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
:
Ch.I: Basic Laws and Models for Flow in Porous Media
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)
Ch. I: Basic Laws and Models for Flow in Porous Media
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
Ch.I: Basic Laws and Models for Flow in Porous Media
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 ) .
Ch.I: Basic Laws and Models for Flow in Porous Media
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 )
Ch.I: Basic Laws and Models for Flow in Porous Media
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.
Ch.I: Basic Laws and Models for Flow in Porous Media
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 :
Ch.I: Basic Laws and Models for Flow in Porous Media
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
:
IV. The Basic Models
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
Ch. I: Basic Laws and Models for Flow in Porous Media
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
Ch. I: Basic Laws and Models for Flow in Porous Media
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
IV. The Basic Models
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
Ch.I: Basic Laws and Models for Flow in Porous Media
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
IV. The Basic Models
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
<
water flow v e c t o r (4.39) can be w r i t t e n , u s i n g t h e c a p i l l a r y law (4.40) the f a c t t h a t P2 = c s t : gz(X)].
We can now d e f i n e K i r c h h o f f ' s p o t e n t i a l :
S
(4.45)
F(S)
k r l ( s ) P',(s)
=
sM its reciprocal function
-
(4.46)
S
=
ds
2
0
(homogeneous t o a p r e s s u r e )
:
c(F)
for F 6 0
and
The s h a p e o f t h e f u n c t i o n s F and c a r e shown i n f i g u r e s 12. With these d e f i n i t i o n s , (4.44) and (4.38) s i m p l i f y t o
In
order
to
write
an
equation
valid
for
11 and
:
both
s a t u r a t e d and
u n s a t u r a t e d c a s e , we e x t e n d , f o r F 2 0 , t h e f u n c t i o n c ( F ) which was d e f i n e d
Ch. I: Basic Laws and Models for Flow in Porous Medh
30
in (4.46)
for F S 0, by the constant value
sM
=
c(0) (see figure 1 2 ) . As
moreover on the boundary separating the saturated an unsaturated zone, both and (4.45)
definitions ( 4 . 4 2 )
of F yield the same value F
continuous by construction, equations (4.43)
and (4.48)
0, and
=
reduce to
6, is
:
which is the sought Richards equation. This equation is of parabolic type for F
O. In the case where c(F)>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
IV. The Basic Models
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
IV. The Basic Models
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 <<< 1 in the stationnary + + case, where q.v = 0 on E, and in the evolution problem coresponding to an increase of the water level in the reservoir, where one expects that q-v 2 0 on E. However, in the case of a decrease of the water level in the
+ +
+ +
reservoir, one expects that q * d 5 0 on M
<<<
which violates (4.34) as
Z,
1.
Moreover, the hypothesis that the air pressure is constant is a reasonnable assumption in the dam problem, where the domain Q2 filled with air is i n contact with the atmosphere on a large part of its boundary. Summing up, we see that Baiocchi's model will give a physically realistic solution both for the stationary case, and for the evolution case when the reservoir water level increases. When the reservoir water level decreases very quickly, larger discrepancies may arise between Baiocchi's solution and the physical one. The success of this model is due in part to the work of BAIOCCHI, FRIEDMAN-TORELLI who, through the use of a clever transformation, have shown that the system of equations (4.52) though (4.56), supplied with ad hoc boundary and initial conditions, could be reduced to a variational (in the simple stationary case) or quasivariational (in the general case) inequality over a
=
a,
U
a 2 , thus making the computation of the free
boundary easy by standard finite elements. Of course, Baiocchi's model can also be seen as the limiting case of the Richards approximation when the capillary pressure goes to zero, i.e. when the S = C(F) function of figure 12 tends to the Heaviside
-
function Y(F)
=
Sm
for
F < 0 and
sM for
F
>
0.
IV.4 - SUMMARY OF DIFFERENT MODELS We have represented on figure 14 the relationships between the different models described in
this chapter, together with the global
pressure equivalent formulation of Muskat's
RPCP
model, which will be
IV. The &sic Modek
35
TWO IMMISCIBLE FLUIDS
TWO FULLY MISCIBLE FLUIDS
CONCENTRATION EQ.
MUSKAT'S RP-CP MODEL
PRESSURE EQ. Cha .I constant air pressure
9IV.2
mathematical transformation
SATURATION EQ. 4-
GLOBAL PRESSURE EQ. Chap.IV, § I incompressibility
incompressibility
constant
RICHARDS EQUATION
D1F.-TRANSPORT
+
4 air pressure
Chap.1, IIV.3.3
( 1 975,B 1)
(1328,31)
SAT. EQ.
E L L I P T I C GLOB. PRES. EQ. Chap. I11
no capillary pressure
no capillary pressure
HYPERBOLIC SAT. EQ. E L L I P T I C PRES. EQ.
?
cross relative permeab. favourable mobility ratio heavier fluid below lighter one
w air pressure
FREE-BOUNDARY MODEL
FREE-BOUNDARY MODEL Chap.1, §IV.3.2
( 1 9702
Chap.1, 8IV.3.4
Figure 14 :
MUSKAT ' S
constant
BAIOCCHI'S
( 1 936)
The different models for immiscible and miscible flows in porous model
media. @ M2
H
m
means
"under hypothesis H,
is a special case of model M1".
Ch.I: Basic Laws and Models for Flow in Porous Medio
36
developped in chapters I11 and IV. As mentioned earlier, the only available mathematical results (existence, uniqueness theorems) concern the global pressure incompressible equivalent of Muskat's RPCP model (diffusion and transport equation + elliptic global pressure equation), the Richards equation and the Baiocchi free boundary model.
V
-
QUALITATIVE
NO-DIFFUSION
BEHAVIOR AND
OF
THE
SOLUTION
IN
THE
NO-CAPILLARY
PRESSURE
CASE
We will consider in this paragraph a typical situation of oil reservoir simulation, where
a resjdent fluid
(fluid C
2),
initially
occupying the porous medium, is displaced by a n injected fluid (fluid # 1 ) . One can think of fluids 1 and 2 as being water and oil (and then an immiscible displacement takes place), or as being some polymer miscible with oil and oil (and then fully miscible displacement takes place). When the reservoir is exploited by a regular net of wells, with a distribution pattern for injection and production well as shown in figure
...
15, one can reduce the problem (if the field is infinite ) to the study of the so called "quarter of a five-spot", where fluid 1 is injected at one
corner of a square, and fluid 2 is produced at the opposite corner. This "quarter of a five-spot" problem already exhibits all the typical features and difficulties of the general problem, and is extensively used in the technical literature for the comparison of numerical methods. Hence we will illustrate with this case the expected behavior of the solutions of the miscible and immiscible equations. However, for explanatory purposes, we will also consider the test problem of figure 16, similar to the quarter a five-spot problem but with a simpler geometry. V.l
-
& case.
Of
THE MISCIBLE OR IMMISCIBLE MODEL
Our main hypothesis will be that we neglect the diffusion term in the miscible case, and the capillary pressure Pc in the immiscible
37
V. Qualitative Behavior of the Solution
0 0
0
0
0
0
0
0
0
0
0
'
0 Y
0
0
0
0
quarter of five
0
0
0
spots
0
0
reservoir Figure 15 : The relationship between the "quarter of fivespot" and a reservoir
with
*production
r
a
regular
r
well,
of
array
=injection
wells
(O=injection wells.
r L=closed
boundary,
=production boundary).
A
,r
-
.. '
..
.
,
.
0 , .
fluid 1
___)
___3
.
.. ' .
.
.. . . .'. . . ., ,
.
.
.
'
.
I
,
.
L
.
, .
.
. . . . . . , . . . ' . . . .. . . . , , . , ' .
.
;fluid 2
. . .. .. . . . ,
. _'
.
.
' ,
I...'
-. I., , (2. . . L.
*
' .
. .
,
,'.
'.
, ..
.. . . . ,. . ' . . ,. , . . . . .. ' , , . : . .. . .. . _ , .. .. . . . . _, . . . . . . . .. . : . .. . . . . , . . . . L
,
I
,
'
rk
.
.
1
\
r*
boundary,
Ch. I: Basic Lavs nndModels for Flow in Porous Media
38
If we then denote by
u either the polymer concentration C (in
the miscible case) or the water saturation
S
(in the immiscible case),
both the miscible equations ( 4 , 4 ) , (4.5) and the immiscible ones (4.12) through (4.15) can be written, for the quarter of a five-spot or for the test problem, as :
(5.1) (5.2)
div
q’
=
0
P = P (5.3)
=
u = l
-d(u) grad P
onre on
r
q . v = o
on
rp.
u = 0
in Q
P +
(5.4)
,
=
Ps
< Pe
+
where the following additional simplifying assumptions have been made in order to retain only the essential features of the equations
*
*
.
:
the fluids and rock are incompressible, the gravity is neglected (i.e the field is horizontal) o = $ = K E 1 , s, = 0, SM = 1
in the immiscible case
(i.e.
zero
residual water and oil saturation). The interpretation of equations (5.1) through (5.4) is then as follows
V. Qualitative Behovior of the Solution
39
In the miscible case :
(5.5)
I
= C = polymer concentration, u b(u) = U,
d(u)
1 u(u) '
=
where ~ ( u )is the viscosity of the mixture for a concentration u. In the immiscible case :
(5.6)
where k.(u) = kr.(u)/p. J J J' = viscosity of fluid # j.
with
kr.(u) J
=
relative permeability and
"lj
In both cases, we define the mobility ratio M of the two fluids :
(5.7)
M = - d(1) =
d(0)
which is given by
1
mobility of injected fluid ( b l ) mobility of resident fluid (#2)
u(O)/u(l)
(5.8)
M =
(krl(l)/kr2(0))
in the miscible case
x
(u / u 2 1
in the immiscible case.
We will. see that this parameter has a strong influence on the solution the equations (5.1) through (5.4). V.2
-
Of
BEHAVIOR OF ONE-DI?lENSIONAL SOLUTIONS
In the test case illustrated on figure 16, the equations (5.1)
through (5.4) reduce to the following 1-D equations (L is the length of the sample) :
Ch.I: Basic Laws and Models for Flow in Porous Medin
40
(5.9)
u = 1
x = o
u = 0
t = o
L Pe - Ps
(5.10)
=
dx d(u(x,t))
q(t) 0
where the pressure drop Pe
- Ps
is constant and given. We analyse now the
behavior of solutions to equations (5.9)-(5.10), case, then in the immiscible case. V.2.1
first in the miscible,
- The miscible case In this case, we see from (5.5) that
the value of
the mobility
ratio M.
function with a discontinuity at V(t)
=
dX
b(u)
=
u, independently of
the solution u(x,t)
of the
(5.9) is, as shown in figure 17, a heavyside
concentration equation
(5.11)
So
=
X(t), travelling with a velocity :
q(t).
Figure 17: The 1-D miscible solution for any mobility ratio.
Y. QualitativeBehavior of the Solution
41
Then the pressure equation reduces to
and determining the miscible solution of (5.9), (5.10) amounts to solving the differential equation (5.11), (5.12) for the front position X(t). We analyse now the influence of the mobility ratio M on the stability of the front position X(t). Suppose that, at time t
=
t
Some perturbation has transfered the
0'
X(tO) to position X' (to). In order to see how this
front from position
perturbation will evolve at subsequent times, we calculate from (5.11), (5.12) the differential equation :
I
(5.13)
*
where (Pe-Ps) d(O) f(X,Xl)
For M
=
CL+ (i-1 )XI CL+ (I;i -1 )X' 1
we see that the perturbation
1
=
2 0.
1
exactly maintained ((XI-x)
=
a constant
XI-X
will be
t L t0 ) , it will neither increase
nor decrease. Moreover, we see in (5.12) that V
is independent of the front position, so that the concentration equation (5.9) and the pressure equation (5.10) are decoupled i n that case.
- For
M
< 1
opposite signs, so that time, going to zero when perturbation
XI-X
=
q(t)
d we see i n (5.13) that =(X'-X)
1
X'-XI t
+
=
(Pe-PS)d(0)/L
and
X'-X
have
will always be a decreasing function of m
:
for mobility ratio lower than 1, the
will resorb when t
+
-;
the 1-D solution is stable in
the sense that, t h e in the presence of perturbations, the position Of the front can be reasonnably predicted. *
For M > 1
similarly we get that
I
X1-X( will always be an
increasing function of time, blowing up to infinity larger than 1, the perturbation
X' -X
:
for mobility ratio
Will explode when t
+
-;
the 1-D
solution is unstable, in the sense that the position of the front cannot be predicted accurately in the presence of small perturbations.
Ch.I: Basic Laws and Models for Flow in Porous Media
42
V.2.2
- The immiscible case Let us make the hypothesis that
:
The shape of the relative permeabilities is independent (5.14) of the mobility ratio M This
shape
can
permeabilit ies
:
which,
hypothesis
under
be
of the fluids.
represented
(5.14),
will
be
by
the
valid
normalized
for any
two
relative
fluids,
independently of their mobility ratio. Then we get, from (5.6) through (5.8), that the function b for the immiscible case can be written as :
M -krl( u ) (5.16)
b(u)
=
M k r l(u)+kr2(u) so that, and this is the main difference between this and the miscible case, the shape of the immiscible fractional flow b(u) will change with the mobility ratio M. This
is
illustrated on
I
functions krl and
-
the
left
of figure 18, for typical
kr2. The corresponding solutions, calculated using the
so-called "Welge tangent" to construct a shock satisfying the entropy condition, are shown on the right part of the figure. for
saturation the 1-D
t4
<
uS1, b(u)
1 +
0
it is clear from (5.16) that for a given when M + 0. Hence uwelge 1 when M + 0 and +
immiscible solution becomes very similar, for small M, to the
miscible solution shown in figure 17. It also shares in that case the stability properties of the 1-D miscible solution, though the calculation are no longer straightforward.
V. Qualitative Behavior of the Sohrtion
43
"t
t1 u
-------
welge
0
M << 1
x(t)
t"T
Figure 18 : The 1-D immiscible solution Por mobility ratios H .
L
X
Ch. I: Basic Laws and Models for Flow in Porous Media
44
u > 0, b(u)
for M +
1
>
we get now from (5.16) that, given a saturation
1
when M
+
Hence
t'.
uwelge
+
when
0
M
+
so that
-i,
the
1-D immiscible solution becomes completely different, for large mobility ratios, from the miscible one : instead of an abrupt front separating the high and low mobility fluids, one gets a large transition zone between the two fluids. At the limit, the height of the front vanishes. In such a displacement, the only instability could come from the change in mobility across the front. Hence we calculate now the mobility ratio of the Mwelge two fluids across the front : d( Uwelge )
(5.17)
Mwelge
'
d(0)
-
which, using (5.151, becomes
Mwelge
=
M ir1 ( Uwelge ) + 'r2(uwelge
But from the definition of uwelge3 we have (see f gure 18)
:
b' (uwelge) = b(uwelge 'welge which, using (5.16) gives for M the following expression welge
When the mobility ratio
M
goes to infinity, we have seen that
Mwelge (normalized) relative permeabilities kr, and
kr2 near
As one typically has :
I-
krl( u )
-
ua
when
u . 0 with a > 1,
1 - kr2(u)
-
u'
when
u
+
we see that, with these conditions : (5.20)
Mwelge
+
0
U
welge depends on the behavior of the
goes to zero, and the limit of
(5.19)
:
when M
+
0 with
@
> 0,
= 0.
45
V. Qualitative Behavior of the Solution
so that the 1-D immiscible interface is stable for sufficiently large mobility ratios M. This will be confirmed by the 2-D numerical results of paragraph V .4.
V . 3 - BEHAVIOR OF TWO-DIMENSIONAL MISCIBLE SOLUTIONS We come back now to the 2-D equations (5.1), ...,(5.4)
with the
interpretation (5.5) of the variables and coefficients. We consider first the test problem of figure 16 which, as we have seen in paragraph V.2.1, admits the 1-D solution of figure 17, and we shall investigate the stability of this solution among truly 2-D solutions. Suppose that, at time t, we replace (see figure 19) the 1-D solution (dashed line) by a perturbed solution (full line). Consider then two flow lines L and L'
crosssing the interface respectively in its
unperturbed and perturbed regions (these flow lines are approximately straight lines if the perturbation is small enough).
- For low mobility ratios, the line
L' spends more time than L in
a low-mobility region, and less in a high mobility one, which results, as I on L' the pressure drop Ap across the slab is given, in a flow-rate I
2
1G1
smaller than the flow-rate on L. As we have seen in paragraph IV.3.2 + -t that q and q' are the velocities of the corresponding points of the interface, the perturbation which we created at time t will resorb at time t + At. This correspond to the property, for 1-D
miscible flow with low
mobility ratio, that any perturbation AX of the front position will resorb as time goes on. For high mobility ratios, the same analysis shows that now 1 i l , SO that the size of the perturbation will increase with time, yielding the so-called fingering process
:
the physically observed fronts
develop many fingers in an unpredictable way, each of them being originated by a small perturbation of the interface, due to small heterogeneities of the porous medium. An exact solution for a one-finger solution has been obtained by JACQUARD-SEGUIN, using a conforming mapping technique, for the case of the
test problem with a slab of infinite length. It is easily understood that the numerical calculation of such high mobility miscible displacements is an extremely difficult task. From the above analysis one can guess that the required tools include
Ch.I: Basic Laws and Models for Flow in Porous Media
46
necessarily
-
a solution scheme for the hyperbolic equation (5.1) with no or
extremely little numerical diffusion, as this diffusion would prevent the fingers to develop,
- a resolution scheme for the pressure equation (5.2) which gives < (as it is the driving term of the hyperbolic
an accurate calculation of
equation (5.1)) taking precisely into account the abrupt variation of the coefficient d(u) across the interface separating the two fluids (as this is the phenomenon which causes the fingers to grow). But in any case the fingers obtained in numerical calculations will always be originated by numerical perturbations. So the predictive value of those models can be at most of a statistical nature. Various authors have computed recently high mobility miscible displacements exhibiting such fingering : SETHIAN-CHORIN-CONCUS, LOTSTED,
GLIMM-MARCHESIN-McBRYAN, BRENIER-COHEN. We show in figure 20 numerical results obtained by BRENIER-COHEN using a level line method for the hyperbolic equation and a mixed finite element method for the pressure equation. These results concern the quarter of a five-spot problem of figure 15. For the high mobility ratio, one sees that the perturbations coming from the grid orientation induce completely different fingers on the fluid interface : when grid (a) is used, on large finger develops toward the production well whereas two fingers cramped along the sides of the domain are seen when grid (b) is used : the displacement is unstable. For the low mobility case the fronts obtained with the two grids are located at the same position, the small fingers (which do not develop) observed in the case of grid (a) come from a poor numerical integration of the pressure equation.
V.4
-
BEHAVIOUR OF 2-D IMMISCIBLE SOLUTIONS We consider now the same equations (5.1)...(5.4)
as above, but
with the interpretation (5.6) for the variables and coefficients. As for the miscible case, the test problem of figure
16 admits 1-D
solutions corresponding to the saturation profiles described in figure 18 for different mobility ratios.
V. QualitativeBehavior of the Solution
-
fluid
5
L
a@ 1 II a
'
-9
q-
low mobility
given c
--
L'
high mobility
a! It a
high mobility
pe-P,
L
a@ a
fluid 1 II
-
7
L.
AP
-
47
p.!
-
-
r
A
q-
II
p.
mobility
Figure 19 : Behavior of 2-D miscible i n t e r f a c e for the test problem.
Ch. I: Basic Laws and Models for Flow in Porous Medin
48
For low mobility (M
<<
1 1 , the 1 - D
immiscible solution (top of
figure 1 8 ) is very close to the miscible one (figure 1 7 ) , so that
the
behavior of the 2-D immiscible and miscible soltions are very similar; they both exhibit a stable fluid interface. -For high mobility ratios
(M >>
1)
however, the immiscible
solution differs completely from the miscible one, as can be seen by comparing figure 18, bottom, with figure 17. As we noticed already in paragraph V.2.2,
when M is large, the transition between the displacing and
displaced fluids is smooth and occurs across a large area. So the driving phenomenon for the instability of the miscible front, namely the steep variation of mobility across the front, is extremely weakened in the immiscible case, and the high mobility immiscible displacements can be expected to be much more stable than the miscible ones. This is confirmed numerically by the results of BRENIER-COHEN (see figure 2 1 ) , where one checks that the change i n the grid orientation has almost no effect on the low saturation contours whereas some effect can be seen on the contours corresponding to higher saturations.
V. Qualitative Behavior of the Solution
49
q;; grid (a)
* .
. . M=20
. .
. .
..
-. -.-/-/,
f”.‘.’
-I+
/ ’
/
’/
‘ I /I
grid (b)
. .. .. .
/ / < 0
I
I
. . . .
.
.
.
I
300
lays
M=- 1
20
600 Figure 20 : Numerical resolution of the quater of five spot problem
of figure 15 with two miscible fluids, for high and low mobility ratios.
50
Ch. I: Basic Laws and Models for Flow in Porous Media
grid b
, , . , I
#
*
*
M=20
400 days
Figure 21
Numerical resolution of the quarter of five spot problem
of figure 15 with two immiscible fluids. Only the case of high mobility ratio is shown, the low-mobility ratio results being similar to those of figure 20. bottom.
51
CHAPTER I1 SLIGHTLY COUPRESSIBLE MONOPHASIC FIELDS
We c o n s i d e r i n t h i s c h a p t e r t h e case o f a p o r o u s medium o c c u p i e d by a s i n g l e f l u i d ; i n o r d e r t o b e d e f i n i t e , we w i l l s u p p o s e t h a t t h i s it c o u l d b e a n y o t h e r f l u i d as well.
f l u i d is o i l ; b u t
We w i l l suppose
t h a t b o t h r o c k and f l u i d are s l i g h t l y c o m p r e s s i b l e . After h a v i n g p r e s e n t e d i n s e c t i o n I t h e (well-known)
parabolic equation describing the evolution
o f t h e p r e s s u r e , we r e c a l l i n s e c t i o n I1 a n e x i s t e n c e a n d u n i q u e n e s s r e s u l t for
equation.
that
a p p r o x i m a t i o n of obtained
either
S e c t i o n I11 w i l l
be
p o i n t sources and sinks.
devoted t o
the
study of
the
Such a n a p p r o x i m a t i o n c a n b e
by u s i n g some f l d i s t r i b u t e d l l a p p r o x i m a t i o n of
t h e Dirac
f u n c t i o n i n t h e r i g h t hand s i d e o f t h e e q u a t i o n , o r by c u t t i n g o f f a small piece
inside
resulting
the
porous
boundary.
point-source
The
medium
and
injecting
the
of
approximations
convergence
solution w i l l justify
in
both
some
sense
fluid
through t h e to
the
t h e a p p r o x i m a t i o n of
i n j e c t i o n o r p r o d u c t i o n wells by a d i s t r i b u t e d r i g h t hand s i d e , which i s a standard procedure i n p r a c t i c a l engineering.
I-
CONSTRUCTION
OF
THE
PRESSURE
EQUATION
We d e f i n e f i r s t some n o t a t i o n ( c f . c h a p t e r I 3111.1)
p
(1.1)
=
po(l+co(P-Pref))
is t h e mass o f a u n i t volume o f t h e o i l is t h e o i l volume f a c t o r
B(P)
=
p(P)/p(Pref)
co
=
o i l compressibility, p
c0
=
d
=[[log
B(P)]
for
P
=
o i l viscosity
=
Pref
$ ( x , P ) = $O(X)(l+cR(P-Pref))
cR c
=
rock c o m p r e s s i b i l i t y , cR
= c0
+ C~
=
:
is t h e p o r o s i t y o f t h e r o c k =
a [log 5
$(x,P)] f o r P
=
Pref
g l o b a l c o m p r e s s i b i l i t y o f t h e r o c k and t h e o i l
Ch.II: Slightly Compressible Monophasic Fields
52
=
v o l u m e t r i c flow v e c t o r of t h e o i l , d e f i n e d by f o r m u l a ( 3 . 6 ) of
is homogeneous t o a v e l o c i t y and is c a l l e d t h e Darcy
c h a p t e r I ("q/o
v e l o c i t y of t h e o i l ) . the
Using
c h a p t e r I,
8IV.1,
of
geometry
we
can
now
chapter I,
s e c t i o n I 1 and t h e model of
write
equations
the
in
Q
and
on
its
boundaries.
-
I n s i d e t h e p o r o u s medium Sl, we have t h e o i l c o n s e r v a t i o n law
and t h e Darcy law
(1.4)
-
-(I(.x) K(x) -B(P) [gradP
=
U
p(P) g grad Z(x)].
A s we s u p p o s e h e r e t h a t t h e f l u i d s a r e o n l y s l i g h t l y c o m p r e s s i b l e ,
we may assume t h a t
(1.5)
co
<<<
1,
CR
<<<
1,
s o t h a t we g e t , by n e g l e c t i n g t h e h i g h e r o r d e r terms i n co
and
cR:
(1.6)
A s u s u a l , we i n t r o d u c e t h e h y d r o s t a t i c p o t e n t i a l
which we t a k e a s unknown i n s t e a d of t h e p r e s s u r e . we g e t t h e e q u a t i o n for
(1.8) (1.9)
TI
=
Q :
aII
+ div
1.I
II.
c a ( x ) $,(x)
"q
inside
-u(x) K ( x ) grad
"s
=
0
II
by :
From ( 1 . 3 ) , (1.41,
(1.6)
I. Construction of the Pressure Equation
53
r II
On t h e l a t e r a l boundary
*
- a no-flow
one can have :
(Neumann) c o n d i t i o n o n t h e p a r t
rLC where
the
f i e l d is closed + + q - v
(1.10)
part
red
r II
of
-
l-IIC
a given p r e s s u r e o r p o t e n t i a l ( D i r i c h l e t ) c o n d i t i o n on t h e
where t h e p r e s s u r e is g i v e n ( f o r i n s t a n c e b y a w a t e r d r i v e )
n
(1.11)
on
r
= I I ~on
(n,
is o f t e n c o n s t a n t o n
rk,
C o n d i t i o n s o n t h e well b o u n d a r i e s
k
=
'id)' 1,2,...,K.
Generally
t h e o n l y a v a i l a b l e measurement a t a w e l l is t h e v o l u m e t r i c o i l p r o d u c t i o n rate
Q k ( t ) a t time t , which c o r r e s p o n d s t o t h e f o l l o w i n g boundary c o n d i t i o n
I
(1.12)
+ + 9-v
Qk(t)
=
aF k = 1,2,...K.
'k
C o n d i t i o n (1.12) ( t h e knowledge o f
is o b v i o u s l y n o t s u f f i c i e n t f o r d e t e r m i n i n g t h e p o t e n t i a l + + q.v a t every point o f r k would b e enough, b u t s u c h a
r e f i n e d measurement
s o we h a v e t o c o p e w i t h o n l y t h e
is n e v e r a v a i l a b l e ,
knowledge of
t h e i n t e g r a l of
G.;
(reasonable)
assumption
the
borehole
is t h e o i l h y d r o s t a t i c p r e s s u r e r e p a r t i t i o n .
Dk
pressure continuity pressure
across
distribution
potential
II
(1.13)
that
on
t h e well
rk
is
fit
rk,
oil-hydrostatic,
is c o n s t a n t ( b u t unknown) o v e r
TI =
Hence we s h a l l make t h e
r e p a r t i t ion
boundary
unknown c o n s t a n t o v e r
rk
inside
t h e well
Using t h e n t h e
we i n f e r t h a t t h e i.e.
that
the o i l
:
'k'
We s h a l l see i n t h e - n e x t p a r a g r a p h (3.2)
(1.13)
rk).
over pressure
t h a t boundary c o n d i t i o n s (1.121,
v e r y well i n t o t h e v a r i a t i o n a l framework a n d l e a d t o a u n i q u e
determination of t h e p o t e n t i a l
T.
