Mathematical models and finite elements for reservoir simulation: single phase, multiphase and multicomponent flows through porous media
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam u.a.
North-Holland
1986
|
| Schriftenreihe: | Studies in mathematics and its applications
17 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 376 S. |
| ISBN: | 0444700994 |
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Datensatz im Suchindex
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| adam_text | vii
CONTENTS
Preface v
CHAPTER I: BASIC LAWS AND MODELS FOR FLOW IN POROUS MEDIA 1
I. GENERALITIES 1
II. THE GEOMETRY OF THE FIELD 4
III. THE BASIC LAWS FOR ONE AND TWO PHASE FLOW 8
111.1. 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
IV. THE BASIC MODELS 17
IV.l. TheMonophasicModel.. 17
IV.2. The Fully Miscible Model 17
IV.3. The Two Phase Immiscible Model 19
IV.3.1. The Relative Permeabilities Capillary Pressure (RPCP) Model 19
IV.3.2. The Muskat Free Boundary Model 22
IV.3.3. The Richards Approximation 27
IV.3.4. The Baiocchi Free Boundary Model 32
IV.4. Summary of the Different Models 34
V. QUALITATIVE BEHAVIOR OF THE SOLUTION IN THE
NO DIFFUSION AND NO CAPILLARY PRESSURE CASE 36
V.I. The Miscible or Immiscible Model 36
V.2. Behavior of One Dimensional Solutions 39
V.2.1. The Miscible Case 40
V.2.2. The Immiscible Case 42
V.3. Behavior of Two Dimensional Miscible Solutions 45
V.4. Behavior of Two Dimensional Immiscible Solutions 46
CHAPTER H: SLIGHTLY COMPRESSIBLE MONOPHASIC FIELDS 51
I. CONSTRUCTION OF THE PRESSURE EQUATION 51
viii Content!
H. EXISTENCE AND UNIQUENESS THEOREMS 56
IH. AN ALTERNATIVE MODEL OF MONOPHASIC WELLS 65
111.1. An Exactly Equivalent Representation of Wells by Source or Sink Terms. ... 67
111.2. An Approximately Equivalent Representation of Wells by Source or
Sink Terms 70
CHAPTER HI: INCOMPRESSIBLE TWO PHASE RESERVOIRS 89
I. INTRODUCTION 89
n. CONSTRUCTION OF THE STATE EQUATIONS 92
11.1. The Equation Inside : The Notion of Global Pressure 93
11.2. The Pressure Boundary Conditions 102
H.3. The Saturation Boundary Conditions 108
II.3.1. Saturation Boundary Conditions on the Injection Boundary F_ 109
IL3.1.1. Dirichlet Condition 109
IL3.1.2. Given Water Injection Rate 110
113.2. Saturation Boundary Conditions on the Production Boundary T+ 112
H.3.2.1. Dirichlet Conditions 112
II.3.2.2. Unilateral Boundary Condition 113
H.3.2.3. Given Water/Oil Production Ratio (WOR) 116
III. SUMMARY OF EQUATIONS OF TWO PHASE FLOWS FOR
INCOMPRESSIBLE FLUIDS AND ROCK 117
111.1. Characteristics Depending Only on the Reservoir J2 117
111.2. Physical Unknowns 117
111.3. Characteristics Depending Only on the Fluids 117
111.4. Characteristics Depending Both on Fluids and Rock 118
111.5. Auxiliary Dependent Variables 119
111.6. Traces on r = 3J2 of the Dependent Variables 119
111.7. Partitions of the Boundary T of the Porous Medium Q, 120
111.8. Functions and Coefficients Depending on Reduced Saturation S Only 120
m.9. Main Dependent Variables 121
m.10. Equations for Pressure, Saturation and Flow Vectors 121
IV. AN ALTERNATIVE MODEL FOR DIPHASIC WELLS 125
V. MATHEMATICAL STUDY OF THE INCOMPRESSIBLE TWO PHASE
FLOWPROBLEMS 130
V.I. Setting of the Problem 131
V.2. Variational Formulations 135
V.3. Some Preliminary Lemmas 139
V.4. Resolution in the Non Degenerate Case 147
V.5. Resolution in the Degenerate Case 155
Contents ix
V.6. The Case of Decoupled Pressure and Saturation Equations 157
V.6.1. Regularity and Asymptotic Behavior for the Non Degenerate Case 159
V.6.2. Regularity and Asymptotic Behavior for the Degenerate Case 171
VI. THE CASE OF FIELDS WITH DIFFERENT ROCK TYPES 177
VI.l. The Different Rock Models 178
VI.2. The Case of a Field with M Different Rock Types 182
CHAPTER IV: GENERALIZATION TO COMPRESSIBLE, THREE PHASE,
