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Petroleum Engineering and Development Studies

Volume 4

Mathematical Theory of Oil and Gas Recovery

With Applications to ex-USSR Oil and Gas Fields

PETROLEUM ENGINEERING AND DEVELOPMENT STUDIES

Advisory Editor John S. Archer

Volume 1 Fundamentals of Casing Design Hussain Rabia

Volume 2 Directional Drilling T. A. Inglis

ISBN 0-86010-863-5

ISBN 0-86010-716-7

Volume 3 Quality Assurance in the Offshore Oil and Gas Industry J. H. Rogerson (editor) ISBN 0-8601 0-947-X

Volume 4 Mathematical Theory of Oil and Gas Recovery. With Applica­tions to ex-USSR Oil and Gas Fields P. Bedrikovetsky ISBN 0-7923-2381-5

Petroleum Engineering and Development Studies

Volume 4

Mathematical Theory of Oil and Gas Recovery

With Applications to ex-USSR Oil and Gas Fields

by

Pavel Bedrikovetsky Moscow State Oil and Gas Academy, Russia

Scientific Editor

eren Rowan Consultant, formerly with British Petroleum

Translator

Ruth Loshak M. Phil., MITJ

SPRINGER-SCIENCE+BUSINESS MEDIA, B.v.

Library of Congress Cataloging-in-Publication Data

Bedrikovetsky, cavei. Mathematical theorv of oii and gas recovery with applications to

ex-USSR oii and gas fields 'by Pavel Bedrikovetsky. p. cm. -- <Petroleum engineer1ng and development studies v.

4> Includes bibliographlcal references <p. > and index. ISBN 978-90-481-4300-9 ISBN 978-94-017-2205-6 (eBook) DOI 10.1007/978-94-017-2205-6 1. Oii fields--Product1on methods--Mathematical models. 2. Gas

fields7·Production methods--Mathematical models. 3. Oii fields­·Former Soviet republics--Production methods. 4. Gas fields--Former Sov1et republics--Production methods. I. Title. II. Series. TN870.8415 1993 622' .33B--dc20 93-14525

ISBN 978-90-481-4300-9

Printed on acid-free paper

Ali Rights Reserved © 1993 Springer Science+Business Media Dordrecht

Originally published by Kluwer Academic Publishers in 1993 No part of the material protected by this copyright notice may be reproduced or

utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

With love to my mother, Rita Klinger

FOREWORD

It is a pleasure to be asked to write the foreword to this interesting new book. When Professor Bedrikovetsky first accepted my invitation to spend an extended sabbatical period in the Department of Mineral Resources Engineering at Imperial College of Science, Technology and Medicine, I hoped it would be a period of fruitful collaboration. This book, a short course and a variety of technical papers are tangible evidence of a successful stay in the UK. I am also pleased that Professor Bedrikovetsky acted on my suggestion to publish this book with Kluwer as part of the petroleum publications for which I am Series Editor.

The book derives much of its origin from the unpublished Doctor of Science thesis which Professor Bedrikovetsky prepared in Russian while at the Gubkin Institute. The original DSc contained a number of discrete publications unified by an analytical mathematics approach to fluid flow in petroleum reservoirs. During his sabbatical stay at Imperial College, Professor Bedrikovetsky has refined and extended many of the chapters and has discussed each one with internationally recognised experts in the field. He received great encouragement and editorial advice from Dr Gren Rowan, who pioneered analytical methods in reservoir modelling at BP for many years.

To the eyes of western petroleum reservoir engineers, used to numerical methods and reservoir simulation, this book is relatively unusual. It is, however, very timely and provides an opportunity to revisit the mathematical basis of analytical continua problems for which there are exact solutions. Of necessity, the reservoir description is relatively simplistic and may be something of a shock to numerical modelling colleagues immersed in permeability characterisation of heterogeneous reservoirs with ten thousand grid cells. Although the source and reliability of charactrerisation data are quite properly a matter of concern, it is not the objective of this book to comment on where these data come from or how effective values may be obtained. Similarly, the appropriateness of particular representations of relative permeability relationships and partition coefficients are not the primary concern. Rather, it is the way such information could be used, if it were available, that is of importance in this analytical mathematics approach.