W e s h a l l r e f e r i n t h e sequel to condition
(1 . 1 2 ) , ( 1 . 1 3 ) , a s t h e t t w e l l - t y p e t t c o n d i t i o n . One o t h e r p o s s i b i l i t y is t o a r b i t r a r i l y d i s t r i b u t e t h e p r o d u c t i o n
Ch.II: Slightly CompressibleMonophasic Fiekis
54
Q ( t ) over t h e well boundary
rate
Tk :
One can show, under a reasonable hypothesis on t h e shape of t h e well
borehole
hereunder), (1.12),
, (cf.
Dk
that
(1.13)
the
DAMLAMIAN-LI
potential
n
and t h a t corresponding t o
diameter of t h e well
TSIEN
TA
corresponding (1.14)
and to
a l s o paragraph
boundary
3
conditions
both converge, when t h e
tends t o zero, t o t h e p o t e n t i a l n corresponding t o
Dk
a Dirac f u n c t i o n with c o e f f i c i e n t Q k ( t ) i n t h e r i g h t hand s i d e of (1.8) a s
ek
soon a s t h e r e p a r t i t i o n functions
satisfy :
(1.15)
\Meas rk
x
Sup s E rk
ek(s)
remains
of
I n t h e case however
bounded
as
the
diameter
Dk
is not a
tends t o zero.
Dk
where t h e l i m i t of t h e well
point b u t a l i n e ( a s f o r example i n t h e v e r t i c a l model of Fig. 3 of chapter I),
conditions
(1.13)
(1.12),
is
prefered
to
as
(1.14),
it
yields
r k compatible with t h a t i n t h e
automatically a pressure d i s t r i b u t i o n on well borehole.
Remark 1 :
It
may
happen
that,
for
measurements a r e a v a i l a b l e . r e p a r t i t i o n hypothesis (constant) value
some
wells,
bottom
pressure
Using then t h e same h y d r o s t a t i c
as above gives t h e knowledge of t h e
lIk of
the
potential
I I
on
rk,
so that
one could think of r e p l a c i n g f o r those wells (1,121, (1.13) by t h e simpler D i r i c h l e t c o n d i t i o n (1.16)
TI
= lIk
on
'k'
However t h i s is not recommended f o r ( a t l e a s t ) two reasons :
-
r e s u l t s from a d i r e c t measure, whereas P r e s u l t s from a k k d i f f i c u l t bottom p r e s s u r e measurement and from a n hypothesis Q
on t h e pressure r e p a r t i t i o n i n s i d e t h e well borehole. Hence
I. Construction of the Pressure Equation
55
t h e u n c e r t a i n l y f o r I l k is g e n e r a l l y h i g h e r t h a n f o r Q
-
k'
When t h e p o t e n t i a l e q u a t i o n i s a p p r o x i m a t e d by v a r i a t i o n a l
finite
or
differences
f i n i t e elements,
standard
("chapeau"
function
basis)
c o n d i t i o n s a r e e q u a l l y e a s y t o take
both
i n t o a c c o u n t . However :
-
i f the w e l l t y p e c o n d i t i o n (1.121,
computed p o t e n t i a l o n
rk
(1.13)
is u s e d , t h e n t h e
c a n b e compared w i t h t h e measured
o n e ( f o r a d j u s t e m e n t p u r p o s e s f o r i n s t a n c e ) , as t a k i n g t h e
rk
trace on
of
t h e computed p o t e n t i a l makes s e n s e for t h e
above m e n t i o n e d a p p r o x i m a t i o n t e c h n i q u e s .
-
i f on t h e c o n t r a r y t h e Dirichlet c o n d i t i o n (1.16)
is u s e d ,
it is t h e n no l o n g e r p o s s i b l e t o compare t h e computed f l o w
rk
rate through
with t h e measured o n e , as t a k i n g t h e t r a c e o n
of t h e n o r m a l d e r i v a t i v e of t h e computed p o t e n t i a l d o e s
rk not
make
sense
for
the
above
mentioned
approximation
techniques. 0
*
Initial
known o v e r a l l
condition
above
the
potential
II,(x)
is
supposed
0,
n(x,O)
(1.17)
:
n,(x)
=
Y x
€
n.
T h i s h y p o t h e s i s is r e a l i s t i c b e c a u s e t h e i n i t i a l p r e s s u r e i s g e n e r a l l y t h e
hydrostatic pressure
i n t h e f i e l d , which is known e v e r y w h e r e o n c e it is
known a t o n e p o i n t , f o r i n s t a n c e a t t h e f i r s t non d r y well t h a t was d r i l l e d
(so t h a t g e n e r a l l y Summing conditions partition
(1.18)
we
up,
have
Dirichlet,
:
r
is a c o n s t a n t ) .
II0 ( x )
on
Neumann,
into three regions
:
r = rD u rN r D = rid,
,
u
rN = rLc,
rw
K =
" k= 1
rki
rw
r
three and
where
possible
well-type
types
of
conditions.
boundary So
we
Ch. !I: Slightly CompressibleMonophasic Fields
56 and d e n o t e by
fl
(5)
Q(s)
(1.19)
=
g i v e n v a l u e of
=
g i v e n v a l u e of
IT
at
s c
rD,
t h e o i l flow a t
s e
rN,
counted
p o s i t i v e l y when o i l i s produced.
The monophasic s t a t e e q u a t i o n s g o v e r n i n g t h e evolutio-
of t h e p o t e n t i a l
d i v [ o ( x ) k ( x ) gradTI
on
rD x
l0,TC
Q
on
c o ( x ) $,(x)
(1.21)
TI =
(1.22)
+ + q-v
(1.23)
I
(1.24)
IT =
So
=
"q-;
r N x]O,T[
= Q k , TI =
no
finally
v
an -
(1.20)
n
n are then :
on
we
get
6
=
0 in
nx
l0,TC
is d e f i n e d i n ( 1 . 9 )
unknow c o n s t a n t o n rkx]0,T[,k=1,2,..,K
at
0
where
]
t=O.
for
the
potential
TI
an
almost
standard
p a r a b o l i c e q u a t i o n , which we s t u d y i n t h e n e x t p a r a g r a p h .
11-
EXISTENCE
W e c o n s i d e r now, into
rD, rN
rw,
and
(2.1 1
O(x)
(2.2)
u
=
AND
UNIQUENESS
i n a domain 62 c Rn w i t h boundary I p a r t i t i o n n e d
t h e following parabolic equation :
- div(g(x) gradu) 0
THEOREMS
on
rD
x
lO,TC,
=
f
i n 62
x
]O,T[,
II. Existence and Uniqueness Theorems
au
I
(2.4)
av
=
57
G,
u
on
il
rw,
(unknown) c o n s t a n t o v e r
=
rwxlo,TC,
on
rW
u = u
(2.5)
Equations (1.24)
(2.1)
a t t=O.
through
(2.5)
c o n t a i n e q u a t i o n s (1.20)
f o r a f i e l d p r o d u c e d by o n l y o n e well
( rw=r 1 ).
through
T h i s is o b v i o u s l y
a c h i e v e d by t a k i n g :
and by s u p p o s i n g t h a t t h e imposed e x t e r i o r p o t e n t i a l
IIe i n t h e D i r i c h l e t
c o n d t i o n ( 1 . 2 1 ) is z e r o :
ne
(2.7)
=
o
rD,
on
which is n o t a r e s t r i c t i v e h y p o t h e s i s ( i f standard s i t u a t i o n , then
(2.7)
reference
if
unknown on
potential, A
r-IIe
A
where
IIe
is c o n s t a n t ,
is
IIe
not
constant,
is a n y g i v e n f u n c t i o n o n
IIe
which is t h e
c a n b e a c h i e v e d by a p r o p e r c h o i c e of t h e 7i
then having
take IIe
as
for t r a c e
rD). W e w i l l u s e some f u n c t i o n s p a c e s f o r t h e m a t h e m a t i c a l s t u d y of
equations (2.1)
t h r o u g h (2.5).
The s p a c e
L2(n)
w i l l b e t h e s p a c e of s q u a r e
i n t e g r a b l e r e a l v a l u e d f u n c t i o n s d e f i n e d o n 0. I t i
a H i l b e r t s p a c e when
equipped w i t h t h e s c a l a r p r o d u c t and norm :
The s p a c e bounded o n
8
Lm(n)
w i l l c o n s i s t o f a class of f u n c t i o n s which are
almost everywhere :
3 v E Lm(n)
<=>
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,
Ch. II: Slightly Compressible Monophasic Fields
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 :
III. An Alternative Model of Monophasic Wells
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
III. An Alternative Model of Monophasic Wells
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
Ch. II: Slightly Compressible Monophasic Fields
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
III. An Alternative Model of Monophasic Wells
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.
Ch. II: Slightly Compressible Monophasic Fields
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
III. An Alternative Model of Monophasic Wells
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.
Ch. II: Slightly Compressible Monophasic Fields
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
III. An Alternative Model of Monophasic Wells
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
III. An Alternative Model of Monophasic Wells
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.
III. An Alternative Model of Monophasic Wells
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
Ch. II: Slightly Compressible Monophasic Fields
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)
Ch. II: Slightly Compressible Monophasic Fields
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,
I1 Construction of the State Equations
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
Ch. III: Incompressible nYo-Phase Reservoirs
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)
I1 Construction of the State Equations
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+$2
S,
P1
P
and P 2 ,
can be expressed in terms of S, P and grad
-P only. Thus the number of unknowns reduces to two. We define first some notation
-
corresponding to Sc Y(S) =
is the reduced saturation
defined in ( 2 . 1 3 1 , so pc(Sc) S
(2.391
(Sc
J
1 dPc (bo(S)- 7 ) d~ ( S ) d s
=
0) :
(dimensionless),
sC
(dimensionless). One has obviously (2.41)
:
‘f(S)+Y1(S) = (bo(S)
- 71 ) pc(S).
Ch.III: Incompressible nYo-Phase Reservoirs
98
A o(=/oSa (T) dT
1 Figure 2
Figure 3
:
:
S b
: 0
1
The functions a(S) and a(S)
The shapes of the functions ,,b
b,
and
b2
s
99
I1 Constructionof the State Equations
pressure P by : P(x,t)
(2.42)
=
7 1 CP,(x,t)
+
P2(x,t)l
One checks then easily, as
pc(Sc)
+
=
Y(S(X,t))
0 and
PCM(X).
O
so
(2.44)
Let us now calculate the gradient of (2.45)
gradP
=
1
P :
grad(P1+P2) + ( b o ( S ) 2
dpc gradS + Y ( S ) grad PCM. 71 ) PCM dS
But, (2.46)
PCM
dPC dS
grad S
=
PCM grad pc(S)
=
grad [PcM P~(S)I-P, grad PCM.
Using (2.46), (2.45) and using (2.24), (2.27), (2.41) we get
:
1
1
grad P = - grad (P +P ) + ( b ( S ) - -1 grad (P1-P2)-Y,(S)grad PCM. 2 1 2 0 2 which using the definition (2.33) of (2.47)
( k l + k 2 ) grad P
Multiplying (2.47) by (2.48)
Go
=
=
bo
reduces to :
k l grad P 1 + k2 grad P2 - ( k l + k 2 ) Y1
(s) grad
PG.
I$, we obtain
il+i2-I$(x) =
d(S) [grad P+Y1(S) grad PCM+Y2(S)grad Pc}
where we have introduced the following functions
:
100
Ch. III: Incompressible nYo4hase Reservoirs
(2.59)
k,Pltk2 P2
I”=
.-1 Pm
k1+k2
I I
I I
I
I
S
I
0 Figure 4 Remark 1 :
We
obtain
global
:
from
The d and Y
-2
equation
gresnure,
in
c a p i l d r y pressure (2.48) redlJCRS t o
the
b
1 functions
(2.48)
the
special
i n t e r p r e t a t i o n of
case
where
PCM(x) is c o n s t a n t o v e r
the
the
maximum
0 ; i n t h a t case
:
(2.51 1 which,
compared
to
the
one-phase
Darcy
Law
(3.3)
of
c h a p t e ? T, shows t h a t t h e g l o b a l p r e s s u r e f i e l d , a p p l i e d t o a f i c t i t i o u s f l u i d of v i s c o s i t y !J s u c h t h a t krl kr 2 k l P l + k 2 P 2 , yields - = d = -+ -and of d e n s i t y p = k l + k2 !J 9 !J2 e q u a l t o t h e sun of t h e o i l a n d water flows. flow
Step 3 : Let u s d e f i n e t h e f o l l o w i n g vect7r’ f i e l d s (2.52) (2.53)
G1
=
-+(x) g r a d PCM
=
- + ( X I g r a d PG
:
m o b i l i t y -1 1,
(flow vector
x
(flow vector
x mobility-’
),
a
101
II. Construction of the State Equations
which are known (as
@,
PCM
and PG are given).
Then equations (2.37). (2.54)
Q
as at
+ div {r
+
(2.55)
r' =
(2.56)
div qo
(2.57)
Go
+
2
1
j=o
(2.38),
+
=
GJ. }
b.(S) J
-$(x) PCM(x) grad
(2.48) can be r e w r i t t e L =
0,
a(S),
0,
2 =
-$(x) d(S) gradP
+
d(S)
1
Gj.
Yj(S)
j=l
which are the sought equations in The
saturation
0
equation
for S and P. (2.541,
(2.55)
is
a
non-linear
diffusion-convection evolution equation. The diffusion term (second term of (2.54) is degenerate (as a' = a vanishes for S=O and S=l as one can see in figure 2) or even missing when the capillary pressure is neglected. The transport term (last term of (2.54) is non linear and not necessarily monotone (the fractional flow bo is monotone increasing, but bl
and
b2
are not as shown in figure 3). The pressure equation (2.56),
(2.57) is a family (one for every
t c lO,T[) of elliptic equations.
The
saturation
equation
(2.54)
and
the
+
pressure
equation
(2.56),(2.57) are coupled through the presence of qo in the 1 in the right hand side of (2.54), and the presence of S in the coefficients of (2.57). +
Remark 2:
One tempting thing to do is to replace in (2.54) qo by its expression (2.57) and to rearrange the terms. This manipulation has to be avoided, in view of the forthcoming numerical resolution of (2.54) through (2.57)
:
in order to
be able to solve properly the pressure equation (2.56) we shall use a mixed finite element method (see chapter V), + which computes an approximation of the global flow qo. Oh
6
Ch. In: Incompressible nvo-t'hase Reservoirs
102
This
approximation
+
qOh
will
be
simply
plugged
in
the
saturation equation (2.54). Hence we shall keep the above form of the equations. Moreover, notice that in the 1-D case,
6o
is
constant
in space
(cf.
(2.56)) and the proposed
manipulation has obviously no interest in that case. 0
+
+
The water and oil flow vectors are then related to S, r, q j , j
=
0,1,2 by +
(2.58)
0,
+
+
+
02 = q0'
(2.59)
and the water and oil pressures (2.62)
P1
=
P - [YG) -
(2.63)
P2
=
P
11.2
-
-
[Y(S)
P,
P2
are then given by
:
1
7 Pc(S)1 PCM, 1
+
and
7 PC(S)I
PCM.
THE PRESSURE BOUNDARY CONDITIONS
III (see figure 1 ) we
As we indicated at the beginning of chapter
have drawn in figure 5 the partition of the boundary into three parts re,
r
of the porous medium
rll and r corresponding respectively to injection well
boundary, lateral boundary and production well boundary.
II. Consmction of the State Equations
103
injection we1I
Figure 5 : The pressure boundary conditions.
As
the global
i~r*~?,si~r’eP
i r governed by a family of elliptic
equations, and is related to the global (water+oil) flow, the corresponding boundary
conditions are
monophasic case
similar to thora [used in chapter I1 for the
:
- Bzundary conditions on the lateral boundary The field being supposed closed on vanishes on
:
the (oilcwater) flow
i.e. using (2.58)
+ + q o - v = O
(2.64)
Remark 3
rR
r,,
TL:
on
rR
(Neumann Condition for ( 2 . 5 6 ) , ( 2 . 5 7 ) ) .
Water drive with given pressure or flow conditions can be taken into account by straightforward modification of
the
boundary conditions.
- Boundary boundaries As
a
conditions
on _the injection and
production well
r e -and rs: for the monophasic case, we shall not distinguish, for the
pressure boundary conditions, bet,.ieerl i rijection and production wells
:
we
Ch. III: Incompressible no-Phase Reservoirs
104
rw
s h a l l d e n o t e by
rw and
re
either
rs,
or
by
rw
s E
t h e c u r r e n t p o i n t on
Iw = rwx]O,T[.
oil
The
injection
and
field
is
production
coupled, wells,
through t h e
for
which
we
boundaries, shall
t o the
suppose
a
that
mathematical model is a v a i l a b l e . Such a well model e n a b l e s u s t o c a l c u l a t e , a t every t i m e
t,
the trace
up t o a c o n s t a n t ,
on
Pe
Tw
of t h e p r e s s u r e i n s i d e t h e well,
knowing o n l y t h e f l o w - r a t e h i s t o r y of t h e two f l u i d s
t h r o u g h every p o i n t of t h e well boundary
rw
:
unknown c o n s t a n t , V ( s , t )
6
Z
W.
We g i v e f i r s t some s i m p l e examples o f such models, which u s e t h e h y d r o s t a t i c p r e s s u r e law i n s i d e t h e well b o r e h o l e
:
- water injection w e l l : t h e d e n s i t y of t h e f l u i d i n t h e well is t h a t of w a t e r ; hence t h e h y d r o s t a t i c p r e s s u r e law y i e l d s P ( s ) = p 1 g Z(x)
(2.66)
-
:
unknown c o n s t a n t .
+
production w e l l : h e r e t h e d e n s i t y o f t h e f l u i d i n s i d e t h e well
depends on t h e l o c a l p r o p o r t i o n of the two f l u i d s . We t a k e f o r d e n s i t y i n t h e well b o r e h o l e t h e water
and o i l d e n s i t i e s weighted by t h e i r
local
p r o d u c t i o n r a t e t h r o u g h t h e boundary : +
+
+
+ + + + Pl$l ' v + P 2 $ 2 " J
+
P1S1 ' k + P 2 O 2 ' V (2.67)
P(X,t)
@,
=
+
+
+
.V+O2.V
Usually and
r',
+ qo
+
-
+ + q0-v +
is very l a r g e n e a r t h e well i n comparison t o ql,
s o t h a t (2.67)
is g e n e r a l l y approximated by ( c f .
(2.601,
+ q2
(2.61)
and (2.33)) : k l ( S (3,
(2.68)
p(s,t)
=
t ) )p l + k 2 ( S ( s , t )
p2
k l ( S ( s , t ) ) + k 2 ( S ( 3 , t) )
Note t h a t t h i s d e n s i t y is t h e d e n s i t y , on t h e well boundary f i c t i t i o u s e q u i v a l e n t f l u i d d e f i n e d i n remark 1 .
rw,
Of
the
II. Constructionof the State Equations
L e t , so e Tw
105
be any r e f e r e n c e p o i n t on t h e well boundary ( a t t h e bottom
f o r example f o r
t h e v e r t i c a l models).
One can t a k e as p r o d u c t i o n well
model :
-
( s is t h e n t h e c u r v i l i n e a r a b s c i s s a on
f o r 2D models
Tw)
S
(2.69)
Pe(s,t)
-
1
=
p ( s , t ) g d Z ( s ) + unknown c o n s t a n t ,
f o r 3D models Z(S)
(2.69bis)
Pe(s,t)
1
=
p ( Z , t ) g dZ
+
unknown c o n s t a n t ,
Z(So)
where
t
(2.70)
p(Z,t)
rz
=
=
1 measr
p ( s , t ) ds,
rz
{ s ~ r ~ I z (= sz)] .
I n summary, we see
t h a t , once t h e c o n s t a n t is f i x e d i n ( 2 . 6 6 ) ,
(2.69) o r ( 2 . 6 9 b i s ) , t h e well model e n a b l e s u s t o c a l c u l a t e t h e p r e s s u r e Pe everywhere on rw assuming t h a t t h e d e n s i t y p i n t h e w e l l is known. T h i s is a c t u a l l y t h e case
for
i n j e c t i o n wells,
where p
is t h e i n j e c t e d f l u i d
d e n s i t y , and f o r 2D-horizontal models, where Z ( s ) is c o n s t a n t and hence Pe t o o , and t h i s is a l s o a p p r o x i m a t e l y t h e case f o r p r o d u c t i o n wells, where p is commonly e v a l u a t e d u s i n g (2.68) w i t h t h e s a t u r a t i o n S e v a l u a t e d a t t h e p r e v i o u s time s t e p . Hence we w i l l assume from now on t h a t t h e p r e s s u r e P up t o a c o n s t a n t , on a w e l l boundary
is known,
r
.
We g i v e now t h e boundary c o n d i t i o n s which c a n be used f o r
Ew
e-
Tw:
i f we know t h e bottom p r e s s u r e
i n t h e w e l l , we know
u s i n g t h e well model
The c o n t i n u i t y of t h e p r e s s u r e i n one
(2.65).
P
everywhere on
i n j e c t e d o r produced f l u i d (water f o r example) a c r o s s t h e well boundary shows t h a t most
the
P,
Irw
=
capillary
Pe.
But t h e g l o b a l p r e s s u r e P d i f f e r s from
pressure
(cf.
(2.44)),
which
is u s u a l l y
P,
by a t
s m a l l when
compared w i t h t h e p r e s s u r e drop a c r o s s t h e f i e l d . Hence we s h a l l n e g l e c t t h i s d i f f e r e n c e and t a k e as boundary c o n d i t i o n
:
106
Ch.III: Incompressible TwoPhase Reservoirs
P Remark 4 :
=
on
P
Tw
, a.e.
on lO,T[
(Dirichlet).
If one does not want to make this approximation, then the above condition has to be replaced, to ensure continuity of the water pressure, by (cf. (2.62)) : p
=
1
pe
+
[Y(S) - 7
PC(S)I PCM. 0
-
if we know at every
(s,t)
E
Zw the (water+oil)
flow rate density
Q(s,t) then we take as boundary condition : + + qo-v
=
, a.e
Q on rw
on
]O,T[
(Neumann).
(This condition can be used practically only for 1-D models, or 2-D models with an hypothesis on the repartition of the well production rate on - if we know only the overall injection or production rate
well, the
boundary condition is + + qo-v
QT(t)
=
'
QT(t) of the
:
a.e. on
lO,T[,
rW
P
rw).
(well-type condition) =
Pe
+
unknown constant,
rw
is given by the well model (2.65).
where P
Finally we have only three types of global pressure boundary conditions boundary rD,
rN
(2.71)
r
of Q (as in the monophasic case) into three corresponding parts
and
i
Dirichlet, Neumann and well-type ; so we shall partition the
:
rw
:
rD
=
the part of
r
rN
=
the part of
r where Neumann conditions hold for P,
rw
=
the part of
r
where Dirichlet conditions hold for P,
where well-type conditions hold for P.
107
I1 Construction of the State Equations The global pressure boundary conditions are then :
(2.72)
where
P
is the given exterior pressure
Q(s,t) (2.73)
=
:
TN
lO,TC,
Z"
overall (water+oil) production rate through at time t
Remark 5
r D X
(water+oil) production rate density at the point
=
(s,t) Q (t) T
on
E
rW
l0,TC.
includes generally the lateral boundary, where Q=O; Q is
positive when something is produced through the boundary, and negative when something is injected through the boundam. Remark 6 :
As
the global pressure equation is stationnary, the global
pressure is well defined only if (2.74)
the r-measure of
rD
is non zero,
which we shall generally assume in the following. If (2.74) is not satisfied, then one has to assume that Q(s) d s
(2.75)
+
QT
=
0
rN
in order to be compatible with the incompressibility of the two fluids; then the pressure
P
is only defined up to a
constant. 0
Ch. III: Incompressible nvuo-phaseReservoirs
108
11.3 - THE SATURATION BOUNDARY CONDITIONS The p o s s i b l e c h o i c e s f o r t h e s a t u r a t i o n boundary c o n d i t i o n a t a point s of
r
are d i f f e r e n t d e p e n d i n g o n w h e t h e r t h e g l o b a l ( o i l c w a t e r ) f l o w + + whether q O * J is n e g a t i v e
u s i n g (3.58),
is d i r e c t e d i n w a r d or o u t w a r d i.e.
or p o s i t i v e . So we i n t r , , i I i - ~ e r i p s t ( c f . f i g . 7 ) :
r-
=
{s e
r
I
r+
=
{s
r
I
+
+
+
+
qo-'j 2 O ] = i n j e c t i o n boundary,
(2.76) E
qo-v
>
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+
space phase @ of figure 9
r+
space phase @ of figure 9
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(<)Zl, Y ,C 2 1 ; ] - ~ , O l u [l,+w[, fF j=l,2;
[O,l], E
a.e. on R.
(5.20)
0 5 uo(x) 5 1
Remark 19 :
If one thinks of u as being the non-wetting phase saturation and of P as being the global pressure, and if one affects the subsript 1
to the non-wetting phase and 2 to the wetting
phase (with the corresponding changes in the definition of the non-linearities), then equations (5.3) through (5.14) correspond, for large 1, to the two-phase flow equations (3.57)
through
(3.69)
of
section I11 with the boundary
conditions indicated in figure 11, 12. One can however notice that the unilateral condition (5.12) is now taken on C
9'
whereas it was taken on Z+ in (3.67) in
section 111. As we noticed in Remark 7, the two boundaries and
r+
r
may differ.
So we take a closer l o o k at the meaning of the unilateral
condition (5.12) on Ts.
133
V. Mathematical Study
+ +
s
A t point
*
Ts where q - v L O(i.e. s
E
model
t h e n we have
t h a t t h e u n i l a t e r a l c o n d i t i o n (5.12) is an
s e e n i n 311.3.2.2 accurate
r,)
E
which
takes
into
account
the
capillary
boundary l a y e r on t h e p r o d u c t i o n boundary.
.At
point s
rs
E
+ +
q-v
where
<
0 (i.e. s
E
r-)
we check now
t h a t , under t h e h y p o t h e s i s (5.21)
<
b(0)
(which
1
satisfied
is
in
practice
where
b(0)
=
O),
the
c o n d i t i o n (5.12) becomes (5.22)
>
u
+
and
0
+
lp2-.o
=
0.
Suppose f o r a w h i l e t h a t +
c o n d i t i o n $I 2'v
+
L 0
and t h e f a c t t h a t b l ( 0 ) +
+ =
Q,-V
u(s)
=
0
f o r such an
s. Then t h e
g i v e s , u s i n g t h e d e f i n i t i o n (5.14) of =
b2(0)
=
+
$2
0 :
+ + + + ( l - b ( ~ ) )q - v - r - b L 0.
Hence from ( 5 . 9 ) and (5.21) we g e t
-
--
i.e.,
6
+ +
( 1 - b ( 0 ) ) q.d
as J, L m > 0 ,
which is a c o n t r a d i c t i o n t o t h e h y p o t h e s i s
u(s)
=
0 because
t h e s o l u t i o n u o f ( 5 . 3 ) t h r o u g h (5.14) s a t i s f i e s , as we s h a l l
see, 0 6 u ( x , t ) 51.
