G.L. Chierici, Principles of Petroleum Reservoir Engineering . Volume 2

Contents of Volume 1

1 Hydrocarbon Reservoirs 2 Reservoir Fluids 3 Reservoir Rocks 4 The Evaluation of Oil and Gas Reserves 5 Radial Flow Through Porous Media:

Slightly Compressible Fluids 6 The Interpretation of Production Tests in Oil Wells 7 The Interpretation of Production Tests in Gas Wells 8 Downhole Measurements for the Monitoring

of Reservoir Behaviour 9 The Influx of Water into the Reservoir

10 The Material Balance Equation Author Index Subject Index

References and Exercises after each Chapter

Gian Luigi Chierici

Principles of ~ Petroleum Reservoir

Engineering

Translated from the Italian by

Peter J. Westaway

With 199 Figures and 42 Tables

Springer-Verlag

Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Professor GIAN LUIGI CHIERICI University of Bologna Chair of Petroleum Reservoir Engineering Faculty of Engineering Viale del Risorgimento, 2 1-40136 Bologna Italy

Translator

PETER 1. WESTAWAY 9 Nile Grove Edinburgh EHIO 4RE, Scotland Great Britain

Title of the Original Italian edition: Principi di ingegneria dei giacimenti petroliferi, Vol. 2

ISBN-13: 978-3-642-78245-9 e-ISBN-13: 978-3-642-78243-5 001: 10.1007/978-3-642-78243-5

Library of Congress Cataloging-in-Publication Data (Revised for vol. 2). Chierici, Gian Luigi. Principles of petroleum reservoir engineering. Includes bibliographical references and indexes. 1. Oil reservoir engineering. 2. Oil fields. 3. Gas reservoirs. I. Title. TN871.C494413 1995 622'.3382 93-26019

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad­casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1995 Softcover reprint of the hardcover 1 st edition 1995

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Typesetting: Macmillan India Ltd., Bangalore 25 SPIN: 10039904 32/3130/SPS - 5 4 3 2 1 0 - Printed on acid-free paper

Preface

Volume 1 of this book dealt with the techniques behind the acquisition, processing and interpretation of basic reservoir data. This second vol­ume is devoted to the study, verification and prediction of reservoir behaviour, and methods of increasing productivity and oil recovery.

I should like to bring a few points to the reader's attention. Firstly, the treatment of immiscible displacement by the method of

characteristics. The advantage of this approach is that it brings into evidence the various physical aspects of the process, especially its dependence on the properties of the fluids concerned, and on the velocity of displacement. It was not until after the publication of the first, Italian, edition of this book (February 1990) that I discovered a similar treatment in the book Enhanced Oil Recovery, by Larry W. Lake, published in 1989.

Another topic that I should like to bring to the reader's attention is the forecasting of reservoir behaviour by the method of identified models. This original contribution to reservoir engineering is based on systems theory - a science which should, in my opinion, find far wider applica­tion, in view of the "black box" nature of reservoirs and their responses to production processes.

Finally, the improvement of productivity and oil recovery factor, to which the last chapter of this volume is devoted. With today's low oil prices (which are expected to persist for several years yet), the use of enhanced oil recovery methods (EOR) is restricted by economic consid­erations to only the most favourable situations, which tend to be few and far between. I have tried to point out how "improved" recovery methods (lOR) - water or non-miscible gas injection - can, on the other hand, if executed using advanced reservoir management techniques, achieve substantial increases in productivity and recovery within a free market framework.

It is really only in the last decade that reservoir engineering has recognised that truly effective field development demands a full and detailed knowledge of the reservoir's internal structure (its "architec­ture"). The first phase in building a picture of this internal structure is the construction of the geological (or "static") model, using information from reservoir rock outcrops, and studying the sedimentology, petrography and petrophysics of the reservoir through the synergetic analysis of cores and well logs, and its stratigraphy and lithology from 3D seismic surveys.

But it should not be forgotten that another - and perhaps more important phase - involves the transformation of the static model to a dynamic one, capable of replicating the observed response of the re­servoir to the production process. The vital information necessary for

VI Preface

this transformation can come only from a proper program of reservoir monitoring and surveillance aimed at determining the distribution of the fluid-dynamic characteristics of the reservoir rock in the interwell regions.

The design of these tests, the interpretation of their results, the modification of the static geological model to optimise the dynamic engineering model and, subsequently, using numerical models to forecast the behaviour of the reservoir under different production scenarios, are all typical activities that occupy the engineer in the domain of reservoir management.

Reservoir management is not, however, a task to be performed in "splendid isolation". Only an approach which combines the expertise of geologists, geophysicists, petrophysicists, geostatisticians, and produc­tion and reservoir engineers can have any hope of achieving the level of understanding of internal reservoir structure that is essential for re­servoir management to be effective.

Hence the need, already being met in the more enlightened ex­ploration and production companies, of multidisciplinary teams of engineers dedicated to all aspects of the management of the reservoir, from discovery to abandonment.

It would not be proper to finish my introduction without expressing my gratitude to those who have collaborated with me in the preparation of this second volume. Firstly, my thanks to the translator, my friend Peter 1. Westaway, who, in addition to producing an impeccable trans­lation, has drawn on his own reservoir engineering experience to suggest improvements to the text. More than a translator, I almost consider him to be the co-author of this book! Once again, thank you Peter.

My thanks also go to Prof. Guido Gottardi, Professor of the Faculty of Engineering at the University of Bologna, and co-lecturer of my course there, whose suggestions for modifications and improvements were so helpful in the preparation of the Italian edition of this book. Prof. Gottardi also provided the computer program that appears in Exercise 12.4. I should also like to thank Mrs. Irma Piacentini and Mr. Carlo Pedrazzini for their meticulous, patient and highly professio­nal work in drawing all of the figures appearing in this volume.

Once again, thank you everybody!

Bologna October 1994

GIAN LUIGI CHIERICI

Contents

Most Commonly Used Symbols . . . . . . . . . . . . . . . . . .

Conversion Factors for the Most Commonly Used Quantities.

