables 2 and 4. The non-linew cmetficients

x-Y

ap

and

aR -

9 aTCM

nv

evaluated with the data of paragraph IV.3.7, are displayed in figure 25.

Coefficients of the

1

d, ^p,

p,

x-9

1

aTCM ap

global pressure equation (4.89 ) avCM aR

?--

IT

continuous and bounded functions of X and R over

the

I

diagram

II

bounded

ternary

over

the

ternary diagram

TABLE 2 Regularity OP the coepficients OP the pressure equation

IV.8.4

-

Regularity of the concentration equation

The

concentration equation

(4.75)

is a

coupled system of

non-linear hyperbolic equations, with, in the regions where two-phase flow takes place, a diffusion term due to capillary forces. As it is well known that the solution of such equations may be discontinuous, we will not

try

to

regularity

give of

paragraph IV.8.2,

any

regularity results, but simply focus on the

coefficients.

With

the

hypotheses

and

notations of

the concentration equation can be rewritten as

:

IV. A ComposirionalModel

305

The c o n t i n u i t y of p and d h a s a l r e a d y been s t u d i e d , and, from

i t s d e f i n i t i o n i n t a b l e , 1 , one checks e a s i l y t h a t

ti

is c o n t i n u o u s over

t h e t e r n a r y diagram, w i t h i t s p a r t i a l d e r i v a t i v e s b e i n g a l s o c o n t i n u o u s ,

e x c e p t on t h e

-

SG = 0 o r 1 and SG = 0 or 1 l i n e s . We t u r n now t o t h e t e r m s i n v o l v i n g t h e c a p i l l a r y p r e s s u r e

II

C

( X , R , P ) , which become :

Using t h e m a j o r a t i o n s (4.88 b i s ) , (4.30 q u a r t o and q u i n t o ) one g e t s e a s i l y

:

is c o n t i n u o u s o v e r t h e t e r n a r y diagram, and v a n i s h e s a t l e a s t as R when R O+. -f

is c o n t i n u o u s o v e r t h e t e r n a r y diagram, and v a n i s h e s at least as when R O+. -f

(4.98) is bounded over t h e t e r n a r y diagram, and c o n t i n u o u s e x c e p t on t h e c r i t i c a l t i e l i n e . is c o n t i n u o u s o v e r t h e t e r n a r y diagram, and v a n i s h e s a t l e a s t as f l when R + O + .

B u t , t h e a . c o e f f i c i e n t s are d e f i n e d by : (4.99)

ai(X,R,P)

=

(CiG-CiL)

* +

* +

.G(SG,R , P ) C1-v G (S G , R , P ) l ,

Ch.IV: Compressible, Three-Phase, Black Oil or CompositionalModels

306

where ( c f . 4.80 b i s ) ) CiG-CiL

+

0 a t l e a s t as f a s t as f i w h e n R

O+,

+

SO

t h a t we get :

[

aa

a* c ,

, ai

ai

ai

an c 7

are c o n t i n u o u s o v e r t h e t e r n a r y

diagram, v a n i s h o u t s i d e of t h e s t r i c t two-phase doindin, and go

(4.100)

t o zero a t least a s f a s t a s r e s p e c t i v e l y R , f i a n d R

With t h e s t r e n g t h e n e d h y p o t h e s i s ( 4 . 9 5 ) on 0

Remark 1 1 :

PC

when R

+

Of remark

10, we g e t

e (R) (4.101)

when R

0

+

R

so t h a t

2

ie

rC

the

+

O+

becomes c o n t i n u o u s o v e r t h e t e r n a r y diagram, function

is

a

continuous

function

of

concentrations with continuous d e r i v a t i v e s

0

We t u r n now t o t h e s t u d y of t h e r e g u l a r i t y of t h e Using t h e one-phase d e n s i t y f u n c t i o n p ( X , R , P )

and

XG

with

XL

defined i n (4.80). bp

-t

0 when R

Using t h e h y p o t h e s i s t h a t t h e

f

XG

a ax

(O,O,P)

XL

XG-xL

of t h e one-phase

both

goes t o

from ( 4 . 8 0 ) ) and on t h e v a l u e of

(which is known t o be

derivative

and

0 . Of c o u r s e , t h e s p e e d w i t h

which 6 p goes t o z e r o w i l l depend on t h e speed w i t h which zero

6p function.