In c o n c l u s i o n t o t h i s remark, we see t h a t t h e e q u a t i o n s ( 5 . 3 ) t o (5.14)
c a n model a n experiment of d i s p l a c e m e n t of a non
w e t t i n g phase by a w e t t i n g phase i n j e c t e d t h r o u g h c l o s e d boundary
r,,
and a n o u t p u t boundary
surrounded by non-wetting f l u i d
:
rs
re,
with a
which is
Ch.III: Incompressible lbo-Phase Reservoirs
134
- on the (global) production part
r+
of
rs
one observes
first production of non-wetting phase only, followed by production of both phases,
r- of rs, the surrounding
- on the (global) injection part
non-wetting f l u i d enters the porous medium, and no wetting phase escapes.
Remark 20 :
0
It is worth noting that equations (5.3) through (5.14) cover exactly")
the problem of the water dam : the wetting phase
(index 2) is the water, the non-wetting phase (index 1 ) is the air, and the boundary conditions are indicated i n figure 14. This problem has been studied by BAIOCCHI in the case where one supposes that the air pressure is constant (i.e. the mobility of air is infinite) and that the air saturation takes only
the values 0 and
1
(which implies that the
capillary pressure is neglected).
Remark 21
:
0
It would be very interesting i f we could know a priori from only the pressure equations (5.3) through (5.7)
rs
of +
q2
=
0
rS
c
-t
and
principle,
r+,
boundary.
the partition
r- and r + . This is for example the case if q1
into
Pe
=
P C P i.e.
all
=
constant, which gives, using a maximum + + on rs and hence q-v 2 0 on r : here
a.e.
the
output
boundary
is
a
production 0
(1) up to the incompressibility hypothesis, uhich can be released - cf. section I of chapter IV.
V.Mathematical Study
135
V.2 - VARIATIONAL FORMULATIONS
Let
re
and
V
{ v
=
rS
be of non-zero measure,
H ' ( Q ) lvlr
=
} ,
0
H=L2(a),
e (5.23)
II VII
=
[v]
=
Ia lgradv12 ( I Igradvl' I v2 a (
J
1/ 2
which is a norm on V, which is a norm on H ' ( Q ) ,
+
rS
equivalent norms on H ,
K
(5.24)
=
{ v
E
lvlr L 0 ]
VI
be a closed convex set of V.
S
We identify H
with
H'
using the scalar product
I I$
associated with the norm
on H ,
so
:
that we have the following
embeddings :
(5.26)
I
VcHcV' with dense inclusions and continuous injections.
We denote by transported from the
11 ll* and norm 11 11
canonical isomorphism from
((
,
))*
the norm and scalar product on V'
and scalar product ( (
V onto V'.
,
))
on V by the
and we remark that, due to the
chosen method for the identification of H with H ' , we have (5.27)
50
4 v
we shall use (
E
V,
4 f
E
H c V'
VIV= (f,V)$
to denote the duality between V
and V'
.
136
Ch.III: Incompressible Tivo-PhaseReservoirs
.
The pressure boundary conditions
...
. . , . . . . . . . . . . . . . .. . .. . . . Q ..' * . .. .:.. . .. .r\ .* .. .., ... . .... . . . . .. . . . . . . . ... ... .. ... .. ... ... .. .. .. .. . .. .. .. . .. . . . . .. .. .' .. . . '.. * .,
- - - / .
1
i.
.
,
A
.
The (air) saturation boundary conditions (the shorn partition of
Figure 14 :
r
into
r-
and
r+
is completely hypothetical)
Boundary conditions for the water dam model
137
V. Mathematical Study
Let u s d e f i n e :
(5.
The s t r o n g v a r i a t i o n a l f o r m u l a t i o n of e q u a t i o n s ( 5 . 3 ) through (5.14) is then : Problem
(
(5.29)
P
(5.30)
-(Ggradw+h
f
L‘(0.T;
(5.31)
I (P-Pe)w=I r
2 =
d(u)
(u,P)
such t h a t
H’(SZ)),
a
4
g ): f i n d
1-1) gradP
re
1
+
a . e on l O , T C ,
Qw aF w f H ’ ( Q ) , Y . ( u ) Gjl,
j=1
J
and
w,
(5.32)
u
(5.33)
(*( t ) , v - u ( t ) ) ,
(5.34) where (5.35)
E
+
dt
u(0)
=
u
Gj-r‘=G
.grad ( v - u ( t ) ) t 0
f f
v E K t lO,TC,
t o get
E
-1
bj(u)Gj-
f i r s t rewrite ( 5 . 8 ) u s i n g
:
i2
=
K,
( 5 . 3 1 ) a r e o b t a i n e d from ( 5 . 3 ) through ( 5 . 5 ) inequality (5.33),
(5.14) and ( 5 . 3 ) which g i v e s
au - d i v oat
2
gradcc(u)+ ( l - b ) ( u ) ) G
j=l
Indeed, t h e equations (5.30),
M u l t i p l y i n g t h e n by v
2
0’
2 + + @ * = q- 1 bj(u) j=O
i n a s t a n d a r d way;
(
a
0.
i n t e g r a t i n g over SZ,
and u s i n g a Green’s formula
and t h e boundary c o n d i t i o n s ( 5 . 1 2 ) , (5.13) we o b t a i n
:
Ch. III: Incompressible Two-Phase Reservoirs
138
d (u,v), +
(5.36)
J
i2.grad v
=
n
J
+2 - + 4 v v L
o
rS
V V E K aF t E lO,T[.
Replacing v by u(t) in (5.36) and substracting it from (5.36) yields the 0 sought inequality (5.33). The resolution of problem ( p
$ )
with the hypotheses (5.15) through
(5.20) is not possible, because the lack of V-ellipticity of the diffusion term in (5.33) (remember that (5.19) says only that c t ' ( c ) the diffusion term u
E
=
a(r,) 2 0, i.e.
is degenerate) is contradictory to the fact that
L'(0,T;V). So we want to weaken the formulation of our problem so that it can
have a solution even in the degenerate case. We remark for that purpose gradu in (5.33) is V-elliptic for the that the diffusion term ,' Ggradu(u)
a
function B(u) where 8 is defined by
:
We define the weak variational formulation of equation (5.3) through (5.14) by : -
Y. Mathematical Study
In problem
139
the
next
( 3 )i n
paragraphs,
we
t h e non-degenerate
s h a l l g i v e e x i s t e n c e theorems f o r c a s e and f o r probleio
( 2 ' )i n
the
d e g e n e r a t e c a s e and t h e n make a d e t a i l e d s t u d y , due t o GAGNEUX, of t h e case where t h e p r e s s u r e and s a t u r a i t i o n e q u a t i o n s can be s o l v e d s e p a r a t e l y . We begin w i t h some p r e l i m i n a r y lemmas.
V.3
- SOHE PRELIMINARY L W S
Lemma 1 : by
Under h y p o t h e s i s (5.19)
on b , t h e f u n c t i o n B d e f i n e d
:
(5.41)
B
6
v(lR)
s.t. B ' ( C )
h a s t h e f o l l o w i n g graph :
=
1 -b(c),
B(0)
=
0
Ch. III: Incompressible Tho-Phase Reservoirs
140
-Lemma 2
:
(compactness lemma for the non-degenerate case)
Suppose that a satisfies the hypothesis (5.19) and
3
(5.42)
>
q
Then for every M>O (5.43)
S,
such that
0
11 B(V)llqy
is relatively compact in L'(Q), L 2 ( Z ) (Y denotes the "trace on
Proof : -
This
r7
>
:
+ 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
<
5
OS5<521
(5.45)
:
M,
llglb
6 M
a.e. on Q,
1
and 'Y(S M ) is relatively compact in (Y still denotes the "trace on rrr operator).
is relatively compact in LZ(Q),
I
Proof
:
in CHAVENT [l].
The relative compactness of
SH
in L'fQ)
was already proven
We give here the proof for the sake of completeness. Define
(cf. 5.37)) -1
F = B
.
V. Mathematical Study
(5.44)
Then [O,
B(1)I
141
just means that F
is an HBlder function of order
v
IF(tl)-F(t2)1
0
on
:
t,, t2
E
CO,B(l)I
e
5 c(a) Itl-t21
.
We divide the proof into three steps : Step 1 : Properties of F ( Y ) for
< s <
~ S ~ P ( Q )o,
Y
1 ,I
< p <
+-
.
The following characterization of WS” holds (cf. LIONS-MAGENES, tome 1 p.59)
and the quantity
is a norm equivalent to the usual one on WS”(n). One checks then easily that, for any function satisfying 0 5 w(x) 2 1 a.e. on 9, F(w) satisfies : F(w)
€
Ws‘ ”‘(n)
with
(5.46)
II F(w) IIwsl*PI(n) Step 2 : Compactness of
Let v v(t)
=
F(w(t))
S,
2 c(a)
s’
es
=
, p’
P e
E
WS”(5l)
’-
e
II WII ws,p(Q)’
in a subspace of L’(Q).
SM, hence w=B(v)
E
9 and 0 5
a.e. on Q. Hence
w 5 B(1)
with w(t) c V c H’(Q) f o r almost every t
As
=
w
H’(R) c Ws’2(Q) f o r any s
E
]O,l[
f
lO,T[.
we get from (5.46)
:
Ch. HI: Incompressible nYo.Phase Reservoirs
142
From (5.47) one gets easily, using the continuity of the injection of H ~ ( R ) into wS'P(n)
which proves that
:
As 0 is bounded and regular, we have, for
ws'
0
<
s"
<
s'
:
( a ) c ws"*p' ( a ) c LP' ( a ) c LZ(Q) c V'
(5.50)
' (a) with compact injection from Ws' "
into Ws"*p' ( a ) ,
so that, from a standard compactness argument (cf. LIONS [21 p.58) we get the relative compactness of WM in Lp' (O,T;WS""' Thus we get (5.51
(0))
for 0 <
s"
<
s'
.
:
SM is relatively compact in L ~ / ~ ( o , T ; w ~ ' " ( a~ )/)~for O<s"<e.
Step 3 : Final compactness results.
First, as
218
>
0 and as Ws "'2'e(a)
c L2(R) with continuous
injection, we get the relative compactness of SM in L2(Q). Secondly, the s"-a/z,2/e ( r ) (cf. wsl* ,2/e ( a ) into W trace operator 'i maps continuously LIONS [l] sff
> 2'
p.87), So
we get from (5.51) used with 0 < 2e < srr<
compactness of Remark 22 :
which itself injects continuously into L2(r) as soon as
'i(S )
M
in L2(z).
8
the relative 0
It is possible to give simple sufficient conditions on the function a so that the condition (5.44) holds. From the HBlder inequality 5 5 e/2]i/edT \ e 5-5 = 1 x dT 2 { [a(?) 5
5 [a(T) 5
5 -812 111-8
1
d.r ]l-e
143
V. Mathematical Study
and the definition (5.44) of c(a) we get :
Hence a first sufficient condition on a for (5.44) to hold is that there exists on c0.11; then e
p>O
such that l/a’ is Lebesgue integrable
]0,1] of (5.44) is given by
e
=
2V 2’+1.
From this we deduce a practical sufficient condition
:
(5.52)
which 0
implies
< e < Min { -2,
that
2+P0
(5.44)
holds
for
any
e
satisfying
2 -1. 2+P1
This contains the physical case of two-phase flow in porous media where a(<) is given by (2.31) and has the aspect shown in figure 2 .
0
+
+
We study now in Lemma 4 the continuity of P. q, b 2
to u on SM.
with respect
144
'
Ch. III: Incompressible T w d h a s e Reservoirs
Lemma 4 : Suppose now that the hypotheses of either lemma 2 (i.e. (5.42) and (5.43)) or lemma 3 (i.e. (5.44) and (5.45)), with the corresponding notation, are satisfied, that (5.12) through (5.17), (5.23) through (5.28) and (5.37) hold, and that :
For a given M > O , (5.54
uk
let uk, u
->
+ +
Gk~$2k
Uk +
(5.56)
'kl'
(5.57)
B(Uk)
(5.58)
duk dt ->
(5.59)
Pk
(5.61) (5.62)
qk
strongly in L2(Q),
-+
strongly in L 2 ( 2 ) ,
'12
->
du dt
weakly in
+
->
q
,
weakly in [L2(Q)In,
+
->
,
weakly in L2(0,T; H'(a)),
->P
lim inf k+a
weakly in
B(u)
+ '2k
:
be the corresponding unique solutions of
u
(5.55)
(5.60)
be given such that
u weakly in L2(Q).
Let pkt (resP. p,q,@,) (5.55) through (5.57). Then we have :
-+
S,
-
weakly in [L'(Q)]",
2 '
I ;2k-graduk 1 ;2-grad u. 2
Q
Q
Proof : (5.55) and (5.56) result directly from the relative compactness of SM and Y(SM) shown in lemma 2 or 3, and (5.58) follows simply from (5.54) (5.59) and the definition of SM. We turn now to the proof of (5.57) through (5.61 ).
145
V. Mathematical Study
- A priori estimates :
From the definition of S,,, we get (5.63)
wk
=
B(uk)
is bounded in (%
.
From the elliptic equations (5.30) and (5.31) and hypotheses (5.12) through (5.16) we get Pk
is bounded in L2(0,T;H1(Q)).
qk
is bounded in [L2(Q)ln.
(5.64)
Rewriting (5.35) as (5.65) and using (5.63) and (5.64) and the hypotheses (5.12) through (5.161, we obtain
i,,is bounded
(5.66)
-
in [L2(Q)ln.
Extraction of subsequence :
From (5.55), (5.56), (5.63), (5.64), (5.66) we get the existence of a subsequence u u
P
such that
:
a.e. on Q,
'U
a.e. on 1, iii)
w
=
B(u
->
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)
V. Mathematical Study
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
:
V. Mathematical Study
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
V. Mathematical Study
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(<) S
4 5
with 6
>
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)
Ch. III: Incompressible nYo-Phase Reservoirs
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
V. Mathematical Study
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. Mathematical Study
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
<
T l e t w be d e f i n e d as :
One checks t h e n t h a t : (5.121)
v
=
cr(u(t))
(5.122)
v
=
s.(w(t))
E
a(a(t))
Using (5.121)
+
E
K f o r a.e.
t s.t.
t , t - T
E
CO,T[.
s ( w ( t ) ) c K.
i n ( 5 , 1 1 9 ) , and (5.122) i n (5.119) w i t h
t-T
and 0
i n s t e a d of t and u, and t a k i n g t h e d i f f e r e n c e y i e l d s :
which t o g e t h e r (5.123) y i e l d s
The
Lebesgue
w i t h t h e Cauchy-Schwartz
inequality i n the
l a s t term of
:
convergence
theorem
shows
that
sc(w)
+
sgn(w)
-
s g n ( u ( t ) - i l ( t + r ) ) i n L 2 , which t o g e t h e r w i t h (5.18) y i e l d s t h e sought r e s u l t (5.18).
V. Mathematical Study
161
. Regularity of u
Let N be a positive integer, t
:
following time discretization of (5.33) 1 n n-1 , - ( u -u
V - U ~ )+ ~
I
n
(5.125)
=
- ,
and consider the
:
$; &ad(v-un)
2 0, aF
v K, n=l ,2,...,N ,
u = u
0'
For uniquely un
E
given in K, the first equation of (5.125) defines
un-'
K (the existence can be proved using a fixed point theorem as
in Theorem 1, and the uniqueness is obtained as above). Taking v=O in (5.125) and using the same majorations as for (5.89), (5.90), we have lun1; 2
N
n=l,2,...,N
c
,
T
1 11 BCU")~~' 2
c/m
n= 1
N
1
lun - u
n-1
1;
5 c
,
where
n=1
(5.1 26)
c
=
~u
4T
+
12
11 ~ 1 1 ,
(Meas
rs) 1 /2 I ( P - P ~ ) - ~ ~ ~ ( ~ ~ )
+
0 0
T
2
-m .1
+
IIBjll:
J=1
As
lq;2(R)-
was done in (5.119) for the continuous case, (5.125) is equivalent to 1 n n-1 , v-a(un))O + j grad (v-a(un)) 2 o - ( u -u
n
aF
Taking v
=
v
c
K
,
n=l ,2,...,N.
a(u"-l) we get
But the b.'s are Lipschitzian and the J
&'
;.Is
J
belong to L"'(Q)
SO
that
:
Ch. III: Incompressible Tbo-Phase Reservoirs
162
Summing then (5.127) (with (5.128)) from n=l to M 5 N we obtain
:
From the bounds (5.126) and (5.129) one proves easily, using the same techniques as in the proof of theorem 2, that Uh
->
weakly in L 5 ( 0 , T ; V) and weakly* in L*(OT; V),
u
weakly in LZ(Q),
is h continuous piecewise linear) function N u ) sequence defined by (5.125). This ends
where u is the solution of (5.32) through (5.35) and where uh (resp. a the
piecewise
constant 0
"interpolating" the (u
(resp.
, u1 ,...,
the proof of theorem 4.
-
THEOREM 5 :
0
(regularity
for
the
non
degenerate
case
with
non
compatible initial data). We assume the hypotheses of theorem 2, plus (5.111) through (5.113) and that 0
(5.20)
I u (x) 4 1. 0
Then the solution u of (5.32) through (5.35) given by theorem 2
is unique and satisfies (5.136)
:
u
e
L"(0,T; V),
$
6
LZ(Q),
(5.137)
fi
(5.138)
f i div
=
-div
[igrad
a ( u ) ] e L 2 ( 0 , T ; H),
and satisfies also (5.118) and (5.118bis) for almost every t,-r f ]OT[ and uo,
0 such that (5.20) holds.
-
163
V. lclathematieal Study
Proof :
and let
Let
K, k=l,2,... be chosen such that
uOk E
be the corresponding solution of (5.32) through (5.35) given
U,
by theorem 4. Using the same techniques as in the proof of theorem 2, one can prove that uk
:
u
+
H) weak* and in L 2 ( Q ) strong,
in L " ( 0 , T ;
a(uk) [ resp 8 ( u k ) ]
+
a(u)
[resp. B(u)]
weakly in L 2 ( 0 , T ; V ) ,
We have now to obtain additional estimations on uk in order to prove (5.136) and (5.137). also
(ak =
We first remark that
etc...)
a(u,)
du (-,
(5.140)
k
V-CL,)~
dt
U,
satisfies (5.119) and hence
:
+
I a
-+ (J
grad v-irad (v-a,)
aF
v
E
+
K, a.e. on l O , T [ .
For any positive integer p, we define then solution of
:
(t)
+
uP (t) =
u,(t),
(5.141 1 Up(0) which, as
dak E dt
u
=
a (0)
=
n ( u o k ) < K,
L Z ( O , T ; H), has the following properties
'a
~k
(5.142)
u;.
k
+
dak dt
strongly in L 2 ( 0 , T ; V ) , strongly in L Z ( O , T ; HI.
U (t) to be the P
Ch.III: Incompressible hv-Phase Reservoirs
164
U (t) E K f o r e v e r y t , one can t a k e v=U ( t ) i n ( 5 . 1 4 0 ) . P P M u l t i p l y i n g t h e n by s and i n t e g r a t i n g between s=O and s = t y i e l d s : As moreover
But,
where
Using
then
(5.145)
in
the
As we know from (5.142)
that
i n t e g r a t i n g by p a r t s y i e l d s
e x i s t s a subsequence, s t i l l denoted by
]O,T[.
Since
right
hand
side
dt
+
(5.143)
U +a s t r o n g l y i n L 2 ( 0 , T ; P k
such t h a t
U
P' One can t h e n p a s s t o t h e l i m i t i n (5.147) when
duk -(s)
of
and
:
dgk
+
H k - g r a d CY
k
=
-(s) dt
P
on
p-'", which y i e l d s :
+
Hk(s)*&ad
V), t h e r e
U ( t ) + a , ( t ) a.e.
gk(S),
we o b t a i n
K Mathematical Study
165
which, u s i n g (5.1391, (5.1441, (5.146) shows t h a t (5.1 49 )
1
f i g B(u) fi
E
:
L'(Q),
a(u) e L"(0,T; V),
E1=a1'2, a'
which p r o v e s ( 5 . 1 3 6 ) , (5.137) as Taking i n t e g r a t i n g over
then
in
C0,TI
yields
(5.33)
v=u(t)
=
a 2
n >
0.
with
B(t)
8
e
.@ ( Q )
and
:
au _ at
(5.150)
in which t o g e t h e r w i t h
is possible i n
a'
(5.137) p r o v e s
@I
(Q),
(5.138) as t h e m u l t i p l i c a t i o n by
fi
(Q).
F i n a l l y , (5.118) and ( 5 . 1 1 8 b i s )
(and hence t h e uniqueness of u )
a r e proved by t h e same t e c h n i q u e s as i n theorem 4 .
-
0
THEOREM 6 : (Asymptotic behaviour of u in t h e non-degenerate case)Let t h e h y p o t h e s i s of theorem 5 h o l d ; i f t h e i n i t i a l data u 0
s a t i s f i e s moreover u
0
E
:
H'(Q) [ r e s p . uo e K 1 ,
G20-&ad v t 0 [ r e s p . 201
s.t. v t o , a.e. on Q,
V V E V
Q where - + i2, J, g r a d =
2
a(uo)
+
C1-b(uo)l
io - 1
j=l
bj(uo)
ij,
+ = - 2 0 [ r e s p . 201 on ?S and (which f o r m a l l y means t h a t d i v $ 20 at t-o + + t h a t $ 2 . . , l t = 0 t 0 [ r e s p . 2 01 on r k u r s ) , t h e n one h a s :
aul
(5.152)
au
(x,t) 2
o
[ r e s p . 2 01
a.e. on nxCO,+nC
I
Ch. MI: Incompressible nvo4hase Reservoirs
166
J u(x,t)dx in the field at time R t is a decreasing [resp. increasing] function of time), (which implies that the amount of oil
(which implies that the amount of oil produced per unit time is a decreasing [resp. increasing] function of time) and (5.154) u(t) + u _ strongly in LP(Q) for every p L 1 and weakly in V where um is among the solutions of
:
the only one which satisfies : (5.156)
ucd=
u', [resp. ub,=
Sup Ul,'Uo
Inf u:,]. u' >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 .
V. Mathematical Study
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<,
: [O,
=
-.[
7
R be d e f i n e d by
I u ( x , t ) dx
f t t O .
a
The f u n c t i o n f is c o n t i n u o u s and r e p r e s e n t s t h e amount of o i l i n t h e f i e l d a t time t. From ( 5 . 1 5 8 ) ,
(5.159) and (5.118) w i t h 9
f ( t ) h f(t+T)
aF
1 u _ ( x ) dx f(t)-f(t+T) f f(t')-f(t'+I) f(t)
(5.161)
+
f_
=
(5.162)
, I
, f'_(t)
+
OSt6t'
exist f t
>
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)
$20(II)
2
0 ; hence
0 and uo is one of the stationary
=
solutions). One checks in this example that the equilibrium profile
0
um
is exactly that of the capillary pressure. This property, which is always true, under condition (5.168), in 1-D samples with both ends insulated, is used as a physical definition of the capillary pressure law. Remark 32
:
0
If we replace, in the last remark, the function b2(U) by
(which corresponds to the (unbounded) capillary pressure law
e),
then the steady-state equation (5.155) has p,(u) = Log o n l y one solution urn E 0. This comes from the fact the "equilibrium profile''
x= Log
boundary condition u(0)
=
0.
1 -u
does not satisfy the 0
V. Mathematical Study
V.6.2
-
171
Regularity and asymptotic behaviour for the degenerate case We turn now to the degenerate case: in order to handle this case,
we had replaced in the case of the coupled system of equations, the
8)by the weaker (8 1, which required that
saturation equations (5.32) through (5.35) of problem ( formulation (5.38)
u,,
E
through (5.40)
of problem
K in order to get a solution (theorem 3 ) . We shall now treat this case in another way, and, still following
CACNEUX [ l ] , [2], we introduce the following variational formulation :
Problem (@)
6%
,
(5.174)
B(u)E
(5.175)
($ (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
V. Mathematical Study
-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
V. Mathematical Study
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
<
0 given) :
to+E -E
for 0
V.
u ( u ) i s , f o r a subsequence, converging toward u(u) i n L m ( 1 6 , T [ ;
V ) weak s t a r , we g e t ( f o r
t
V')
6 (t)dt = 1. E
6 ~ ( t - t o ) ( ( a ( u ( t ) ) , v ) ) d=t & f E ( n )
t
0
-E
+
n -to
0.
Ch. III: Incompressible %@Phase Reservoirs
116
E>O be g i v e n ; l e t t i n g
Let
which is t r u e f o r any
q-0 we o b t a i n :
L 0 ; hence we have proven t h a t
I
The f i r s t a s s e r t i o n o f (5.187) f o l l o w s t h e n from ( 5 . 1 8 9 ) , from t h e
f a c t t h a t a ( u ) is bounded i n L"'([G,T];
V ) and from t h e f a c t t h a t
n
D(A) is
dense i n V . The remainder of (5.187) f o l l o w s t h e n immediately. From (5.151) and (5.181) w e see t h a t uo s a t i s f i e s , f o r a l l (5.151) w i t h a
6 applies t o u
n
17
i n s t e a d of a (where a ( c )
n
for all
+t,T
(5.1 90 1
>
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
supp
CI
U
-
is in the interior of
'ilm
u QRR'
Equation ( 6 . 2 4 ) through ( 6 . 2 8 ) and ( 6 . 3 1 ) through ( 6 . 3 4 ) are the sought equations governing the evolution of S and P over a field containing different types of rock. We
see
that
taking
into
account more
than
one rock
type
introduces, besides the forseeable dependance on x of all non linearities, integrals over the boundaries
rmII separating different rock types in the
right-hand side of the variational equations (6.31) and (6.32) determining
Go
and
r'
(caution : the flux of
r'
across the boundaries
contiquous ! ) , plus additional equations ( 6 . 2 7 1 , associated with the boundaries TmR
.
rmII is not
( 6 . 2 8 ) and (6.331, (6.34)
VI. The Case of Fields with Different Rock Types
187
We shall see in chapter V, for the case of a single rock type (i.e.
when the last term in (6.31) and (6.32)
disappears), that equations
(6.24) through (6.26) and (6.31), (6.32) are the basis for the numerical calculation of P and S by mixed and discontinuous finite elements. The numerical techniques described in chapter V can be generalized'to the case of multiple rock-types, provided that the finite element mesh is chosen in such a way that the TmR boundaries consist of edges of elements of the mesh,
This Page Intentionally Left Blank
189
CHAPTER
IV
GENERALIZATION TO COMPRESSIBLE,
THREE-PHASE.
BLACKOIL OR COMPOSITIONAL MODELS
The aim of this chapter is to see to what extent the formulation introduced in chapter I11 for two-phase, incompressible flows, using the concept of a global pressure, can be generalized to more realistic field conditions
appearing
compressible
or
in
black
secondary oil
(compositional displacements).
models) As
recovery as
techniques
well
as
in
(three-phase, tertiary
ones
this formulation is, for two phase
incompressible flow, the starting point for the mathematical study of the equations (chapter 111, paragraph V) and for their numerical approximation (chapter V),
we believe that the models described in this chapter can
serve
further
for
theoretical
and/or
numerical
developments
for
multiphase, compressible or compositionnal models. So we will make no attempt in this chapter to give any existence theorem, leaving this matter for further study. Each paragraph will be concerned with deriving a set of partial differential equations describing the type of flow under study. We will also make no attempt to write down the equations for the more general case, compressible, three phase, N
component compositionnal
flow, but will preferably investigate one new feature at a time. We will focus our attention on the equation inside the porous medium
R , as the boundary conditions are similar to those described in
chapter 111, paragraph 11.2 and 11.3. I -
THE
TWO-PHASE
COMPRESSIBLE
MODEL
We consider here the direct generalization of the two-phase incompressible model of chapter I11 to compressible fluids and rock. The material developped in this section has been introduced in CHAVENT [31 where a detailed comparison with the miscible equation can be found.