BLACK OIL OR COMPOSITIONAL MODELS 189
I. THE TWO PHASE COMPRESSIBLE MODEL. 189
1.1. Equations from the Physics 190
1.1.1. Conservation Laws 190
1.12. Muskat Law (Relative Permeabilities) 191
1.1.3. Capillary Pressure Law 191
1.2. Simplifying Hypotheses 191
1.2.1. Pressure Dependent Coefficients 191
1.2.2. Choice of a Rock Model 192
1.2.3. Limitation of the Pressure Range 194
1.2.4. Summary of the Resulting Equations 195
1.3. The Global Pressure Equation 196
1.4. The Saturation Equation 198
1.5. Summary of the Two Phase Compressible Model 199
1.6. The Case of Slightly Compressible Rock and Fluids 202
II. THE THREE PHASE COMPRESSIBLE MODEL 203
11.1. Equations from the Physics 204
11.1.1. Conservation Law 204
11.1.2. Muskat Law (Relative Permeabilities) 204
II.l .3. Capillary Pressure Law 205
11.2. Simplifying Hypotheses 205
II.2.1. Pressure Dependent Coefficients 205
H.2.2. Choice of a Rock Model 206
11.2.3. The Total Differential (TD) Condition on the Rock Model 209
11.2.4. limitation of the Pressure Range 213
11.2.5. Summary of the Resulting Equations 214
O.3. The Global Pressure Equation 214
11.4. The Saturation Equation. 216
11.4.1. Determination of the Equations 216
11.4.2. A Hyperbolicity Condition 218
11.5. Construction of Three Phase Data Satisfying the TD Condition (2.24) 220
II.5.1. The Practically Available Data 220
n.5.2. Continuation of Capillary Pressures 222
11.5.3. Continuation of Relative Permeabilities 222
x Contents
11.5.4. Numerical Algorithm for the Computation of TD Three Phase
Relative Permeabilities 225
11.5.5. Examples of TD Three Phase Data 229
11.5.6. The Hyperbolicity Condition 234
II.6. Summary of the Three Phase Compressible Model 235
III. THE BLACK OIL MODEL 237
HI.l. Range of Validity 237
III.2. Components and Phases 238
m.3. Description of Phases Equilibrium 239
111.4. Description of Phases Characteristics 242
111.5. Governing Equations from the Physics 243
IH.6. Global Pressure and Pressure Equation 244
111.7. Saturations/Dissolution Factors Equations 250
111.8. Summary of the Black OilModel 253
IV. A COMPOSITIONAL MODEL 255
IV.l. Range of Validity 255
IV.2. Description of the Thermodynamic Equilibrium 256
IV.3. Description of Phase(s) Characteristics 260
W3.1. In the One Phase Domain 260
IV.3.2. In the Two Phase Domain (General) 261
IV.3.3. Compositional Residual Saturations 262
IV.3.4. Compositional Relative Permeabilities 263
IV.3.5. Compositional Capillary Pressure 267
IV.3.6. Hypothesis on Pressure Dependent Data 269
IV.3.7. An Example of Compositional Two Phase Data 270
IV.4. Governing Equations from the Physics 274
IV.4.1. In the One Phase Domain 274
IV.4.2. In the Two Phase Domain 275
IV.4.3. Matching of Two Phase and One Phase Equations 276
IV.5. Introducing a Global Pressure 277
IV.5.1. Some Preliminaries 277
IV.5.2. Definition of the Global Pressure P 280
IV.6. The Global Pressure Equation 284
IV.7. The Concentration Equations 290
IV.8. Regularity of the Equations 291
IV.8.1. Coupling Between Pressure and Concentration Equations 294
IV.8.2. Some Preliminaries for the Study of the Regularity 295
IV.8.3. Regularity of the Pressure Equation 300
IV.8.4. Regularity of the Concentration Equation 304
CHAPTER V: A FINITE ELEMENT METHOD FOR INCOMPRESSIBLE
TWO PHASE FLOW 311
I. INTRODUCTION 311
Contents xi
1.1. Introductory Remarks 311
1.2. Equations of Incompressible Two Phase Flow 313
1.3. Discretization 314
II. APPROXIMATION OF THE PRESSURE VELOCITY EQUATIONS 315
II.l. Approximation Spaces 315
H.2. Approximation Equations 318
III. RESOLUTION OF THE ALGEBRAIC SYSTEM FOR
PRESSURE VELOCITY 322
III.l. Introduction 322
III .2. The Mixed Hybrid Formulation of the Pressure Velocity Equations 323
II1.3. The Algebraic System Derived from the Mixed Hybrid Formulation 326
IV. APPROXIMATION OF THE ONE DIMENSIONAL SATURATION
