TEAM LinG - Live, Informative, Non-cost and Genuine!Principles of AppliedReservoir SimulationTEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!Second Edition_£ Gulf Professional Publishingii an imprint of Butterworth-HeinemannAmsterdam Boston Heidelberg London New York Oxford Paris San DiegoSan Francisco Singapore Sydney TokyoPrinciples of AppliedReservoir SimulationTEAM LinG - Live, Informative, Non-cost and Genuine!Gulf Professional Publishing is an imprint of Elsevier.Copyright €> 2001 by Elsevier (USA).All rights reserved.Originally published by Gulf Publishing Company, Houston, TX.No part of this publication may be reproduced, stored in a retrieval system, ortransmitted in any form or by any means, electronic, mechanical,photocopying, recording, or otherwise, without the prior written permission ofthe publisher.Permissions may be sought directly from Elsevier's Science & TechnologyRights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44)1865 853333, e-mail: permissions@elsevier.co.uk. You may also completeyour request on-line via the Elsevier Science homepage(http://www.elsevier.com), by selecting 'Customer Support' and then'Obtaining Permissions'./~"^%^* This book is printed on acid-free paper.Library of Congress Cataloging-in-Publication DateFanchi, John R.Principles of applied reservoir simulation/John Fanchi. - 2nd editionp. cm.Includes bibliographical references and index.ISBN 0-88415-372-X(alk. paper)1. Oil fields-Computer simulation. 2. Petroleum-Geology-Mathematical models.I. TitleTN870.53.F36 2000622'.3382'0113-dc21 00-064650British Library Cataloguing-in-Pulication DataA catalogue record for this book is available from the British Library.The publisher offers special discounts on bulk orders of this book.For information, please contact:Manager of Special SalesElsevier Science200 Wheeler RoadBurlington, MA 01803Tel: 781-313-4700Fax: 781-313-4802For information on all Gulf publications available, contact our World WideWeb homepage at http://www.bh.com/gulf10 9 8 7 6 5 4 3 2Printed in the United States of America.Disclaimer:Some images in the original version of this book are not available for inclusion in the eBook. TEAM LinG - Live, Informative, Non-cost and Genuine!To my parents,John A. and Shirley M. FanchiTEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!About the Author xivPreface to Second Edition xvPreface to First Edition xvi1 Introduction to Reservoir Management 1.,1 Consensus Modeling 21.2 Management of Simulation Studies 41.3 Outline of the Text 6Exercises 7Part I - Reservoir Engineering Primer2 Basic Reservoir Analysis 112.1 Volumetrics 112.2 Material Balance 122.3 Decline Curve Analysis 16Exercises 173 Multiphase Flow Concepts 193.1 Basic Concepts 193.2 Capillary Pressure 223.3 Mobility 243.4 Fractional Flow 26Exercises 30VIICONTENTSTEAM LinG - Live, Informative, Non-cost and Genuine!4 Derivation of the Flow Equations 314.1 Conservation of Mass 314.2 Flow Equations for Three-Phase Flow 334.3 Flow Equations in Vector Notation 36Exercises 375 Fluid Displacement 395.1 Buckley-Leverett Theory 395.2 Welge's Method 425.3 Miscible Displacement 44Exercises 466 Frontal Stability 486.1 Frontal Advance Neglecting Gravity 486.2 Frontal Advance Including Gravity 516.3 Linear Stability Analysis 53Exercises 557 Pattern Floods 567.1 Recovery Efficiency 567.2 Patterns and Spacing 587.3 Pattern Recovery 61Exercises 638 Recovery of Subsurface Resources 648.1 Production Stages 648.2 Enhanced Oil Recovery 698.3 Nonconventional Fossil Fuels 71Exercises 739 Economics and the Environment 759.1 SPE/WPC Reserves 759.2 Basic Economic Concepts 779.3 Investment Decision Analysis 819.4 Environmental Impact 82Exercises 85vinTEAM LinG - Live, Informative, Non-cost and Genuine!Part II - Reservoir Simulation10 Overview of the Modeling Process 8910.1 Basics Reservoir Analysis 8910.2 Prerequisites 9010.3 Computer Modeling 9010.4 Major Elements of a Reservoir SimulationStudy 92Exercises 9411 Conceptual Reservoir Scales 9511.1 Reservoir Sampling and Scales 9511.2 Integrating Scales - the Flow Unit 971 i .3 Geostatistical Case Study 101Exercises 10412 Reservoir Structure 10612.1 Giga Scale 10612.2 Mega Scale 11112.3 Reservoir Description Using Seismic Data 115Exercises 11913 Fluid Properties 12013.1 Fluid Types 12013.2 Fluid Modeling 12413.3 Fluid Sampling 128Exercises 12814 Rock-Fluid Interaction 13114.1 Porosity, Permeability, Saturation andDarcy'sLaw 13114.2 Relative Permeability and CapillaryPressure 13514.3 Viscous Fingering 139Exercises 141TEAM LinG - Live, Informative, Non-cost and Genuine!15 Fundamentals of Reservoir Simulation 14215.1 Conservation Laws 14215.2 Flow Equations 14315.3 Well and Facilities Modeling 14515.4 Simulator Solution Procedures 14615.5 Simulator Selection 15 3Exercises 15416 Modeling Reservoir Architecture 15616.1 Mapping 15616.2 Grid Preparation 15816.3 Model Types 16416.4 Basic Simulator Volumetrics 166Exercises 16617 Data Preparation for a Typical Study 16817.1 Data Preparation 16817.2 Pressure Correction 17017.3 Simulator Selection and Ockham's Razor 172Exercises 17518 History Matching 17618.1 Illustrative History Matching Strategies 17718.2 Key History Matching Parameters 18018.3 Evaluating the History Match 18218.4 Deciding on a Match 18318.5 History Match Limitations 184Exercises 18519 Predictions 18619.1 Prediction Capabilities 18619.2 Prediction Process 18719.3 Sensitivity Analyses 18819.4 Economic Analysis 19019.5 Validity of Model Predictions 191Exercises 192TEAM LinG - Live, Informative, Non-cost and Genuine!Part III - Case Study20 Study Objectives and Data Gathering20.120.220.320.420.520.6Study ObjectivesReservoir StructureProduction HistoryDrill Stem TestFluid PropertiesReservoir Management ConstraintsExercises21 Model Initialization21.121.221.321.421.5VolumetricsMaterial BalanceRelative PermeabilityFluid ContactsGrid PreparationExercises22 History Matching and Predictions22.122.222.3Well Model PreparationFull Field (3D) Model History MatchPredictionsExercises197197197199201203207207208208209212214215216, 218218222223224Part IV - WINB4D User's Manual23 Introduction to WINB4D 22923.1 Program Configuration 23123.2 Input Data File - WTEMP.DAT 23223.2 Data Input Requirements 23323.4 Example Input Data Sets 23424 Initialization Data 23924.1 Grid Dimensions and Geometry 23924.2 Seismic Velocity Parameters 24524.3 Porosity, Permeability, and TransmissibilityDistributions 249XITEAM LinG - Live, Informative, Non-cost and Genuine!24.4 Rock and PVT Regions 25524.5 Relative Permeability and CapillaryPressure Tables 25724.6 Fluid PVT Tables 25824.7 Pressure and Saturation Initialization 26224.8 Run Control Parameters 26424.9 Solution Method Specification 26524.10 Analytic Aquifer Models 26725 Recurrent Data 27025.1 Timestep and Output Control 27025.2 Well Information 27226 Program Output Evaluation 27826.1 Initialization Data 27826.2 Recurrent Data 279Part V: Technical Supplements27 Simulator Formulation 28527.1 Equations 28527.2 Coordinate Orientation 28727.3 Petrophysical Model 28827.4 Material Balance 29128 Rock and Fluid Models 29228.1 Three-Phase Relative Permeability 29228.2 Transmissibility 29428.3 Terminology and General Comments 29528.4 Extrapolating Saturated Curves 30028.5 Gas PVT Correlation Option 30129 Initialization 30429.1 Pressure Initialization 30429.2 Gravity Segregated Saturation Initialization 30529.3 Aquifer Models 30730 Well Models 31030.1 Rate Constraint Representation 310xiiTEAM LinG - Live, Informative, Non-cost and Genuine!30.2 Explicit Pressure Constraint Representation 31430.3 GOR/WOR Constraints 31530.4 Fluid Withdrawal Constraints 31630.5 Fluid Injection Constraints 31631 Well Flow Index (PID) 31831.1 Productivity Index 31831.2 Vertical Wells 31931.3 Horizontal Wells 32032 The IMPES Formulation 32232.1 Flow Equations and Phase Potentials 32232.2 Introduction of the Capillary PressureConcept 32332.3 The Pressure Equation 325REFERENCES 333INDEX 347XlllTEAM LinG - Live, Informative, Non-cost and Genuine!About the AuthorJohn R. Fanchi is a Professor of Petroleum Engineering at the ColoradoSchool of Mines. He has worked in the technology centers of three major oilcompanies (Marathon, Cities Service, and Getty), and served as an internationalconsultant. His oil and gas industry responsibilities have revolved aroundreservoir modeling, both in the areas of simulator development and practicalreservoir management applications. Dr. Fanchi's publications include softwaresystems for the United States Department of Energy and numerous articles. Heis the author of four books, including Math Refresher for Scientists andEngineers, Second Edition and Integrated Flow Modeling. He has a Ph.D. inphysics from the University of Houston.xivTEAM LinG - Live, Informative, Non-cost and Genuine!Preface to the Second EditionThe second edition of Principles of Applied Reservoir Simulation has beenexpanded to include background material on reservoir engineering. The chaptersin Part I - Reservoir Engineering Primer are intended to make the book moreaccessible to people from such disciplines as geology, geophysics, and hydrology.The material should serve as a review for petroleum engineers. Chapters in PartII - Modeling Principles have been substantially revised and updated whereappropriate. Exercises have been added or modified to improve their usefulness.Much of the material in the program technical supplement has been integratedinto the main body of the text because it is relevant for flow simulators in general,and not just for the accompanying software.The simulator WINB4D accompanying the text is a version of the originalBOAST4D flow simulator modified for use in a Windows operating environmentwith a dynamic memory management system. The dynamic memory managementsystem expands the range of applicability of the program. A visualization program(3DVIEW) is included on the accompanying CD. It lets the reader obtain a 3Dperspective of the reservoir using WINB4D output.I would like to thank my students in the undergraduate senior reservoirengineering course at the Colorado School of Mines for their comments andsuggestions. I would also like to thank Kathy Fanchi for helping complete therevisions to the second edition, and David Abbott for providing the originalversion of 3DVIEW. Any written comments or suggestions for improving thematerial are welcome.John R. Fanchi, Ph.D.Golden, ColoradoJune 2000xvTEAM LinG - Live, Informative, Non-cost and Genuine!PrefacePrinciples of Applied Reservoir Simulation is a vehicle for widelydisseminating reservoir simulation technology. It is not a mathematical treatiseabout reservoir simulation, nor is it a compendium of case histories. Both ofthese topics are covered in several other readily available sources. Instead,Principles of Applied Reservoir Simulation is a practical guide to reservoirsimulation that introduces the novice to the process of reservoir modeling andincludes a fully functioning reservoir simulator for the reader's personal use.Part I explains the concepts and terminology of reservoir simulation. Theselection of topics and references is based on what I have found to be most usefulover the past two decades as both a developer and user of reservoir simulators.I have provided advice gleaned from model studies of a variety of oil, gas, andcondensate fields.Participation is one of the best ways to learn a subject. The exercises inPart I give you an opportunity to apply the principles that are discussed in eachchapter. As a means of integrating the material, the principles of reservoirsimulation are applied to the study of a particular case in Part II. By the timeyou have completed the case study, you will have participated in each technicalphase of a typical model study.Parts III and IV are the User's Manual and Technical Supplement,respectively, for the three-dimensional, three-phase black oil simulatorBO AST4D that accompanies the text. BOAST4D is a streamlined and upgradedversion of BOAST II, a public domain black oil simulator developed for the U. S.Department of Energy in the 1980's. As principal author of BOAST II, I haveadded several features and made corrections to create BOAST4D. For example,you can now use BOAST4D to model horizontal wells and perform reservoirgeophysical calculations. The latter calculations are applicable to an emergingtechnology: 4D seismic monitoring of fluid flow. The inclusion of reservoirxviTEAM LinG - Live, Informative, Non-cost and Genuine!geophysical calculations is the motivation for appending "4D" to the programname. In addition, BOAST4D includes code changes to improve computationalperformance, to allow the solution of material balance problems, and to reducematerial balance error.BOAST4D was designed to run on DOS-based personal computers with486 or better math co-processors. The simulator included with this book is well-suited for learning how to use a reservoir simulator, for developing an understand-ing of reservoir management concepts, and for solving many types of reservoirengineering problems. It is an inexpensive tool for performing studies thatrequire more sophistication than is provided by analytical solutions, yet do notrequire the use of full-featured commercial simulators. Several example datasets are provided on disk to help you apply the simulator to a wide range ofpractical problems.The text and software are suitable for use in a variety of settings, e.g. inan undergraduate course for petroleum engineers, earth scientists such asgeologists and geophysicists, or hydrologists; in a graduate course for modelers;and in continuing education courses. An Instructor's Guide is available fromthe publisher.I developed much of the material in this book as course notes for acontinuing education course I taught in Houston. I would like to thank BobHubbell and the University of Houston for sponsoring this course and Tim Calkof Gulf Publishing for shepherding the manuscript through the publicationprocess. I am grateful to my industrial and academic employers, both past andpresent, for the opportunity to work on a wide variety of problems. I would alsolike to acknowledge the contributions of Ken Harpole, Stan Bujnowski, JaneKennedy, Dwight Dauben and Herb Carroll for their work on earlier versionsof BOAST. I would especially like to thank my wife, Kathy Fanchi, for her moralsupport and for the many hours at the computer creating the graphics and refiningthe presentation of this material.Any written comments or suggestions for improving the material arewelcome.John R. Fanchi, Ph.D.Houston, TexasAugust 1997XVllTEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 1Introduction to Reservoir ManagementReservoir modeling exists within the context of the reservoir managementfunction. Although not universally adopted, reservoir management is oftendefined as the allocation of resources to optimize hydrocarbon recovery froma reservoir while minimizing capital investments and operating expenses[Wiggins and Startzman, 1990; Satter and Thakur, 1994; Al-Hussainy andHumphreys, 1996; Thakur, 1996]. These two outcomes - optimizing recoveryand minimizing cost - often conflict with each other. Hydrocarbon recoverycould be maximized if cost was not an issue, while costs could be minimizedif the field operator had no interest in or obligation to prudently manage a finiteresource. The primary objective in a reservoir management study is to determinethe optimum conditions needed to maximize the economic recovery of hydrocar-bons from a prudently operated field. Reservoir modeling is the most sophisti-cated methodology available for achieving the primary reservoir managementobjective.There are many reasons to perform a model study. Perhaps the mostimportant, from a commercial perspective, is the ability to generate cash flowpredictions. Simulation provides a production profile for preparing economicforecasts. The combination of production profile and price forecast gives anestimate of future cash flow. Other reasons for performing a simulation studyfrom a reservoir management perspective are listed in Table 1 -1. Several of theitems are discussed in greater detail in later chapters.TEAM LinG - Live, Informative, Non-cost and Genuine!2 Principles of Applied Reservoir SimulationTable 1-1Why Simulate?Corporate Impact+ Cash Flow Prediction0 Need Economic Forecast of Hydrocarbon PriceReservoir Management4 Coordinate Reservoir Management Activities4 Evaluate Project Performance0 Interpret/Understand Reservoir Behavior4 Model Sensitivity to Estimated Data0 Determine Need for Additional Data4 Estimate Project Life+ Predict Recovery vs Time4 Compare Different Recovery Processes4 Plan Development or Operational Changes4 Select and Optimize Project Design0 Maximize Economic Recovery1.1 Consensus ModelingReservoir modeling is the application of a computer simulation systemto the description of fluid flow in a reservoir [for example, see Peaeeman, 1977;Aziz and Settari, 1979; Mattax and Dalton, 1990]. The computer simulationsystem is usually just one or more computer programs. To minimize confusionin this text, the computer simulation system is called the reservoir simulator, andthe input data set is called the reservoir model.Many different disciplines contribute to the preparation of the input dataset. The information is integrated during the reservoir modeling process, andthe concept of the reservoir is quantified in the reservoir simulator. Figure 1-1illustrates the contributions different disciplines make to reservoir modeling.TEAM LinG - Live, Informative, Non-cost and Genuine!Introduction to Reservoir Management 3Figure 1-1. Disciplinary contributions to reservoir modeling(after H.H. Haldorsen and E. Damsleth, ©1993; reprinted bypermission of the American Association of PetroleumGeologists).The simulator is the point of contact between disciplines. It serves as afilter that selects from among all of the proposed descriptions of the reservoir.The simulator is not influenced by hand-waving arguments or presentation style.It provides an objective appraisal of each hypothesis, and constrains the powerof personal influence described by Millheim [1997]. As a filter of hypotheses,the reservoir modeler is often the first to know when a proposed hypothesis aboutthe reservoir is inadequate.One of the most important tasks of the modeler is to achieve consensusin support of a reservoir representation. This task is made more complex whenavailable field performance data can be matched by more than one reservoirmodel. The non-uniqueness of the model is discussed in greater detail throughoutthe text. It means that there is more than one way to perceive and representavailable data. The modeler must sort through the various reservoir represen-tations and seek consensus among all stakeholders. This is often done byrejecting one or more proposed representations. As a consequence, the humanelement is a factor in the process, particularly when the data do not clearlysupport the selection of a single reservoir representation from a set of competingrepresentations. The dual criteria of reasonableness and Ockham's RazorTEAM LinG - Live, Informative, Non-cost and Genuine!4 Principles of Applied Reservoir Simulation[Chapter 9.3; Jefferys and Berger, 1992] are essential to this process, as is anunderstanding of how individuals can most effectively contribute to the modelingeffort.1.2 Management of Simulation StudiesIdeally, specialists from different disciplines will work together as a teamto develop a meaningful reservoir model. Team development proceeds in wellknown stages [Sears, 1994]:+ Introductions: Getting to know each other4 "Storming": Team members disagree over how to proceed0 Members can lose sight of goals4 "Norming": Members set standards for team productivity4 "Performing": Team members understand0 what each member can contribute<> how the team works bestProper management recognizes these stages and allows time for the teambuilding process to mature.Modem simulation studies of major fields are performed by teams thatfunction as project teams in a matrix management organization. Matrixmanagement is synonymous here with Project Management and has two distinctcharacteristics:4 "Cross-functional organization with members from different work areaswho take on a project." [Staff-JPT, 1994]+ "One employee is accountable to two or more superiors, which cancause difficulties for managers and employees." [Staff-JPT, 1994]To alleviate potential problems, the project team should be constituted such that:+ Each member of the team is assigned a different task.4 All members work toward the same goal.Team members should have unique roles to avoid redundant functions. If theresponsibilities of two or more members of the team overlap considerably,confusion may ensue with regard to areas of responsibility and, by implication,of accountability. Each team member must be the key decision maker in aparticular discipline, otherwise disputes may not get resolved in the time avail-TEAM LinG - Live, Informative, Non-cost and Genuine!Introduction to Reservoir Management 5able for completing a study. Teams should not be allowed to flounder in anegalitarian Utopia that does not work.Effective teams may strive for consensus, but the pressure of meetingdeadlines will require one team member to serve as team leader. Deadlinescannot be met if a team cannot agree, and there are many areas where decisionsmay have to be made that will not be by consensus. For this reason, teams shouldhave a team leader with the following characteristics:4 Significant technical skills4 Broad experienceTeam leaders should have technical and monetary authority over the project. Ifthey are perceived as being without authority, they will be unable to fulfill theirfunction. On the other hand, team leaders must avoid authoritarian control orthey will weaken the team and wind up with a group.According to Maddox [1988], teams and groups differ in the way theybehave. Group behavior exhibits the following characteristics:+ "Members think they are grouped together for administrative purposesonly. Individuals work independently, sometimes at cross purposes."4 "Members tend to focus on themselves because they are not sufficientlyinvolved in planning the unit's objectives. They approach their job simplyas hired hands."By contrast, the characteristics of team behavior are the following:4 "Members recognize their interdependence and understand bothpersonal and team goals are best accomplished with mutual support. Timeis not wasted straggling over territory or seeking personal gain at theexpense of others."^ "Members feel a sense of ownership for their jobs and unit because theyare committed to goals they helped to establish."Similar observations were made by Haldorsen and Damsleth [1993]:4 "Members of a team should necessarily understand each other, respecteach other, act as a devil's advocate to each other, and keep each otherinformed."Haldorsen and Damsleth [1993] argue that each team member should have thefollowing focus:4 Innovation and creation of value through the team approachTEAM LinG - Live, Informative, Non-cost and Genuine!6 Principles of Applied Reservoir Simulation4 Customer orientation with focus on "my output is your input"Mclntosh, etal. [ 1991] support the notion that each team member shouldfulfill a functional role, for example, geoscientist, engineer, etc. A corollary isthat team members can understand their roles because the roles have been clearlydefined,Proper management can improve the likelihood that a team will functionas it should, A sense of ownership or "buy-in" can be fostered if team membersparticipate in planning and decision making. Team member views should in-fluence the work scope and schedule of activity. Many problems can be avoidedif realistic expectations are built into project schedules at the beginning, and thenadhered to throughout the project. Expanding work scope without alteringresource allocation or deadlines can be demoralizing and undermine the teamconcept,Finally, one important caution should be borne in mind when performingstudies using teams: "Fewer ideas are generated by groups than by individualsworking alone - a conclusion supported by empirical evidence from psychology[Norton, 1994]." In describing changes in the work flow of exploration anddevelopment studies, Tobias [ 1998, pg. 38] observed that "asset teams have theirdrawbacks. The enhanced teamwork achieved through a team approach oftencomes at the expense of individual creativity, as group dynamics can and oftendoes inhibit individual initiative [Kanter, 1988]." Tobias recommended thatorganizations allow "the coexistence of both asset teams and individual workenvironments." His solution is a work flow that allows the "simultaneouscoexistence of decoupled individual efforts and recoupled asset team coordina-tion."1.3 Outline of the TextThe remainder of the text is organized as follows. Part I presents a primeron reservoir engineering. The primer is designed to provide background conceptsand terminology in the reservoir engineering aspects of fluid flow in porousmedia. Material in Part II explains the concepts and terminology of reservoirsimulation. A typical exercise in Part II asks you to find and change data recordsin a specified example data file. These records of data must be modified basedTEAM LinG - Live, Informative, Non-cost and Genuine!Introduction to Reservoir Management 1on an understanding of the reservoir problem and a familiarity with theaccompanying computer program WINB4D, WINB4D is a three-dimensional,three-phase reservoir simulator. These terms are discussed in detail in subsequentchapters.The exercises in Part II use different sections of the user's manualpresented in Part IV. If you work all the exercises, you will be familiar with theuser's manual and WINB4D by the time the exercises are completed. Much ofthe experience gained by running WINB4D is applicable in principle to othersimulators.Successful completion of the exercises in Part II will prepare you for thecase study presented in Part III. The case study is designed to integrate thematerial discussed in Parts I and II. By the time Part III is completed, you willhave participated in each technical phase of a typical model study.Parts IV and V are the User's Manual and Technical Supplement,respectively, for WINB4D. Supplemental information in Part V provides moredetailed descriptions of the algorithms coded in WINB4D,ExercisesExercise 1.1 WINB4D Folder: A three-dimensional, three-phase reservoirsimulator (WINB4D) is included on a disk with this book. The WINB4D user'smanual is presented in Part IV, and a technical supplement is provided in PartV. Prepare a folder on your hard drive for running WINB4D using the procedureoutlined in Chapter 23. What is the size of the file WINB4D.EXE in kilobytes(KB)?Exercise 1.2 WINB4D Example Data Sets: Several example data sets areprovided on the WINB4D disk. Copy all files from your disk to the \WINB4Dfolder on your hard drive. Make a list of the data files (files with extension"dat"). Unless stated otherwise, all exercises assume WINB4D and its data setsreside in the \WINB4D directory.Exercise 1.3 The program 3D VIEW maybe used to view the reservoir structureassociated with WINB4D data sets. 3DVIEW is a visualization program thatTEAM LinG - Live, Informative, Non-cost and Genuine!8 Principles of Applied Reservoir Simulationreads WINB4D output files with the extension "arr". To view a reservoirstructure, proceed as follows:Use your file manager to open your folder containing the WINB4D files.Unless stated otherwise, all mouse clicks use the left mouse button.Start 3DVIEW (double click on the application 3DVIEW.EXE)Click on the button "File".Click on "Open Array File".Click on "CSJRim.arr" in the File List.Click on "OK".At this point you should see a structure in the middle of the screen. The structureis an anticlinal reservoir with a gas cap and oil rim. To view different perspec-tives of the structure, hold the left mouse button down and move the mouse. Withpractice, you can learn to control the orientation of the structure on the screen.The gridblock display may be smoothed by clicking on the "Project"button and selecting "Smooth Model Display". The attribute shown on the screenis pressure "P". To view other attributes, click on the "Model" button, set thecursor on "Select Active Attribute" and then click on oil saturation "SO". Theoil rim should be visible on the screen.To exit 3DVIEW, click on the "File" button and then click on "Exit",TEAM LinG - Live, Informative, Non-cost and Genuine!TEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 2Basic Reservoir AnalysisThe tasks associated with basic reservoir analyses provide informationthat is needed to prepare input data for a simulation study. These tasks includevolumetric analysis, material balance analysis, and decline curve analysis. Inaddition to providing estimates of fluids in place and forecasts of fieldwideproduction, they also provide an initial concept of the reservoir which can beused to design a model study. Each of these tasks is outlined below,2.1 VolumetricsFluid volumes in a reservoir are values that can be obtained from a varietyof sources, and therefore serve as a quality control point at the interface betweendisciplines. Geoscientists use static information to determine volume in aprocess that is often referred to as volumetric analysis [see, for example, Mian,1992; Tearpock and Bischke, 1991 ]. Material balance and reservoir simulationtechniques use dynamic data to obtain the same information. Consequently, anaccurate characterization of the reservoir should yield consistent estimates offluid volumes originally in place in the reservoir regardless of the method chosento determine the fluid volumes. In this section, we present the equations forvolumetric estimates of original oil and gas in place.Original oil in place (OOIP) in an oil reservoir is given byAr 7758 (|> Ah S.N = 2—2!- (2 1}B l j11TEAM LinG - Live, Informative, Non-cost and Genuine!1.2 Principles of Applied Reservoir SimulationwhereN original oil in place [STB]<f) reservoir porosity [fraction]A reservoir area [acres]h0 net thickness of oil zone [feet]Soi initial reservoir oil saturation [fraction]Boi initial oil formation volume factor [RB/STB]Associated gas, or gas in solution, is the product of solution gas-oil ratio Rso andoriginal oil in place N.Original free gas in place for a gas reservoir is given by775844^ S .G = ^_£. (22)giwhereG original free gas in place [SCF]hg net thickness of gas zone [feet]Sg initial reservoir gas saturation [fraction]Bgi initial gas formation volume factor [RB/SCF]Equation (2.2) is often expressed in terms of initial water saturation Swi bywriting Sgi = 1 - Swi. Initial water saturation is usually determined by well logor core analysis.2.2 Material BalanceThe law of conservation of mass is the basis of material balance calcula-tions. Material balance is an accounting of material entering or leaving a system.The calculation treats the reservoir as a large tank of material and uses quantitiesthat can be measured to determine the amount of a material that cannot bedirectly measured. Measurable quantities include cumulative fluid productionvolumes for oil, water, and gas phases; accurate reservoir pressures; and fluidproperty data from samples of produced fluids.Material balance calculations may be used for several purposes. TheyTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 1 3provide an independent method of estimating the volume of oil, water and gasin a reservoir for comparison with volumetric estimates. The magnitude ofvarious factors in the material balance equation indicates the relative contributionof different drive mechanisms at work in the reservoir. Material balance can beused to predict future reservoir performance and aid in estimating cumulativerecovery efficiency. More discussion of these topics can be found in referencessuch as Dake [1978] and Craft, et al. [1991],The form of the material balance equation depends on whether thereservoir is predominately an oil reservoir or a gas reservoir. Each of these casesis considered separately.Oil Reservoir Material BalanceThe general material balance equation for an oil reservoir is the Schilthuisequation [1961] expressed in a form given by Guerrero [1966]:(2.3)-TV R B -{W+W.-W)Bp so g \ e i p J wAll of these terms are defined in the Nomenclature at the end of this chapter. Theunit of each quantity is presented in square brackets in the Nomenclature. Thephysical significance of the terms in Eq. (2.3) can be displayed by first definingthe termsB -B .D _"l-Swlg Btwi (2.4)BticAP.« /TEAM LinG - Live, Informative, Non-cost and Genuine!14 Principles of Applied Reservoir SimulationSubstituting Eq. (2.4) in Eq. (2.3) gives the general material balance equationin the form(2,5)The terms in Eq. (2.5) have a physical significance. The terms on the righthand side of Eq. (2.3) represent fluid production and injection, while the termson the left hand side represent volume changes. The significance of the termsIs summarized in Table 2-1.Table 2-1Physical Significance of Material Balance TermsTermND0ND,oMA. + Arw)NDr#A*AAGAGptPgcGfi,'WJBWwpw*APhysical SignificanceChange in volume of initial oil and associated gasChange in volume of free gasChange in volume of initial connate waterChange in formation pore volumeCumulative oil productionCumulative gas produced in solution with oilCumulative solution gas produced as evolved gasCumulative gas cap gas productionCumulative gas injectionCumulative water influxCumulative water injectionCumulative water productionEquation (2.3) is considered a general material balance equation becauseit can be applied to an oil reservoir with a gas cap and an aquifer. The derivationof the material balance equation is based on several assumptions: the system isin pressure equilibrium; the system is isothermal; available fluid property dataare representative of reservoir fluids; the reservoir has a constant volume;TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 1 5production data is reliable; and gravity segregation of phases can be neglected.A discussion of the relative importance of drive mechanisms obtained from Eq,(2.3) is presented in Chapter 8.Gas Reservoir Material BalanceThe general material balance equation for a gas reservoir can be derivedfrom Eq. (2.3) by first recognizing the relationshipGBgi = NmBH (2,6)defines original gas in place G. Substituting Eq. (2.6) into Eq. (2.3) gives thegeneral material balance equationN(B, - B,.'BgiSwlg ( B» - B\ ( NBti GBgi } (2.7)' — __ _j. ^ urI- S.__( 5., J 1- 5..... 1- S...J f*A + fas, + G^ - G,jy] - ]VpJRso5gEquation (2.7) is further simplified by recognizing that the material balance fora gas reservoir does not include oil in place so that N - 0 and Np = 0. Theresulting material balance equation isB~B - GBWater compressibility and formation compressibility are relatively smallcompared to gas compressibility. Consequently, Eq. (2.8) is often written in thesimplified form( B - B }GB ^^ = B« - G<B*' - (w- + w< - W^B- (2-9)TEAM LinG - Live, Informative, Non-cost and Genuine!16 Principles of Applied Reservoir Simulation2.3 Decline Curve AnalysisArps [1945] studied the relationship between flow rate and time forproducing wells. Assuming constant flowing pressure, he found the relationship;^ =-«?"" (2-10)where a and n are empirically determined constants. The empirical constant nranges from 0 to 1,Solutions to Eq. (2.10) show the expected decline in flow rate as theproduction time increases. A fit of an equation of the form of Eq. (2.10) to flowrate data is referred to as decline curve analysis. Three decline curves have beenidentified based on the value of n.The Exponential Decline curve corresponds to n — 0. It has the solutionq=qte'at (2.11)where qi is initial rate and a is a factor that is determined by fitting Eq. (2.11)to well or field data.The Hyperbolic Decline curve corresponds to a value of n in the range0 < n < 1. The rate solution has the formq~n = nat + q:n (2.12)where qi is initial rate and a is a factor that is determined by fitting Eq. (2.12)to well or field data.The Harmonic Decline curve corresponds to n - 1. The rate solution isequivalent to Eq. (2.12) with n = 1, thusq~l = nat + q:1 (2.13)where ql is initial rate and a is a factor that is determined by fitting Eq. (2.13)to well or field data.Decline curves are fit to actual data by plotting the logarithm of observedrates versus time t. The semi-log plot yields the following equation forexponential decline:tnq=tnqi-at (2.14)Equation (2.14) has the form y - mx + b for a straight line with slope m andintercept b. In the case of exponential decline, time / corresponds to theTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 1 7independent variable*, $n q corresponds to the dependent variable y, $n qi is theintercept b, and -a is the slope m of the straight line. Cumulative production fordecline curve analysis is the integral of the rate from the initial rate qt at t - 0to the rate q at time t. For example, the cumulative production for the exponentialdecline case isN f<*(2J5)The decline factor a is for the exponential decline case and is found by re-arranging Eq. (2. 11), thusfl=--ln — (2.16)tExercisesExercise 2.1 Copy file EXAM1 .DAT to file WTEMP.DAT and run WINB4D.What are the volumes of initial fluids in place in the model? Hint: Open the runoutput file WTEMP.ROF to find initial fluids in place.Exercise 2.2 Derive the material balance equation for a system with no gas capbeginning with Eqs. (2.3) and (2.4).Exercise 2.3 Use Eq. (2.9) to show that the material balance equation for adepletion drive gas reservoir is" (P/Z).where G is original free gas in place, G^ is cumulative free gas produced, P isreservoir pressure, and Z is the real gas compressibility factor. Subscript tindicates that the ratio P/Z should be calculated at the time / that correspondsto Gpc, and subscript / indicates that the ratio P/Z should be calculated at theinitial time. The units of GP and G must agree for the equation to be consistent.TEAM LinG - Live, Informative, Non-cost and Genuine!18 Principles of Applied Reservoir SimulationExercise 2.4 Derive Eq. (2.15) for the exponential decline case by using Eq.(2.11) as the integrand and performing the integration.Nomenclaturefor Equation (2.3)BcfGG,mNRywR,s.'wigWIOwwAPgas formation volume factor (FVF) [RB/SCF]gas cap FVF [RB/SCF]injected gas FVF [RB/SCF]oil FVF [RB/STB]B0 + (Rsi- Rso)Bg = composite oil FVF [RB/STB]Bw + (Rswi - Rsw)Bg = composite water FVF [RB/STB]formation (rock) compressibility [1/psia]initial gas in place [SCF]cumulative gas injected [SCF]cumulative gas cap gas produced [SCF]cumulative solution gas produced as evolved gas [SCF]ratio of gas reservoir volume to oil reservoir volumeinitial oil in place [STB]cumulative oil produced [STB]solution gas-oil ratio [SCF/STB]initial solution gas-oil ratio [SCF/STB]solution gas-water ratio [SCF/STB]initial solution gas-water ratio [SCF/STB]gas saturation [frac.]oil saturation [frac.]water saturation [frac.]initial water saturation [frac.]initial water saturation in gas cap [frac.]initial water saturation in oil zone [frac.]cumulative water influx [STB]cumulative water injected [STB]cumulative water produced [STB]P,'P = reservoir pressure change [psia]initial reservoir pressure [psia]reservoir pressure corresponding to cumulative fluid times [psia]TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 3Multiphase Flow ConceptsThis chapter summarizes the basic concepts of multiphase flow includinginterfacial tension, wettability, and contact angle. These concepts lead naturallyto a discussion of capillary pressure, mobility, and fractional flow.3.1 Basic ConceptsSome basic concepts must be introduced as prerequisites for understandingcapillary pressure. The concepts are interfacial tension, wettability, and contactangle. They are defined here.Interfacial TensionOn all interfaces between solids and fluids, and between immiscible fluids,there is a surface free energy resulting from electrical forces. These forces causethe surface of a liquid to occupy the smallest possible area and act like amembrane. Interfacial tension (IFT) refers to the tension between liquids at aliquid/liquid interface. Surface tension refers to the tension between fluids ata gas/liquid interface.Interfacial tension is energy per unit of surface area, or force per unitlength. Interfacial tension is often abbreviated as IFT. The units of IFT aretypically expressed in milli-Newtons/meter or the equivalent dynes/cm. Thevalue of IFT depends on the composition of the two fluids at the interfacebetween phases. Table 3-1 lists a few examples:19TEAM LinG - Live, Informative, Non-cost and Genuine!20 Principles of Applied Reservoir SimulationTable 3-1Examples of Interfacial TensionFluid PairAir-BrineOil-BrineGas-OilIFT Range (niN/m or dyne/cm)72-10015-4035-65Interfacial tension (IFT) can be estimated using the Macleod-Sugdencorrelation. The Weinaug-Katz variation of the Macleod-Sugden correlation isNe'MLy,(3.1)whereo1 chiMLMyPiPKinterfacial tension [dyne/cm]parachor of component i [(dynes/cm) 1/4/(g/cm3)]molecular weight of liquid phasemolecular weight of vapor phaseliquid phase density [g/cm3]vapor phase density [g/cm3]x: mole fraction of component / in liquid phaseyt mole fraction of component i in vapor phaseParachors are empirical parameters. The parachor of component i can beestimated using the molecular weight Mi of component i and the empiricalregression equationPchi = 10.0 +2.92 Ml (3.2)This procedure works reasonably well for molecular weights ranging from 100to 500. A more accurate procedure for a wider range of molecular weights isgiven by Fanchi [1990].WettabilityWettability is the ability of a fluid phase to preferentially wet a solidsurface in the presence of a second immiscible phase. The wetting, or wettability,TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 21condition in a rock/fluid system depends on IFT. Changing the type of rock orfluid can change IFT and, hence, wettability of the system. Adding a chemicalsuch as surfactant, polymer, corrosion inhibitor, or scale inhibitor can alterwettability.Contact AngleWettability is measured by contact angle. Contact angle is alwaysmeasured through the more dense phase. Contact angle is related to interfacialenergies by(3.3)oswsa ow cos0whereoos interfacial energy between oil and solid [dyne/cm]om interfacial energy between water and solid [dyne/cm]oow interfacial energy, or IFT, between oil and water [dyne/cm]0 contact angle at oil-water-solid interface measured throughthe water phase [degrees]Examples of contact angle are presented in Table 3-2 for different wettingconditions.Table 3-2Examples of Contact AngleWetting ConditionStrongly Water- wetModerately Water-wetNeutrally WetModerately Oil-wetStrongly Oil-wetContact Angle, degrees0-3030-7575-105105-150150-180Wettability is usually measured in the laboratory. Several factors canaffect laboratory measurements of wettability. Wettability can be changed bycontact of the core during coring with drilling fluids or fluids on the rig floor,and contact of the core during core handling with oxygen and/or water from theTEAM LinG - Live, Informative, Non-cost and Genuine!22 Principles of Applied Reservoir Simulationatmosphere. Laboratory fluids should also be at reservoir conditions to obtainthe most reliable measurements of wettability. Based on laboratory tests, mostknown reservoirs have intermediate wettability and are preferentially water wet.3.2 Capillary PressureCapillary pressure is the pressure difference across the curved interfaceformed by two immiscible fluids in a small capillary tube:PC = Pm - Pw (3.4)wherePc capillary pressure [psi]Pnw pressure in nonwetting phase [psi]Pw pressure in wetting phase [psi]Capillary Pressure TheoryEquilibrium between fluid phases in a capillary tube is satisfied by therelationships/ores up = force down. These forces are expressed in terms of theradius r of the capillary tube, the contact angle 6, and the interfacial tension 0,The forces are given byforce up - IFT acting around perimeter of capillary tube= O cos 0 x 2Krandforce down = density gradient difference x cross-sectionalarea x height h of capillary rise in tubeThe density gradient F is the weight of the fluid per unit length per unit cross-sectional area. For example, the density gradient of water Tw is approximately0.433 psia/ft at standard conditions. If we assume an air-water system, the forcedown isforce down = (Tw - Tair)Ttr2hwhere the cross-sectional area of the capillary tube is Tlr2. Capillary pressureP. is defined as the force/unit area, thusTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 23Pc = force up I Tlr2 = force down / Tlr2.Capillary Pressure and Pore RadiusExpressing capillary pressure in terms offeree up per unit area gives:n 27crocos0 2ocos0PC = _ - - = ----- (3>5)71 r4 rwherer pore radius [cm]a interfacial (or surface) tension [mN/m or dynes/cm]6 contact angle [degrees]Equation (3.5) shows that an increase in pore radius will cause a reduction incapillary pressure while a decrease in IFT will cause a decrease in capillarypressure.Equivalent HeightExpressing Pc in terms of force down leads to the expressionPC = - 2_- = h(rw - Tair) (3.6)Tir2whereh height of capillary rise [ft]Pc capillary pressure [psi]Tw water, or wetting phase, density gradient [psi/ft]rai> air, or nonwetting phase, density gradient [psi/ft]Solving for h yields the defining relationship between capillary pressure andequivalent height, namelyThe equivalent height provides an estimate of the height of the transition zonebetween immiscible phases. A more precise definition of transition zone is givenTEAM LinG - Live, Informative, Non-cost and Genuine!24 Principles of Applied Reservoir Simulationin the following section. Equivalent height is inversely proportional to thedifference in densities between two immiscible phases. The relatively largedensity difference between gas and liquid results in a smaller transition zoneheight than the relatively small difference between two liquid phase densities.Oil-Water Capillary PressureOil is the nonwetting phase in a water-wet reservoir. Capillary pressurefor an oil-water system isPC,* = po - pw (3-8)whereP0 pressure in the oil phase [psi]Pw pressure in the water phase [psi]Capillary pressure increases with height above the oil-water contact (OWC) aswater saturation decreases,Gas-Oil Capillary PressureIn gas-oil systems, gas usually behaves as the nonwetting phase and oilis the wetting phase. Capillary pressure between oil and gas in such a systemisP = P - P (39}ego g o \J.y)wherePg pressure in the gas phase [psi]P0 pressure in the oil phase [psi]Capillary pressure increases with height above the gas-oil contact (GOC) as gassaturation decreases.3.3 MobilityA measure of the ability of a fluid to move through interconnected porespace is the concept of mobility. It is defined here for single phase andmultiphase flow. The multiphase flow definition is based on the concept ofrelative permeability, which is presented next.TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 25Relative PermeabilityThe general definition of relative permeability iskr = -^ (3.10)kabswherekr relative permeability between 0 and 1 ,keff effective permeability [md]kabs absolute permeability [md]Fluid phase relative permeabilities for oil, water and gas phases, respectively,arekro-kjktk^kjktkr^kjk (3.11)where k$ is the effective permeability of phase (!, &ri is the relative permeabilityof phase d, and k is absolute permeability.MobilityFluid phase mobility is defined as the ratio of effective phase permeabilityto phase viscosity. Mobility for oil, water and gas phases respectively areV — • ^ = — • V- (3.12)U LI U,~0 r*W 'gwhere |lf is the viscosity of phase 1 Relative mobility is defined as relativepermeability divided by viscosity [Dake, 1978]. Absolute permeability is nota factor in the definition of relative mobility.Mobility RatioMobility ratio is defined as the mobility of the displacing fluid "kD behindthe front divided by the mobility of the displaced fluid Kd ahead of the front,thus(3.13)TEAM LinG - Live, Informative, Non-cost and Genuine!26 Principles of Applied Reservoir SimulationAn example of mobility ratio is the mobility ratio of water to oil for a waterflood:In this case, relative permeability to water is evaluated at residual oil saturationSor, and relative permeability to oil is evaluated at connate water saturation Swc.Notice that absolute permeability factors out of the expression for mobility ratio.Consequently, mobility ratio can be calculated using either mobilities or relativemobilities.3.4 Fractional FlowThe fractional flow of water is the ratio of water production rate to totalproduction rate. In the case of an oil-water system, the fractional flow of wateris given by/, = ~ = -^— (3.15)*r <?w + <lowherefw fractional flow of waterqw water volumetric flow rate [RB]q0 oil volumetric flow rate [RB]qt total volumetric flow rate [RB]Notice that the flow rates are expressed in terms of reservoir volumes. Thefractional flow of oil/, and the fractional flow of water are related byfw= 1 -f0for an oil-water system. Based on the definition of fractional flow, we see thatfractional flow should be a value between 0 and 1.Simplified Fractional Flow EquationA simplified fractional flow equation is obtained by replacing flow rateswith Darcy's Law in the definition of fractional flow. If we neglect capillarypressure and gravity for simplicity, we obtainTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 27/. - - - (3.16)kkAdP klA dPwhere ^ is cross-sectional area and /^ is the pressure of phase L Since capillarypressure is neglected, we have the equality of phase pressures Pw = P0 so thatEquation (3.17) can be expressed in terms of mobilities as1 1^£ ¥z i+jjl (3;l8)k.rwThe construction of Eq. (3.18) is based on the following simplifying assump-tions: Darcy's Law adequately describes flow rate, and capillary pressure andgravity are negligible. Given these assumptions, we can calculate/,, at reservoirconditions.Fractional Flow Equation with GravityGravity can be included in the fractional flow equation as follows. First,let us consider the two-phase flow of oil and water in a tilted linear system.Darcy's Law including capillary pressure and gravity effects for linear flow iskkAro^dx• u ^where(3.19)P0gsinaTEAM LinG - Live, Informative, Non-cost and Genuine!28 Principles of Applied Reservoir Simulationa dip angle of formationg gravitational constantIf we differentiate capillary pressure for a water-wet system with respect toposition x along the dipping bed, we finddx dx dxCombining Eqs. (3.19) and (3.20) givesAkkru M»where we have used qt = q0 + qw. If we write the density difference asAp = Pw - P0, (3.22)collect terms, and simplify we obtainqt li BP~t >o , cowAk\ k kro rw /Akk dxsina (3.23)Rearranging and collecting terms gives the fractional flow to water fw inconventional oilfield units:1+0.001127Akkdp0.433(Y ~Yjsina(3.24)k urw POA cross-sectional area of flow system [ft2]k absolute permeability [md]kro relative permeability to oilkm relative permeability to water|J,0 oil viscosity [cp]\lw water viscosity [cp]Pcow oil-water capillary pressure [psi] = P0 - PwTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 29x direction of linear flow [ft]a dip angle of formation [degrees]Y0 oil specific gravity (water =1)jw water specific gravity (water = 1 )The general expression forfw includes all three terms governing immiscibledisplacement, namely the viscous term (kro/krw) (|iw/ (10) , the capillary pressureterm d PcoJ$x, and the gravity term (YW~Y0) sin a,It is interesting to note that the capillary pressure and gravity terms aremultiplied by II qt in Eq. (3.24). Most waterfloods have sufficiently high flowrates that capillary pressure and gravity effects can be neglected, leaving thesimplified expression:f a _ L __, ^ kro V« (3.25)••-- ••Equation (3.25) is in agreement with Eq. (3.18), as it should be.Gas Fractional FlowA similar analysis can be performed to determine the fractional flow ofgasjC The result for a gas-oil system isf _-Akk1+0.001127U Q 'dP— ^ - 0.433 (Y ~Y0dx° t}(3.26)l+£k u.rg ~owherekrg relative permeability to gasM-g gas viscosity [cp]Pcgo gas-oil capillary pressure = Pg-P0 [psi]Yg gas specific gravity [water = 1]qg gas volumetric flow rate [RB/D]qt' total volumetric flow rate = q0 + qg [RB/D]Immiscible displacement of oil by gas is analogous to water displacing oil withTEAM LinG - Live, Informative, Non-cost and Genuine!30 Principles of Applied Reservoir Simulationthe water terms replaced by gas terms. In general, the gravity term in^ shouldnot be neglected unless q,' is very high because of the specific gravity differencebetween gas and oil.ExercisesExercise 3.1 Estimate the parachors for butane and decane.Exercise 3.2 Derive the relationship between the equivalent height of a transitionzone and pore radius by using Eq. (3.5) to eliminate capillary pressure from Eq.