Ch.N:Compressible, Three-Phase,Black Oil or Compositional Models
190
We consider a field boundary
re
n, with closed boundary
and production boundary
rs.
rR,
injection
The boundary conditions, being
identical to those of the incompressible case, will not be considered
In this field, we consider the displacement of two immiscible 1 as being water and fluid 2 a3 being oil). All notations introduced in this paragraph are compatible with those used for the incompressible case, i.e. the notations are here.
compressible fluids (one can think of fluid
identical when rock and fluid compressibilities vanish.
1.1
EQUATIONS FROM THE PHYSICS. Given
a
reference pressure
Po,
pressure, we consider the following unknowns
-
for example the surface
:
S
=
actual saturation of fluid 1,
P.
=
pressure in fluid j, j=l,2,
=
volumetric flow vector of fluid j,
J
j '
If we introduce the volume factor B.(P) of each fluid by J
j=1,2.
:
we get the following equations (compare with paragraph 11.1
of chapter
111) : 1.1.1
- Conservation Laws
Here the porosity
4
depends on the pressure; however, it is physically
or anything else) should be used to 2 Practically this does not cause any problem, as 4 is a
not clear which pressure evaluate
a.
slowly varying function of
(P1
or
P
P, and the difference between p,
and
P2 is
191
I. The no-Phase Compressible Model
small. We shall take advantage of this fact in the next paragraph to simplify the equations. 1.1.2
(1.4)
-
Muskat law (Relative Permeabilities)
i. = -oK(x) J
kr .(x,s,P.) B .( P . ) [grad P.-p.(P.) g grad Zl,
!J+(P)
J
J
The relative permeability
kr.
J’
J
J
J=1,2.
eventually depend on the
which may
!J. are, logically, pressure, the density Pj’ and the viscosity J evaluated at the pressure P . of the fluid under consideration.
J
1.1.3 - Capillary pressure law P 1 - P2
(1.5)
=
PC(S,X).
1.2 -SIMPLIFYING HYPOTHESES
We
list now the hypotheses we make
in order to obtain our
simplified model. 1.2.1
- Pressure dependent coefficients We have seen that
P. J
P. B. J’
3’
u . , logically depend on the pressure J
of the corresponding fluid whereas there is no reason to use
rather than P 2
to evaluate the porosity
p1
L$.
Taking advantage of the fact that all these functions vary slowly with
P
and that the difference between P1
and P2
(i.e. the capillary
pressure) is small, we shall decide to evaluate all those coefficient using a pressure field the presssures P ,
and
P
(to be determined later) intermediate between
P2.
Hence we shall write
where P
is, for the time being, any pressure field satisfying
Ch.IV: Compressible, Three-Phase,Black Oil or Compositional Models
192
1.2.2
-
Choice of a rock model
Using the terminology of section VI of chapter 111, we suppose
n
that the porous medium now
contains only a single rocktype. We describe
the two rock models which can be used to modelize the spatial
dependance of relative permeabilities and capillary pressure. Suppose that for some point x0 -of R , we know the following quantities
:
-
S,(Xo),
l-FM(xo)
5E
[0,1] -$ kry (21, krE(5)
krl(S) 0 = 0
(1.8)
-
kr0(0) = 1 2
0 6
-
residual saturations of fluid 1 and 2.
=
relative permeability as function of non reduced saturation, and satisfying
-
-
S 5 S m (xO ) ,
-
, kr02(S) =
kry ( 1 )
=
0,
(x) < S 6 1, M O
0 S
0
kro 1
=
-
increasing, kr2 decreasing,
=
reduced capillary pressure as function of reduced saturation and satisfying (3.20) in chapter 111.
Then, recalling the relationship between the non-reduced saturation and the reduced saturation S
-
S(x,S)
=
Sm(x)
S(x,S)
=
(S-
+
S(SM(X) -
s,(x)
Sm(x))
/ (SM(X) -
xo,
functions of the reduced saturation S
s,(x))
+x,+s$,
-
u x , u s,
the relative permeabilities as
:
L
kr.(S) 0 J
=
kr. 0 (S(Xo,s)), J
S
:
we can determine, at the point
(1.9)
-
j = l,2.
I. The no-Phase Compressible Model
193
Similarly, from the relative permeabilities at the point xo and the mobility factors B j (P)
d.(P)
Pj(P)
=
3
of each fluid, we define the global mobility at xo d(S,P) (1.10-a)
-
=
dl(P)kry(z)
=
d(z(xo,S),P)
:
0 dz(P)krz(S)
+
(as function of
S)
or d(S,P)
0
dl (P)kr, ( S )
=
0
+
d2(P)kr2(S) (as function of S )
and the fractional flow at xo
:
(as function of S )
The rock model enables us then to define the capillary pressure and the relative permeabilities at any point
x of
R.
In all rock models, the capillary pressure at point x is given by (1.11-a)
P1-P2 = PCM(X)Pc(S)
where S=S(x,?), (1.81, and
S
-*
p (S)
is the reduced capillary pressure given in
PCM(x) is an amplitude factor depending only on x.
Then the relative permeabilities at point x will be determined by using the same fractional flow v(S,P) function which has been determined
_-
at point xo,
and the same global mobility function (either d(S,P) Or
d(S,P)). The relative permeabilities are then :
dl(P 1
1
Ch. IV: Compressible, Three-Phase,Bhck Oil or Compositional Models
194
i
d(S ’)
..... f o r
=
k r0 ( S ) 2
where S = S ( x , s )
a rock model of t h e f i r s t k i n d ,
where S = S ( x , s )
..... f o r
a rock model of t h e second kind.
Whatever r o c k model is u s e d , we d e f i n e t h e f u n c t i o n s : bO(S,P)
\;(S,P)
=
(1.12bis)
f o r a rock model of t h e f i r s t kind
d(S,P) d(x,S,P)
=
--
f o r a rock model of t h e second k i n d
d ( S ( x , S ) ,PI
will
which
enable us
to
not d i s t i n g u i s h between t h e two c a s e s i n t h e
sequel.
1.2.3
- Limitation of t h e pressure range The
P
CM
fractional
flow
bo
and
t h e maximum c a p i l l a r y pressure
( x ) being now g i v e n , we suppose t h a t t h e p r e s s u r e s
f i e l d belong t o a n a - p r i o r i by g i v e n i n t e r v a l
[Pmin,
P,
and P 2 i n t h e
PMax 1 s a t i s f y i n g
I. The Ikro-Phase CompressibleModel
3
8
195
~]0,1[ such that
(1 -13) .y X E J&
a,
aF S E CO,ll,
P rz CPMin’PMaxl.
This is a kind of compatibility condition between the Volume factors and viscosities of each fluid and the magnitude of the capillary pressure.
results from (1.12),(1 .12bis), a sufficient condition for
As
(1.13) to hold is 2
a P (x) !-Log CM ap
1
j=l
d.(P) I S 48, J
or equivalently, with the differential notation
1 6dl I + Id]d2 s
(1.14)
&P
=
pCM(x).
2
1
This
48 for
means that, for a change in pressure of the order of
magnitude of the capillary pressure, the relative change in the volume factor over viscosity ratio has to be small compared to 2. This is always satisfied
in practice (even with
e <<<
1)
in the range of pressures
encountered in oil fields, since violating (1.14) would probably require that the pressures P
be of the same order of magnitude as PCM(X).
Thus in practice (1.13) holds with :
(1 .15)
1.2.4.
e <<< I. - Summary of r e s u l t i n g equations Summing up the above simplifications, equations (1.3) to (1.5)
become [IJ
g(x,P) B1(P)
51
+
(1 .16)
where bo and d are given by (1 .12 bis).
div
;1 0 , =
Ch. IV: Compressible, Three-Phase, Black Oil or Compositional Models
196
One can also notice that we have kept the actual saturation S in the two first equations of (1.16), as replacing it by an expression written in terms of the reduced saturations no longer simplifies tne equations as it did in the incompressible case. 1.3
- THE GLOBAL PRESSURE EQUATION +
terms of the gradient of some global pressure
P,
i.e.
+ Q1
+ $, in without the gradient
We first try to express the global flow vector q0
=
of the saturation S. Using the same idea as in the incompressible case, we define the functions
-I
and
Y
of the reduced saturation S, which now depend also on
1
the pressure P , by
(1.17) n
One obviously has (remember that p ( S ) c c (1.19)
Y(S,P)
+
Y1(S,P)
=
=
0 by definition of S
[bO(S,P) -
1
~1
pc(S),
r
We can now define the global pressure P ‘b X E
a, Y
by :
S E [O,ll, ‘b P,,P2€ [Pmin’Pmax1
(1.22)
with P1-P2 = PCM(x)pc(S), P
=
1
7 (Pl+P,)
+
PCM(X) Y(S,P).
:
197
I. The no-Phase Compressible Model Equation ( 1 . 2 2 ) u n i q u e l y d e f i n e s
P
is a c o n t r a c t i o n from t h e [ P 1 , P 2 ]
PCM(x) Y(S,P)
P
s i n c e t h e mapping
+
71
(P1+P2)
+
i n t e r v a l t o i t s e l f , as
f o l l o w s from (1.20) and (1.21 1. Taking t h e g r a d i e n t i n t h e d e f i n i t i o n (1.22) of
gradP
(1.23)
=
-12
ay g r a d ( P 1 + P 2 ) + YgradPCM + PCM 5 gradS
P
+
yields, ay
PCM 5 gradP,
i.e. (l-PCM $1
gradP
=
1
7
3
1 dp g r a d ( P 1 + P 2 ) + YgradPCM + PCM(bo- 7 )
gradS,
but, PCM zdPC g r a d S
=
grad(PCMpc)
=
g r a d ( P 1 - P,)
-
pc gradPCM
- pc gradPCM.
Hence P +P
( 1 -PCM $1
gradP
=
grad
+
i.e.,
'
l2
+
YgradPCM +
( b - -1) g r a d ( P -p ) - (bo-\)pcgradPCM, 0 2 1 2
using (1.19),
From (1.24)
and t h e f d
and 4 t h e q u a t i o n s
of ( 1 . 1 6 ) one o b t a i n s
immediately t h a t
where Y 2 (1.69
,
and P G a r e (1.3l),
defined s i m i l a r l y t o t h e incompressible case (see
( 1 . 3 2 ) h e r e u n d e r ) , and where
(1.26
is a d i m e n s i o n l e s s f u n c t i o n which is i n p r a c t i c e e x t r e m l y c l o s e t o 1 , as it r e s u l t s from (1.13) and (1.151, and from t h e d e f i n i t i o n ( 1 . 1 7 ) of Y. The s o u g h t p r e s s u r e e q u a t i o n is o b t a i n e d by a d j o i n i n g t o (1.25) t h e sum of t h e two f i r s t e q u a t i o n s of ( 1 . 1 6 ) :
198
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
a at I
(1.27)
1.4
I
o O(x,P) [B1(P)S+B2(P)(1-s)1
+
div
Go
0.
=
- THE SATURATION EQUATION Using the identity
il
(1.28)
(l-bO)$l
=
-
+
boi2
+
bO(al
+ +
$2) + $1
in the first equation of (1.16) and replacing
and
+
by their
$2
expressions given in the third and fourth equations of (1.16) we obtain (1.29)
2 { o$B1s]
+
div[-bO(l-bO)oKd
[grad(P1-P2) (p1-P2)ggradZ1+b0G 1 = o w +
Using the capillary pressure law we have
(1.30)
a
{o$BIS}+ div {-b (l-bo)~Kd[grad(PCMpc)-(pl-p2)ggradZl
+
boG ]
=
0.
0
If we introduce a mean gravity potential PG by
( 1 -32)
P,o
- 1
-
;. (p 1(P0)
+
p (P 1) 2
0
=
mean density at references
pressure then the saturation equation (1.30) can be rewritten so as to let appear a +
capillary diffusion term r , a transport term caused by the global flow, a second transport term caused by capillary heterogeneity and a third one caused by gravity (1.33)
:
T~ a {o$B1s}
+
+ div { + r+boqo
-
-
oKdbO(l-bO) [pcVPCM+ p 1 - p 2 VPGl mO
where, (1.34)
(1.36)
+
r
%(S,P) Expression
= =
- PcM(x)o(x)K(x)d(x,S,P) bo(S,P) [l-bO(S,P)I
(1.34),(1.36)
for
a(S,P) grad S,
dpc
d~ (s). the
capillary
flow
vector
+
r
correspond to the formula (6.13) of Chapter 111 in the incompressible case.
I. The Two-PhaseCompressibleModel
199
In the special case of a rock model of the first kind, where
does not depend on x (cf.(l .12bis)),
r'
one can get for
d
the analogues of
formulae (3.621, (3.28, (3.29) of Chapter I11 in the incompressible case :
1.5 - SUMUARY FOR THE TWO-PHASE COMPRJBSIBLE
MODEL
The notation introduced below is fully compatible with that of section I11 of chapter I11 on incompressible fluids. Characteristics depending only on the reservoir D : (1.40)
@(x,P)
(1 .41)
(1.42)
=
rock porosity at point x and pressure P
K(x)
=
rock permeability
o(x)
=
section of the field R at point x
( 1 .Q3)
$(XI
=
o(x) K(x)
(1.44)
Z(x)
=
depth.
Physical Unknowns (1.45) (1.46)
P.
transmissivity
:
=
pressure in the jth fluid,
=
saturation in fluid # 1.
J
S
=
j = 1 ,2
Characteristics depending only on the fluids (1.47)
=
:
reference pressure (surface condition pressure for example)
(1.48)
p.(P)
=
density (mass per unit volume) of jth fluid
(1.49)
B.(P)
=
3 = P.(P,)
J
P (PI
(1 - 5 0 )
J
OP,
volume factor of jth fluid
J
=
-21
(p (P )+p (P ) ) = mean density of the two fluids 1 0 2 0
at reference pressure
200
( 1 .51
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
1
u.(P)
=
viscosity of jth f l u i d
d.(P) J
=
J lJj(P)
3
B.(P)
(1.52)
=
m o b i l i t y of j t h f l u i d
global mobility ( 1 -61 )
d(S,P)
z(s!xO,S)P)
=
=
d l ! P ) k r l0( S ) + d 2 ( P ) k r ,0( S )
Non l i n e a r i t i e s i n t h e p r e s s u r e and s a t u r a t i o n e q u a t i o n s d ( S , P ) !indep.of (1.63)
d(x,S,P)
=
-
d(S(x,S),P)
(1.64)
b (S,P) 0
=
x) f o r a r o c k model of f i r s t kind f o r a r o c k model of second k i n d
b(S,P)
(1.66)
(1.67)
(1.68)
(1.69)
( 1 .70)
( c a n be used o n l y w i t h a r o c k model o f f i r s t k i n d )
20 1
I. The -0-Phase Compressible Model (1.73)
B(S,P)
bO(S,P) [ l - b o ( S . P ) l
=
dpC ( S )
( c a n be used w i t h a r o c k model o f f i r s t o r second k i n d )
aY (1 -76)
X(X,S,P)
1 - PCM(X)
=
(S,P).
Main dependant v a r i a b l e s
(1.77)
(1.78)
S
=
a c t u a l s a t u r a t i o n of f l u i d C 1
P
=
(P1+P2)/2
PCM(x) 'I(S,P)
+
global pressure.
=
A u x i l i a r y dependant v a r i a b l e s
(1.79)
i. flow J
(1 -80)
+ r
(1 -81)
=
=
p a r t of
=
0,
+
qo
v e c t o r of t h e j t h f l u i d
+
+
+
0, and of -9 caused by c a p i l l a r y d i f f u s i o n 2
+
+ $2 =
global flow vector.
u Pressure equation i n Cl :
( 1 .82)
a at
(1.83)
{o
=
Q
[BIS +
-JI d
{x
B2(1-z]
}
+
div
o;
= -$
d
x
0
gradP + f 1 gradPCM + Y2 g r a d P c ]
or equivalently:
Go
=
2 grad P
+
d
1
j=1
+ Y . 9.. J
J
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
202
Saturation equation in il
a
(1.84)
{o @
i++ $l
=
r
+
B1
5 )
bo
Go
div
+
:
i,
=
0
- JI { blgrad PCM
+
b2grad Pc]
{ or equivalent by
(1.85)
+
where r is given by : (rock model of first kind only)
(-PcM JI a(S,P) grad S (1.86)
=I
f
(PcM JI d(x,S,P)9(S,P) gradS (rock model of first o r second kind) Equations for separate phase pressures and flows
They are identical to equations (3.70) to ( 3 . 7 3 ) of chapter 111.
1.6
-
THE CASE OF SLIGHTLY COIBRESSIBLE ROCK AND FLUIDS When the compressibility of rock and fluid is small, we will
write, as in formulae (1.1) factors B. under the form
of chapter 11, the porosity $ and volume
:
J
@(X,P)
=
f$o(x) [l
B.(P)
=
1
+
cr(P-Po)1,
(1.87) J
+
C.(P-Po), J
j
=
1,2,
where we suppose that the compressibility coefficients c r' c, and c2 are "small", i.e. that
(1.88)
I c.(P-Po)I J
where CPminl Pmax1 functions
<<< 1,
+pe
CPMin' Pmaxl
t
is the range of pressure of interest. Then the
a , bo, bl, b2, Y, Y 1 , Y2, a, 9 can be considered as independant
of P (as in the incompressible case), the function identical to 1 , and the equations (1.82),
x
can be taken
...,(1.86) are approximated by
:
II. The Three-Phase Compressible Model
203
o Pressure equation in il :
(1.89)
- ap a2 O ~ X ~ ~ ~ ~ X ~ ~ C C+ ~(c,-c,)(P-P,) + C ~ ~ + C 1 ~ div ~ ~ - = S0 ~ ~
(1.90)
Go
+
Go
2
=
-@(x)d(x,S)
gradP
d(x,S)
+
1 Vj(S) ;j. j=l
Saturation equation i n SZ :
(rock model of first kind only)
-PCM(x)@(x)a(S)gradS (1 . 9 2 ) r
=
-P
CM
model of first or second kind)
(x)$(x)d(x,S)2(S)gradS(rock
which reduce exactly, when cr
=
c1
=
c
2
=
0, to the incompressible
equations obtained in chapter 111. 11-THE
THREE-PHASE
COMPRESSIBLE
MODEL
We consider in this section the flow in a porous medium of three immiscible compressible fluids
:
water (fluid d 1 ) oil
(fluid f/ 2)
gas
(fluid f/ 3 ) ,
numbered in order of decreasing wettability. We suppose moreover that no exchange takes place between phases (this case will be considered in .?I11 and 3IV for oil-gas flow). We do not specify the conditions on the boundary of the field
SZ
or at the injection and production wells, as they are similar to those described in chapter I11 for the two phase incompressible case.
204
Ch. I K Compressible, Three-Phase,Black Oil or Compositional Models
I n order to handle the compressibility of the fluids, we
bill1
make the same kind of hypotheses as in the two-phase case. We refer to paragraph I for the discussion of these hypotheses. - EQUATIONS
11.1
nion THE
PHYSICS
Given a reference pressure P o , for example the surface pressure, the main unknowns of the problem are j
P., j j
=
1,2,3
:
actual saturation of fluid
=
1,2,3
:
pressure in fluid d j
=
1,2,3
:
volumetric flow vector of fluid # j evaluated at reference pressure P0'
We introduce i,he
S -1
J
s2
= 1 , we shall in the sequel use only 3 as arguments in saturation dependant functions.
As obviously
and s3 -
S1
+
+
The required equations are then 11.1.1
:
- Conservation law
a {
(2.3)
a Q
B. 5 . } J
J
where B . is evaluated at P. J
at P 1 as at P2 or at P 11.1.2
j
volume factor B.(P) of each fluid by
-
-
:
J'
3
+
div
$J.
=
0
and the porosity
j @
=
1,2,3,
can be evaluated as well
or at any neighbouring pressure P.
- Huskat law (relative permeabilities)
where the relative permeabilities kr. depend, at point x, on both J S1 and S 3 ("three-phase relative permeabilities") and
saturations
possibly on the pressure P.. J
I1 The Three-Phase CompressibleModel
11.1.3.
205
- Capillary pressure law
The water-oil and oil-gas capillary pressures for three-phase S, and flow may depend, at point x, on both saturations s3 ("three-phase capillary pressures") :
Various classes of the above three-phase relative permeabilities and capillary pressures have been proposed in the literature (see for example STONE), and used in numerical models. We will define, in the next paragraph, a new class of relative permeabilities and capillary pressures, such that the flow equations can be written in terms of only one pressure (the "global pressure"), which leads to simplifications in numerical models (cf.Chapter V). This new class and its relations with the previous ones will be discussed in paragraph 11.5 11. 2 - SIMPLIFYING HYPOTHESES
We list in this paragraph all the simplifications we make in the general equations of paragraph 11.1 11.2.1
:
- Pressure dependant coefficients
In the same way as in the two phase compressible case, we decide to evaluate all pressure dependant coefficients in equations (2.3) to (2.5) at a pressure
P
"close" to
P1,P
2
and P
3
;
hence we shall write
where P is, for the time being, any pressure field satisfying
:
206
Ch. IV: Compressible, Three-Phase,Black Oil or Compositioml Models
Min
(2.7)
P.(x,t) 5 P(x,t) 5 J
j=1 ,2,3 11.2.2
-
+x
Max P.(x,t) j=1,2,3
c R,
+ t>O
Choice of a rock model
* Reduced Saturations : at every point x e R , and for every P , one supposes that one is given the residual saturation S (x,P) j=1,2,3 Rj of each fluid. The residual saturation SR,(x,P) for example is the
-
-
largest saturation SR1 such that
-
-
- -
SR1a>kr1 (X,Sl,S3,P)= 0, + and similarly for SR2 and SR3 (usually, the independant of P). (2.a)
S1 2
J
where P
=
:
R3
(X,P) '
(P +P +P ) / 3 . Of course 1 2 3
s1 + s2 + s3 =
1.
As long as only displacement phenomena occur,
than S for all Rj unknowns.
are taken
sR j (x,P)
1-SR1(x,P)-SR2(X,P)-S
(2.9)
SRj's
SRj have been given we can define
the reduced saturations S., j=1,2,3, by
s. -
-
-
Once the residual saturations J
S3'
j,
-
S. J
remains always larger
and the reduced saturations
S. may be used as J
- Shape of relative permeabilities Let
be the mobilities of the fluids, and define the global mobility d and the fractional flow functions
dj,
j=1,2,3 of each fluid by :
207
II. The Three-Phase Compressible Model
so t h a t , (2.12)
v1
v2
+
krl
,
+
Knowing
v3
f o r two-phase
1.
and k r
kr2.
two of t h e three f u n c t i o n s As
=
3
t h e n c l e a r l y amounts knowing
d
and
v l , v2, \ s 3 . f l o w , t h e hypc heses on t h e s p a t i a l depen snce
of t h e r e l a t i v e p e r m e a b i l i t i e s a r e c o n v e n i e n t l y e x p r e s s e d i n terms of d , v1
v2 and
v
For
3'
the
s a k e of
simplicity,
we
suppose t h a t
the f i e l d
c o n t a i n s o n l y one r o c k t y p e . Hence we s h a l l c o n s i d e r t h a t t h e f o l l o w i n g
are
expressions
valid
at
every
point
x
E
n
(S1,
S3
are
reduced
saturations): v. J
(2.13)
=
v.
J
(S1,
s3, P )
j=1,2,3,
d(S ,S , P ) if one u s e s a rock model of t h e f i r s t kind 1 3 (2.14)
d
=
d
(?1 ,s3, P ) i f one u s e s a rock model
(1
of t h e second kind.
1 (1)
When a rock model of t h e second kind is u s e d , we d e f i n e d ( x , S s , P ) = 1' 3 a r e e v a l u a t e d a t p o i n t x from S 1 , S t h r o u g h d(S , P ) where 5 1 , S3 3 1 3 formula ( 2 . 9 ) and s h a l l i n t h e s e q u e l c o n s i d e r o n l y t h e f u n c t i o n s
--
,s
Remark 1
Of c o u r s e , t h e f u n c t i o n s d and v . c a n n o t be chosen e n t i r e l y arbitrarily
if
one
wants
J the
corresponding
relative
p e r m e a b i l i t i e s , c a l c u l a t e d from ( 2 . 1 1 ) , t o be r e a l i s t i c ! A minimum s e t o f n e c e s s a r y c o n d i t i o n s is
(1)
c f . paragraph V I of c h a p t e r 111.
:
208
Ch. IV: Compressible, Three-Phase, Black Oil or Compositional Models
-
d(S1 ,S P)
>
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
II. The Three-Phase Compressible Model
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 <
1
0
belong to an apriorily
ap
I $ (sl,
s3,p)) 5
f S1' f
where
Pc
s3
E
[O,ll,
[Pmin'Pmaxl,
p
is the global capillary function, whose existence is asserted
by the TD condition and which is given by formula (2.29). Condition (2.32) is always satisfied in practical situations
:
ap
by calculating the derivative
in (2.29) and using the hypothesis ap (2.21) and (2.22) on the capillary pressure curves we get a suffficient
condition for (2.32) to hold, namely
:
But, one checks easily that
1 %avl
(sl,s
3
,P)I
d' 1 $1 4 Max{ I dl
(2.34)
(PI -
dl
2
- (P)), l d2
and one obtains a similar formula for \
av 1 - 1 ap3
dl
1
~ 1
P12 cM
=
Max s1 rS3
(2.35) Pz;
=
Max
s1 p S 3
p
d' 3 ) -(p)l~
1,
3
.
So, if we define the maximum capillary pressures by
(
PLE
and P32 cM
Ch.IV: Compressible, Three-Phase,Bkck Oil or Compositional Models
214
then a sufficient condition for ( 2 . 3 3 ) , and hence for (2.32) to hold, is (in differential notation) :
This is the same condition as in the two-phase case, (compare with ( 1 . 1 4 ) ) , and is always satisfied in practical situations. 11.2.5
- Sumnary of resulting equations With the hypotheses and notations introduced in this paragraph,
the physical equations ( 2 . 3 ) to
a [
(2.38)
(2.41)
( 2 . 5 ) can be rewritten as :
a @ (x,P) Bj(P)
p3 - P2
2. } J
+
div
$.J
=
0,
j=1,2,3,
32 P c 6,’ S3),
=
We will start form the above equations and from the TD condition (2.23)
on
the rock
model to get the sought pressure an saturation
equations. 11.3 - THE GLOBAL PRESSURE EQUATION
Suming for j = l , 2 , 3 (2.42)
a at {
equations ( 2 . 3 8 ) yields
-
a Q ( B ~s1
+
B~ s2
+
B
3
S )} 3
+
div
(2.43)
and from (2.39) we get
:
3 (2.44)
=
- oKd
1
j=1
\ . [gradP. J
J
p . ggrad Z
J
which can be written, using ( 2 . 4 0 ) and (2.411,
1
4
=
0,
215
II. The Three-PhaseCompressible Model
4
(2.45)
=
-oKd [grad P2
+
v1 gradPA2+ v3 gradP:2
- p g gradZl
where
3 (2.46)
P(S,,S3,P)
=
1
Uj(S1,S3,P) Pj(P).
j=l
Using then the TD condition (2.24) we get ap
(2.47) where
=
P
-oKd [gradP ( S ,S P) - ap ( S 1 ,S 3 ,P)gradP-pg gradZ] c 1 3
is still any pressure field satisfying (2.7).