11 Immiscible Displacement in Homogeneous Porous Media

11.1 11.2 11.3 11.3.1 11.3.2 11.3.2.1 11.3.2.2 11.3.2.3 11.3.3

11.3.4

11.3.5 11.3.6 11.4 11.4.1 11.4.2 11.5 11.5.1

11.5.2

11.5.3

Introduction ................... . Basic Assumptions ............... . Displacement in One-Dimensional Systems .. The Fractional Flow Equation ........ . The Buckley-Leverett Displacement Equation Case 1 - Fractional Flow Curve fw(Sw) Concave Downwards . Case 2 - Fractional Flow Curve fw(Sw) Concave Upwards. Case 3 - Fractional Flow Curve fw(Sw) S-Shaped ..... Calculation of the Average Saturation Behind the Front: Welge's Equation .................... . Calculation of Oil Recovery as a Function of Time, and the Fraction of Displacing Fluid in the Production Stream ............... . The Effect of Oil Viscosity and Flow Rate on Displacement. Frontal Instability: Fingering .................. . Displacement in a Two-Dimensional System (x, z) . ..... . Segregated Flow - the Concept of Vertical Equilibrium (VE) Gravity Stabilisation of the Front in Segregated Flow .... Relative Permeability and Capillary Pressure Pseudo-Curves Total Layer Thickness ht Much Less Than the Height he . . . of the Capillary Transition Zone ................ . Total Layer Thickness ht of the Same Order as the Height he of the Capillary Transition Zone ................ . Total Layer Thickness hI' Much Greater Than the Height he of the Capillary Transition Zone.

xv

xxv

1 2 4 4 7 9

10 13

16

18 20 22 24 24 27 34

35

35

38

References . . . . . . . 40

Exercises . . . . . . . . 41 11.1 Calculation of Sw.f and ER,o at water breakthrough for three different water/oil viscosity ratios, p. 41. - 11.2 Calculation of Sw,f, (fw)BT and ED for the vertical displacement of oil by water, at three different frontal velocities, p. 44. - 11.3 Calculation of the critical velocity for the displacement of oil by gas, for five different sets of rock and fluid properties. p. 47. - 11.4 Calculation of the oil recovery factor for vertical displacement by gas, for five different velocities of the gas front. p. 48.

12 The Injection of Water into the Reservoir

12.1 12.2

Introduction ................ . The Development of Water Injection Technology.

55 56

VIII

12.3 12.4 12.5 12.5.1 12.5.2 12.5.2.1 12.5.2.2 12.6

12.6.1 12.6.2 12.7

12.7.1 12.7.2

Contents

Factors Influencing Oil Recovery by Water Injection. ED: Microscopic Displacement Efficiency Ev: Volumetric Efficiency .. . EA : Areal Sweep Efficiency ........ . E1: Vertical Invasion Efficiency ..... . Layers Not in Vertical Communication. Layers with Vertical Communication .... Deformation of the Water/Oil Contact in the Near-Wellbore Region: "Water Coning" ...... . Calculation of the Critical Flow Rate for Water Coning. Production Above the Critical Rate ............ . Surveys and Tests in Reservoirs Produced Under Water Injection ....... . Monitoring the Advance of the Water Front. Interwell Testing.

59 61 62 63 65 65 71

75 77 82

86 87 89

References . . . . 97

Exercises . . . . . 98 12.1 Calculation of fw as a function of ER.o, and ER.o(t) for the injection of water in a direct line drive pattern. p. 98. ~ 12.2 Calculation of the areal sweep efficiency EA(fw) for water injection in direct and staggered line drive patterns, at different values of Mwo. p. 101. ~ 12.3 Calculation of the pseudo-relative permeability curves for a multilayered reservoir, with vertical interlayer communication. p. 104. ~ 12.4 Program to calculate the pseudo-curves of relative permeability and capillary pressure in segregated and dispersed flow regimes for reservoirs consisting of several layers in vertical communication. p. 107. ~ 12.5 Calculation of the vertical invasion efficiency curve E1(fw) as a function of the fraction of water produced, for a reservoir consisting of several layers not in vertical communication. p. 114. ~ 12.6 Calculation of the critical oil flow rate for water coning for different penetration ratio and reservoir anisotropy. p. 118. ~ 12.7 Calculation of the time to water breakthrough, and the subsequent behaviour of fw in a well producing oil above the critical rate for water coning. p. 119.

13 The Simulation of Reservoir Behaviour Using Numerical Modelling

13.1 13.2 13.3 13.3.1

13.3.2 13.3.3

13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.10.1 13.10.2 13.10.3 13.11

Introduction ............................... . The Philosophy and Methodology Behind Numerical Modelling. Types of Numerical Model ...................... . Classification Based on the Way in Which the Flow Equations Are Discretised Classification Based on Reservoir Geometry . Classification Based on the Number and Nature of the Mobile Phases ... The Continuity Equation ............. . The Flow Equation ................. . Single Phase Flow of a Slightly Compressible Fluid. Single Phase Flow of Gas . . . . . . . . . . . . . . . . Multiphase Flow .................... . Generalised Compositional Equation for Multiphase Flow Discussion of the Flow Equations Monophasic Flow. Biphasic Flow . . . . . . . . . . . . Triphasic Flow. . . . . . . . . . . . Basic Principles of the Finite Difference Method

123 123 125

125 126

129 130 132 134 136 137 139 141 141 141 142 142

Contents

13.11.1 13.11.2 13.11.3 13.11.4

13.11.5 13.12 13.12.1 13.12.2

13.13

13.13.1 13.13.2 13.13.2.1 13.13.2.2 13.13.2.3 13.13.2.4 13.13.3 13.13.3.1 13.13.3.2 13.13.3.3 13.13.3.4 13.13.3.5 13.13.4 13.13.4.1 13.13.4.2 13.13.4.3 13.14 13.14.1 13.14.2 13.14.3 13.14.3.1 13.14.3.2 13.14.3.3 13.14.3.4 13.15

13.16 13.16.1 13.16.2 13.16.3 13.16.4 13.16.5 13.16.6 13.16.7 13.16.8 13.16.9 13.16.10 13.16.11