The author has been ambitious in the breadth of the field that he has taken on. He has divided the book into seven parts that broadly reflect his experience in tackling problems in the petroleum reservoirs of the oil USSR.

In Part I he deals with the mathematics of waterflooding, and describes two phase incompressible and compressible flow in I-D in porous media of various complexity.

vii

viii FOREWORD

In Part II he deals with chemical flooding and is particularly interested in the Riemann problem. Continuous injection and slug injection of one or two chemicals are treated, and the 1-0 models are extended to 2-D for the case of 2-phase displacement. The author tantalisingly speculates on a 3-D analytical model for chemical flooding in a stratified reservoir.

Thermal waterflooding is treated in Part III and the approaches presented are utilised in an example from the Kharyagi field.

In Part IV we find a treatment of gas and solvent injection into oil and gas condensate reservoirs. In oil fields the injection process is considered both as continuous and with slugs, and the experience with the Vuktyl field is used as an example. An exact analytical solution for the WAG process is developed and utilised in an example from the Kharyagi field. Gas condensate initially in place is estimated from phase equilibrium assumptions and an interesting case of displacement of retrograde condensate by a LPG slug is discussed.

A single chapter comprises Part V, and ideas of in situ chemical refining as an example of chemical reactions during flow in porous media are explored. The development of gas fields containing hydrogen sulphide provide the background, and examples are drawn from Gugurtly and Sovetabad.

In Part VI the effects of temperature, gravity and capillarity on immiscible fluid stratification in porous reservoirs are developed.

Finally, in Part VII, the author develops relationships for gravity stabilised gas injection into thick gas condensate reservoirs, and provides an example with the Karachaganak field in Kazakhstan.

The book should be of interest to many petroleum reservoir engineers, and it will provide an opportunity to validate a range of numerical reservoir simulation models by comparison with the exact results of simple analytical methods. It also provides an insight into the ways in which petroleum reservoir engineering has been considered in the ex-USSR and it offers, through its examples, a glimpse of some of the reservoirs which may become the subject of future East-West collaborative developments.

Professor John Archer, F.Eng Pro Rector and Professor of Petroleum Engineering Imperial College of Science, Technology and Medicine London SW7 2AZ 5th April 1993

Table of Contents

Foreword by Professor John Archer Preface

PART I. WATER·FLOODING

CHAPTER 1. ONE-DIMENSIONAL MOTION OF A TWO-PHASE SYSTEM OF IMMISCIBLE LIQUIDS IN A POROUS MEDIUM

1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. l.l0.

Equations of two-phase flow through a porous medium Formulation of the displacement problem. Discontinuous solutions Method of characteristics Discontinuities The problem of stabilized zones Solution of tbe Riemann problem (decay of a discontinuity) Contour integration metbod. Motion of discontinuity Graphical-anal ytical calculation of displacement coefficient. Welge' s method Inverse problems of two-phase displacement Features of water-drive displacement of a rbeologically anomalous oil

CHAPTER 2. PERCOLATION MODELS OF FLOW THROUGH A POROUS MEDIUM

2.1. 2.2. 2.3.

Basic relations of percolation theory Percolation model of a porous medium Conduction parameters of a porous medium

CHAPTER 3. ANALYTICAL MODELS OF WATER-FLOODING OF STRATIFIED RESERVOIRS

3.1.

3.2. 3.3.

Asymptotic analysis of tbe effect of capillary and gravitational forces on the two-dimensional transport of two-phase systems through porous media

Water-flooding of a stratified reservoir in the viscous dominated case Displacement from stratified reservoirs in the case of capillary-gravitatiooal

equilibrium

CHAPTER 4. EFFECTS OF COMPRESSIBILITY ON TWO-PHASE DISPLACEMENT

4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7.

4.8.