i n t r o d u c e d i n ( 4 . 4 6 ) we g e t

one-phase d e n s i t y f u n c t i o n is c o n t i n u o u s one s e e s , as t e n d toward z e r o , t h a t

the

the

density along t h e c r i t i c a l t i e

line. So we w i l l suppose t h a t : (4.103) which

(O,O,P)

is

for

*

0

example t h e c a s e when p

is c a l c u l a t e d , a s i n IV.3.7

l i n e a r i n t e r p o l a t i o n of t h e components p r o p e r t i e s .

by


307

One gets then, under hypothesis (4.103) (4.1 04)

Gp(R,P)

is equivalent to- 2

(o,o,p)

when R

+ O+

so that the 6 p function is continuous over the ternary diagram and vanishes over the fluid domain. We study now the derivatives of ie of

only in the ( X , R , P ) unknowns

+*

ax

over the ternary diagram,

6p

:

axG ( X ,R+,P) - G aR

2 ax

dR ' 1dR

axL ( X ,R+,P) L aR

The first line of the right-hand side of (4.105) goes to zero as XG and XL+

0 and

f

is continuous. Differentiating then

with respect to R , one gets

(O,O,P),

shows that

(4.80)

(XG,

R+,P) and

ap ( X , , R + , P )

:

a6

2 ( R , P ) is equivalent to - 2a(P) aR

(4.107)

XL i n

and

:

which, together with the fact that both tend towards

XG

*ax

(O,O,P) when R+O+

so the derivatives of 6 p over the ternary diagram are not bounded. In order to check if the a. 6 p function could not be more regular, we compute the partial derivatives of the a.(X,R,P) defined in

(4.99)

:

*

asG

av,

(1-2vc)

7

q

av,

ax *

asG au,

(1-2v ) [-*-+G asG a R

aR

Ch. IVC Compressible, Three-Phase.Black Oil or Compositional Models

308

P, T

Coefficients

,. C:, ~

of

ai,

I

I bounded and conti-

d, arc

a x , ai

I

ai

-'

the concentration equation (4.97).

I

nuous functions of X and R over

I

diagram

a

bounded over the

;iii (ai6p)

ternary di ayrain.

TABLE

3

Regularity of the coefficients of the concentration equations

TABLE 4 (next page)

Summary of detailed regularity properties of nonlinearities

Symbols

:

B C CP CTL

= = = =

TD

=

FD

=

bounded E continuous Q critical point critical tie line ternary diagram fluid domain

= =

unbounded discontinuous

t

0

e L

n m

m 0

L1

m

lo)

w

L

-

0

0

m " Y m

lo)

'ai

m v

m


m c 0 3

0 L

m

Y

n 0

-

a

,

m m

C

U

t

0

- -

e

,

m

m

C

*n

0 c

n

-

e

t

0

- -

x

,

m m

C

n

t

0

e

C

n

C 0 0 .3 Y

0 0

E C ZL. m 5

u

0 C X

a

J

,

X Y O L aJ

ki-5 3

m 5 0m. n.-.-n z o m c

3 0 0

I

0

n

x c

a

-

m

c

-

cC

F

-

m

0

c

B

0

LI:

111

0

a

m

309


310

For fixed R and P , Cic-CiL is independant of X, so that

a +C ax

.

-C ) iC iL

0. From (4.24) through (4.25) we see that (1-2vc) 1s aa bounded, and from (4.80 bis) and 4.88 bis) we see that is bounded

ax

when R+O. From (4.106) infinity as

1

one would check easily that

Moreover, as

aR

and

aR

Remark 13

a z(ai6p)

:

1

(see

(see (4.80 bis))

is continuous over the ternary diagram, and vanish as

(4.109)

(Cic-CiL) tends to

may go to infinity as

(4.88 bis) and (4.25)) and CiG-CiL goes to zero as aa . 1 we obtain that 2 aR blows up as JR when R+O+.

ax a (ai6p)

a

a t,G

as:

when R+O,

is bounded over the ternary diagram, discontinuous at tile orit,ical point.

and

If by chance the density function has a zero derivative at

the critical point along the critical tie-line, (4.103)

iS

replaced by (4.110)

3

(O,O,P)

0.

=

a6

Then 6p

+

0 as R (at least) and

5

is (at least) bounded

when R+O+, so that already 6 p has bounded derivatives over the ternary diagram. Multiplying by a. shows then that a. 6p has continuous derivatives over the ternary diagram. If moreover, (4.111)

3 Then

(O,O,P) 6p

=

0

itself has continuous derivatives over the

ternary diagram.