13.17 13.17.1 13.17.2 13.17.3 13.17.4 13.17.5 13.18

Discretisation ...................... . Types of Grid (or Mesh) ................ . Discretisation of Derivatives into Finite Differences. Choice of Time for the Spatial Derivative, and the Conditions for Stability ...... . Rounding and Truncation Errors ...... . Numerical Simulation of Single Phase Flow The Finite Difference Equation for Single Phase Flow Matrix Form of the Finite Difference Equation for Single Phase Flow . . . . . . . . . . . . . . . . . . . Solution of the Linear Algebraic Equations Derived from the Discretisation of the Flow Equation Introduction ............. . Direct Methods . . . . . . . . . . . . The Gaussian Elimination Method. Triangular Matrices. . . . . . . . . . . . Factorisation of the Coefficient Matrix The Ordering of Sparse Matrices. Iterative Procedures ........... . Introduction ............... . The Point Jacobi Method (Simultaneous Displacement) The Point Gauss-Seidel Method (Successive Displacement) Point Successive Over-Relaxation (SOR) ...... . A Simple Example of the Use of Iterative Methods . Methods for Alternating Directions Introduction .... The AD! Method . . . . . . . . . . . The lAD! Method . . . . . . . . . . Numerical Simulation of Multiphase Flow. The Finite Difference Equation for Multiphase Flow. Numerical Dispersion and Its Treatment in Multiphase Flow. Solution of the Finite Difference Equation for Multiphase Flow Choice of Solution Method ....... . The IMPES Method .......... . The Method of Simultaneous Solution .. The Fully Implicit Method ....... . Introduction to the Study of Reservoir Behaviour Through Numerical Modelling ........... . Gathering, Editing and Preprocessing the Basic Data. Choice of Gridding for Reservoir Discretisation . Gross and Net Pay Thickness for Each Layer. Porosity and Pore Compressibility. Horizontal Permeability .. Vertical Permeability .... The Correlation Parameter Capillary Pressure Curves . Relative Permeability Curves PVT Properties . . . . . . . . Initial Gas/Oil and Water/Oil Contacts. Initial Pressure and Temperature at the Reference Datum, and the Temperature Gradient .... . Initialisation of the Model ....... . Calculation of Geometrical Parameters Calculation of Phase Pressures ..... Calculation of Capillary Pressures and Saturations Calculation of the Initial Volume of Fluids in the Reservoir. Plotting the Iso-Value Maps. . ........ . Choice of Model and Method of Simulation ......... .

IX

142 143 144

146 149 150 150

153

155 155 155 155 157 157 159 159 159 160 161 161 162 164 164 164 165 167 167 170 173 173 174 178 179

180 182 182 183 184 184 185 187 187 188 189 190

190 191 191 191 192 192 193 194

x

13.18.1 13.18.2 13.18.3

13.19

13.19.1 13.19.2 13.19.3 13.19.4 13.20 13.20.1 13.20.2 13.20.2.1 13.20.2.2 13.20.2.3 13.20.3 13.20.3.1 13.20.3.2 13.20.3.3 13.21

14

14.1 14.2 14.2.1 14.2.2 14.2.3 14.3 14.3.1 14.3.2 14.3.2.1 14.3.2.2 14.3.3 14.3.4 14.3.4.1 14.3.4.2 14.3.4.3 14.3.5

14.3.5.1

Choice of Model .................. . Convergence and Stability Criteria ........ . Simulation of the Displacement Process: Choice of Relative Permeability and Capillary Pressure Pseudo-Curves. Validation of the Numerical Model by Matching the Reservoir History. . . . . . . . . . Scope and Methodology of the Validation Process Data Needed for History Matching .. Use of the Model in History Matching Checking the History Match . . . Forecasting Reservoir Behaviour. Introduction ...... . Production Constraints . . . Well Constraints ....... . Surface Facility Constraints. Reservoir Constraints .... Calculation of Well Flow Rates. Introduction . . . . . . . . . . . . Wells Producing at Fixed Rates Wells Producing at Deliverability Summary.

Contents

194 194

195

198 198 198 199 201 202 202 202 202 203 204 204 204 206 209 209

References . . . . . . . . . . . . . . 210

Exercises . . . . . . . . . . . . . . . 213 13.1 Derivation of the continuity equation in radial coordinates. p. 213. - 13.2 Derivation of an expression for the coefficient a in Eq. (13.16b) for one- and two-dimensional geometries. p. 214. -13.3 Derivation of equations to describe the behaviour of a reservoir containing a moderately volatile oil, and a black oil, from the generalised compositional equation (Eq. (13.41)] for polyphasic flow. p. 217. - 13.4 Derivation of Eq. (13.77), for the z-axis component of the transport term. p. 220. - 13.5 Derivation of the finite difference equations for three-dimensional polyphasic flow. p. 222. - 13.6 Expression of the general transport term in fully implicit form, eliminating non-linearity. p. 226.

Forecasting Well and Reservoir Performance Through the Use of Decline Curves and Identified Models

Introduction .................. 231 Production Decline Curves ......... 231 Conditions for the Use of Decline Curves. 231 The Characteristics of Decline Curves . . . 232 Practical Applications of the Decline Curve Method 234 Identified Models ..................... 236 The Reservoir and Aquifer as a Single Dynamic System 236 The Basics of Systems Theory. 237 Definitions .......... 237 The "z-Transform" ....... 239 Discrete Dynamic Systems ... 241 The Model Identification Procedure. 242 Ordering of the Model ......... 242 Determination of the Parameters of the Model 244 Evaluation of Model Stability ........... 244 Practical Example: Identification of a Gas Reservoir with an Aquifer .... 244 Initial Considerations. . . . . . . . . . . . . . . . . . . 244

Contents

14.3.5.2 14.3.5.3 14.3.5.4 14.3.5.5

The Minerbio Gas Reservoir . . . . . . . . . . . . . . Determination of the Order of the Model ...... . Investigation of the Conditions for Time Invariance Validation of the Identified Model .

References . . . . . . . . . . . . . . . . . . . . . . . . .

Exercises ......................... . 14.1 Use of decline curves to determine the incremental cumulative production at abandonment of a reservoir where infill wells have been drilled. p. 251.