We now define the global pressure P by P(x,t)
=
P2(x,t)
V X f
n,
+
:
PC(S1(X,t), s (x,t), P(x,t))
3
(2.48) Vt>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,
II. The Three-Phase Compressible Model
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
II. The Three-Phase Compressible Model
(2.61)
I
kr
(S )
=
gas
32 3 kr ( S )
=
oil
=
capillary pressure curve
3
23
P:~ \ 32
relative permeabilities
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
relative permeabilities
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
:
Ch.IV: Compressible, Three-Phase,Black Oil or Compositional Models
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
II. The Three-Phase Compressible Model
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
Ch.IV: Compressible, Three-Phase,Black Oil or Compositional Models
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.
II. The Three-PhaseCompressible Model
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<
0
, - >o,
O
as1
but, also
as3
a\,
as1
:
II. The Three-PhaseCompressible Model
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
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
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 ) .
Ch.IV: Compressible, Three-Phase,Black Oil or Compositional Models
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
III. The Black Oil Model
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
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
proportions of the oil and gas components.
As we mentionned at the end of paragraph 111.1, we restrict our P <<< Perit, as the unknowns rs, Rs used for the
study to the case where
formulation of the black-oil model rely essentially on the existence of at least one liquid or one gaseous phase, and hence make no sense for P > Pcrit. This condition on P amounts to the following condition on and
r(P)
(3.3)
R(P)
:
R(P) r(P)
<<<
1.
111.4 - DESCRIPTION OF PHASES CHARACTERISTICS
As we have seen in the last paragraph, the liquid and gaseous phases tend to become indistinguishable when the pressure P approaches the critical pressure Pcrit. Hence the maximum capillary pressure and the residual saturations should tend toward zero when P tends toward Pcrit' However, as we suppose that P < P crit' we shall neglect the influence of P on the maximum capillary pressure and residual saturations, but this is
not an essential simplification, since it could be accounted for in the same way as will be done in paragraphe IV for compositional models. c
be
the
At every point x in the field, let then, s ( x ) [resp. SoR(x)] gR residual gas [resp.oil] saturation, i.e. the largest gas
[resp.oil] saturation for which the gas [resp.oil] relative permeability
-
(x) = 0, but we will write the equations under the gRS (x) t 0 and S (x) 2 0. gR OR - As the gas may dissolve in the oil phase, the 0 5 S 2 S (x) g gR saturation range can actually be attained, though the corresponding relative permeabilities are equal to zero. Similarly, the l-SoR(x) 6 S g 5 1 saturation range can also be attained, as the immobile oil can vanishes. Usually
S
sole assumption that
vaporize into the gaseous phase. Finally, the gas saturation
1.
5g may
range
in the whole interval [0,1
Thus if we define, as usual, the reduced gas saturation Sg by (3.4)
III. The Black Oil Model
243
then S may take values smaller than zero and greater than one. -g As pointed out in paragraph V . l of chapter I11 one is obliged to use a rock model of the second kind, in order to generate relative permeabilities which meet continuously with 1 when the corresponding (not reduced) fluid saturation tends toward 1. reservoir
made
of
only
one rock
Hence, if we suppose the
type, the
permeabilities are given by (cf. 1.11-b and c)
where
dg, do
gas
and
oil
relative
:
are the mobilities of the liquid and gaseous phases at
pressure P (the composition of these phases changes with P), and where and the global mobility i are given g (1.10) (typical shape are shown in figure 11). Notice that
the fractional flows L ~ , -= ~1 by (1.81,
...,
functions are continued by 0 or 1 outside the g -0 interval of reduced saturations.
the
and
1,
The capillary pressure is of the form (cf figure 1 1 )
[0,11
:
P -P =P ( S ) g o c g as we have supposed that its maximum value is constant through out the reservoir. Contrarily to the fractional flow functions g and v 0 ' the capillary function Pc is not continued outside the interval [O,ll of reduced saturations. It vanishes for the value S of the reduced gas gc saturation (usually S = 0). gc
111.5
-
GOVERNING EQUATIONS FROM THE PHYSICS
As in the previous compressible problems, porosity, volume P to be defined reservoir for example, the following
factors and viscosities are evaluated at a pressure later. Then we get, for a 3-D equations
:
Conservation of gas and oil components (3.7)
a
5[
@(x,P) [Bg(P)SG
+
RS Bo(P)So]}
:
+
div
g
=
0,
Ch. IV: Compressible, Three-Phase,Black Oil or CompositionalModels
244
a
5{
(3.8)
$ ( x , P ) [rg Bg(P)s I3
+
Bo(P)so]}
+ div
io 0. =
Muskat law :
Capillary pressure law :
(3.11 1
P
g
- Po
Pc(Sg).
=
Phase equilibrium
Plus, of course
-
sg
(3.14) (3.15)
111.6
./
-
+
so
+ L J
g
o
(see paragraph 111.3)
:
:
=
1,
=
1.
GLOBAL PRESSURE AND PRESSURE EQUATION We would like to express the global flow vector
in term of the gradient of only one pressure P. From (3.9) (3.10) we get (3.17)
=
-Kd { ( l + r s )
"g
:
gradP
g
+
(l+Rs) vo grad P
1
III. The Black OilModel
245
0
(I
Figure 1 1 : The data for t h e rock model of secand kind.
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
246
which can be s p l i t i n t o :
(3.18)
-Kd [
=
[ (rs-r(P))
-kd
By function
(l+r(P))v
analogy
with
the
g
v
g
gradP
+
g
gradP
3
strictly
( l + R ( P ) ) 'bo gradPo + (RS-R(P))
diphasic
] }.
vo gradPo
case,
we
1
dP C
define
Y by S
( 3 . 1 9 ) Y(S , P ) g
g
=
( l + r ( P ) ) vg(s,p)
I [ S
gc
-
( 1 + r( P ) ) L ~ ( ~) +, (Pl + R ( P ) )i o ( s , P )
one f i r s t checks t h a t , g i v e n
upper bounds
P
.
min
and
e
s
g
E CO,ll,
r]O,l[
tF 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
III. The Black Oil Model
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 <
+m.
for
5
:
1 P - - P ( S )+Y(S P) g 2 c g g
I3
g
Equation (3.25) has a unique solution as soon as Pg
and Po
as it results from (3.20) and the
lie in the interval [Pmin,
Pmaxl' implicit function theorem. As usual, this solution satisfies
Min (P P ) S P 2 Max (P P ) . g' 0 g' 0 We calculate now the gradient of the global pressure P
:
(3.26)
for 0 2 S
:
we get, using the same calculations as
2 1
( i
in paragraph 1.3
:
(3.27)
-
gradP
=
(l+r(P))v gradP + (l+R(P))vogradPo g (l+r(P))vg (l+R(P))Jo
+
2 ap
gradP.
+
-
of
-21 P c
(3.28)
+ Y
for S 6 0 we get, using the regular continuation 6 defined i n (3.23) : grad P
=
grad P
0
+
(0,P) gradP ap
Ch. IV: Compressible, Three-Phase, Black Oil or Compositional Models
248
grad P
(3.29)
we g e t , similarly
2 1 -
for S g-
*
gradP
=
g
+
av ap
(1 , P ) grad P
Let :
S*
(3.30)
g
=
Max { 0 , Min { S
=
Y(Sg. P ) .
g'
l]},
and
(3.31)
Y
*
*
Then e x p r e s s i o n s ( 3 . 2 8 ) ,
(3.29)
can be s e e n , u s i n g t h e n a t u r a l
n t i n u t i o n s of v and v as s p e c i a l cases of (3.27) where is * g 0' * ay is v a l i d f o r ap , s o t h a t e q u a t i o n ( 3 . 2 7 ) w i t h r e p l a c e d by -
C
-
-m
< s < g--
+m.
-*
Hence e q u a t i o n (3.18) f o r
(3.32)
-Kd [
=
q
reduces t o :
[(l+r(P))dg+ (l+R(P))*
-Kd [ ( r
-r(P)
u gradP g
+
g
]
(1-
(Rs-R(P))
* a? ap )gradP ) vogradPo
}.
I n o r d e r t o s i m p l i f y ( 3 . 3 2 ) w e u s e now t h e phase e q u i l i b r i u m c o n d i t i o n s (3.12),
(3.13) :
oil
phase
saturation
may
if S = 0 , t h e n Rs may be smaller t h a n R ( P ) , ( t h e -g be u n s a t u r a t e d ) , b u t as t h e c o r r e s p o n d i n g reduced
S
satisfies
g
-
gradPo
=
(1-
S
g
<
* av ) ap
0 , we g e t from ( 3 . 2 8 ) t h a t :
gradP.
Moreover, w e have from (3.13) :
and, of course,
\'
63
=
0.
III. The Black Oil Model
*
249
<5 <
if 0
t h e n we g e t from (3.121,
1
g-
(3.13)
:
and t h e second term i n t h e r i g h t hand s i d e of ( 3 . 3 2 ) v a n i s h e s .
-
g
then
= 1 -’
rs
may be s m a l l e r t h e n
r ( P ) (the
gaseous phase may be u n s a t u r a t e d i n t h e heavy component), but a s
we g e t from (3.29)
S
g
>
1
:
* grad P and from (2.12)
g
( 1 - -a )- t
=
ap
grad P ,
:
Summing up t h e t h r e e c a s e s , we s e e t h a t t h e g l o b a l f l o w v e c t o r -b
q
is always g i v e n by :
Adding ( 3 . 7 ) and ( 3 . 8 ) y i e l d s
Equations (3.33), the dissolution f a c t o r s
that
the
pressure
depending on
S
g’
r
equation r
s
and
are t h e sought p r e s s u r e e q u a t i o n . A s
(3.34) and
Rs
are always p o s i t i v e numbers, we s e e
is
of
parabolic type,
Rs.
with
coefficients
Ch.IV: Compressible, Three-Phase,Black Oil or Compositional Models
250
111.7
-
SATURATIONS / DISSOLUTION FACTORS EQUATIONS
-
We search now for equations for the saturation S and the g dissolution factors Rs and rs. From the phase equilibrium condition (3.12). we see that sg, Rs and rs are never simultaneously unknown :
Hence we will need only one saturations/dissolution factor equation. In order to find that equation, we look at equations (3.7) (3.8) with so replaced by 1-5 g' and that We see that (3.7) does not contain any at aR S (3.8) does not contain any at term. Hence if we want to get an
ars
equation which is valid in each of the three cases quoted in (3.35), we have to use a combination of (3.7) and (3.8). So we take the difference of (3.7) and (3.8) which yields
a
(3.36)
+
{ $(x,P) [(l-rs)B g (P)sg - (l-Rs)Bo(P)So]
and from (3.91, (3.10) we get +
(3.37)
@ g-
->
$o
In order to express
identity
:
=
+
t
+
+
div {I$g - @
o}=O,
:
-Kd { ( l - r s )
q g - <jo
}
'g
gradP +
in terms of q
g
- (l-Rs)
vo
gradPo }.
and of grad S g'
we use the
III. The Black Oil Model
which, using (3.9
25 1
,
(3.10) and (3.17), yields
- 2KZ(l-r
Rs) "g
J~
[gradP - gradPo} g
+
Using the capillary pressure law (3.1 1 ) , the sought expression
for
+ @
+
- 4 is g o
Remark 3
:
r coefficient of grad S
One could replace
:
only when
1)
g
f
0,
by
r(P) and Rs by R(P) in the as this coefficient is non-zero
g' i.e. when
"0
0
< S < 1. g
0
Equations (3.36),
(3.40) are the sought saturation/dissolution
factors equations. They are valid everywhere in the reservoir :
reduce to
:
in domains where
-
S g
> =
(no gaseous phase), they
252
Ch.IV: Compressible, Three-Phase,Black Oil or Compositional Models
(3.41) which is a transport equation for Rs
inside the liquide phase;
-
in domains where 0 < 3 < 5 (x) (one immobile gaseous g -gRphase plus a mobile oil phase), the equations reduce to :
a
(3.42)
{ Q (x,P) i(l-r(P)) B ( P ) s - (l-R(p))Bo(P)(l-Sg)]} g
g
l-R(P) - div { l+R(P)
c;
]
=
0,
5g' which fixes the rate at which the gaseous phase is dissolved into the liquid one;
which is an ordinary differential equation for
-
or S (x) < 5 < l-SoR(x) gR g < 1 (one mobile gaseous phase and one mobile liquid
in domains where
0 < S g equations (3.36),
equivalently phase),
(3.40)
have exactly the same form as the
saturation equation of the two-phase compressible case studied in section
I, but with coefficients modified to account for the bubble point and dew point
curves
R(P)
and
diffusion-transport equation for
r(P).
S
g
Namely,
have
a
non-linear
with a degenerated diffusion term;
* in domains where 1-5 (x) OR g phase plus a mobile gaseous phase), we get
<
s
in domains where S = 1 g-
< 1 (one -
immobile oil
:
which is an ordinary differential equation for
.
we
S
g;
(no liquid phase), we get
which is a transport equation for rs inside the gaseous phase.
:
253
III. The Black Oil Model
111.8 - SUMMARY FOR THE BLACK OIL MODEL
We
list here only the notations specific to the black-oil
-
-
model : PHASES :
saturations of gaseous and liquid phases
S g , So
corresponding residual saturations at point x s ( x ) , %(XI gR S reduced saturation (may range out of the [0,1] interval) g * S = Max (0,Min ( S ,l ) ) B
g
P
g'
P
pressures in the gaseous and liquid phases 0
Rs, r
dissolution factors (see figure 9 for the definition)
R(P), r(P) maximum values of Rs and r and dew point curves).
COMPONENTS
at pressure P (bubble
:
+
+
volumetric flow vectors of gas and oil components.
$g, $o
GLOBAL : +
+
+
q
=
Qg
+
9,
global flow vector
P global pressure (see (3.25) for the definition).
FUNCTIONS
:
_v
g
,P) global mobility
(S ,P), v o ( S P) gaseous and liquid phase fractional flow g g' functions
P ( S ) = P -P capillary pressure curve c g g o reduced gas saturation for which Pc vanishes S gc Y(S ,P) see (3.19) for definition g Y* = r(syP, P).
254
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
SATURATION/DISSOLUTION FACTORS EQUATION
PLUS
:
s g
+so
=
1
u
+
=
1
g
v
o
:
IV. A CompositionalModel
255
IV - A COMPOSITIONAL MODEL W e s t u d y h e r e t h e s i m u l t a n e o u s f l o w t h r o u g h t h e p o r o u s medium of t h r e e h y d r o c a r b o n components :
component # 1
heavy molecular weight
component # 2
medium m o l e c u l a r w e i g h t
component # 3
l i g h t molecular weight.
Under
reservoir
conditions,
components
and 2 a r e u s u a l l y
1
l i q u i d , a n d component 3 is u s u a l l y g a s e o u s . Moreover, components 1 and 3 a l o n e , a r e n o t m i s c i b l e i n a l l p r o p o r t i o n s , b u t components 1 and 2 a r e ,
as a r e components 3 and 2.
Their flow,
would
black-oil
modelled
be
by
the
i n t h e a b s e n c e o f component 2,
model d e s c r i b e d
i n t h e previous
The i n t r o d u c t i o n of a medium m o l e c u l a r w e i g h t component w i l l
paragraph.
p e r m i t u s t o a c h i e v e m i s c i b i l i t y of components 1 and 3 w i t h o u t h a v i n g t o s e t t l e f o r p r e s s u r e s a b o v e t h e c r i t i c a l p r e s s u r e of t h e 1-3 c o u p l e o r f o r a h i g h e r t e m p e r a t u r e . S w h m i s c i b i l i t y is h i g h l y d e s i r e d , a s it s u p r e s s e s the
residual
oil
saturation
-
( t h e c o r r e s p o n d i n g o i l c a n n o t be
SOR
r e c o v e r e d by i m m i s c i b l e f l o o d i n g ) , and r e q u i r e s a smaller p r e s s u r e d r o p f o r t h e flooding of t h e r e s e r v o i r .
For a b a s i c d e s c r i p t i o n of m i s c i b l e
d i s p l a c e m e n t s and t h e i r a p p l i c a t i o n t o o i l r e c o v e r y , see STALKUP and a l , LATIL.
IV.1 - RANGE OF VALIDITY We a r e s t u d y i n g t h e i s o t h e r m a l f l o w o f t h r e e p u r e components
( e a c h component c o n s i s t s of o n l y o n e c h e m i c a l s p e c i e s ) , and we suppose t h a t t h e r e is no m o b i l e w a t e r i n t h e p o r o u s medium ( t h i s case c o u l d b e handled
by
putting
together
results
of
sections
I1
and IV of
this
neglecting
the
chapter). Nonessential
simplifications
consist
in
d i f f u s i o n of t h e components i n s i d e t h e p h a s e s , and i n a s s u m i n g a c o n s t a n t thickness
a(x)
1 for the reservoir.
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
256
Unlike f o r t h e p r e v i o u s models, we w i l l write t h e c o n s e r v a t i o n e q u a t i o n s u s i n g masses r a t h e r t h a n volumes, as t h i s t u r n s o u t t o be more convenient
f o r c o m p o s i t i o n a l models. Hence we w i l l n o t u s e t h e volume
f a c t o r s B , but only t h e d e n s i t i e s p . IV.2 - DESCRIPTION OF THE THERMODYNAMIC EQUILIBRIUM Let us be g i v e n a p r e s s s u r e
c r i t i c a l p r e s s s u r e of t h e 111.3
for
the definition)
P
<
where
Perit,
is t h e
Perit
components e q u i l i b r i u m ( s e e paragraph
1-3
(we r e c a l l t h a t t h e t e m p e r a t u r e is a g i v e n
c o n s t a n t and we s h a l l n o t mention i t f u r t h e r ) . Take t h e n ( s e e f i g u r e 12) t h r e e masses
C 1 , C2, C3
them t o p r e s s u r e
of components 1,2,3
with
bring
C1+C2+C3 = 1,
and l e t them i n t e r a c t . What happens t h e n depends on
P
,
C2, C ) on t h e t e r n a r y diagram of 3 f i g u r e 12. B a r y c e n t r i c c o o r d i n a t e s a r e used i n t h i s diagram, s o t h a t t h e
t h e p o s i t i o n of t h e p o i n t C vertex
=
t h e component # 1
of
(C,
corresponds t o C
(1 ,O,O), t h e 2-3 s i d e
=
c o r r e s p o n d s t o C = (0,C C ) e t c . . . . T h i s diagram is d i v i d e d by t h e 2' 3 bubble p o i n t l i n e and dew p o i n t l i n e , which meet a t t h e c r i t i c a l p o i n t , i n t o two main r e g i o n s :
- t h e two-phase-region
l ' i n s i d e " t h e bubbleldew p o i n t l i n e ,
- t h e one phase r e g i o n " o u t s i d e " t h e bubbleldew p o i n t l i n e . Hence we w i l l c o n s i d e r two c a s e s : CASE 1
C
:
components
=
split
concentrat ions three
( C l , C2, C3) f a l l s i n t h e two-phase
CL
components
into =
in
a
(C,,, each
liquid
phase
c2L' c3L) phase
are
and
and
a
C,
=
located
~
region:
gaseous
(C,,,
C2,,
the three phase.
C
3, respectively
)
The
Of t h e
on
the
bubble-point and dew-points l i n e . They n e c e s s a r i l y s a t i s f y :
-i n S,,
o r d e r t o comply w i t h t h e mass c o n s e r v a t i o n r u l e . (Here P,, p L and SL a r e t h e d e n s i t i e s and s a t u r a t i o n s of t h e gaseous and l i q u i d
phases.)
Hence t h e t h r e e p o i n t s
C,,
C, CL
t e r n a r y diagram. The s t r a i g h t l i n e t h a t t i e s
tie line.
have t o be a l i g n e d on t h e c,,
C
and CL
is c a l l e d a
IV, A Compositional Model
257
I f we knew t h e d i r e c t i o n of t h e t i e l i n e going through C , t h e n t h e composition
CG,
CL
o f t h e two phases would be known !
By chance, I
i f we s t a r t a g a i n t h e same experiment w i t h a d i f f e r e n t c o n c e n t r a t i o n b u t s t i l l l y i n g on t h e t i e l i n e c o r r e s p o n d i n g t o C , t h i s w i l l n o t a f f e c t t h e c o m p o s i t i o n s of t h e phases b u t o n l y t h e p r o p o r t i o n j n which t h e y are g e n e r a t e d Hence a l l t h e p o i n t s composition C
G-
This
C
on
CG
cL s p l i t
C,
it turns out t h a t
CL, c_c (sL sL, -SG
(EL -
=
#
= f
CG)
-
SG).
i n t o phases having t h e same
and CL. implies
that
ties
lines
cannot
intersect
inside
the
two-phase domain. I f we f i r s t d e c r e a s e t h e p r o p o r t i o n of component # 2 , t h e p o i n t C moves toward t h e l e f t , and t h e c o r r e s p o n d i n g t i e l i n e n e c e s s a r i l y t e n d s
toward t h e 1-3 s i d e of t h e t e r n a r y diagram. I f now we i n c r e a s e t h e p r o p o r t i o n of component # 2 , t h e p o i n t C moves t o
the right
i n t h e t e r n a r y diagram, t h e c o n c e n t r a t i o n s CG and
C t e n d b o t h toward t h e c r i t i c a l c o n c e n t r a t i o n C c ,
f o r which t h e gaseous
and
t h e t h r e e components
L
liquid
become
p h a s e s become
miscible.
The
i n d i s t i n g u i s h a b l e , and
critical
tie
line
is
then
the tangent
t o the
bubble/dew p o i n t l i n e a t t h e c r i t i c a l p o i n t . It
parameter
r e s u l t s from t h i s a n a l y s i s t h a t t h e t i e l i n e s form a one
f a m i l y of s t r a i g h t l i n e s , which do n o t
intersect inside the
two-phase domain. Moreover, we s h a l l suppose, i n view of t h e forthcoming developments ( c f .
(4.58),
t h a t t h e t i e l i n e s do n o t i n t e r s e c t i n s i d e t h e
whole t e r n a r y diagram. We s h a l l d e n o t e by
R
t h e parameter a s s o c i a t e d t o each t i e
l i n e . I t w i l l be c o n v e n i e n t t o d e f i n e
R
as :
(4.2)
where s ( r e s p . s13) is t h e s l o p e of t h e t i e l i n e ( r e s p . of t h e 1-3 s i d e
of t e r n a r y diagram) i n t h e Oxy orthonormal system of a x i s w i t h O x on t h e c r i t i c a l t i e l i n e . Hence R R
=
=
0 c o r r e s p o n d s t o t h e c r i t i c a l t i e l i n e , and
1 c o r r e s p o n d s t o t h e 1-3 s i d e of t h e t e r n a r y diagram.
258
Ch.IV: Compressible, Three-Phase,Black Oil or Compositional Models
Question:
0 (pressure PI
Answer:
Figure 12: The ternary diagram a t a given pressure P (the
fi
represent masses).
IV. A CompositionalModel
259
Using equation ( 4 . 1 ) and the properties of tie lines, we obtain another parametrization of the two-phase domain
(4.3)
:
inside the two-phase domain, knowing knowing
C
(R,SG).
=
( c ,c c 1
#
2'
3
amounts
So the thermodynamical equilibrium in the two-phase domain can be represented by considering as given R as a function of
C -G
and
CL as functions of R
and P
v: c ,
C
and P and
:
(4.4)
R
Remark 4 :
Often, all tie lines seem to converge to the same point of
=
R(C,P),
Y P,
the 1 - 3 side of ternary diagram (cf. figure 1 2 ) . This is the Hand's rule approximation, which has been used for the computation of the figures illustrating this paragraph. CASE 2 :
C
=
Then
(C,,C2,C3)
the
falls in the one-phase domain.
three components remain in a single phase, with
composition C. This one phase domain has been divided, on the ternary diagram of figure 1 2 , rather arbitrarily in three regions
:
- the liquid region (below the bubble point line and left of the critical tie line). - the gaseous region (above the dew point line and left of the critical tie line). - the fluid region (right of the critical tie line), where the generated phase has characteristics intermediate between those of a gas and those of a liquid. In fact, the transition from a liquid to a gas (through a "fluid") can be done fully progressively. However the above partition of
01. IV: Compressible, Three-Phase,Black Oil or Compositional Models
260
the one-phase domain will be convenient for the mathematical treatment of
the equations. We
now the influence of pressure changes on
describe
the
ternary diagram. If we decrease the pressure, the size of the two-phase domain usually increases, the critical point moves to the right, and may even disappear (then the bubble point line and dew point line do not meet any longer, the ternary diagram contains only liquid, two-phase and gas domains). If we increase the pressure, the size of the two-phase domain usually decreases, the critical point moves to the left; the two-phase domain disappears when the critical point meets the 1 - 3
side of the
ternary diagram, which happens when the pressure P becomes equal to the critical pressure Pcrit of the 1-3 components system. Above critical pressure the three components are miscible in any proportion, and the "FLUID" zone of figure 12 covers the whole diagram.
The thermodynamical equilibrium hereabove described is supposed to be attained instantaneously. Hence one needs to be able to calculate the
concentrations C,
and
corresponding to
CL
any given overall
Y
concentration C (with C
=
CL
=
C if C happens to be in the one-phase
region) and pressure P. This is the role of the so called "flash routine", which may work either using the above geometrical description of the ternary diagram and the Hand's rule approximation, or more often by solving for CG , CL a system of non linear equations, based on the equation
of
state
representation
of
the
phases
and
on
routine
is
behavior
thermodynamical considerations. We
will
suppose
hereafter
that
such a
flash
available, and hence consider as known, in the two-phase domain, the functions R, C, saturation
sc,
"
and CL
defined in
(4.4) and
(4.51, and the gas
defined by (4.1).
IV.3 - DESCRIPTION OF PHA.SE(S) CHARACTERISTICS IV.3.l
- In the one-phase domain
In this domain, the unique existing phase is characterized by its density p and viscosity u , both taken as given continuous functions
N.A CompositionalModel
26 1
of the concentration C and the pressure P, with (bounded) continous partial derivatives (4.6)
P
=
:
P(C.P)
P
=
!J(C,P).
It will be convenient to define a mobility function d as d
(4.7)
=
d(C,P)
For each
=
p(c’p) !J(C,P)
:
for C in the one-phase domain, p > 0.
P < Perit, d(C,P)
d(Cc,P)
+
when
the one-phase domain, to the critical concentration Cc
C
tends, inside associated with
the pressure P. - In the two-phase domain (General)
IV.3.2
Of course, the densities and mobilities of each phase are given
where
d
is given by ( 4 . 7 )
and
CL, Cc
are the liquid and gaseous phase
compositions, given by (11.10, (4.5). But, we have also to define the relative permeabilities and the capillary pressure curves, which are necessary for the description of the simultaneous
flow
of
the
two
conditions, (either because R that the residual saturations
+
As
phases. 0 or P
SLR and
-t
-
one
approaches critical
Perit), the experiments show
ScR and the capillary pressure
Pc - PL decrease to zero, and that the relative permeability curves tend to cross straight lines with slopes
+
1
and -1
corresponding to a
piston-type displacement (see for example BARDON-LONCERON). This first implies that the total mobility curve
-
->
Sc
->
krG
dC (where C stands for critical conditions), as dC and d tend separately toward dC and kr C -L (resp. krL) tends toward Sc (resp. l-Sc). dc
+
krLdL tends toward the constant function Sc
Ch.IV: Compressible. Three-Phase,Bhck Oil or Compositional Models
262
Secondly, this implies that the fractional flow curve krG dG SG -> krGdG+krLdL tends toward the linear function
-
s G -> R
+
0.