I3

31 1

V

CHAPTER

A

F I N I T E

ELEMENT

METHOD

FOR

TWO-PHASE

-

I

1.1

-

INCOMPRESSIBLE

FLOW

INTRODUCTION

INTRODUCTORY REMARKS

S t a n d a r d f i n i t e e l e m e n t methods a r e n o t a p p r o p r i a t e for r e s e r v o i r s i m u l a t i o n f o r two b a s i c r e a s o n s . The f i r s t is t h a t t h e y were d e s i g n e d for p r o b l e m s w i t h smooth s o l u t i o n s ( d i f f u s i o n p r o c e s s e s , s t r u c t u r a l m e c h a n i c s ) . whereas i n r e s e r v o i r s i m u l a t i o n , s o l u t i o n may d e v e l o p s t e e p f r o n t s when t h e

e q u a t i o n s a r e d o m i n a t e d by c o n v e c t i o n e f f e c t s . The s e c o n d r e a s o n is t h a t t h e velocity

of

the

fluids,

which

is a n

important coupling f a c t o r

in the

r e s e r v o i r e q u a t i o n s , is p o o r l y a p p r o x i m a t e d by t h e s t a n d a r d p i e c e w i s e l i n e a r c o n t i n u o u s f i n i t e e l e m e n t s commonly u s e d f o r smooth e l l i p t i c and p a r a b o l i c problems. F i n i t e e l e m e n t methods t a k i n g i n t o a c c o u n t t h e s e two a s p e c t s of the

reservoir

simulation

were

[l],

in

JAFFRE

[l],

a n d EWING-RUSSEL-WHEELER

for t h e m i s c i b l e

c a s e . I n a l l of t h e s e works a c c u r a t e v e l o c i t i e s are c a l c u l a t e d u s i n g finite

elements

d e s c r i b e d below.

for

the

and

f o r t h e i m m i s c i b l e case, and i n DOUGLAS

CHAVENT-COHEN-JAFFRE-DUPUY-RIBERA,

r11, DOUGLAS-EWING-WHEELER

introduced

pressure

equation,

F o r t h e problem of

the

mixed

and t h i s p r o c e d u r e w i l l be

a p p r o x i m a t i o n of

s h a r p , moving

f r o n t s , we s h a l l p r e s e n t a n upwind scheme b a s e d o n a d i s c o n t i n u o u s f i n i t e e l e m e n t a p p r o x i m a t i o n associated w i t h a s l o p e limiter. T h i s method l e a d s t o a more a c c u r a t e scheme t h a n t h e s t a n d a r d , f i r s t o r d e r , o n e p o i n t , upstream

w e i g h t e d , f i n i t e d i f f e r e n c e scheme and t h u s r e d u c e s t h e n u m e r i c a l d i f f u s i o n . Reservoir

e n g i n e e r s are

a p p e a r more d i f f i c u l t t o implement

r e t i c e n t t o u s e f i n i t e e l e m e n t s which than f i n i t e differences.

Therefore we

Ch. V: A Finite Element Method for Incompressible Two-Phase Flow

312

shall try to be as clear as possible in the description of our method. Actually finite

element methods (F.E.M.'s)

methods (F.D.M.'s)

differ from finite difference

in two aspects. First F.E.M.'s

approximations of various degrees while F.D.M.'s approximations. Usually for F.E.M.'s,

use piecewise polynomial

use only piecewise constant

one increases the accuracy of the

scheme by increasing the degree of the polynomials. Second, F.E.M.'s

are

formulated for irregular meshes where triangles as well as rectangles can be used while F.D.M.'s

are formulated only for regular meshes. It is this

second aspect of F.E.M.'s

which makes the programming more difficult. It is

the price one must pay to locally refine a mesh, or to build a mesh following the boundary of the domain or between two rock types. If one thinks the complication is not worthwhile, one can always write a fast, easier-to-code version of the method on rectangular meshes.