(3.7).Exercise 3.3 Suppose kJiS^ ~ kro(Swc) in Eq. (3.14) and water viscosity is 1cp. Plot Mw 0 versus oil viscosity for oil viscosity ranging from 0.1 cp to 100 cp.Exercise 3.4 Derive Eq. (3.21) from Eqs. (3.19) and (3.20).TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 4Derivation of the Flow EquationsMany derivations of the oil, water, and gas fluid flow equations exist inthe literature [for example, see Crichlow, 1977; Peaceman, 1977]. Consequently,only a brief discussion will be presented here. It closely follows the presentationoriginally published in Fanchi, et al. [1982].4.1 Conservation of MassWe begin by considering the flow of fluid into and out of a single reservoirblock (Figure 4-1). Let the symbol J denote fluid flux. Flux is defined as the rateA /j, — **ii7JX+AX y^^- fmnm—^nm^^. j£TzFigure 4-1. Reservoir block: the coordinateconvention follows Sawyer and Mercer [1978].of flow of mass per unit cross-sectional area normal to the direction of flow,which is the x direction in the present case. Assume fluid flows into the blockatx (Jx) and out of the block at ;c + A x (Jx+&x)- By conservation of mass, we havethe equality:31TEAM LinG - Live, Informative, Non-cost and Genuine!32 Principles of Applied Reservoir Simulationmass entering the block - mass leaving the block= accumulation of mass in the block.If the block has length A*, width Ay, and depth Az, then we can write the massentering the block in a time interval A/ as| (',/ ) AvAz+ (j } AxAz + (j } AjcAv A/ = Mass in (4,1)I '< x ' x \ y I v \ z I : " \where we have generalized to allow flux in the y and z directions as well, Thenotation (Jx)x denotes the x direction flux at location*, with analogous meaningsfor the remaining terms,Corresponding to mass entering is a term for mass exiting which has theformlW*.*,*y*z + (^WA*A* + (^)2+A,A*A^A'(4.2)+ gAxAyAzA/ = Mass outWe have added a source/sink term q which represents mass flow into (source)or out of (sink) a well. A producer is represented by q > 0, and an injector byq<0.Accumulation of mass in the block is the change in concentration of phase<! (C4) in the block over the time interval A?. If the concentration Ct is definedas the total mass of phase 0 (oil, water, or gas) in the entire reservoir blockdivided by the block volume, then the accumulation term becomes[(^X+A* ~ (C^JAjtAyAz = Mass accumulation (43)Using Eqs. (4.1) through (4.2) in the mass conservation equalityMass in - Mass out = Mass accumulationgives[(/^AyAz H- (J^AjcAz + 0/Z)ZA*A7]A?- lWx.***y*z + (^WA*A*+ Wz^z&x&y]&t (4.4)- ^AxAyAzA/ = [(C()^Ar - (Cf),]A*A;pAzDividing Eq. (4.4) by AxAyAzA/ and rearranging givesTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 33A*A* A 7=Az ArIn the limit as A*, Aj, Az, and A? go to zero, Eq. (4.5) becomes thecontinuity equationdJx dJv dJz dC,y (A &}\*T,Ojdx dy dz dtThe oil, water, and gas phases each satisfy a mass conservation equation havingthe form of Eq. (4.6).4.2 Flow Equations for Three-Phase FlowThe flow equations for an oil, water, and gas system are determined byspecifying the fluxes and concentrations of the conservation equations for eachof the three phases. A flux in a given direction can be written as the density ofthe fluid times its velocity in the given direction. Letting the subscripts o, w, andg denote oil, water, and gas, respectively, the fluxes become:DB(A, - -^ (4.8)gB B B(where Rso and Rsw are gas solubilities; B0, Bw, and Bg are formation volumefactors; the subscript sc denotes standard conditions (usually 60°F and 14.7 psiain oilfield units); and p denotes fluid densities. The velocities v are assumedto be Darcy velocities and their x components areTEAM LinG - Live, Informative, Non-cost and Genuine!34 Principles of Applied Reservoir Simulation® P - '"_- (A 1 A\a ro ... (4,10)a^w w_a_dxP - W /A 1 1 \rw ,,> (4.11 j(4.12)where g is the acceleration of gravity in ft/sec2, andgc is 32.174 ft/sec2 (WINB4Dassumes g - gc). These equations should be valid for describing fluid flow inporous media even if g and gc change, such as on the Moon, Mars, or the spaceshuttle. Similar expressions can be written for the>> and z components.The phase mobility Ae is defined as the ratio of the relative permeabilityto flow of the phase divided by its viscosity, thusA, = V»*i (4.13)The phase densities are related to formation volume factors and gas solubilitiesbyPo = — [Po*c + ^oPff*J' (4.14)(4.15)(4.16)Besides fluxes, we also need concentrations. These are given byCo = 4»Po,A/*0» (4.17)(4.18)TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 35c c cso£„ B,(4.19)where <j> is the porosity and St is the saturation of phase fi. The saturations satisfythe constraintO 4. O 4. P — 1 //i OA\5o + ^w ^g ~ l (4,20)Combining Eqs. (4.6), (4.7) through (4.9), and (4.17) through (4.19) gives a massconservation equation for each phase:Oildx_a_dz£££vZ0(4.21)a f A 5 'Water\dxaxwpwsc-V.yw'w /dzB.(4.22)--Up ~1q»~ dt( W*CB0)Gas" I 8&c -. , so"gsc SY/* gsc Iav I B B B i* \ K O W I(4.23)O 'gSC SO^gSC SW'gSC Ia_ _ ^ _ ^^ _ z^ ^O2T JD jD /> 1a/TEAM LinG - Live, Informative, Non-cost and Genuine!36 Principles of Applied Reservoir SimulationThe densities at standard conditions are constants and can be divided out of theabove equations. This reduces the equations to the following form:Oild*BWater/ \d v,yw I u zw9* M dy(B») dz(B»s<?wPw.c dtGas- Aa*By ( Bg Bft w\ (4.26)Zff" JO JWo -i. 11 J_ ti(c c c N-JL + R __2. + R _JL^ "^ ^w,4.3 Flow Equations in Vector NotationEquations (4.10) through (4.16), (4.20), and (4.24) through (4.26) are thebasic fluid flow equations which are numerically solved in a black oil simulator.A glance at Eqs. (4.24) through (4.26) illustrates the computational complexityTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 3?of the basic three-dimensional, three-phase black oil simulator equations.Equivalent but much simpler appearing forms of the flow equations are presentedin terms of vector operators as- V(427)q™ 3 - - • (4,28)and- VPgscdt(4.29)+1? * "}_- v = —-vx + —v + —-vz. (430)where the symbol V • v for the divergence of the velocity vector is shorthandfor the expressiond d d— v + —v + —dx dy y dzA review of vector analysis can be found in many references, such as Kreyszig[1999] and Fanchi [2000].ExercisesExercise 4.1 Suppose the unit of density posc is mass per volume at standardconditions, and the unit of Darcy velocity is length per time. Use dimensionalanalysis to determine the unit of flux in Eq. (4.7).Exercise 4.2 The densities in Eqs. (4.14) and (4.15) include gas dissolution.Rewrite Eqs. (4.19), (4.23), and (4.29) for a system with no gas dissolved ineither the oil or water phases.TEAM LinG - Live, Informative, Non-cost and Genuine!38 Principles of Applied Reservoir SimulationExercise 4.3 Run EXAM3.DAT and record the time, pressure, oil rate, waterrate, gas rate, and GOR at the end of the run. These values are obtained fromthe one line timestep summary file WTEMP.TSS (also see Chapter 26.2). Is gassignificant in this model?TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 5Fluid DisplacementFluid displacement processes require contact between the displacing fluidand the displaced fluid. The movement of the interface between displacing anddisplaced fluids and the breakthrough time associated with the production ofinjected fluids at producing wells are indicators of sweep efficiency. This chaptershows how to calculate such indicators using two analytical techniques: Buckley-Leverett theory with Welge's method for immiscible fluid displacement, andsolution of the convection-dispersion equation for miscible fluid displacement.5.1 Buckley-Leverett TheoryOne of the simplest and most widely used methods of estimating theadvance of a fluid displacement front in an immiscible displacement process isthe Buckley-Leverett method. Buckley-Leverett Theory [1942] estimates therate at which an injected water bank moves through a porous medium. Theapproach uses fractional flow theory and is based on the following assumptions:• Flow is linear and horizontal• Water is injected into an oil reservoir• Oil and water are both incompressible• Oil and water are immiscible• Gravity and capillary pressure effects are negligibleThe following analysis can be found in a variety of sources, such as Collins[1961], Dake [1978], Wilhite [1986], and Craft, et al. [1991].39TEAM LinG - Live, Informative, Non-cost and Genuine!40 Principles of Applied Reservoir SimulationFrontal advance theory is an application of the law of conservation ofmass. Flow through a small volume element (Figure 5-1) with length Ax and//Porous Material(e.g. Rock)v/AFigure 5-1. Flow Geometrycross-sectional area A can be expressed in terms of total flow rate qt as9* = QO + <?w (s.i)where q denotes volumetric flow rate at reservoir conditions and the subscripts{o, w, t} refer to oil, water, and total rate, respectively. The rate of water enteringthe element on the left hand side (LHS) isqtfw = entering LHS (5.2)for a fractional flow to water fw. The rate of water leaving the element on theright hand side (RHS) isqt(fw + A/w) = leaving RHS (53)The change in water flow rate across the element is found by balancing massfor an immiscible, incompressible system, thusrate change = water entering - water leaving(54)The change in water saturation per unit time is the rate change in Eq. (5.4)divided by the pore volume of the element, thusAxTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 41In the limit as A/ -> 0 and Ax -* 0, we pass to the differential form of Eq. (5.5)for the water phase:dt -4(j) dxA similar equation applies to the oil phase:(5.7)dt A$ dxSince/, depends only on Sw, we can write the derivative of fractional flow as"a7 = Ist "a7 (5'8)Substituting dfjdx into dSJdx yieldsdSw -qt dfwdt A dS dx(5,9)It is not possible to solve for the general distribution of water saturation S^x,t) in most realistic cases because of the nonlinearity of the problem. For example,water fractional flow is usually a nonlinear function of water saturation. It istherefore necessary to consider a simplified approach to solving Eq. (5.9).We begin by considering the total differential of Sw(x, t):w w QX wdt dx dt dt(5.10)Equation (5.10) can be simplified by choosing x to coincide with a surface offixed Sw so that dSJdt = 0 and( dS.}dx\ ( dt )_____ I — _ \ / ff 1 1 \.J / \ (5.H)dxSubstituting Eqs. (5.8) and (5.9) into Eq. (5. II) gives the Buckley-Leverettfrontal advance equation:TEAM LinG - Live, Informative, Non-cost and Genuine!42 Principles of Applied Reservoir Simulationdx\ -'jfl ,<,K ^c I (5,12)* iThe derivative (dxidi)Sw is the velocity of the moving plane Sw, and (dfJdSw)Swis the slope of the fractional flow curve. The integral of the frontal advanceequation givesr - W< ( dL\s-~7* \7s~l (5J3)\ ' $„wherexSw distance traveled by a particular Sw contour [ft]Wj cumulative water injected [cu ft](dfw/dsj\ slope of fractional flow curveWater Saturation ProfileA plot of Sw versus distance using Eq. (5.13) and typical fractional flowcurves leads to the physically impossible situation of multiple values of Sw ata given location. A discontinuity in Sw at a cutoff location xc is needed to makethe water saturation distribution single valued and to provide a material balancefor wetting fluids. The procedure is described by Collins [ 1961 ] and summarizedbelow.5.2 Welge's MethodIn 1952, Welge published an approach that is widely used to perform theBuckley-Leverett frontal advance calculation. Welge's approach is bestdemonstrated using a plot offw vs Sw (Figure 5-2).A line is drawn from the water saturation Sw before the waterflood -irreducible water saturation Swirr - and tangent to a point on thefw curve. Theresulting tangent line is called the breakthrough tangent, or slope. It is illustratedin Figure 5-2. Water saturation at the flood front S^is the point of tangency onthefw curve. Fractional flow of water at the flood front is/^and occurs at thepoint of tangency S^ on the/w curve. In Figure 5-2, Swf\s 65% and/w/is 95%TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 43Average water saturation behind the flood front Swbt is the intercept of the maintangent line with the upper limiting line where/,, = 1.0. In Figure 5-2, average5^ is 67%.0 0.2 0.4 0.6 0.8 1.0Figure 5-2. Welge's MethodIn summary, when water reaches the producer, Welge's approach givesthe following results:• Water saturation at the producing well is Swf• Average water saturation behind the front is Swbt• Producing water cut at reservoir conditions isfwfOther useful information about the waterflood can be obtained from Welge'sapproach.The time to water breakthrough at the producer is. LA*" " 9, (4W)V (5J4)whereqt injection rate(dfw I dS^\ slope of main tangent lineV 'SwfL linear distance from injection well to production wellCumulative water injected is given byTEAM LinG - Live, Informative, Non-cost and Genuine!44 Principles of Applied Reservoir SimulationQi =where Qi is the cumulative pore volume of injected water. The slope of the waterfractional flow curve with respect to water saturation evaluated at the watersaturation at breakthrough gives Q( at breakthrough.Effects of Capillary Pressure and GravityIn the absence of capillary pressure and gravity effects, the flood frontpropagates as a "sharp" step function, or piston-like displacement. The presenceof capillary pressure leads to the imbibition of water ahead of the front. Thiscauses a change in the behavior of produced fluid ratios. Rather than an abruptincrease in WOR associated with piston-like displacement, the WOR willincrease gradually as the leading edge of the mobile water reaches the well andis produced. In addition, the WOR will begin to increase sooner than it wouldhave in the absence of capillary pressure. By contrast, gravity causes high Swvalues to lag behind the front. The result is a smeared or "dispersed" flood front.5.3 Miscible DisplacementBuckley-Leverett theory treats the displacement of one fluid by anotherunder immiscible, piston-like conditions. An immiscible displacement occurswhen the displaced and displacing fluids do not mix. The result is a readilydiscernible interface between the two fluids. In a miscible displacement, thefluids mix and the interfacial tension approaches zero at the interface. A miscibledisplacement system is described by a convection-dispersion (C-D) equation.As an illustration, consider the one-dimensional C-D equation for the concentra-tion C of the displacing fluid:n d2C BC dCD v = — f5 16\dx2 dx dt IJ'10JWe assume here that dispersion D and velocity v are real, scalar constants. Thediffusion term has the Fickian form D'd2C/dx2 and the convection term isvdC/dx. When the diffusion term is much larger than the convection term, theTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 45vdC/dx. When the diffusion term is much larger than the convection term, theC-D equation behaves like the heat conduction equation, which is a parabolicpartial differential equation (PDE). If the diffusion term is much smaller thanthe convection term, the C-D equation behaves like a first-order hyperbolic PDE.The C-D equation is especially valuable for studying numerical solutionsof fluid flow equations because the C-D equation can be solved analytically andthe C-D equation may be used to examine two important classes of PDEs(parabolic and hyperbolic). To solve the C-D equation, we must specify twoboundary conditions and an initial condition. The two boundary conditions areneeded because the C-D equation is second-order in the space derivative. Theinitial condition satisfies the need for a boundary condition in time associatedwith the first-order derivative in time. The boundary conditions for the miscibiedisplacement process are that the initial concentration of displacing fluid is equalto one at the inlet (x = 0), and zero for all other values of x. The mathematicalexpressions for these boundary conditions are concentration C(0, t) = I at theinlet, concentration C(°°, f) = 0 at the edge of the linear system for all times tgreater than the initial time t = 0, and the initial condition C(x, 0) = 0 for allvalues of x greater than 0.The propagation of the miscibie displacement front is calculated bysolving the C-D equation. The analytical solution of the one-dimensional C-Dequation is [Peaceman, 1977]C(jc, 0 = -U erfc2lJC - VtX + Vt2]/Dtwhere the complementary error function erfc(y) is defined as(5.17)2 r 2erfc(j) = 1 - — je z rfz (5,18)V* oAbramowitz and Stegun [ 1972] have presented an accurate numerical algorithmfor calculating the complementary error function erfcO/). A comparison of theanalytical solution of the C-D equation with numerical solutions is given inFanchi [2000].TEAM LinG - Live, Informative, Non-cost and Genuine!46 Principles of Applied Reservoir SimulationExercisesExercise 5.1 Consider an oil-water system in which oil viscosity is 0.64 cp andwater viscosity is 0.5 cp. Oil relative permeability (krow) and water relativepermeability (Arw) are given in the following table as a function of watersaturation (Sw). Complete the table by using the viscosity and relative permeabil-ity information to calculate oil mobility (A,0), water mobility (A.w), total mobility(1,), water fractional flow (/"„,), and oil fractional flow (£,). Total mobility is thesum of oil mobility and water mobility. Assume absolute permeability is 100md.sw -0.300.350.400.450.500.550.600.650.700.80*n.0.0000.0050.0100.0170.0230.0340.0450.0640.0830.120^row1.0000.5900.3200.1800.0800.0300.0100.0010.0000.000^k*,LLExercise 5.2 Plot X0, Xw, and A, in Exercise 5.1 as a function of Sw. What is themobility ratio of the oil-water system? Hint: See Eq. (3.14).Exercise 5.3 Piotf0 and/,, in Exercise 5.1 as a function of Sw. Use the plot offw versus Sw and Welge's method to determine water saturation at the producingwell, average water saturation behind the front, and producing water cut atreservoir conditions.TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 47Exercise 5.4 Run EXAM3.DAT and plot water saturation as a function ofdistance between wells at the midpoint of the ran and at the end of the ran. Hint:water saturation is reported in the run output file WTEMP.ROF.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 6Frontal StabilityThe stability of a flood front can influence the efficiency of fluiddisplacement. A front is stable if it retains the shape of the interface betweendisplaced and displacing fluids as the front moves through the medium. Ananalysis of frontal stability is presented in this chapter in terms of a specificexample - the advance of a water-oil displacement front in the absence of gravityand in the presence of gravity. The stability of the front is then studied usinglinear stability analysis.6.1 Frontal Advance Neglecting GravityThe displacement of one phase by another may be analytically studiedif a linear, homogenous porous medium is assumed. Let us first consider thedisplacement of oil by water in a horizontal porous medium of length L. Weassume piston-like displacement of a front located atxf. Application of Darcy'slaw and the continuity equation leads to a pressure distribution described byPoisson's equation. The absence of sources or sinks in the medium reducesPoisson's equation to the Laplace equation for the water phase pressure:£3 D= 0, 0<x<xr (6.1)dxThe corresponding equation for oil phase pressure is48TEAM LinG - Live, Informative, Non-cost and Genuine!oxPart I: Reservoir Engineering Primer 493f = 0, xf < x < L (62)Equations (6.1) and (6.2) apply to those parts of the medium containing waterand oil respectively. They assume that the fluids are incompressible, and thatthe oil-water interface is a piston-like displacement in the ^-direction. The piston-like displacement assumption implies a discontinuous change from mobile oilto mobile water at the displacement front. This concept differs from the Buckley-Leverett analysis presented in Chapter 5. Buckley-Leverett theory with Welge'smethod shows the existence of a transition zone as saturations grade from mobileoil to mobile water. The saturation profile at the interface between the immisciblephases depends on the fractional flow characteristics of the system. The presentmethod has less structure in the saturation profile, but is more readily suited foranalyzing the stability of the displacement front.Boundary conditions at the displacement front are given by continuity ofphase pressureP0 = Pw at x=xf(t) (6.3)and continuity of phase velocity(64)where A.J is the mobility of phase 0. Equation (6.3) is valid when we neglectcapillary pressure, and the effect of gravity has been excluded from Eq. (6.4).The exclusion of gravity corresponds physically to flow in a horizontal medium.Boundary conditions at the edges of the porous medium arePw=Platx=0 (6.5)andP0 = P2 at*= L (6.6)Equations (6.1) through (6.6) may be solved analytically. We begin byintegrating Eqs. (6.1) and (6.2) to find the general solutionsPw = Awx+Bw (6.7)andTEAM LinG - Live, Informative, Non-cost and Genuine!50 Principles of Applied Reservoir SimulationP0 = A0x+B0 (6.8)where the constant coefficients {Ae Bs} are determined by applying the boundaryconditions. Substituting Eq. (6.5) in Eq. (6.7) givesBw = P{ (6.9)The remaining coefficients are found by simultaneously solving Eqs. (6.4), (6,7),and (6.8) subject to Eqs. (6.3), (6.5), and (6.6). The resulting coefficients areAPAw = -~rz—-——— (6.10)ML + (1 - M}xfA0=MAW (6.11)B0 = PI = (Aw - A0)xf = P, + (1- M)Awxf (6.12)where M is the mobility ratioM=-~- (6.13)koand the pressure difference isAP=P,-P2 (6.14)The frontal velocity Vyis given bydx vwhere Sor is residual oil saturation, Swc is connate water saturation, and vw is theDarcy velocity:Vw ~ ~^w~A = ~^w^w (6.16)Substituting Eq. (6.16) into (6.15) givesdt ~ An~ s - ? ^F i>rr . /, m~l (6.17)- A/)*,]TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 51The integral of Eq. (6.17) with respect to time gives the frontal advance.6.2 Frontal Advance Including GravityGravity is included in the analysis of frontal advance in a dipping reservoir(Figure 6-1) by replacing phase pressure in Eqs. (6.1) through (6.6) with phaseFigure 6-1. Geometry of Frontal Advancepotential<J>; = Pf - p (gxsm&The resulting equations for phase potentials are0,0<x<xf (6.18)dx2d2®(6.19)-> — \J) A f 'X J*. ^, JL^5jc 7The phase potentials at the flood front are related bywith continuity of phase velocitiesTEAM LinG - Live, Informative, Non-cost and Genuine!52 Principles of Applied Reservoir SimulationThe boundary conditions for the phase potentials areQ (6.22)and0>0 = 02 atx = L (6.23)Capillary pressure is still neglected in this formulation. Equation (6.20) is theanalog of Eq. (6.3).The solutions of the second-order ordinary differential equations Eqs,(6.18) and (6.19) are the linear relationships®w = A'wx+B'w (6.24)<S>0 = A'0x+Bi (6.25)The coefficients are evaluated by substituting Eqs. (6.24) and (6.25) into Eqs.(6.18) and (6.19) and applying the boundary conditions. The coefficients areAL = -(6.26)ML + (1 - M)xfA'0 = MAI (6-27)*; = 0, (6.28)(6.29)The Darcy velocity of the water phase isvw = ~dw ;~" = ~A A (6.30)dxThe velocity of frontal advance in a dipping reservoir is found by substitutingEq. (6.30) into Eq. (6.15) to finddt W-Sor-Swc) ML+(l-M)xf (' }TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 536.3 Linear Stability AnalysisThe stability of frontal advance is determined by considering the rate ofgrowth of a perturbation at the front. We first express the frontal advancevelocity Eqs. (6.17) and (6.31) in the general formdxf a + $xf./ ^L (632)di j + 8jtfwhere the coefficients are independent of time and frontal location. Equation(6.32) is a nonlinear, first-order differential equation. Imposing a slightperturbation on the front location givesd(xf + e) a + $(xf + e)' = _i__ (6J3)dt y +b(xf + K)The velocity of propagation of the perturbation is given by the differencebetween Eqs. (6.33) and (6.32):de a + $xf + pe a + $xf= • f -*- (6.34)dt j + oxf + 6s y + bxfCombining fractions and simplifying yieldsj£ (6.35)dtFurther simplification is achieved by recognizing that the perturbation is slightso that we have the approximation1 8s c5 w 1 — for 6s « y + ox fi + . S£ T + 5^/ f (6.36)y +8^Substituting Eq. (6.36) into Eq. (6.35) gives1-8s(637)TEAM LinG - Live, Informative, Non-cost and Genuine!54 Principles of Applied Reservoir SimulationKeeping only terms to first order in e and simplifying givesdt ~ (yEquation (6.38) has the solution(6.38)(6.39)where e0 is an integration constant, andBy — 8otT = — ^6 40^(y+5*7)2If T is negative, the perturbation decays exponentially. If T is greater than zero,the perturbation grows exponentially. Finally, if T equals zero, the perturbationdoes not propagate because de/dt - 0 in Eq. (6.38).We can now examine the stability of a displacement front. ComparingEq. (6.32) with (6.31) lets us make the identificationsa =Xw(p0-pw)gsine(6-42)T = ML (6.43)8 = (1-M) (6.44)The resulting expression for the growth of a perturbation is& M (l-MX$,-<D2)+M.(p0-pJgsin0(6'45)Equation (6.45) agrees with Eq. (7-104) in Collins [1961].Zero growth rate of a perturbation is determined by setting the derivativede/dt = 0 in Eq. (6.45). The resulting condition for zero growth rate is(1- AfX*i-*2)+ M,(pe-pw)gsine = 0 (6.46)TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 55If the medium is horizontal, the condition for a system without gravity is(1 ~ M)AP = 0 (6.47)To see the effect of mobility ratio Mon finger growth for the gravity-free case,we set g =• 0 in Eq, (6.45) to getfife X W6 (1 -dt <j>(l- Sor - S-.J [ML(6.48)The finger grows exponentially if M> 1, decays exponentially if M< I, and doesnot propagate if M = 1.ExercisesExercise 6.1 Show that Eq. (6.7) is a solution of Eq. (6.1).Exercise 6.2 Use Eq. (6.45) to determine the rate of finger growth of a unitmobility flood in a horizontal medium. Hint: Set M = 1 in Eq. (6.45) andsimplify.Exercise 6.3 Use Eq. (6.48) to explain why the mobility ratio condition M< 1is considered "favorable" for a displacement flood.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 7Pattern FloodsThe effectiveness of a displacement process depends on many factors.These factors include reservoir and fluid characteristics that are beyond ourcontrol, such as depth, structure, and fluid type. Other factors that influencedisplacement efficiency can be controlled, however. They include the numberand type of wells, well rates, and well locations. The distribution of wells isknown as the well pattern. The impact of well pattern on displacement effective-ness is discussed after definitions of recovery efficiency are presented.7.1 Recovery EfficiencyRecovery efficiency is quantified by comparing initial and final volumesof fluid in place. It takes into account volumetric and displacement efficiencies.The different aspects of recovery efficiency are defined and then combined toform overall recovery efficiency.Displacement efficiency accounts for the efficiency of recovering mobilehydrocarbon. To be specific, we define displacement efficiency for oil as theratio of mobile oil to original oil in place at reservoir conditions:V S - V S S - SF - P oi P or - oi or /"? i \° ^ ^r (7-°whereVp initial pore volumeSoi initial oil saturation56TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 57Sor residual oil saturationDisplacement efficiency can approach 100% if residual oil saturation can bedriven to zero. One of the goals of enhanced oil recovery processes such asmicellar-polyrner flooding or miscible flooding is to reduce residual oil satura-tion and increase displacement efficiency.The definition of displacement efficiency can be modified to include theeffects of swelling. Swelling is represented by using surface volume rather thanreservoir volume in the definition of displacement efficiency. The volumeconversion is achieved by dividing reservoir volume by formation volume factor.For example, the displacement efficiency of a waterflood isV S . V S S. Sp oi p or 01 or17 _ oi __ oa ._ oi oaE° ~ ~ <7'2)whereBoi oil FVF at the beginning of waterfloodBoa oil FVF at the waterflood pressureNotice that oil formation volume factor is a maximum at the bubble pointpressure of the oil. If the waterflood is conducted at or just above bubble pointpressure, the value of B^ will be maximized and the residual oil term will beminimized. The resulting displacement efficiency for a waterflood is thenmaximized.Displacement efficiency is a measure of how effectively mobile hydrocar-bons can be recovered. Although the above definitions of displacementefficiency have been given for oil, similar definitions can be provided for gas.In addition to displacement efficiency, volumetric factors are needed todetermine overall recovery efficiency. Areal and vertical sweep efficiencies aredefined by„ swept areaEA = - , - (7.3)total areaandTEAM LinG - Live, Informative, Non-cost and Genuine!58 Principles of Applied Reservoir Simulation„ _ swept thickness(7,4)total thicknessReservoir flow models are useful tools for quantifying both swept area and sweptthickness. The product of areal and vertical sweep efficiency is the volumetricsweep efficiency Evol:Evor^EF (7.5)whereEA areal sweep efficiencyEy vertical sweep efficiencyOverall recovery efficiency must account for both volumetric anddisplacement effects. It is therefore defined as the product of volumetric sweepefficiency and displacement efficiency:^-ED^EvorED^EAxEy (7,6)whereRE recovery efficiencyNotice that each of the efficiency factors in recovery efficiency can be relativelylarge,, and yet recovery efficiency will be relatively small. For example, supposethe areal and vertical efficiencies are each 70% and displacement efficiency is80%, the product of these efficiencies is approximately 39%. This means thateven the reservoirs with the best recovery efficiency often have a substantialvolume of unrecovered hydrocarbon remaining in the ground. The mostimportant goal of improved recovery techniques is to recover this remainingresource.7.2 Patterns and SpacingThe displacement processes discussed in Chapters 5 and 6 study fluiddisplacement between one injection well and one production well. The alignmentof the injector-producer pair represents a linear displacement process. It is thesimplest pattern involving injection and production wells. A variety of otherTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 59patterns may be defined. Several examples are shown in Figure 7-1. A repre-sentative pattern element for the five-spot pattern is shown using lines betweenwells to denote boundary wells.Direct Line Drive Patterna — distance between neighboring wellsd — distance between rows of wellsStaggered Line Drive Patterna = distance between neighboring wellsd ~ distance between rows of wellsFive-Spot Patternd = distance between neighboring producers= distance between neighboring injectors• • • * •4 A A A AA A A A A* * • * *0 • • * *• • • * *» A * A *A • A • A/ \* A * A *\ /A • A * A• A • A «Figure 7-1. Well Locations in Selected Well Patterns. Production Well •;Injection Well *.TEAM LinG - Live, Informative, Non-cost and Genuine!60 Principles of Applied Reservoir SimulationThe ratio of the number of producing wells to the number of injection wells isshown in Table 7-1. The patterns depicted in Table 7-1 and Figure 7-1 aresymmetric patterns that are especially effective for reservoirs with relativelysmall dip and large areal extent. The injectors and producers are generallyinterspersed. Other patterns in which injectors and producers are groupedtogether may be needed for reservoirs with significant dip. For example, aperipheral or flank injection pattern may be needed to effectively flood ananticlinal or monoclinal reservoir.Table 7-1Producer-to-Injector Ratios for Common Well PatternsWell PatternFour-SpotFive-SpotDirect Line-driveStaggered Line-driveSeven-SpotNine- SpotProducer : Injector Ratio21111/21/3The location of injection wells depends on factors such as reservoirstructure, injected fluid type, and displacement mechanism. For example,upstructure gas injection can be an effective displacement process for producinga monoclinal reservoir containing oil. It relies on the movement of a gas-oilcontact and the displacement of oil to downstrucrure production wells. On theother hand, downstructure peripheral injection of water can be used to displaceoil to upstructure producers in an anticlinal reservoir. In this case, downstructurewater injection is used to move the oil-water contact upstructure and displaceoil to upstructure production wells. The same displacement concept applies toproduction of an anticlinal oil reservoir with strong aquifer support.In addition to reservoir geometry and displacement process, the wellpattern depends on the distribution of existing production wells and the desiredspacing of wells. Well spacing is an estimate of the area being drained by aTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 61production well. A reduction in well spacing requires an increase in the densityof production wells. The density of production wells is the number of productionwells in a specified area. Well density can be increased by drilling additionalwells in the space between wells in a process called infill drilling. Infill drillingis an effective means of altering flow patterns and improving recovery efficiency,but can be more expensive than a fluid displacement process. The selection ofa development plan depends on a comparison of the economics of alternativedevelopment concepts. Reservoir models are especially useful tools forperforming these studies.7.3 Pattern RecoveryOptimum performance may be achieved with the patterns defined in theprevious section by controlling the rates of injectors and producers. Thesecalculations can be performed analytically if we assume the displacing anddisplaced fluids are incompressible, the mobility ratio is one, and the reservoirhas uniform properties. Values of injection rates for the three patterns shownin Figure 7-1 are presented in Table 7-2 [ Wilhite, 1986]. Units and nomenclaturefor the rate equations in Table 7-2 are barrels per day for rate q; darcies forpermeability k; feet for thickness h; well separations a and d, and wellbore radiusrw; pounds per square inch for pressure change A/3; and centipoise for viscosity|i. The well separations are defined in Figure 7-1.Table 7-2Analytical Injection Rates for Selected Well PatternsPatternDirect Line DriveStaggered Line DriveRate3.541 khbP dq = —HIn — + 1.571- -1.838\rw) a-,-> 1a3541 kh^PMIn — + 1.571- -1.838\rwj aTEAM LinG - Live, Informative, Non-cost and Genuine!62 Principles of Applied Reservoir SimulationTable 7-2Analytical Injection Rates for Selected Well PatternsPatternFive-SpotRate3541 khbP*fa]In — - 0.619\rw)The calculation of analytical injection rates, even under a set of restrictiveassumptions, provides a methodology for designing well patterns without usinga reservoir simulator. More accurate estimates of injection rates under a lessrestrictive set of assumptions are obtained using reservoir simulators. Forexample, simulators have been used to correlate volumetric sweep efficiencywith mobility ratio and permeability variation in a reservoir that is beingsubjected to a pattern flood [Wilhite, 1986]. One measure of permeabilityvariation is the Dykstra-Parsons coefficient of permeability variation.The Dykstra-Parsons coefficient can be estimated for a log-normalpermeability distribution asVDp = 1- exp - jertk*Hwhere kA is the arithmetic average permeability for n samplesk --f kKA ~ L Kin /=!and kH is the harmonic average permeabilityJL_ ly _LkH « ~ ktThe Dykstra-Parsons coefficient should be in the range 0 < VDP < 1. For aperfectly homogeneous reservoir, VDP = 0 because kA = kH. An increase inreservoir heterogeneity increases VDP. Typical values of the Dykstra-Parsonscoefficient are in the range 0.4 < VDP < 0.9.Correlations of volumetric sweep efficiency with mobility ratio andpermeability variation show that volumetric sweep efficiency declines asTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 63reservoir heterogeneity increases or mobility ratio increases, particularly formobility ratios greater than one. This makes sense physically if we recall thedefinition of mobility ratio.Mobility ratio is the mobility of the displacing fluid behind the frontdivided by the mobility of the displaced fluid ahead of the front. If the mobilityof the displacing fluid is greater than the mobility of the displaced fluid, thenthe mobility ratio is greater than one. On the other hand, if the mobility of thedisplacing fluid is less than the mobility of the displaced fluid, then the mobilityratio is less than one. Mobility ratios less than or equal to one are consideredfavorable, while mobility ratios greater than one are considered unfavorable.Unfavorable mobility ratios often occur when gas is displacing oil or water isdisplacing high viscosity oil. An example of a flood with a favorable mobilityratio is the displacement of a low-viscosity oil by water.ExercisesExercise 7.1 Core floods show that the waterflood of a core with 80% initialoil saturation leaves a residual oil saturation of 30%. If the same core isresaturated with oil and then flooded with carbon dioxide, the residual oilsaturation is 10%. What are the displacement efficiencies for the waterflood andthe carbon dioxide flood?Exercise 7.2 Assuming a log-normal distribution, estimate the Dykstra-Parsonscoefficient for three sample permeabilities: k{ = 35 md; k2=48 md; k3 -126 md.Exercise 7.3 (A) Run EXAM6.DAT and record the time, pressure, oil rate, waterrate, gas rate, cumulative oil produced, and cumulative gas produced at the endof the run. (B) What is the oil recovery efficiency at the end of the run? Hint:original oil in place is output in the run output file WTEMP.ROF.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 8Fluid recovery concepts during the life of a reservoir are summarized inthis chapter. A review of the various production stages during the life of aconventional reservoir is followed by a discussion of recovery mechanisms forenhanced oil recovery and non-conventional fossil fuels.8.1 Production StagesThe production life of a reservoir begins when reservoir fluid is withdrawnfrom the reservoir. Production can begin immediately after the discovery wellis drilled, or several years later after several delineation wells have been drilled.Delineation wells are used to define the reservoir boundaries, while developmentwells are used to optimize resource recovery. Optimization criteria are definedby management and should take into account relevant governmental regulations.The optimization criteria may change during the life of the reservoir for a varietyof reasons, including changes in technology, economic factors, and newinformation obtained during various stages of reservoir production. The stagesof reservoir production are described below.Primary ProductionPrimary production is ordinarily the first stage of production. It reliesentirely on natural energy sources. To remove petroleum from the pore spaceit occupies, the petroleum must be replaced by another fluid, such as water,natural gas, or air. Oil displacement is caused by the expansion of in situ fluids64TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 65as pressure declines during primary reservoir depletion. The natural forcesinvolved in the displacement of oil during primary production are calledreservoir drives. The most common reservoir drives for oil reservoirs are waterdrive, solution or dissolved gas drive, and gas cap drive.The most efficient drive mechanism is water drive. In this case, waterdisplaces oil as oil flows to production wells. An effective reservoir managementstrategy for a water drive reservoir is to balance oil withdrawal with the rate ofwater influx. Water drive recovery typically ranges from 35% to 75% of theoriginal oil in place (OOIP).In a solution gas drive, gas dissolved in the oil phase at reservoirtemperature and pressure is liberated as pressure declines. Some oil moves withthe gas to the production wells as the gas expands and moves to the lowerpressure zones in the reservoir. Recovery by solution gas drive ranges from 5%to 30% OOIP.A gas cap is a large volume of gas at the top of a reservoir. Whenproduction wells are completed in the oil zone below the gas cap, the drop inpressure associated with pressure decline causes gas to move from the higherpressure gas cap down toward the producing wells. The gas movement drivesoil to the wells, and eventually large volumes of gas will be produced with theoil. Gas cap drive recovery ranges from 20% to 40% OOIP, although recoveriesas high as 60% can occur in steeply dipping reservoirs with enough permeabilityto allow oil to drain to downstructure production wells.Gravity drainage is the least common of the primary production mecha-nisms. In this case oil flows downstructure to a producing well. This is the resultof a pressure gradient that favors downstructure oil flow to oil movementupstructure due to gravity segregation. Gravity drainage can be effective whenit works. It is most likely to happen in shallow, highly permeable, steeplydipping reservoirs.A schematic comparison of primary production mechanisms on reservoirpressure and recovery efficiency is sketched in Figure 8-1. In many cases, oneor more drive mechanisms are functioning simultaneously. The behavior of thefield depends on which mechanism is most important at various times duringthe life of the field. The best way to predict the behavior of such fields is withTEAM LinG - Live, Informative, Non-cost and Genuine!66 Principles of Applied Reservoir Simulationsophisticated reservoir flow models.100K^-^E00 60Recovery Efficiency, % OOIPA: Liquid and Rock ExpansionB: Solution Gas DriveC: Gas Cap ExpansionD: Gravity DrainageE: Water InfluxFigure 8-1. Comparison of primary productionmechanismsIf we rearrange the terms in the general material balance equation for anoil reservoir, Eq. (2.3), we can estimate the relative importance of different drivemechanisms. The indices representing different drives are given in Table 8-1relative to the hydrocarbon production given byDHC = NpB0 + [Cps - NpRso]Bg + G^ (8.1)Table 8-1Drive Indices from the Schilthuis Material Balance EquationDriveSolution GasGas CapWaterInjected FluidsConnate Water and Rock ExpansionIndexIsg = ND0/DHCIgc = NDgo/DHCIw = [(We-Wp)Bw]IDHC^[Wfo + GpllDneI^mDr + D^+NDJ/DHcTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 67The sum of the drive indices equals one, thusIv + Igc + Iw + 1, + I* = 1 (8,2)Equation (8.2) can be derived by rearranging Eq. (2.3). A comparison of themagnitudes of the drive indices indicates which drive is dominating the perform-ance of the reservoir.Although the above discussion referred to oil reservoirs, similar commentsapply to gas reservoirs. Water drive and gas expansion with reservoir pressuredepletion are the most common drives for gas reservoirs. Gas reservoir recoverycan be as high as 70% to 90% of original gas in place (OGIP) because of therelatively high mobility of gas.Gas storage reservoirs have a different life cycle than gas reservoirs thatare being depleted. Gas storage reservoirs are used as a warehouse for gas. Ifthe gas is used to as a fuel for power plants, it will also need to be periodicallyproduced and replenished. The performance attributes of a gas storage reservoirare [Tek, 1996, pg. 4]:* Verification of inventory* Assurance of deliverability* Containment against migrationThe gas inventory consists of working gas and cushion gas. Gas deliverabilitymust be sufficient to account for swings in demand. Demand swings arise fromsuch factors as seasonal variations. Gas containment is needed to conserve theamount of stored gas. For more discussion of natural gas storage in reservoirs,see references such as Tek [1996], Smith [1990], and Katz and Lee [1990].Secondary ProductionPrimary depletion is usually not sufficient to optimize recovery from anoil reservoir. Oil recovery can be doubled or tripled by supplemental naturalreservoir energy. The supplemental energy is provided using an external energysource, such as water injection or gas injection. The injection of water or naturalgas may be referred to as pressure maintenance or secondary production. Thelatter term arose because injection usually followed a period of primary pressuredepletion, and was therefore the second production method used in a field. ManyTEAM LinG - Live, Informative, Non-cost and Genuine!68 Principles of Applied Reservoir Simulationmodern reservoirs incorporate pressure maintenance early in the production lifeof the field, sometimes from the beginning of production. In this case thereservoir is not subjected to a conventional primary production phase. The term"pressure maintenance" is a more accurate description of the reservoirmanagement strategy for these fields than the term "secondary production."Alternative ClassificationsBoth primary and secondary recovery processes are designed to produceoil using immiscible methods. Additional methods may be used to improve oilrecovery efficiency by reducing residual oil saturation. The reduction of residualoil saturation requires a change in such factors as interfacial tension orwettability. Methods designed to reduce residual oil saturation have been referredto in the literature as:• Tertiary Production• Enhanced Oil Recovery• Improved Oil RecoveryThe term tertiary production was originally used to identify the third stage ofthe production life of the field. Typically the third stage occurred after water-flooding. The third stage of oil production would involve a process that wasdesigned to mobilize waterflood residual oil. An example of a tertiary productionprocess is a chemical flood process such as surfactant flooding. Tertiaryproduction processes were designed to improve displacement efficiency byinjecting fluids or heat. They were referred to as enhanced recovery processes.It was soon learned, however, that some fields would perform better if theenhanced recovery process was implemented before the third stage in the lifeof the field. In addition, it was found that enhanced recovery processes wereoften more expensive than just drilling more wells in a denser pattern.The drilling of wells to reduce well spacing and increase well density iscalled infill drilling. The birth of the term "infill drilling" was coincident withthe birth of another term, "improved recovery." Improved recovery includesenhanced oil recovery and infill drilling. Some major improved recoveryprocesses are waterflooding, gasflooding, chemical flooding, and thermalrecovery, [Dyke, 1997]. They are discussed in more detail below.TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 698.2 Enhanced Oil RecoveryImproved recovery technology includes traditional secondary recoveryprocesses such as waterflooding and immiscible gas injection, as well asenhanced oil recovery (EOR) processes. EOR processes are usually classifiedas one of the following processes: chemical, miscible, thermal, and microbial.A brief description of each of these processes is presented here. The literatureon EOR processes is extensive. For more detailed discussions of EOR processes,including screening criteria and analyses of displacement mechanisms, see suchreferences as Taber and Martin [1983], Lake [1989], Martin [1992], Taber, etal. [1996], and Green and Willhite [1998].ChemicalChemical flooding methods include polymer flooding, micellar-polymeror surfactant-polymer flooding, and alkaline or caustic flooding. Polymerflooding is designed to improve the mobility ratio and fluid flow patterns of adisplacement process by increasing the viscosity of injected water containingpolymer, Micellar-polymer flooding uses a detergent-like solution to lowerresidual oil saturation to waterflooding. The polymer slug injected after themicellar slug is designed to improve displacement efficiency. Alkaline floodinguses alkaline chemicals that can react with certain types of in situ crude. Theresulting chemical product is miscible with the oil and can reduce residual oilsaturation to waterflooding.MiscibleMiscible flooding methods include carbon dioxide injection, natural gasinjection, and nitrogen injection. Miscible gas injection must be performed ata high enough pressure to ensure miscibility between the injected gas and in situoil. Miscibility is achieved when interfacial tension (IFT) between the aqueousand oleic phases is significantly reduced. The desired IFT reduction is typicallyfrom around 1 dyne/cm to 0.001 dyne/cm or less. Any reduction in IFT canimprove displacement efficiency, and a near miscible process can yield muchof the incremental oil that might be obtained from a miscible process. If reservoirTEAM LinG - Live, Informative, Non-cost and Genuine!70 Principles of Applied Reservoir Simulationpressure is not maintained above the minimum miscibility pressure (MMP) ofthe system, the gasflood will be an immiscible gas injection process.Immiscible gas can be used as the principal injection fluid in a secondarydisplacement process, or it can be used as the injection fluid for a tertiaryprocess. Two improved recovery processes based on immiscible gas injectionare the double displacement process (DDP) and the second contact waterdisplacement (SCWD) process [Novakovic, 1999]. Both processes require theinjection of immiscible gas into reservoirs that have been previously waterflood-ed. Oil remaining after waterflood can coalesce into a film when exposed to animmiscible gas. The processes require favorable gas-oil and oil-water interfacialtensions. The oil film can be mobilized and produced by down-dip gravitydrainage (the DDP) process or by water influx from either an aquifer or waterinjection (SCWD) following the immiscible gas injection period.ThermalThermal flooding methods include hot water injection, steam drive, steamsoak, and in situ combustion. The injection or generation of heat in a reservoiris designed to reduce the viscosity of in situ oil and improve the mobility ratioof the displacement process. Electrical methods can also be used to heat fluidsin relatively shallow reservoirs containing high-viscosity oil, but electricalmethods are not as common as hot-fluid injection methods. Steam injectionmethods work by injecting steam into the reservoir, while in situ combustionrequires compressed air injection after in situ oil has been ignited. Steam andhot water injection processes are the most common thermal methods becauseof the relative ease of generating hot water and steam. The in situ combustionprocess is more difficult to control than steam injection processes and it requiresan in situ oil that can be set on fire. Hot gases and heat advance through theformation and displace the heated oil to production wells.MicrobialMicrobial EOR uses the injection of microorganisms and nutrients in acarrier medium to increase oil recovery and/or reduce water production inpetroleum reservoirs. Dietrich, et al.