(4.3),
-
SG when
We now define precisely these two-phase characteristics. Using we will express aj.1 i,hede caracteristics in term of (R,SG)rather
than C.
IV.3.3
- Compositional
residual Saturations
We take as given the two functions :
which satisfy : -
1
(4.10)
Remark 5 :
SGR -t
SLR
+
0 when 0 when
R R
a S ~ constant ~
a R s
+
0 or 0
,
P
or
P
2% a R <
'crit P
J
9
.
crit,
constant
= -
9
YR.
The compositional residual saturation satisfying (4.10) can easily be generated from two-phase data relative to the
1-3 components by setting
:
(4.11)
where ?13
GR
and :? ;
are the residual saturations for the 1-3
which are supposed to tends toward zero where P
-
Perit'
components,
and eRS(R)
interpolation function satisfying (4.12)
eRS
(0)
=
0,
aRS (1) =
1
which is left to the choice of the user.
deRS
dR (R) , -
S
constant R '
is an
IV. A Compositional Model
263
We can now define the reduced saturation in the gaseous phase by : SG-zGR(x,R ,P )
(4.14)
SG
=
SG (2G'X' . R,P)
Notice that the
-
=
-
1 -SGR(x,R , P )-SLR(x,R , P
-
SG S,
mapping depends on x,
in the black-oil model, the reduced saturation outside of the interval [O,l].
SG
R
and P. As
will take values
Hence we will consider (cf. figure 13),
inside the two-phase domain of the ternary diagram, a strict two-phase domain, defined as the set of all points S belongs to [O,l].
C
whose reduced saturations
-G
From the physical definition of the residual saturations, we see that the strict two-phase domain is the set of overall compositions
C for which both the liquid and gaseous phases can be displaced. This domain depends on
IV.3.4
x
and
2, 4s do the residual saturations.
- Compositional Relative permeabilities In order to generate krG (x,SG,R.P) and krL(x,SG,R,P) satisfying
the conditions stated in paragraph IV.3.2, we will use a compositional rock-model of the second kind, which will continue, on the two-phase part of the ternary diagram, a black-oil rock model of the second kind defined
for the 1-3 component system as in paragraph 111.3. Hence we take as given the two functions
:
[0,13
->
d(SG , R , P )
global mobility
[O,ll
->
LG(SG,R,P)
fractional flow
(4.15)
satisfying the following conditions : i) For R
=
1 (no d 2 component) d(SG,l,P) and uG(SG,l,P) coincide with
the global mobility and fractional flow curves i n the 1-3
component
black-oil model, as defined in paragraph 111.4, with the difference that we are considering here mass flow instead of volume flow.
264
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
N
, \ \ q
%
Critical Point
DOMAIN
\
1, sG
t
Figure 13 :The two-phase domain and strict two-phase domain in the C coordinates (top) and in the
(R,SG)coordinates
(bottom)
IV. A CompositionalModel
Hence
i
(4.16)
and (cf. (4.5)
i
(4.18)
ii)
265
d and
d(SG,l ,P)
vG
have to satisfy -13 d (SG,P)
=
--
and ( 4 . 8 ) )
-13
:
-
k r G ( S G ) (resp. krA3 ( S G ) ) ,
GL3(sG)(resp.
the relative permeabilities at a given expressed as functions of the saturation
kr13 ( S G ) ) are point x O c R , (resp. the
zG
reduced saturation S G at the point x o ) .
For 1 > R > 0 one wants that *
:
:
outside the strict two-phase domain
(4.19)
inside the strict two-phase domain
(which
is
and (1.12)),
already
satisfied
forR=l as
can
be
seen
from
(4.16)
Ch.IV: Compressible, Three-Phase,BIack Oil or Composition01 Models
266
- On the bubble point and dew point lines, the global mobility meets continuously with p h ~ s emobilities, as defined in (4.7) and (4.8) :
-
d(O,R,P)
(4.21)
iii)
for R
d(CL,P)
=
d L
on the bubble point line,
d(1 ,R,p) = d(CG,P)
=
dG
on the dew point line.
=
=
0, ie at the critical point, the global mobility is
independant of the saturation and equal to the critical mobility, and the fractional flow curve is linear
--
d(SG,O,P)
=
:
sG
d(Cc,P)
+f
sG
fF SG
[O,ll,
(4.22) \IG(SG’O,P) = This
ends
the
C0,ll.
list of conditions to be
--
compositional rock-model d(SG,R,P),
,G(SG,R,P).
The compositional relative permeahilities then given by
satisfied by any
krG
and
krL
are
:
From the conditions (4.22) satisfied by d and v G and the and S L R , we
condition (4.10) satisfied by the residual saturations see that the relative permeabilities towards the Remark 6 :
-
krGand krL will tend, as R
sGand 1-2,functions, as required by A
full
scR
+
0,
experimental evidence.
set of compositional relative permeabilities is
usually not available to the user, as its experimental
IV. A Compositional Model
267
determination is extremely delicate and time consuming (see
for example).
BARDON-LONGERON
If by chance such
compositional relative permeabilities are available, then the
relations
(4.23)
can
corresponding total mobility
be d
used
to
determine
and fractional flow
the vG*
But, in the usual case where only the d 1-3 component data are available, one can easily construct functions d and vc
satisfying all the conditions (4.16) through (4.22) by
interpolating between the # 1-3 component data (R=l) and d = cste and ' I G = SG :
the critical data ( R = O ) where
d(SG,R,P)
-
13 , db3, dL3 d13, v G
where
{ d(CG,P)
sc
-
+
d(CL,P)(l-SG) }
x
1
(4.24)
and
=
are defined in (4.17), (4.18) and where
are interpolation functions, satisfying,
8"
i
(4.25)
ed(o)
=
0,
ii ( 1 )
=
1
ev(0)
=
0,
0
,(1)
=
1,
d
d8
constant % (R) 2 R
and left to the user's choice.
IV.3.5
Bd
-
0
Compositional capillary pressure
The capillary pressure can be defined only at places where both the gas and liquid pressures
PG
and
PL
are defined, ie at places
where both phases are mobile; hence the compositional capillary pressure has to be defined only on the strict two-phase domain of the figure 13, ie for reduced saturation
SG
ranging between 0 and 1 . A s the capillary
pressure must vanish when one approaches critical conditions, we set :
Ch.IVC Compressible, Three-Phase,Black Oil or Compositional Models
268
where the maximum capillary pressure PPM ” and the reduced capillary pressure curve p (S ) satisfy : c G PCM(x,R,P)
-
0 as R
-
or
0
P
7
‘wit is an increasing function of S G , vanishing for SG=SGc, p, and satisfying :
(4.29)
I Pc(SG)I
5 1 , and
or
pc(0)
pc(l)
=
1
aP
Concerning the derivatives of PCM and pc, we will suppose that
2 which asG
usually tends to infinity when SG rapidly, and that
aPCM aR
+
0 or 1 ,
will not blow up too
remains always bounded, even when one approaches
the critical conditions. More precisely, we suppose that
i 1%
dpC v ~ ( S ~ , R , P) (S ) - 0 dSG G
(4.30)
(l-vG(SG,R,P)) dPC ( S G ) dSG aR
(x,R,P) 2
when SG
+
+
0,
when SG * 1 ,
0
,
constant
aL
R E CO,ll,
dP where R is defined in (4.2). As nobody knows how quickly 2 blows up dSG to zero, to infinity and how uG (i.e. the relative permeabilities) goes the two first assumptions of (4.30) are not practically restrictive. As for the last assumptions in (4.301, it seems to be reasonable in view of the available experimental data. Remark 7
:
Let the capillary pressure law for the 1-3 component system be given by :
13 where PCM(x,P)
+
0 when P
+
Pcrit . and pb3 satisfies the
IV. A CompositionalModel
269
second part of ( 4 . 2 9 ) .
Then the compositional capillary
pressure data PCM(x,R,P) and pc(Sc)
of
(4.28)
can be
defined by :
where : d0 (4.30 quarto)
9
PC ( 0 ) = 0,
a PC ( 1 )
=
0,
p c (R) dR
2 constant.
From (4.30 quarto) we see that necessarily (4.30quinto)
IV.3.6
-
0 2 0
PC
(R) S constant
x
:
R.
Hypothesis on pressure dependent data
We will suppose that all pressure dependent data
:
- the residual saturations SGR(x,R,P) and SLR(x,R,P) in the two-phase domain, defined in ( 4 . 9 ) , - the phase mobilities
dh3 (P)
and
q3(P)
in the 1-3
component system, defined in ( 4 . 1 8 ) , - the maximum capillary pressure
...vary
P
CM
(x,R,P), defined in ( 4 . 2 8 )
slowly enough with the pressure P so that
the differential of f for a variation 6P of P ) there exist G ,
0
<
0
<
:
1 , such that
(6f
denotes
Ch. IV: Compressible, Three-Phase,Bhzck Oil or CompositionalModels
210
where
:
. Pmin' . 'max are given lower and upper bounds for pressure in the model range of validity,
*
ASmin=
As
Min x E 9
I1-scR (x
Min P
€
I 1
,P)-sLR(xI 1 ,P ) I .
CPMin,PmaxI P
for the t w - p i l d s e
compressible model and the black-oil
model, condition (4.30sexto) is always satisfied in practice for the range of pressure used in reservoir simulation.
IV.3.7
- An exemple of compositional two-phase data We consider a system made of decane ( C l o ) , n-butane (C,) and at a temperature of 7 1 ° C and a pressure of 2750 psia. We
methane ( C l ) ,
want to determine the compositional relative permeabilities and capillary pressure using the technique described in Remarks 5 to 7, which consists of interpolating between
the 8 1 -3 component system and the critical
values. The required physical data, taken from REAMER et al, and LEE et al, are listed in figure 14, with the exception of the decanelbutane relative permeabilities and the capillary pressure curve which are the R = l curves of figure
16.
The one phase mobility function d defined in (4.7) has been obtained by
interpolating linearly for p
and
p
between the single
component values. The set of tie lines has been described using Hand's rule (cf. remark 4), and each tie-line is indexed by R = l (resp. s ) is the slope of the (resp. characteristic) tie 1:ne Oxy
system of
axis
shown
in
compositional
in the
figure 14 (this parametrization is
equivalent to that indicated in ( 4 . 2 ) ) . The
, where s
residual
-
saturations SGR and
mobility function Z(SG,R,P), the fractional flow
G
(S
G
SLR, the
,R,P), and the
capillary pressure PCM(x,R,P) have been determined using respectively formula ( 4 . 1 1 1 ,
(4.24),
(4.30 ter) with the interpolation functions
whose derivatives gently vanish for R=O
and 1.
271
IV. A Compositional Model
b'
H a n d ' s Point
(methane)
~
/
ONE
PHASE
\
2
-butane)
1
2
3
decane
n-butane
methane
DENSITY
.a9148
.65695
.1101
VISCOSITY
.59031
.07814
.011
CRITICAL CONCENTRATION
.048
.242
.7 1
decane/methane r e s i d u a l s a t u r a t i o n s -l3
'LR
Figure 14
:
=
.1,
-1 3 SGR
=
:
.01
The data for the determination of the compositional relative
permeabilities in the decane-n butane-methane system.
212
Figure
Ch.IF Compressible, mree-Phase, Bhck Oil or Compositional Models
15
:
The
compositional
fractional
flow
(left) and
global
mobilities (right) for decaneh-butanehethane at a temperature of 71 OC and a pressure P
=
193 kg/cm2.
decane/methane alone, the R
=
The R=l
curves corresponds to the
0 curves correspond to critical point.
IV. A CompositionalModel
..................
--------
-.-.-.-.-
R r O R = .33 Rs.66 R= 1
Figure 16 : The compositional relative permeability and capillary pressure curves for decane /n-butane / methane.
Ch.IV: Compressible, Three-Phase,Black Oil or Compositional Models
214
The f u n c t i o n s relative
vG, d
permeahilities,
are shown i n f i g u r e 1 5 , t h e c o r r e s p o n d i n g
computed
through
(4.23),
and
the
capillary
pressure appear i n f i g u r e 16.
IV. 4- GOVERNING EQUATIONS FROH THE PHYSICS Our main unknowns b e i n g t h e c o n c e n t r a t i o n
C = (C,,C2,C3)
and
the p r e s s u r e ( s ) , we w i l l write mass b a l a n c e e q u a t i o n for e a c h component,
and u s e IV.4.1
mass -
f l o w v e c t o r f i e l d s as a u x i l i a r y unknowns.
In the one-phase domain Let u s d e n o t e by
(4.31)
p r e s s u r e i n t h e s i n g l e e x i s t i n g p h a s e (IIw i l l a l s o b e
II =
called PL, P
F
or PG a c c o r d i n g t o t h e n a t u r e o f t h i s s i n g l e
p h a s e (see f i g u r e 1 2 ) , + q
(4.32)
f l u i d mass f l o w v e c t o r .
=
Then t h e Darcy law, p l u s t h e mass b a l a n c e f o r e a c h component, y i e l d the equations
a [
(4.33)
:
@ ( x , P ) p(C,P)Ci]
+
div {Ci
q’ }
=
0,
where @ ( x , P ) is t h e p o r o s i t y , K(x) t h e p o r o s i t y , and h a v e been d e f i n e d i n (4.6)
i=1,2,3,
p(C,P)
and d ( C , P )
and ( 4 . 7 ) .
3 As
t h e i r sum :
(4.35
1 C;=
1 , one c a n r e p l a c e a n y one o f t h e e q u a t i o n s ( 4 . 3 3 ) by
i=l
a
1
P(X,P)
P(C,P
] + d i v ( G } = O
which is a pr’essure e q u a t i o n i n t h e one-phase domain.
IV, A CompositionnlModel
275
Though t h e r e is f o r t h e time b e i n g no r e a s o n f o r t h a t , we have allowed o u r s e l v e s , i n e q u a t i o n s ( 4 . 3 3 ) t h r o u g h ( 4 . 3 5 ) , porosities,
d e f i n e d ) d i f f e r e n t from t h e a c t u a l p r e s s u r e l e g i t i m a t e i f P is n o t t o o d i f f e r e n t from
IV.4.2
t o evaluate the
d e n s i t i e s and v i s c o s i t i e s u s i n g a p r e s s u r e P ( s t i l l t o be I I
i n t h e f i e l d . T h i s w i l l be
and
P L (one i n t h e gaseous
IT.
- In t h e two-phase domain We have now two p r e s s u r e s
PG
p h a s e , one i n t h e l i q u i d p h a s e ) . L e t u s d e f i n e +
Q
(4.36)
G
=
mass flow v e c t o r of t h e gaseous phase
=
mass flow v e c t o r of t h e l i q u i d phase.
+
QL
:
Then t h e Muskat Law f o r each phase, t h e inass 114lanCe e q u a t i o n f o r e a c h component and t h e c a p i l l a r y p r e s s u r e law y i e l d :
iG
(4.38)
where
pG,
=
-K(x)
d(SG,R,P)bG ( S G , R , P ) {
p L are d e f i n e d i n ( 4 . 8 ) ,
3 Once a g a i n , a s
1
(FG,R) i n
(4.3).
3
CiG
=
1
and
e q u a t i o n (4.37) by t h e i r sum
1 i=1
i=1
(4.42)
and
gradP G- p G g g r a d 2 } ,
:
CiL
=
1 , one r e p l a c e any one of t h e
Ch. IV: Compressible, Three-Phase,Black Oil or CompositionalModels
216
As u s u a l i n two-phase f l o w s , p o r o s i t i e s , d e n s i t i e s , v i s c o s i t i e s
a r e e v a l u a t e d a t a p r e s s u r e P , t o be chosen such a s t o r a n g e between and P
IV.4.3
- Matching
- On CL
=
pG
L'
C,
of two-phase and one-phase e q u a t i o n s
t h e bubble p o i n t l i n e
5,
=
0 , one has vc
=
0,
i,
=
0 , and
and t h e o n l y mobile f l u i d is t h e l i q u i d phase; hence c o n t i n u i t y
o f p r e s s u r e and f l u x r e q u i r e s t h a t (4.43)
PL + $L
(4.44)
:
= TI,
+
9.
=
Moreover,
t h e c o e f r i c i e n t s of t h e two phase e q u a t i o n s ( 4 . 3 7 ) ,
(4.39) r e d u c e t o tho4? of t h v (4.45)
p,
(4.47)
iL
sc
CiC
+
iJrl~'-1JhASe
PL CiL
SL =
PL
equation (4.33), (4.34),
ci
%=
P ( C , P ) Ci'
-K(x) d ( C , P ) { g r a d PL - p ( c ) g g r a d
=
Z
1.
Comparing (4.47) and (4.34) g i v e s , u s i n g ( 4 . 4 3 ) , (4.48)
vll
VP
=
-
:
L'
Hence we s e e t h a t , equations (4.3'0,
(4.44)
on t h e
( 4 . 3 9 ) match nicc1.y
On t h e dew
bubble p o i n t dii-,ii
l i n e , t h e two-phase
t h e one-pllase e q u a t i o n .
p o i n t l i n e , one g e t i n a s i m i l a r way t h a t t h e
two-phase eqiuations ( 4 . 3 7 ) , (4.38) match w i t h t h e one phase e q u a t i o n . Hence
we
will
try
to
replace
the
two-phase
equations
by
e q u a t i o n s similar t o t h e one-phase e q u a t i o n s (4.33 t o 3 5 ) , and w'iich w i l l match them on t h e bubble and dew p o i n t l i n e s , t h u s e n a b l i n g us t o o b t a i n e q u a t i o n s v a l i d t h r o u g h o u t t h e i r r . m i r y diagram. I n t h e s e e q u a t i o n s , g o i n g from a one-phase
l o c a t i o n t o a two phase l o c a t i o n w i l l a l t e r o n l y t h e
c o e f f i c i e n t s o f the e q u d t i o n s , but n o t t h e form of t h e e q u a t i o n .
IV. A Compositioml Model
211
IV.5 - INTRODUCING A GLOBAL PRESSURE
The fractional flow function vG defined in (4.15) being of the same form as the one defind in paragraph 1.2.2 f o r the zo:nprc?ssible case, the calculations will be basically the same. However,
b!c
will use in this
paragraph and in the next one, a different presentation, which it is hoped
will
give a
better
intuitive feeling and
result
in
easier
calculations. IV.5.1
- Some Preliminaries
For every reduced saturation
SG c
CO,ll, and
R
E
C0,ll we
define: Y(SG,R,P)
I
sG
=
1 dPc 2 ] dSG ( s ) ds
[ v (s,R,P)-
C
sGc
(4.49)
-IG(SG,R,P)
Y ( S ,R,P) L
where
C
=
=
Y(S , R , P ) -
G
Y(SG,R,P)+
r
1
2 pc ( SG )
=
pc ( S c )
=
2
1
vG(s,R,P)
sGc have been defined in (4.28), ( 4 . 2 9 ) .
pc, SGc
I Y(SG,R,P)I Y
C1-vG(s,R,P)]
G
5
3 I P,
dPC -
dSG One checks easi ly
(SG)l,
%'
(4.51) G
Y
L
S -p ( 0 )
, -p
(1)
is an increasing function of
5 Y (1,R,P) 2 0 , G
sG
(4.52) pc(0) 5 YL(O,R,P) 5 0
(4.53)
ds
dPC ( s ) ds
:
is a decreasing function of
0 5 Y (O,R,P)
(3)
dSG
the f o l lowing properties of these functions (4.50)
sG
, 0 5 Y ( l , R , P ) 5 p (l), L
278
Ch.IV: Compressible, Three-Phase.Black Oil or Compositional Models
(4.54)
(4.55)
A
corollary of ( 4 . 5 4 ) , using assumption ( 4 . 3 0 ) , is that ? 3,
aYG (l,R,P) =
(4.56)
O i L (O,R,P) = 0.
0
asG
asG
One can notice that the function Y1 defined i n (4.53) is a continuation, to the ternary diagram, of the function Y, defined in
(1.18) for the case of two-phase compressible flow. From (4.55)
where
we get, u s i n g assumptions (4.20) and ( 4 . 3 0 sexto):
SC(SG;x,R,P) is defined in (4.14) We have illustrated in figure 17 the shape of the -:G and yL
functions for a capillary pressure curve p
vanishing for a non-zero
value SGc of the reduced saturation. This value of SGc was chosen for illustrative purposes, but SGc notice is the zero derivative of
=
L-
0
works as well. The main thing to
at SG
=
0 and of Y
-G
at SG 5. -
We continue now tu the whole ternay diagram the functions S
G
and R , which have until now been defined only for the two-phase domain. To any point x
E
R , any concentration C in the ternary diagram, and any
*
pressure P , we associate SG figure 13)
:
=
*
SG(x,C,P) and R
*
I =
R (C,P) defined by (cf.
N.A
F i g u r e 17 : The
219
,p
Yc
and
YL
functions of reduced saturations SG
Ch. IV: Compressible, Three-Phase,Black Oil or Compositioml Models
280
if C belongs to the enlarged liquid domain
0
at pressure P , SG
(4.14)
defined
when
C
belongs
to
the
strict two phase domain at pressure P , if
1
C
belongs to the enlarged gas domain
at pressure P , and any continuation of
the above function,
meeting continuously on the fluid-liquid and
fluid-gas
exception of
boundaries
(with
the
the critical point), if C
belongs to the fluid domain.
I*
if
0
C
belongs
to
the
fluid
domain
at
pressure P
R (C,P) =
the
parameter
of
the
tie-line
going
through C if
C
belongs to the liquid, two-phase or
gas domain. Remember that the strict two-phase domain, the enlarged liquid and gas domain depend on the spatial position x c
0,
as the residual saturations
may vary from one spatial position to the other. This explains why depends also on x. Now, for a fixed x, the function C ->
*
: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.
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
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
IV. A CompositionalModel
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
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
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
:
IV. A Compositional Model
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
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
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
IV. A Compositional Model
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).
IV. A CompositionalModel
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
:
IV. A Compositional Model
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.
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
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
IV. A CompositionalModel
a { q' (4.89)
=
$(PI p(X,R,P) ]
div
+
{GI
=
0
-Kd(X,R,P { [x(X,R,P)-?(X,R,P)
aTCM
(R,P)] grad P
-fi(X,R,P) g grad 2
aT -[?(X,R,P) S ( R , P ) - n(X,R,P)] grad R'].
One checks easily, from their definitions in table 1, that p, d,
5
are
continuous functions of X a n d R over the whole ternary diagram, with bounded continuous partial derivatives everywhere, except possibly on the SG = 0 and SG = 1 lines for p, d, 5 and on the SG = 0 and SG = 1 lines for
5.
Moreover, though p. d and
5
are defined via
?iGo r
SG
1 ), they are easy to calculate near the critical point, where
(cf. Table
sc and SC -
may vary very abruptly, as they tend to become independant of SG or SG when
R+O.
For
x
we get, using the remarks 5 through 7 *
*
We see that, despite the fact that
SG
:
is a discontinuous function of
at the critical point (see (4.5811, the x function is continous as
(X,R)
the integral is multiplied by
the product of the two interpolation
and e,,, each of them vanishing when functions e PC Similarly, we get for 7 :
R
+
0, R
> 0.
which is discontinuous at the critical point. But, on the other hand we have (4.92)
:
%
:A
dP
(R;P) =
- (P) dP
B
PC
(R')
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
302
From (4.92), (4.93) and (4.30 quarto) we see that
an
Y
continuous and
A
,"," bounded
?
aT
-$
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.
IV. A CompositionalModel
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
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
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
IV. A CompositionalModel
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
IV. A Compositional Model
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
Ch. IV: Compressible, Three-Phase,Black Oil or Compositional Models
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
II. Approximation of the Pressure-Velocity Equations
+
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
:
II. Approximation of the Pressure-Velocity Equations
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
Ch. V: A Finite Element Method for Incompressible no-Phase Flow
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:<
A basis
is the outer
are (QZ,E, K E " ~ , E c aK) and
they are the total flows of the fluids through the edges of an element K in the direction of the outer normal. The dimension of 3xNT+I(xNQ, where NT
(resp.NQ)
?
is
equal to
denotes the number of triangles (resp.
parallelograms). Let us introduce N o the NED-dimensional space of functions defined only on the edges of
,
which are constant on each edge. The lagrangian
multipliers of the continuity constraints on the normal components of the velocity lie in No and are denoted Ah.
A convenient basis of N o is the set
fg}
{pE, E such that pE restricted to the edge E' is equal to the Kronecker E are symbol 6 E , . The degrees of freedom of Ah, denoted (AE, E €
Z)
approximations of the pressure on the edges of the mesh. We consider now p; e M o , Ah f No such that
the
following
problem.
Calculate
+* qOh €
z*,
aK>E dx, (3.1 .d)
(3.1 .a)
=
J
n
P:~
on
div G1Sh*v dx
c
?*,
TS'
=
0,
v e
MO,
dY,
p e
N o , p=O on
Proposition 2 gives the relation between equations (2.2) and (3.1).
r 5'
In.Resolution of the Algebraic System for &essure-Velocity
325
Proposition 2 : Equations and +I qOh
=
(3.1)
+n qOh, P;
=
Pt
have a
u n i q u e s o l u t i o n ( +* q o h , PR, Ah)
P,") where (qOh, +n
E
E
?*xM0xNo
?xMo i s t h e u n i q u e s o l u t i o n o f
e q u a t i o n s (2.2).
Proof: Equation i n t e r i o r edge E
=
is
(3.l.c)
i.e.
K. n K.
J'
1
across t h e e d g e s a n d
+*
+ +* + qoh-wKi+qoh-vK, = 0 on any J t h e normal components o f a r e continuous
equivalent
to
i:h
+
"q"Oh' x.
On t h e o t h e r hand f o r a n y e d g e E n o t i n c l u d e d i n Ts, we h a v e
s i n c e normal components of
+
+ X a r e c o n t i n u o u s across t h e e d g e s . For t h e
s
p r o d u c t i o n b o u n d a r y , (3.1 . d ) i m p l i e s
+
t",
+
Since ? C we c a n write e q u a t i o n ( 3 . 1 . b ) f o r s e X a n d i t f o l l o w s t h a t +* qOh, PL s a t i s f y e q u a t i o n s ( 2 . 2 ) . Hence from p r o p o s i t i o n 1 , t h e y e x i s t , a r e
*; Oh -- +'qOh' P*h = Pnh ' Now we c h e c k t h a t A h e x i s t s and is u n i q u e . h h i s g i v e n by e q u a t i o n where and P t a r e now known. I t is a l i n e a r s y s t e m , s o i t is
u n i q u e and
(3.l.b)
G:h
enough t o show t h a t
1
1 E E Z
h
+ + h
S'VK
=
0,
;E
jis
K E %
aK
3
E
i m p l i e s A h E 0. To d o s o , t a k e f o r
which t e r m i n a t e s t h e p r o o f .
t h e b a s i s f u n c t i o n s of
2".