To simplify, we consider only the case of incompressible two-phase flow though the method described below can be extended to other cases as shown in ROBERTS-SALZANO, JAFFRE-ROBERTS C11,

BRENIER-JAFFRE. We shall

emphazise the spatial approximation, using a simple explicit, one point forward difference time stepping. The source terms (injection and production wells) will be modelled by specifying boundary conditions on boundaries surrounding each well. Usually the size of these boundaries will be of the same order of magnitude as the size of the edges of tbe mesh (several hundred meters)

whereas

the

actual well

diameter

is

only

20

to

30

centimeters. We will not describe here how the boundaries of the finite element mesh can be linked to the boundaries of the well. This can be done by

using

macroelement

as

explained

in

CHAVENT-COHEN-JAFFRE,

and

CHAVENT-COCKBURN-COHEN-JAFFRE. Also in the latter paper, it is shown how to handle the case of an oil field with several rock types, but, for the sake of simplicity, in the following, the oil field is assumed to be

01

one rock

type. The main features of the finite element method we are going to describe are the following : - mixed finite elements to approximate the pressure equat on, - a higher order scheme using discontinuous finite elements to approximate the convection effects in the saturation equation, so that numerical diffusion is reduced,

-

a slope limiter to preserve stability and prevent overshoots in

this higher order scheme,

I. Introduction

313

- mixed finite elements again to approximate capillary diffusion effects. 1.2

-

EQUATIONS OF INCOI!PRESSIBLE TWO-PHASE FLOW We recall from chapter 111 ~111.10 that the formulation of the

equations of incompressible two-phase flow are given by the following first order system of equations, set in 62

pressure equations

x

]O.T[,

where 62 denotes the oil field

:

:

saturation equations : 0

(1.2.a)

as +

div ;,(S)

=

0,

at

(1.2.b)

+)

=

(1.2.c)

?1 ( S )

=

r' =

(1.2.d)

-$

r' +

f' 1 ( S )

=

0,

2

L

j=o

;.

b.(S), J

PCM grad

J

a(S).

First we note tinat the pressure itself does not appear in the saturation equation and that only the velocity

+

qo is present. Thus a mixed

finite element method is particularly suitable for the pressure equation (1.1)

since it provides a way to directly approximate the velocity

Go.

This

method w i l l be described in section I1 and an efficient method for solving the resulting linear system will be given in section 111. For

the

saturation

equation,

we

shall

study

first

the

one-dimensional case in sections IV and V, and then extend the method to the two-dimensional case in section VI. The saturation equation (1.2) is a parabolic equation of diffusion-convection type. The convective terms are

Ch. V: A Finite Element Method for Incompressible nYo-Phase Flow

314

usually dominant and the saturation developes stiff fronts which are smeared out by numerical diffusion in first order schemes. Thus we shall build a new higher order scheme which can work with or without capillary diffusion, so that the fronts can be represented in a more accurate way. This scheme will be obtained by using a discontinuous finite element method. To complete the formulation of our problem, we have to add to equations (1.11, (1.2) conditions on the boundary

r of the oil field

condition. The boundary

Cl

r

of

Cl

and an initial

is assumed to be made up of

three parts, r the injection boundary, rs the production boundary and rR the closed boundary (see figure 5 of chapter 111). For our exposition, we choose, among the boundary conditions described in gII.2 and 811.3 of chapter 111 the following ones

/

where

qd

+

+

qo*v

=

s=

1

on

re

x

lO,TC,

is a given total oil+water flow rate and Pd a given pressure. The initial condition is, of course : S(q.0)

So

,

qd

:

=

So

in Q ,

being the given saturation at time

t=O.

1.3 - DISCRETIZATION

In the following, the oil field domain. It is discretized by a mesh

%

Cl

is a two-dimensional polygonal

of triangles and quadrangles in such

a manner that no angle is too small or too large. Common bounds are 30 and 120 degrees. In practice quadrangles are restricted to be parallelograms since these can be generated from a reference square by affine transformations, like

triangles

from

a

reference

triangle.

This

property

makes

the

calculation of integrals over them less expensive since fewer points of numerical integration are needed than when general quadrangles are used.

II. Approximation of the Pressure-Velocity Equations

We shall denote by the domain, by E an edge of

% ”&/

315

the set of edges of the discretization of and by K an element of

% . NED

(resp. NEL)

will denote the number of edges (resp. elements) in the discretization.

Figure 1 : The discretized domain

11- APPROXIMATION

11.1

-

Q with its boundaries.

OF THE PRESSURE-VELOCITY

EQUATIONS

APPROXIMATION SPACES

Since the pressure and the saturation equations are coupled by the velocity and not by the pressure itself, we would like accurate velocities. The RAVIART-THOMAS

mixed finite element method has been designed for this

purpose and we shall use it here with the lowest index, index 0, to approximate the pressure equations. The pressure and the velocity are approximated in two finite dimensional spaces, M o and

f

respectively, which are defined as follows.

Mo is the space of functions which are constant on each element of

%.

Ch. V: A Finite Element Method for Incompressible TWO-Phase How

316

A

simple basis

of

characteristic functions of

Mo

functions of dimension of M o

is the set {lK, K

Mo

the

elements.