15 Techniques for Improving the Oil Recovery

15.1 15.2 15.3 15.3.1 15.3.2

15.3.2.1 15.3.2.2 15.3.3

15.3.3.1 15.3.3.2 15.3.3.3 15.3.3.4 15.4

15.4.1 15.4.1.1 15.4.1.2 15.4.1.3 15.4.2 15.4.2.1

15.4.2.2

15.4.2.3

15.5

15.5.1 15.5.2 15.6 15.6.1

15.6.2 15.6.3 15.6.4 15.7 15.7.1 15.7.2 15.7.3 15.8 15.8.1

Introduction ............................. . The Current Status of the World's Oil Reserves ........ . Quantity and Distribution of Oil Remaining in the Reservoir. Introduction ............................. . Methods for Evaluating the Overall Quantity of Remaining Oil . . . . . . . . The Material Balance Method ......... . Single Well Tracer Test ............. . Determination of the Distribution of Remaining Oil Throughout the Reservoir . . . . . . . . . . . Retrieval and Analysis of Downhole Cores. Resistivity Logs . . . . . . . . . . Pulsed Neutron Logs (PNL) ........ . Nuclear Magnetism Log (NML) ...... . ARM - Advanced Reservoir Management: Improvement of the Oil Recovery Factor by Optimisation of Water or Gas Injection .................. . ARM for Water Injection .................. . The Effect of Well-Spacing on Oil Recovery by Water Injection Why Infill Drilling Can Lead to an Increase in Oil Recovery . Optimising Infill Well Location by ARM Techniques. Crestal Injection of Non-Miscible Gas ............. . Unidirectional Displacement of Oil by Gas in the Presence of Siw' • . . . . • . . . . .. ..•....•..•..

Vertical Displacement of Oil by Gas in a Gravity-Stabilised Regime ......... . Production of Oil Through Vertical Displacement by Gas Under Gravity-Dominated Conditions. Enhanced Oil Recovery (EOR): Definition and Classification of Methods . . . . Introduction .................. . Types of Enhanced Oil Recovery Methods . . . . . Steam Drive . . . . . . . . . . . . . . . . . . . . . . . The Influence of Temperature on Reservoir Rock and Fluid Properties ........... . Steam Drive Technology. . . . . . . . . . Problems Encountered in the Reservoir. Summary .......... . Huff'n'puff, or Steam Soak Origins of the Method . . . Steam Soak Technology .. Reservoir Engineering Considerations . In Situ Combustion .... Principles of the Method. . . . . . . . .

XI

245 246 247 249

250

251

255 255 257 257

258 258 258

261 262 262 263 265

269 269 269 272 273 276

276

277

278

280 280 282 284

284 287 291 293 294 294 294 296 297 297

XII

15.8.2 15.8.3 15.8.4 15.8.5 15.9 15.9.1 15.9.2

15.9.3

15.9.3.1 15.9.3.2 15.9.3.3 15.9.3.4 15.9.4 15.9.4.1 15.9.4.2

15.9.4.3

15.9.5

15.9.5.1 15.9.5.2 15.9.5.3 15.9.5.4 15.9.6 15.9.7 15.9.8 15.9.9 15.9.10 15.10 15.10.1 15.10.2 15.10.3 15.10.3.1 15.10.3.2 15.10.4 15.10.5 15.10.6 15.11 15.11.1 15.11.2 15.11.3 15.11.3.1 15.11.3.2 15.11.3.3 15.11.4 15.11.5 15.11.6 15.11.7 15.12 15.12.1 15.12.2 15.12.3 15.12.4 15.13

Contents

What Happens in the Reservoir. . . . . . . . . 299 Wet Combustion . . . . . . . . . . . . . . . . . 301 Injection of Oxygen or Oxygen-Enriched Air. 302 Summary. . . . . . . . . 302 Miscible Gas Flooding. . . . . . . . . . . . . . 303 Introduction . . . . . . . . . . . . . . . . . . . . 303 Pseudo-Ternary Diagrams for the Reservoir Oil/Injection Gas Mixture. . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Phase Behaviour of a Mixture of Reservoir Oil/Gaseous Hydrocarbon and/or Nitrogen . . . . . . . . . . . . . . . . 304 Use of Pseudo-Ternary Diagrams to Describe Phase Behaviour. 304 First Contact Miscibility. . . . . . . . . . . . . . . . . . 305 Multiple Contact Miscibility: Condensing Gas Drive. . . . . . . 305 Multiple Contact Miscibility: Vaporising Gas Drive . . . . . . . 306 Phase Behaviour of a Mixture of Reservoir Oil and Carbon Dioxide 308 Thermodynamic Properties of Carbon Dioxide. . 308 Type I Behaviour of a Mixture of Reservoir Oil and Carbon Dioxide . . . . . . . . . . . . . . . . . 309 Type II Behaviour of a Mixture of Reservoir Oil and Carbon Dioxide . . . . . . . . . . . . . Correlations for the Estimation of Minimum Miscibility Pressure (MMP) . Vaporising Gas Drive .......... . Condensing and Vaporising Gas Drives. . . . Miscibility of Oil and Nitrogen ........ . Miscible Displacement with Carbon Dioxide. Flow Regimes Encountered in Miscible Gas Drive Displacement Under Gravity-Stabilised Conditions. Miscible Gas Drive Technology ........... . Immiscible Flooding and Huff'n puff Using Carbon Dioxide Summary ..... . Polymer Flooding. Introduction ... . Polymers Used .. . Rheology of Polymer Solutions. Overview of Rheology . . . . . . Polymer Solutions ....... . Improvement of the Permeability Profile with Polymer Gels Polymer Technology ... . Summary .......... . Micellar/Polymer Flooding Introduction ........ . Water/Oil/Surfactant Micellar Solutions Phase Behaviour of Water/Oil/Surfactant + Cosurfactant Systems Type 11(-) Systems Type II( +) Systems . . . . . . . . . . . . . . . . . . . . . . . . . Type III Systems . . . . . . . . . . . . . . . . . . . . . . . . . . The Effect of Water Salinity on the Properties of the System Adsorption by the Reservoir Rock (Surfactant Retention) Micellar-Polymer Flooding Technology. Summary ..... Caustic Flooding . . . . . . . . . . . . Introduction .............. . The Interactions Behind the Process. Caustic Flooding Technology .... . Summary ................ . Monitoring EOR Performance in the Reservoir.