Statement of the one-dimensional problem Conditions at discontinuities Self-similar solutions of one-dimensional problems Solutions for displacement of a liquid by gas and a gas-liquid mixture Decay of a discontinuity Equations of the displacement of gas by water from stratified reservoirs Derivation of the equations for the displacement of oil (water) by gas from

stratified reservoirs 'Quasi-one-dimensional' displacement from a heterogeneous reservoir

IX

vii xvii

3

3 7 9

12 15 16 20 22 23 25

27

27 31 33

40 48

57

60

60 61 63 65 67 71

71 73

x CONTENTS

4.9. 4.10. 4.11. 4.12. 4.13. 4.14.

Mixing criterion Phase portrait of the system Phase portrait of system during displacement of gas by water Phase portrait for displacement of water by gas Calculation of displacement efficiency Quantitative effects of compressibility

Conclusions: Part I

PART II. CHEMICAL FLOODING

CHAPTER 5. ONE-DIMENSIONAL DISPLACEMENT OF OIL BY CHEMICAL SOLUTIONS

5.1. 5.2.

5.3. 5.4. 5.5. 5.6.

5.7. 5.8.

Model of the displacement of oil by chemical soalutions The effect of different chemicals on the motion of the water-oil system in a

porous medium Formulation of the problem of oil displacement by a chemical solution Decay of a discontinuity for a hyperbolic system of quasi-linear equations Solution of the problem of the displacement of oil by a chemical solution Admissibility and stability of discontinuities in two-phase flow in a porous

medium by a chemical solution Decay of an arbitrary discontinuity Graphical-analytical calculation of the displacement coefficieIJt

CHAPTER 6. THE EFFECT OF NON-EQUILIBRIUM SORPTION AND SOLUTION ON THE DISPLACEMENT OF BY CHEMICAL FLOODING

6.1. 6.2. 6.3. 6.4. 6.5.

Analysis of equations of motion Construction of the solution Approximate calculation of the motion of the displacement front Displacement of oil from a water-encroached reservoir Effects of non-eqUilibrium of the solubility function

75 76 77 78 80 82

85

86

88

89

93 97 98

105

114 120 126

127

127 129 132 133 134

CHAPTER 7. DISPLACEMENT OF OIL BY A CHEMICAL SLUG WITH WATER DRIVE 138

7.1. Structure of displacement zone: linear sorption isotbenn and solubility function of added chemical

7.2. Computation of efficiency during displacement of oil by a chemical slug from an 139

undeveloped reservoir 149 7.3. Algorithm for determining the necessary slug volume corresponding to a

specified displacement efficiency 152 7.4. Slug dynamics: concave sorption isotherm 153 7.5. Slug dynamics: convex sorption isotherm 157 7.6. Oil displacement by a chemical slug: irreversible sorption of chemical 161 7.7. Preliminary evaluation of use of polymers for development of the Chaivo More reservoir

(Sakhalin) 170

CHAPTER 8. OIL DISPLACEMENT BY A COMBINATION OF MULTI-CHEMICAL

8.1.

8.2.

SLUGS 174

The displacement of oil by a surfactant slug driven through the reservoir by a buffer polymer slug and water

Calculation of the displacement efficiency with the use of multi-chemical slugs 176 187

CONTENTS

8.3.

8.4.

Enhanced oil recovery for flooding by a soluble surfactant slug driven by polymer slug and water

Dynamics of two-chemical slugs with irreversible sorption

CHAPTER 9. MOTION OF A THIN SLUG OF CHEMICAL IN lWO-PHASE FLOW IN A POROUS MEDIUM

9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7.

Formulation of the problem Derivation of self-similarity conditions Construction of the self-similar solution Plane-parallel displacement Radial displacement Some generalizations Conclusions

CHAPTER 10. THE INVERSE PROBLEM OF DETERMINING THE DEGREE OF

xi

190 193

199

199 202 203 205 208 208 209

SORPTION OF A CHEMICAL FROM LABORATORY DATA 210

CHAPTER 11. AN ANALYTICAL MODEL OF lWO-DIMENSIONAL DISPLACEMENT OF OIL FROM RESERVOIRS IN A SYSTEM OF WELLS 214

11.1. 11.2. 11.3. 11.4. 11.5. 11.6.