The

€

%]

degrees of

of the

freedom of

are their constant values on the elements and the

is the number of elements NEL.

a point of 0 and by P k ( K ) To define 3, we denote by x=(xl, x,) the set of polynomials of total degree K defined on K. Let T be the I

reference triangle with vertices reference quadrangle

(O,O),

vector-valued polynomials

x"?)

(resp.

x'(^Q))

(O,l),

(O,O),

(1 , O )

and

^Q

be the

(O,l), (l,l), (1,O). We define the set of

:

is a three-dimensional (resp. four-dimensional) vector

space; its elements are vector-valued functions whose normal components are

?

constant on each edge of

.Q)

(resp.

^Q).

Any triangle (resp. parallogram) K of under an affine transformation x

two matrix and

bK

For any K c % ,

let

if K

a point of

=

FK ( ? )

=

%

is the image of

AK~ *

+

0. We denote by

bK, with

AK

?

(resp.

a two by

J K the jacobian of

FK.

is a triangle (resp. parallelogram) }. f

+

Such a transformation mapping s to s is chosen in order to preserve the integrals along the edges of the normal components across the edges. +

Now we define X as the space of vector valued functions that

s such

:

i)

the restriction of

s'

to any element K of

?(K), ii) (2.1)

+

the

normal

on E.

lies in

+

s across the edges are is such that E = K. n K. and + + 4 +J Ki, then SIKi*.Ji S I K , ' V j = o

components

continuous, i.e., if E E + 0 . is the outer normal to

%

of

+

J


+

X

is a s p a c e of

Condition

f u n c t i o n s d i s c o n t i n u o u s a c r o s s t h e edges of

is

ii)

317

lies i n H(div,Q).

equivalent t o requiring that

Another u s e f u l p r o p e r t y is t h a t t h e o p e r a t o r

t h e mesh.

x'

d i v maps

onto

Mo

(see

RAVIART-THOMAS).

Let u s describe a b a s i s of

u n i t normal

CE

o n e a c h edge

of

E

%

t.

We choose a r b i t r a r i l y a p o s i t i v e

, and i n

t h e f o l l o w i n g i t should be

understood t h a t t h e normal components of f u n c t i o n s of taken with respect t o of f u n c t i o n s of function

1

+ +

s'

t

of

x'

:E.

such t h a t

1

+

+

s E s V E ,1= i f E=E'

x'

across

has one d e g r e e of

of

?

E

are

is t h e s e t {:EIE€%}

and 0 o t h e r w i s e . Thus a

E'

and t h e dimension

S*wE,

A convenient b a s i s f o r

+

X

freedom per edge

E

E

't;

which is

is t h e number of edges NED.

E

Degrees of freedom

Basis functions lK, K e

%

such t h a t

1 if X E K

v

E

Mo

lK(X)

=

0 if x L

+

K.

+

S € X

where

tiE

is

t h e a r b i t r a r i l y chosen

p o s i t i v e u n i t normal t o E i

-P

Figure 2 Remark 1 :

:

Degrees of freedom and basis functions for M" and X

I n t h e c a s e of r e c t a n g u l a r g r i d s , t h e d e f i n i t i o n o f g r e a t l y s i m p l i f i e d (see DOUGLAS

Ell).

x'

. can be

Ch. V: A Finite Element Method for Incompressible Tho-Phase Flow

318

11.2 - APPROXIMATION EQUATIONS W e d e n o t e by +

qOh

an

an a p p r o x i m a t i o n t o t h e p r e s s u r e i n

Ph

approximation

to

the

velocity

+

in

There

X.

Mo

and by

is a one-to-one

c o r r e s p o n d a n c e between them and the v e c t o r s of t h e i r d e g r e e s of freedom

We remark t h a t

:

h a s a n a c t u a l p h y s i c a l meaning s i n c e i t is t h e

Q,

t o t a l flow rate of t h e f l u i d s a c r o s s

i n t h e d i r e c t i o n of t h e p o s i t i v e

E

normal t o E.

Now we c a n write t h e v a r i a t i o n a l form of t h e a p p r o x i m a t e p r e s s u r e e q u a t i o n s . We assume t h a t we a r e g i v e n a s a t u r a t i o n

a t time

St

nAt

i n an

a p p r o x i m a t i o n s p a c e which w i l l be d e f i n e d l a t e r and we c a l c u l a t e P E i n M o +n + and qOh i n X i n t h e f o l l o w i n g way. +n qOh i n e q u a t i o n ( 1 .1 . a ) and we m u l t i p l y F i r s t we r e p l a c e q+o by by t e s t f u n c t i o n s

v

div

(2.2.a)

of

.v

hG:

and i n t e g r a t e over

Mo

dx

=

,

0

Q

:

v E Mo.