[1996] summarized the results of fiveTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 71successful commercial microbial EOR projects. The projects reflected a diversityof locations, lithologies, depths, porosities, permeabilities, and temperatures.Two of the projects were in the U.S., two in China, and one in Argentina, andincluded sandstone, fractured dolomite, siltstone/sandstone, and fracturedsandstone reservoirs. Reservoir depths ranged from 4450 to 6900 feet, tempera-tures from 110° to 180° F, porosity from 0.079 to 0.232, and effective permeabil-ity from 1.7 to 300 md. Evidence from laboratory research and case/field studiesshows that microbial EOR processes can result in the incremental recovery ofoil and also reduce water production from high permeability zones. However,more research needs to be done to maximize the potential for microbial EOR.Some effort in this direction has been conducted. A microbial transport simulatorwas developed under the auspices of the U.S. Department of Energy as amodification to the black oil simulator BOAST.8.3 Nonconventional Fossil FuelsClean energy refers to energy that is generated with little environmentalpollution. Natural gas is a source of clean energy. Oil and gas fields areconsidered conventional sources of natural gas. In the following, we discuss twononconventional sources of natural gas: coalbed methane, and gas hydrates.(oalbed MethaneCoalbeds are an abundant source of methane [Selley, 1998; Rogers, 1994].The presence of methane gas in coal has been well known to coal miners as asafety hazard, but is now being viewed as a source of natural gas. The gas isbound in the micropore structure of the coalbed. It is able to diffuse into thenatural fracture network when a pressure gradient exists between the matrix andthe fracture network. The fracture network in coalbeds consists of microfractures.The microfractures allow Darcy flow and are called "cleats."Gas recovery from coalbeds depends on three processes [Kuuskraa andBrandenburg, 1989]. Coalbed methane exists as a monomolecular layer on theinternal surface of the coal matrix. Its composition is predominately methane,but can also include other constituents, such as ethane, carbon dioxide, nitrogenTEAM LinG - Live, Informative, Non-cost and Genuine!72 Principles of Applied Reservoir Simulationand hydrogen [Mavor, et al., 1999]. Gas content can range from approximately20 SCF gas per ton of coal in the Powder River Basin of Wyoming [Mavor, etal., 1999] to 600 SCF/ton in the Appalachian Basin [Gaddy, 1999]. Gas recoverybegins with the desorption of gas from the internal surface to the coal matrixand micropores. The gas then diffuses through the coal matrix and microporesinto the cleats. Finally, gas flows through the cleats to the production well. Theflow rate depends, in part, on the pressure gradient in the cleats and the densityand distribution of cleats. The controlling mechanisms for gas production fromcoalbeds are the rate of desorption from the coal surface to the coal matrix, therate of diffusion from the coal matrix to the cleats, and the rate of flow of gasthrough the cleats.The production performance of a coalbed methane well typically exhibitsthree stages. The reservoir dewaters and methane production increases duringthe first stage of pressure depletion. Methane production peaks during the secondstage. The amount of water produced is relatively small compared to gasproduction during the second stage because of gas-water relative permeabilityeffects, and desorption of natural gas provides a counterbalance to permeabilityloss as a result of formation compaction. The third stage of production is similarto conventional gas field production in which gas rate declines as reservoirpressure declines.Gas HydratesThe entrapment of natural gas molecules in ice at very low temperaturesforms an ice-like solid. The ice-like solid substance is a metastable complexcalled a gas hydrate. Gas hydrates are clathrates. A clathrate is a chemical com-plex that is formed when one type of molecule completely encloses another typeof molecule in a lattice. In the case of gas hydrates, hydrogen-bonded watermolecules form a cage-like structure in which mobile molecules of gas areabsorbed or bound.The presence of gas hydrates can complicate field operations. Forexample, the existence of hydrates on the ocean floor can affect drillingoperations in deep water. The simultaneous flow of natural gas and water intubing and pipelines can result in the formation of gas hydrates that can impedeTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 73or completely block the flow of fluids through pipeline networks. Heating thegas or treating the gas-water system with chemical inhibitors can prevent theformation of hydrates, but increases operating costs,Gas hydrates are generally considered a problem for oil and gas fieldoperations, but their potential commercial value as a clean energy resource ischanging the industry perception. The potential as a gas resource is due to therelatively large volume of gas contained in the gas hydrate complex. Inparticular, Makogon, et al. [1997] have reported that one cubic meter of gashydrate contains 164.6 m3 of methane. This is equivalent to one barrel of gashydrate containing 924 ft3 of methane, and is approximately six times as muchgas as the gas contained in an unimpeded gas-filled pore system [Selley, 1998,pg, 25]. The gas in gas hydrates occupies approximately 20% of the volume ofthe gas hydrate complex. The remaining 80% of gas hydrate complex volumeis occupied by water.Gas hydrates can be found throughout the world [Selley, 1998; Makogon,et al., 1997]. They exist on land in sub-Arctic sediments and on seabeds wherethe water is near freezing at depths of at least 600 to 1500 feet. For instance,favorable conditions for gas hydrate formation exist at sea floor temperaturesas low as 39°F in the Gulf of Mexico and as low as 30°F in some sections of theNorth Sea. According to Makogon, et al. [ 1997], over 700 trillion m3 in exploredreserves of methane in the hydrate state exist. Difficulties in cost-effectiveproduction have hampered development of the resource.ExercisesExercise 8.1 Use the definitions in Table 8-1 and Eq. (8-1) to derive Eq. (8-2)fromEq. (2.3).Exercise 8.2 (A) Which drive index in Table 8-1 will be largest in a fieldcontaining a dead oil that is subjected to pressure depletion? (B) Suppose a deadoil reservoir is subjected to a peripheral waterflood. Identify the two driveindices in Table 8-1 that will have the greatest influence on oil recovery.TEAM LinG - Live, Informative, Non-cost and Genuine!74 Principles of Applied Reservoir SimulationExercise 8.3 EOR simulators can be found on the internet. Access the internetand search for a website containing public domain EOR simulators. Hint: TheUnited States Department of Energy is one governmental agency that hasdistributed EOR software using a website.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 9Economics and the EnvironmentEconomic analyses are an essential aspect of a reservoir managementstudy. The economic performance of a prospective project is often the decidingfactor in determining whether or not a project is undertaken. Consequently, itis important to be aware of basic economic concepts and factors that may effectthe economic performance of a project. These topics are introduced here. Furtherdetails can be found in references such as Thompson and Wright [1985] andSatter and Thakur [1994].9.1 SPEAVPC ReservesThe analysis of a petroleum project depends on the amount of commer-cially valuable resource that is available. According to the Society of PetroleumEngineers and the World Petroleum Congress [Staff-JPT, 1997], reserves arethose quantities of petroleum which are anticipated to be commerciallyrecoverable from known accumulations from a given date forward. Table 9-1summarizes the SPEAVPC definitions of reserves. The definitions of reservesinclude both qualitative and quantitative criteria. Although the SPEAVPCdefinitions have been adopted in many parts of the world, they are not universal.For example, a different, yet analogous, set of definitions exists in the RussianFederation [Nemchenko, et al., 1995; Grace, et al., 1993].75TEAM LinG - Live, Informative, Non-cost and Genuine!76 Principles of Applied Reservoir SimulationTable 9-1SPE/WPC Reserves DefinitionsProvedreservesUnprovedreservesProbablereservesPossiblereserves4 Those quantities of petroleum which, by analysis of geologicaland engineering data, can be estimated with reasonable certaintyto be commercially recoverable, from a given date forward, fromknown reservoirs and under current economic conditions,operating methods, and government regulation.4 In general, reserves are considered proved if the commercialproducibility of the reservoir is supported by actual productionor formation tests.4 There should be at least a 90% probability (P90) that thequantities actually recovered will equal or exceed the estimate.Those quantities of petroleum which are based on geologicand/or engineering data similar to that used in estimates ofproved reserves; but technical, contractual, economic, or regula-tory uncertainties preclude such reserves being classified asproved.4 Those unproved reserves which analysis of geological andengineering data suggests are more likely than not to be recover-able.4 There should be at least a 50% probability (P50) that thequantities actually recovered will equal or exceed the estimate.4 Those unproved reserves which analysis of geological andengineering data suggests are less likely to be recoverable thanprobable reserves. I4 There should be at least a 10% probability (P]0) that thequantities actually recovered will equal or exceed the estimate.The probability distribution associated with the SPE/WPC reservesdefinitions can be estimated with relative ease if the modeling team hasperformed a sensitivity analysis that generates a set of cases that yield low,medium, and high reserves estimates. In the absence of data to the contrary, areasonable first approximation is that each case is equally likely to occur. Giventhis assumption, an average p, and standard derivation o may be calculated fromthe sensitivity analysis results to prepare a normal distribution of reserves. Fora normal distribution with mean (J, and standard deviation o, the SPE/WPCTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 11reserves definitions are quantified as follows:Proved reserves = P^ = |i - 1.28OProbable reserves = P50 = |iPossible reserves = P10 = [I + 1.28OThe normal distribution can be used to associate an estimate of the likelihoodof occurrence of any particular prediction case with its corresponding economicforecast.9.2 Basic Economic ConceptsThe cash flow of a project is the net cash generated or expended on theproject as a function of time. The time value of money is included in economicanalyses by applying a discount rate to adjust the value of money to the valueduring a base year. The discount rate is the adjustment factor, and the resultingcash flow is called the discounted cash flow. The net present value (NPV) ofthe cash flow is the value of the cash flow at a specified discount rate. Thediscount rate at which NPV is zero is called the discounted cash flow return oninvestment (DCFROI) or Internal Rate of Return (IRR).A typical plot of NPV as a function of time is shown in Figure 9-1. Theearly time part of the figure shows a negative NPV and indicates that the projectTime (Years)Figure 9-1. Typical cash flowTEAM LinG - Live, Informative, Non-cost and Genuine!78 Principles of Applied Reservoir Simulationis operating at a loss. The loss is usually associated with initial capital invest-ments and operating expenses that are incurred before the project begins togenerate revenue. The reduction in loss and eventual growth in positive NPVis due to the generation of revenue in excess of expenses. The point in time onthe graph when the NPV is zero after the project has begun is the payout time.The concept of payout time applies to either discounted or undiscounted cashflow. Payout time on Figure 9-1 is approximately 1.5 years.The discounted cash flow return on investment (DCFROI) and payout timeare measures of the economic viability of a project. Another measure is theprofit-to-investment ratio. The profit-to-investment (PI) ratio is a measure ofprofitability. It is defined as the total undiscounted cash flow without capitalinvestment divided by total investment. Unlike DCFROI, the PI ratio does nottake into account the time value of money. The definitions of several commonlyused economic measures are presented in Table 9-2. Useful plots include a plotof NPV versus time and a plot of NPV versus discount rate.Table 9-2Definitions of Selected Economic MeasuresDiscount RateNet Present Value(NPV)DCFROI or IRRPayout TimeProfit-to-Investment(PI) RatioFactor to adjust the value of money to a baseyear.Value of cash flow at a specified discount rate.Discount rate at which NPV = 0.Time when NPV = 0.Undiscounted cash flow without capital invest-ment divided by total investment.The ideas discussed above are quantified as follows. Net present valueis the difference between the present value of revenue R and the present valueof expenses £, thusNPV = R-E (9.1)If we define AE(&) as the expenses incurred during a time period k, then E maybe written asTEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 79where i 7is the annual inflation rate, N is the number of years of the expenditureschedule, and Q is the number of times interest is compounded each year. Asimilar expression is written for revenue R:where A/?(fc) is revenue obtained during time period k, and i is the annual interestor discount rate. Equations (9.2) and (9.3) include the assumptions that i and / /are constants over the life of the project, but i and / 'are not necessarily equal.These assumptions let us compute the present value of money expended relativeto a given inflation rate i ' and compare the result to the present value of revenueassociated with a specified interest or discount rate i.Illustration: Application to an Oil Production ProjectThe net present value and break-even oil price for an oil production proj ectcan be obtained from the above analysis as an illustration of the concepts. Wespecify the base year for present value calculations as the year when the projectbegins. In this case, we have no initial revenue and the initial expense is justinitial investment //, thusAfl(O) = 0 andA£(0) = // (9.4)Substituting Eqs. (9.2) through (9.4) into Eq. (9.1) givesQ;Revenue from the sale of oil during period k has the formTEAM LinG - Live, Informative, Non-cost and Genuine!80 Principles of Applied Reservoir Simulation(9,6)where P0 is the present price of oil, and AN° (k) is the incremental oil productionduring period k. Notice that we are assuming the value of produced gas isnegligible in this example. An inflation factor on the price of oil is included inEq. (9.6). Combining Eqs. (9.4), (9.5), and (9.6) yields net present value for thisproject:O(9.7)Q QThe incremental oil production in Eq. (9.7) is typically obtained as aforecast using reservoir engineering methods. Some of the most frequently usedmethods include decline curve analysis, material balance analysis, or reservoirsimulation. The oil production profile used in the economic analysis mayrepresent both historical and predicted oil recovery. Th& predicted oil recoveryis used to determine project reserves. Several different production profiles maybe required to determine the probabilistic distribution of reserves and associatedeconomic sensitivity.A break-even oil price Poe for a specified rate of return / = ROR andproduction profile is calculated by setting NPV= 0 as the break-even conditionin Eq. (9.7). Rearranging the resulting equation gives the following estimate ofbreak-even oil price:QN^Q(9.8)~.f ROR1 TQA plot of Pw versus ROR shows the sensitivity of break-even oil price todifferent rates of return.TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 819.3 Investment Decision AnalysisEconomic analyses are performed to provide information about theeconomic performance that can be expected from a project relative to alternativeinvestment options. The decision to invest in a project depends on many factors.Thompson and Wright [1985, pg. 3-2] list the following set of characteristicsfor measures of investment worth that can be used to compare and rankcompeting projects:* Consistent with corporate goals.* Easy to understand and apply.«Permits cost-effective decision making.«Provides a quantitative measure for acceptance or rejection,* Permits alternatives to be compared and ranked.* Incorporates the time value of money.The economic measures that are used in investment decision analysisdepend on the experience of the decision makers who will use the economicmeasures. Some of the most commonly used economic measures are payout,present worth, net present value, discount rate, profit-to-investment ratio, andinternal rate of return. The relative importance of each economic measure isdetermined by the decision makers. For example, a proposed project with anearly payout but relatively low discount rate may be more attractive to acompany that needs to maintain a positive cash flow than another project witha higher discount rate but which does not payout as soon. The criteria foracceptance or rejection of a project may change, even within a company, as theeconomic environment changes.Combinations of economic measures are often used as economic criteriafor making decisions about projects. For example, a project may be consideredeconomically viable if the internal rate of return (IRR) is greater than 30% andthe profit-to-investment ratio (PI) is greater than 0.5. Economic viability isinfluenced by both tangible and intangible factors. Intangible factors such asenvironmental and socio-political concerns are relatively difficult to quantify,yet may have a greater influence on the final decision than tangible factors.Tangible factors, such as well costs and reserves, are relatively easy to quantify.TEAM LinG - Live, Informative, Non-cost and Genuine!82 Principles of Applied Reservoir Simulation9.4 Environmental ImpactEnvironmental issues must always be considered when developing areservoir management strategy. For example, the Louisiana Offshore OilProduction (LOOP) facility is designed to keep hydrocarbon transfer operationsfrom pipelines to tankers away from sensitive coastal areas. Periodic watersampling of surface and produced waters can assure the fresh water sources arenot contaminated. In addition, periodic testing for the excavation or productionof naturally occurring radioactive materials helps assure environmentalcompliance,A well-managed field should be compatible with both the surface andsubsurface environment. The advantages of operating a field with prudentconsideration of environmental issues can pay economic dividends. In additionto improved public relations, a sensitivity to environmental issues can minimizeadverse environmental effects that may require costly remediation and financialpenalties. Remediation is often in the form of clean-up, such as the clean-uprequired after the oil spill from the Exxon-Valdez oil tanker in Alaska. Newtechnologies are being developed to improve our ability to clean-up environmen-tal pollutants. For example, bioremediation uses living microorganisms or theirenzymes to accelerate the rate of degradation of environmental pollutants[Westlake, 1999].SubsidenceAn issue of special importance to reservoir characterization is subsidence.Subsidence is a compressibility effect that depends on the geomechanics of theproduced interval and its overburden. Subsidence, or the change in thicknessA& of the reservoir, can be estimated from the compressibility and pressuredepletion of the system using the equationA/z = cBhAP = <kcfhAPwhereCB bulk compressibility [psia"1]cf formation compressibility [psia"1]TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 83h net thickness of reservoir [ft]4> porosity [frac]AF pressure depletion [psia]If properties like compressibility are measured hydrostatically, they should becorrected to uniaxial compressibilities [Teeuw, 1971] so that the subsidenceestimate becomesA/z =where V is Poisson's ratio and the subscript u denotes uniaxial compressibility.The correction for uniaxial compaction recognizes that reservoirs with largelateral dimensions relative to their vertical thickness deform mainly in thevertical direction.In many cases, subsidence has little or no adverse environmental effects.In some cases, however, subsidence can be a significant concern. For example,a pressure maintenance program in a field where surface subsidence is a likelyconsequence of pressure depletion can improve resource recovery and help avoideconomic liabilities resulting from damage caused by surface subsidence.Subsidence in the Long Beach, California, area due to production of theWilmington field had to be mitigated with a pressure maintenance program.Subsidence has been responsible for production induced seismicity inareas such as the Rocky Mountain Arsenal near Denver, Colorado, whereproduction induced seismicity was identified as the cause of earthquakes.Earthquakes due to natural causes have led to fatalities in tectonically activeareas like the Sea of Okhotsk, offshore Sakhalin Island, Russia. Developmentactivities in tectonically active areas, such as offshore Sakhalin Island, need toanticipate the impact of subsidence and production induced seismicity as partof their reservoir management plans. Examples of compaction studies arepresented by Fredrich, et al. [1998] and Settari and Walters [1999].Sustainable DevelopmentFailure to adequately consider environmental issues can lead to bothtangible and intangible losses. Intangible losses are difficult to quantify, but caninclude loss of public support for an otherwise economically viable project.TEAM LinG - Live, Informative, Non-cost and Genuine!84 Principles of Applied Reservoir SimulationTangible losses have more readily quantifiable economic consequences. Forexample, near- and long-term economic liabilities associated with potable watercontamination can adversely effect project economics. It becomes a questionof business ethics whether a practice that is legal but can lead to an adverseenvironmental consequence should nonetheless be pursued because a cost-benefitanalysis showed that economic liabilities were less than economic benefits.Typically, arguments to pursue an environmentally undesirable practicebased on cost-benefit analyses do not adequately account for intangible costs.For example, the decision by Shell to dispose of the Brent Spar platform bysinking it in the Atlantic Ocean led to public outrage in Europe in 1995.Reversing the decision and disassembling the platform for use as a quay inNorway resolved the resulting public relations problem, but the damage had beendone. The failure to anticipate the public reaction reinforced a lack of publicconfidence in the oil and gas industry, and helped motivate government actionto regulate the decommissioning of offshore platforms in northwest Europe[Offshore Staff, 1998].The problem facing the industry is to learn how to achieve sustainabledevelopment. One industry response to environmental and social concerns inthe context of sustainable development is the "triple bottom line" [Whittaker,1999]. According to this view, sustainable development must integrate socialand environmental concerns into a development plan that optimizes economicprofitability and value creation. The three components of sustainable develop-ment, and the three goals of the triple bottom line (TBL), are economicprosperity, social equity, and environmental protection. The focus of TBL is thecreation of long-term shareholder value by recognizing that corporations aredependent on licenses provided by society to do business. Whittaker [ 1999, pg.25] reports that "After a period of serious introspection following the Brent Spardebacle, Royal Dutch/Shell is perhaps the most enthusiastic supporter of TBL."Although TBL is in its infancy, key elements of TBL policy are beginning toemerge. They include [Whittaker, 1999, pg. 25]:« Performance measurements that include qualitative social indicators andecoefficiency measures (such as energy consumption and recycling) inaddition to compliance and pollutant emissions.TEAM LinG - Live, Informative, Non-cost and Genuine!Part I: Reservoir Engineering Primer 85• Development and implementation of strategies that will enable theindustry to meet both future global energy needs and environmentalobjectives.• Investment in natural gas, low or zero-emissions fuels, and renewableforms of energy.• Improved communications with communities affected by operations,Global Climate ChangeOne of the most pressing environmental concerns is global climate change.A purported cause of adverse global climate change is due to the "greenhouseeffect." Increasing levels of carbon dioxide in the atmosphere absorb infraredradiation rather than letting it escape into space. The resulting atmosphericheating is attributed to excessive emissions of carbon dioxide into the atmo-sphere. Government and industry are undertaking programs to address this issue.For example, one possible solution is to collect and store carbon dioxide inreservoirs in a process known as CO2 sequestration. The goal of CO2 sequestra-tion and similar programs is to provide economically competitive and environ-mentally safe options to offset all projected growth in baseline emissions ofgreenhouse gases.ExercisesExercise 9.1 Five independent studies determined the following reserves forReservoir A:StudyOil Recovery (MSTBO)13202150348042605370Assuming a normal distribution of reserves, estimate proved, probable andpossible reserves. Hint: Calculate the average and standard deviation for the oilrecoveries reported above.Exercise 9.2 Suppose a reservoir has an average porosity of 20%, a formationcompressibility of 20><10~6 psia"1, and a net thickness of 500 feet; and thereservoir is subjected to a pressure depletion of 3000 psia. (A) Plot subsidenceTEAM LinG - Live, Informative, Non-cost and Genuine!86 Principles of Applied Reservoir Simulationas a function of Poisson's ratio for a Poisson's ratio ranging from 0.1 to 0.35.(B) If you are operating the field from a platform that is built with a deck thatis 10 feet above maximum wave height, discuss the possible impact of subsidenceon operations? (C) Discuss the possible impact of subsidence on wellborefor deviated wells drilled from the platform.TEAM LinG - Live, Informative, Non-cost and Genuine!Part IIReservoir SimulationTEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!Overview of the Modeling ProcessThe process of applying a reservoir flow simulator to the study of aphysical system is outlined here. The best technology for making reservoirperformance predictions today is to model fluid flow in porous media usingcomputer programs known as simulators.10.1 Basic Reservoir AnalysisReservoir characterization and reservoir engineering evaluations areusually performed as a part of standard business practice independent of areservoir simulation study. The tasks associated with basic reservoir analysisare described in Chapter 2 and in such references as Craft, et al. [1991], Mian[1992], and Tearpock and Bischke [1991]. They provide information that isneeded to prepare input data for a simulation study. For example, materialbalance studies require the acquisition of fluid property data, field pressures, andproduction volumes. This information is also needed to conduct a model studyusing a reservoir simulator. Volumetric analyses provide independent appraisalsof reservoir volume that can be used to check the original fluid volumescalculated by a reservoir model. In addition, basic reservoir analysis can providean initial concept of the reservoir and associated drive mechanisms. Theseconcepts can be used to design the model study. The modeling team needs tobe aware of existing studies and should relate model performance to previousstudies whenever possible.89TEAM LinG - Live, Informative, Non-cost and Genuine!90 Principles of Applied Reservoir Simulation10.2 PrerequisitesSeveral prerequisites should be satisfied before a model study is under-taken [Coats, 1969]. The most important, from a business perspective, is theexistence of a problem of economic importance. At the very least, the objectivesof a model study should yield a solution to the economically important problem,Once the objectives of a study are specified, the modeler should gatherall available data and reports relating to the field. The term "modeler" is usedin the remainder of the text as a synonym for "modeling team" unless an explicitdistinction must be made. If necessary data is not available, the modeler shoulddetermine if the data can be obtained, either by analogy with other reservoirsor by correlation. Values for all model input data must be obtained because thesimulator will not run without a complete set of data. In some cases, simplifyingassumptions about the reservoir may have to be made because there is notenough data available to quantitatively represent the system in greater detail,In addition to clearly defined objectives, another prerequisite that mustbe satisfied before committing to a simulation study is to determine that theobjectives of the study cannot be achieved using simpler techniques. If lessexpensive techniques, such as decline curve analysis or the Buckley-Leverettwaterflood displacement algorithm [Collins, 1961; Craig, 1971; and Dake, 1978],do not provide adequate results, then more sophisticated and costly methods arejustified.10.3 Computer ModelingA comprehensive reservoir management model can be thought of as fourinteracting models: the reservoir model, the well model, the wellbore model, andthe surface model. The spatial relationship between these models is illustratedin Figure 10-1. The reservoir model represents fluid flow within the reservoir.The reservoir is modeled by subdividing the reservoir volume into an array, orgrid, of smaller volume elements (Figure 10-2). Many names are used to denotethe individual volume elements: for example, gridblock, cell, or node. The setof all volume elements is known by such names as grid or mesh.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 91Surface Model• Wellbore ModelFigure 10-1. Reservoir management system.Every practical reservoir simulator includes both a reservoir model anda well model. The well model is a term in the fluid flow equations that representsthe extraction of fluids from the reservoir or the injection of fluids into thereservoir. Full-featured commercial simulators also include a wellbore modeland a surface facility model. The wellbore model represents flow from thesandface to the surface. The surface model represents constraints associated withsurface facilities, such as platform and separator limitations.UnconformityFigure 10-2. Subdivide reservoir.The mathematical algorithms associated with each model depend onphysical conservation laws and empirical relationships. Computer simulatorsare based on conservation of mass, momentum, and energy. The most widelyused simulators assume the reservoir is isothermal, that is, constant temperature.If we are modeling a reservoir where thermal effects matter, such as a secondaryTEAM LinG - Live, Informative, Non-cost and Genuine!92 Principles of Applied Reservoir Simulationrecovery process where heat has been injected in some form, then we need touse a simulator that accounts for temperature variation and associated thermody-namic effects. The set of algorithms is sufficiently complex that high-speedcomputers are the only practical means of solving the mathematics associatedwith a reservoir simulation study. These topics are discussed in more detail inlater chapters.10.4 Major Elements of a Reservoir Simulation StudyThe essential elements of a simulation study include matching fieldhistory; making predictions, including a forecast based on the existing operatingstrategy; and evaluating alternative operating scenarios [Mattax and Dalton,1990; Thomas, 1982], During the history match, the modeler will verify andrefine the reservoir description. Starting with an initial reservoir description,the model is used to match and predict reservoir performance. If necessary, themodeler will modify the reservoir description until an acceptable match isobtained. The history matching phase of the study is an iterative process thatmakes it possible to integrate reservoir geoscience and engineering data.The history matching process may be considered an inverse problembecause an answer already exists. We know how the reservoir performed; wewant to understand why. Our task is to find the set of reservoir parameters thatminimizes the difference between the model performance and the historicalperformance of the field. This is a non-unique problem since there is usuallymore than one way to match the available data.Once a match of historical data is available, the next step is to make a basecase prediction, which is essentially just a continuation of existing operatingpractice. The base case prediction gives a baseline for comparison with otherreservoir management strategies.Model users should be aware of the validity of model predictions. Oneway to get an idea of the accuracy of predictions is to measure the success offorecasts made in the past. Lynch [1996] looked at the evolution of the UnitedStates Department of Energy price forecast over a period of several years forboth oil and gas. The quality of price forecasts is illustrated in Figure 10-3,TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 93120if 100I. 80•S 60Q.R ^OCL.== 20OO. . , . .; '• • ,• • :' ' : ; ! ' ..-•-*"• : _.' :._!_!./ . V - ". / . ! ^ w .'. i „ * r **— ' - - '• • \ ..***.-**V"Is»~"~-"'~"~*-- * * . .: ii ' 'Actual1981"1984•1987•19911975 1981 1986 1991 1996YearFigure 10-3. Price forecasting.Forecasts that were made in years 1981, 1984, 1987, and 1991 are comparedto the actual prices. Even though price forecast is essential to a commercialenterprise, it is clear from Lynch's study that there is considerable uncertaintyassociated with the price forecast. The wide swing in oil price in the late 1990'swhere oil price varied by a factor of two indicates the volatility of economicfactors that are needed in forecasts.In addition to uncertainty in economic parameters, there is uncertaintyin the forecasted production performance of a field. Forecasts do not accountfor discontinuities in historical patterns that arise from unexpected effects. Thisis as true in the physical world as it is in the social [Oreskes, et al, 1994].Simulators do not eliminate uncertainty; they give us the ability to assess andbetter manage the risk associated with the prediction of production performance.A valuable but intangible benefit of the process associated with reservoirsimulation is the help it provides in managing the reservoir. One of the criticaltasks of reservoir management is the acquisition and maintenance of an up-to-date data base. A simulation study can help coordinate activities as a modelingteam gathers the resources it needs to determine the optimum plan for operatinga field. Collecting input data for a model is a good way to ensure that everyimportant technical variable is considered as data is collected from the manydisciplines that contribute to reservoir management. If model performance isespecially sensitive to a particular parameter, then a plan should be made todetermine that parameter more accurately, for example, from either laboratoryor appropriate field tests.TEAM LinG - Live, Informative, Non-cost and Genuine!94 Principles of Applied Reservoir SimulationExercisesExercise 10.1 Original Volume In Place: Data file EXAM 1 .DAT is a materialbalance model of an undersaturated oil reservoir undergoing pressure depletion.Run EXAM 1 .DAT and find the volume of oil and gas originally in place.Exercise 10.2 Gas Reservoir Material Balance: Suppose a gas reservoir has thefollowing production history:GP(Bscf)0.0150.1230.3120.6521.3822.2102.9733.3554.0924.4474.822P(psia)19461934191318731793170216171576149014531413Z0.8130.8130.8140.8150.8190.8230.8280.8300.8350.8380.841P/Z(psia)23932378235022972190206819531899178317341680where G> is cumulative gas production, P is pressure, and Z is gas compress-ibility factor. Draw a straight line through a plot of GP vs P/Z to find originalgas in place (OGIP). OGIP corresponds to PIZ- 0. These results were obtainedfrom data file EXAM8.DAT. Verify that the OGIP for the model is about 15.9Bscf by running EXAM8.DAT and finding the OGIP in WTEMP.ROF. Howmuch oil and water are originally in place?TEAM LinG - Live, Informative, Non-cost and Genuine!Conceptual Reservoir ScalesOne of the most important goals of modeling is to reduce the riskassociated with making decisions in an environment where knowledge is limited.The range of applicability of acquired data and the integration of scale-dependentdata into a cohesive reservoir concept are discussed below.11.1 Reservoir Sampling and ScalesA sense of just how well we understand the reservoir can be obtained byconsidering the fraction of reservoir area sampled by different techniques. Asan example, suppose we want to find the size of the area sampled by a wellborethat has a six-inch radius. If we assume the area is circular, we can calculate thearea as TC r2 where r is the sampled radius. The resulting sampled area is less thana square foot. To determine the fraction of area sampled, we normalize thesampled area with respect to the drainage area of a well, say a very modest fiveacres. What fraction of the area is directly sampled by the wellbore? The drain-age area is 218,000 square feet. The fraction of the area sampled by the well isthree to four parts in a million. This is a tiny fraction of the area of interest.A well log signal will expand the area that is being sampled. Suppose awell log can penetrate the formation up to five feet from the wellbore, whichis a reasonably generous assumption. The fraction of area that has been sampledis now approximately four parts in ten thousand. The sample size in a drainagearea of five acres, which is a small drainage area, is still a fraction of a percent.95TEAM LinG - Live, Informative, Non-cost and Genuine!96 Principles of Applied Reservoir SimulationCore and well log information gives us a very limited view of the res-ervoir. A seismic section expands the fraction of area sampled, but the interpreta-tion of seismic data is less precise. Seismic data is often viewed as "soft data"because of its dependence on interpretation. The reliability of seismic interpreta-tion can be improved when correlated with "hard data" such as core and welllog measurements.The range of applicability of measured data depends on the samplingtechnique. Did we take some core out of the ground, measure an electricalresponse from a well log, or detect acoustical energy? The ranges are illustratedin Figure 11-1. Payers and Hewett [ 1992] point out that scale definitions are notuniversally accepted, but do illustrate the relative scale associated with reservoirproperty measurements. Scale sizes range from the very big to the microscopic.To recognize variations in the range of data applicability, four conceptual scaleshave been defined (Figure 11-2) and will be adopted for use in the followingdiscussion.WELL COR ELECTRIC LOG SEISMIC SECTION100*-150'-ISO'49mFigure 11-1. Range of data sampling techniques (afterRichardson, et al., 1987a; reprinted by permission of theSociety of Petroleum Engineers).The Giga Scale includes information associated with geophysicaltechniques, such as reservoir architecture. Theories of regional characterization,such as plate tectonics, provide an intellectual framework within which GigaScale measurement techniques, like seismic and satellite data, can be interpreted.The Mega Scale is the scale of reservoir characterization and includes wellTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 91logging, well testing, and 3D seismic analysis. The Macro Scale focuses on datasampling at the level of core analysis and fluid property analysis. The MicroScale includes pore scale data obtained from techniques such as thin sectionanalysis and measurements of grain-size distribution. Each of these scalescontributes to the final reservoir model.MICROGIGAFigure 11-2. Reservoir scales (afterHaldorsen and Lake, 1989; reprinted bypermission of the Society of PetroleumEngineers).11.2 Integrating Scales - the Flow UnitAll of the information collected at various scales must be integrated intoa single, comprehensive, and consistent representation of the reservoir. Theintegration of data obtained at different scales is a difficult issue that is oftenreferred to as the "scale-up" problem [for example, see Oreskes, et al., 1994].Attempts to relate data from two different scales can be difficult. For example,permeability is often obtained from both pressure transient testing and routinecore analysis. The respective permeabilities, however, may appear to beuncorrelated because they represent two different measurement scales. Animportant task of the scale-up problem is to develop a detailed understandingof how measured parameters vary with scale. The focus on detail in one or moreTEAM LinG - Live, Informative, Non-cost and Genuine!98 Principles of Applied Reservoir Simulationaspects of the reservoir modeling process can obscure the fundamental reservoirconcept in a model study. One way to integrate available data within the contextof a "big picture" is to apply the flow unit concept.A flow unit is defined as "a volume of rock subdivided according to geo-logical and petrophysical properties that influence the flow of fluids through it"[Ebanks, 1987]. Typical geologic and petrophysical properties are shown inTable 11 -1, A classic application of the flow unit concept is presented in a paperby Slatt and Hopkins [1990],Table 11-1Properties Typically Needed to Define a Flow UnitGeologicTextureMineralogySedimentary StructureBedding ContactsPermeability BarriersPetrophysicalPorosityPermeabilityCompressibilityFluid SaturationsA reservoir is modeled by subdividing its volume into an array of repre-sentative elementary volumes (REV). The REV concept is not the same as theflow unit concept. A flow unit is a contiguous part of the reservoir that hassimilar flow properties as characterized by geological and petrophysical data.Several flow unit identification techniques are proposed in the literature, suchas the modified Lorenz plot used by Gunter, et al. [1997].A simplified variation of the modified Lorenz plot technique is to identifya flow unit by plotting cumulative flow capacity as a function of depth.Cumulative flow capacity Fm is calculated asFm = cum flow capacity = ]£ kth. /£ kihi ', m= \,,..,n/=! / /=!where n is the total number of reservoir layers. The layers are numbered in orderfrom the shallowest layer / = 1 to the deepest layer i = m for a cumulative flowcapacity Fm at depthTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 99mZ = Z0 + Y hm U fa*/ ii=\where Z0 is the depth to the top of layer 1 from a specified datum. A flow unitwill appear on the plot as a line with constant slope. A change in slope isinterpreted as a change from one flow unit to another, as illustrated in Figure11-3. Slope changes in Figure 11-3 occur at depths of 36 feet, 76 feet, 92 feet,108 feet, 116 feet, 124 feet, 140 feet, 152 feet, and 172 feet. The largest slopeis between 108 feet and 116 feet, and corresponds to a high permeability zone.It is followed immediately by a low permeability zone at a depth of approxi-mately 120 feet.1,000Depth (feet)Figure 11-3. Identifying flow units.Flow units usually contain one or more REVs. By contrast, the REV isthe volume element that is large enough to provide statistically significantaverage values of parameters describing flow in the contained volume, but smallenough to provide a meaningful numerical approximation of the fundamentalflow equations [for example, see Bear, 1972]. As noted by Payers and Hewett[1992], "It is somewhat an act of faith that reservoirs can be described byrelatively few REV types at each scale with stationary average properties."The flow unit concept is an effective means of managing the growing baseof data being provided by geoscientists. Increasing refinement in geoscientifkTEAM LinG - Live, Informative, Non-cost and Genuine!100 Principles of Applied Reservoir Simulationanalysis gives modelers more detail than they can use. Even today, with 100,000to one million gridblock flow models, modelers cannot use all of the informationthat is provided by computer-based geologic models. Computer-based geologicmodels often have in excess of one million grid points. It is still necessary tocoarsen detailed geologic models into representative flow units.An understanding of the big picture, even as a simple sketch, is a valuableresource for validating the ideas being quantified in a model. Richardson, et al.