311

313 313 314 314 315 316 318 319 321 323 324 324 324 326 326 328 329 330 332 333 333 333 335 335 336 336 338 340 341 341 343 343 343 344 344 346

Contents

IS.13.1 IS.13.2 IS.13.2.1

IS.13.2.2

15.13.3 15.13.3.1 IS.13.3.2 IS.13.3.2.1 IS.13.3.2.2 IS.13.4 IS.14 15.1S

Classical Methods .............. . Seismic Methods. . . . . . . . . . . . . . . . Spatial Distribution of Fluid Saturations by 3D Seismic Surveys ........... . Interwell Distribution of Fluid Saturations by Cross-Well Seismics. . . . . . . . . . . Electrical and Electromagnetic Methods DC Electrical Techniques ........ . Electromagnetic (EM) Methods. . . . . . Controlled Source Audio Magnetic Tellurics (CSAMT). High-Frequency Electromagnetic (HFEM) Method .. . Summary ........................... . Selection of the Appropriate EOR Process for a Given Field The Future of EOR.

References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Exercises ............................ . 15.1 Derivation of the equations relating oil recovery at breakthrough and N gV' and the Corey n-exponent, for vertical non-miscible gas displacement. p. 369.

Author Index .

Subject Index .

XIII

346 347

348

349 349 350 350 350 351 351 353 355

363

369

377

381

Most Commonly Used Symbols

Letter Quantity Dimensions Symbol

A area U A matrix of coefficients

b well penetration from top of reservoir, also: L hw/h,

bD dimensionless well penetration Br formation volume factor of fluid "i" (f = g, 0, w) Bg gas formation volume factor Bgi gas formation volume factor, under initial reservoir

conditions Bo oil formation volume factor Bod oil formation volume factor, for differential gas liberation Bor oil formation volume factor, for flash gas liberation Boi oil formation volume factor, under initial reservoir

conditions Bw water (brine) formation volume factor

C compressibility m- I Ltl Cr coefficient of porosity variation with pressure

(pore compressibility) m-ILtl cg gas compressibility m-ILtl

Co oil compressibility m- I Lt2

c, total compressibility (pore volume + fluids) m-ILtl Cw water (brine) compressibility m-ILtl C tracer concentration in the injected water, also: mL -3

aquifer constant [Eq. (9.25)] m- 1 L4 t2 Ci • g mass fraction of component "i" in the gaseous

hydrocarbon phase Ci •o mass fraction of component "i" in the oil phase Ci •w mass fraction of component "i" in the water phase Co tracer concentration in the oil phase mL -3

Cr specific heat capacity of the rock UC 2 T- 1

C, solvent concentration in the miscible phase (volume ratio) Cw tracer concentration in the water phase mL -3

d diameter, also: L distance L

d column vector of constants various D depth L D(t) rate of oil production decline [Eq. (14.1)] t- I

DI longitudinal dispersion coefficient Ut- I

Do molecular diffusion coefficient UC I

XVI Most Commonly Used Symbols

Letter Quantity Symbol

J

areal sweep efficiency microscopic displacement efficiency microscopic displacement efficiency at breakthrough of displacing fluid vertical (invasion) efficiency recovery factor oil recovery factor final recovery factor at reservoir abandonment volumetric (invasion) efficiency

average number of shale lenses, per unit thickness of reservoir rock signal sampled at time t, in a discrete system displacing fluid fraction gas fraction Larmor frequency of proton-free precession oil fraction water fraction

fw.a water fraction in produced liquids, at reservoir abandonment

9 G

G(z) GOC GOR Gp

GWC GWR

h

he ho.v ho ht

hv hw llH

Ho.v (HO.V)BT He HK Hp

iw

force retardation factor [Eq. (15.4)J fraction of reservoir rock volume consisting of impermeable lenses

acceleration due to gravity gas in place in the reservoir, total initial, under standard conditions transfer function in a discrete system gas/oil contact gas/oil ratio gas produced, cumulative, under standard conditions gas/water contact gas/water ratio

thickness, also: Planck's constant, n (6.625 ·10- 34 J. s) thickness of capillary transition zone dimensionless height of water cone oil column thickness at the well total layer thickness height of water cone, above static water/oil contact water column thickness at the well specific enthalpy of steam, relative to original reservoir temperature correlation parameter [Eq. (12.52a)J Ho. v at water breakthrough by coning [Eq. (12.53b)] Earth's magnetic field, tesla normalized iteration parameter [Eq. (13.96b)] polarizing magnetic field, tesla

unit x-direction Cartesian vector injected water rate

Dimensions

L -I

various

L

L mL2 t- 1

L

L L L L

Ut- 2

mt-1q-1 m -I L3t mt- I q-I

L3 t- I

Most Commonly Used Symbols XVII

Letter Symbol

Quantity Dimensions

j J(S) Jg

Jgo(Sg) J I

Jo

Jw

Jwo(Sw)

unit y-direction Cartesian vector Leverett J -function gas productivity index Leverett J-function, gas-oil systems productivity index of fluid "1" oil productivity index water productivity index Leverett J-function, water-oil systems

m- I L 4 t m- I L 4 t m- I L 4 t

k permeability U k average permeability L 2

k unit z-direction Cartesian vector [k] permeability tensor L 2

k (superscript) iteration counter: pk = pressure value computed at the k-th iteration in the iterative ADI (IADI) method

kg permeability, effective, to gas L 2

kh horizontal permeability L 2

ko permeability, effective, to oil L 2

kr radial permeability, also: L 2

relative permeability krg permeability, relative, to gas k:g normalized relative permeability to gas [Eq. (13.124b)] kro permeability, relative, to oil k:'., normalized relative permeability to oil [Eqs. (13.123a)

and (13.124a)] krw permeability, relative, to water k:w normalized relative permeability to water [Eq. (13.123b)] kro pseudo-relative permeability to oil, in a multilayer

pay zone krw pseudo-relative permeability to water, in a multilayer

pay zone krg.or endpoint relative permeability to gas (Sg = 1 - Siw - Sor) kro,iw endpoint relative permeability to oil (So = 1 - Siw) krw,or endpoint relative permeability to water (Sw = 1 - Sor) kro,iW endpoint pseudo-relative permeability to oil in a

multilayer pay zone kv vertical permeability L 2

kx average permeability in the Cartesian x-direction L 2

ky average permeability in the Cartesian y-direction L 2

kw permeability, effective, to water U K thermal diffusivity, also: L 2 t - I

tracer partition equilibrium constant [Eq. (15.2)] Kani permeability anisotropy coefficient Ke dispersion coefficient in miscible displacement U t - I