Formulation of the problem Description of streamlines The analytical simulation of two-dimensional displacement Calculation of the integral flow characteristics Analytical model of water-flooding Conclusion

CHAPTER 12. CHEMICAL FLOODING IN STRATIFIED RESERVOIRS

12.1. 12.2. 12.3.

Equations of displacement of oil from stratified reservoirs by a chemical solution Tracer analyses in stratified reservoirs Inverse problems; determination of the permeability profile

CHAPTER l3. METIIOOOLOGY OF THE APPLICATION OF 3D ANALYTICAL MODELS TO FEASmILITY STUDIES AND DESIGN OF CHEMICAL FLOODING SCHEMES

13.1. l3.2. l3.3.

13.4. 13.5.

Experimental results Justification for the initial data used in the calculation Analytical calculation of one-dimensional displacement from a homogeneous

reservoir Analytical modelling of polymer flooding in five-spot pattern Analytical calculation of displacement from stratified reservoir

214 215 218 220 221 222

223

223 227 231

233

233 234

235 236 238

Conclusions and Recommendations: Part II 239

PART ill. HOT WATER FLOODING 242

CHAPTER 14. DISPLACEMENT OF NON-NEWTONIAN OIL BY HOT WATER WITH

14.1.

14.2.

HEAT LOSSES TO ADJACENT LAYERS 244

Solution of the problem of one-dimensional displacement with continuous hot water injection

Determining the coefficient of heat exchange 244 248

xii CONfENTS

14.3. 14.4. 14.5.

Displacement by a non-active liquid Approximate method of calculating flow in the heated zone Field applications: Uzen field

CHAPTER 15. HOT WATER FLOODING OF WAXY CRUDE WITH PARAFFIN SEPARATION

15.1. 15.2. 15.3. 15.4. 15.5.

15.6. 15.7.

15.8. 15.9.

15.10.

Derivation of equations of displacement Analysis of the displacement equations The construction of self-similar solutions The effect of paraffin separation on permeabilities Investigation of the effect of paraffin separation on the efficiency of hot water

flooding The determination of the flow characteristics from displacement data The displacement of waxy crude by a hot water slug from a reservoir, without

heat losses to adjacent layers Comparison of hydrodynamics of flow in different displacement regimes Hot water displacement of waxy crude with separation of the solid phase and

hysteresis of its solution Graphical-analytical calculation of oil recovery with displacement of waxy crude

by hot water from a non-insulated reservoir

Conclusions: Part III

PAR.T IV. THE INJECTION OF GASES AND SOLVENTS INTO GAS CONDENSATE AND OIL RESER.VOIRS

CHAPTER 16. THE DISPLACEMENT OF RETROGRADE CONDENSATE AND OIL BY GASES AND SOLVENTS

16.1. 16.2. 16.3. 16.4. 16.5. 16.6. 16.7.

Equations of multi-phase mUlti-component flow in a porous medium Two-phase three-component displacement of incompressible fluids Self-similar solutions of displacement problems The injection of lean gas into a depleted gas-condensate reservoir The displacement of gas-condensate mixture by rich gas Quasi-piston-like displacement The displacement of a gas-liquid hydrocarbon mixture by a gas with which it is

in equilibrium

CHAPTER 17. THE DISPLACEMENT OF RETROGRADE CONDENSATE BY SLUGS

14<1 250 252

257

257 261 262 265

272 274

276 278

281

285

291

293

295

296 298 303 304 308 310

312

OF RICH GAS 314

17.1. 17.2. 17.3.

17.4.

The displacement of condensate by a slug of lean gas driven by dry gas The displacement of condensate by a slug of rich gas The displacement of oil and gas condensate by solvent slugs with components

with non-linear distribution isotherms Graphical-analytical technique for calculating the recovery of a condensate that

has separated in the reservoir: displacement by solvent slugs

CHAPTER 18. ANALYTICAL WATER-ALTERNATE GAS MODELLING

18.1. 18.2.