[ 1987b] sketched several common types of reservoir models: a deep-water fan;a sand-rich delta; a deltaic channel contrasted with a deltaic bar, etc. Theirsketches illustrate what the reservoir might look like for a specified set ofassumptions. A sketch such as Figure 11-4 is a good tool for confirming thatpeople from different disciplines share the same concept of a reservoir; it is asimple visual aid that enhances communication. In many cases, especially thecase of relatively small fields, the best picture of the reservoir may only be aqualitative picture. When a more detailed study begins, the qualitative picturecan be upgraded by quantifying parameters such as gross thickness in the con-text of the conceptual sketch of the reservoir.Figure 11-4. Mississippi Delta.Confidence in model performance is acquired by using the model to matchhistorical field performance. History matching and model validation will bediscussed in greater detail later. From a technical perspective, flow modelsshould be updated and refined as additional information is obtained from theTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 101field. In practice, the frequency of model updates depends on the importanceof the resource being modeled to the enterprise.11.3 Geostatistical Case StudyThe process of characterizing a reservoir in a format that is suitable foruse in a reservoir simulator begins with the gathering of data at control pointssuch as wells. Once this occurs, the data can be contoured and digitized. Theresulting set of digitized maps becomes part of the input data set for a reservoirsimulator.The contouring step in the process outlined above is changing. Contouringis the step in which reservoir parameters such as thickness and porosity arespatially distributed. The spatial distribution of reservoir parameters is afundamental aspect of the reservoir characterization process. Two methods forspatially distributing reservoir parameters are emerging: geostatistics andreservoir geophysics.Many modelers view geostatistics as the method of choice for sophisti-cated reservoir flow modeling [for example, see Lieber, 1996; Haldorsen andDamsieth, 1993; and Rossini, et al, 1994], even though the resulting reservoircharacterization is statistical. By contrast, information obtained from reservoirgeophysics is improving our ability to "see" between wells in a deterministicsense. Are these methods competing or complementary? This section presentsa case study that demonstrates several points about geostatistics. A reservoirgeophysical case study is presented in the next chapter. A review of these studiescan help you decide whether either method is appropriate for a particularapplication.An example of a ftill field model study using a geostatistical reservoirrealization is the reservoir management study of the N.E. Nash Unit in Oklahoma[Fanchi, et al., 1996]. The goal of the study was to prepare a full field reservoirmodel that could be used to identify unswept parts of the field. We knew, basedon the history of the field, that water was breaking through at several wells. Thestudy was designed to look for places where an additional production well couldbe economically drilled.TEAM LinG - Live, Informative, Non-cost and Genuine!102 Principles of Applied Reservoir SimulationThe N.E. Nash Unit has a gradual dip from north to south. The Misenersandstone reservoir is bounded above by the Woodford shale, on the flanks bythe Sylvan shale, and below by the Viola limestone. The Viola limestone doesallow some aquifer support for the Misener sandstone.One of the primary tasks of the study was to map the N.E. Nash Unit. Twosets of maps were prepared: conventional hand-drawn maps, and a set of mapsbased on a geostatistical analysis of the field. The hand-drawn maps correspondto the deterministic approach in which a single realization is used, while thegeostatistical maps correspond to a stochastic image of the reservoir.A geostatistical analysis was performed using 42 well control points tocalculate structural tops, gross thickness, net-to-gross ratio, and porosity. Across-plot between porosity and core permeability yielded a relationship forcalculating permeability from porosity. From this data, directional semi-variograms (Table 11 -2) were prepared to describe the spatial continuity of eachparameter. The semi-variograms represent parameter changes as functions ofdistance and direction. For a detailed technical discussion of geostatistics, seea text such as Isaaks and Srivastava [1989]. Hebert, et al. [1993] have publishedsome geostatistical software that is compatible with BOAST II.Table 11-2Semi-Variogram ModelGoal: Model spatial correlation of data with semi-variance y(h)Semi-VarianceValue of spatially distributed property at point xit for example, (j),Spatial vector or "lag" distance between data point at xt + h and datapoint at xf. "Lag" A is a vector with length and direction.N(h)Number of data pairs approximately separated by vector h.When two sets of maps were compared, the hand-drawn maps were foundto be more homogenous than the geostatistical maps. The geostatistical mapsTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 103exhibited the large-scale trends shown in the hand-drawn maps, but containedmore local variability. This was not surprising, since additional heterogeneityis expected to arise as a result of geostatistical mapping.The choice of final maps was based on management priorities: minimizethe risk of drilling a dry hole on the flanks of the field, and complete the studybefore water breakthrough occurred in the remaining oil producers. Thegeostatistical model satisfied both of these criteria. The main flow path in thereservoir was narrower in the geostatistically generated maps than in the hand-drawn maps, and the geostatistical realization could be modified in a day or two.Once a set of maps was chosen, the history match process could begin.Tracer information in the form of salinity changes was useful in helping identifysources of injection water as the water was produced. This was valuable indefining flow channels that could not otherwise be inferred. In some areas,transmissibility and porosity changes were needed to match water cut andreservoir pressure.The geostatistical realization used in the N.E. Nash study was just a singlerealization. It was selected because it satisfied constraints imposed by previousvolumetric and material balance studies. If these constraints were not availableor were less reliable, which would be the case early in the life of a field, ageostatistical study would require the use of multiple realizations to characterizethe reservoir. This raises the question of how many realizations are necessary.Value —*-AvgFigure 11-5. Running average.TEAM LinG - Live, Informative, Non-cost and Genuine!104 Principles of Applied Reservoir SimulationFigure 11-5 shows a random sampling from a discrete probability-distribution. A running average is also plotted. The figure shows that the runningaverage does not stabilize, or approach a constant value, until at least 20 trialshave been completed. This is a large number of realizations if history matchingis needed for each realization. Indeed, it would be an unacceptably large numberof realizations, in most cases, because of the time it takes to perform a historymatch,Multiple realizations can also confuse people who are not closely involvedwith the modeling process because they do not have a single picture of thereservoir. On the other hand, the use of multiple realizations makes it possibleto quantify the uncertainty associated with our limited knowledge of propertiesdistributed spatially throughout the field. Table 11-3 summarizes the advantagesand concerns associated with geostatistics. There is no established procedurefor selecting one or more realizations for history matching from a set ofgeostatistically derived realizations. One procedure is described by Rossini, etal. [1994], An application of reservoir geostatistics in the context of amultidisciplinary study is presented by Wang, et al. [1998].Table 11-3GeostatisticsAdvantages+ Realism4 Quantifies uncertaintyConcerns+ Cost and confusion of multiple realizations+ History matching still necessary to accountfor model discontinuities such as channeling4 History matching complicated by factors suchas probabilistically generated heterogeneityExercisesExercise 11.1 (A) Run EXAM1.DAT and record the final time, final pressureand initial oil volume (B) Multiply the volume of the reservoir in EXAM 1 .DATby 0.5,10 and 100. This can be done by altering the gridblock size (see Chapter24.1.1). Make a table showing the final time, final pressure, and initial oilvolume for each case. (C) How does the change in volume affect the pressureTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 105performance of the model as a function of time?Exercise 11.2 Repeat Exercise 11.1, but make the volume changes by modifyingthe grid dimensions using the modification option presented in Chapter 24.1,2.Exercise 11.3 Roll a pair of dice 50 times and record the results. Calculate arunning average by calculating a new average after each trial (roll of the dice),Plot the running average for each trial. How many trials are necessary beforethe average stabilizes, that is, the average approaches a constant value?TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 12Reservoir StructureThe physical size and shape of the reservoir may be inferred from severalmethods that serve as sources of information for defining the large-scalestructure of the reservoir. These information sources are briefly reviewed below.12.1 Giga ScaleSeismic measurements discussed in the literature by authors such asAusburn, et al. [ 1978], McQuillin, et al. [ 1984], Sheriff [ 1989] and Dorn [ 1998]provide much of the Giga Scale information that can be directly used tocharacterize a reservoir. Historically, seismic analyses have been of interestprimarily as a means of establishing the structural size of the reservoir. Peopledid not believe that seismic data could resolve sufficient detail to provideinformation beyond overall reservoir structure. But that view has changed withthe emergence of 4-D seismic monitoring and reservoir geophysics [for example,see Richardson, 1989;Ruijtenberg,etal., 1990; Anderson, 1995;He,etal., 1996;Johnston, 1997;; Fanchi, et al. 1999]. It is therefore worthwhile to introducesome basic geophysical concepts within the context of the reservoir managementfunction.Seismic waves are vibrations that propagate from a source, such as anexplosion, through the earth until they encounter a reflecting surface and arereflected into a detector, such as a geophone. Figure 12-1 shows a seismic trace.Each trace represents the signal received by a detector. Changes to the amplitude106TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 107of seismic waves occur at reflectors. A seismic reflection occurs at the interfacebetween two regions with different acoustic impedances.Model100ms —200 ms —Seismic response-51015__.0<i2Figure 12-1. Seismic trace for a sand wedge (afterRuijtenberg, 1990; reprinted by permission of the Societyof Petroleum Engineers).Acoustic impedance is a fundamental seismic parameter. Acousticimpedance is defined as Z = p V where p is the bulk density of the medium andV is the compressional velocity of the wave in the medium. Figure 12-23.0n 28E•S2 2.6o>2'4C 2.20)Q2.01.81.5 2.0 3 4 56789Velocity (km/sec)Figure 12-2. Seismic wave velocity and bulk density ofrock (after Telford, et al., 1976; reprinted by permissionof Cambridge University Press; after Gardner, et al.,1974).TEAM LinG - Live, Informative, Non-cost and Genuine!108 Principles of Applied Reservoir Simulationillustrates a correlation between seismic wave velocity and the bulk density ofdifferent types of rock. Further discussion of rock properties and their relation-ship to seismic variables can be found in the literature [for example, Schon1996],A change in acoustic impedance will cause a reflection of the sound wave.The ability to reflect a sound wave by a change in acoustic impedance isquantified in terms of the reflection coefficient. The reflection coefficient R atthe interface between two contiguous layers is defined in terms of acousticimpedances asZ2 - Zn _ / £ _where subscripts 1 and 2 refer to the contiguous layers.Reflection coefficient magnitudes for typical subsurface interfaces areillustrated in Table 12-1. Values of reflection coefficients at the sandstone/lime-stone interface show that reflection coefficient values can be relatively smallIn addition to reflection coefficient, a transmission coefficient can be defined.The transmission coefficient is one minus the reflection coefficient.Table 12-1Typical Reflection CoefficientsInterfaceSandstone on limestoneLimestone on sandstoneOcean bottomReflection Coefficient0.040- 0.0400.11 (soft) to 0.44 (hard)Nonzero reflection coefficients occur when a wave encounters a changein acoustic impedance, either because of a change in compressional velocity ofthe wave as it propagates from one medium to another, or because the bulkdensities of the media differ. If the change in acoustic impedance is large enough,the reflection can be measured at the surface. That is why gas tends to show upas bright spots on seismic data - there is a big change in the density of the fluid.By contrast, the presence of an oil/water contact is harder to observe with seismicTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 109measurements because density differences between the oil and water phases arerelatively small and result in small changes in acoustic impedance.The seismic trace plots seismic amplitude versus two-way travel time, orthe time it takes the seismic wave to propagate from the source to the receiver.One of the central problems in seismic data processing is to determine thetime/depth conversion. The conversion of travel time data to formation depthrequires that the velocity associated with each geologic zone be known or canbe inferred as the wave evolves with time. When the time/depth conversion isapplied to seismic data, it can change the relative depths of seismic amplitudesassociated with adjacent traces.Figure 12-3 shows the amplitude and wavelength of a seismic wave [afterde Buyl, et al., 1988]. The sonic log response shown in Figure 12-3 illustratesthe relationship between seismic amplitude and the sonic log. Sonic logs aretypically used to calibrate seismic data when seismic data is used in reservoircharacterization. The sonic log response in Figure 12-3 delineates the top andbase of a geologic section.Sonic LogSeismic WaveFigure 12-3. Seismic wave and sonic logresponse.The wavelength of the seismic wave is the velocity of the wave dividedby its frequency. Alternatively, the wavelength is the velocity in a given mediumTEAM LinG - Live, Informative, Non-cost and Genuine!110 Principles of Applied Reservoir Simulationtimes the period of the wave. The frequency of the wave is a measure of theenergy of the wave and is conserved as the wave propagates from one mediumto another. The wavelength, however, can vary from one medium to another.When waves overlap - or superpose - they create a wavelet, as shownin Figure 12-4. The time duration associated with the wavelet disturbance isdenoted Af. The wavelet has a velocity Fin a medium, and the period Tis thewidth of the wavelet when plotted as a trace on a time-map of seismic data. Thelength of the wave is equal to the velocity V times the period T. Thus, if thewavelet has a 10 millisecond period and the velocity is 5000 feet per second ina particular medium, then the length L of that wavelet is 50 feet.WaveletV= velocity in mediumT = A / = period of waveletFigure 12-4. Seismic wavelet.If seismic data has enough resolving power to show the reflecting bound-aries of a geologic layer, then the amplitudes of the seismic waves may be usefulfor further characterizing petrophysical properties of the reservoir. For example,suppose a reservoir region is characterized by a porosity <f>, permeability K, netthickness hnet, and oil saturation S0. Seismic amplitude may be correlatable withrock quality (for example, Khnet or §khnet) or oil productive capacity (forexample, S0 <j> khnet). When a correlation does exist between seismic amplitudeand a grouping of petrophysical parameters, the correlation may be used to helpguide the distribution of reservoir properties in areas between wells.Figures 12-5a and b show two approaches to contouring a set of valuesat control points. The smooth contour lines shown in Figure 12-5a are preferredby mappers [Tearpock and Bischke, 1991 ] unless the undulating contour linesin Figure 12-5b are supported by additional data. Seismic correlations can beused to justify the more heterogeneous contouring style shown in Figure 12-5b.A growing body of literature provides additional discussion of this applicationTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 111in the context of an emerging discipline known as reservoir geophysics. Forexample, see de Buyl, et al. [1988], Evans [1996], Blackwelder, et al. f 1996],Beasley [1996], and Jack [1998].Control PointFigure 12-5a. Smooth contour lines.Figure 12-5b. Undulating contour lines.12.2 Mega ScaleThe Giga Scale helps define reservoir architecture, but is too coarse toprovide the detail needed to design a reservoir development plan. The MegaScale is the scale at which we begin to integrate well log and well test data intoa working model of the reservoir. Table 12-2 illustrates the type of informationthat can be obtained at the Mega Scale level from well log data. The mostcommon interpretations of each log response are included in the table. Forexample, a high gamma ray response implies the presence of shales, while a lowgamma ray response implies the presence of clean sands or carbonates. Acombination of well logging tools is usually needed to minimize ambiguity inlog interpretation, as discussed by Brock [1986].TEAM LinG - Live, Informative, Non-cost and Genuine!112 Principles of Applied Reservoir SimulationTable 12-2Well Log ResponseLogGamma rayResistivityDensityAcoustic(sonic)NeutronSpontaneouspotentialVariableRock typeFluid typePorosityPorosityHydrogencontentPermeablebedsResponseDetects shale from in situ radioactivity.4 High GR -» shales+ Low GR =* clean sands or carbonatesMeasures resistivity of formation water.+ High resistivity =* hydrocarbons4 Low resistivity =* brineMeasures electron density by detectingCompton scattered gamma rays. Electrondensity is related to formation density.Good for detecting hydrocarbon gas withlow density compared to rock or liquid.+ Low response ==» low HC gas content^ Large response => high HC gas contentMeasures speed of sound in medium.Speed of sound is faster in rock than influid.4 Long travel time =* slow speed => largepore space4 Short travel time =* high speed =» smallpore spaceFast neutrons are slowed by collisions tothermal energies. Thermal neutrons arecaptured by nuclei, which then emitdetectable gamma rays. Note: Hydrogenhas a large capture cross-section forthermal neutrons. Good for detecting gas.+ Large response =» high H content+ Small response =* low H contentMeasures electrical potential (voltage)associated with movement of ions.+ Low response =* impermeable shales4 Large response => permeable bedsTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 113Table 12-3 from Kamal, et al. [1995] illustrates the type of informationthat can be obtained at the Mega Scale level from well test data. The table alsonotes the time in the life of the project when the well test is most likely to beran. It is usually necessary to run a variety of well tests as the project matures.These tests help refine the operator's understanding of the field and oftenmotivate changes in the way the well or the field is operated. Additionalinformation about well testing can be found in literature sources such asMatthews and Russell [1967], Earlougher [1977], and Sabet [1991],Table 12-3Reservoir Properties Obtainable from Transient TestsType of TestDrill stem testsRepeat-formationtests / Multipleformation testsDrawdown testsBuildup testsPropertiesReservoir behaviorPermeabilitySkinFracture lengthReservoir pressureReservoir limitBoundariesPressure profileReservoir behaviorPermeabilitySkinFracture lengthReservoir limitBoundariesReservoir behaviorPermeabilitySkinFracture lengthReservoir pressureReservoir limitBoundariesDevelopment StageExploration andappraisal wellsExploration andappraisal wellsPrimary, secondary andenhanced recoveryPrimary, secondary,and enhanced recoveryTEAM LinG - Live, Informative, Non-cost and Genuine!114 Principles of Applied Reservoir SimulationTable 12-3 (cont.)Reservoir Properties Obtainable from Transient TestsStep-rate testsFalloff testsInterference andpulse testsLayered reservoirtestsFormation partingpressurePermeabilitySkinMobility in variousbanksSkinReservoir pressureFracture lengthLocation of frontBoundariesCommunicationbetween wellsReservoir type behaviorPorosityInterwell permeabilityVertical permeabilityProperties of individuallayersHorizontal permeabilityVertical permeabilitySkinAverage layer pressureOuter boundariesSecondary andenhanced recoverySecondary andenhanced recoveryPrimary, secondary,and enhanced recoveryThroughout reservoirlifeTables 12-2 and 12-3 illustrate a few of the methods used to gather MegaScale information. Advances in technology periodically add to a growing listof transient tests and well log tools [for example, see Kamal, 1995; Felder,1994]. In many cases, budgetary constraints will be the controlling factor indetermining the number and type of tests run. The modeling team must workwith whatever information is available. Occasionally, an additional well test orwell log will need to be run, but the expense and scheduling make it difficultto justify acquiring new well log or well test information once a simulation studyis underwav.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 11512.3 Reservoir Description Using Seismic DataReservoir geophysics has the potential to image important reservoirparameters in regions between wells. This potential has limitations, but beforediscussing these limitations, let us first consider how reservoir geophysics maybe used and review an example where the potential of reservoir geophysics wasrealized.The reservoir geophysical procedure requires the correlation of seismicdata with reservoir properties. Correlations are sought by making crossplots ofseismic data with reservoir properties. Some correlation pairs are listed below.4 Seismic Amplitude vs Rock Quality0 Rock Quality = khnet, $khnet, etc.^ Seismic Amplitude vs Oil Productive Capacity (OPC)4 Acoustic Impedance vs PorosityIf a statistically significant correlation is found, it can be used to guide the dis-tribution of reservoir properties between wells. Ideally, the property distributionprocedure will preserve reservoir properties at wells.De Buyl, et al. [ 1 988] used reservoir geophysics to predict reservoir pro-perties of two wells. They correlated well-log-derived properties with seismicallycontrolled properties, for example, porosity, then used the correlation todistribute properties. Maps drawn from seismically controlled distributionsexhibited more heterogeneity than conventional maps drawn from well-log-derived properties. Unlike geostatistics, where additional heterogeneity isobtained by sampling from a probability distribution, heterogeneity based onseismically controlled distributions represents spatial variations in reservoirproperties determined by direct observation, albeit observation based oninterpreted seismic data.An indication of the technical success of the reservoir geophysicaltechnique is given in Table 12-4. Actual values of reservoir parameters at twowell locations are compared with values predicted using both well-log-derivedproperties and seismically controlled properties. This work by De Buyl, et al.[1988] is notable because it scientifically tests the seismic method: it makesTEAM LinG - Live, Informative, Non-cost and Genuine!116 Principles of Applied Reservoir Simulationpredictions and then uses measurements to assess their validity. In this particularcase, a reservoir characterization based on seismically controlled propertiesyielded more accurate predictions of reservoir properties than predictions madeusing a reservoir characterization based only on well data.Table 12-4Predictions at New Wells from Seismic and Well Data[de Buyl, et al., 1988]WellIJTop of Reservoir (m)Gross Porosity (vol %)Net <j)h (m)Top of Reservoir (m)Gross Porosity (vol %)Net (|)h (m)MeasuredValues-178.015.01.78-182.013.91.08SeismicPredicted-175.015.51.53-179.010.61.05Well DataPredicted-181.015.41.96-174.08.00.15Although reservoir geophysical techniques are still evolving, it is possibleto make some general statements about the relative value of this emergingtechnology. Table 12-5 summarizes the advantages and concerns associated withreservoir geophysics.Table 12-5Reservoir GeophysicsAdvantages4 Able to "see" betweenwells4 Single realizations enhance0 communication0 understandingConcerns4 Cost of data acquisition and analysis4 Limited applicability4 Validity of realization unknown withoutsensitivity analysisTo demonstrate the limits of applicability of reservoir geophysics, thereservoir geophysical algorithm in WINB4D was used to study a hypotheticalreservoir system in which we could expect to see significant changes in seismicproperties as a function of field performance over time. In particular, a dippinggas reservoir with aquifer influx was studied. The reservoir grid is shown inFigure 12-6. The reservoir has an initial gas saturation of 70% and an initialTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 117Dipping Gas Reservoirwith Aquifer Influx-88004 5 6 7 8 9 10WINB4D Index IFigure 12-6. Cross-section of dipping gas reservoir.irreducible water saturation of 30%. The initial ratio of compressional velocityto shear velocity (Vpl JQ was 1.684. A downdip aquifer provides pressure supportand water invasion as the reservoir is produced.Figure 12-7 shows the results after one year of depletion with aquiferAquifer Influx at 1 YearSgr = 0%WINB4D Block IndexFigure 12-7. Reservoir performance with Sgr = 0%.TEAM LinG - Live, Informative, Non-cost and Genuine!118 Principles of Applied Reservoir Simulationinflux for a system with an irreducible gas saturation (Sgr) of 0%. The changein gas saturation shows the influx of aquifer water. The change in fluid contentchanges fluid bulk modulus. As a consequence, the ratio VpIVs changessignificantly in the waterflooded part of the reservoir.If we rerun the example with an irreducible gas saturation of three percent,we obtain the results shown in Figure 12-8. The large change in VpIVs is noAquifer Influx at 1 YearSgr = 3%10WINB4D Block IndexFigure 12-8. Reservoir performance with Sgr = 3%.longer observed because the presence of a small amount of gas significantlychanged the compressibility of the system.Time-lapse seismic tomography, or 4-D seismic, could be used in ourhypothetical example to track the movement of invading aquifer water, but thepresence of a small amount of gas in the invaded zone increases the difficultyof detecting the gas-water contact. Calculations of 4-D seismic performancebased on algorithms like the one coded in WINB4D can predict 4-D seismicresponses [Fanchi, 1999], but such algorithms are not yet widely available incommercial simulators.Although it is risky to predict technological developments, it is possibleto infer trends by extrapolating ongoing research activities in the industry.Thakur [1996], for example, wrote that data management and the integrationTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 119of disciplines will play an increasingly important role in the future of reservoirmodeling. Many modelers have predicted that the integration of disciplines willmanifest itself in reservoir modeling as finer 3-D models with more seismic andgeological detail [He, et al,, 1996; Kazemi, 1996; Uland, et al., 1997]. Thisprediction is being borne out with growing interest in shared earth models[Tippee, 1998], model-centric working environments [Tobias, 1998; Fanchi, etal, 1999], and reservoir simulation models with a million or more gridblocks[Dogra, 2000].ExercisesExercise 12.1 Seismic Parameters: Data set EXAM 11.DAT is a cross-sectionmodel of a two-layer gas reservoir undergoing depletion with aquifer influx intothe lower layer. (A) Run EXAM 11 .DAT and find the initial water saturation (5W),compressional velocity (Yp), reflection coefficient (RC), and ratio ofcompressional velocity to shear velocity (Vp/Vs) in block 1=1 of layer k = 2.(B) Verify the maps IVPMAP, IRCMAP and IVRMAP are activated using theinformation given in Chapter 25.1. Record Sw, VP, RC, and VpIVs in block I =1 of layer K = 2 at the end of the run. Notice how the attributes change as watermoves into the layer.Exercise 12.2 Repeat Exercise 12.1 using a critical gas saturation of 0. Thisshould be achieved by setting the relative permeability of gas to 0.01 at a gassaturation of 0.03.Exercise 12.3 Repeat Exercise 12.1 using a grain bulk modulus KG that is equalto the frame bulk modulus KB. Use Eq (27.13) to explain your results.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 13Fluid PropertiesProperties of petroleum fluids must be quantified in a reservoir simulator.The range of applicability of a reservoir simulator is defined, in part, by thetypes of fluids that can be modeled using the mathematical algorithms codedin the simulator. For these reasons, it is worth considering the general types offluids that may be encountered in a commercial reservoir environment [forexample, see Pedersen, et al., 1989; Koederitz, et al., 1989; McCain, 1973; andAmyx, etal., I960].13.1 Fluid TypesAn estimate of the elemental composition (by mass) of petroleum isgiven in the following chart:CarbonHydrogenSulphurNitrogenOxygenMetals84% -11%-0.6% -0.02%0.08%0% - 087%14%8%- 1.7%- 1.8%.14%It can be seen from the table that petroleum fluids are predominantly hydro-carbons. The most common hydrocarbon molecules are paraffins, napthenes,and aromatics because of the relative stability of the molecules. A paraffin is120TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 121a saturated hydrocarbon, that is, it has a single bond between carbon atoms.Examples include methane and ethane. Paraffins have the general chemicalformula CnH2n+2. Napthenes are saturated hydrocarbons with a ringed structure,as in cyclopentane. They have the general chemical formula CnH2n. Aromaticsare unsaturated hydrocarbons with a ringed structure that have multiple bondsbetween the carbon atoms as in benzene. The unique ring structure makes aro-matics relatively stable and unreactive.A general PVT diagram of a pure substance displays phase behavior asa function of pressure, volume, and temperature. The types of properties ofinterest from a reservoir engineering perspective can be conveyed in a pressure-temperature (P-T) diagram of phase behavior like the one shown in Figure 13-1(after Craft, et al. [1991]). Most reservoir fluids do not exhibit significant tem-perature effects in situ, although condensate reservoirs in thick sands may displaya compositional gradient that can influence condensate yield as a function ofwell perforation depth.Single-Phase RegioniI| Critical PointTwo-Phase RegionFigure 13-1. P-T diagram.The P-T diagram includes both single-phase and two-phase regions. Theline separating the single-phase region from the two-phase region is called thephase envelope. The black oil region is at low temperature and in the highpressure region above the bubble point curve separating the single-phase andTEAM LinG - Live, Informative, Non-cost and Genuine!122 Principles of Applied Reservoir Simulationtwo-phase regions. If we consider pressures in the single-phase region and moveto the right of the diagram by letting temperature increase towards the criticalpoint, we encounter volatile oils. At temperatures above the critical point butless than the cricondentherm, reservoir fluids behave like condensates. Thecricondentherm is the maximum temperature at which a fluid can exist in boththe gas and liquid phases. When reservoir temperature is greater than the cri-condentherm, we encounter gas reservoirs. A summary of these fluid types isgiven in Table 13-1. Notice that separator gas-oil ratio (GOR) is a usefulindicator of fluid type.Table 13-1Rules of Thumb for Classifying Fluid TypesFluidTypeDry gasWet gasCondensateVolatile oilBlack oilHeavy oilSeparator GOR(MSCF/STB)No surface liquids>1003-1001.5-30.1 - 1.5~0Pressure DepletionBehavior in ReservoirRemains gasRemains gasGas with liquid drop outLiquid with significant gasLiquid with some gasNegligible gas formationLet us consider a reservoir containing hydrocarbons that are at a pressureand temperature corresponding to the single-phase black oil region. If reservoirpressure declines at constant temperature, the reservoir pressure will eventuallycross the bubble point pressure curve and enter the two-phase gas-oil region.Similarly, starting with a single-phase condensate and letting reservoir pressuredecline at constant temperature, the reservoir pressure will cross the dew pointpressure curve to enter the two-phase region. In this case, a free-phase liquiddrops out of the condensate gas. Once liquid drops out, it is very difficult torecover. One recovery method is dry gas cycling, but the recovery efficiencywill be substantially less than 100%. If we drop the pressure even further, it isTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 123possible to encounter retrograde condensation for some hydrocarbon composi-tions.The P-T diagram also applies to temperature and pressure changes in awellbore. In the case of wellbore flow, the fluid moves from relatively highreservoir temperature and pressure to relatively low surface temperature andpressure. As a result, it is common to see fluids that are single-phase in thereservoir become two-phase by the time they reach the surface.Figure 13-2 is a P-T diagram that compares two-phase envelopes for fourtypes of fluids. A reservoir fluid can change from one fluid type to anotherdepending on how the reservoir is produced. A good example is dry gas injec-tion into a black oil reservoir. Dry gas injection increases the relative amountof low molecular weight components in the black oil. The two-phase enveloperotates counter-clockwise in the P-T diagram as the relative amount of lowermolecular weight components increases. Similarly, dry gas injection into acon-densate can make the phase envelope transform from one fluid type to another.Thus, the way the reservoir is operated has a significant impact on fluid be-havior in the reservoir and at the surface.TemperatureFigure 13-2. Typical two-phase P-T envelopes fordifferent fluid types.Table 13-2 shows different compositions for typical fluid types. Dry gasusually contains only the lower molecular weight components. Gas condensatesstart to add higher molecular weight components. Volatile oils continue to addhigher molecular weight components. The addition of higher molecular weightcomponents and the reduction of lower molecular weight components eventuallyTEAM LinG - Live, Informative, Non-cost and Genuine!124 Principles of Applied Reservoir Simulationyields a black oil. If we monitor methane content (C,), we see that it tends todecrease as fluids change from dry gas to black oil.Table 13-2Typical Molar Compositions of Petroleum Fluid Types[after Pedersen, et al., 1989]ComponentN2C02C,C2€3iC4+nC4iC5+nC5iC6+nC6C7C8C9C10c,,C12C13C14Cssc,;CPv-qg1 9^20Gas0.31.190.04.91.91.10.4C6+: 0.3Gas Condensate0.718.6570.868.534.952.000.810.460.610.710.390.280.200.150.110.100.070.05C17+: 0.37Volatile Oil1.672.1860.517.524.744.122.971.992.452.411.691.421.02C!2+:5.31Black Oil0.672.1134.937.007.825.483.803.044.394.713.211,791.721.741.741.351,341.061.021.000.90C20.:9.1813.2 Fluid ModelingIn general, fluid behavior is best modeled using an equation of state. Table13-3 shows some cubic equations of state (EoS) used in commercial com-positional simulators. In addition to pressure (P), volume (V), and temperature(T), the EoS contains the gas constant R and a set of adjustable parameters (a,b} which may be functions of temperature. The EoS in Table 13-3 are called"cubic" because they yield a cubic equation for the compressibility factor Z =PVIRT. In the case of an ideal gas, Z - 1.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 125Table 13-3Examples of Cubic Equations of StateRedlich-KwongSoave-Redlich-KwongPeng-RobinsonZudkevitch-JoffePPPPRT aiTVlV-b V(V+b)RT a(T)V-b V(V+b)RT a(T)V-b V(V+b) + b(V-b)RT a(T)ITV2V-b(T) V[V+b(T)]Equations of state are valuable for representing fluid properties in manysituations. For example, suppose we want to model a system in which productionis commingled from more than one reservoir with more than one fluid type. Inthis case the most appropriate simulator would be a compositional simulatorbecause a black oil simulator would not provide as accurate a representation offluid behavior.The two most common types of reservoir fluid models are black oilmodels and compositional models. Black oil models are based on the assumptionthat the saturated phase properties of two hydrocarbon phases (oil and gas)depend on pressure only. Compositional models also assume two hydrocarbonphases, but they allow the definition of many hydrocarbon components. Unlikea black oil simulator, which can be thought of as a compositional simulator withtwo components, a compositional simulator often has six to ten components. Bycomparison, process engineering simulators that are used to model surfacefacilities typically require up to 20 components or more. The cost of runninga compositional simulator increases dramatically with increases in the numberof components modeled, but the additional components make it possible to moreaccurately model complex fluid phase behavior. If compositional model resultsare to be used in a process engineering model, it is often necessary to compro-mise on the number of components to be used for each application.TEAM LinG - Live, Informative, Non-cost and Genuine!126 Principles of Applied Reservoir SimulationEquations of state must be used to calculate equilibrium relations in acompositional model. This entails tuning parameters such as EoS parameters{a, b} in Table 13-3. Several regression techniques exist for tuning an EoS. Theyusually differ in the choice of EoS parameters that are to be varied in an attemptto match lab data with the EoS.Pressure—^- Pressure—^- Pressure-Figure 13-3. Gas phase properties.Figures 13-3 and 13-4 show typical fluid property behavior of gas andoil properties for a black oil model. Gas phase properties are gas formationvolume factor (Bg), gas viscosity (flg), and liquid yield (rs). Oil phase propertiesare oil formation volume factor (50), oil viscosity (|I0), and solution GOR (Rso).Saturated UndersaturatedABPressureFigure 13-4. Oil phase properties.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 127Both saturated and undersaturated curves are included as functions of pressureonly. Phase changes occur at the saturation pressures. Single-phase oil becomestwo-phase gas-oil when pressure drops below the bubble point pressure (P6),and single-phase gas becomes two-phase gas condensate when pressure dropsbelow the dew point pressure (Pd).Simulators run most efficiently when fluid property data are smoothcurves. Any discontinuity in a curve can cause numerical difficulties. Ordinarily,realistic fluid properties are smooth functions of pressure except at points wherephase transitions occur. As a practical matter, it is usually wise to plot input PVTdata to verify the smoothness of the data. Most simulators reduce the nonlinearityof the gas formation volume factor Bg by using the inverse bg = l/Bg to interpo-late gas properties.Oil properties from a laboratory must usually be corrected for use in ablack oil simulator [Moses, 1986]. Flow in the reservoir is a relatively slowprocess that corresponds to a differential process in the laboratory. A differentialprocess is one in which pressures are allowed to change in relatively smallincrements. For comparison, a flash process allows pressures in the experimentto change by relatively large increments. The production of oil up the wellboreto surface facilities is considered a flash process. Oil is flashed to the surfacethrough several pressure and temperature regimes. The corrections applied tooil property data are designed to adjust the data to more adequately representfluids as they flow differentially in the reservoir prior to being flashed to surfaceconditions. The corrections alter solution gas-oil ratio and oil formation volumefactor. The effect of the correction is illustrated by the case study in Chapter 20,The oil property correction is often significant.Water properties must also be entered in a simulator. Ideally waterproperties should be measured by performing laboratory analyses on producedwater samples. If samples are not available, correlations are often sufficientlyaccurate for describing the behavior of water.In the absence of reliable fluid data for one or more of the reservoir fluids,it may be necessary to use correlations. McCain [ 1991 ] reviewed the state of theart in the use of correlations to describe fluid properties. New correlations forTEAM LinG - Live, Informative, Non-cost and Genuine!128 Principles of Applied Reservoir Simulationestimating bubble point pressure, formation volume factor, and isothermal oilcompressibility have been proposed by Levitan and Murtha [1999].13.3 Fluid SamplingAll laboratory measurements of fluid properties and subsequent analysesare useless if the fluid samples do not adequately represent in situ fluids. Thegoal of fluid sampling is to obtain a sample that is representative of the originalfluid in the reservoir. It is often necessary to condition the well before the sampleis taken. A well is conditioned by producing any nonrepresentative fluid, suchas drilling mud, from within and around the wellbore until it is replaced byoriginal reservoir fluid flowing into the wellbore. Fluid samples may then betaken from either the surface or subsurface.Subsurface sampling requires lowering a pressurized container to theproduction interval and subsequently trapping a fluid sample. This is routinelyaccomplished by drill stem testing, especially when access to surface facilitiesis limited. It is generally cheaper and easier to take surface samples fromseparator gas and oil.If a surface sample is taken, the original in situ fluid, that is, the fluid atreservoir conditions, must then be reconstituted by combining separator gas andseparator oil samples. The recombination step assumes accurate measurementsof flow data at the surface, especially gas-oil ratio. Subsurface sampling froma properly conditioned well avoids the recombination step, but is more difficultand costly than surface sampling, and usually provides a smaller volume ofsample fluid. The validity of fluid property data depends on the quality of thefluid sampling procedure.ExercisesExercise 13.1 Data set EXAM9.DAT models depletion of a gas reservoir withaquifer support. Initial reservoir pressure is approximately 1947 psia. (A) Runthe model at a temperature of 226°F and record time, pressure, gas rate, andwater rate at the end of the run. Report the gas viscosity in the gas PVT tableTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 129at 2015 psia pressure. (B) Repeat A at a temperature of 150°F. (C) Explain thedifferences in model performance. For this example, neglect the temperaturedependence of water properties. Refer to Chapter 24.6 for a description ofWINB4D fluid property input data.Exercise 13.2 Data file CS-VC4.DAT is a vertical column model with fourlayers. Layers K = 1, 3,4 are pay zones, and layer K = 2 is a shale layer. Thedata set is a model of primary depletion of an initially undersaturated oilreservoir. (A) Run CS-VC4.DAT for three years and show gas saturation in all4 layers at the end of the run. You should see gravity segregation and theformation of a gas cap in layer K = 3. (B) By referring to Chapter 25 and fileWTEMP.WEL, determine which model layers are being depleted throughwellbore perforations.Exercise 13.3 Replace solution gas-oil ratio in CS-VC4.DAT with the followingdata. Run the modified data set for a period of three years, and then comparethe results with the results of Exercise 13.2.Pressure(psia)14.7514.71014.71514.72014.72514.73014.74014.75014.76014.7Solution Gas-Oil Ratio(SCF/STB)1.054.0105.0209.0292.0357.0421.0486.0522.0550.0TEAM LinG - Live, Informative, Non-cost and Genuine!130 Principles of Applied Reservoir SimulationExercise 13,4 Run the data set prepared in Exercise 13.3 with the assumptionthat no fluids can flow between model layers (multiply z direction transmissi-bility by zero).Exercise 13.5 Run data file CS-VC4.DAT with the bubble point pressurereduced by 500 psia. What effect does this have on solution gas-oil ratio andmodel performance?TEAM LinG - Live, Informative, Non-cost and Genuine!Rock-Fluid InteractionThe previous two chapters described the data needed to model the solidstructure of the reservoir and the behavior of fluids contained within the solidstructure. Small-scale laboratory measurements of fluid flow in porous mediashow that fluid behavior depends on the properties of the solid material. Theinteraction between rock and fluid is modeled using a variety of physicalparameters that include relative permeability and capillary pressure [Collins,1961; Dake, 1978; Koederitz, et al., 1989]. Laboratory measurements provideinformation at the core scale (Macro Scale) and, in some cases, at the micro-scopic scale (Micro Scale). They are the subject of the present chapter.14.1 Porosity, Permeability, Saturation, and Darcy's LawPorosity, permeability, and saturation can be obtained from Mega Scalemeasurements such as well logs and well tests, and by direct measurement inthe laboratory. Comparing values of properties obtained using methods at twodifferent scales demonstrates the sensitivity of important physical parametersto the scale at which they were measured. Ideally there will be good agreementbetween the two scales; that is, well log porosity or well test permeability willagree with corresponding values measured in the laboratory. In many cases,however, there are disagreements. Assuming measurement error is not the sourceof disagreement, differences in values show that differences in scale can impactthe measured value of the physical parameter. A well test permeability, forexample, represents an average over an area of investigation that is very large131TEAM LinG - Live, Informative, Non-cost and Genuine!132 Principles of Applied Reservoir Simulationcompared to a laboratory measurement of permeability using a six-inch coresample. The modeling team often has to make judgements about the relativemerits of contradictory data. The history matching process recognizes this sourceof uncertainty, as is discussed in subsequent chapters.The most common types of reservoir rock are listed in Table 14-1. Oneof the most fundamental properties of rock that must be included in a reservoirmodel is porosity. Porosity is the fraction of a porous medium that is void space.If the void space is connected and communicates with a wellbore, it is referredto as effective porosity, otherwise the void space is ineffective porosity. Theoriginal porosity resulting from sediment deposition is called primary porosity.Secondary porosity is an incremental increase in primary porosity due to thechemical dissolution of reservoir rocks, especially carbonates. Primary andsecondary porosity can be both effective and ineffective. Total porosity is acombination of ineffective porosity and effective (interconnected) porosity.Table 14-1Common Reservoir RocksSandstonesShalesCarbonatesCompacted sedimentConglomerateLaminated sedimentPredominantly clayProduced by chemical and biochemical sourcesLimestonePorosity values depend on rock type, as shown in Table 14-2. There aretwo basic techniques for directly measuring porosity: core analysis in thelaboratory and well logging. Laboratory measurements tend to be more accurate,but sample only a small fraction of the reservoir. Changes in rock properties mayalso occur when the core is brought from the reservoir to the surface. Well logmeasurements sample a much larger portion of the reservoir than core analysis,but typically yield less accurate values. Ideally, a correlation can be establishedbetween in situ measurements such as well logging and surface measurementssuch as core analysis.