Ki equilibrium flash vaporization ratio for component "i" Ki,go mass partition coefficient between gas and hydrocarbon

liquid phase, for component "i" Ki,gw mass partition coefficient between gas and water phase,

for component "i"

unit radial-direction vector L ~ L L lower triangular matrix

XVIII Most Commonly Used Symbols

Letter Symbol

m

ms mtot

mw M M Mgo Ms Mwo MMP

n

Quantity

mass, also: real gas pseudo-pressure function, also: cementation factor [Archie's Eq. (15.8a)], also: ratio of the initial hydrocarbon pore volume of the gas cap to that of the oil, both under reservoir conditions, also: system order, in a discrete, sampled-data system tracer mass flow rate resulting from dispersion mass, gas phase mass of component "i" mass, liquid hydrocarbon phase, also: tracer mass flow rate in absence of dispersion mass of injected steam total mass flow rate of tracer = mo + mdis

mass of water (brine) phase molecular weight (mass) nuclear magnetization mobility ratio, gas/oil mobility ratio, diffuse-front approximation [Eq. (11.33)] mobility ratio, water/oil minimum miscibility pressure

number of components in a system, also: saturation exponent [Archie's Eq. (l5.8a)], also: reciprocal of decline-curve exponent [Eq. (14.2a)]

n (superscript) time level (pO means "pressure at time to") N oil in place in the reservoir, initial, under standard

P P Pb Pg

Pi pL.k PK Po Ptf Pw Pwf PwLU

conditions gravity-to-viscous force ratio [Eq. (15.11)] cumulative oil production, under standard conditions cumulative oil production, fraction of mobile oil in place Peclet number [Eq. (12.64)] mobile oil in place [ = VR <)1(1 - Siw - Sor)] volume of remaining oil in the reservoir, under standard conditions capillary number (viscous-to-capillary force ratio) gravity number (viscous-to-gravity force ratio) modified gravity number [Eq. (15.27)] number of grid blocks in the Cartesian x-direction number of grid blocks in the Cartesian y-direction number of grid blocks in the Cartesian z-direction

pressure average pressure pressure, bubble point (saturation) pressure, in the gas phase initial pressure, static pressure in model block (i, j, k), at time to average pressure in model block K, at well bore pressure, in the oil phase tubing head pressure, flowing pressure, in the water (brine) phase bottomhole pressure, flowing bottomhole pressure, flowing, in the uppermost model block flowing into the well

Dimensions

m t- 1

m m m mt- 1

m m t- 1

m m L-1t-1q

mL -It- 2

mL- 1 t- 2

mL- 1 t- 2

mL- 1 C 2

mL- 1 t- 2

mL- 1 t- 2

mL -It- 2

mL- 1 t- 2

mL- 1 t- 2

mL- 1 t- 2

mL- 1t- 2

mL- 1 C 2

Most Commonly Used Symbols

Letter Symbol

PWQC

Pws Pc Pc p;1

Pc, go PC,OW

qcrit

qg qg,sc qi q]

lim lim,g

lim, 0

lirn,w qo qo,crit

qo,sc qsc qt

Ii, Ii"g Ii" 0

Ii"w

qw

r rh

R Rk+ 1

Ri,j,k

Ro.i

ROS Rs Rsd Rsf RSi Rsw Rt.R

Quantity

pressure, at water/oil contact bottomhole pressure, static capillary pressure average capillary pressure inverse function of Pc capillary pressure between oil and gas phases capillary pressure between oil and water phases

critical (maximum) volumetric flow rate for front stabilization, reservoir conditions volumetric gas flow rate, reservoir conditions volumetric gas flow rate, standard conditions initial volumetric flow rate volumetric flow rate of fluid "/" (l = g, 0, w) produced from a model block mass rate (source/skin) term for single-phase flow mass rate (source/skin) term for flow of gas in equilibrium under standard conditions with oil mass rate (source/skin) term for flow of oil in equilibrium under standard conditions with gas mass rate (source/skin) term for flow of water volumetric oil flow rate, reservoir conditions critical (maximum) volumetric flow rate of oil, standard conditions, for water coning volumetric flow rate of oil, standard conditions flow rate, standard conditions volumetric flow rate of total fluid produced, also: volumetric flow rate at time "t" volumetric flow rate of fluids produced volumetric flow rate of gas produced, standard conditions volumetric flow rate of oil produced, standard conditions volumetric flow rate of water produced, standard conditions volumetric flow rate of water

radius radius of reservoir rock heated by the injected steam gas constant average of the absolute value of residuals after the (k + l)th iteration [Eq, (l3,90a)] residual in the material balance equation for model block (i, j, k) [Eqs. (13.98a) and (13.111a)] resistivity, formation 100% saturated with injected water of resistivity Rw, i remaining oil saturation (also Rso) gas solubility in oil gas solubility in oil, from differential liberation gas solubility in oil, from flash liberation gas solubility in oil, at initial pressure gas solubility in water true resistivity of reservoir rock (external to the flushed zone)

XIX

Dimensions

mL- I t- 2

mL- I t- 2

mL- I t- 2

mL- I t- 2

L3 t- I

L3t- 1

L3C l

L3t- 1

L3t- 1

m t- I

mC I

m t- I

mC I

L3 t- I

L3t- 1

L3 t- I

L3t- 1

L3 t- I

L3t- 1

L3t- 1

L3t- 1

L3 t- I

L3C I

L3 t- I

L L mL2C 2T- I

mL3t- I q-2

mUt- I q-2

xx Most Commonly Used Symbols

Letter Quantity Dimensions Symbol

RT summation of the absolute value of material balance residuals of all model blocks [Eq. (13.113)J