Mathematical model of WAG process under conditions of complete miscibility Exact analytical solution of the ID WAG problem

314 322

324

324

327

328 330

CONTENTS

18.3. 18.4. 18.5.

The results of the WAG calculation: fonnation II of Kharyagi field The results of the WAG calculation: fonnation III of Kharyagi field Conclusions

CHAPTER 19. THE TWO-PHASE DISPLACEMENT OF BINARY MIXTURES AT LARGE PRESSURE GRADIENTS

CHAPTER 20. THE EFFECT OF CAPILLARY FORCES ON PHASE EQUILIBRIA AND DISPLACEMENT IN POROUS MEDIA

20.1. 20.2. 20.3. 20.4. 20.5. 20.6. 20.7. 20.8.

Conditions of phase equilibrium Model of a porous medium Detennination of the Leverett function Phase diagrams of capillary systems Calculation of phase equilibrium in a porous medium Examples Estimation of reserves of heavy components Two-phase displacement in a porous medium

CHAPTER 21. INVERSE PROBLEMS OF LABORATORY MULTI-PHASE DISPLACEMENT WITH PHASE TRANS mONS

21.1. 21.2. 21.3.

21.4. 21.5. 21.6.

Formulation of the laboratory displacement problem Solution of the inverse problem in the two-phase multi-component case Identifying the equilibrium conditions of a two-phase mUlti-component system in

a porous medium Three-component displacement with constant flow Design of the laboratory experiment Range of application of the method

CHAPTER 22. FEASIBILITY STUDY AND PLANNING OF ENHANCED CONDENSATE RECOVERY: APPLICATION OF ANALYTICAL MODELS TO VUKTYL OIL-GAS-CONDENSATE FIELD

22.1. 22.2. 22.3. 22.4. 22.5.

22.6.

Analysis of displacement efficiency at reservoir pressure 10 MPa Investigation of displacement efficiency at low reservoir pressures Estimates of sweep efficiency in area of 'Condensate-I' pilot test The characterization of a reservoir from pilot test results Perfonnance indices of the displacement of retrograde condensate by LPG slug

in new pilot area Methodology of the application of analytical models in feasibility studies and

planning of field development

Main conclusions and results: Part IV

PART V. THE THEORY OF IN SITU SWEETENING OF NATURAL GASES

CHAPTER 23. THE THEORY OF IN SITU SWEETENING OF NATURAL GASES

23.1.

23.2. 23.3.

Development of gas fields with non-hydrocarbon components by natural and artificial in situ sweetening

Mathematical model of in situ gas sweetening Wave dynamics of reaction fronts during the flow of sour gas through an iron­

bearing reservoir

xiii

338 343 347

348

353

353 354 356 358 360 361 362 366

371

372 373

376 377 380 383

385

385 388 392 394

396

398

401

403

403

404 406

410

xiv CONTENTS

23.4.

23.5.

23.6.

Evolution of concentration waves as hydrogen sulphide moves to H2S-free part of reservoir (with reference to North Balkui field)

Determination of reactive capacity of reservoirs from displacement data: inverse problems

Travelling concentration waves with allowance for local effects in laboratory sweetening

415

417

421

Conclusions and recommendations: Part V 427

PART VI. THE GRAVITATIONAL STRATIFICATION AND SEGREGATION OF TWO·PHASE MULTI·COMPONENT FLUIDS IN THICK OIL·GAS. CONDENSATE RESERVOIRS 428

CHAPTER 24. STRATIFICATION OF MUL TI·COMPONENT MIXTURES IN THE EARTH'S THERMAL AND GRAVITATIONAL FIELDS 430

24.1. 24.2. 24.3. 24.4. 24.5. 24.6. 24.7. 24.8. 24.9. 24.10. 24.11. 24.12. 24.13.