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 133Table 14-2Dependence of Porosity on Rock TypeRockTypeSandstoneUnconsolidated sandstoneCarbonate« Intercrystalline limestone• Oolitic limestone• DolomitePorosityRange (%)15-3520-355-2020-3510-25TypicalPorosity (%)2530152520Darcy's Law is the basic equation describing fluid flow in a simulator.Darcy's equation for single-phase flow isQ = -0.001127Axwhere the physical variables are defined in oil field units asQ = flow rate (bbl/day)A = cross-sectional area (ft2)|l = fluid viscosity (cp)K = permeability (md)P = pressure (psi)x = length (ft)Darcy's Law says that rate is proportional to cross-sectional area times pressuredifference AP across a distance A jc, and is inversely proportional to the viscosityof the fluid. The minus sign shows that the direction of flow is opposite to thedirection of increasing pressure; fluids flow from high pressure to low pressurein a horizontal (gravity-free) system.The linearity of Darcy's Law is an approximation that is made by virtuallyall commercial simulators. Fluid flow in a porous medium can have a nonlineareffect that is represented by the Forcheimer equation [Govier, 1978]. Thenonlinear effect becomes more important in high flow rate gas wells.TEAM LinG - Live, Informative, Non-cost and Genuine!134 Principles of Applied Reservoir SimulationPermeability is a physical constant describing flow in a given sample fora given fluid and set of experimental conditions. If those conditions are changed,the permeability being measured may not apply. For example, if a waterfloodis planned for a reservoir that is undergoing gravity drainage, laboratorymeasured permeabilities need to represent the injection of water into a core withhydrocarbon and connate water. The permeability distribution and the relativepermeability curves put in the model need to reflect the type of processes thatoccur in the reservoir.Permeability has meaning as a statistical representation of a large numberof pores. A Micro Scale measurement of grain-size distribution shows thatdifferent grain sizes and shapes affect permeability. Permeability usuallydecreases as grain size decreases. It may be viewed as a mathematical conve-nience for describing the statistical behavior of a given flow experiment. In thiscontext, transient testing gives the best measure of permeability over a largevolume. Despite its importance to the calculation of flow, permeability and itsdistribution will not be known accurately. Seismic data can help define thedistribution of permeability between wells if a good correlation exists betweenseismic amplitude and a rock quality measurement that includes permeability.It is not unusual to find that permeability has a directional component:that is, permeability is larger in one direction than another [for example, seeFanchi, et al., 1996]. When a model is being designed, the modeling team shouldaccount for the direction associated with permeability. In principle, simulatorscan take all of these effects into account. In practice, however, the tensorpermeability discussed in the literature by, for example, Bear [1972] and Lake[1988] is seldom reflected in a simulator. The usual assumption is that perme-ability is aligned along one of three orthogonal directions known as the principalaxes of the tensor. This assumption has implications for model studies thatshould be considered when assessing model results (see Chapter 15 and Fanchi[1983]).In many cases vertical permeability is not measured and must be assumed.A rule of thumb is to assume vertical permeability is approximately one tenthof horizontal permeability. These are reasonable assumptions when there is nodata to the contrary.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 13514.2 Relative Permeability and Capillary PressureReservoir models calculate saturation as a function of time. Consider thecase of water displacing oil. Initially, oil occupies the interior of pore spaces,and connate water is adjacent to the rock surface of a water-wet reservoir. Whenthe flood begins, water displaces oil through the interconnected pore space. Themeasure of interconnectedness is permeability. The oil left behind after thewaterflood is residual or irreducible oil saturation. Similar behavior is seen forother combinations of multiphase flow, for example, gas-oil, gas-water, and gas-oil-water. Multiphase flow is modeled by including relative permeability curvesin the simulator. Saturation end points for the relative permeability curves areused to establish initial fluids-in-place in addition to modeling flow behavior.A typical set of relative permeability curves is shown in Figure 14-1.Relative permeability curves represent flow mechanisms, such as drainage orimbibition processes, or fluid wettability. Relative permeability data should beobtained by experiments that best model the type of displacement that is thoughtto dominate reservoir flow performance. For example, water-oil imbibitioncurves are representative of waterflooding, while water-oil drainage curvesdescribe the movement of oil into a water zone. The modeling team needs toWater Saturation (fraction)krw (Imb,) -o Kro (drainage) -*- Kro (1mb.)Figure 14-1. Typical water-oil relative permeability curves.TEAM LinG - Live, Informative, Non-cost and Genuine!136 Principles of Applied Reservoir Simulationrealize that the relative permeability curves used in a flow model are mostrepresentative of the type of experiment that was used to measure the curves.Applying these curves to another type of displacement mechanism can introducesignificant error.Several procedures exist for averaging relative permeability data [forexample, Schneider, 1987; Mattax and Dalton, 1990; Blunt, 1999]. In practice,relative permeability is one of the most useful physical quantities available forperforming a history match. The curves that are initially entered into a reservoirmodel are often modified during the history matching process. The rationale forchanging relative permeability curves is based on the observation that relativepermeability curves are usually obtained by flooding core in the laboratory.Laboratory floods correspond to a much smaller scale than flow through thedrainage area of a well. Therefore, it is easy to argue that the laboratory curvesare not representative of flow on the reservoir scale. In the absence of measureddata, correlations such as Honarpour, et al. [1982] give a reasonable startingpoint for estimating relative permeability. Relative permeability hysteresiseffects can also be included in reservoir simulation using a procedure presentedbyKillough[1976].Capillary pressure is usually included in reservoir simulators. The relation-ship between capillary pressure and elevation is used to establish the initialtransition zone in the reservoir. The oil-water transition zone, for example, isthe zone between water-only flow and oil-only flow. It represents that part ofthe reservoir where 100% water saturation grades into oil saturation withirreducible water saturation. Similar transition zones may exist at the interfacebetween any pair of immiscible phases.Capillary pressure data is used primarily for determining initial fluidcontacts and transition zones. It is also used in fractured reservoir models forcontrolling the flow of fluids between the fracture and the rock matrix. Ifcapillary pressure is neglected, transition zones are not included in the model.This is illustrated in Figure 14-2. Figure 14-3 shows the effect of neglectingcapillary pressure when a grid is used to represent the reservoir. The fluid contentof the block is determined by the location of the block mid-point relative to acontact between two phases. The block mid-point is shown as a dot in the centerTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 137of the blocks in Figure 14-3. Thus, if the block mid-point is above the gas-oilcontact (GOC), the entire block is treated as a gas cap block (single-phase gaswith irreducible water saturation), even if much of the block extends into theoil column. A more accurate representation may be obtained by decreasing thethickness of the gridblocks, but this often results in a substantial increase in thecost of making computer runs. The relative benefits of incremental accuracyversus incremental cost must be considered when modeling transition zones.Gas Cap -^^Oil ColumnWater LegFigure 14-2. Case 1: Neglect transition zones.sOil Col.Water LegOil Col.Water LegGas CapOil Col.Gas Cap*Oil Col.GOCwoeFigure 14-3. Initial fluid distribution in model without transitionzone.The inclusion of a transition zone in the model requires specifying acapillary pressure (Pc) curve as a function of saturation for whatever transitionzone is being modeled: oil-water, gas-oil, or gas-water. The height htz of thetransition zone above the free water level (the level corresponding to Pc=0 psia)TEAM LinG - Live, Informative, Non-cost and Genuine!138 Principles of Applied Reservoir Simulationis proportional to the capillary pressure and inversely proportional to the densitydifference between the two fluids (Eq. (3.7)). The height of the transition zoneis a function of saturation because capillary pressure depends on saturation. Theoil-water transition zone is typically the thickest transition zone because thedensity difference between oil and water is less than the density differencebetween gas and an immiscible liquid.Figures 14-4 and 14-5 illustrate the initialization of a model containinga nonzero capillary pressure curve. First, the height htz above a specified contact,such as the water-oil contact (WOC), is calculated from Pc and Ap, Thesaturation of a block with a mid-point at height htz above the contact is thencalculated from the relationship between capillary pressure and saturation.- WOCA. Gas-Oil Transition B. Oil-Water TransitionFigure 14-4. Case 2: Include transition zone inmodel.SL, = 0.50Sw = 0.20= 0.80 __WOC0.2 0.5 0.8Figure 14-5. Initial gridblock saturations in model with transition zone.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 139Transition zones complicate the identification of fluid contacts becausethe definition of fluid contact is not universaily accepted. For example, water-oilcontact may be defined as the depth at which the capillary pressure is zero (thefree water level). The WOC depth can be identified using a Repeat FormationTest by finding the point of intersection between the oil-phase pressure and thewater-phase pressure. By contrast, water-oil contact may be defined as thedeepest point in the reservoir at which a well can still produce water-free oil.The different definitions of contact result in differences in the transition zonemodel, so it is important to know which definition is applicable and who has theauthority to judge the validity of the model. In some cases, it may be necessaryto prepare models with both definitions and treat one definition as the base casewhile the other definition is viewed as a sensitivity.The proper way to include capillary pressure in a model study is to correctlaboratory measured values to reservoir conditions. This is done by applyingthe correction:where y is interfacial tension (IFT) is wettability angle [Amyx, et al., 1 960]. Theproblem with the correction is that it requires data that are often poorly known,namely interfacial tension and wettability contact angle at reservoir conditions.Rao and Girard [1997] have described a laboratory technique for measuringwettability using live fluids at reservoir temperature and pressure. Alternativeapproaches include adjusting capillary pressure curves to be consistent with welllog estimates of transition zone thickness, or assuming the contact angle factorsout. If laboratory measurements of IFT are not available, IFT can be estimatedfrom the Macleod-Sugden correlation for pure compounds orthe Weinaug-Katzcorrelation for mixtures [Fanchi, 1990].14.3 Viscous FingeringViscous fingering is the unstable displacement of a more viscous fluidby a less viscous fluid. The fingering of an injection fluid into an in situ fluidcan influence reservoir flow behavior and adversely impact recovery. It isTEAM LinG - Live, Informative, Non-cost and Genuine!140 Principles of Applied Reservoir Simulationimportant to note, however, that fingering occurs even in the absence of a porousmedium. If a low viscosity fluid is injected into a cell containing a high viscosityfluid, the low viscosity fluid will begin to form fingers as it moves through thefluid. It will not uniformly displace the higher viscosity fluid. These fingers canhave different shapes. Figure 14-6 shows an example of a "skeletal" fingerFigure 14-6. "Skeletal" viscousfinger (after Daccord, et al. 1986;reprinted by permission of theAmerican Physical Society).[Daccord, et al., 1986] while Figure 14-7 illustrates "fleshy" fingers [forFigure 14-7. Viscous fingering (Fanchiand Christiansen, 1989; reprinted bypermission of the Society of PetroleumEngineers).example, see Paterson, 1985; Fanchi and Christiansen, 1989]. If we watch fingersevolve in a homogeneous medium (Figure 14-7), we see fingering display aTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 141symmetric pattern. The symmetry can be lost if there is some heterogeneity inthe system.Fingering can be a reservoir heterogeneity problem or a fluid displacementproblem. Most reservoir simulators do not accurately model lingering effects.It is possible to improve model accuracy by using a very fine grid to cover thearea of interest, but the benefits associated with such a fine grid are seldomsufficient to justify the additional cost.ExercisesExercise 14.1 Data set EXAM3.DAT is a model of a Buckley-Leverett water-flood. (A) Multiply horizontal permeability by 0.5 and run the model. Plot oilrate as a function of time and WOR as a function of time. (B) Repeat A bymultiplying horizontal permeability in the original data set by 10. (C) Explainthe difference between parts A and B. Consider breakthrough times (time whenwater production begins), water-oil ratio, and cumulative oil produced at the endof the run. See Chapters 24.3.1 and 24.3.2 for a description of permeability inputdata. Cumulative production can be found in WTEMP.PLT.Exercise 14.2 Repeat Exercise 14.1, but modify horizontal transmissibilityinstead of horizontal permeability. See Chapter 24.3.3. for details.Exercise 14.3 Double water relative permeability in EXAM3.DAT anddetermine the effect on water-oil ratio and breakthrough times.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 15Fundamentals of Reservoir SimulationPrevious chapters describe much of the data that is needed by a reservoirsimulator. Our goal here is to outline the physical, mathematical and computa-tional basis of reservoir flow simulation. For a more detailed technical presenta-tion, consult one of the many sources available in the literature [for example,see Aziz and Settari, 1979; Bear, 1972; Mattax and Dalton, 1990; Peaceman,1977; and Thomas, 1982]. The set of equations used in WINB4D is derived inChapter 32.15.1 Conservation LawsThe basic conservation laws of reservoir simulation are the conservationof mass, energy, and momentum. Mass balance in a representative elementaryvolume (REV) or gridblock is achieved by equating the accumulation of massin the block with the difference between the mass leaving the block and the massentering the block. The set of equations used in WINB4D are derived from themass conservation principle in Chapter 4. A material balance is performed foreach block. What makes a simulator different from a reservoir engineeringmaterial balance program is the ability of the simulator to account for flowbetween blocks.A material balance calculation is actually a subset of the simulator cap-ability. This is an important point because it means a reservoir simulator can beused to perform material balance work. The advantage of using a simulatorinstead of a material balance program is that the simulation model can be142TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 143enlarged to include position-dependent effects by modifying the grid represent-ing the reservoir architecture. Thus, a single block material balance calculationin a reservoir simulation model can be expanded with relative ease to Includeflow in one, two, or three spatial dimensions. This procedure is used in the casestudy presented in Part III.Most reservoir simulators assume reservoirs are produced underisothermal conditions. They also assume complete and instantaneous phaseequilibration in each cell. Thus, most simulators do not account for eithertemperature gradients or the time it takes a mixture to reach equilibrium. Theyassume, instead, that reservoir temperature remains constant throughout the lifeof the field and that equilibration is established instantaneously. These are oftenreasonable assumptions.Momentum conservation is modeled using Darcy' s Law. This assumptionmeans that the model does not accurately represent turbulent flow in a reservoiror near the wellbore. Some well models allow the user to model turbulent flow,especially for high flow rate gas wells. Turbulent flow models relate pressurechange to a linear flow term, as in Darcy's Law, plus a term that is quadratic inflow rate. This quadratic effect is not usually included in the reservoir model,only in the well model.15.2 Flow EquationsThe general equations for describing fluid flow in a porous medium areshown in Table 15-1 and associated nomenclature is presented in Table 15-2.The molar conservation equation includes a dispersion term, a convection term,a source/sink term representing wells, and the time varying accumulation term.The dispersion term is usually neglected in most workhorse simulators such asblack oil and compositional simulators. Neglecting dispersion simplifies programcoding and is justified when dispersion is a second-order effect. In somesituations, such as miscible gas injection, physical dispersion is an effect thatshould be considered. Further discussion of dispersion is presented in Chapter16.TEAM LinG - Live, Informative, Non-cost and Genuine!144 Principles of Applied Reservoir SimulationTable 15-1Molar Conservation Equation for Component kPhysical SourceDispersionConvectionSource/SinkAccumulationDarcy's LawTermnpV • S 4> St D. p{ • V*^n-V • E p(*t|Ff+ fi*n= _ (h S p ^ 515/ [ i»i f *{ j*rlF, = -K— - (VPf - Y{VzTable 15-2Terminology of Molar Conservation EquationVariable£*/A'r=rs^/«c«/;^»5,^1*klY»^^P«*MeaningDispersion tensor of component k in phase 0Permeability tensorRelative permeability of phase 0Number of componentsNumber of phasesPressure of phase 0Saturation of phase Q.Darcy's velocity for phase 0Mole fraction of component k in phase 0Pressure gradient of phase 0Viscosity of phase £Density of phase £PorosityTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 145The molar flow equations were derived using the conservation lawsintroduced in Chapter 15.1. An energy balance equation can be found in thethermal recovery literature [Prats, 1982]. The energy balance equation is morecomplex than the flow equations because of the presence of additional nonlinearterms. Energy loss to adjacent non-reservoir rock must also be computed. Theresulting complexity requires substantial computation to achieve an energybalance. In many realistic systems, reservoir temperature variation is slight andthe energy balance equation can be neglected by imposing the isothermalapproximation. The result is a substantial savings in computation expense witha reasonably small loss of accuracy.Several supplemental - or auxiliary - equations must be specified tocomplete the definition of the mathematical problem. There must be a flowequation for each modeled phase. Commercial black oil and compositionalsimulators are formulated to model up to three phases: oil, water, and gas. Theinclusion of gas in the water phase can be found in some simulators, though itis neglected in most. The ability to model gas solubility in water is useful forCO2 floods or for modeling geopressured gas-water reservoirs. Some black oilsimulator formulations include a condensate term. It accounts for liquid yieldassociated with condensate reservoir performance.In addition to modeling reservoir structure and PVT data, simulators mustinclude rate equations for modeling wells, phase potential calculations, and rock-fluid interaction data such as relative permeability curves and capillary pressurecurves. Saturation-dependent rock-fluid interaction data are entered in eithertabular or analytical form. More sophisticated simulators let the user representdifferent types of saturation change processes, such as imbibition, drainage, andhysteresis. Applying such options leads to additional computation and cost,15.3 Well and Facilities ModelingWell and surface facility models are simplified representations of realequipment [Williamson and Chappelear, 1981]. The well model, for example,does not account for flow in the wellbore from the reservoir to the surface. Thiseffect can be taken into account by adding a wellbore model. The wellbore modelTEAM LinG - Live, Informative, Non-cost and Genuine!146 Principles of Applied Reservoir Simulationusually consists of a multivariable table relating surface pressure to suchparameters as flow rate and GOR. The tables are often calculated using a separateprogram that performs a nodal analysis of wellbore flow. Well models typicallyassume that fluid phases are folly dispersed and that the block containing thewell is perforated throughout its thickness. Some commercial simulators willlet the user specify a perforation interval under certain conditions,The different types of well controls include production and injection wellcontrols, and group and field controls for a surface model. The production wellmodel assumes the user specifies one option as the primary control, but may alsospecify other options as targets for constraining the primary control. Forexample, if oil rate is the primary control, then the produced GOR may berestricted so that the oil rate is decreased when GOR exceeds the specified value.This provides a more realistic representation of actual field practice.Injection well controls assume that initial injection well mobility is givenby total gridblock mobility. This makes it possible to inject a phase into a blockthat would otherwise have zero relative permeability to flow.Allocation of fluids in a well model depends on layer flow capacity andfluid mobility. The fluid allocation procedure in WINB4D is discussed inChapter 30. Simulators can also describe deviated or horizontal wells dependingon how the well completions and parameters are specified.Well, group and field controls can be specified in commercial simulatorswith a surface facilities model. The user specifies a hierarchy of controls thatmost realistically represent how the field is being operated. For example, wellproduction may be constrained by platform separator and storage capacity, whichin turn is constrained by pipeline flow capacity. The ability to integrate reservoirand surface flow technology using a single simulator is an area of research thatis receiving increasing attention [for example, see Heinemann, et al., 1998].15.4 Simulator Solution ProceduresFluid flow equations are a set of nonlinear partial differential equationsthat must be solved by computer. The partial derivatives are replaced with finitedifferences, which are in turn derived from Taylor's series [for example, see AzizTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 147and Settari, 1979; Peaceman, 1977; Rosenberg, 1977; Fanchi, 2000]. Thisprocedure is illustrated in Table 15-3. The spatial finite difference interval AJCalong the jc-axis is called gridblock length, and the temporal finite differenceinterval Ads called the timestep. Indices ij, k are ordinarily used to label gridlocations along the*,;;, z coordinate axes, respectively. Index n labels the presenttime level, so that n + 1 represents a future time level. If the finite differencerepresentations of the partial derivatives are substituted into the original flowequations, the result is a set of equations that can be algebraically rearrangedto form a set of equations that can be solved numerically. The solution of theseequations is the job of the simulator.Table 15-3Finite Difference ApproximationFormulate fluid flow equations, such as,dxKka8*1 BApproximate derivatives with finite differences0 Discretize region into gridblocks AJC:dx x.+l - jc. AJC0 Discretize time into timesteps A/:BS Sn*1 - Sn _dt tn + l - tn AfNumerically solve the resulting set of linear algebraic equationsThe two most common solution procedures in use today are IMPES andNewton-Raphson. The terms in the finite difference form of the flow equationsare expanded in the Newton-Raphson procedure as the sum of each term at thecurrent iteration level, plus a contribution due to a change of each term withrespect to the primary unknown variables over the iteration. To calculate theseTEAM LinG - Live, Informative, Non-cost and Genuine!148 Principles of Applied Reservoir Simulationchanges, it is necessary to calculate derivatives, either numerically or analyti-cally, of the flow equation terms. The derivatives are stored in a matrix calledthe acceleration matrix or the Jacobian. The Newton-Raphson technique leadsto a matrix equation J • $X = R that equates the product of the accelerationmatrix /and a column vector 6Xof changes to the primary unknown variablesto the column vector of residuals R. It is solved by matrix algebra to yield thechanges to the primary unknown variables $X. These changes are added to thevalue of the primary unknown variables at the beginning of the iteration. If thechanges are less than a specified tolerance, the iterative Newton-Raphsontechnique is considered complete and the simulator proceeds to the next timestep.The three primary unknown variables for an oil-water-gas system are oil-phase pressure, water saturation, and either gas saturation or solution GOR. Thechoice of the third variable depends on whether the block contains free gas,which depends, in turn, on whether the block pressure is above or below bubblepoint pressure. Naturally, the choice of unknowns is different for a gas-watersystem or a water only-system. The discussion presented here applies to the mostgeneral three-phase case.A simpler procedure is the IMplicit Pressure-Explicit Saturation (IMPES)procedure. It is much like the Newton-Raphson technique except that flowcoefficients are not updated in an iterative process. The Newton-Raphsontechnique is known as a fully implicit technique because all primary variablesare calculated at the same time; that is, primary variables at the new time levelare determined simultaneously. By contrast, the IMPES procedure solves forpressure at the new time level using saturations at the old time level, and thenuses the pressures at the new time level to explicitly calculate saturations at thenew time level. WINB4D, the program provided with this book, is an implemen-tation of a noniterative IMPES formulation [Fanchi, et al., 1982; Fanchi, et al.,1987]. The formulation is outlined in Chapters 27 and 32. A variation of thistechnique is to iteratively substitute the new time level estimates of primaryvariables in the calculation of coefficients for the flow equations. The iterativeIMPES technique takes longer to run than the non-iterative technique, butgenerates less material balance error [Ammer and Brummert, 1991],TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 149A flow chart for a typical simulator is shown in Figure 15-1 (see Crichlow,1977). The simulation program begins by reading input data and initializing thereservoir. This part of the model will not change as a function of time. Informa-tion for time-dependent data must then be read. This data includes well and fieldcontrol data. The coefficients of the flow equations and the primary unknownvariables are then calculated. Once the primary variables are determined, theprocess can be repeated by updating the flow coefficients using the values ofthe primary variables at the new iteration level. This iterative process canimprove material balance. When the solution of the fluid flow equations iscomplete, flow properties are updated and output files are created before the nexttimestep calculation begins.Read InputInitializeg50s±3.IMPLICITI Read Rates IrI Calculate Flow Coefficients IIMPES ^™M™*™**™^MM^*^*^y**M^^M**^^'^'*^*'I Solve Node Unknowns \\ Update Physical Properties \vI Create Output FilesFigure 15-1. Typical simulator flow chart.Fully implicit techniques do more calculations in a timestep than theIMPES procedure, but are stable over longer timesteps. The unconditionalstability of the fully implicit techniques means that a fully implicit simulator cansolve problems faster than IMPES techniques by taking significantly longertimesteps.TEAM LinG - Live, Informative, Non-cost and Genuine!150 Principles of Applied Reservoir SimulationA problem with large timesteps in the fully implicit technique is theintroduction of a numerical effect known as numerical dispersion [Lantz, 1971;Fanchi, 1983]. Numerical dispersion is introduced when the Taylor seriesapproximation is used to replace derivatives with finite differences. The resultingtruncation error introduces an error in calculating the movement of saturationfronts that looks like physical dispersion, hence it is called numerical dispersion.Numerical dispersion arises from time and space discretizations that leadto smeared spatial gradients of saturation or concentration [Lantz, 1971] and gridorientation effects [Fanchi, 1983; and Chapter 16]. The smearing of saturationfronts can impact the modeling of displacement processes. An illustration offront smearing is presented in Figure 15-2 for a linear Buckley-Leverettwaterflood model. The numerical front from an IMPES calculation does notexhibit the same piston-like displacement that is shown by the analyticalBuckley-Leverett calculation [for example, see Collins, 1961; Wilhite, 1986;Craft, etal, 1991]..o52CDCO"S.m10 r — Buckley-Leverett0.6060.40.2o, x IMPES^*-*&.j-» » ii i^n n """••Ajt^z" »' * *|, 120 days x 360 daysi1 ° X0,0 i 0Distance from InjectorFigure 15-2. Numerical dispersion (after Fanchi,1986; reprinted by permission of the Society ofPetroleum Engineers).Numerical dispersion Dnum in one spatial dimension has the formIt depends on gridblock size A*, timestep size A/, velocity v of frontal advance,porosity (|>, and numerical formulation. The "+" sign applies to the fully implicitTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 151formulation, and the "-" sign applies to IMPES. Notice that an increase in A?in the fully implicit formulation increases D"um while it decreases DHum whenthe IMPES technique is used. Indeed, it appears that a judicious choice of Al-and A/ could eliminate Dnum altogether in the IMPES method. Unfortunately,the combination of AJC and A t that yields D" um = Q violates a numerical stabi litycriterion. In general, IMPES numerical dispersion is not as large as thatassociated with fully implicit techniques.As a rule of thumb, timestep sizes in folly implicit calculations shouldnot exceed a quarter of a year, otherwise numerical dispersion can dominate frontmodeling. By contrast, the maximum timestep size in an IMPES simulator canbe estimated by applying the rale of thumb that throughput in any block shouldnot exceed 10% of the pore volume of the block. Throughput is the volume offluid that passes through a block in a single timestep. IMPES timestep sizes areoften on the order of a month or less. An example of a throughput calculationis given in Chapter 22.The IMPES timestep limitation is less of a problem than it might other-wise seem, because it is very common to have production data reported on amonthly basis. The reporting period often controls the frequency with which wellcontrol data is read during a history match. Thus, during the history match phaseof a study, simulator timestep sizes are dictated by the need to enter historicaldata. Large timestep sizes reduce the ability of the model to track variations ofrate with time because historical data must be averaged over a longer period oftime. As a result, the modeler often has to constrain the fully implicit simulatorto run at less than optimum numerical efficiency because of the need to moreaccurately represent the real behavior of the physical system.Fully implicit techniques represent the most advanced simulationtechnology, yet IMPES retains vitality as a relatively inexpensive means ofmodeling some problems. Unless a folly implicit model is readily available, itis not always necessary nor cost-effective to employ the most advancedtechnology to solve every reservoir simulation problem. The wise modeler willrecognize that you do not have to use a sledge hammer to open a peanut!Simulators also differ in their robustness, that is, their ability to solvea wide range of physically distinct problems. Robustness appears to depend asTEAM LinG - Live, Informative, Non-cost and Genuine!152 Principles of Applied Reservoir Simulationmuch on the coding of the simulator as it does on the formulation technique. Thebest way to determine simulator robustness is to test the simulator with data setsrepresenting many different types of reservoir management problems. Theexamples provided with WINB4D are designed to demonstrate the robustness,or range of applicability, of the simulator.Simulator technology is generally considered proprietary technology, yetit has an economic impact that takes it out of the realm of the research laboratoryand makes it a topic of importance in the corporate boardroom. Nevertheless,numerical representations of nature are subject to inaccuracies [for example, seeMattax and Dalton, 1990; Saleri, 1993; and Oreskes, et al., 1994]. This pointhas been illustrated in several simulator comparison projects sponsored by theSociety of Petroleum Engineers beginning with Odeh [1981] and continuingthrough Killough [1995]. Each comparison project was designed to allowcomparisons of proprietary technology by asking participating organizations tosolve the same pre-determined problem. Figure 15-3 is taken from the firstcomparison project [Odeh, 1981]. The first project compared the performanceof simulators modeling the injection of gas into a saturated black oil reservoir.Figure 15-3 shows that differences in the formulations of several reservoirsimulators lead to differences in predictions of economically important quantitiessuch as oil rate production.O4 6 8 10Time, yearsFigure 15-3. Oil rate from first SPE comparativesolution project (after Odeh, 1981; reprinted bypermission of the Society of Petroleum Engineers).TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 153In summary, a representation of the reservoir is quantified in the reservoirflow simulator. The representation is validated during the history matchingprocess, and forecasts of reservoir performance are then made from the validatedreservoir representation.15.5 Simulator SelectionThe selection of a reservoir simulator depends on such factors as theobjectives of the study, fluid type, and dimensionality of the system. Forpurposes of illustration, we focus our attention on a study which uses either ablack oil or a compositional simulator. Standard black oil and compositionalsimulators assume isothermal flow and mass transfer within a block is instanta-neous. A compositional simulator represents the fluid as a mixture of hydrocar-bon components. Black oil simulators may be viewed as compositionalsimulators with two components. They can have gas dissolved in the oil phase,as well as oil dissolved in the gas phase. Black oil simulators need both saturatedand under-saturated fluid property data, as discussed in Chapter 13.Black oil and compositional simulators usually assume fluids have aminimal effect on rock properties. Thus, standard versions of the simulators willnot model changes in rock properties due to effects like grain dissolution, tarmat formation, or gel formation resulting from a vertical conformance treatment.Special purpose simulators or special options within a standard simulator mustbe obtained to solve such problems.Fluid type is needed to decide if the reservoir should be modeled usingeither a black oil simulator or a compositional simulator. Well logs candistinguish between oil and gas, but are less useful in further classifying fluidtype. A pressure-temperature diagram is useful for determining reservoir fluidtype, but its preparation requires laboratory work with a fluid sample. A simplerway that is often sufficient for classifying a fluid is to look at solution gas-oilratio. Table 13-1 shows typical solution GOR ranges for each fluid type. As arule of thumb, compositional models should be used to model volatile oil andcondensate fluids, while black oil and dry gas fluids are most effectivelyTEAM LinG - Live, Informative, Non-cost and Genuine!154 Principles of Applied Reservoir Simulationmodeled with a black oil simulator. The applicability of this rule depends on theobjectives of the study.The pressure range associated with fluid property data should cover theentire range of pressures expected to be encountered over the life of the field.The data should be smooth to enhance computational efficiency and to ensuredata consistency. A check on data consistency is a calculation of fluid compress-ibility. If a negative compressibility is encountered, the data need to be corrected.The problem of negative compressibility occurs most often when data isextrapolated beyond measured pressure ranges.Flow units should be determined by reviewing geological and petro-physical data. It is possible to represent the behavior of a flow unit by defininga set of PVT and Rock property tables for each flow unit. PVT property tablescontain data that describe fluid properties, while Rock property tables representrelative permeability and capillary pressure effects. Each set of PVT or Rockproperty tables applies to a particular region of gridblocks, hence the collectionof blocks to which a particular set of PVT or Rock property tables applies isreferred to as a PVT or Rock region. The number of flow units, and thecorresponding number of PVT and Rock regions, should be kept to the minimumneeded to achieve the objectives of the study. This statement is anotherapplication of Ockham's Razor (Chapter 17).ExercisesExercise 15.1 Data file EXAM8.DAT has a gas well under LIT control.Determine the effect of doubling the turbulence factor on gas rate, cummulativegas production, and reservoir pressure at the end of the run. See Chapters 25.2and 30 for more discussion.Exercise 15.2 WINB4D contains a few fieldwide controls (see Chapter 24.8).Data file EXAM4.DAT is a 2D areal model of an undersaturated oil reservoirundergoing primary depletion. Modify data file EXAM4.DAT so that fieldwidepressure is not allowed to drop below the initial bubble point pressure using theTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 155run controls in Chapter 24,8. The initial bubble point pressure is also describedin Chapter 24.6. What effect does this have on the duration of the ran?Exercise 15.3 Data set EXAM3.DAT can be used to study the numericaldispersion associated with a Buckley-Leverett type waterflood of an under-saturated oil reservoir. Run EXAM3.DAT with constant timesteps of 5 days,10 days, and 15 days. Rerun the problem with timestep size beginning at 5 daysand allowed to vary from 5 days to 15 days. Make a table showing waterbreakthrough time (time when the model reaches a water-oil ratio of 0.1) for eachcase. Timestep controls are discussed in Chapters 24.8 and 25.1.Exercise 15.4 Data set EXAM7.DAT is one version of the Odeh [1981] SPEcomparative solution problem. Run EXAM7.DAT and compare the results tothose reported by Odeh. What is the WINB4D material balance error? Thematerial balance error associated with this data set provides a good test of thequality of WINB4D relative to other programs based on the original version ofBOAST [for example, Fanchi, et al., 1982; Fanchi, et al., 1987; Louisiana StateUniversity, 1997].Exercise 15.5 Data set EXAM10.DAT illustrates the use of PVT and Rockregions in WINB4D (see Chapter 24.4). Run EXAM 10.DAT and determine thenumber of regions in the data set.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 16Modeling Reservoir ArchitectureReservoir architecture is modeled by contouring and digitizing geologicmaps. The mapping/contouring process is the point where the geological andgeophysical interpretations have their greatest impact on the final representationof the reservoir. This process has been discussed by several authors, includingHarpole [1985], Harris [1975], and Tearpock and Bischke [1991]. Methods fornumerically representing reservoir architecture are discussed in this chapter,16.1 MappingThe different parameters that must be digitized for use in a grid includeelevations or structure tops, permeability in three orthogonal directions, porosity,gross thickness, net to gross thickness, and where appropriate, descriptions offaults, fractures, and aquifers. The resulting maps are digitized by overlayinga grid on the maps and reading a value for each gridblock. The digitizing processis sketched in Figures 16-la through 16-Id.The resolution of the model depends on the resolution of the grid. A finegrid divides the reservoir into many small gridblocks. It gives the most accuratenumerical representation, but has the greatest computational expense. A coarsegrid has fewer gridblocks, but the coarse gridblocks must be larger than the finegridblocks to cover the same model volume. As a result, the coarse grid is lessexpensive to run than a fine grid, but it is also less accurate numerically. Theloss of accuracy is most evident when a coarse grid is used to model the interfacebetween phases such a fluid contacts and displacement fronts. Thus, fine grid156TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 157• 70• 90• 80• 60Figure 16-1 a. Gather data. Figure 16-lb. Contour data.7077777470Figure 16-lc. Overlay grid. Figure 16-ld. Digitize data.modeling is often the preferred choice to achieve maximum numerical accuracy.It is important to recognize, however, that a fine grid covering an area definedby sparse data can give the illusion of accuracy. Sensitivity studies can helpquantify the uncertainty associated with the model study.The gridding process is most versatile when used with an integrated 3Dreservoir mapping package. Modern mapping techniques include computergenerated maps that can be changed relatively quickly once properly set up.Dahlberg [ 1975] presented one of the first analyses of the relative merits of handdrawn and computer generated maps. Computer generated maps may not includeall of the detailed interpretations a geologist might wish to include in the model,particularly with regard to faults, but the maps generated by computer in a 3Dmapping program do not have the problems so often associated with the stackingof 2D plan view maps, namely physically unrealistic layer overlaps. Layeroverlaps need to be corrected before the history match process begins.Another problem with computer generated maps is the amount of detailthat can be obtained. Computer generated maps can describe a reservoir withTEAM LinG - Live, Informative, Non-cost and Genuine!158 Principles of Applied Reservoir Simulationa much finer grid than can be used in a reservoir simulator. For example, acomputer mapping program such as that described by Englund and Sparks [ 1991 ]or Pannatier [1996] may use a grid with a million or more cells to represent thereservoir, yet reservoir simulation grids are usually 100,000 blocks or less. Thismeans that the reservoir representation in the computer mapping program mustbe scaled up, or coarsened, for use in a reservoir simulator. Although manyattempts have been made to find the most realistic process for scaling up data,there is no widely accepted up-scaling method in use today [for example, seeChristie, 1996; Dogru, 2000],16.2 Grid PreparationReservoir grids may be designed in several different ways. For a reviewof different types of grids, see Aziz [1993]. Definitions of coordinate systemorientation vary from one simulator to another and must be clearly defined foreffective use in a simulator. Reservoir grids can often be constructed in one-,two-, or three-dimensions, and in Cartesian or cylindrical coordinates. HorizontalID models are used to model linear systems that do not include gravity effects.Examples of horizontal ID models include core floods and linear displacementin a horizontal layer. Core flood modeling has a variety of applications, includingthe determination of saturation-dependent data such as relative permeabilitycurves, A dipping ID reservoir is easily defined in a model by specifyingstructure top as a function of distance from the origin of a grid.Figure 16-1 is an example of a 2D grid. Grids in 2D may be used to modelareal and cross-sectional fluid movement. Grid orientation in 2D is illustratedby comparing Figure 16-lc and Figure 16-2. Although Figure 16-lc has fewerblocks, which is computationally more efficient, Figure 16-2 may be useful insome circumstances. For example, Figure 16-2 is more useful than Figure 16-1 cif the boundary of the reservoir is not well known or an aquifer needs to beattached to the flanks of the reservoir to match reservoir behavior.The use of 2D grids for full field modeling has continued to be populareven as computer power has increased and made large 3D models practical.Figure 16-3 shows a simple 3D grid that is often called a "layer cake" grid.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 159Figure 16-2. Grid orientation.Techniques are available for approximating the vertical distribution of fluidsin 2D cross-sectional and 3D models by modifying relative permeability andcapillary pressure curves. The modified curves are called pseudo curves. Taggart,et al. [1995] present a discussion of several pseudoization techniques and theirlimitations. An example of a pseudoization technique is the vertical equilibrium(VE) approximation. The principal VE assumption is that fluid segregation inthe vertical dimension is instantaneous. This assumption is approximated inw/TkFigure 16-3. Example of a 3D "layercake" grid.TEAM LinG - Live, Informative, Non-cost and Genuine!160 Principles of Applied Reservoir Simulationnature when vertical flow is rapid relative to horizontal flow. This situationoccurs when the vertical permeability of the reservoir is comparable inmagnitude to its horizontal permeability, and when density differences aresignificant, such as in gas-oil or gas-water systems. For more discussion ofspecific pseudoization techniques, see Taggart, et al. [ 1995] and their references,One reason for the continuing popularity of 2D grids is that the expecta-tion of what is appropriate grid resolution has changed as simulation technologyevolved. Thus, even though 3D models could be used today with the gridresolution that was considered acceptable a decade ago for 2D models, modernexpectations often require that even finer grids be used for the same types ofproblems. This is an example of a task expanding to fit the available resources.It is not obvious that increased grid definition is leading to better reservoirmanagement decisions. Indeed, it can be argued that the technological abilityto add complexity is making it more difficult for people to develop a "bigpicture" understanding of the system being studied because they are too busyfocusing on the details of a complex model. Once again, a judicious use ofOckham's Razor is advisable in selecting a reservoir grid. The grid should beappropriate for achieving study objectives.Near-wellbore coning models may be either 2D or 3D grids, but aredefined in cylindrical rather than Cartesian coordinates. Coning (or radial)models are designed to study rapid pressure and saturation changes. An exampleof a radial grid is shown in Figure 16-8. High throughput, that is, large flow ratethrough relatively small, near-wellbore gridblocks is most effectively simulatedby a fully implicit formulation. IMPEScan be used to model coning, but timestepsmust be very small, possibly on the orderof minutes or hours. Small timesteps arenot a problem if the duration of the mod-eled history is short, as it would be in theO Corner Point• Block Centeredcase of a pressure transient test.Gridblocks may be defined in termsof comer-point geometry or block-cen- Figure 16-4. Gridblock represen-tered geometry (Figure 16-4). Block-cen- tation.