R

functions parts

:j,

equations

j=l ,2

+

(2.52),

j = l ,2 t o t h e given

X by d i v i d i n g by 6 and m u l t i p l y i n g by t e s t

in

(2.53)

in

chapter

111, and

integrating

by

:

j

n

+

$-'

+

q l h * s dx +

+

dx

R Thus

Gjh,

we c a l c u l a t e t h e a p p r o x i m a t i o n

Then vector f i e l d s

Glh =

a

dx -

PCM d i v

div

P

=

Q

s'

dx

Iaa PCM

-

G

P

aa

G

+ + s.v dY,

+ + s - L IdY,

+

s

+

E

X,

E

x.

+

s

+

+

and q2h are s o l u t i o n s o f two l i n e a r s y s t e m s which a r e s o l v e d a t t h e

b e g i n n i n g of t h e s i m u l a t i o n . edge E

=

K n L,

For t h e r i g h t - h a n d

sides, n o t e t h a t , for a n

+ i n t e r s e c t i o n o f the e l e m e n t s K and L , i f sE is t h e b a s i s

f u n c t i o n whose f l u x p o i n t s o u t s i d e K , we h a v e s i m p l y

I1 Approximation of the Pressure-VelocityEquations

PCM div

n

$

dx

=

P

319

CMK

-

‘CML’

where PCMK and PCML are averages of PCM on K and L , and similarly for PG’ At time nAt we calculate an approximation h:q to the given injected flow rate qd and an approximation Pndh to the given pressure ’d on the production boundary such that qdh and Pdh are constant on each edge of

re

and Ts respectively.

+ qo, P , S , and qj, j=1,2 j=l,2 respectively; we multiply by test functions in

Finally, in equation (1.l.b) we replace +n by qOh, P,:

x’

SE,

such that

G.

+ +Jh’

S.\J=O

on

Teu rQ;

and we integrate by parts taking into

account the boundary conditions (1.3 left)

+n + qOh-v

(2.2.c)

Equations

=

qndh

(2.2)

on

+

re,

:

+n qOh-o +

=

o

rQ.

on

are the variational forms of the approximate

pressure equations. Proposition 1 : Equations ( 2 . 2 ) have a unique solution

+n P:) (qh,

E

x’

X

Mo.

Proof : Since equations (2.2) form a finite dimensional linear system, it

is sufficient to prove that, for all data equal to zero, the unique solution

is the zero solution. Thus, assume that Ph: = 0, +n = q in (2.2.b), we obtain from (2.2.a) Oh

Gjh

=

0,

qndh

= 0.

+

Setting s

-

1 From the physics, d and that

=

Oh -

0.

I ) ’

are stricty positive functions, so

3 20

Ch. V: A Finite Element Method for Incompressible no-Phase Flow

Now equation (2.2.b) reduces to

For

2, we take elements of the base of

hence P K. - PKj= 0

for any interior edge E

=

1

element with one edge i n c l l i d e d PK

=

+ +

such that s-vl

K. n K. 1 J'

and P

=

K

in the production boundary

r

=

0,

'eU r~ 0 for any

.

Therefore

0 for any K and P E 3 0. 0

The discrete by

form of the pressure-velocity equations is obtained

writing equations (2.2) in terms of

degrees of freedom and basis

functions. Equations (2.2) are equivalent to (2.3.a)

1

QE

E c aK

j div

E:

dx

=

:

K

0,

E

"%: ;

K

For convenience, we plug (2.3.~) into (2.3.b) so that the first sum in (2.3.b)

is taken now for

equations in matrix form

The matrix AQ

D

C

re

u

rL.

Then we can write our

:

is sparse, symmetric, positive definite and of

dimension NED. Its coefficients are

:


The matrix coefficients

321

is sg-ir'se t o o and of dimension

DIil

1 if E c aK and

DIV

K,E

=

NEL

x

NED, with

:

K div

s'E

dx

=

-1

'

J

if E c aK and E:

0 if

The vector

+

is pointing invard toward K,

E C aK.

FQ has dimension NED

as in (2.4)

is pointing outward from K,

~

if E a' T e u T

and its components are

S'

F Q =~

if E c reu

re.