Thermal diffusion flows of muIti-component mixtures Formulation of the thermogravitational stratification problem The segregation of mUlti-component mixtures Asymptotic expansion of the solution Stratification of binary mixtures Stratification of binary mixtures of ideal gases Example calculation for a binary mixture Stratification of binary mixtures of incompressible liquids Qualitative investigation of the system of stratification equations of ideal mixtures Stratification of a multi-component mixture of ideal gases Stratification of weak solutions Stratification of a mixture of ideal gases and an incompressible liquid Stratification in gas-oil systems

CHAPTER 25. CAPILLARY -GRA VIT A TlONAL STRA TIFlCA TION OF TWO-PHASE

431 433 434 438 443 443 447 448 449 454 456 457 458

MIXTURES IN THICK RESERVOIRS 460

25.1. 25.2. 25.3. 25.4. 25.5. 25.6.

The equations of two-phase equilibrium in a gravitational field Determination of the Leverett function Method of solution of phase equilibrium equations Example calculation Estimation of reserves Two-phase vertical displacement in a porous medium

CHAPTER 26. ANALYSIS OF CONVECTIVE INSTABILITIES IN BINARY MIXTURES IN POROUS MEDIA

26.1. 26.2. 26.3. 26.4. 26.5. 26.6.

Equations of flow of a binary mixture in a porous medium Linearization in the neighbourhood of mechanical equilibrium Normal perturbations The stability paradox A layer with impermeable boundaries Determination of critical permeability

460 461 463 465 467 468

475

475 477 479 482 483 484

CONTENTS

CHAPTER 27. THE DYNAMIC GRA VIT ATIONAL SEPARATION OF OIL AND WATER IN RESERVOIRS OF LIMITED THICKNESS

27.1. 27.2. 27.3. 27.4. 27.5.

Fonnulation of the problem Separation of a system described as a 'liquid layer above a gas layer' Separation of 'layer of liquid above layer of gas' in a semi-infinite reservoir Drainage of retrograde condensate to the bottom of a thick reservoir Conclusions

Main conclusions and results: Part VI

PART VII. GRAVITY-STABILIZED GAS INJECTION

CHAPTER 28. VERTICAL DISPLACEMENT OF GRA VITY -STRATIFIED TWO-PHASE THREE-COMPONENT FLUIDS

28.1.

28.2. 28.3. 28.4. 28.5.

28.6.

28.7.

Statement of the problem of vertical displacement of two-phase three-component mixtures

Solution of problem of vertical displacement by a solvent slug Hydrodynamic picture of displacement Calculation of component recovery Vertical displacement of gravity-stratified oil-gas-condensate mixtures by

methane Picture of two-phase flow with phase transitions during vertical displacement of

an oil-gas-condensate fluid by methane Vertical displacement of super-critical single-phase oil-gas-condensate mixture

CHAPTER 29. ANALYTICAL MODEL OF GRAVITY-STABILIZED GAS INJECTION

xv

487

488 489 496 498 501

502

504

506

506 507 511 514

516

522 524

IN A THICK HETEROGENEOUS RESERVOIR 527

29.1. 29.2. 29.3.

Derivation of equations of motion Solution of displacement problem Analysis of calculated results

527 529 533

Conclusions: Part VII 536

APPENDIX A: Admissibility of discontinuities in two-phase flow in a porous medium with chemical flooding 537

APPENDIX B: Stability of discontinuities in two-phase flow in a porous medium with chemical flooding 545

APPENDIX C: Classification of decay configurations of an arbitrary discontinuity for two-phase flow in a porous medium with chemical flooding 551

REFERENCES 557

NOMENCLATURE 570

INDEX 572

PREFACE

The subject of this book is the analytical modelling of the processes of secondary and tertiary recovery of oil and gas condensate. Mathematical models of multi-dimensional processes of two-phase multi-component displacement in heterogeneous reservoirs (based on exact and asymptotic solutions of problems of motion in porous media) are an effective tool for use in feasibility studies and development planning of actual oil and gas­condensate fields. Also, they are valuable in the search for and synthesis of optimal schemes of recovery and for the investigation of the physical mechanisms of oil and condensate recovery using different flooding techniques.