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 161tered geometry is the most straight forward technique, but comer-point geometryhas gained popularity because it yields more visually realistic representationsof reservoir architecture. This is valuable when making presentations to peoplewho are nonspecialists. The different geometric representations are illustratedfor a two-layer dipping reservoir in Figure 16-5. Although corner-point geometryis visually more realistic, it is easier to define a grid with block-centeredgeometry. Block-centered geometry requires the specification of the lengths ofeach side of the block and the block center or top. Corner-point geometryrequires specifying the location of all eight corners of the block. This is mostreadily accomplished with a computer program.Conventional Grid with Dip-Aligned Grid with Dip-Aligned Grid withRectangles Rectangles ParallelogramsFigure 16-5. Geometric representations of a dipping reservoir.There is little computational difference between the results of corner-pointand block-centered geometry. One caution should be noted with respect tocomer-point geometry. It is possible to define very irregularly shaped grids usingcorner-points. This can lead to the distortion of flood fronts and numericalstability problems. Flood front distortions caused by gridding is an example ofthe grid orientation effect discussed by many authors, including Aziz and Settari[1979], and Mattax and Dalton [1990].The grid orientation effect is exhibited by looking at a displacementprocess in 2D (Figure 16-6). Each producer is equidistant from the single injectorin a model that has uniform and isotropic properties. If grid orientation did notmatter, the symmetry of the problem would show that both wells would produceinjected water at the same time. The figure shows that production is not the same.Injected fluids preferentially follow the most direct grid path to the producer.Thus, even though the producers are symmetrically located relative to the in-TEAM LinG - Live, Informative, Non-cost and Genuine!162 Principles of Applied Reservoir Simulationjector and each other, the grid orientation altered the expected flow pattern,Figure 16-6 shows the effect on frontal advance. In this case, the front arrivessooner at the producer in the upper right than the producer in the upper left. Ifthese results are incorporated in a reservoir management plan, they can reducethe overall effectiveness of the plan.Figure 16-6. Grid orientation effect (after Hegre, et al.1986; reprinted by permission of the Society ofPetroleum Engineers).Another example of the grid orientation effect arises in connection withthe modeling of pattern floods.Figure 16-7 illustrates two gridsthat can be used to model flowin a five-spot pattern. The paral-lel grid results in earlier break-through of injected fluids thanthe diagonal grid. This effectcan be traced to the finitedifference representation of thefluid flow equations.Most finite differencesimulators only account for Figure 16-7. Parallel and diagonal grids (afterflow contributions from blocks T°d.' * al 19^2; rfrinted ^ Permission ofthe Society of Petroleum Engineers).TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 163that are nearest neighbors to the central block along orthogonal Cartesian axes.In Table 16-1, the central block is denoted by "C" and the nearest neighbor blockcontributing to the standard finite difference calculation in 2D are denoted byan asterisk. The five blocks comprise the five-point differencing scheme of the2D Cartesian grid.Table 16-1Finite Difference StencilsBlockJ-lJJ + l1-19*9I#C*1 + 19*9Reservoir simulators are usually formulated with the assumption thatdiagonal blocks do not contribute because the grid is aligned along the principalaxes of the permeability tensor. Diagonal blocks are denoted by "9" in Table16-1. The nine-point stencil includes all nine blocks in the calculation of flowinto and out of the central block. Grid orientation effects can be minimized, atleast in principle, if the diagonal blocks are included in the nine-point finitedifference formulation [for example, see Young, 1984; Hegre, etal, 1986: Lee,et al., 1997]. This option is available in some commercial simulators. In 3Dmodels, the number of blocks needed to represent all adjacent blocks, includingdiagonal terms, is 27. By contrast, only seven blocks are used in the conventionalformulation of a 3D finite difference model.Local grid refinement (LGR)is used to provide additional griddefinition in a few selected regionsof a larger grid. Raleigh [1991] com-pared local grid refinement with aradial grid (Figure 16-8) and showedthat the results are comparable.When LGR is used, it typically in- Figure 16-8. LGR and radial grids,creases computer processor time for a run because of increased throughput inLGR Radial GridTEAM LinG - Live, Informative, Non-cost and Genuine!164 Principles of Applied Reservoir Simulationsmall blocks. An LGR grid is an example of a flexible or unstructured grid. Aflexible grid is made up of polygons in 2D (polyhedra in 3D) whose shape andsize vary from one subregion to another in the modeled region.Although many grid preparation options are available, improving gridpreparation capability is an on-going research and development topic. Forexample, not all flow simulators use a finite difference formulation. Some arebased on a control volume finite element formulation that use triangular meshesin 2D (tetrahedral meshes in 3D). Finite difference grids typically display globalorthogonality in which the grid axes are aligned along orthogonal coordinatedirections. Examples of globally orthogonal coordinate systems include theCartesian x-y-z system and the cylindrical r-Q-z system. Grids with globalorthogonality may be distorted to fit local irregularities such as faults usingcorner-point geometry. By contrast, finite element grids display orthogonalityin which gridblock boundaries are perpendicular to lines joining gridblock nodeson opposite sides of each boundary. An example of a locally orthogonal gridis a perpendicular bisector (PEBI) grid. Aziz [ 1993], Chin [1993], Heinemann[ 1994], Verma and Aziz [ 1997], and Heinemann and Heinemann [ 1998] provideadditional discussion of grid preparation research.16.3 Model TypesModels may be classified into three different types: full field models,window area models, and conceptual models. Full field models are used to matchperformance of the entire field. They take into account the interaction betweenall wells and layers. The results of full field models are already matched to fieldscale and require no further scaling. The disadvantage of using full field modelsis that the number of grid blocks may need to be large or the grid size may needto be relatively coarse to include the entire field.Window area models are designed to look at smaller areas of the field.These models are often constructed from a full field description. Window areamodels allow finer grid resolution or shorter turnaround time if the model runsfaster than a full field model. The window area models are useful for studyingrecovery mechanisms and for determining reasonable grid preparation criteriaTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 165for use in full field models, especially with regard to layering. Full field modelsrequire sufficient layering to track fluid contact movement or other depth de-pendent information that is needed to achieve study objectives. Window areamodels have the disadvantage of not being able to accurately model flux acrosswindow area boundaries. This means that effects of wells outside the windowarea are not taken into account except through boundary conditions. Somecommercial simulators will output time-dependent boundary conditions for usein window area models. Although this information is helpful, the process iscumbersome and does not necessarily yield accurate results. Field history canbe used to guide development of the window area model, but has only limitedutility as a criterion for validating window model performance. Heinemann[1995] has discussed further concepts and applications of a dynamic windowingtechnique that is designed to minimize the difficulties of preparing and applyingwindow area models in conjunction with full field models.One of the most useful types of models is the conceptual model.Conceptual models can be built quickly and require only an approximatedescription of that part of the reservoir that is relevant to the conceptual study.Computer resource requirements are relatively small when compared with fullfield or window area models. Results of the conceptual model are qualitativeand best used for comparing concepts such as vertical layering. They can alsobe used to prepare pseudo curves for use in full field or window area models.For example, the saturation of a block in a model with a transition zone dependson the depth of the center-point of the block (see Chapter 6). As a result, a gridthat is vertically coarse may have only a rough approximation of the transitionzone. More accurate modeling of saturation gradient in a transition zone requiresvertical grid refinement or use of pseudo curves. Conceptual models are usefulfor preparing such pseudo curves. The disadvantage to conceptual models is thattheir results do not apply directly to the description of a particular field. Sincethere is no history match, conceptual model results should be viewed asqualitative rather than quantitative estimates of field performance. They doprovide useful qualitative information that can be applied to specific fields inwindow area and full field models.TEAM LinG - Live, Informative, Non-cost and Genuine!166 Principles of Applied Reservoir Simulation16.4 Basic Simulator VolumetricsReservoir simulators calculate reservoir volume using a procedure similarto the procedure described in this section. Bulk volume VB of each gridblockdefined in a Cartesian coordinate system {x, y, z} is calculated from the grossthickness Az = h of each gridblock and the gridblock lengths Ax, Ay along thex andy axes:Porosity <j) and net-to-gross ratio T) are then used to calculate gridblock porevolumeVp = §r\VB = $T\hAxAy = §hnelAxAywhere net thickness is defined by hnet = r\h. The volume of phase $ in the grid-block at reservoir conditions is the product of the gridblock pore volume andphase saturation, thusV = SV,Pwhere S'? is the saturation of phase 1 Total model volumes are calculated bysumming over all gridblocks.Many commercial simulators provide optional variations on the simpleprocedure outlined above. A comparison of reservoir simulator calculated volu-metrics with volumetrics from another source, such as a material balance studyor a computer mapping package, provides a means of validating volumetric esti-mates using independent sources.ExercisesExercise 16.1 Sketch the model grids for data sets EXAM 1 .DAT, EXAM2.DAT,EXAM3.DAT, EXAM5.DAT, and EXAM7.DAT using information from eachdata set,Exercise 16.2 Repeat Exercise 16. 1 for case study data sets CS-MB.DAT, CS-VC.DAT, and CS-XS.DAT.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 167Exercise 16.3 Modify the grid in EXAM3.DAT so that it has only five blocksin the x direction, but the model volume is unchanged. Be sure to relocate thewells relative to the grid to keep them in their appropriate physical locationsUse the equations in Chapter 31.2 to correct the PID index. How does the coarsergrid affect model performance?Exercise 16.4 Modify the grid in EXAM2.DAT so that it has 5x5x4 grid-blocks. The well should be in the center of the reservoir and the reservoir volumeshould be unchanged by the redefinition of the grid. Use the equations in Chapter31.2 to correct the PID index. How does the finer grid affect model performancewhen the model is run for three years?TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 17In a typical study it is necessary to first specify project objectives. Theobjectives help define the level of detail that will be incorporated in the reservoirmodel. Once objectives are defined, it is helpful to think of the study proceedingin three phases [Saleri, 1993 ]: the history match phase; a calibration phase, whichprovides a smooth transition between the first and third phases; and theprediction phase. The first step toward obtaining a history match is the collectionand analysis of data.17.1 Data PreparationData must be acquired and evaluated with a focus on its quality and theidentification of relevant drive mechanisms that should be included in the model[for example, see Crichlow, 1977; Saleri, et al., 1992; Raza, 1992]. Given thatinformation, it is possible to select the type of model that will be needed for thestudy: conceptual, window area, or full field model. In many cases all three ofthese models may need to be used, as illustrated in Fanchi, et al. [1996]. Datamust be acquired for each model.Some of the data that is required for a model study can be found inexisting reports. The modeling team should find as many reports as it can fromas many disciplines as possible. Table 17-1 lists the types of data that are neededin a model study. A review of geophysical, geological, petrophysical,, andengineering reports provides a background on how the project has beendeveloped and what preconceived interpretations have been established. During168TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 169the course of the study, it may be necessary to develop not only a new view ofthe reservoir, but also to prepare an explanation of why the new view is superiorto a previously approved interpretation. If significant gaps exist in the reports,particularly historical performance of the field, it is wise to update them.Table 17-1Data Required for a Simulation StudyPropertyPermeabilityPorosity, RockcompressibilityRelative permeability andcapillary pressureSaturationsFluid property (PVT) dataFaults, boundaries, fluidcontactsAquifersFracture spacing, orientation,connectivityRate and pressure data,completion and workover dataSourcesPressure transient testing, Coreanalyses, Correlations, Well performanceCore analyses, Well logsLaboratory core flow testsWell logs, Core analyses, Pressure cores,Single-well tracer testsLaboratory analyses of reservoir fluidsamplesSeismic, Pressure transient testingSeismic, Material balance calculations,Regional exploration studiesCore analyses, Well logs, Seismic,Pressure transient tests, Interferencetesting, Wellbore performanceField performance historyA review of rock and fluid property may show that the amount of availabledata is limited. If so, additional data should be obtained when possible. This mayrequire special laboratory tests, depending on the objectives of the study. Ifmeasured data cannot be obtained during the scope of the study, then correlationsTEAM LinG - Live, Informative, Non-cost and Genuine!170 Principles of Applied Reservoir Simulationor data from analogous fields will have to be used. Values must be entered intothe simulator, and it is prudent to select values that can be justified.Well data should be reviewed. If additional field tests are needed, theyshould be requested and incorporated into the study schedule. Due to the costsand operating constraints of a project, it may be difficult to justify the expenseof acquiring more data or delaying the study while additional data is obtained.The modeling team should take care to avoid underestimating the amount ofwork that may be needed to prepare an input data set. It can take as long tocollect and prepare the data as it does to do the study.17.2 Pressure CorrectionWhen pressures are matched in a model study, the calculated and observedpressures should be compared at a common datum. In addition, pressures fromwell tests should be corrected for comparison with model block pressures. Awidely used pressure correction is the Peaceman [1978, 1983] correction.Figure 17-1 illustrates a pressure buildup curve as a function of radialdistance from the center of a wellbore with radius rw. To obtain a well blockpressure P0 from a pressure buildup (PBU), Peaceman used a Cartesian grid tomodel the PBU performance of a well to find an equivalent well block radiusr0. A Homer plot of a PBU test is illustrated in Figure 17-2.RadiusFigure 17-1. Pressure buildup.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 171Peaceman showed that the shut in pressure Pws of an actual well equalsthe simulator well block pressure P0 at a shut in time A/5 given by(17.1)fKwhere K is permeability, <f) is porosity, |l is viscosity, and CT is total compress-ibility. Units for all variables are given in Table 17-2 at the end of this section,Log (T;, + AO/A/Figure 17-2. Horner plot of PBU.The relationship between gridblock pressure P0 and flowing pressure Pwfat the wellbore isP..., = P. - 141.2In — + Sr.(17,2)where Q is the flow rate, B is formation volume factor, and S is skin. For anisotropic reservoir, that is, a reservoir in which x-direction permeability equalsy direction permeability (Kx = Ky), the equivalent well block radius is given interms of the block lengths {A*, Ay}, thusro = 0.14 (A*2 + A>>2)'/2 (17.3)shut in time can be masked by wellbore storage effects. If it is, the shut inpressure P^ may have to be obtained by extrapolation of another part of thecurve, such as the radial flow curve. Table 17-2 summarizes the parametersinvolved in the Peaceman correction for a consistent set of units. An applicationTEAM LinG - Live, Informative, Non-cost and Genuine!172 Principles of Applied Reservoir Simulationof the Peaceman correction is presented in Chapter 22 as part of a case study.Peaceman's work with 2D models was extended to 3D by Odeh [1985].Table 17-2Oilfield Units for the Peaceman CorrectionParameterBCThKP P P1 0' l Wf> * WSQro>rwSA/,A*, A>>4>M-UnitsRB/STB•-ipsiftmdpsiaSTB/Dftfractionhrftfractioncp17.3 Simulator Selection and Ockham's RazorSeveral requirements must be considered when selecting a simulator.These requirements can be classified into two general categories: reservoir andnon-reservoir. From a reservoir perspective, we are interested in fluid type,reservoir architecture, and the types of recovery processes or drive mechanismsthat are anticipated.Reservoir architecture encompasses a variety of parameters that have amajor impact on model design. Study objectives and the geologic model mustbe considered in establishing the dimensionality of the problem (1 D, 2D, or 3D)TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 173and the geometry of the grid. Do we need special grid options, such as radialconing or local grid refinement, or will Cartesian coordinates be satisfactory?If the study is designed to investigate near wellbore flow, it would be wise toselect a grid that provides good spatial resolution near the wellbore, for example,radial coordinates. On the other hand, if the study is intended to provide anoverview of field performance, a coarse Cartesian grid may be satisfactory.The level of complexity of the geology will influence grid definition, andin the case of fractured reservoirs, the type of flow equations that must be used[for example, see Reiss, 1980; Aguilera, 1980; Golf-Racht, 1982; and Lough,et al, 1996]. A highly faulted reservoir or a naturally fractured reservoir is moredifficult to describe numerically than a homogeneous sand.Model selection will be influenced by the types of processes and drivemechanisms that dominate flow in the reservoir. Processes range from gas capdrive and water drive under primary depletion, through water or gas injectionin pressure maintenance programs, to miscible or thermal flooding in enhancedrecovery projects. The choice of model will vary depending on the anticipatedprocess. For example, dry gas injection in a condensate reservoir is typicallymodeled with a compositional simulator, while steam flooding a heavy oilreservoir should be modeled with a thermal simulator.A few guidelines are worth noting with regard to simulator selection.Many novice modelers make the mistake of selecting models that are much morecomplex than they need to be to satisfy the objectives of the study. Accordingto Coats [1969], the modeler should "select the least complicated model andgrossest reservoir description that will allow the desired estimation of reservoirperformance." This is a restatement of Ockham's Razor.William of Ockham, a fourteenth-century English philosopher, said"plurality must not be posited without necessity" [Jefferys and Berger, 1992],Today this is interpreted to mean that an explanation of the facts should be nomore complicated than necessary. We should favor the simplest hypothesis thatis consistent with the data.Ockham's Razor should be applied with care, however, because one ofthe goals of a model study is to establish a consensus about how the reservoirbehaves. This consensus is political, to an extent, because the model must satisfyTEAM LinG - Live, Informative, Non-cost and Genuine!174 Principles of Applied Reservoir Simulationthe people who commissioned the study. Their views may require using a modelthat has more complexity than required from a technical modeling perspective.Non-reservoir requirements include personnel, simulator availability, andcost effectiveness. Personnel will be needed to gather and evaluate data, prepareinput data, perform the history match, and then make predictions. Data gatheringmay take a few days or several months depending on the quality and extent ofthe data base for a particular field. The history matching and prediction phasesdo not necessarily have to be done by the same modeler. In some companies,history matching is done in a collaborative effort between a specialized tech-nology center and a field office, while most of the prediction work is completedin the field office. This takes advantage of specialized expertise: technologycenters, including outside consultants, routinely set up and ran models, whileday-to-day changes that impact production operations are handled in the fieldoffice. The division of labor between history matching and prediction makessense in some circumstances.A wide variety of simulators are available for a price. The work horsesimulators - black oil and compositional - can often be leased on an as-neededbasis or are available through computer networks. More specialized simulatorsmay be obtained from software vendors, or as publicly available research codesdeveloped at university and government laboratories.As complexity increases, so also does cost. A good economic argumentto support Ockham's Razor is to remember that the latest technology is notalways the best technology for a project, and its use comes with a cost. Modelingteams are often tempted to apply the latest technology, even if it is not warranted.An example is the use of local grid refinement (LGR) to model horizontal wells.LGR is an innovative grid preparation technique that can improve spatialresolution, but at a substantial increase in computer cost and simulator sophisti-cation. It is very common to find LGR used to model horizontal wells. In somecases, such as feasibility studies, this level of technical detail exceeds the needsof the study objectives and simply adds cost to the project without adding thecorresponding value. A wise modeling team will match the level of technologywith the objectives of the study. The result will be the selection of the most costeffective method for achieving study objectives.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 175The cost of a simulation study can be estimated based on previous experi-ence with similar studies. As an example of how to estimate the cost for a blackoil simulation study, begin by calculating the product of the number ofgridblocks and the number of timesteps denoted by GBTS. Once GBTS isknown, it should be related to the computer processing (cpu) time needed tomake a ran. The amount of cpu time per GBTS is determined by dividing thecpu time needed to make previous model runs by the number of GBTS in thoseruns. The product of GBTS and cpu time per GBTS gives total cpu time neededfor a run. The cost of the study then depends on the number of runs that needto be made. The number of runs can be estimated by assuming that approxi-mately 100 runs will be needed to obtain a history match. A similar approachis applied to estimating the cost of making predictions. Personnel cost Isapproximately equal to computer cost for the study, This does not include thecost of data collection and evaluation.ExercisesExercise 17.1 Data set EXAM10.DAT uses multiple Rock and PVT regions.Review EXAM 10.DAT and simplify the data set without altering model results.List the changes you make to the data set. Chapter 24.4 presents a descriptionof Rock and PVT region data records.Exercise 17.2 A model has 10 * 10 x 4 gridblocks and takes 5 minutes to ran100 timesteps. Calculate cpu time per GBTS. Estimate how long it would taketo make 100 runs with 200 timesteps each.Exercise 17.3 (A) Use Eq. (17.3) to calculate the equivalent well block radiusof a block with Ax = Ay = 200 ft. (B) Estimate shut in time for the Peacemancorrection using Eq. (17.1). Assume <J> = 0.15, CT= I * 10~5 psia"1, jl = 2 cp andK=10md.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 18History MatchingThe history matching process begins with clearly defined objectives.Given the objectives, it is necessary to acquire model input data, especially thehistory of field performance. One of the essential tasks of the data acquisitionstage is to determine which data should be matched during the history matchingprocess. For example, if a gas-water reservoir is being modeled, gas rate isusually specified and water production is matched. By contrast, if an oil reservoiris being modeled, oil rate is specified and water and gas production are matched.Data acquisition is an essential part of model initialization. Modelinitialization is the stage when the data is prepared in a form that can be usedby the simulator. The model is considered initialized when it has all the data itneeds to calculate fluids in place. The reservoir must be characterized in a formatthat can be put in a simulator and that is acceptable to the commissioners of thestudy. Reservoir characterization includes the selection of a grid and associateddata for use in the model. It may also require the study of multiple reservoirrealizations in the case of a geostatistical model study [for example, seePannatier, 1996; Lieber, 1996; Rossini, et al, 1994; Englund and Sparks, 1991;Haldorsen and Damsleth, 1990; and Isaaks and Srivastava, 1989]. All fluid datacorrections, such as flash corrections applied to differential PVT data in a blackoil simulation, must be completed during the model initialization process.In many cases, simple conceptual models may be useful in selecting a finalgrid for the model study, especially when determining the number of layers. Asan illustration, suppose we want to track flood front movement in a very largefield. In this case, we want as much areal definition as possible (at least 3 to 5176TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 177gridblocks between each gridblock containing a well), but this may mean lossof vertical definition. A way to resolve the problem is to set up one or morecross-section models that represent different parts of the field. Vertical confer-mance effects in these regions are modeled in detail by calculating flowperformance with the cross-section models. The flow performance of a detailedcross-section model is then matched by adjusting relative permeability curvesin a model with fewer layers. The resulting pseudo-relative permeability curvesare considered acceptable for use in an areal modelAnother aspect of model initialization is equilibration. This is the pointat which fluid contacts are established and fluid volumes are calculated.Resulting model volumes should be compared with other estimates of fluid inplace, notably volumetric and material balance estimates. There should bereasonable agreement between the different methods (for example, within twopercent). Finally, the history match can begin.18.1 Illustrative History Matching StrategiesA universally accepted strategy for performing a history match does notexist. History matching is as much art as science because of the complexity ofthe problem. Nevertheless, there are some general guidelines that can help movea history match toward successful completion. These guidelines have beenpresented by such authors as Crichlow [1977], Mattax and Dalton [1990],Thomas [1982], and Saleri, et al. [1992]. One set of guidelines is presented inTable 18-1. The first two steps in the table take precedence over the last two,If the first two steps cannot be achieved, there is a good chance the model isinadequate and revisions will be necessary. An inadequate model may be dueto a variety of problems: for example, the wrong model was selected, thereservoir is poorly characterized, or field data is inaccurate or incomplete.Among the data variables matched in a typical black oil or gas study arepressure, production rate, water-oil ratio (WOR), gas-oil-ratio (GOR), and tracerdata if it is available. More specialized studies, such as compositional or thermalstudies, should also match data unique to the process, such as well streamcomposition or the temperature of produced fluids.TEAM LinG - Live, Informative, Non-cost and Genuine!178 Principles of Applied Reservoir SimulationTable 18-1Suggested History Matching ProcedureStepRemarksMatch volumetrics with material balance and identify aquifer support.IIMatch reservoir pressure. Pressure may be matched both globally andlocally. The match of average field pressure establishes the globalquality of the model as an overall material balance. The pressuredistribution obtained by plotting well test results at given points in timeshows the spatial variation associated with local variability of fieldperformance.IllMatch saturation dependent variables. These variables include water-oil ratio (WOR) and gas-oil ratio (GOR). WOR and GOR are often themost sensitive production variables in terms of both breakthrough timeand the shape of the WOR or GOR curve.IVMatch well flowing pressures.The pressure is usually the first dynamical variable to be matched duringthe history matching process. A comparison of estimated reservoir pressuresobtained from well tests of a single well on successive days shows that errorsin reported historical pressures can be up to 10 percent of pressure drawdown.This error may be as large or larger than the Peaceman correction discussed inChapter 17. As a first approximation, it is sufficient to compare uncorrectedhistorical pressures directly with model pressures, particularly if your initialinterest is in pressure trends and not in actual pressure values. Pressurecorrections should be applied when fine tuning the history match.Production rates are usually from monthly production records. Themodeler specifies one rate or well pressure, and then verifies that the rate isentered properly by comparing observed cumulative production with modelcumulative production. After the rate of one phase is specified, the rates of allother phases must be matched by model performance. In many cases, observedrates will be averaged on a monthly or quarterly basis and then compared withTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 179model calculated rates. If the history of reservoir performance is extensive, thenit is often wise to place a greater reliance on the validity of the most recent fielddata when performing a history match.Phase ratios, such as GOR and WOR, are sensitive indicators of modelperformance. Matching ratios provides information about pressure depletion andfront movements. Tracers are also useful for modeling fluid fronts. Tracers neednot be expensive chemicals; they can even be changes in the salinity of producedwater. Salinity changes can occur as a result of mixing when injected brine andin situ brine have different salinities. Water sample analysis on a periodic basisis useful for tracking salinity variation as a function of time.Time-Lapse Seismic History MatchingAn emerging history matching strategy is to combine time-lapse seismicreservoir monitoring with traditional flow modeling in a process referred to asseismic history matching [Lumley and Behrens, 1997]. Seismic history matchingis an iterative process, as illustrated in Figure 18-1.The ovals in the figurerepresent model preparation, while the rectangles correspond to the historymatching process.Update Reservoir ModelMake Reservoir ManagementDecisions 4D Seismic DataCompare withAV/^^ ^"N. f Seismicf Reservoir \ ( ModelingV Modeling ) X^ ImagingRock PhysicsElastic PropertiesFigure 18-1. Seismic history matching [after Lumley and Behrens,SPE 38696, 1977].TEAM LinG - Live, Informative, Non-cost and Genuine!180 Principles of Applied Reservoir SimulationThe seismic history matching process includes steps for incorporatingtime-lapse seismic monitoring information. Time-lapse seismic monitoring isthe comparison of two or more 3-D seismic surveys over the same region atdifferent points in time. WINB4D includes algorithms for providing informationthat can facilitate all of the tasks shown in Figure 18-1. This has been madepossible by the inclusion of a petrophysical model in the flow simulator,18.2 Key History Matching ParametersA fundamental concept of history matching is the concept of a "hierarchyof uncertainty." The hierarchy of uncertainty is a ranking of model input dataquality that lets the modeler determine which data is most and least reliable.Changes to model input data are then constrained by the principle that the leastreliable data should be changed first. The question is: which data are leastreliable?Data reliability is determined when data are collected and evaluated forcompleteness and validity [Raza, 1992; Saleri, et al., 1992]. This is such animportant step in establishing a feel for the data that the modeler should beclosely involved with the review of data. Relative permeability data are typicallyplaced at the top of the hierarchy of uncertainty because they are modified moreoften than other data. Relative permeability curves are often determined fromcore floods. As a consequence, the applicability of the final set of curves to therest of the modeled region is always in doubt.Initial fluid volumes may be modified by changing a variety of inputparameters, including relative permeability endpoints and fluid contacts. Model-calculated, original fluid volumes in place are constrained by independenttechniques like volumetrics and material balance studies.Attempts to match well data may require changing the producing intervalor the productivity index of a perforation interval. If it is difficult to match wellperformance in a zone or set of zones, the modeler needs to look at a variety ofpossibilities, including unexpected completion and wellbore problems. In onestudy, for example, an unexpectedly high GOR from a perforation interval thatwas known to be below the gas-oil contact was due to gas flow in the annulusTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 181between the tubing and the casing. This result was confirmed by running acement bond log and finding a leak in the wellbore interval adjacent to the gascap. Gas from the gas cap was entering the welibore and causing the larger thanexpected production GOR. This effect can be modeled by a variety of options,depending on the degree of accuracy desired: for example, it could be modeledby altering productivity index (PI) in the well model or by designing a nearwellbore conceptual model and preparing pseudorelative permeability curves,The choice of method will influence the predictive capability of the model. Thus,a pseudo-relative permeability model will allow for high GOR even if the wellis recomputed, whereas the PI could be readily corrected at the time of wellrecompletion to reflect the improvement in wellbore integrity.Map adjustments may also be necessary. This used to be considered a lastresort change because map changes required substantial effort to redigitize themodified maps and prepare a revised grid. Pre-processing packages andcomputer-aided geologic modeling are making map changes a more acceptablehistory match method. In the case of geostatistics, a history matching processmay actually involve the use of several different geologic models. Each geologicmodel is called a stochastic image or realization. Additional discussion ofgeostatistics is presented in Chapter 11.Toronyi and Saleri [1988] present a detailed discussion of their approachto history matching. It is noteworthy because they provide guidance on howchanges in some history match parameters affect matches of saturation andpressure gradients. A summary is presented in Table 18-2. It shows, for example,that a change in pore volume can effect pressure as it changes with time. Asanother example, relative permeability changes are useful for matching saturationvariations in time and space. Notice that fluid property data are seldom changedto match field history. This is because fluid property data tend to be moreaccurately measured than other model input data.History matching must not be achieved by making incorrect parametermodifications. For example, matching pressure may be achieved by adjustingrock compressibility, yet the final match value should be within the set of valuestypically associated with the type of rock in the formation. In general, modifiedparameter values must be physically meaningful.TEAM LinG - Live, Informative, Non-cost and Genuine!182 Principles of Applied Reservoir SimulationTable 18-2Influence of Key History Matching ParametersParameterPore volumePermeability thicknessRelative permeabilityRock compressibilityBubble-point pressure* Avoid changing if possiblePressurematchAP/A/AP/A*Not used*AP/A/ *Saturationmatch*AS/A*AS/ A* and AS/A/Not used*18.3 Evaluating the History MatchOne way to evaluate the history match is to compare observed andcalculated parameters. Typically, observed and calculated parameters arecompared by making plots of pressure vs time, cumulative production (orinjection) vs time, production (or injection) rates vs time, and GOR, WOR, orwater cut vs time. Other comparisons can and should be made if data areavailable. They include, for example, model saturations versus well logsaturations, and tracer concentration (such as salinity) versus time. In the caseof compositional simulation, dominant components (typically methane) shouldbe plotted as a function of time.In many studies, the most sensitive indicators of model performance areplots of GOR, WOR, or water cut vs time. These plots can be used to identifyproblem areas. For example, suppose we plot all high/low WOR and GOR wellsor plot all high/low pressure wells. A review of such plots may reveal a groupingof wells with the same problem. This can identify the presence of a systematicerror or flaw in the model that needs to be corrected. If the distribution israndom, then local variations in performance due to heterogeneity should beconsidered.TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 18318.4 Deciding on a MatchThere are several ways to decide if a match is satisfactory. In all cases,a clear understanding of the study objectives should be the standard for makingthe decision. If a coarse study is being performed, the quality of the matchbetween observed and calculated parameters does not need to be as accurate asit would need to be for a more detailed study. For example, pressure may beconsidered matched if the difference between calculated and observed pressuresis within ±10% draw down. The tolerance of ±10% is determined by estimatingthe uncertainty associated with measured field pressures and the required qualityof the study. A study demanding greater reliability in predictions may need toreduce the tolerance to ± 5% or even less, but it is unrealistic to seek a toleranceof less than one percent. The uncertainty applies not to individual well gaugepressures, which may be measured to a precision of less than one percent, butto estimates of average field or region pressure from two or more well tests. Thelatter error is generally much larger than the precision of a single well test. Inany event, model-calculated pressure trends should match field or region pres-sure performance.Another sensitive indicator of the quality of a history match is the matchof WOR, GOR, or water cut. Three factors need to be considered: breakthroughtime, the magnitude of the difference between observed and calculated values,and trends. Adjustments in the model should be made to improve the quality ofeach factor. Saleri [1993] has observed that a match of the field is more easilyobtained than a match of individual well performance. Indeed, he notes thatmatching every well is virtually impossible. As a rale of thumb, the field matchmay be valid for a year or more without updating, and we can expect the wellmatch to be valid for up to six months without updating. Deviations from thisrule will vary widely, and will depend on the type of system modeled and thealignment of the interpreted model with reality. Indeed, gas reservoirs withoutaquifer influx may be accurately modeled for the life of the field, while a gasreservoir with complex lithology and water influx may never be satisfactorilymatched.TEAM LinG - Live, Informative, Non-cost and Genuine!184 Principles of Applied Reservoir SimulationModelers must resist being drawn into the "one more run" syndrome. Thisoccurs when a modeler (or member of the study team) wants to see "just onemore run" to try an idea that has not yet been tried. In practice, a final match isoften declared when the time or money allotted for the study is depleted.18.5 History Match LimitationsHistory matching may be thought of as an inverse problem. An inverseproblem exists when the dependent variable is the best known aspect of a systemand the independent variable must be determined [Oreskes, et al, 1994]. Forexample, the "dependent variable" in oil and gas production is the productionperformance of the field. Production performance depends on input variablessuch as permeability distribution and fluid properties. The goal of the historymatch is to find a set of input variables that can reconstruct field performance.In the context of an inverse problem, the problem is solved by finding aset of reasonable reservoir parameters that minimizes the difference betweenmodel performance and historical performance of the field. As usual, we mustremember that we are solving a non-unique problem whose solution is often asmuch art as science. The uniqueness problem arises from many factors. Mostnotable of these are unreliable or limited field data and numerical effects.Advances in hardware and software technology have made it possible tominimize the effects of numerical problems, or at least estimate their influenceon the final history match solution. Data limitations are more difficult to resolvebecause the system is inherently underdetermined: we do not have enough datato be sure that our final solution is correct.Test of ReasonablenessA model may be considered reasonable if it does not violate any knownphysical constraints. In many cases, a model may be acceptable if it is reason-able. In other situations, not only must physical constraints be satisfied, butapproved processes for evaluating data must also be followed. Thus a model maybe reasonable, but if it is based on an innovative technique that is reasonable butnot approved, the model will be unacceptable. The modeler may use a methodTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 185that is in the literature, but the commissioner of the study may have a philosophi-cal or empirical objection to the method. Window area modeling is a goodexample of a method that may be reasonable but not acceptable because failureto adequately describe flux across window area boundaries can yield poor results.If someone in a position of authority or influence has had a bad experience withthe modeling method, they may refuse to accept results from the model.Similarly, the modeler needs to be aware that some modeling methods are notuniversally accepted. At the very least, alternative methods may be needed tocorroborate the disputed method as part of a sensitivity analysis or modelvalidation exercise.ExercisesExercise 18.1 (A) Run EXAM6.DAT and plot average reservoir pressure as afunction of time. (B) Multiply the pore volume of data set EXAM6.DAT by 0.9and repeat part A. (C) How does the change in pore volume affect pressure asa function of time?Exercise 18.2 Double the horizontal permeability in layer K = 1 of data setEXAM6.DAT. (A) Plot the average reservoir pressure as a function of time. (B)What is the effect on production, by layer, at the end of two years? FileWTEMP.WEL provides rate information by layer for all wells.Exercise 18.3 Set the x direction transmissibility to 0 between 1 = 2 and 1 = 3for blocks ranging from J = 1 to J = 4 in layers K = 1 and K = 2 of data setEXAM6.DAT. This transmissibility barrier represents a flow barrier such as asealing fault. How does the barrier alter flow patterns and the distribution ofreservoir pressure?TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 19PredictionsThe previous chapters have shown how to build a working model of thereservoir and establish a level of confidence in the validity of model results. Itis time to recall that modeling was undertaken to prepare a tool that would helpus develop recommendations for a reservoir management program. The primaryreservoir management objective is to determine the optimum operatingconditions needed to maximize the economic recovery of hydrocarbons. Thisis accomplished, in principle, by marshaling accessible resources to4 optimize recovery from a reservoir, and+ minimize capital investments and operating expenses.The commercial impact of the simulation study is the preparation of a cash flowprediction from projected field performance. Thus, the model study is oftencompleted by making field performance predictions for use in economic analysisof possible operating strategies.19.1 Prediction CapabilitiesPerformance predictions are valuable for a variety of purposes. Predictionscan be used to better interpret and understand reservoir behavior and theyprovide a means of determining model sensitivity to changes in input data. Thissensitivity analysis can guide the acquisition of additional data for improvingreservoir management.Predictions enable people to estimate project life by predicting recoveryvs time. Project life depends not only on the flow behavior of the reservoir, but186TEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 187also on commercial issues. Models let the user impose a variety of economicconstraints on future reservoir performance during the process of estimatingproject life. These constraints reflect a range of economic criteria that willinterest management, shareholders, and prospective investors.Commercial interests are clearly important to the future of a project, andso are technical issues. It is often necessary to compare different recoveryprocesses as part of a study. Since there is only one field, it is unrealistic tobelieve that many different recovery processes can be evaluated in the field, evenas small-scale pilot projects. Pilot projects tend to be substantially moreexpensive to run than simulation studies. In some cases, however, it might beworthwhile to confirm a simulation study with a pilot project. This is especiallytrue with expensive processes such as chemical and thermal flooding,Yet another use for model predictions is the preparation of a reservoirmanagement plan. Reservoir management plans have been discussed in previoussections. Their preparation is often the single most important motivation forperforming a simulation study.19.2 Prediction ProcessThe prediction process begins with model calibration. It is usuallynecessary to ensure continuity in well rate when the modeler switches from ratecontrol during the history match to pressure control during the prediction stageof a study. This is illustrated in Figure 19-1 where the solid curve is the predictedrate based on the productivity index (PI) used in the history match. A clearHistory -< >• PredictionRate --,„ :~"~~"~~ •—^ _ VadjustPITimeFigure 19-1. Model calibration.TEAM LinG - Live, Informative, Non-cost and Genuine!188 Principles of Applied Reservoir Simulationdiscontinuity in rate is observed between the end of history and the beginningof prediction. The rate difference usually arises because the actual well PI,especially skin effect, is not accurately modeled by the model PL An adjustmentto model PI needs to be made to match final historical rate with initial predictedrate.The next step is to prepare a base case prediction. The base case predictionis a forecast assuming existing operating conditions apply. For example, the basecase for a newly developed field that is undergoing primary depletion shouldbe a primary depletion case that extends to a user-specified economic limit. Bycontrast, if the field was being waterflooded, the waterflood should be the basecase and alternative strategies may include gas injection and WAG (water-altemating-gas).The base case prediction establishes a basis from which to comparechanges in field performance resulting from changes in existing operatingconditions. In addition, a sensitivity analysis should be performed to provideinsight into the uncertainty associated with model predictions. A procedure forconducting a sensitivity analysis is outlined below.19.3 Sensitivity AnalysesSensitivity analyses are often needed in both the history matching andprediction stages [for example, see Crichlow, 1977; Mattax and Dalton, 1990;Saleri, 1993; and Fanchi, et al, 1996]. Any method that quantifies the uncer-tainty or risk associated with selecting a particular prediction case may be viewedas a sensitivity analysis. An example of a sensitivity analysis technique that iscost-effective in moving a history match forward is conceptual modeling. It canbe used to address very specific questions, such as determining the impact offluid contact movement on hydrocarbon recovery. Similarly, window modelsthat study such issues as the behavior of a horizontal well in a fault block provideuseful information on the sensitivity of a model to changes in input parameters.Another example of a sensitivity analysis technique is risk analysis.Murtha [1997] defines risk analysis as "any form of analysis that studies andhence attempts to quantify risks associated with an investment." Risk in thisTEAM LinG - Live, Informative, Non-cost and Genuine!Pan II: Reservoir Simulation 189context refers to a potential "change in assets associated with some chanceoccurrences." Risk analysis generates probabilities associated with changes ofmodel input parameters. The parameter changes must be contained within rangesthat are typically determined by the range of available data, information fromanalogous fields, and the experience of the modeling team. Each model run usinga complete set of model input parameters constitutes a trial. A large number oftrials can be used to generate probability distributions. Alternatively, the resultsof the trials can be used in a multivariable regression analysis to generateanalytical expressions, as described below.One of the most widely used techniques for studying model sensitivityto input parameter changes is to modify model input parameters in the historymatched model. The following procedure combines multivariable regression andthe results of model trials to generate an analytical