Rv oil solubility in the gas phase RW.i resistivity, injected water mL3t- I q-Z

RW.R resistivity, reservoir water mL3CI q-Z

s spin quantic number S cross-sectional area, also: U

skin factor So saturation, displacing fluid Sr saturation, fluid "I" (f = g, 0, w) Sg saturation, gas S* g normalized saturation, gas [Eq. (13.124c)J Sgr saturation, residual gas Si saturation, fluid ";" SiW saturation, irreducible (critical) water

~w average irreducible water saturation

~w average irreducible water saturation in a multilayer pay zone

So saturation, oil S* 0 normalized saturation, oil [Eq. (15.13b)] Sor saturation, residual oil

~or average residual oil saturation Sor average residual oil saturation in a multilayer pay zone Sor.g residual oil saturation after oil displacement by gas SOR steam/oil ratio Sw saturation, water S* w normalized saturation, water [Eq. (13.123c)]

~w average saturation, water Sw average water saturation in a multilayer pay zone Swe water saturation at the outlet end of the porous medium Sw.r water saturation at the displacing fluid front Swi saturation, initial, water

t time M time step tBT time to breakthrough to dimensionless time (to)BT dimensionless time to water breakthrough in a well,

owing to water coning T temperature, also: T

transmissibility m- I L4 t

Tc temperature, critical T Tr., transmissibility to fluid "/" (f = g, 0, w) in the "s"

direction (s = x, y, z) in multiphase flow m- I L4 t

Tr temperature, reduced TR temperature, reservoir T T, single-phase transmissibility in the "s" direction

(s = x, y, z) m- I L4 t

TI longitudinal spin-lattice relaxation time constant t Tz transversal spin-spin relaxation time constant Tf free-precession signal decay time constant

Most Commonly Used Symbols XXI

Letter Quantity Dimensions Symbol

U column vector of unknowns various u(t) input signal, in a discrete system various ug Darcy velocity of gas L t- I

Uj vector "Darcy velocity", fluid "i" Lt- I

U ix Cartesian x-direction component of vector Uj Lt- I

Ujy Cartesian y-direction component of vector Uj L t- I

Uiz Cartesian z-direction component of vector Uj Lt- I

Uo Darcy velocity of oil Lt- I

up vector "Darcy velocity" of primary tracer (Sect. 15.3.2.2) L t- I

Us, critical (maximum) Darcy velocity for miscible displacement under stabilized front conditions L t- 1

U, Darcy velocity of total fluid = qJ A L t- I

(u,)crit critical (maximum) Darcy velocity for front stabilization, reservoir conditions L t- 1

Uw Darcy velocity of water (brine) Lt- 1

Uw vector "Darcy velocity" of water (brine) Lt- 1

Uz Cartesian z-direction component of Darcy velocity L t- I

U upper triangular matrix U(z) z-transform of the input signal, in a discrete system

v velocity Lt- I

v(S) velocity of fluid saturation S moving through the porous medium L t- 1

Vf front velocity L t- 1

Vw vector displacing water velocity, at microscopic level L t- I

V volume L3

Vb model block volume ~Xj~Yj~Zk L3 Vg volume of gas phase, reservoir conditions L3 Vrn Darcy mass velocity mL -2t- 1

Vo volume of oil phase, reservoir conditions L3 Vp pore volume L3

~VP.j.j.k pore volume in model block (i, j, k) L3 VR volume, reservoir rock L3 Vv reservoir rock volume flooded by the injected fluid L3

Vw volume, connate water, reservoir conditions L3

W width, also:

We Wj

Wp WAR WC (WC)lirn

WOC WOR

y(t) Y(z)

reservoir water volume, standard conditions L3 encroached water volume, standard conditions L 3 cumulative volume of injected water, standard conditions L 3

cumulative volume of produced water, standard conditions L 3 water/air ratio, in wet combustion water cut in well stream limiting value for the water cut, well perforated throughout the oil-bearing and water-bearing intervals of reservoir rock (Eq. (12.58)] water-oil contact L water/oil ratio

output signal, in a discrete system various z-transform of the output signal, in a discrete system

XXII Most Commonly Used Symbols

Letter Quantity Dimensions Symbol

Z vertical depth below datum, also: L gas compressibility factor

Z gas compressibility factor at average pressure p ZD dimensionless height above the static water/oil

contact [Eq. (12.45b)J ZGOC vertical depth of the gas/oil contact, below datum L Zi gas compressibility factor at initial reservoir pressure Pi Zi.j.k vertical depth below datum of the centre of model block

(i, j, k) L ZWOC vertical depth below datum of the water/oil contact L

Greek (1 geometrical factor [Eq. (13.16b)J various (11 dispersivity, longitudinal L (1MA dispersivity, at macroscopic scale L (1ME dispersivity, at megascopic scale L

f3 geometrical factor [Eq. (13.30)J various

Y proton gyromagnetic ratio m- 1 q

Y shear rate [Eq. (15.34b)J t- 1

Yg gas relative density (air = 1.00), standard conditions

b(t - kt!t) unit step function, in sampled-data systems

e an infinitesimal, or very small, quantity various

f} dip angle of rock layer e c contact angle

Ag gas mobility = kkrg /Ilg m- 1 L3 t Ao oil mobility = kkro/J.!o m- 1L3 t Aw water mobility = kkrw/J.!w m- 1 L3 t

J1 viscosity mL- 1 t- 1

J1app viscosity, apparent, non-Newtonian fluid mL- 1 t- 1

J1g viscosity, gas mL-1t- 1

J10 viscosity, oil mL- 1 t- 1

J1s viscosity, solvent, in miscible displacement mL- 1 t- 1

J1w viscosity, water mL -It- 1

Ph rock bulk density mL -3

Pr density, pore fluid mL -3

Pg density, gas, reservoir conditions mL -3

Pg.sc density, gas, standard conditions mL -3

Pi density, fluid "i" mL -3

Po density, oil, reservoir conditions mL -3

Po,sc density, oil, standard conditions mL -3

Pr density, rock grains mL -3

Ps' density, solvent, in miscible displacement mL -3

Pw density, water, reservoir conditions mL -3

Pw,sc density, water, standard conditions mL -3

Most Commonly Used Symbols XXIII

Letter Symbol

Quantity Dimensions

Xo

interfacial tension iteration parameter, in the iterative alternating direction implicit (lADI) method [Eq. (13.96b)] interfacial tension, microemulsion/oil interfacial tension, microemulsion/water in terfacial tension, oilj gas interfacial tension, microemulsion at the invariant point (type III systems)/oil interfacial tension, microemulsion at the invariant point (type III systems}/water interfacial tension, water/oil neutron capture cross section, per unit volume