EQUATION: THE CASE WITH NEITHER CAPILLARY PRESSURE
NOR GRAVITY 330
IV.l. Introduction 330
IV.2. A General Discontinuous Finite Element Scheme 332
IV.3. The Case k = 0: Piecewise Constant Approximation 335
IV.4. The Case k = 1: Piecewise Linear Approximation 335
IV.5. A Slope limiter 338
V. APPROXIMATION OF THE ONE DIMENSIONAL SATURATION
EQUATION IN THE GENERAL CASE 345
V.I. The Gravity Effects 345
V.2. The Capillary Pressure Effects . 347
VI. APPROXIMATION OF THE SATURATION EQUATION IN
TWO DIMENSIONS 355
VI.l. Approximation Spaces 355
VI.2. Approximation Equations 356
VI.3. Integration Formulas 358
VI.4. A Slope Limiter 359
VI.5. Some Theoretical Results 362
Vn. NOTES AND REMARKS 362
Vn.l. The Pressure Equation 362
VII.2. The Saturation Equation 363
VII.3. The Coupled System 364
REFERENCES 365
|
| any_adam_object | 1 |
| author | Chavent, Guy Jaffré, Jérôme |
| author_facet | Chavent, Guy Jaffré, Jérôme |
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| callnumber-subject | TN - Mining Engineering and Metallurgy |
| classification_rvk | SK 950 |
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| dewey-full | 622/.1882 |
| dewey-hundreds | 600 - Technology (Applied sciences) |
| dewey-ones | 622 - Mining and related operations |
| dewey-raw | 622/.1882 |
| dewey-search | 622/.1882 |
| dewey-sort | 3622 41882 |
| dewey-tens | 620 - Engineering and allied operations |
| discipline | Bergbau / Hüttenwesen Mathematik |
| format | Book |
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| id | DE-604.BV000499115 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:14:54Z |
| institution | BVB |
| isbn | 0444700994 |
| language | English |
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| oclc_num | 14002735 |
| open_access_boolean | |
| owner | DE-12 DE-384 DE-739 DE-706 DE-83 DE-188 |
| owner_facet | DE-12 DE-384 DE-739 DE-706 DE-83 DE-188 |
| physical | XI, 376 S. |
| publishDate | 1986 |
| publishDateSearch | 1986 |
| publishDateSort | 1986 |
| publisher | North-Holland |
| record_format | marc |
| series | Studies in mathematics and its applications |
| series2 | Studies in mathematics and its applications |
| spelling | Chavent, Guy Verfasser aut Mathematical models and finite elements for reservoir simulation single phase, multiphase and multicomponent flows through porous media Guy Chavent ; Jérôme Jaffré Amsterdam u.a. North-Holland 1986 XI, 376 S. txt rdacontent n rdamedia nc rdacarrier Studies in mathematics and its applications 17 Fluides, Dynamique des - Modèles mathématiques Fluides, Dynamique des - Modèles mathématiques ram Gisements pétrolifères, Étude des - Modèles mathématiques Randwaardeproblemen gtt Wiskundige modellen gtt dynamique fluide inriac milieu poreux inriac modélisation mathématique inriac méthode élément fini inriac pétrole inriac réservoir inriac simulation inriac Éléments finis, Méthode des Éléments finis, Méthode des ram Mathematisches Modell Finite element method Fluid dynamics Mathematical models Oil reservoir engineering Mathematical models Rohstoffreserve (DE-588)4425009-5 gnd rswk-swf Erdöl (DE-588)4015179-7 gnd rswk-swf Simulation (DE-588)4055072-2 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Strömungsmechanik (DE-588)4077970-1 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Erdöl (DE-588)4015179-7 s Rohstoffreserve (DE-588)4425009-5 s Simulation (DE-588)4055072-2 s DE-604 Strömungsmechanik (DE-588)4077970-1 s Mathematisches Modell (DE-588)4114528-8 s Finite-Elemente-Methode (DE-588)4017233-8 s Jaffré, Jérôme Verfasser aut Studies in mathematics and its applications 17 (DE-604)BV000000646 17 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000309430&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chavent, Guy Jaffré, Jérôme Mathematical models and finite elements for reservoir simulation