The displacement of oil and condensate by gases, hot liquids and chemicals involves more complex physico-chemical processes than do water-flooding or pressure depletion of reservoirs. Non-isothermal phase transitions and interphase mass transfer take place in the porous medium and the composition of the injected fluid has an important bearing on the behaviour of the reservoir system. The mathematical models of the recovery processes are correspondingly more complicated.

Moreover, the number of parameters involved in the recovery process is increased when gas injection or improved water-flooding is applied. The formulation of an optimal recovery scheme requires that choices be made. Decisions have to be taken with regard to the composition of the injected fluid, the injection pressure and temperature, the volume of slugs and the order in which they are to be injected. The number of cases to be evaluated, for a comparison of the effectiveness of different techniques in field development, is thereby increased.

At the same time, the decision to use new technology brings with it greater responsibilities, as it involves higher capital investment than the conventional techniques of water-flooding or depletion of reservoirs. Additional sensitivity studies of the proposed enhancement techniques must be carried out in cases where the geological and production information is scanty.

The modelling of field development with multi-variant calculations is usually performed by numerical simulation of multi-dimensional, two-phase, multi-component displacement from heterogeneous reservoirs. Although recent years have seen some progress in the development of efficient, universal, numerical schemes, these use an excessive amount of computer time, even with the powerful modem workstations that are now available. Numerical simulation of highly heterogeneous reservoirs, results in a reduction in accuracy of the computation and an increase in the amount of time that it takes, thus inhibiting extensive comparisons of the various recovery methods and

xvii

xviii PREFACE

sensitivity studies of the critical variables. The difficulties that arise in describing the reservoir from the production history or from the results of pilot tests are magnified: for instance, tuning of the reservoir model demands that several simulator runs be made before agreement can be achieved between the field data and the simulation results.

An analytical approach to reservoir simulation is followed in this book. Both direct and inverse problems are simple to solve with the use of models constructed from explicit formulae (the exact and asymptotic solutions of problems of motion in a heterogeneous porous medium). The widespread availability of personal computers has enhanced the role of analytical models.

While the computations they entail are relatively simple, analytical models possess a complex structure within a rich conceptual framework. They can provide for discontinuities of the parameters, different types of continuous non-linear waves, the interaction of fronts and waves, and different sequences of shocks and waves in 'profiles' and in 'history'. They encompass all the physical processes that take place in heterogeneous reservoirs during multi-phase mUlti-component displacement: phase transitions, chemical reaction and interphase mass transfer, which take place on concentration and temperature fronts; breakthrough of the displacing fluid in high­permeability zones and fractures; delayed sweep of zones of low permeability, and so forth. The analytical solutions can therefore be used for the classification of regimes of condensate and oil displacement by different gases; the determination of the laws of evolution of chemical slugs in reservoirs with sorption isotherms and solubility curves of different shapes, or with irreversible sorption; the investigation of chromatographic separation of the components of the displacing and injected fluids in the porous medium as a result of phase transitions; the analysis of the laws of interaction of water and gas slugs during WAG processes, and so on. The interpretation of results of laboratory experiments on the basis of one-dimensional analytical models provides a clear example of their worth.

Each method of enhanced recovery, usually, involves several physico-chemical or hydrodynamic mechanisms determining its efficiency. For instance, the injection of gases and solvents involves evaporation and condensation; polymer flooding leads to an increase in viscosity of the displacing phase; hot water flooding lowers the viscosity of oil; surfactant flooding causes a diminution of surface tension. One-dimensional models of displacement account for the most important of these factors. and thus permit an answer to the question: "How effective is enhanced oil recovery when a particular recovery technique is employed in a specific field?" Thus, extensive comparisons can be made between the alternative recovery methods and suitable candidates for employment in the particular conditions of the field under consideration can be selected.