expression for quantifyingthe effect of changing model parameters.Assume a dependent variable F has the formF = K n XjJj*\where {Xj} are n independent variables and K is a proportionality constant thatdepends on the units of the independent variables. Examples of Xj are wellseparation, saturation end points, and aquifer strength. Taking the logarithm ofthe defining equation for F linearizes the function F and makes it suitable formultivariable regression analysis, thusInF = InK + £ e/m Xj7=1A sensitivity model is constructed using the following procedure:4 Run a model with different values of {Xj}4 Obtain values of F for each set of values of {Xj}The constants K, {e-\ are obtained by performing a multivariable regressionanalysis using values of F calculated from the model runs as a function of {Xj} .In addition to quantifying behavior, the regression procedure providesan estimate of fractional change of the dependent variable F when we makeTEAM LinG - Live, Informative, Non-cost and Genuine!1 90 Principles of Applied Reservoir Simulationfractional changes to the independent variables {Xj} . The fractional change inF is given bydF dXThis lets us compare the relative importance of changes to the independentvariables. Notice that the proportionality constant K has been factored out of theexpression dF/Ffor the fractional change in F. Thus, the quantity dF/Fdoes notdepend on the system of units used in the sensitivity study.19.4 Economic AnalysisIn addition to providing technical insight into fluid flow performance,model predictions are frequently combined with price forecasts to estimate howmuch revenue will be generated by a proposed reservoir management plan. Therevenue stream is used to pay for capital and operating expenses, and theeconomic performance of the project depends on the relationship betweenrevenue and expenses [see, for example, Bradley and Wood, 1 994; Mian, 1 992;Thompson and Wright, 1 985]. A discussion of basic economic concepts is givenin Chapter 9. It is sufficient to note here the role of economic analysis in thecontext of a model study.In a very real sense, the reservoir model determines how much money willbe available to pay for wells, compressors, pipelines, platforms, processingfacilities, and any other items that are needed to implement the plan representedby the model. For this reason, the modeling team may be expected to generateflow predictions using a combination of reservoir parameters that yield betterrecoveries than would be expected if a less "optimistic" set of parameters hadbeen used. The sensitivity analysis is a useful process for determining thelikelihood that a set of parameters will be realized. Indeed, modern reservesclassification systems are designed to present reserves estimates in terms of theirprobability of occurrence. A probabilistic analysis is discussed in Chapter 9. Theprobabilistic representation of forecasts gives decision-making bodies such asTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 191corporate managements and financial institutions the information they need tomake informed decisions.19.5 Validity of Model PredictionsThe validity of model predictions was studied by Saleri [1993] whocompared actual field performance with predicted performance. Figure 19-2sp ~l 'ts"'o80-&^ "QO 40-,g> -2 0HistoryPressure* ^1j\n.J\J_J .vJt _XtyrCrdlr'O 82 $4'k-^teT"^1f^^^ w..O/7 RateGOR_,_ -x^K^v^r-**ForecastArt aJSSp"^» "H3 ^"^ ^^"**f"9^ *"- --"^r^^ii-* Watercut32005Oto_7600 Q;07600•2o:800OO 80 £rv7/meFigure 19-2. Quality of field performance match (after Saleri,1993; reprinted by permission of the Society of PetroleumEngineers).illustrates his results. The overall match of field performance, such as total rateand pressure performance, is reasonable. The field match is somewhat deceptivehowever, because the validity of individual well performance forecasting varieswidely. Indeed, the match of water and gas performance for about half of thewells was deemed a "bust" by the author. This is not unusual in a model study,Saleri arrived at the following conclusions:4 "Barring major geologic and/or reservoir data limitations, fieldwidecumulative production forecast accuracies would tend to range from 10%to 40%." [Saleri, 1993]4* "Well performance forecasts are bound to be less successful thanfieldwide predictions." [Saleri, 1993]TEAM LinG - Live, Informative, Non-cost and Genuine!192 Principles of Applied Reservoir SimulationThese points underscore the need to recognize that the history match processdoes not yield a unique solution. Forecasts of reservoir behavior depend on thevalidity of the history match.Despite the uncertainty associated with simulator-based forecasts,reservoir simulation continues to be the most reliable method for makingperformance predictions, particularly for reservoirs that do not have an extensivehistory or for fields that are being considered as candidates for a change inreservoir management strategy. Other methods, such as decline curve analysisand material balance analysis, can generate performance forecasts, but not tothe degree of detail provided by a reservoir model study. As Saleri [ 1993] noted,4 "While a 10% to 40% forecast uncertainty may appear alarming in anabsolute sense, the majority of reservoir engineering decisions requirechoices based solely on comparative analyses (for example, peripheralvs. pattern flood). Thus, in selecting optimum management strategies,finite-difference models still offer the most effective tools."Saleri's view is similar to that of Oreskes, et al. [1994], Even thoughmodels are non-unique representations of nature, they still have many uses. Insummary, models can be used to4 corroborate or refute hypotheses about physical systems;4 identify discrepancies in other models; and4 perform sensitivity analyses.Part IV integrates the ideas presented above in the context of a case study.ExercisesExercise 19.1 Data set EXAM4.DAT is a 2D areal model of an undersaturatedoil reservoir undergoing primary depletion. (A) Run EXAM4.DAT anddetermine oil recovery at the end of the run. (B) Set the bottomhole pressure(BMP) in well P-l ofEXAM4.DAT to 150 psia and run the model. How muchoil is recovered in the modified model?Exercise 19.2 Beginning at the end of year one, add a water injection well ineach of the four corner gridblocks in data set EXAM4.DAT with the BHPTEAM LinG - Live, Informative, Non-cost and Genuine!Part II: Reservoir Simulation 193modification described in Exercise 19.1. The maximum allowable BHP for aninjection well is 5000 psia. Assume the target rate for the oil production wellis 600 STB/D. Maximize oil recovery by varying the amount of water injected,Data set EXAM6.DAT is an example of a data set with the injection wells added.TEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!TEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 20Study Objectives and Data GatheringThe first step in a study is to identify its objectives. Study objectivesprovide information about resource requirements. After stating the objectivesof the study, the data for characterizing the reservoir is then described.20.1 Study ObjectivesTwo objectives of the case study are to increase your understanding ofthe reservoir simulation process, and to acquire experience working with asimulator. The experience you gained working the exercises in Parts I and II isa transferable skill. Many of the tasks performed with WINB4D may differ indetail from other simulators, but are conceptually universal. Although the aboveobjectives are important from a pedagogical point of view, they are secondarywithin the context of the case study.The reservoir management objective of the case study is to optimizeproduction from a dipping, undersaturated oil reservoir. There will beconstraints imposed on the case study objective. Before discussing the con-straints, however, it is first necessary to gather some background informationabout the field.20.2 Reservoir StructureA seismic line through an east-west cross-section of the field is shownin Figure 20-1. The single well (P-l) has been producing from what appears to197TEAM LinG - Live, Informative, Non-cost and Genuine!198 Principles of Applied Reservoir Simulationbe a fault block bounded upstructure and to the east by an unconformity;downstructure and to the west by a fault or aquifer; and to the north and southby sealing faults.Depth A(ft) A""*^~ DoO ""•"•HP*1"' W *• " r "T * „Well_920o P- 1 .-",:'•"Seismic Reflectors(Processed with time-_9600 .•.'•" . • " depth conversion) _i » "i i i i i i 110 400 800 1200 1600 2000Distance from Western Fault (ft)Figure 20-1. East-west seismic line.A well log trace is shown in Figure 20-2. An analysis of the well log datashows that two major sands are present and are separated by a shale section. Welllog results are presented in Table 20-1. The table headings refer to porosity <|>,water saturation Sw, gross thickness h, and the net-to-gross ratio NTG.WellP-lLogTraceSand' Shale&! Sand with ShaleFigure 20-2. Well log trace.TEAM LinG - Live, Informative, Non-cost and Genuine!Part III: Case Study 199Table 20-1Well Log Analysis SummaryLithology(FromCuttings)SandstoneShaleSandstone withShale StringerDepth (ft) toTop ofFormation933094109430*(fr.)0.20—0.25sw(fr.)0.30—0.30h(ft)8020120NTG(fr.)0.9—0.8A conceptual sketch of the reservoir cross-section is shown in Figure 20-3,Notice that we have adopted an unconformity as our geologic model.ImpermeableCap RockOilWaterFigure 20-3. Conceptual sketch of reservoir cross-section(after Clark, 1969; reprinted by permission of the Societyof Petroleum Engineers).20.3 Production HistoryWell P-1 has produced for a year. Its production history is shown in Tables20-2a and 20-2b.TEAM LinG - Live, Informative, Non-cost and Genuine!200 Principles of Applied Reservoir SimulationTable 20-2aProduction HistoryTIMEDAYS51324416691122153183214245274305336365RATESOILSTB/D500500500500500500500500500500500500500500500GASMSCF/D228228228228228228228228228228228228228228228WATERSTB/D000001112223345GORSCF/STB457457457457457457457457457457457457457457457WOR000000000000000A review of Tables 20-2a and 20-2b shows that oil rate has remainedconstant. Real data would show some variability, of course, but we are usingan idealized data set to simplify the problem. Gas production has also remainedconstant, and there has been no change in the gas-oil ratio. This suggests thatthe reservoir is undersaturated; that is, reservoir pressure is above bubble pointpressure. Only one hydrocarbon phase - the liquid phase - is produced atreservoir conditions from an undersaturated reservoir. The fact that GOR doesnot change over the life of the field is interpreted to mean that the reservoir wasundersaturated at initial conditions.TEAM LinG - Live, Informative, Non-cost and Genuine!Part III: Case Studv 201Table 20-2bProduction HistoryTIMEDAYS51324416691122153183214245274305336365AVGRESPRESSUREPSIA392939153906390138993898389738973897389638953895389438933892CUM PRODOILMSTB3612203346617791107122137152168183GASMMSCF13591521283542495663707783WATERMSTB00000000000001120.4 Drill Stem TestWell P-l logs and cores showed the presence of two major sands. A drillstem test (DST) was subsequently run in each major sand. Basic facts from theDST are summarized in Table 20-3.Table 20-3Summary of Well P-l DST ResultsWellbore RadiusWellbore SkinInitial PressureNo-Flow Boundary0.25 ft-0.53936 psia at 9360 ftWithin 700 ftTEAM LinG - Live, Informative, Non-cost and Genuine!202 Principles of Applied Reservoir SimulationPermeability was estimated from the DST data for each sand. The results,together with average water saturation (5W) values and calculated oil saturation(S0) values, are presented in Table 20-4 for both major sands.Table 20-4Saturation and Permeability Values for Each Major Sand20.4.1Sand!2s»00.3.3S. = 1-SW0.70.7Permeability (nn75250d)DST Radius of InvestigationThe radius of investigation for a DST can be estimated at various shut-intimes by using the formular. - 0.029where K is permeability in md,Nf<i>CAr»CT<j> is fractional porosity, |l is viscosity in cp, CTis total compressibility in 1/psia, and Ads shut-in time in hours. The followingphysical properties apply to the case study DST:K<t>M-CTpermeabilityporosityviscositytotalcompressibility250 md0.2280.71 cp13 x lO^psia1Substituting values for the physical parameters givesr. =0.029N0.02y^Kkt4>Hcr250 x Af.228 x 0.71 x 13 x 1Q~6= 316Table 20-5 shows values of ri for shut-in times of 0.25 day, 0.5 day, and 1 day.TEAM LinG - Live, Informative, Non-cost and Genuine!Part III: Case Study 203Table 20-5Estimating the Radius of InvestigationShut-in timedays0.250.501.00hrs61224Radius of Investigation[ft]77011001550The DST showed that a no-flow boundary exists within approximately 700 ftof production well P-l.20.5 Fluid PropertiesIn addition to pressure, flow capacity, and boundary information, the DSTwas used to acquire a fluid sample. Table 20-6 presents fluid properties froma laboratory analysis of the DST fluid sample.Table 20-6Fluid PropertiesPressurepsia14.7514.71014.71514.72014,72514.73014.74014.75014.76014.7OilViscp1.040.9100.8300.7650.6950.6410.5940.5100.4500.410FVFRB/STB1.061.2071.2951.3651.4351.5001.5501.6001.6201.630RsoSCF/STB1150280390480550620690730760GasViscp00.01120.01400.01650.01890.02080.02280.02600.02850.0300FVFRCF/SCF0.93580.03520.01800.01200.00910.00740.00630.00490.00400.0034WaterViscp0.50.50050.50100.50150.50200.50250.50300.50400.50500.5060FVFRB/STB1.0191.01751.01601.01451.01301.01151.01001.00701.00401.0010TEAM LinG - Live, Informative, Non-cost and Genuine!204 Principles of Applied Reservoir SimulationInitial reservoir pressure was estimated from the DST to be 3936 psia ata depth of 9360 ft below sea level. This pressure is over 1400 psia greater thanthe laboratory measured bubble point pressure of 2514 psia. Table 20-6 presentsfluid properties for undersaturated oil that must be corrected for use in a reservoi rsimulator,20.5.1 Black Oil PVT CorrectionFluid properties for the oil phase are shown in Table 20-7a.Table 20-7aCorrected Oil Phase PropertiesPressure(psia)14.7514.71014.71514.72014.72514.73014.74014.75014.76014.7Oil Vis(cp)1.0400.9100.8300.7650.6950.6410.5940.5100.4500.410OUFVF(KB/STB)1.0621.2071.2951.3651.4351.5001.5501.6001.6201.630OilRso(SCF/STB)1150280390480550620690730760GasVis(cp)0.00800.01120.01400.01650.01890.02080.02280.02600.02850.0300GasFVF(RCF/SCF)0.93580.03520.01800.01200.00910.00740.00630.00490.00400.0034WaterVis(cp)0.50000.50050.50100.50150.50200.50250.50300.50400.50500.5060WaterFVF(RB/STB)1.01901.01751.01601.01451.01301.01151.01001.00701.00401.0010The corrections for adjusting laboratory-measured differential liberationand separator data to a form suitable for use in a black oil simulator are givenby the conversion equations [Moses, 1986]:BRso(P) = Rsofbp ~ [Rsodbp - R*od(P)\ ~-"odbpwhere B0 is the oil formation volume factor and R^ is the solution gas-oil ratio.The subscripts are defined as d = differential liberation data;/= flash data; andbp = bubble point. For the case study, laboratory measurements include a flashTEAM LinG - Live, Informative, Non-cost and Genuine!Part HI: Case Study 205from 6000 psig to 0 psig. The separator test conditions and results are presentedin Table 20-7b.Table 20-7bSeparator Test (Flash)Sep.P(psig)100I0GOR(SCF/STB)57278Total GOR = 650FVF(RB/STB)1.5The values needed to perform the differential to flash conversion are thefollowing:Bojbp"odbp1.50 RB/STB1.63 RB/STBR-sodbp™sojbp760 SCF/STB650 SCF/STBThe corresponding correction factors areB (D) = B J(D) x —'-— = B ,(D) x 0.920W odW} j^ 0WRso(p) = 650 - [760 - Rsod(p)} x 0.92Applying these corrections to the differential data yields the corrected resultsshown in Table 20-7c.Table 20-7cCorrected Oil-Phase PropertiesPressure(psia)14.7514.71014.71514.72014.72514.7Oil FVF(RB/STB)1.0621.1101.1911.2561.3201.380OURso(SCF/STB)189208310392457TEAM LinG - Live, Informative, Non-cost and Genuine!206 Principles of Applied Reservoir SimulationTable 20-7cCorrected Oil-Phase PropertiesPressure(psia)3014.74014.75014.76014.7Oil FVF(RB/STB)1.4261.4721.4901.500Oil Rso(SCF/STB)52158662265020.5.2 Undersaturated Oil PropertiesSlopes for undersaturated oil properties are calculated from Table 20-The slopes are discussed in Chapter 24.6.Table 20-8Undersaturated Oil PropertiesPressure(psia)25153935Corrected Bopb(RB/STB)1.38001.3473Ho(cp)0.6410.706RemarksBubble PointUndersaturated ValuesThe rate of change of oil FVF with respect to pressure for the under-saturated oil is approximated by the differenceA B0 1.3473-1.3800 n _ RB/STB« - U.A/> 3935-2515 psiaThis linear approximation is reasonable in many cases. Nonlinear, undersaturatedslopes can be modeled by some simulators.Similarly, the rate of change of oil viscosity with respect to pressure forthe undersaturated oil isA 0.706-0.641AP 3935-2515 psiaThe rate of change of solution GOR (Rso) is zero in the pressure regime abovethe bubble point pressure.TEAM LinG - Live, Informative, Non-cost and Genuine!Part III: Case Studv 20720.6 Reservoir Management ConstraintsReservoir management constraints are presented in Table 20-9, Theyinclude limits on capital expenditures, such as the number of wells that can bedrilled, and operating constraints. In this case, for example, it is consideredimportant to keep water-oil ratio (WOR) less than five STB water/STB oil. Theseconstraints are typically imposed by considerations ranging from technical tocommercial. The constraints are especially important in the prediction phase ofthe study.Table 20-9Reservoir Management Constraints4 One additional well may be drilled.4 Completion interval in existing well may be changed.0 The well is presently completed in entire pay interval.4 Target oil rate ~ 1000 STB/D4 Water is available for injection if desired.4 Limit WOR < 54 Minimum allowed BMP = 2600 psia4 Maximum allowed injection pressure = 5000 psia4 Minimum economic oil rate =100 STB/DExercisesExercise 20.1 Plot FVF, viscosity, and solution GOR versus pressure for satu-rated and undersaturated oil.Exercise 20.2 Verify that the PVT values are properly entered in data set CS-MB.DAT. What is the bubble point pressure in the model?TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 21Model InitializationThe initial reservoir fluid saturation and pressure distributions are basedon data presented in this chapter. These distributions provide a volumetricestimate of initial fluids in place that can be compared with basic reservoiranalysis of field performance. The basic reservoir analysis described belowincludes a geologic estimate of volumetrics and a material balance determinationof initial fluids in place.21.1 VolumetricsA volumetric estimate of oil volume is a useful number for checking theaccuracy of the numerical representation of the reservoir in a reservoir model.The volume of oil in the reservoir may be estimated as the product of bulkvolume Fj, porosity <j>, and oil saturation S0.The bulk volume of the reservoir is estimated by writing bulk volume VBas the product A* A_yAz where A*, Ay, and Az approximate the length, width,and net thickness of the pay interval, respectively.4 From maps: A* = 2000' and A>> = 1200' for an area ~ 55 acres4 From well logs: Az = 72' + 961 = 168'The resulting estimate of bulk volume VB is 4.03 x 108 ft3.Pore Volume VP is the product §VB. Porosity (j) is estimated as thethickness weighted average porosity from well logs:* . 72 x Q.20 + 96 x Q.25 m168208TEAM LinG - Live, Informative, Non-cost and Genuine!Part HI: Case Study 209Taking the product of porosity and bulk volume gives the following estimateof pore volume:Vp = $VB * 9.18 x io7ft3 * 16.4 x W6RBThe product of oil saturation and pore volume gives an estimate of oil volumein reservoir barrels. Dividing this volume by an average oil formation volumefactor B0 for the reservoir gives an estimate of oil volume in stock tank barrels.The value of oil FVF at an initial average reservoir pressure of 3935 psia is1.3473 RB/STB. This value is obtained from laboratory data that has beencorrected for use in a reservoir simulator (Chapter 20.5). The resulting oilvolume isSVP 0.7Vp 11 5 x 106RBV = -2_£ « £ ~ -ii£ iu *** ~ 8.5 x 106STB0 5 B. 1.3473 RB/STB21.2 Material BalanceVolumetrics provides one measure of the quality of a reservoir model,but it is based on information that does not change with time. Another estimateof original oil volume can be obtained from a material balance study if areasonable amount of production data is available, such as the historical datapresented in Chapter 20. At this point we have surmised that the reservoir wasinitially undersaturated, but it may not have aquifer support.The presence of a few barrels of water during the latter months of the firstyear of production indicates that mobile water is present, but its source isunknown. The volume of produced water is small enough to be water mobilizedby swelling as reservoir pressure declines, or it could be the first indication ofwater production from aquifer influx. Both of these scenarios can be assessedif we consider the possibilities of depletion with and without aquifer influx.We begin by deriving the material balance equation for the more generalcase: depletion of an undersaturated oil reservoir with water influx. Thederivation is simplified by assuming formation compressibility is negligible andthen setting the decrease in oil volume at reservoir conditions equal to theTEAM LinG - Live, Informative, Non-cost and Genuine!210 Principles of Applied Reservoir Simulationincrease in water volume at reservoir conditions as oil is produced and reservoirpressure decreases.1 . Calculate the decrease in oil volume AF0 (RB):LetN = original oil in place = OOIP (STB)Boi = oil FVF (RB/STB) at initial pressure P(Af, = oil produced (STB) at pressure P and time tB0 = oil FVF (RB/STB) at pressure P and time tThenNBoi = OOIP (RB) at initial reservoir pressure Pf(N - Np) B0 = OIP (RB) at pressure P and time tTherefore the change in oil volume is given byAF0 = NBoi- (N-Np)Bo2, Calculate the increase in water volume AFW (RB):LetW = original water in place = OWIP (RB) at initial pressure PfBw = water FVF (RB/STB) at pressure P and time tWp = water produced (STB) at pressure P and time /We = water influx (RB)ThenW Bw = cumulative water produced (RB) at pressure P and time /Therefore the change in water volume is given byW- WB - W = W - WB3 . The assumption that the volume of the reservoir remains constant implies A V0- A Vw. Combining results from steps 1 and 2 above gives the material balanceequation for depletion of an incompressible, undersaturated oil reservoir withaquifer influx:TEAM LinG - Live, Informative, Non-cost and Genuine!Part III: Case Study 2! 1NBoi - (N - Np)Bo = We - WpBwThe two unknowns in the equation are N and We.The simplest production scenario is to assume that water influx isnegligible, that is, We - 0. If we further observe that water production Wp is in-significant, we haveN BN =pwhere Boi = 1.3473 RB/STB at P( = 3935 psia. Oil FVF has been corrected foruse in this calculation (see Chapter 20 for details). The corresponding estimatefor OOIP is N ~ 1500 Np with B0 - Boi ~ 0.0009 RB/STB. The results of thecalculation are presented in Table 21-1.Table 21-1Results Assuming No Water InfluxTime(days)91183274365Pressure(psia)3898389738953892B.(RB/STB)1.34821.34821.34821.3483N,(MSTB)4691137183N(MMSTB)69136205274The value of/^increases at each time. This implies that the material balancemodel does not account for all of the pressure support and suggests that anaquifer influx model should be considered.If we use a volumetric estimate ofN, namely Nvol = 8.5 MMSTB fromChapter 21.1, we can calculate We. Again recognizing that Wp ~ 0, the materialbalance equation becomesW€ = N(Boi - Bo) + NpB0Results of the calculation are shown in Table 21-2.TEAM LinG - Live, Informative, Non-cost and Genuine!212 Principles of Applied Reservoir SimulationTable 21-2Results Assuming Water Influx with Volumetric OOIPTime(days)90180270365Pressure(psia)3898389738953892B.(RB/STB)1.34821.34821.34821.3483^(MSTB)4691137183w.(MMSTB)54 (52)115(113)177(174)239 (234)Notice that We increases as a function of time. The values in parentheses areWINB4D values when the correct aquifer model is used.21.3 Relative PermeabilityAs we continue our preparation of a three-dimensional simulation model,we observe that not all of the data needed by the simulator is available. Sincewe cannot ignore data and still perform a credible model study, we mustcomplete the data set. Several options are available, such as ordering additionalmeasurements or finding reasonable correlations or analogies for the missingdata. In this case, our commercial interests are best served by moving the projectforward without additional expense or delays.We do not have laboratory-measured relative permeability data. We couldattempt to construct relative permeability data from production data, but ourproduction history is essentially single-phase oil. Since we must specify relativepermeability to run the model, we can turn to analogous reservoirs or correlationsfor guidance. Let us choose the Honarpour, et al. [ 1982] correlation for a water-wet sandstone as a starting point for determining relative permeability curves.Well logs provide some information about saturation end points such as initialand irreducible water saturation. Core floods and capillary pressure measure-ments could provide information about residual hydrocarbon saturations, butthey are not available. For that reason, end points like residual oil saturation mustbe estimated. Results of the calculation are shown in WINB4D format (Chapter24.5) in Table 21-3 and Figure 21-1. The acronyms in Table 21-3 are definedas follows:TEAM LinG - Live, Informative, Non-cost and Genuine!Part HI: Case Studv 213RELATIVE PEIWEABtLITYRelative Permeability vs Saturation0.2O.2 0.4 0.6 0.8Saturation-E3- Krow vs So ~*-KigvsSgFigure 21-1. Correlation based on Honarpour, et al. [1982].4 SAT is the saturation associated with each phase4 KROW is the relative permeability of oil in the presence of waterexpressed as a function of oil saturation4 KRW is the relative permeability of water in a water-oil systemexpressed as a function of water saturation4 KRG is the relative permeability of gas in a gas-oil system ex-pressed as a function of gas saturation4 KROG is the relative permeability of oil in the presence of gasexpressed as a function of liquid saturationTable 21-3Relative PermeabilitySAT0.0000.0300.0500.1000.1500.200KROW0.0000.0000.0000.0000.0000.000KRW0.0000.0000.0000.0000.0000.000KRG0.0000.0000.0200.0900.1600.240KROG0.0000.0000.0000.0000.0000.000TEAM LinG - Live, Informative, Non-cost and Genuine!214 Principles of Applied Reservoir SimulationTable 21-3 (cont.)Relative PermeabilitySAT0.2500.3000.3500.4000.4500.5000.5500.6000.6500.7000.8000.9001.000KROW0.0000.0000.0010.0100.0300.0800.1800.3200.5901.0001.0001.0001.000KRW0.0000.0000.0050.0100.0170.0230.0340.0450.0640.0830.1200.1200.120KRG0.3300.4300.5500.6700.8101.0001.0001.0001.0001.0001.0001.0001.000KROG0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.000If our choice of relative permeability correlations does not match field perfor-mance, we will have to change the relative permeability curves. In any event,we recognize that in this case study relative permeability is poorly known andshould be considered uncertain.21.4 Fluid ContactsA water-oil contact (WOC) was not seen on either well logs or seismicdata. The production of a small amount of water suggests that there may be aWOC in the vicinity of the reservoir. The data are not compelling, however. Wecould assume the oil zone extends well below the bottom depth of our well, butthis would be an optimistic assumption that could prove to be economicallydisastrous. In the interest of protecting our investment, let us make the moreconservative assumption that a WOC does exist and is just beyond the range ofour observations, namely well log and seismic data. We assume WOC ~ 9600ft, which is near the bottom of the seismically observed reservoir structure. Thepressure at this WOC depth is estimated to be about 4000 psia.TEAM LinG - Live, Informative, Non-cost and Genuine!Part III: Case Study 21521.5 Grid PreparationFigure 21-2 is a sketch of the well location relative to the interpretedreservoir boundaries. Based on seismic data shown in Chapter 20.2, the reservoiris thought to be bounded to the east by a facies change.Figure 21-2. Plan view,A cross-section through points B and B' is shown in Figure 21-3. Thesides of the reservoir appear to be bounded by faults. Without evidence to thecontrary, we assume that the faults are sealing. This assumption is subject toverification during the history match phase of the study.BFigure 21-3. BB' cross-section.A cross-section through points A and A' is sketched in Figure 21-4. Itillustrates the dip of the reservoir and the layering. The structure of the reservoirTEAM LinG - Live, Informative, Non-cost and Genuine!216 Principles of Applied Reservoir Simulationis based on well log and seismic interpretation. The downdip fault is speculative.It is based on the assumption that the fault shown on the western side of Figure21-2 extends down through the formation. This is not obvious from seismic data.Indeed, if the reservoir is receiving aquifer support, the aquifer influx will comefrom downdip as the reservoir is depleted. Bear in mind, however, that both thefault and the aquifer may be present. This could happen, for example, if the faultis not sealing. The fault could be providing a flow path for water influx fromanother horizon.Figure 21-4. AA' cross-section.ExercisesExercise 21.1 Verify the calculations reported in Tables 21-1 and 21-2.Exercise 21.2 Data file CS-MB.DAT is an input file for a material balanceanalysis of the case study. It represents the reservoir as a single gridblock, or"tank" model. The tank model is equivalent to a material balance calculation.Run WINB4D with the file CS-MB.DAT. Verify that the original volume of oilin the model agrees with the volumetric estimate presented in Chapter 21.1.Exercise 21.3 Use data file CS-MB.DAT to study the effect of aquifer influxon material balance performance. This is done by modifying the input data setto include an aquifer model, then adjusting aquifer parameters until model poreTEAM LinG - Live, Informative, Non-cost and Genuine!Part III: Case Study 217volume weighted average reservoir pressures match the pressures reported inChapter 20. Note: the pore volume weighted average reservoir pressure Pav isgiven byp -where N is the total number of gridblocks in the model grid, P, is the oil phasepressure in gridblock/, and Vpj is the pore volume of gridblock/. Chapter 24. 1 0contains details on how to set up an analytic aquifer. For an example of a dataset with an analytic aquifer model, see data file EXAM9.DAT.Exercise 21.4 Data set CS-VC.DAT is a vertical column model of the case study.Sketch the grid to scale, locate the contacts on the sketch, and match reservoirpressure.Exercise 21.5 Repeat Exercise 2 1 .4 beginning with the cross-section model dataset CS-XS.DAT.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 22History Matching and PredictionsThe history match is now well under way. The models discussed in theexercises of Chapter 21 are conceptual models designed to provide you with asense of how fluids move in the reservoir. This is the art of modeling. As youwork with various models of the reservoir, you should begin to develop aknowledge base for determining how changes to model parameters will helpachieve a match for a particular observable. This knowledge base is valuableas you develop your feel for the study.The previous chapters set the stage for preparing a 3D model of the casestudy reservoir. A 3D model should provide enough reservoir definition to letus make meaningful performance predictions. Before matching the 3D model,we discuss how to incorporate well information into the model. Once the wellmodel has been prepared, we proceed to history matching and performancepredictions.22.1 Well Model PreparationWell model calculations require estimates of productivity index andflowing bottomhole pressure. These calculations are illustrated here.Productivity Index EstimateWell model calculations in WINB4D need to have the quasi-stationaryproductivity index factor (PID) specified by the user. PID is estimated from theexpression (Chapter 30)218TEAM LinG - Live, Informative, Non-cost and Genuine!Part HI: Case Stud 2190.00708 £wherere = drainage radius (ft)rw = wellbore radius (ft)S = skinKe = kroKabs ~ effective permeability (md)^«« = net thickness (ft)Given S = -0.5, rw = 0.25 ft andro - 0.14(Ax2 + A^2)'/2 - 40ftwith A* = A.y = 200 ft., we findP/D = 1.55 x 10-3 Kabshnetwhere re ** r0. Table 22- 1 presents the calculation of PID for each layer identifiedby well log analysis.Table 22-1Estimate of PID by LayerLayer1234Kal* [md]750250250hn«m72206432PID8.4024.812.4Oil Well FBHP EstimateThe production well model needs a flowing bottomhole pressure (FBHP).Assuming an oil column in the wellbore, we can prepare a quick estimate ofFBHP for a single-phase oil well that is completed at a 9500 ft depth byassuming FBHP ~ oil head. Consequently, oil head is approximated byY0Az ~ FBHPTEAM LinG - Live, Informative, Non-cost and Genuine!220 Principles of Applied Reservoir Simulationwhere J0 is the oil pressure gradient and Az is the height of the oil column. Anestimate of average oil pressure gradient for the oil column is found by averagingthe pressure gradient at surface and reservoir conditions:4 Approximate pressure gradient at surface conditions:p = 46.244 A ~~ 0.321ft3 ftwhere oil density at surface conditions (ps) is 46.244 Ibm/SCF.Approximate pressure gradient at reservoir conditions:p = £i * 34.3 A « 0.238 £™#0 ft3 ftwhere oil FVF (B0) at bottomhole conditions is 1.3482 RB/STB.The resulting FBHP for use in WINB4D isFBHP = l/20.321 . -, 0.238ft ftx 9500ft « 2660psiaA more accurate estimate can be obtained from wellbore correlations or nodalanalysis as discussed by such authors as Brown and Lea [1985].Well Block Pressure from PBUIn Chapter 1 7 we saw that a pressure correction was needed to properlyrelate the pressure buildup (PBU) curve to simulator well block pressures. Toillustrate this correction, suppose a well is in a block with grid dimensions Ax= 200 ft and Ajy = 200 ft. We want to compare the simulator well block pressurewith a pressure from a PBU. Peaceman [1978, 1983] showed that shut-inpressure Pws of the actual well should equal the simulator well block pressureP0 at a shut-in time A/, given byKTEAM LinG - Live, Informative, Non-cost and Genuine!Part HI: Case Study 22!For an isotropic reservoir in which horizontal permeability does not depend ondirection, that is, Kx = Ky, we estimate the equivalent radius of a well in thecenter of a gridblock asro « 0.14(A*2 + Ay2)!/2The shut in time Af, at which the PBU pressure should be obtained is calculatedfrom the following physical parameters:crC0cwS0Sw^0d>K3x lO^psia"113 x lO^psia13 x lO^psia10.70.30.71 cp0.2075 mdThe equivalent radius of the well block is estimated to be r0 ~ 0.14 (2002+ 2002)'/2 = 39.6 ft, while the total compressibility is given by CT = cr + S0 c0 +Swcw = 3 x 10'6 + 0.7 (13 x 10'6) + 0.3 (3 x ]0'6) * 13 x 10'6psia'1. The PBU shutin time corresponding to these values isAf - 1688 (0-20) (0.71) (13 x IQ-«) (39.6)275= 0.065 hr. « 4 minutesThis early time part of the PBU curve could be masked by wellbore storageeffects. Since the shut in pressure Pws of the actual well equals the simulator wellblock pressure P0 at a shut in time A^, the shut in pressure Pws may have to beobtained by extrapolation of the radial flow curve.Throughput EstimateModel timestep size is estimated by calculating pore volume throughputfrom well flow rates. In our case, pore volume throughput is given byTEAM LinG - Live, Informative, Non-cost and Genuine!222 Principles of Applied Reservoir SimulationVPT = - (5.6146)VP = <|) AJC AJF Az = pore volume (ft3)Q = volumetric flow rate at reservoir conditions (RB/day)Af = timestep size (day)Timesteps for an IMPES simulator should correspond to about 1 0% throughputor less. The maximum timestep is estimated as follows.Suppose (]> = 22.5%, A* = A>> = 200', Az = hnet, and Q = 400 RB/day.Then A? is found by setting VPT = 0.10 and rearranging the pore volumethroughput equation to give5.6146Q 5.61460= QAhnet (days)If hnet = 100 ft, then A? « 40 days is an estimate of the maximum IMPEStimestep size.22.2 Full Field (3D) Model History MatchData file CS-HM.DAT is the three-dimensional model used to preparethe production history presented in Chapter 20. The grid in Figure 22- 1 was usedFigure 22-1. Plan view of grid.TEAM LinG - Live, Informative, Non-cost and Genuine!Part III: Case Study 223to model the reservoir shown in Figure 21-2. Each gridblock is a square withlengths A* = A>> = 200 ft. The dark areas of the grid are outside the reservoirarea. The pore volume in the dark area is made inactive in data file CS-HM.DATby using porosity multipliers.The depth and thickness of each gridblock depend on reservoir architec-ture. The model grid should approximate the structure depicted in Figure 21 -4,which is based on Figures 20-1 and 20-2. The dip of the reservoir is includedby specifying the tops of each gridblock. The gridblock length modificationsare designed to cut off those parts of the block that continue the grid beyond thesurface of the unconformity sketched in Figure 21-4.Transmissibility multipliers in the vertical direction are set to 0 to simulateimpermeable shale barriers. This includes the shale streak that divides the secondmajor sand into two thinner sands with a shale break. The interpretation ofseismic data was unable to resolve this feature, but the well log shown in Figure22-2 does indicate the presence of a shale streak.Figure 22-2. Overlay of seismic and well logdata.The water-oil contact is at 9600 ft. A steady-state aquifer is in communica-tion with all three oil layers at this depth. It is the source of water productionshown in Table 20-2.22.3 PredictionsNow that we have a history match model, we are ready to make predic-tions. The first step is to establish a base case prediction which assumes thereTEAM LinG - Live, Informative, Non-cost and Genuine!224 Principles of Applied Reservoir Simulationwill be no changes in operating strategy. Given a base case prediction, severalruns should be made to optimize reservoir performance within the constraintsimposed by the commissioners of the study. If the model is run with well P-lswitched from oil rate to bottom hole pressure control, the PI for well P-1 needsto be calibrated to assure continuity in the oil rate. The following exercises aredesigned to guide you through the prediction process.ExercisesExercise 22.1 Repeat the shut in time calculation using AJC = 1000 ft and Ay= 1000 ft. The new shut in time Af5 should be less than one hour.Exercise 22.2 Run data set CS-XS.DAT with maximum timestep sizes rangingfrom 15 days to 60 days. Select a maximum timestep size by monitoring thematerial balance error and the stability of the solution. A solution is unstableif it oscillates, that is, variables like GOR or WOR vary between a high and lowvalue from one timestep to the next.Exercise 22.3 What is the effect of doubling the PID in data set CS-XS.DAT?Exercise 22.4 How does model performance change if skin 5 = 0?Exercise 22.5 What is the effect of reducing the well FBHP by 1000 psia? Thereduction in FBHP is one way to simulate gas lift or pumping.Exercise 22.6 Data set CS-HM.DAT was used as the basis of the case study.Run data set CS-HM.DAT and verify that it matches the data shown in Table20-2.Exercise 22.7 Several sensitivity runs may be made by varying model parame-ters and noting reservoir performance. As an example of a sensitivity study, varythe WOC by ±100 ft. How does this variation affect water breakthrough and oilrecovery during the history match period?TEAM LinG - Live, Informative, Non-cost and Genuine!Part HI: Case Study 225Exercise 22.8 Run data set CS-HM.DAT for five years (four years into thefuture) with Well P-1 under oil rate control. This run establishes a base caseprediction.Exercise 22.9 Data set CS-PD.DAT is the base case prediction. Beginning withthis data set, maximize oil recovery given the constraints listed in Table 20-9.Two ideas to consider are downdip water injection after drilling an updipproducer; and downdip production after drilling an updip gas injector.TEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!Part IVWINB4D User's ManualTEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!Introduction to WINB4DWINB4D simulates isothermal, Darcy flow in up to three dimensions. Itassumes reservoir fluids can be described by up to three fluid phases (oil, gas,and water) with physical properties that depend on pressure only. Gas is allowedto dissolve in both the oil and water phases, A feature unique to WTNB4D is theinclusion of compressional velocity and acoustic impedance calculations. Thesereservoir geophysical calculations make it possible to track changes in seismicvariables as a function of time, which is the basis for 4D seismic analysis.WINB4D was designed to run on Windows-based personal computerswith 486 or better math co-processors. This size simulator is well-suited forlearning how to use a reservoir simulator, developing an understanding ofreservoir management concepts, and for solving many types of reservoirengineering problems. It is an inexpensive tool for performing studies that callfor more sophistication than is provided by analytical solutions, yet do notrequire the use of full-featured commercial simulators.WINB4D is a modified version of the black oil simulator BOAST II thatwas published by the U.S. Department of Energy in 1987 [Fanchi, et al, 1987].BOAST II was an improved version of BOAST, an implicit pressure-explicitsaturation (IMPES) simulator published by the U.S. Department of Energy in1982 [Fanchi, et al., 1982]. There have been several modifications of BOASTII published by the Bartlesville Project Office of the U.S. Department of Energy.WINB4D is based on BOAST II.A comparison of differences between BOAST II and WINB4D is givenin Tables 23-1 and 23-2.The first table shows that a variety of useful geophysical229TEAM LinG - Live, Informative, Non-cost and Genuine!230 Principles of Applied Reservoir Simulationand reservoir engineering features have been added to WINB4D, including theability to perform material balance studies with a tank model, the representationof horizontal or deviated wells, and the calculation of important reservoirgeophysical information.Table 23-1Comparison of Reservoir Modeling DifferencesFEATUREMaterial balance tankmodel (1 gridblock)Well completionsHorizontal wellSlanted wellCompressional velocityShear velocityAcoustic impedanceReflection coefficientModify (|>, KModify transmissibilitySaturation initializationBOAST IINot availableVertically contiguousNot availableNot availableNot availableNot availableNot availableNot availableInput <J>, KInput transmissibilityUser specifiedWINB4DNewFlexible - may skiplayersNewNewNewNewNewNewAdded multiply byfactorMultiply trans, byfactorAdded gravitysegregated optionTable 23-3 presents WINB4D enhancements designed to improve com-putational performance. For example, a more accurate algorithm for interpolatinggas formation volume factor Bg.BOAST II has been tested under a wide range of conditions. Detailedcomparisons with other simulators were made for four types of problems: oiland gas depletion, waterflooding, gas injection with constant bubble pointpressure, and gas injection with variable bubble point pressure. Favorablecomparisons were observed with respect to oil rates, GORs, gas saturations, andpressures. The one exception is the reservoir pressure comparison for the variableTEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 231bubble point pressure case. In this case, BOAST II reservoir pressures wereconsistently lower than other simulator values. A mass conserving expansionof accumulation terms can improve the accuracy for variable bubble pointpressure problems [Fanchi, 1986], but the mass conserving expansion optionrequires additional run time and is not included in WINB4D.Table 23-2Comparison of Computational DifferencesFEATUREInterpolationSaturation table endpointsTimestepping andreportsDebug codesRestartStabilized IMPESBOAST IIBSSet to -0.1 and 1.1Counter and userspecifiedOptionalAvailableAvailableWINB4Dbg=\IB,Improves materialbalanceSet to 0.0 and 1.0Simplify to userspecified onlyDeletedDeleted - restart byspecifying arraysDeleted - not robustWINB4D retains the robustness of BOAST II while substantiallyincreasing program accuracy. WINB4D has an improved interpolation algorithmthat reduces material balance error for some problems by as much as a factorof ten relative to the DOE versions BOAST and BOAST II. This featureincreases the range of applicability of WINB4D and is especially valuable forgas and gas-oil systems. The algorithm does not degrade program speed.23.1 Program ConfigurationThe user needs to have at least 1 Megabyte RAM to run WINB4D. Theversion of WINB4D accompanying this book allows the user to define grids withup to 1000 gridblocks. Parameters that may be dimensioned by the user at thetime of the run are listed below.TEAM LinG - Live, Informative, Non-cost and Genuine!232 Principles of Applied Reservoir SimulationTable 23-3Standard Configuration of WINB4DDimensioning ParametersMaximum number of blocks in modelMaximum number of Rock regionsMaximum number of entries in a Rock region tableMaximum number of PVT regionsMaximum number of entries in a PVT region tableMaximum number of wellsMaximum number of connections per well1000330330255WINB4D must be copied to a folder on your hard drive before running.The following procedure is recommended for a CD drive D and hard drive Crunning Windows 95/98/NT:+ Open Windows Explorer and select your CD drive.4 Use a Windows-based Unzip program to extract all of the filesfrom the WINB4D file on the CD to a folder on your hard drive.4 Respond to questions.23.2 Input Data File - WTEMP.DATWINB4D reads a file called WTEMP.DAT and outputs to filesWTEMP.TSS, WTEMP.PLT, WTEMP.WEL, WTEMP.ROF, and WTEMP.-ARR. The output files are described in Chapter 26. You should rename any runsyou wish to save because WINB4D overwrites the WTEMP.