single phase, multiphase and multicomponent flows through porous media Studies in mathematics and its applications Fluides, Dynamique des - Modèles mathématiques Fluides, Dynamique des - Modèles mathématiques ram Gisements pétrolifères, Étude des - Modèles mathématiques Randwaardeproblemen gtt Wiskundige modellen gtt dynamique fluide inriac milieu poreux inriac modélisation mathématique inriac méthode élément fini inriac pétrole inriac réservoir inriac simulation inriac Éléments finis, Méthode des Éléments finis, Méthode des ram Mathematisches Modell Finite element method Fluid dynamics Mathematical models Oil reservoir engineering Mathematical models Rohstoffreserve (DE-588)4425009-5 gnd Erdöl (DE-588)4015179-7 gnd Simulation (DE-588)4055072-2 gnd Mathematisches Modell (DE-588)4114528-8 gnd Strömungsmechanik (DE-588)4077970-1 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd |
| subject_GND | (DE-588)4425009-5 (DE-588)4015179-7 (DE-588)4055072-2 (DE-588)4114528-8 (DE-588)4077970-1 (DE-588)4017233-8 |
| title | Mathematical models and finite elements for reservoir simulation single phase, multiphase and multicomponent flows through porous media |
| title_auth | Mathematical models and finite elements for reservoir simulation single phase, multiphase and multicomponent flows through porous media |
| title_exact_search | Mathematical models and finite elements for reservoir simulation single phase, multiphase and multicomponent flows through porous media |
| title_full | Mathematical models and finite elements for reservoir simulation single phase, multiphase and multicomponent flows through porous media Guy Chavent ; Jérôme Jaffré |
| title_fullStr | Mathematical models and finite elements for reservoir simulation single phase, multiphase and multicomponent flows through porous media Guy Chavent ; Jérôme Jaffré |
| title_full_unstemmed | Mathematical models and finite elements for reservoir simulation single phase, multiphase and multicomponent flows through porous media Guy Chavent ; Jérôme Jaffré |
| title_short | Mathematical models and finite elements for reservoir simulation |
| title_sort | mathematical models and finite elements for reservoir simulation single phase multiphase and multicomponent flows through porous media |
| title_sub | single phase, multiphase and multicomponent flows through porous media |
| topic | Fluides, Dynamique des - Modèles mathématiques Fluides, Dynamique des - Modèles mathématiques ram Gisements pétrolifères, Étude des - Modèles mathématiques Randwaardeproblemen gtt Wiskundige modellen gtt dynamique fluide inriac milieu poreux inriac modélisation mathématique inriac méthode élément fini inriac pétrole inriac réservoir inriac simulation inriac Éléments finis, Méthode des Éléments finis, Méthode des ram Mathematisches Modell Finite element method Fluid dynamics Mathematical models Oil reservoir engineering Mathematical models Rohstoffreserve (DE-588)4425009-5 gnd Erdöl (DE-588)4015179-7 gnd Simulation (DE-588)4055072-2 gnd Mathematisches Modell (DE-588)4114528-8 gnd Strömungsmechanik (DE-588)4077970-1 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd |
| topic_facet | Fluides, Dynamique des - Modèles mathématiques Gisements pétrolifères, Étude des - Modèles mathématiques Randwaardeproblemen Wiskundige modellen dynamique fluide milieu poreux modélisation mathématique méthode élément fini pétrole réservoir simulation Éléments finis, Méthode des Mathematisches Modell Finite element method Fluid dynamics Mathematical models Oil reservoir engineering Mathematical models Rohstoffreserve Erdöl Simulation Strömungsmechanik Finite-Elemente-Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000309430&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000646 |
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