We o b t a i n :
Ch. V: A Finite Element Method for Incompressible Wo-Phase Flow
326
The main feature of formulation (3.1) is that the equation (3.l.b) for the basis of X*
gives a set of local equations, i.e.
connection between degrees of
freedom of
there is no
two different elements. This
important property, which was not true for equation (2.2.b), will be used to solve efficiently the linear system derived from formulation (3.1). To see that Ah is an approximation of the pressure o n the edges of the mesh, one rewrites equations (3.l.b) in the form
:
Thus A h appears as the trace of the pressure on the edges when we multiply by test functions in
"x*
the equation expressing the total velocity
in terms of the pressure, and we integrate by parts. 111.3 - THE ALGEBRAIC SYSTEM DERIVED FROM THE MIXED-HYBRID FORMULATION
As before we write equations (3.1) in terms of degrees of freedom and basis functions. The resulting linear system reads
:
L Here AQ* is a symmetric positive definite matrix with dimension 3xNT+4xNQ. It is bloc-diagonal, each block corresponding to an element of and being a 3x3 o r 4x4 matrix. Thus AQ* is easy to invert. The non zero coefficients of AQ* are, precisely for K c EcaK:
=I-d(sn)
-1 +
1
A~;K,D),M , E )
J,
%,
D
cg
,
E c
g,
D c aK,
+
'K,D"K,E
dx.
The matrix DIV* has 3xNT+4xNQ columns and NEL rows. Its nonzero coefficients are for K
E
,
E c aK :
327
III. Resolution o f the Algebraic System for Pressure-Velocity
The matrix B has 3xNTillxNQ columns and NED rows. Its nonzero coefficients are for K c
%
,
E c aK, E ct
rs
:
-'.
B ~ , ( ~ ,=~ )
The matrix Is is a diagonal matrix with dimension NED and its nonzero E c rs : coefficients are for E c
z,
The structures of the matrices A Q * , DIV*, B, I
are illustrated i n figure 3 .
K
El
E*
E3
1
1
1
X
x
K['
X
1 X
X
X
X
x
x
1
1
]
The matrix DIV*
X
X
x
The matrix AQ*
K
D F E
K' * D .
ryv--
-1
D C KnK'
E c r-rs
_-
F
I I I
0
0
1
I
An example OP the matrices
F
The matrix Is
The matrix B
:
E
-
FcrS
Figure 3
:i D
AQ*,
DIV*, B, Is in case of triangles.
Ch. V: A Finite Element Method for Incompressible no-Phase How
328
For the right-hand side, the vector FQ* has dimension 3xNT+QxNQ and its components are, for K c
%,
E c aK
:
The components for the NED-dimensional vector FA are
FAE
=
:
0
if
E is an interior edge of if E c ril
P:~
if
E c
rs
-q:h
if
E c
re.
Linear system (3.2) is larger than (2.5) solve. Since AQ*
but it is easier to
is a bloc-diagonal, symmetric, definite positive matrix,
one can easily eliminate Q* : (3.3)
Q*
=
A Q * - ~ [FQ*
+
t ~ ~ +~ t~.nl * . ~
and (3.2) becomes :
Then it
is easy to check that DIV*-AQ*-l-tDIV* is a diagonal
matrix of dimension NEL. Hence it is easy to eliminate P in (3.4)
and we obtain, from (3.4)
Linear system (3.6)
:
is a sparse symmetric system of dimension NED. Its
nonzero coefficients are those connecting two edges which are faces of same element. From proposition 2 we know that it has a unique solution. Moreover, we have the following proposition.
III. Resolution of the Algebraic System for hessure-VeIocity
Proposition 3
329
:
The matrix
is positive definite. Proof : -
For any A , we set
:
so that we have
But equality (3.7) implies
DIV*.AQ*-’.(~DIV*P
+
t ~ =~ 0, )
hence (AQ*-’
. t ~ ~ t~ *~ ~~ ,+~ (AQ*-’ * ~ . t) ~ A ,t ~ ~ =~ 0 .* ~ )
Plugging this equality in ( 3 . 8 ) , we obtain :
is positive definite and I
Since AQ*-’
is semi positive definite, the
matrix R is semi positive definite and ( R A , A ) = 0 implies t tDIV*P + BA = 0. (IsA, A ) = 0 and The first equality implies AE implies pK
=
AE
Y K c
%, 4
0, 4 E E
=
E
c
g,E
c
g ,E c rs,
and the second equality
a ~ .
Therefore P=O, and A=O, and R is positive definite. 0
Ch. V: A Finite Element Method for Incompressible live-Phase Now
330
Thus equation (3.6)
is easy to solve. One can use for instance
conjugate gradient methods with preconditionning. To conclude, we give the steps for the calculation of the pressure and the velocity
:
1. Solve the linear system (3.6) which is symmetric positive definite and
has dimension the number of edges. This gives values of the pressure on the edges. 2. Solve the diagonal linear system (3.5) with dimension the number of
elements. This gives values of the pressure inside each element.
3. Solve the block-diagonal symmetric positive definite linear system (3.3). This gives the total (oil+water) flow rate across the edges.
Remark 4
:
The introduction of A can be seen only as a trick to solve linear system (2.5) gives also more
for pressure and velocity. However it
information about the pressure, since it
calculates it on the edges, and this information is more
is calculated more accurately than P ; cf.
accurate as A
ARNOLD-BREZZI
.
IV - A P P R O X I U A T I O N O F T H E O N E - D I M E N S I O N A L EQUATION
:THE C A S E W I T H N E I T H E R P R E S S U R E NOR
IV.1
-
SATURATION
CAPILLARY
GRAVITY
INTRODUCTION
Now
we
turn
our
attention to
the saturation equations. For
simplicity, we first consider the one-dimensional case which can be viewed as modelling experiments in a core sample. In this section, we separate the difficulties by neglecting capillary pressure and gravity effects. Thus the saturation equation reduces to
:
W.Approximation of the One-Dimensional Saturation Equation
where the interval ]a,b[ equation (l.l.a),
33 1
represents a core sample. From the pressure
we see that qo is a constant, which we assume is given by
a boundary condition and is nonzero. The
fractional flow bo
is an
increasing function of S and
therefore the direction of the water flow is given by the sign of qo. If qo is positive (resp. qo
< O), the fluids are moving from the upstream boundary
x=a (resp. x=b) to the downstream boundary x=b (resp. x=a). condition is necessary only at the injection boundary S(a,t)
Sa
=
=
1
(4.2)
S(b,t)
=
Sb = 1
A
boundary
:
if qo > 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.
Ch. V: A Finite Element Method for Incompressible no-Phase Flow
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
=
value of
sh
on
:
X~+~,~C,
X. 0. = 1
h. 1
J l C 1 I 2Q dx. x.
1-1 1 2
Assuming for instance that q
0
is positive, from (4.5) we have
$lh(xi+l,2)=$l(Si-l), i=l ,..,I so that (4.7) can be rewritten
Thus, for k=O, the scheme (4.7) reduces to the standard one point upstream weighted finite difference scheme. It is well known that its accuracy is O(At+h) and it is stable for
(4.10)
It produces neither overshoots nor spurious oscillations. However, its numerical diffusion is large, and we are going to design a scheme with less numerical
diffusion
and
more
accuracy
by
increasing
the
degree
of
approximation. IV.4
-
THE CASE k=l :PIECEWISE LINEAR APPROXIMATION
The saturation will now be approximated
in MI, the space of
-
discontinuous piecewise linear functions. We introduce the functions vi+1/2, V. :-,,2,
i=l,...,I which form a basis of the vector space M':
Ch. V: A Finite Element Method for Incompressible 1suo-Phase Flow
336
( ~ - x ~ - , / ~ ) / hif ~
if
=to
x
+
1-1
12
if
Ki=
[ X ~ - ~ , ~ , X ~1+ , / ~ , ,
x d Ki,
( ~ ~ + ~ / ~ - x ) /ifh ~ x
V.
E
E
Ki.
x Z Ki.
Also we shall use the following notation
:
=
sh (xi+1/2)
, s ; , ~ , ~= ~,,(xfi+~/~)
f o r i=o,...,~,
0
Figure 6
:
The basis functions of M' and the degrees OP Preedom of Sh.
IV. Approximation of the One-Dimensional Saturation Equation
Taking f o r v q0LO
i n (4.7)
337
t h e b a s i s f u n c t i o n of M'.
we o b t a i n i f
:
To c a l c u l a t e t h e
integral arising in (4.11),
a good choice of
i n t e g r a t i o n formula is one t h a t is exact f o r polynomials of degree 2 ( s e e COHEN [ l ] ) ,
f o r i n s t a n c e , Simpson's formula
Analysis of nonlinear (4.11)
case.
the
However
becomes, f o r i
=
which can be r e w r i t t e n
1,
scheme ( 4 . 1 1 )
:
has not
i n t h e l i n e a r case with
...,I
:
been completed i n t h e constant coefficients,
Ch. V: A Finite Element Method for Incompressible no-Phase Flow
338
I n LESAINT-RAVIART, solution,
L2
the
error
it was proved t h a t , f o r a r e g u l a r c o n t i n u o u s
for
discretization
the
in
space
is
O(h2).
CHAVENT-COCKBURN have s t u d i e d t h e c o n s i s t e n c y of the scheme and showed t h a t 1 t h e t r u n c a t i o n e r r o r f o r t h e a v e r a g e S i = 2 (Si-1,2+S:+1/2) of t h e
s a t u r a t i o n o v e r the i n t e r v a l s is O ( A t + h 2 ) . These r e s u l t s confirm t h a t t h e scheme we have d e s i g n e d is
d
'nigher o r d e r scheme.
Also i n CHAVENT-COCKBURN.
s t a b i l i t y of t h e scheme ( 4 . 1 2 ) has been
s t u d i e d w i t h Von Neumann t e c h n i c s .
" 6 h3/2
condition
However stability,
I t is shown t h a t t h e scheme (4.12)
s t a b l e f o r a f i n i t e time i n t e r v a l l O , T [ ,
L"(O,T;L'(a,b))
one
is
provided that t h e
C is s a t i s f i e d .
it
should
been
has
use
observed
in
practice
that
for
better
trapezoydal r u l e t o calculate the f i r s t
the
i n t e g r a l i n ( 4 . 7 ) . Then the scheme r e a d s
:
F i g u r e 7 shows comparisons between the s t a n d a r d one p o i n t upstream weighted scheme ( 4 . 9 ) ( c u r v e ( 1 ) ) and t h e p i e c e w i s e l i n e a r upstream weighted u s i n g Simpson's formula t o c a l c u l a t e t h e i n t e g r a l i n ( 4 . 1 3 )
scheme (4.13)
( c u r v e ( 2 ) ) . The d a t a are t h e same as i n f i g u r e 4 and t h e p i c t u r e is drawn for t
=
0.3.
One can o b s e r v e t h a t scheme (4.13) g i v e s a s h a r p f r o n t b u t i t
produces o v e r s h o o t s and o s c i l l a t i o n s . A l s o t h e c a l c u l a t e d f r o n t is l a g g i n g behind t h e f r o n t of
t h e e x a c t s o l u t i o n , which shows t h a t t h e scheme is
a n t i d i f f u s i v e . F i n a l l y scheme (4.13) is less s t a b l e t h a n scheme ( 4 . 9 ) .
IV.5
-
A SLOPE LIMITER
Slope overshoots
and
limiters
have
oscillations
been in
introduced
accurate
finite
by
VAN
LEER
to
prevent
d i f f e r e n c e methods
for
h y p e r b o l i c e q u a t i o n s . We s h a l l f i r s t a d a p t one of t h e s e s l o p e limiters t o
339
Iv.Approximation of the One-Dimensionai Saturation Equation
discontinous finite elements methods and then we shall modify it. st time level We denote now by SE, the saturation at the (n+l) and we shall calculate from Sh* a new saturation which will be slope limited.
calculated with (4.13),
V
Sn+! ha
h
1
1
1
.
1
,
#
1
0
Figure 7 :
Saturations calculated with ( 1 ) finite differences (k=O).
1
,
I1
one-point upstream weighted
(2) upstream weighted piecewise
linears without a slope limiter, and ( 3 ) with a first slope and compared to (4) the exact solution. In
limiter ( e = 1 ) .
each case. the largest time step, preserving stability, has been used.
We introduce the average values of S;
on the intervals
Assuming that the flow is moving from left to right ( q o > O ) . boundary we set for S i the boundary data (4.2) S.
i
=
S
a
= 1,
at the injection
:
i=O
and at the production boundary
,
i=I+l.
Ch. V: A Finite Element Method for Incompressible Tbo-Phse Flow
340
Since there is no boundary condition at the production boundary, the value set for SI+l is chosen so that it has no effect during the process of slope limiting.
. calculated to be “as closetTas possible The new saturation Sn+l h is to s: i=l
with respect to the L’-norm, to have the same average values S. 1’ as s,: so that the modified scheme remains conservative, and to
,...,I
satisfy the inequalities : +
Min(Si-l, S . ) 5 (Si-1,2
)n+l
2 Max(Si-l ,’i
)n+l 5 Max (Si,Si+ ) , i=l Min(S., Si+l)5 (Si+1/2
,...,I.
These inequalites are limiting the slopes of S r ’ on each intervals. Precisely S r ’ is defined by (4.14)
(4.15)
sn+ 1
E
:
M‘, - ~ ; I I ~ = ~ vh~;nM, ( ~ ~ )
IIS;+’
i=1$-..#1~
satisfying ( 4 . 1 6 ) , (4.17 ) (4.16)
(4.17)
v.1
=
1 (V.-1 + 1 / 2 2
+
v;-1/2)
=
Si’
i=l,...,I,
v1-1/2 5 Max (Si-l,si)I Min (Si, S i + l ) 2 V .1+1/2 2 Max (Si, Si+l),i=1,
Min (Si-l,S 1. )
=
z‘
...,I.
Thus Sn+‘ h is derived from S*h by solving a series of two-dimensional minimization problems. On each interal K i t the pair ( (S+-l /2)n, ( S-i + l / 2 )“I
is the projection in lRz of ( ( S i - l , 2 ) * , ,(Xl+X2) 1 = si and the line
(Si-1/2)*)on the intersection of
the
rectangle
defined
by
Min(Si-l, S. 1 ) 5 x1 5 Max(Si-l, S . 1, Min(Si, Si+l)5 x2 5 Max(Si, Si+l). Examples of such slope limitations are presented on figure 8 . In one dimension, the calculation of Sn” from S: 1
considering all the situations which can occur.
can actually be programmed by
341
IV. Approximation of the One-Dimensional Saturation Equation
-
_ _ - -saturation before
slope limitdng
saturation after s l o p e limiting
Figure 8
: Slope
limiting the saturation.
Condition (4.16) guarantees mass conservation for the modified scheme and, since the average values on the intervals are preserved, the n+ 1 given S* is a local and thus inexpensive procedure. A l s o calculation of Sh h we remark that this slope limiting proceedure acts only around the fronts where the saturation tends to oscillate, but not in the area where the saturation is smooth. Figure 7 shows the comparison between the saturations calculated with and without the slope limiter ( 4 . 1 4 ) .
...,(4.17).
Curves ( l ) , ( 2 ) and
(3) have been calculated with the same spatial discretization. Introducing the slope limiter has suppressed overshoots and oscillations, and while the numerical diffusion has increased, it is still much smaller than for curve (1).
It is important to note that the stability has also increased
:
fewer
time steps were needed to calculate curve (3) than curve ( 2 ) . The stability condition for scheme (4.13) with the slope limiter (4.14),. ..(4.17)
is
However one can observe that, like without slope limiting, the scheme is still antidiffusive
:
in the smooth region behind the front the
calculated solution is above the exact solution, and the calculated front is lagging behind the exact front.
342
Ch. V: A Finite Element Method for Incompressible nvo-Phase How
This (4.14),
defect can be corrected by modifying the slope limiter
...(4.17)
in such a way that it adds more numerical diffusion in the
scheme. This can be achieved by reducing in ( 4 . 1 7 ) the range of permissible + )n+l n+ 1 values for (Si-1,2 and (Si+1,2) I which reduces the slopes of the linears of the slope limited satiut-aLion. Precisely we replace (4.17) by :
(4.19)
I
(l-B)Si+eMin(Si-l ,Si) SVl-,,2 5 (1-e) Si+B Max (SiS1,Si),
(l-e)si+e Min(Si,~i+l)< v ; + ~ , ~I (1-e) si+eMax
( s ~ , s ~ + ~i=l,.., ), I,
Figure 9 : Saturation calculated with the slope limiter (4.14),
(4.19)
with e=o (equivalent to k = O ) ,
((4.19)
equivalent to (4.1511,
...,(4.16),
with e=0.4.
with e=1
and compared with the exact
solution. Figure 9 shows the results obtained with 8=1, 8=0.4 compared to the exact solution. A l l
and B = O ,
curves are calculated for the same
discretization in space and with the largest At satisfying (4.18). 6 = 1, (4.19) becomes (4.17),
antidiffusive. For a=O, (4.19)
becomes V:-1,2
point
upstream weighted
=
When
the slope limiter is loose and the scheme is
the slope limiter is the strictest possible since Vi+1,2 = si; then the scheme reduces to the one finite difference scheme
(k=O) which
is
too
diffusive. By moving e from 0 to 1, we obtain a family of schemes whose
343
IV. Approximation of the One-Dimensional Saturation Equation
numerical
diffusion
is
intermediate
scheme
obtained
between
d i f f u s i o n of
the
d i f f u s i o n of
t h e scheme o b t a i n e d w i t h 8=1.
changing 8 w i t h t h e s t e p of 0.1,
with
O=O,
the
too
large
numerical
and t h e n e g a t i v e
numerical
Experiments show t h a t ,
when
t h e b e s t answer is o b t a i n e d f o r 8=0.4.
About s t a b i l i t y , s i n c e f o r 8=0 t h e s t a b i l i t y c o n d i t i o n is ( 4 . 1 0 ) and f o r 8=1 i t is ( 4 . 1 8 ) , one g u e s s e s t h a t t h e s t a b i l i t y c o n d i t i o n is i n t h e g e n e r a l case : V+Sh(X)) (4.20) x
Max 6
O(x)
l a , bC
At
h
4- 1
1 4 i L 1 ,
'
1+8
which is confirmed by experiments. f o r a given h,
One s h o u l d a l s o n o t e t h a t , depends
on
the
time s t e p A t ,
diffusion decreases,
s i n c e when
s o t h a t t h e b e s t v a l u e of
that, with t h e l a r g e s t A t s a t i s f y i n g (4.20),
t h e best value of 8
increases,
At 8'
the
numerical
d e c r e a s e s . Thus we found
t h e best r e s u l t s a r e obtained
f o r 8=O.3 ( f o r 0=0.4 t h e scheme h a s become s l i g h t l y a n t i d i f f u s i v e i n t h e s e The c a l c u l a t e d s o l u t i o n is shown on f i g u r e 10 where it is
conditions). compared w i t h calculated largest
t h e one p o i n t upstream weighted f i n i t e d i f f e r e n c e S o l u t i o n t w i c e as many
with
intervals
time s t e p s a t i s f y i n g (4.10)
i n space.
h a s been used.
r e f i n e d d i s c r e t i z a t i o n s i n s p a c e and
i n time f o r
For
the latter,
the
I n s p i t e o f t h e more t h e f i n i t e difference
s o l u t i o n , t h e a c c u r a c y is r o u g h l y t h e same f o r t h e two c a l c u l a t e d s o l u t i o n s . T h i s comparison shows what we g a i n e d by d e s i g n i n g scheme ( 4 . 1 2 ) , ( 4 . 1 4 ) , (4.16),
(4.19)
:
...,
t h e same a c c u r a c y is o b t a i n e d when u s i n g l a r g e r i n t e r v a l s
i n s p a c e and l a r g e r time s t e p s . I n COCKBURN-JAFFRE, the
slope
limiter
(4.141,
i t is proved t h a t t h e scheme (4.71, ( 4 . 8 ) with
...( 4 . 1 6 ) ,
(4.19)
f o r any 8, 02821, is t o t a l
v a r i a t i o n d i m i n i s h i n g and t h a t t h e approximate s o l u t i o n c o n v e r g e s t o t h e weak s o l u t i o n of ( 4 . 1 1.
The convergence t o t h e e n t r o p y s a t i s f y i n g s o l u t i o n ,
though demonstrated n u m e r i c a l l y , h a s not been proved ma t h e m a t i c a l l y .
Remark 5
:
I t is n o t d i f f i c u l t t o f o r m u l a t e a f i n i t e d i f f e r e n c e v e r s i o n o f scheme ( 4 . 1 2 ) ,
(4.14)
,...,( 4 . 1 6 ) ,
d i f f e r e n c e approximation S y , through t h e f o l l o w i n g s t e p s :
Given a f i n i t e n+ 1 l S i S I , we o b t a i n Si , 1LiS1, (4.19).
Ch. V: A Finite Element Method for Incompressible nYo-Phase Flow
344
1 - Calculate the slopes py, 1SiSI by :
pn
=
0 if S n 2 Min ( S r - l ,
pn
=
28 - Min (S.-S. 1-1, Si+l-Si) if hi
Si-l
=
28 Max h.
Si-l 2 Si 2 Si + l '
1
ri
p. 1
and
approximation
2
s ni
or S r 2 Max ( S r - l I S y + l ) ,
(S.-Si-l, S i + l - S i ) if
define
=
Sn l+l
+
the
discontinuous
S Si S S i r l ,
piecewise
linear
:
n Pi (x
-
Xi+1/2+xi-1/2 2
).
n+ 1 Calculate Si by
-
1 SiSI.
can be found in VAN LEER.
Such schemes with 8=1/2 and 8=1
-
(2): h=1/50, A t =1/104
-/
3
I * I
-,
01 0
*is-
I
I
I
Figure 10 : Comparison
(4.14)
l
I
of
I
I
saturations
,...,(4.161,
(4.19)
I
I
I
calculated with
e=0.3
upstream weighted finite differences on
~
~
(1
by
and
by
~
~
1
~
(4.12).
scheme (2) one
point
a more refined mesh.
with ( 3 ) the exact solution. The largest time step possible have been used.
~
~
V. The One-DimensionalEquation in the General Case
v -
A P P R O X I ~ A T I O NO F
SATURATION
V.l
-
EQUATION
345
THE IN
ONE-DIMENSIONAL
THE
GENERAL
CASE
THE GRAVITY EFFECTS With gravity effects included, the saturation equation reads on la,bC
(5.1 .a) with (5.1 .b)
@l(x,s)
=
qo bo(S)
+
x
:
lO,TC,
ql(x) bl(S).
In this case $l is no longer a monotone function of the saturation (see figure 3 of chapter 111) as the flow of water can now change direction in the interval ]a,b[.
There are places where the notion of upstream and
downstream are no longer meaningful, namely all points where the saturation makes the derivative of 4, vanish. Therefore we must give a new definition of the approximate flow rate $,h at points of discretization, which is consistant with the previous one stated in (“1.5).
This definition is based
on ideas of GODUNOV who designed a finite difference scheme for first order
nonlinear hyperbolic equations. To equation (5.11, we add the following boundary conditions (see
LEROUX). Given S (t) and Sb(t), we write $,(S(i,t)) I(i,t)
=
:
c(i,t) Min {E(i,t) a l ( k ) } , kcI( i,t)
=
i
=
[Min(S.(t), S(i,t)), Max(Si(t), S(i,t))],
a,b, i=a,b,
(5.2)
~ ( a , t )= sign of S(a,t) - Sa(t), E(b,t)
We extremity
=
sign of Sb(t) - S(b,t).
recall that boundary condition (5.2) means that, at each x=i, i=a,b, either S is continuous i.e. S(i,t)=S.(t),
discontinuous i.e. S(i,t)
t
or S is
Si(t) and is such that for every k in I(i,t) the
346
Ch. V: A Finite Element Method for Incompressible Two-Phase Flow
Rankine-Hugoniot velocity of propagation of the discontinuity between S(i,t) Actually (5.2)
and k points outward from the interval ]a,b[.
is the entropy
condition written at the extremities of la,b[. Now we turn to the definition of the discretized water flow rate For that purpose, we solve at the point the
$:h(xi+l/2),i=0,..,I. Riemann problem
:
-au+ - a at ax
$1 =
x
O*
E
IR, t
Itn, TC,
<
if
x
if
x > x.
1+1/2'
n with ( S l I 2 ) = Sa(tn), (S;+1/2)n = S b (t"). Then we take as @yh(xi+l/2) the value of the water flow rate for the entropy solution u at x=x. and 1+1/2 t=t"+ :
As shown by LEROUX, this value can be easily calculated exactly by solving the minimization problem:
and this is what will be done in practice. It
is
not
difficult
to
check
that
when
the
function
+@l(xi+1/2,S) is monotone then (5.3) reduces to the calculation of the water flow rate with the upstream saturation : S
@fh(xi+1/2)
=
@ , ( x ~ + , / ~ , ( S ~ + ~ / ~if ) ~S)
$l(xi+,,2,s)
is increasing on the
$1 (xi+l/ 2 , s )
is decreasing on the
-f
interval I ( X ~ +/2), ~ $yh(Xi+l / 2 ) = $ 1 (Xi+l/2, ( S I + l /2)n)
interval I ( X ~ + / 2~) .
if S
-f
K The One-DimensionalEquation in the General Gzse
I
Figure 1 1
:
:
341
II
An example of calculation of the approximated water flow #lh(xi+l,2) with formula (5.3).
Thus the definition (5.3) is consistant with the definition (4.5). Remark 6
:
Definition (5.3) for the numerical water flow rate is not the only possible choice. Instead of Godunov flux, one could use ENGQUIST-OSHER flux, or the upstream weighted flux of finite difference methods used in reservoir simulation (see BRENIER-JAFFRE).
V.2
-
THE CAF'ILLARY PRESSURE EFFECTS
Now the saturation equation is n the one-dimens m a 1 case on la,bCxlO,T[,
:
Ch. V: A Finite Element Method for Incompressible 1Evo-PhaseFlow
348
2 (5.4.c)
f(x,S)
1
=
j=O
(5.4.d)
r(x)
-K(x
=
The f u n c t i o n f is t h e c o n v e c t i v e p a r t o f t h e w a t e r f l o w r a t e
$,.
I t s a p p r o x i m a t i o n h a s b e e n d e s c r i b e d i n t h e p r e v i o u s s e c t i o n s . The f u n c t i o n
r
is t h e p a r t of 0,
d u e t o c a p i l l a r y d i f f u s i o n and t h e d a t a f u n c t i o n
CY
d e p e n d s on t h e c a p i l l a r y p r e s s u r e a s d e s c r i b e d i n c h a p t e r I11 s e c t i o n 1 1 . 1 . Plugging
(5.4.b),
(5.4.c),
(5.4.d)
in
we
(5.4.a)
see
that
capillary
d i f f u s i o n p r o d u c e s s e c o n d order terms i n t h e s a t u r a t i o n e q u a t i o n . c a s e of
Consider t h e water,
where x
displacement of
o i l by t h e i n j e c t i o n o f
a is t h e i n j e c t i o n boundary and x
=
=
b t h e production
boundary. We assume t h a t o n t h e i n j e c t i o n boundary t h e w a t e r s a t u r a t i o n is maximum a n d t h a t o n t h e p r o d u c t i o n boundary t h e w a t e r a n d o i l f l o w r a t e s a r e p r o p o r t i o n a l t o t h e m o b i l i t i e s ( s e e c h a p t e r 111, 311.3.2.3)
:
2 Sa(t)
(5.5)
,
1
=
r(b,t)
+
1
b.(Sb(t)) q.(b) J J
= 0,
t
c 10,Tc.
j =1
- Approximation spaces
V.2.1
S i n c e S is a p p r o x i m a t e d by d i s c o n t i n u o u s p i e c e w i s e p o l y n o m i a l s , a g a i n mixed f i n i t e e l e m e n t s p r o v i d e a s u i t a b l e method f o r t h e a p p r o x i m a t i o n o f c a p i l l a r y d i f f u s i o n terms. In
the
r
is
polynomials
of
function
discretization rh
E
one-dimensional the
vector
degree
E. If
Xo, t h e
k+l
on
case, Xk
the of
each
approximation
continuous interval
S were a p p r o x i m a t e d i n M’,
s p a c e of
a p p r o x i m a t e d i n M’
space
continuous
piecewise
space f o r
functions
Ki,
i-1
which
,...,I
Of
the are the
r would b e a p p r o x i m a t e d by linears.