The two-dimensional analytical models of displacement from stratified reservoirs and in systems of wells can be used as the basis for investigation of the recovery mechanisms of each selected recovery technique for the specific heterogeneity of an actual field (reduced sweep over the cross-section and the area of the reservoir when gas injection is used, improved sweep with polymer flooding, crossflow between layers for different degrees of heterogeneity) with a variety of systems of injection and producing wells. It is

PREFACE xix

at this stage that a definite decision is taken about the method of development of a particular field and preliminary estimates are made of the basic production parameters (injection pressure and temperature, composition of the injected gas or liquid, slug volume).

The combination of cross-section and areal models of two-dimensional displacement assists the formulation of an analytical quasi-three-dimensional model of displacement from a heterogeneous reservoir in a system of wells. This model is presented as a set of explicit analytical formulae, which permit a sensitivity study to be made of the chosen method, refinements to be made to the basic production parameters, and oil recovery estimated.

The exact analytical solutions of inverse problems are intended to be used to determine the water-saturation distribution and the distribution of residual oil-saturation from displacement histories.

Numerical methods have the obvious advantage over analytical models that they are universal, allowing simulations to be performed in reservoirs with three-dimensional heterogeneities of any type. This is particularly helpful for accurate recovery forecasting and detailed reservoir characterization.

Analytical models are, therefore, important in the initial stage when decisions are being taken on the technology and in optimization studies when the geological and physical data are inadequate. The analytical solutions of inverse problems provide primary reservoir characterization from development history. Naturally, more detailed modelling of the development process and better forecasts of oil recovery will be made on the basis of three-dimensional numerical models. The method chosen as a result of analytical modelling is then subjected to further numerical investigation. Analytical solutions of the inverse problem give the preliminary sketch of heterogeneity for reservoir characterization. This picture is then filled out by tuning the numerical model in the 3D simulator, which allows for more complex heterogeneity. Thus, exact and asymptotic solutions of direct and inverse problems can be used to augment and optimize the numerical studies.

Clearly, feasibility studies and development planning can benefit greatly from the combined use of analytical and numerical models. The experience of their use in the development planning of a number of oil and gas-condensate fields in Russia, Kazakhstan, Turkmenistan and Uzbekistan is described in the chapters that follow.

The nomenclature used in this volume accords with the conventions applied in this area in the CIS (former USSR), and differs in some respects from the SPE specifications. The reader should refer to the list given at the end of the book.

Much of the book was written on the basis of results obtained in co-authorship and in close collaboration with colleagues and students at the Department of Oil and Gas Fluid Mechanics of the Moscow State Oil and Gas Academy, with researchers from the VNIINeft and VNIIGaz, and the Hungarian Institute of Hydrocarbons. I wish especially to thank Acad. Prof. Kaplan Basniev, Russian Academy of Sciences, for his support and fruitful cooperation over the course of many years. I am indebted to my

xx PREFACE

Ph.D. student A.A. Shapiro for his invaluable help in preparation of the book: he actually wrote Chapter 2 , and contributed to discussions of many of the chapters.

I offer my heartfelt thanks to colleagues who kindly agreed to edit particular chapters and to discuss in detail the content of the book as a whole: Prof. Jan de Haan (Delft University of Technology), Prof. John Fayers and Prof. Martin Blunt (Stanford University), Prof. James Glimm (Stony Brook University), Dr. Michael King and Peter Briggs (British Petroleum), Dr Richard Wheaton (British Gas), Prof. Dabir Tehrani (Heriot-Watt University), Dr. John Barker (Elf Aquitaine), Dr Richard Dawe and Dr. Raad Issa (Imperial College of Science and Technology). The quality of exposition has profited greatly from their suggestions. I would also like to thank Prof. 1.S. Archer for his help and support in the preparation and promotion of the book.

I am especially grateful to Dr. Gren Rowan, who has willingly devoted a great deal of time and effort to discussions of terminology and improvement of the English version. The responsibility for any errors that remain is, of course, mine.

My sincere thanks are due also to Ruth Loshak for her patience, perseverance and commitment to the translation of the volume.

The translation was generously sponsored by British Petroleum, British Gas, Agip, Shell and Chevron, whom I thank for their assistance and support.

Pavel Bedrikovetsky Imperial College of Science and Technology, London March 1993