Thus to calculate the global velocity and the pressure, we have to solve the linear system (2.5) which has dimension NED

Remark 2

:

Consider the case where the mesh

+

NEL.

is regular like a finite

difference mesh and number the elements K. . and the unknowns 1s.J

P. .,

as in block-centered finite difference Qi+1/2,j+1/2 methods. Using the vertices of K. . as integration points, in i,j

19.I

the calculation of the f i r s t integral of (2.3.b), one gets for a vertical interior edge K. n K. . indexed by i+1 ,j 1,J i+l/2,j : (2.6)

Ch. V: A Finite Element Method for Incompressible Two-Phose Flow

322

Similar formulas are obtained for Qi-l,2,j, Qi,j+l/2

and

Qi,j-1/2' Equations (2.3.a) with the finite difference notations can be rewritten for K=K

(2.7)

i,j

Qi+1/2,j + Qi,j+1/2 - Qi-1/2,j - 'i,j-1/2

=

'9

Thus, with the trapezoidal quadrature rule using the vertices of the rectangles as nodes, equations (2.3.a)

and (2.3.b)

reduce to the block-centered finite difference scheme (2.6), (2.7) where the coefficients d$ are calculated by harmonic averaging. For more details, see RUSSEL-WHEELER. Remark 3

:

In

the

incompressible

case,

since

the

velocity

is

divergence-free, it can be calculated in the subspace Of

x'

of

divergence-free

vectors,

CHAVENT-COHEN-JAFFRE-DUPUY-RIBERA.

cf.

JAFFRE

131

and

This method enables us to

calculate velocities without calculating the pressure, by solving a linear system of dimension equal to the number of vertices domain).

minus

one

Thus,

pressure-velocity

(in the case such

a

equations

of

method is

much

a

simply connected

for

solving

cheaper. However

the it

cannot be extended to the compressible case so we do not describe it i n detail.

111- RESOLUTION

OF THE ALGEBRAIC SYSTEM FOR

PRESSURE-VELOCITY

111.1

-

INTRODUCTION The linear system (2.5) is not positive definite and we shall pay

some attention to its resolution. Several methods have been investigated to solve systems like (2.5). BERCOVIER described a penalty method; AD1 methods

III. Resolution of the Algebraic System for Pressure-Velocity

323

have been designed by BROWN, DOUGLAS-DURAN-PIETRA. Another method is a lagrangien method as described in CHAVENT-COHEN-JAFFRE, for a reservoir simulation problem. For a general presentation of this method we refer to FORTIN-GLOWINSKI, or HESTENES. We introduce the affine manifold

and the lagrangian bilinear form

sn x' on

x

Mo :

t

F Q ~ ( ~-) J R

v div

s'

dx

where n + + (sh) q. - S dx. Jh One can show that equations (2.2) are equivalent to the min-max problem

and Pi can be +n div qOh = 0.

seen as

the

lagrangian multiplier

of

:

the constraint

Therefore we can solve equations (2.2) by solving the equivalent min-max problem which can be done by an augmented lagrangian method. This method is an iterative method for which there is a parameter to adjust and at each iteration we have to solve an NED-dimensional linear system. 111.2

-

THE MIXED-HYBRID F0R)IIILATION OF THE PRESSURE-VELOCITY EQUATIONS An other improved lagrangien method

is based on the mixed-hybrid

formulation of the pressure equation. It is obtained by dualizing not only the

incompressibility constraint inside each element as in the above

lagrangien method, but also the continuity constraint on the flow across the edges. This last method is very efficient and we shall describe it below. It follows the analysis of ARNOLD-BREZZI. With this method the pressure and the velocity are calculated by solving an equivalent linear system which reduces to solving only one non diagonal symmetric positive definite system of dimension NED, one diagonal

3 24


system of dimension NEL and one block diagonal system of dimension NED. Also there is no parameter to adjust. Moreover this last method gives more information about the pressure since it calculates also degrees of freedom Of

the pressure on the edges. Let

?* be the space of vector valued functions

such that only

(2.1) i) is required (and not continuity of the normal components across the

edges). The velocity is now calculated in for

?*

is the set of functions

x'"

and is denoted by.,:'q

{ g K,E, K f %,

E c aK} such that s+K,Ehas

E s ~ , ~dx- =J 6E, ~ where K:

+

its support in K, lies in %(K) and E normal to K. The degrees of freedom

+

Of

h:
0, if qo < 0.