tortuosity, hydraulic, also: shear stress Coats' vertical equilibrium parameter [Eq. (11.49)]

porosity free fluid porosity, NML dimensionless fluid potential [Eq. (12.45a)] IPD value at top of water cone fluid potential per unit mass, at the external boundary of the drainage area gas potential per unit mass oil potential per unit mass oil potential per unit mass at top of water cone water potential per unit mass bottomhole flowing potential, per unit mass

static nuclear spin susceptibility

Ut- 2

Ut- 2

L 2 t- 2

Ut- 2

Ut- 2

Ut- 2

w iteration parameter, SOR Q function [Eq. (12.47)] used in calculating the critical oil

rate in water coning

Mathematical Symbols, and Operators

div

F(s) F(z)

grad

£U(t)] z-m v V'

~tU

o Dx 0 Dy 0 Dz divergence: div " = - + - + -ox oy oz Laplace transform of f(t) z-transform of sampled signal f*(t), in a discrete system

oIP oIP oIP gradient: grad IP = ~i + ~j + ~k ox oy az Laplace transform of f(t) delay operator for m time steps, in a sampled-signal discrete system del, or nabla: VIP = grad IP divergence: V'" = div "

a2 IP a2 IP a2 IP Laplacian: V2 IP = --2 + --2 + --2 ax ay az change of quantity "u" over a time step

finite difference operator for the net inflow in the s-direction (s = x, y, z), single-phase flow (transport term)

XXIV Most Commonly Used Symbols

Letter Quantity Dimensions Symbol

17sLlsTsLl s cP finite difference operator for the net inflow into a model block, single phase flow (transport term)

Ll.Tr,s (LlsPr - 9 Pr Ll.D) finite difference operator for the net inflow of fluid 'j' (f = 0, g, w) in the s-direction (s = x, y, z)

17s Ll.Tr,s (LlsPr - 9 Pr LlsD) finite difference operator for the net inflow of fluid "f" (f = g, 0, w) into a model block, in muItiphase flow

LlsR.To,s (LlsPo - gpo LlsD) finite difference operator for the net inflow in the s-direction (s = x, y, z) of gas in solution in reservoir oil

17sLlsRsTo,s(LlsPo - 9 Po LlsD) finite difference operator for the net inflow into a model block of gas in solution in reservoir oil

17N summation over all model blocks in a zone, in a lease or in the entire reservoir

Conversion Factors for the Most Commonly Used Quantities

Quantity SI Unit Practical unit Conversion factor, F (1) (2) (1) = Fx(2)

Space, time, speed

Length m in 2.54* E-02 ft 3.048* E -01 mile (intI) 1.609344 E +03

Area m2 sq in 6.4516* E-04 sq ft 9.290304* E-02 acre 4.046856 E + 03 ha 1.0* E +04

Volume, capacity m3 cu ft 2.831685 E -02 US gal 3.785412 E - 03 bbl (42 US gal) 1.589873 E -01 acre-ft 1.233489 E + 03

Time s min 6.0* E + 01 h 3.6* E + 03 d 8.64* E +04 yr 3.15576* E +07

Speed m/s ft/min 5.08* E-03 ft/h 8.466666 E - 05 ft/day 3.527777 E-06 mile/year 5.099703 E - 05 km/year 3.168809 E-05

Mass, density, concentration

Mass kg Ibm (pound mass) 4.535924 E - 01 US ton (short) 9.071847 E +02 US cwt 4.535924 E + 01

Density kg/m3 Ibm/ft3 1.601846 E + 01 Ibm/US gal 1.198264 E +02 lbm/bbl 2.853010 E + 00 °API (1.415 E + 05)/

(131.5 + API)

Concentration kg/m3 lbm/bbl 2.853010 E +00 kg/kg ppm (wt) 1.0* E-06

Pressure, compressibility, temperature

Pressure Pa psi 6.894757 E +03 kg/cm2 9.80665* E +04 atm 1.01325* E + 05 mm Hg = torr 1.333224 E +02

* Exact conversion factor

XXVI Conversion Factors for the Most Commonly Used Quantities

Quantity SI Unit Practical unit Conversion factor, F (1) (2) (1) = F x (2)

Pseudo-pressure Pals (psi)2/cP 4.753767 E + 10 of real gases (kg/cm 2)2 /cP 9.617038 E + 12

Pressure gradient Palm psi/ft 2.262059 E + 04 kg/cm2 x m 9.80665* E + 04

Com pressi bili ty Pa- 1 psi -1 1.450377 E -04 cm2/kg 1.019716 E - 05

Temperature K of (OF + 459.7)1 1.8

°C °C + 273.2

Temperature K/m °F/ft 1.822689 E + 00 gradient °C/m 1.0* E + 00

Flow rate, productivity index

Flow rate m3/s ft3/h 7.865791 E -06 (volume basis) ft 3/d 3.277413 E -07

bbl/h 4.416314 E - 05 bbl/d 1.840131 E -06 m3/h 2.777778 E - 04 m3/d 1.157407 E - 05

Mass flow rate kg/s lbm/h 1.259979 E - 04 lbm/d 5.249912 E -06 US ton/h 2.519958 E - 01 US ton/d 1.049982 E -02

Gas/oil ratio m3/m3 cu ft/bbl 1.781076 E - 01

Productivity index m3/(s.Pa) bbl/( d . psi) 2.668884 E -10 m3/(d . kg.cm - 2) 1.180227 E -10

Transport properties

Permeability m2 md 9.869233 E -16 D 9.869233 E - 13

Viscosity Pa·s IlP 1.0* E - 07 cP 1.0* E - 03

Mobility (k k r/Il) m2/Pa·s md/cp 9.869233 E -13 D/cp 9.869233 E -10

Diffusi vi ty m2/s ftl/h 2.58064* E - 05 cm2/s 1.0* E -04

Surface tension N/m dyne/em 1.0* E - 03