But
s i n c e S is
, t h e space of discontinuous piecewise l i n e a r functions,
t h e c o r r e s p o n d i n g a p p r o x i m a t i o n s p a c e f o r r i n t h e mixed f i n i t e e l e m e n t method is X ’ , V.2.2
t h e s p a c e of continuous piecewise q u a d r a t i c functions.
- Approximation equations I n order t o write down t h e a p p r o x i m a t i o n e q u a t i o n s we p l u g (5.4.b)
i n t o (5.4.a),
we r e p l a c e S and r by t h e a p p r o x i m a t e f u n c t i o n s ShQ M’ and
V. The One-Dimensional Equation in the General Gzse
rh
E
349
we multiply by test functions in M' and integrate. The convective
X',
terms are approximated as described in the previous sections.
T
rh I I I
I
I I
1
1
I I I
I
I
I
I
I I
I
1
Figure 12 : Approximation of r in X D and in X ' .
If we integrate the capillary terms by parts, the approximation equations of (5.4.a),
(5.4.b), (5.4.~) become
where ~ ~ ( x ~ and + ~ I/" (~X )~ + ~ / are ~ ) as in (5.3) and To approximate (5.4.d)
STl2
=
Sl, S;+1/2 =
s;.
and calculate rh, we replace r and S in
(5.4.d) by their approximations rh and S and we multiply by test functions h'
Ch. V: A Finite Element Method for Incompressible i%o-Phase Flow
350
SEX'
such t h a t s ( b ) = O . T h i s l a s t c o n d i t i o n is due t o t h e boundary c o n d i t i o n
(5.5) r i g h t . Then we i n t e g r a t e o v e r l a , b [ and i n t e g r a t e by p a r t s . Thus we obtain, using (5.5) l e f t ,
To approximate ( 5 . 5 ) r i g h t , we write (5.7) where E n is a v a l u e o f I n ( b ) where t h e minimum i n ( 5 . 6 . c )
is r e a c h e d .
Because Sb is a n unknown, e q u a t i o n ( 5 . 6 ) and ( 5 . 7 ) a r e coupled i n a n o n l i n e a r way when g r a v i t y e f f e c t s a r e t a k e n i n t o a c c o u n t . An e a s y way t o
decouple t h e s e e q u a t i o n s is t o approximate ( 5 . 5 ) r i g h t by s u b s t i t u t i n g f o r gn
:
(5.8) For n=O we s i m p l y take E,
-1
=g
0
. , we f i r s t s o l v e t h e l i n e a r system ( 5 . 8 ) ,
Thus, t o c a l c u l a t e S:+' (5.6.d)
with
test
functions
vanishing a t
c a l c u l a t e SE by s o l v i n g ( 5 . 6 . 6 )
x=b, which g i v e s u s r n . we h' u s i n g t e s t f u n c t i o n s s such t h a t s ( b ) = l ,
which amounts t o s o l v i n g one e q u a t i o n with one unkmwn. c a l c u l a t e t h e approximate water flow r a t e $7h u s i n g (5.6.b)
Second, we can a f t e r having
s o l v e d t h e s i m p l e m i n i m i z a t i o n problems i n o n e v a r i a b l e ( 5 . 6 . ~ ) . T h i r d , we n+ 1 F i n a l l y we s o l v e t h e s e r i e s o f I 2 x 2 l i n e a r systems ( 5 . 6 . a ) t o o b t a i n Sh
.
a p p l y t h e s l o p e limiter d e s c r i b e d i n IV.5.
V.2.3
-
The a l g e b r a i c l i n e a r s y s t e m t o c a l c u l a t e rh 'i+1/2+'i-1/2 t h e mid-point of t h e i n t e r v a l K i , 2 convenient basis of X' is t h e s e t of functions
Denote by i = 1 , . ..,I.
[
Si+1/2,
A
i=O
x.
=
,...,I ] u (
si,
i = l ,..., I ] l y i n g i n X ' ,
q u a d r a t i c s and s u c h t h a t supp s i + 1 / 2= si+l/2(xi+1/2) suppsi
=
=
1,
si+l,2(xi)
[xi-1/2,xi+l/2],
=
si+,/2'
which a r e p i e c e w i s e n
(Xi+l)
=
[a,b],
0,
s i ~ x i ~ = ~ , s i ~ x i - 1 ~ 2 ~ == so ,i ~i =xl i, ..., + l I~. 2 ~
V. The One-Dimensional Equation in the General Case
351
We shall use the following notation r.1+1,2 am R
=
r.=r ( x 1, i=l... i h i
rh(xi+l/2), i=O,...,I
=I
1
sR
p
Sm
:
with L, m
dx
E
{i, i-l/Z
b K'PCM
m 1+1/2 a ~ + =~ am/ ~
1+1/2 6m
(6
.I. i=l ,
...,I} ,
Kronecker symbol),
Figure 13 : Basis function of X'.
In (5.6.d), we take for test functions s the basis functions of X ' , s e , &*I+1/2, and we express r in terms of the sR so that (5.6.d) with s ( b ) = O , h together with (5.8) reduces to the following positive definitive symmetric linear system of dimension 21+1 and five non zero diagonals :
Ch. V: A Finite Element Method for Incompressible nYo-Phse Flow
352
T h i s p o s i t i v e d e f i n i t e system w i l l b e s o l v e d a t e a c h time s t e p t o c a l c u l a t e
r t , b u t t h e matrix d o e s n o t depend on t h e time s t e p . Moreover it is e a s y t o eliminate r .
1'
positive
i=l,...,I
i n t h e system s o t h a t t h e s y s t e m is reduced t o a
d e f i n i t e symmetric
system o f
dimension 1 + 1 with t h r e e nonzero
diagonals.
V.2.4
-
Calculating r. i n Xo instead of X ' A s we saw i n s e c t i o n I V , we c o n s t r u c t e d a more a c c u r a t e scheme f o r
convective
terms
approximated,
by
i.e.
increasing t h e
by t a k i n g Sh i n
index
of
s p a c e i n which S is
the
S i n c e we used mixed
instead of Mo.
MI
f i n i t e e l e m e n t s i n t h e s t a n d a r d way, t h i s o b l i g e d u s t o i n c r e a s e a l s o t h e a c c u r a c y i n t h e a p p r o x i m a t i o n o f t h e d i f f u s i o n terms by t a k i n g r h i n XI i n s t e a d o f t a k i n g rh i n X 0 a s we would have done i f S had been approximated h i n MO. Actually,
because,
for
the
same
degree of
approximation,
the
a c c u r a c y is lesser f o r t h e a p p r o x i m a t i o n o f c o n v e c t i v e terms t h a n f o r t h e approximation accuracy f o r
of
d i f f u s i o n terms,
these,
it
is n o t n e c e s s a r y t o have such a n
e s p e c i a l l y when d e a l i n g w i t h problems w i t h a
amount o f c o n v e c t i o n . T h e r e f o r e ,
and c o n v e c t i v e terms, i t is s u f f i c i e n t t o c a l c u l a t e r h
k e e p Sh i n M ' .
large
t o balance the accuracy i n the diffusion E
Xo,
even though we
T h i s remark is a c t u a l l y more i m p o r t a n t i n t h e two-dimensional
c a s e where X 0 h a s more t h a n t w i c e a s few d e g r e e s o f freedom t h a n X', s o t h a t t h e computation time saved i n t h i s way is s i g n i f i c a n t . (4.81,
(5.9)
Thus we c a l c u l a t e S E M I , rh h (5.6.a) ( 5 . 6 . c ) , ( 5 . 8 ) and
E
X o and S:
a r e a l number such t h a t
,...,
32
b
1
as
rt*s a(Sn) h ax a K * P ~ ~ a
are s a t i s f i e d .
+
a ( 1 ) s(a)-a(S:)
s(b) = 0 ,
s < XO,
V. The One-Dimensional Equation in the General Case
V.2.5
-
The algebraic linear s y s t e m t o c a l c u l a t e '&r A
i=O,
353
convenient
...,I] l y i n g
basis
for
X0
is
the
set
of
i n X o and s a t i s f y i n g
Figure 14 :
Basis functions of X o .
We i n t r o d u c e t h e f o l l o w i n g n o t a t i o n , r.1 + 1 / 2
=
rh(xi+l/2)'
i=O,
...,I ,
functions
{s~+,/~,
Ch. V: A Finite Element Method for Incompressible nvo-phase Flow
354
To c a l c u l a t e rh, we u s e ( 5 . 9 ) w i t h t e s t f u n c t i o n s s i n t h e basis sL + l / 2 , %*I, and we e x p r e s s r h i n terms of these
of X" a n d v a n i s h i n g a t x-b.
s o t h a t (5.8) and (5.9) produce t h e f o l l o w i n g symmetric l i n e a r
functions,
system of dimension 1 + 1 with t h r e e non z e r o d i a g o n a l s \
\
\
\
\ \ \
:
\
\
a i-1 .
\
\
i
a.
1\
\ai+l I\
\ 1 \
\
\
\
\
\
\
\
S i n c e we t a k e rh
E
\
X" t o b u i l d a scheme which a c t s , i n t h e a b s e n c e
of c o n v e c t i v e terms (qoEq133), as i f t h e s a t u r a t i o n had been approximated i n all
M",
integrals
involving
a
should b e c a l c u l a t e d w i t h
t h e one-point
i n t e g r a t i o n formula X. l 1 + l I 2f dx
=
(x.1+1/2-xi-1/2)
f(xi).
X.
1-1 / 2
Then,
i n t h e absence o f c o n v e c t i v e terms, t h e c a l c u ited s a
p i ecew i se c o n s t a n t
Concerning t h e m a t r i x c o e f f i c i e n t a;, Simpson's r u l e ,
Ira
3n Sh
S
. i f they a r e c a l c u l a t e d w i t h
o n e c a n o b s e r v e t h a t t h e s a t u r a t i o n Sh can t a k e v a l u e s
o u t s i d e t h e i n t e r v a l [ O , l ] . T h i s d e f e c t c a n b e e a s i l y c o r r e c t e d by u s i n g t h e t r a p e z o i d a l r u l e i n s t e a d . With t h e use of t h e t r a p e z o i d a l r u l e , t h e mixed method
for
c a l c u l a t i n g rh becomes
the
block-centered
f i n i t e difference
method a s a l r e a d y commented i n remark 2 f o r t h e p r e s s u r e e q u a t i o n .
V.2.6
- Time
stepping
It is well know t h a t , f o r e x p l i c i t schemes as d e s c r i b e d above, t h e At p 6 c o n s t a n t and is t h e r e f o r e At more r e s t r i c t i v e t h a n t h e o n e due t o c o n v e c t i o n terms (?;S c o n s t a n t ) .
s t a b i l i t y c o n d i t i o n due t o d i f f u s i o n terms is To
get
rid
of
this
s t r o n g s t a b i l i t y requirement,
c a l c u l a t e t h e d i f f u s i o n terms a t time ( n + l ) , i . e . 1
i n e q u a t i o n (5.6.6)
or (5.9).
one
should
one s h o u l d r e p l a c e S:
by
Doing so c o u p l e s t h e s e e q u a t i o n s w i t h
( 5 . 6 . a ) i n a n o n l i n e a r system, and more work h a s t o be done t o f i n d t h e b e s t way t o s o l v e t h i s n o n l i n e a r system.
VI. The Saturation Equation in lWo Dimensions
-
VI
OF
APPROXIUATION IN
355
TWO
THE
SATURATION
EQUATION
DIMENSIONS
In this section, we show how we can derive in a straightforward way a two-dimensional scheme for the saturation equation using the methods we described in the one-dimensional case.
-
VI.1
APPROXIUATIOB SPACES
The
saturation
is
approximated
by
Sh
in
the space M'
of
% , v IK) is a polynomial defined by its values at the vertices and linear on the
discontinuous functions v whose restriction to each element K of
edges of K. Since v is discontinuous, for one vertex A of the mesh, there are as many degrees of freedom as there are elements having A as a vertex. The number of degrees of freedom is thus 3xNT+4xNP where NT (resp.NP) is the number of triangles (resp. parallelograms) of basis function v
the discretization
%.
A
of M' vanishes outside K, is equal to 1 at the vertex A
K,A of K and is equal to 0 at the other vertices of K.
Degrees of freedom
Basis functions
vK,A' A a vertex of K, such that
v
E
M' A
K
a vertex of K,
E%.
A any vertex and K'any element of the discretization.
Figure 15 :
Degrees of freedom and basis functions of U'.
Ch. V: A Finite Element Method for Incompressible Tivo-Phase Flow
356
According to section V.2.4,
the function
r'
at nAt is approximated
x'"
by :n~ h
which has been described in 311.1. + + In two dimensions, the total velocity q is approximated by qh in
+
and calculated as described in sections I1 and 111. A l s o the vector
X"
+
fields q. j = 1 , 2 of gravity J' j=1,2 in approximated by Jh'
-
VI.2
and
heterogenous capillary pressure
;" as i n 3
G.
are
11.2.
APPROXIMATION EQUATIONS There is no difficulty in writting the analogues of equations
(5.6) and (5.9) to approximate equations (1.2) in the two-dimensional case. Thus we are calculating S t M' and r*h E ;", such that : h
where
Gyh
+n
+
is defined in f;, the interior of K, and @ l h - ~ v is defined on aK as
follows. R and any real number k, we set :
For any x (6.2)
Then we define, in the interior of K,
Since S t is discontinuous on the edges of the elements of B
S:
given K
E
% , we
on aK. If an edge of aK is included in
interior trace of S t on
r,
and S:
r,
Sy
on this edge will denote the
determined by the saturation boundary conditions on
s:
On aK
n
=
(r-re), S:
for
will represent an exterior trace
r.
To approximate the
Dirichlet saturation boundary condition given in (1.3) we set
(6.4)
%,
denote by Sfl and ST, the interior and exterior traces of
1
on aK n
re.
is an unknown.
VI. The Saturation Equation in mo Dimensions
357
+n + Taking into account boundary conditions (1.31, Olh*v is defined on aK as fOllOW3 :
This means that for each x direction orthogonal to
E
aK.
aK, x 6
rR,
we solve a Riemann problem in the
In practice we shall do this only at the
integration points of the edges. +n Concerning the calculation of rh, we approximate equation (1.2.d) with the boundary condition (1.3) by the analogues in two dimensions of +O (5.8) and (5.9). Thus we calculate E X and Sr, linear on the edges of
;:
rR
u
rs,
by
:
where gn is a value of In(x) where the minimum in (6.5) is reached. Again to decouple equations (6.51, (6.6),
gn-l in (6.7)
:
(6.8)
r'"
and we set 5-l
2
h
=
+
1
j=1 0
5
.
bj(C
n-1
) = O
on P R u
(6.7) we replace
r S'
En by
Ch. V: A Finite Element Method for Incompressible no-Phase Flow
358
n+ 1 Let us recapitulate what we have to do to calculate S h ' First we calculate as described in sections I1 and 111. Next we
:;
calculate
s'
E
X''
E
2'
G:
equations (6.8) and (6.6) with test functions
using + +
such that s - u
s'
functions
to calculate
K
X'' En
z , using
r n. cl
0 on
=
system of dimension NED. such that
Ts. This requires the solution of a linear
Then we calculate S:
s'
+
o
ril
using
rS. Actually
(6.6) with test
will serve only +n + for the next time step. Now we can calculate O l h - v on aK, -.J
f
on
u
S:
(6.5).
This requires the solution of simple minimization n+ 1 problems. Also we calculate inside K using ( 6 . 3 ) . Finally we obtain Sh E
Gyh
by solving equations (6.1).
They reduce to a series of 3 x 3 or 4 x 4 symmetric
linear systems according to whether K is a triangle or a parallelogram.
v1.3
- INTEGRATION FORHULAS In
(6.1)
and
(6.6), there are several integrals which need
numerical integration. The first integral in the right-hand side of (6.1)
can be easily
calculated exactly. aut, since in the one-dimensional case the trapezordal rule gives better results (see 81V.4),
one should use the integration
formula using the vertices as integration points, and this diagonalizes the matrix. For the integrals over the edges arising in (6.1) and (6.61,
the
two-point Gauss formulas is adequate since it is exact for polynomials of degree 3 . Concerning integrals over
the elements K
involving nonlinear
functions of the saturation, an adequate choice is shown in table 6.2. These are the least expensive formulas, exact for polynomials in P 3 (resp. Q3 ) for triangles (resp. quadrangles), since they use only four points. We remark that, in practice, the formula for triangle given in table 6.2 does not present any problem in spite of its use of negative weights. Alternative
choices,
for
triangles,
are
the
three
point
integration formula using the mid-points of the edges which is exact only for Pc polynomials, or an integration formula using nine points which is the smallest number of points necessary f o r the formula to be exact for P 3 polynomals with positive weights. The latter formula is of course more expensive.
359
VI. The Saturation Equation in nvo Dimensions
Concerning the first integral in the left-hand side of ( 6 . 6 1 , extend an observation made in 811.2.5
we
in the one-dimensional case : one
should use the integration formula which has f o r nodes the vertices of the element, in order to prevent the saturation from having values outside the interval Cot 11.
Unit rectangle
Triangle
Cartesian coordinates
weight
(1+1/6)/2,(1-1fi)/2
1/4
(1+1/6)/2,(1+1/n)/2
1/11
25/48
Table 1 : The most precise four-point integration formulas.
VI.4 - A SLOPE LIMITER VI.4.1
-
Formulation
In
this
section,
we
formulate
a
slope
limiter
for
the
two-dimensional case which is an extension of the one described in section IV.5 for the one-dimensional case. First
let
us introduce some notation.
We
denote by S;
the
saturation at the (n+lIst time level calculated by (6.1) and previously n+ 1 denoted by Sh , and now ”S: is a slope limited saturation obtained from S; as will be described below. Given K
E
%,
nv(K) denotes the number of its vertices, 3 if K is
a triangle, 4 if K is a parallelogram. For vh between the restrictions to K of vh and S* h
(6.9)
1 J(v)=K h 2
nv(K1
€
MI, we introduce a distance
Ch. V: A Finite Element Method for Incompressible nYo-Phase Flow
360
and we denote by vK, the average of vh on K,
v
nv(K) =-
1
'K,Ai.
nv(K) i=l
Now, given A a vertex of the mesh, we denote by %(A)
the set of
elements having A as a vertex,
%(A)
=
{K
A is a vertex of K
E
and we calculate the minimum and the maximum of
1, the averages of S*h
On
elements of %(A),
n+1 The new saturation Sh will be calculated as to be as close as possible to S;
with respect to the distance JK, as to have the same average -n+l = ?$, K E %, which preserves the value on K as S; , i.e. sK conservativity of the scheme, and as to satisfy, for 0 2 0 L 1.
3;
(1-e)
+
1 e SMIN(A~) s s n+ K.Ai
L
(i-e)s*K
+
e SMAX(A~), i=l,...,nv(~), K
6 % .
n+l This latter inequalities limit the variations of S h' Precisely, for a given S*h'. SnC1 is defined so that its restriction h to each K ,p3(K) .
I
(6.10)
6
%,
is the solution of the following minimization problem in
Find Sit'
=
...,nv(K))
(Sn+l i=l, K.Ai'
where
x. 1
(PK) n (QK) such that
nv(K)
(6.11)
of
E
I
1 xi = nv(K).Si, i=l nv(K) (QK) is the hypercube II [(l-e)S;+e SM i=l
(PK) is the hyperplane
Problem (6.10) has a unique solution since it is the minimization
a convex function on a convex non empty set (PK) n (QK) (the point =
-*S
K'
...,nv(K)
i=l,
lies in (PK) n (QK)).
361
VI. The Saturation Equation in n o Dimensions
VI.4.2
-
Implementation Problem
c o n s t r a i n t VK
is
(6.10)
easy
to
solve.
We
suggest
Then problem (6.10) is e q u i v a l e n t t o t h e s a d d l e p o i n t problem
i
(6.13)
Find (SKn+l , A ) L(SKn+l,h)
=
u
(QK)
E
i
Max E IR
Find V(u)
Min
L ( V ( u ) , p)
=
I
L(VK,p).
E
W, t h e m i n i m i z a t i o n problem
:
such t h a t
(QK)
E
:
VK c ( Q K )
Min
L(VK,
u).
VK E ( Q K )
Then we s o l v e t h e maximization problem (6.15)
the
W such t h a t
x
T h e r e f o r e we f i r s t s o l v e , f o r a g i v e n u (6.14)
dualizing
(PK) by i n t r o d u c i n g t h e l a g r a n g i a n i n W n v ( K ) x W
E
Find h
t
L(V(A),A)
:
IR such t h a t =
Max
L(V(u),
u),
u EW
and SKn+l s a t i s f i e s SK"+l
= V(A).
S i n c e SKY = ( S ~ , * i , i ~ l , . . . , n v ( K ) )e ( P K ) , t h e Lagrangian L d e f i n e d i n (6.12) c a n be r e w r i t t e n (6.16) where
I( 1)
d e n o t e s t h e E u c l i d e a n norm i n RnS(K)and U is t h e v e c t o r such
that U.=l, i = l ,
that V(p),
...,nv(K),
which is normal t o ( P K ) .
E x p r e s s i o n ( 6 . 1 6 ) shows
t h e s o l u t i o n t o problem ( 6 . 1 4 ) is t h e p r o j e c t i o n o f SK*-uU o n t o
t h e hypercube ( Q K ) ,
s o t h a t V ( p ) is simply o b t a i n e d by t r u n c a t i o n o f t h e
components of S K * - ~ U . Thus t h e f u n c t i o n u+F(p)
=
L(V(u),u)
is e a s y t o c a l c u l a t e , and
f i n d i n g SKn+' r e d u c e s t o s o l v i n g t h e one-dimensional ( 6 . 1 5 ) . One c a n check t h a t t h e d e r i v a t e s of F a r e :
maximization problem
Ch. V: A Finite Element Method for Incompressible Tko-Phase How
362
Thus slope limiting the saturation reduces to maximizing, for each element of the mesh, a one-dimensional concave function which has piecewise constant second derivates. Therefore slope
limiting
the
saturation is not
an
expensive process, especially since it is trivial except in the vicinity of the fronts. VI.5
-
SOkE THEORETICAL RESULTS
The analysis of the method described above has not been completed. Only partial results have been obtained in the linear case, when the solution of the continuous equation is smooth and the slope limiter is not used. For the discretization in space of first order terms, LESAINT has shown that the L2-error between the true solution and the approximate solution is O(h2) for meshes of the type used in finite difference methods. JOHNSON-PITKARANTA have
proved
that
it
for general meshes
is O(h3")
including triangles. In JAFFRE 141, the whole semi-discretization in space has been studied and it has been showed that the L2-error is O(h), an estimate which remains valid when the capillary pressure vanishes.
However this last
estimate is not optimal.
-
VII
V 11.1
-
AND
NOTES
REHARKS
THE PRESSURE EQUATION Mixed
RAVIART-THOMAS.
finite The
DOUGLAS-ROBERTS [21.
elements were most
recent
and
first described complete
and
analysis
analyzed
by
is
to
due
The method has been used for incompressible two-phase
flow first in JAFFRE [ l ] ,
[ 2 ] , and more recently
in CHAVENT-COHEN-JAFFRE and
VII. Notes and Remarks
363
CHAVENT-JAFFRE-COHEN-DUPUY-RIBERA. In this last paper, a fast version of the mixed finite element method which uses a divergence-free basis for the velocities is described. Mixed finite elements have also been used in the simulation of miscible displacements see DOUGLAS-EWING-WHEELER 111 , [ 2 ] and DARLOW-EWING-WHEELER. In this last paper, numerical results are given when the method is used with index k = l for more accurate velocities. The compressible displacements
mixed
finite
problems and
three-dimensional
as
element shown
ROBERTS-SALZANO problems,
method
in
can
also
DOUGLAS-ROBERTS for
NEDELEC
immiscible
has
designed
be
applied for
[l]
displacements. and
for
miscible For
analyzed
a
method
by
three-dimensional version of the mixed finite element method. We
also
mention
a
new
mixed
finite
element
BREZZI-DOUGLAS-MARINI which gives more accurate velocities for the same approximation to the pressure and which could be used with profit to formulate the approximate pressure equations. The use of the mixed-hybrid formulation to solve the linear system derived from the mixed formulation, as described in section 111, has been also presented in MARINI. There is no difficulty to extend the idea to higher
indexes k>O,
to the BREZZI-DOUGLAS-MARINI mixed
finite element
method, and to three dimensional problems. VII. 2
THE SATURATION EQUATION Discontinous
finite
elements
were
first
used
in
the
one
dimensional case in CHAVENT-COHEN and COHEN [ l ] for the case without gravity and in CHAVENT-SALZANO for the case with gravity. For
the
linear
case
and
for
a
smooth
true
solution, the
discretization in space of the convective terms was first introduced and analyzed in LESAINT and LESAINT-RAVIART, in one and two dimensions. More recently JOHNSON-PITKARANTA has improved this analysis. In JAFFRE C41, the whose semi-discretization in space has been studied but the error estimates are not optimal. In the nonlinear case, the analysis is under way for smooth solutions
;
see JAFFRE-ROBERTS [2].
In one dimension, the idea of slope limiting has been first introduced by VAN LEER. Numerical comparison of the discontinuous finite element method with other schemes can be found in CHAVENT-COCKBURN. In one dimension, for the non-smooth case, only convergence to the weak solution
364
Ch. V: A Finite Element Method for Incompressible Two-Phase Flow
has been proved in COCKBURN-JAFFRE. In more than one dimension, the analysis remains a difficult problem for higher order schemes. Results have been obtained
only
for
first
order
finite
difference
schemes,
see
KUZNETSOV-VOLOSIN, CRANDALL-MAJDA and SAUNDERS. This discontinuous finite element method can be easily extended to three-dimensional
problems;
see COCKBURN-JAFFRE. It can be applied to
multiphase flow as in BRENIER-JAFFRE. Also
it can be associated with
implicit and/or more accurate discretizations in time; see VEERAPPA GOWDA.
VII.3
-
THE COUPLED SYSTEM The first numerical results for the coupled system were in JAFFRE
[2]
for the case without gravity and in COHEN [ 2 1 when gravity effects are
taken
in
account.
More
recent
numerical
results
are
presented
in
CHAVENT-COHEN-JAFFRE-DUPUY-RIBERA and in CHAVENT-COHEN-JAFFRE. However in these papers, no slope limiter was used. Numerical results with slope limiting will be provided in later publications. The numerical methods Tor two-phase flow have not been analyzed. More work has been done for miscible displacements. See DOUGLAS C31 for a review of this work. In EWING-WHEELER Galerkin methods are studied. In DOUGLAS-EWING-WHEELER [ l ] ,
[2]
a Galerkin method for the concentration is
associated with a mixed finite element for the pressure. In RUSSEL, a characteristics method is used for the concentration. The discontinuous finite element method for the concentration and the mixed finite element method for the pressure have been also applied to miscible displacements and analyzed in JAFFRE-ROBERTS [ I ] .
All these studies have been done for a
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to
of
compressible
the problems
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be
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NEDELEC, as
in
ROBERTS-SALZANO and to multiphtise flow as in BRENIER-JAFFRE. The first analysis of a finite element method for compressible flow can be found in DOUGLAS-ROBERTS [11 f o r the miscible case.
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