With the initial condition

(4.3)

S(m.0)

=

so

and the boundary condition (4.2), equation (4.1) has a unique physical (i.e. entropy satisfying) solution which usually develops a discontinuous front (see figure 4). This sharp front is difficult to approximate numerically. For example it is well known that, on one hand, centered differencing in space (with forward differencing in time) yields an unstable scheme, and on the other hand one point upstream weighting gives a stable scheme with too much numerical diffusion smearing the front. Therefore, in order to decrease the numerical diffusion, we are going to describe a finite element upstream weighted scheme of higher order.


332

....

Figure 4 : Solution of equations (4.1). (4 . 3 ) at various times with S2 +,(S) = s2+(1-s)1/2 , so G 1.111.. @ 1 . sa = 1 .

=

IV.2 - A GENERAL DISCONTINUOUS FINITE ELEkENT SCHEFE

Discretize the space interval [a,b] with a set K.

=

[x.

i=l

,...,I

%

of intervals

such that xlI2 = a < . . . < ~ ~ + ~ / ~ < . . . < x ~ + ~ , ~ = b

S u p hi. We approximate the saturation in the ,,1 SiSI finite dimensional space M" of functions which are discontinuous at the and h.

=

xi+l/2-xi-l,2, h

=

points of discretization and restrict to polynomials of degree k , k>O, on each interval of the mesh. For practical purposes, k will be 0 or 1. We denote by Sh the approximate saturation. At each point x.

1+

i=O,...I,

/2'

it is discontinuous and we have a left-hand side limit Sh (x-i+ 12)

and a right-hand side limit S (x+ h i+1/2)'

IV. Approximation of the One-DimensionalSaturation Equation

I

I

\!

I

I

t

. -

.

333

I

I

I

I ! I a

1

I

I

I I

I

X.

I I

1 I

b X

X.

1-112

Piyre 5 :

'J

I

I

I

0

I

1+1/2

k The approximate saturation Sh lying in I4

.

In reservoir simulation, it is essential to design schemes which preserve the mass balance of the fluids. Thus on each interval K i , we want to satisfy the water conservation law

:

X.

(4.4)

J1+ll2

0

"h at

dx

=

@1h(xi+1,2) -

@lh(Xi-l/2)~

14iSI

X.

1-1 / 2

which says that the infinitesimal difference quotient of the quantity of water in K. is equal to the difference of the approximated water flow rates @lh at the extremities of Ki.

However since

Q l h depends

on

the

saturation, and since the

approximation saturation is discontinuous at the points of discretization, we have to define an approximation of the water flow rate $,h at these points. We

introduce upstream weighting in the scheme by defining the

approximate water flow rate as

:

Ch. V: A Finite Element Method for Incompressible Tbo-PhaseFlow

334

In (4.5) the water flow rate at the points of discretization is calculated with the upstream value of the saturation at these points. At the injection boundary, this upstream value is the boundary data (4.2). To obtain a conservative upstream weighted scheme for equation k

(4.11, we multiply equation (4.1.a) by test functions v in M , integrate over the domain ]a,b[, and integrate by parts on each subinterval the terms involving the first derivative in space. In the boundary terms arising from this integration by parts, we use the approximate water flow rate. This yields the semi-discretized in space scheme X.

X.

(4.6)

1

j1c1/2 0 3 at

1Si6I x.

1-1

:

J1+1'2

v dx - 1

ol(Sh)av dx

1 SiSI xi-1/ 2

12

where the approximate water flow rate

olh

is defined by (4.5).

Noting that the characteristic function

l K of the interval

K is

k

in the test space M , we can substitute it for v in (4.6) and we check that the conservation equality (4.4) is a consequence of (4.6). Let to = 0 < tl < tn < tN = T a partition of the interval

...

[O,T] with a constant time step At

=

in time yields the explicit scheme

:

(4.8)

0

Sh

=

t

n+1

- tn, OSnS-1. Forward differencing

Soh.

The approximate water flow rate

$rh

is calculated by (4.5) with

the saturation SE at the nth time level, and the saturation Soh is an approximation in Mk projection of S

0

into M

of k

the

initial saturation So,

for

instance the

.

Now, we describe more explicitly the procedure when k=O and k = l .

IV. Approximation of the One-Dimensional Saturation Equation

IV.3

-

335

THE CASE k=O : PIECEWISE CONSTANT APPROXIMATION

M o is the space of functions which are constant on the intervals xi+,/2[, i=l,.. . , I and the set of characteristic functions of these intervals is a basis of M'. We introduce the following notations for each i, i

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