* files when it isexecuted.The easiest way to prepare a new data file is to edit an old one. This willgive you an example of the formats needed for most options. If you start withan old data set, make sure that you check all applicable data entries and makechanges where appropriate.TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 23323.3 Data Input RequirementsWINB4D input data is divided into two parts: initialization data, andrecurrent data. Initialization data is described in Chapter 24. It includes data thatis set at the beginning of the study and is not expected to change during a modelran. Such data includes the reservoir description and fluid properties. Recurrentdata is described in Chapter 25 and refers to data that is expected to changeduring the course of a simulation. It includes well schedules and timestep controlinformation. Additional discussion of WINB4D is presented in Part V: TechnicalSupplement.Title or heading records are read before each major and many minorsections. These records are designed to make the input data file easier to readand edit.All input data, with the exception of well names, is entered as free formatdata. Two free format data entries must be separated by a comma or a space ifthey are entered on the same line.In many cases, codes are read that will specify the type of input to followand the number of values that will be read. These codes increase the efficiencyand flexibility of entering input data.Input tabular data should cover the entire range of values expected to occurin a simulation. Examples of tabular data include fluid property data entered asfunctions of pressure and relative permeability data entered as functions ofsaturation. The linear table interpolation algorithms in WINB4D will returntabulated endpoint values if the independent variable goes outside the range ofthe input tabular values. No message will be printed if this occurs.If an array of input values must be read, the following input order mustbe followed. Layer 1 (K = 1) is read first. The data in each layer are read byrows, starting with row 1 (J = 1). Values of the array element are read for thefirst row starting with column 1 (I = 1) and proceeding to the end of the row(column I = II). After II values are read, the next row (J = 2) of values areentered. These values must begin on a new line. This data entry procedure isrepeated for all rows and, subsequently, for all layers until the complete set ofarray elements has been entered.TEAM LinG - Live, Informative, Non-cost and Genuine!234 Principles of Applied Reservoir Simulation23.4 Example Input Data SetsSeveral example input data sets are included with the book. A few arelisted below.FILEEXAM 1. DATEXAM2.DATEXAM3.DATEXAM4.DATEXAM5.DATEXAM6.DATEXAM7.DATEXAM8.DATEXAM9.DATEXAM10.DATEXAM 11. DATGRIDII x J J x KKIxlxJ1x1x410 x I x 19x9x]10 x 1 x49x9x210 x 10x39x9x29x9x210 x 8 x410 x 1 x2MODELTYPEMaterialBalanceIDVerticalIDHorizontal2DAreal2D Cross-section3D3D3D3D3D2D Cross-sectionREMARKSPrimary depletion of an under-saturatedoil reservoir (high GOR)Primary depletion of an under-saturated oil reservoir (moderate GOR)Buckley-Leverett waterfloodPrimary depletion of an undersaturatedoil reservoir (high GOR)Multi-layer waterflood of anundersaturated oil reservoir (high GOR)5-spot waterflood of an under-saturated oil reservoir (high GOR)Gas injection into undersaturated oilreservoir (high GOR) - OdehexampleDepletion of gas reservoirDepletion of gas reservoir with aquifersupportDepletion of a faulted oil reservoir withmultiple PVT and ROCKregionsDepletion of gas reservoir withaquifer supportExample Input Data SetThe following data set is presented to illustrate the WINB4D input fileformat. Additional spacing has been provided between some lines to improvedata set readability. The actual WINB4D data set should contain no blank linesbetween records.TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 235BEGINNING OF DATA SET —PRIMARY DEPLETION OF AN OIL RESERVOIR - VERTICAL COLUMN MODELGRID DIMENSIONS1, 1, 4,3,3,30,10,10Gridblock LENGTHS-I -1 002000.01200.02*50.0 2*60.02*36.0 2*38.0Gridblock LENGTH MODIFICATIONS0, 0, 0, 0, 0DEPTH TO TOP OF UPPER SAND29330938094309490MODULI AND ROCK DENSITY-1 -1 -1-13E63E63E6168MODULI AND ROCK DENSITY MODIFICATIONS00000POROSITY AND PERMEABILITY DISTRIBUTIONS00002*0.20 2*0.252*75 2*2502*75 2*2502*7.5 2*25POROSITY AND PERMEABILITY MODIFICATION CARDS0, 0, 0, 0, 0TRANSMISSIBILITY MOD. - NO FLOW BETWEEN LAYERS 2 AND 30, 0, 1, 0111133 0.0ROCK AND PVT REGIONS1, !TEAM LinG - Live, Informative, Non-cost and Genuine!236 Principles of Applied Reservoir SimulationSAT KRO KRW KRG KROG PCOW PCGO0,00 0.00 0.00 0.00 0.0 0.0 0.00,03 0.00 0.00 0.00 0.0 0.0 0.00.05 0.00 0.00 0.02 0.0 0.0 0.00.10 0.00 0.00 0.09 0.0 0.0 0.00.15 0.00 0.00 0.16 0.0 0.0 0.00.20 0.00 0.00 0.24 0.0 0.0 0.00,25 0.00 0.00 0.33 0.0 0.0 0.00.30 0,0001 0.00 0.43 0.0 0.0 0.00.35 0.001 0.005 0.55 0.0 0.0 0.00.40 0.01 0.010 0.67 0.0 0.0 0.00.45 0.03 0.017 0.81 0.0 0.0 0.00.50 0.08 0.023 1.00 0.0 0.0 0.00.55 0.18 0.034 1.00 0.0 0.0 0.00.60 0.32 0.045 1.00 0.0 0.0 0.00.65 0.59 0.064 1.00 0.0 0.0 0.00.70 1.00 0.083 1.00 0.0 0.0 0.00.80 1.00 0,12 1.00 0.0 0.0 0.00.90 LOO 0.12 1.00 0.0 0.0 0.01.00 1.00 0.12 LOO 0.0 0.0 0.0ITHREE SW(IRR.)0, 0.30PBO PBODAT PBGRAD2514.7, 9200.0, 0.0VSLOPE BSLOPE RSLOPE PMAX REPRS0.000046, -0.000023, 0.0, 6014.7, 0OIL:P14.7514.7,1014.7,1514.7,2014.7,2514.7,3014.7,4014.7,5014.7,6014.7,MUO1.04000.9100,0.8300,0.7650,0.6950,0.6410,0.5940,0.5100,0.4500,0.4100,BO1.06201.1110,1.1920,1.2560,1.3200,1.3800,1.4260,1.4720,1.4900,1.5000,RSO1.089.0208.0309.0392.0457.0521.0586.0622.0650.0TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 237WATER: P MUW BW RSW14.7, 0.5000, 1.0190, 0.0514.7, 0.5005, 1.0175 0.01014.7, 0.5010, 1.0160, 0.01514.7, 0.5015, 1.0145, 0.02014.7, 0.5020, 1.0130, 0.02514.7, 0.5025, 1.0115, 0.03014.7, 0.5030, 1.0100, 0.04014.7, 0.5040, 1.0070, 0.05014.7, 0.5050, 1.0040, 0.06014.7, 0.5060, 1.0010, 0.0GAS AND ROCK PROPERTIES0P MUG BG PSI CR14.7, 0.008000, 0.935800, 0.0, 0.000003514.7, 0.011200, 0.035200, 0.0, 0.0000031014.7, 0.014000, 0.018000, 0.0, 0.0000031514.7, 0.016500, 0.012000, 0.0, 0.0000032014.7, 0.018900, 0.009100, 0.0, 0.0000032514.7, 0.020800, 0.007400, 0.0, 0.0000033014.7, 0.022800, 0.006300, 0.0, 0.0000034014.7, 0.026000, 0.004900, 0.0, 0.0000035014.7, 0.028500, 0.004000, 0.0, 0.0000036014.7, 0.030000, 0.003400, 0.0, 0.000003RHOSCO RHOSCW RHOSCG46.244, 62.238, 0.0647EQUILIBRIUM PRESSURE INIT. / CONSTANT SATURATION INIT.1, 1, 0, 04000, 9600, 0, 80000.70, 0, 0.25NMAX FACT1 FACT2 TMAX WORM AX GORMAX PAMIN PAMAX1000, 1.50, 0.50, 365, 5.0, 500000, 1500, 6000KSOL MITR OMEGA TOL TOL1 DSMAX DPMAX NUMDIS1, 100, 1.50, 0.1, 0.001, 0.05, 100.0, 1AQUIFER MODEL0RECURRENT DATA*** DATA SET 1 - HISTORY ***1, 491.25 182.5 273.75 365.01, 1, I, 0, 0, 10, 0, 0, 0, 05.0, 1.0, 10.0WELL INFORMATIONI 0TEAM LinG - Live, Informative, Non-cost and Genuine!238 Principles of Applied Reservoir SimulationWELL P-1P-lI 4I, 1, 1, 2.7 26001, 1, 2, 2.7 26001, 1, 3, 9.4 26001, 1, 4, 9.4 26001, 500.0, 0.0, 0.0, 0.0END OF DATA SETTEAM LinG - Live, Informative, Non-cost and Genuine!Initialization DataInitialization data records are read once at the beginning of the simulation.They must be read in the order presented below.1. Title Up to 80 characters; this record will appear as run title.24.1 Grid Dimensions and Geometry24.1.1 Grid Dimensions1. Heading Up to 80 characters.2. II, JJ, KK, IOROK, IOPVT, IOTBL, IONWL, IOCONCodeIIJJKKIOROKIOPVTMeaningnumber of gridblocks in the x directionnumber of gridblocks in they directionnumber of gridblocks in the z directionmaximum number of Rock regions (such as 3)maximum number of PVT regions (such as 3)239TEAM LinG - Live, Informative, Non-cost and Genuine!240 Principles of Applied Reservoir SimulationCodeIOTBLIONWLIOCONMeaningmaximum number of entries in PVT and Saturation tables(such as 30)maximum number of wells (such as 10)maximum number of connections per well (such as 10)3. Heading Up to 80 characters.4. KDX, KDY, KDZ, KDZNETKDX Control code for input of x direction grid size.KDY Control code for input of y direction grid size.KDZ Control code for input of z direction gross gridblock thick-nesses.KDZNET Control code for input of z direction net gridblock thick-nesses.CodeKDXKDYValue-101-10MeaningThe x direction grid dimensions are the same for allblocks in the grid. Read only one value.The x direction dimensions are read for each block inthe first row (J = 1) of layer one (K = 1). These samevalues are assigned to all other rows and all other layersin the model grid. Read II values.The x direction dimensions are read for each block inlayer one (K = 1 ). These same values are assigned to allother layers in the grid. Read II x JJ values.The y direction grid dimensions are the same for allblocks in the grid. Read only one value.The y direction dimensions are read for each block inthe first column (I = 1) of layer one (K = 1). Thesevalues are assigned to all other columns and all otherlayers in the model grid. Read JJ values.TEAM LinG - Live, Informative, Non-cost and Genuine!PartIV: WINB4D User's Manual 241CodeKDZKDZNETValue1-101-101MeaningThe j; direction dimensions are read for each block inlayer one (K = 1 ). These same values are assigned to allother layers in the grid. Read II x JJ values.The z direction gross thickness is the same for all blocksin the grid. Read only one value.A constant value of gross thickness is read for eachlayer in the grid; each layer may have a different, butconstant value. Read KK values.The z direction gross thickness is read for each block inthe grid. Read II x JJ x KK values.The z direction net thickness is the same for allblocks in the grid. Read only one value.A constant value of net thickness is read for eachlayer in the grid; each layer may have a different, butconstant value. Read KK values.The z direction net thickness is read for each block inthe grid. Read II x JJ x KK values.5. DXDX Gridblock size in x direction (ft).If KDX = -1, read one constant value.If KDX = 0, read II values (one for each row).If KDX = +1, read II x JJ values (one for each K = 1 block).6. DYDY Gridblock size in y direction (ft).If KDY = -1, read one constant value.If KDY = 0, read JJ values (one for each column).If KDY = +1, read II x JJ values (one for each K = 1 block).7. DZDZ Gross gridblock thickness in z direction (ft).If KDZ = -1, read one constant value.If KDZ = 0, read KK values (one for each layer).TEAM LinG - Live, Informative, Non-cost and Genuine!242 Principles of Applied Reservoir SimulationIf KDZ = +1, read II * JJ * KK values (one for each block).8. DZNETDZNET Net gridblock thickness in z direction (ft).If KDZ = -1, read one constant value.If KDZ = 0, read KK values (one for each layer).If KDZ = +1, read II x JJ x KK values (one for each block).24.1.2 Modifications to Grid Dimensions1. Heading Up to 80 characters.2. NUMDX, NUMDY, NUMDZ, NUMDZN, IDCODENUMDX Number of regions where x direction grid size (DX) ischanged.NUMDY Number of regions where y direction grid size (DY) ischanged.NUMDZ Number of regions where z direction gross thickness (DZ)is changed.NUMDZN Number of regions where z direction net thickness (DZN)is changed.IDCODE = 0 means do not print the modified distributions;= 1 means print the modified distributions.3 II, 12, Jl, J2, Kl, K2, DXOmit this record if NUMDX = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.Kl Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.DX New value of x direction grid size for region (ft).TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 243NOTE: NUMDX records must be read.4 I1,I2,J1,J2,K1,K2,DYOmit this record ifNUMDY = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction,K1 Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.DY New value of y direction grid size for region (ft).NOTE: NUMDY records must be read.5 I1,I2,J1,J2,K1,K2,DZOmit this record ifNUMDZ = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.KI Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.DZ New value of z direction gross thickness for region (ft).NOTE: NUMDZ records must be read.6 II, 12, Jl, J2, Kl, K2, DZNETOmit this record ifNUMDZN = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.Kl Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.DZNET New value of z direction net thickness for region (ft).TEAM LinG - Live, Informative, Non-cost and Genuine!244 Principles of Applied Reservoir SimulationNOTE: NUMDZN records must be read.24.1.3 Depths to Top of GridblocksThe coordinate system used in WINB4D is defined so that values in thez (vertical) direction increase as the layer gets deeper. Thus, depths must be readas depths below the user-selected reference datum. Negative values will be readas heights above the datum.1. Heading Up to 80 characters.2. KELKEL Control code for input of depth values.KEL0123MeaningA single constant value is read for the depth to the top of all grid-blocks in layer 1 (horizontal plane). Each layer is contiguous in thisoption. Depths to the top of gridblocks in layers below layer 1 arecalculated by adding the layer thickness to the preceding layer top;thus Top (I, J, K + 1) = Top (I, J, K) + DZ (I, J, K)A separate depth value must be read for each gridblock in layer 1 .Read II x JJ values. Each layer is contiguous in this option. Depthsto the top of gridblocks in layers below layer 1 are calculated byadding the layer thickness to the preceding layer top; thus Top (I, J,K + 1 ) = Top (I, J, K) + DZ (I, J, K)A separate depth value is read for each layer. Read KK values. Eachlayer is horizontal (layer cake) in this option.A separate depth value is read for each gridblock. Read II x JJ x KKvalues.3. ELEVELEV Depth to top of gridblock (ft).If KEL = 0, read one constant value.If KEL = 1, read II x JJ values (one for each block in layer 1).If KEL = 2, read KK values (one for each layer).If KEL = 3, read II x JJ x KK values (one for each block).TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 24524.2 Seismic Velocity Parameters24.2.1 Moduli and Grain Densities1, Heading Up to 80 characters.2 KKB, KKG, KMU, KRHOKKB Control code for input of the frame bulk modulus(evacuated porous rock).KKG Control code for input of the grain bulk modulus (solidmatrix material).KMU Control code for input of the shear modulus (evacuatedporous rock).KRHO Control code for input of the grain density (solid matrixmaterial).CodeKKBKKGValue-101-101MeaningFrame bulk moduli are the same for all blocks in the grid.Read only one value.A constant value of frame bulk modulus is read for eachlayer in the grid; each layer may have a different, butconstant value. Read KK values.Frame bulk moduli are read for each block in the grid.Read II x JJ x KK values.Grain bulk moduli are the same for all blocks in the grid.Read only one value.A constant value of grain bulk modulus is read for eachlayer in the grid; each layer may have a different, butconstant value. Read KK values.Grain bulk moduli are read for each block in the grid.Read II x JJ x KK values.TEAM LinG - Live, Informative, Non-cost and Genuine!246 Principles of Applied Reservoir SimulationCodeKMUKRHOValue-101-101MeaningShear moduli are the same for all blocks in the grid. Readonly one value.A constant value of shear modulus is read for each layerin the grid; each layer may have a different, but constantvalue. Read KK values.Shear moduli are read for each block in the grid. Read IIx JJ x KK values.Grain densities are the same for all blocks in the grid.Read only one value.A constant value of grain density is read for each layer inthe grid; each layer may have a different, but constantvalue. Read KK values.Grain densities are read for each block in the grid. ReadII xjjxKK values.3. KBKB Frame bulk modulus (psia).If KKB = -1, read one constant value.If KKB = 0, read KK values (one for each layer).If KKB = +1, read II x JJ x KK values (one for each block).NOTE: In the absence of relevant data, a value of 3 x 106psia is a reasonable estimate.4. KGKG Grain bulk modulus (psia).If KKG = -1, read one constant value.If KKG = 0, read JJ values (one for each layer).If KKG = +1, read II x JJ values (one for each block).NOTE: In the absence of relevant data, a value of 3 x 106psia is a reasonable estimate.5. MUMU Shear modulus (psia).TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 247If KMU = -1, read one constant value.If KMU = 0, read KK values (one for each layer),If KMU = +1, read II x Jj x KK values (one for each block).NOTE: In the absence of relevant data, a value of3 x 106 psia is a reasonable estimate.6 RHOMARHOMA Grain density (lbf/ft3).If KRHO = -1, read one constant value.If KRHO = 0, read KK values (one for each layer).If KRHO = +1, read II x JJ x KK values (one for each block).NOTE: In the absence of relevant data, a value of 168lbf/ft3 (corresponding to 2.7 g/cm3) is a reasonable esti-mate.24.2.2 Modifications to Moduli and Grain Densities1. Heading Up to 80 characters.2 NUMKB, NUMKG, NUMMU, NUMRHO, IDCODENUMKB Number of regions where frame bulk modulus (KB) ischanged.NUMKG Number of regions where grain bulk modulus (KG) ischanged.NUMMU Number of regions where shear modulus (MU) ischanged.NUMRHO Number of regions where grain density (RHO) ischanged.IDCODE = 0 means do not print the modified distributions;= 1 means print the modified distributions.3 II, 12, Jl, J2, Kl, K2, KBOmit this record if NUMKB = 0.TEAM LinG - Live, Informative, Non-cost and Genuine!248 Principles of Applied Reservoir Simulation11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.Kl Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.KB New value of frame bulk modulus (psia).NOTE: NUMKB records must be read.4 II, 12, Jl, J2, Kl, K2, KGOmit this record ifNUMKG = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.K1 Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.KG New value of grain bulk modulus (psia).NOTE: NUMKG records must be read.5 II, 12, Jl, J2, Kl, K2, MUOmit this record ifNUMMU - 0.11 Coordinate of first region block in I direction,12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.K1 Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.MU New value of shear modulus.NOTE: NUMMU records must be read.6 I1,I2,J1,J2,K1,K2, RHOOmit this record ifNUMRHO = 0.TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 24911 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction,K1 Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.RHO New value of grain density (lbf/ft3).NOTE: NUMRHO records must be read.24.3 Porosity, Permeability, and Transmissibility Distributions24.3.1 Porosity and Permeability1. Heading Up to 80 characters.2. KPH, KKX, KKY, KKZKPH Control code for input of porosity.KKX Control code for input of x direction permeability.KKY Control code for input of y direction permeability.KKZ Control code for input of z direction permeability.CodeKPHKKXValue-101-101MeaningThe porosity is constant for all gridblocks. Read only onevalue.A constant value is read for each layer in the grid. Read KKvalues.A value is read for each block in the grid. Read II x JJ x KKvalues.The x direction permeability is constant for all gridblocks.Read only one value.A constant value is read for each layer in the grid. Read KKvalues.A value is read for each block in the grid. Read II x JJ x KKvalues.TEAM LinG - Live, Informative, Non-cost and Genuine!250 Principles of Applied Reservoir SimulationCodeKKYKKZValue-10I-i01MeaningThe y direction permeability is constant for all gridblocks.Read only one value.A constant value is read for each layer in the grid. Read KKvalues.A value is read for each block in the grid. Read II x Jj x KKvalues.The z direction permeability is constant for all gridblocks.Read only one value.A constant value is read for each layer in the grid. Read KKvalues.A value is read for each block in the grid. Read II x Jj x KKvalues.3. PHIPHI Porosity (fraction).If KPH = -1, read one constant value.If KPH = 0, read KK values (one for each layer).If KPH = +1, read II x Jj x KK values (one for each block).4. PERMXPERMX Permeability in x direction (md).If KKX = -1, read one constant value.If KKX = 0, read KK values (one for each layer).If KKX = +1, read II x Jj x KK values (one for each block).5. PERMYPERMY Permeability in y direction (md).If KKY = -1, read one constant value.If KKY = 0, read KK values (one for each layer).If KKY = +1, read II x Jj x KK values (one for each block).TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 2516 PERMZPERMZ Permeability in z direction (md).If KKZ = -1, read one constant value.If KKZ = 0, read KK values (one for each layer).If KKZ = +1, read II x JJ x KK values (one for each block).24.3.2 Modifications to Porosities and Permeabilities1. Heading Up to 80 characters.2 NUMP, NUMKX, NUMKY, NUMKZ, IPCODENUMP Number of regions where porosity (PHI) is changed.NUMKX Number of regions where x direction permeability(PERMX) is changed.NUMKY Number of regions where y direction permeability(PERMY) is changed.NUMKZ Number of regions where z direction permeability(PERMZ) is changed.IPCODE = 0 means do not print the modified distributions;= 1 means print the modified distributions.3 II, 12, Jl, J2, Kl, K2, VALPHIOmit this record if NUMP = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.Kl Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.Code•\J1 TA/fDiNUMrValue<0>0MeaningNew value of porosity for region (fr).Multiply value of porosity by VALPHI.TEAM LinG - Live, Informative, Non-cost and Genuine!252 Principles of Applied Reservoir SimulationNOTE: NUMP records must be read where ... de-notes the absolute value.4. II, 12, Jl, J2, Kl, K2, VALKXOmit this record ifNUMKX = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.K1 Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.CodeNUMKXValue<0>0MeaningNew value of x direction permeability for region(md).Multiply value of x direction permeability byVALKX.NOTE: | NUMKX | records must be read.5. II, 12, Jl, J2, Kl, K2, VALKYOmit this record ifNUMKY = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.Jl Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.Kl Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.CodeNUMKYValue<0MeaningNew value ofy direction permeability for region(md).TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 253CodeValue>0MeaningMultiply value of y direction permeability byVALKY.NOTE: NUMKY records must be read.6. II, 12, Jl, J2, Kl, K2, VALKZOmit this record ifNUMKZ = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.Jl Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.Kl Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.CodeNUMKZValue<0>0MeaningNew value of z direction permeability for region(md).Multiply value of z direction permeability byVALKZ.NOTE: NUMKZ records must be read.24.3.3 Modifications to TransmissibilitiesIt is important to keep in mind the directional convention used inspecifying transmissibility modifications. For example, in gridblock (I, J, K):TX(I, J, K) refers to flow across the boundary between blocks 1-1 and I,TY(I, J, K) refers to flow across the boundary between blocks J-l and J, andTZ(I, J, K) refers to flow across the boundary between blocks K-l and K.1. Heading Up to 80 characters.TEAM LinG - Live, Informative, Non-cost and Genuine!254 Principles of Applied Reservoir Simulation2, NUMTX, NUMTY, NUMTZ, ITCODENUMTX Number of regions where x direction transmissibility (TX)is changed.NUMTY Number of regions where y direction transmissibility (TY)is changed.NUMTZ Number of regions where z direction transmissibility (TZ)is changed.ITCODE = 0 means do not print the modified distributions;= 1 means print the modified distributions.3 II, 12, Jl, J2, Kl, K2, VALTXOmit this record if NUMTX = 0.Coordinate of first region block in I direction.Coordinate of last region block in I direction.Coordinate of first region block in J direction.Coordinate of last region block in J direction.Coordinate of first region block in K direction.Coordinate of last region block in K direction.Multiplier of x direction transmissibility for region.NOTE: NUMTX records must be read.II, 12, Jl, J2, Kl, K2, VALTYOmit this record if NUMTY = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.Jl Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.Kl Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.VALTY Multiplier of y direction transmissibility for region.NOTE: NUMTY records must be read.TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 2555 II, 12, Jl, J2, Kl, K2, VALTZOmit this record ifNUMTZ = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction,J2 Coordinate of last region block in J direction,K1 Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.VALTZ Multiplier of z direction transmissibility for region.NOTE: NUMTZ records must be read.24.4 Rock and PVT Regions1. Heading Up to 80 characters.2. NROCK, NPVTNROCK Number of distinct Rock regions. A separate set of satu-ration-dependent data must be entered for each Rock region.NPVT Number of distinct PVT regions. A separate set of pressure-dependent data must be entered for each PVT region.3. Heading Up to 80 characters.Omit this record if NROCK = 1.4 NUMROKOmit this record if NROCK = 1.NUMROK = 0 Enter Rock region value for each block.NUMROK > 0 Number of regions where the Rock region defaultvalue of 1 is changed.5. IV ALOmit this record if NROCK = 1 or NUMROK > 0.IVAL Array of Rock region values. Read II x Jj x KK values.TEAM LinG - Live, Informative, Non-cost and Genuine!256 Principles of Applied Reservoir Simulation6. II, 12, Jl, J2, Kl, K2, IVALOmit this record ifNROCK = 1 or NUMROK = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.JI Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.Kl Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.IVAL Number of the saturation-dependent data set to be assignedto this Rock region and IVAL z NROCK.NOTE: NUMROK records must be read.7. Heading Up to 80 characters.Omit this record ifNPVT = 1.8. NUMPVTOmit this record ifNPVT = 1.NUMPVT = 0 Enter PVT region value for each block.NUMPVT > 0 Number of regions where the PVT region defaultvalue of 1 is changed.9. IVALOmit this record ifNPVT = 1 or NUMPVT > 0.IVAL Array of PVT region values. Read II x JJ x KK values.10. II, 12, Jl, J2, Kl, K2, IVALOmit this record ifNPVT = 1 or NUMPVT = 0.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.Kl Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 257IVAL Number of the pressure-dependent data set to be assignedto this PVT region and IVAL s NPVT.NOTE: NUMPVT records must be read.24.5 Relative Permeability and Capillary Pressure TablesThe following saturation-dependent data should be entered a total ofNROCK times - one set of records for each Rock region defined in Section 24,4.1, Heading Up to 80 characters.2 SAT1 KROW1 KRW1 KRG1 KROG1 PCOW1 PCGO1SAT Phase saturation (fr). Set SAT1 = 0.0 and SATn = 1.0,KROW Oil relative permeability for oil-water system (fr).KRW Water relative permeability for oil-water system (fr).KRG Gas relative permeability for gas-oil system (fr).KROG Oil relative permeability for gas-oil system (fr).PCOW Oil/water capillary pressure (psi).PCGO Gas/oil capillary pressure (psi).NOTE: SAT refers to the saturation of each particularphase. For example, in a data line following SAT = 0.2 wehaveKROW Oil relative permeability at 20% oil saturation.KRW Water relative permeability at 20% water saturation.KRG Gas relative permeability at 20% gas saturation.KROG Oil relative permeability at 20% liquid (water plus oil)saturation.PCOW Oil/water capillary pressure at 20% water saturation.PCGO Gas/oil capillary pressure at 20% gas saturation.NOTE: KROG is used only when a three-phase oil relativepermeability is calculated (ITHREE= 1 in Record 4 below).TEAM LinG - Live, Informative, Non-cost and Genuine!258 Principles of Applied Reservoir SimulationCapillary pressures are defined as PCOW = Po - Pw andPCGO = Pg - Po where Po, Pw, and Pg are the oil-, water-,and gas-phase pressures, respectively.3. Heading Up to 80 characters.4. ITHREE, SWRITHREE Code specifying desired relative permeability option.SWR Irreducible water saturation (fraction).CodeTTUTQpTn1 i JnLKJiDValue01MeaningOil relative permeability read from the relative perme-ability data for the two-phase water/oil system.Oil relative permeability calculated from Stone's three-phase relative permeability modelRepeat records 1 to 4 a total of NROCK times.24.6 Fluid PVT TablesThe following pressure-dependent data should be entered a total of NP VTtimes - one set of records for each PVT region defined in Section 24.4.1. Heading Up to 80 characters.2. PBO, PBODAT, PBGRADPBO Initial bubble point pressure (psia).PBODAT Depth at which PBO applies (ft).PBGRAD Constant bubble point pressure gradient (psia/ft).3. Heading Up to 80 characters.TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 2594 VSLOPE, BSLOPE, RSLOPE, PMAX, IREPRSVSLOPE Slope of the oil viscosity versus pressure curve forundersaturated oil, i.e. for pressures above PBO. The slope(A|J.0/A/)0) should be in cp/psia,BSLOPE Slope of the oil formation volume factor versus pressurecurve for undersaturated oil. The slope (A50/AP0) shouldbe in RB/STB/psia and should be negative or zero. BSLOPEis not the same as the undersaturated oil compressibility.RSLOPE Slope of the solution gas-oil ratio versus pressure curve. Theslope (A#W/AP0) should be in SCF/STB/psia and is nor-mally zero.PMAX Maximum pressure entry for all PVT tables (psia).IREPRS = 0; constant bubble point pressure.= 1; estimate variable bubble point pressure.5. Heading Up to 80 characters; oil table follows.6 PI MUO1 BO1 RSO1PMAX MUO(PMAX) BO(PMAX) RSO(PMAX)P Pressure (psia). Pressures must be in ascending order fromPI (normally 14.7 psia) to PMAX. The last table entry mustbe PMAX.MUO Saturated oil viscosity (cp).BO Saturated oil formation volume factor (RB/STB).RSO Saturated oil solution gas-oil ratio (SCF/STB).NOTE: Oil properties must be entered as saturated oil overthe entire pressure range.7. Heading Up to 80 characters; water table follows.8 PI MUW1 BW1 RSW1TEAM LinG - Live, Informative, Non-cost and Genuine!260 Principles of Applied Reservoir SimulationPMAX MUW(PMAX) BW(PMAX) RSW(PMAX)P Pressure (psia). Pressures must be in ascending order fromP1 (normally 14.7 psia) to PMAX. The last table entry mustbe PMAX.MUW Water viscosity (cp).BW Water formation volume factor (RB/STB).RSW Water solution gas-water ratio (SCF/STB).NOTE: It is usually assumed in black oil simulations thatthe solubility of gas in water can be neglected. In this case,set RSW = 0.0 for all pressures.9. Heading Up to 80 characters.10 KGCORCodeKGCORValue01MeaningRead gas and rock properties tableActivate gas correlation option andpressibility vs pressure tableread rock com-11. Heading Up to 80 characters; gas table follows.12 PI MUG1 BG1 PSI1 CR1PMAX MUG(PMAX) BG(PMAX) PSI(PMAX) CR(PMAX)Omit this record if KGCOR = 1P Pressure (psia). Pressures must be in ascending order fromP1 (normally 14.7 psia) to PMAX. The last table entry mustbe PMAX.MUG Gas viscosity (cp).BG Gas formation volume factor (RCF/SCF).PSI Gas pseudo-pressure (psiaVcp).CR Rock compressibility (1 /psia).TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 26!13 KODEA, MPGT, TEM, SPGOmit this record ifKGCOR = 0.KODEA Gas composition option (see Chapter 28.3).MPGT Number of gas PVT table entries (1 < MPGT < 25).TEM Reservoir temperature (°F).SPG Gas specific gravity (air = 1.0).14, FRCIOmit this record ifKGCOR = 0.FRCI Component mole fraction of gas. Read 12 entries in thefollowing order.FRCI(I)123456Component IH2SCO2N2c,C2C3FRCI(I)789101112Component IiC4nC4iC5nC5C6C7+15. PRSCI, TEMCI, RMWTIOmit this record ifKGCOR = 0 or if KODEA * 4.PRSCI Critical pressure (psia).TEMCI Critical temperature (°R).RMWTI Molecular weight.16. Heading Up to 80 characters; rock compressibility table follows.Omit this record ifKGCOR = 0.17. PI CR1PMAX CR(PMAX)Omit this record ifKGCOR = 0.TEAM LinG - Live, Informative, Non-cost and Genuine!262 Principles of Applied Reservoir SimulationOptionConstant rock compressibilityNOTE: Enter 1 record.Pressure-dependent rock com-pressibilityNOTE: Enter MPGTrecords.CodePMAXCRPCRMeaningMaximum table pressure (psia)from record 4.Rock compressibility (1/psia)Pressure (psia). Pressures mustbe in ascending order from PI(normally 14.7 psia) to PMAX.The last table entry must bePMAX.Rock compressibility (1/psia)18. Heading Up to 80 characters.19. RHOSCO, RHOSCW, RHOSCGRHOSCO Stock tank oil density (Ib/cu ft).RHOSCW Stock tank water density (Ib/cu ft).RHOSCG Gas density at standard conditions (Ib/cu ft).NOTE: At standard conditions (14.7 psia and 60 degreesF for oilfield units) pure water has a density of 62.4 Ib/cu ftand air has a density of 0.0765 Ib/cu ft.Repeat records 1 through 19 a total of NPVT times.24.7 Pressure and Saturation Initialization1, Heading Up to 80 characters.2, KPI, KSI, PDATUM, GRADKPI Pressure initialization code.KSI Saturation initialization code.PDATUM Depth to pressure datum (ft).GRAD Estimated pressure gradient (psia/ft) for pressure correctionsto PDATUM. If GRAD = 0, a map of pressures correctedto PDATUM will not be printed. If GRAD * 0, a map ofTEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 263pressures corrected to PDATUM will be printed usingpressure gradient GRAD,CodeKPIKSIValue0101MeaningRead II x JJ * KK pressures (one for each block).Equilibrium pressure initialization. Requires pressures anddepths at the OWC and GOC.Read II x JJ x KK oil saturations (one for each block) andII x JJ x KK water saturations. Gas saturations will becalculated by the program.Gravity segregated oil, water and gas saturation initializa-tion.NOTE: Options KPI and KSI may be used to prepare arestart data file.3. POOmit this record if KPI = 1.PO Oil-phase pressure (psia). Read II x JJ x KK values.4. PWOC, WOC, PGOC, GOCOmit this record if KPI = 0.PWOC Pressure at the water-oil contact (psia).WOC Depth to the water-oil contact (ft below datum).PGOC Pressure at the gas-oil contact (psia).GOC Depth to the gas-oil contact (ft below datum).NOTE: Repeat this record a total of NROCK times - onerecord for each Rock region.5. SOOmit this record if KSI = 1.SO Oil saturation array (fraction). Read II x JJ x KK values,6. SWOmit this record if KSI = /.SW Water saturation array (fraction). Read II x JJ x KK values.TEAM LinG - Live, Informative, Non-cost and Genuine!264 Principles of Applied Reservoir Simulation7. SOI, SGI, SOROmit this record ifKSI = 0.SOI Initial oil saturation for the oil-water zone to be assigned toall blocks in the rock region (fraction). Initial water satura-tion in the oil-water zone is I - SOI.SGI Initial gas saturation for the gas-water zone to be assignedto all blocks in the rock region (fraction). Initial watersaturation in the gas-water zone is 1 - SGI.SOR Irreducible oil saturation to be assigned to all blocks in therock region (fraction). If SOR > 0, calculated So will be setto 0 when So < SOR. Water and gas saturations are thenrenormalized.NOTE: Repeat this record a total of NROCK times - onerecord for each Rock region.24.8 Run Control ParametersI, Heading Up to 80 characters.2 NMAX, FACT1, FACT2, TMAX, WORMAX, GORMAX, PAMIN,PAMAXNMAX Maximum number of timesteps allowed.FACT 1 Factor for increasing timestep size using automatic timestepcontrol. FACT1 = 1.0 for fixed timestep size. A commonvalue for FACT1 is 1.25.F ACT2 Factor for decreasing timestep size using automatic timestepcontrol. FACT2 = 1.0 for fixed timestep size. A commonvalue for FACT2 is 0.5.TMAX Maximum elapsed time to be simulated (days); the run willbe terminated when the time exceeds TMAX.WORMAX Maximum allowed water-oil ratio for a producing oil well(STB/STB); WORMAX > 0.TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 265GORMAX Maximum allowed gas-oil ratio for a producing oil well(SCF/STB); GORMAX ;> 0.PAMIN Minimum field average pressure (psia); the run will beterminated when the pore volume weighted average reser-voir pressure < PAMIN.PAMAX Maximum field average pressure (psia); the run will beterminated when the pore volume weighted average reser-voir pressure > PAMAX.NOTE: PAMIN and PAMAX should be within the rangeof pressures covered by the fluid PVT tables discussed inChapter 24.6.3. WOROCKOmit this record ifWORMAX * 0.WOROCK Maximum WOR allowed in the corresponding Rock region.NOTE: If a well is completed in more than one Rock region,the largest maximum WOR which applies to the Rockregions penetrated by the well will be used as the WORcontrol for that well. Enter NROCK records - one for eachRock region.4. GOROCKOmit this record if GORMAX * 0.GOROCK Maximum GOR allowed in the corresponding Rock region.NOTE: If a well is completed in more than one Rock region,the largest maximum GOR which applies to the Rockregions penetrated by the well will be used as the GORcontrol for that well. Enter NROCK records - one for eachRock region.24.9 Solution Method Specification1. Heading Up to 80 characters.TEAM LinG - Live, Informative, Non-cost and Genuine!266 Principles of Applied Reservoir Simulation2,KSOL Solution method code.MITR Maximum number of LSOR iterations per timestep. Atypical value is 100.OMEGA Initial LSOR acceleration parameter. Values of OMEGAshould be between 1.0 and 2.0. A typical initial value is 1.5.TOL Maximum acceptable pressure change for convergence ofLSOR iterations (psia). A typical value is 0.1.TOL1 Parameter for determining when to change OMEGA. Atypical value is 0,001. If TOL1 = 0.0, the initial value ofOMEGA will be used for the entire run.DSMAX Maximum saturation change allowed per timestep (traction).The timestep size will be reduced by FACT2 if the saturationchange of a phase in any gridblock exceeds DSMAX duringa timestep. A typical value for DSMAX is 0.05.DPMAX Maximum pressure change allowed per timestep (psia). Thetimestep size will be reduced by FACT2 if the pressurechange in any gridblock exceeds DPMAX during a timestep.A typical value of DPMAX is 100 psia.NUMDIS Code for controlling numerical dispersionCode1C SOTValue12345MeaningID Tridiagonal Algorithm. Use with ID problemsand OD (tank) problems, i.e. when II = JJ = KK = 1 .Direct solution band algorithm. Use with 2D and3D problems.LSORX - Iterative matrix solver with direct solver inx direction.LSORY - Iterative matrix solver with direct solver iny direction.LSORZ - Iterative matrix solver with direct solver inz direction.TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 267CodeMT T"NjfT"YFQfNUMUloValue12MeaningSingle-point upstream weighting.Two-point upstream weighting.24.10 Analytic Aquifer Models1. Heading Up to 80 characters.2 IAQOPTIAQOPT Analytic aquifer model code.CodeIAQOPTValue01234567891011MeaningNo analytic aquifer modelPot aquifer model (small and boundedaquifer)Steady-state aquifer model (constant aquifer pres-sure)Carter-Tracy aquifer model: Re/Rw =1.5Carter-Tracy aquifer model: Re/Rw = 2.0Carter-Tracy aquifer model: Re/Rw = 3.0Carter-Tracy aquifer model: Re/Rw = 4.0Carter-Tracy aquifer model: Re/Rw = 5.0Carter-Tracy aquifer model: Re/Rw = 6.0Carter-Tracy aquifer model: Re/Rw = 8.0Carter-Tracy aquifer model: Re/Rw = 10.0Carter-Tracy aquifer model: Re/Rw = °°NOTE: Only one aquifer model option (IAQOPT) may beselected for a given run. Different aquifer influx strengthsmay be specified for a given aquifer.3. NAQENOmit this record if IAQOPT * 1.TEAM LinG - Live, Informative, Non-cost and Genuine!268 Principles of Applied Reservoir SimulationNAQEN Number of regions containing a pot aquifer.4 I1,I2,J1,J2,K1,K2,POTOmit this record iflAQOPT * 1.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.J1 Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.Kl Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.POT Pot aquifer strength (SCF/psia).NOTE: NAQEN records must be read.5. NAQENOmit this record iflAQOPT *2,NAQEN Number of regions containing a steady-state aquifer.6. II, 12, Jl, J2, Kl, K2, SSAQOmit this record iflAQOPT * 2.Coordinate of first region block in I direction.Coordinate of last region block in I direction.Coordinate of first region block in J direction.Coordinate of last region block in J direction.Coordinate of first region block in K direction.Coordinate of last region block in K direction.Steady-state aquifer strength (SCF/day/psia).NOTE: NAQEN records must be read.7. NAQREGOmit this record iflAQOPT < 3.NAQREG Number of Carter-Tracy aquifer parameter regions.TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 2698 AQCR, AQCW, AQMUW, AQK, AQPHI, AQH, AQS, AQREOmit this record iflAQOPT < 3.Aquifer rock compressibility (1/psia).Aquifer water compressibility (1/psia).Aquifer water viscosity (cp).Aquifer permeability (md).Aquifer porosity (fraction).Aquifer net thickness (ft).Aquifer to reservoir boundary interface (fraction). A valueof 0 implies there is no boundary (hence no influx); a valueof 1 implies that the aquifer surrounds the gridblock.AQRE External aquifer radius (ft).9. NAQENOmit this record iflAQOPT < 3.NAQEN Number of regions containing a Carter-Tracy aquifer.10 I1,I2,J1,J2,K1,K2Omit this record iflAQOPT < 3.11 Coordinate of first region block in I direction.12 Coordinate of last region block in I direction.Jl Coordinate of first region block in J direction.J2 Coordinate of last region block in J direction.Kl Coordinate of first region block in K direction.K2 Coordinate of last region block in K direction.NOTE: NAQEN lines must be read. Repeat records 8through 10 a total of NAQREG times.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 25Recurrent DataRecurrent data records are read periodically during the course of thesimulation run. These data include the location and specification of wells in themodel, changes in well completions and field operations over time, a scheduleof well rate and/or pressure performance over time, timestep control informationfor advancing the simulation through time, and controls on the type andfrequency of printout information provided by the simulator.1. Major Heading Up to 80 characters,NOTE: This record signifies the start of the recurrentdata section.25.1 Timestep and Output ControlTimestep and output control records must be read to start the simulation.1. Heading Up to 80 characters.2. IWLCNG, IOMETHIWLCNG Controls reading of well information.IOMETH Controls program output and well scheduling.270TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 271CodeIWLCNGlOMETHValue01> 1MeaningDo not read well informationRead well informationNumber of elapsed time values to be read on record 3 .The program will print results to output files at theseelapsed times and allow you to change well character-istics after the last elapsed time entered during thisrecurrent data period,3. FTIOFTIO Array containing total elapsed times at which output will occur(days). Up to 50 monotonically increasing values may be entered.The first entry must be greater than 0 and greater than the last entryof any previously completed recurrent data periods.NOTE: When the elapsed time of a run equals an FTIO value, thewell and basic summary reports will be printed. Maps will also beprinted according to the instructions given in record 4,4 IPMAP, ISOMAP, ISWMAP, ISGMAP, IPBMAP, IAQMAPIPMAP Control code for printing pressure array.ISOMAP Control code for printing oil saturation array.ISWMAP Control code for printing water saturation array.ISGMAP Control code for printing gas saturation array.IPBMAP Control code for printing bubble point pressure array.IAQMAP Control code for printing aquifer influx array.Code Value012MeaningDo not print the arrayPrint the arrayPrint the array and a digital contour plot5 IVPMAP, IZMAP, IRCMAP, IVSMAP, IVRMAPTEAM LinG - Live, Informative, Non-cost and Genuine!272 Principles of Applied Reservoir SimulationIVPMAP Control code for printing seismic compressional velocity(Vp) array.IZMAP Control code for printing seismic acoustic impedance array.IRCMAP Control code for printing seismic reflection coefficient array.IVSMAP Control code for printing seismic shear velocity (Vs) array,IVRMAP Control code for printing seismic velocity ratio Vp/Vs array.Code Value012MeaningDo not print the arrayPrint the arrayPrint the array and a digital contour plotNOTE: If IVRMAP > 0, time-dependent arrays will beappended to file WTEMP.ARR.6 DT, DTMIN, DTMAXDT Starting timestep size (days). DT may vary between DTMINand DTMAX when automatic timestep control is invoked.DTMIN Minimum timestep size allowed (days). A typical value is1 day.DTMAX Maximum timestep size allowed (days). A typical value is30 days.25.2 Well InformationOmit this section iflWLCNG = 0.1. Heading Up to 80 characters.2. NWELLN, NWELLONWELLN Number of new wells for which complete well informationis entered.NWELLO Number of previously defined wells for which new ratesand/or rate controls are entered.TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 2733. Heading Up to 80 characters.Omit this record ifNWELLN = 0.4. WELLIDOmit this record ifNWELLN = 0.WELLID Well name with up to five characters.5 IDWELL, KONECTOmit this record ifNWELLN = 0.IDWELL Well identification number. Each well should have a uniqueIDWELL number. If two or more wells have the sameIDWELL number, the characteristics of the last well enteredwill be used.KONECT Total number of gridblocks connected to well IDWELL.6. I, J, K, PID, PWFOmit this record ifNWELLN = 0.I x coordinate of gridblock containing well.J y coordinate of gridblock containing well.K z coordinate of gridblock containing well.PID Layer flow index for gridblock.PWF Flowing bottomhole pressure for block (psia). This value isused only if KIP is negative for this well.NOTE: KONECT records must be read. PID for a verticalwell can be estimated asKhPID = 0.00708Inwherer0 - 0.14 (DX2 + DY2)l/2andTEAM LinG - Live, Informative, Non-cost and Genuine!274 Principles of Applied Reservoir SimulationK = layer absolute permeability (md)h = layer thickness (ft)DX = x direction gridblock length (ft)DY = y direction gridblock length (ft)rw ~ wellbore radius (ft)r0 = equivalent well block radius (ft)S = layer skin factorDeviated (slanted) and horizontal wells may be represented bycalculating an appropriate PID and specifying gridblock locations thatmodel the expected well trajectory. For example, a horizontal well thatis aligned in the x direction will have constant J and K indices, and indexI will vary if there is more than one connection.To shut in a connection, set that connection PID to 0. To shut ina well, set all of its connection PID values to zero.7. KIP, QO, QW, QG, QTOmit this record ifNWELLN = 0.KIP Code for specifying well operating characteristics.Rate Controlled Well (KIP > 0):QO Oil rate (STB/D).QW Water rate (STB/D).QG Gas rate (MSCF/D).QT Total fluid voidage rate (RB/D).NOTE: The total fluid rate given by QT is the oil plus water plusgas production for the well or the total reservoir voidage rate atreservoir conditions. For multi-layer systems, QT is a target rate.BMP Controlled Production Well with Optional Rate Constraints(KIP = -1):QO Minimum oil production rate required (STB/D).QW Maximum oil production rate allowed (STB/D).QG 0.0QT Maximum liquid withdrawal rate allowed (STB/D).TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User '$ Manual 275NOTE: Rate constraints are not activated if the corresponding rateequals zero.BHP Controlled Water Injection Well with Optional Rate Constraints(KIP = -2):QO 0.0QW Maximum water injection rate allowed (STB/D).QG 0.0QT 0.0NOTE: QW should be a negative number or zero. The rateconstraint is not activated if QW = 0.BHP Controlled Gas Injection Well with Optional Rate Constraints (KIP= -3):QO 0.0QW 0.0QG Maximum gas injection rate allowed (MSCF/D),QT 0.0NOTE: QG should be a negative number or zero. The rateconstraint is not activated if QG = 0.Gas Production Well (KIP = -4):QO 0.0QW 0.0QG 0.0QT 0.0NOTE: Sign conventions for rates:Negative rates indicate fluid injection.Positive rates indicate fluid production.Summary of KIP ValuesCode32MeaningGas well - injection rate specifiedWater well - injection rate specifiedTEAM LinG - Live, Informative, Non-cost and Genuine!276 Principles of Applied Reservoir SimulationSummary of KIP ValuesCode1-1-2-3.4MeaningProduction well - rate specified+ Oil rate specified: QO > 0, QW = QG = QT = 04 Water rate specified: QW > 0, QO = QG = QT = 0* Gas rate specified: QG > 0, QO = QW = QT =0+ Total rate specified: QT > 0, QO = QW = QG =0Oil and/or water production well - PI and FBHPcontrolWater well - PI and FBHP controlGas injection well - PI and FBHP controlGas production well - LIT representation8. ALIT, BLITOmit this record if NWELLN = 0 or KIP * -4.ALIT "a" coefficient of LIT gas well analysis.BLIT "b" coefficient of LIT gas well analysis.NOTE: Records 4 through 8 should be repeated NWELLN times.9. Heading Up to 80 characters.Omit this record ifNWELLO = 0.10. WELLIDOmit this record ifNWELLO - 0.WELLID Well name with up to five characters.11 IDWELL, KONECTOmit this record ifNWELLO = 0.IDWELL Well identification number. Each well should have a uniqueIDWELL number. If two or more wells have the sameTEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 27?ID WELL number, the characteristics of the last well enteredwill be used.KONECT Total number of gridblocks connected to well IDWELL12. I, J, K, PID, PWFOmit this record if NWELLO = 0.I x coordinate of gridblock containing well.J y coordinate of gridblock containing well.K z coordinate of gridblock containing well.PID Layer flow index for gridblock.PWF Flowing bottomhole pressure for block (psia). This value isused only if KIP is negative for this well.NOTE: KONECT records must be read.13 KIP, QO, QW, QG, QTOmit this record if NWELLO = 0.KIP Code for specifying well operating characteristics. Seerecord 6 for a description of the KIP options.14. ALIT, BLITOmit this record if NWELLO = 0 or KIP * -4.ALIT "a" coefficient of LIT gas well analysis.BLIT "b" coefficient of LIT gas well analysis.NOTE: Records 10 through 14 should be repeated NWELLO times.TEAM LinG - Live, Informative, Non-cost and Genuine!Program Output EvaluationYou are given the option at the start of a WINB4D run to direct outputto either the screen or to a set of files. It is often worthwhile to send output tothe screen when first building and debugging a data set. WINB4D will abort atthe point in the data set where it encounters improperly entered data. Forevaluating run results, it is preferable to send output to files. In this case, a oneline timestep summary is sent to the screen each timestep so that you can monitorthe progress of a run. All output files are in text format.A run may be aborted by typing <Cntl> C. You may then choose toterminate the job.26.1 Initialization DataThe reservoir flow simulator W1NB4D outputs the following initializationdata in text file WTEMP.ARR:+ Gridblock sizes+ Node midpoint elevations4 Porosity distributions+ Permeability distributions4 Rock and PVT region distributions+ Relative permeability and capillary pressure tables+ PVT tables4 Slopes calculated from PVT data4 Timestep control parameters278TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 279$ Analytic aquifer model selection4* Initial fluid volumes-in-place+ Initial pressure and saturation arrays+ Initial seismic velocities array4 Initial acoustic impedance array4 Initial well informationOther output can be obtained at your request. For example, if a modificationoption is invoked, you may print out the altered array. It is worthwhile to do thisas a check on the input changes.26.2 Recurrent DataAll output files are text files so that they may be read by a variety ofcommercially available spreadsheets. WINB4D output may then be manipulatedusing spreadsheet options. This is especially useful for making plots ordisplaying array data. Different output files are defined so that simulator outputfile sizes are more manageable. The output files are designed to containinformation that is logically connected, e.g. well data in one file, reservoirproperty distributions in another file. The different output files are describedbelow.26.2.1 Timestep Summary File - WTEMP.TSSA one line timestep summary is automatically printed out as a record ofthe progress of the ran. This summary provides you with necessary informationfor evaluating the stability of the solution as a function of time. Significantoscillations in GOR or WOR, or large material balance errors are indicative ofsimulation problems and should be corrected. A smaller timestep through thedifficult period is often sufficient to correct IMPES instabilities.26.2.2 Run Summary And Plot File - WTEMP.PLTThe run summary file contains a concise summary of total field productionand injection and fieldwide aquifer influx. The WOR and GOR are ratios of totalTEAM LinG - Live, Informative, Non-cost and Genuine!280 Principles of Applied Reservoir Simulationproducing fluid rates. Consequently these ratios are comparable to observedfieldwide ratios.The output quantities include: cumulative production of oil, water andgas; cumulative injection of water and gas; pore volume weighted averagepressure; aquifer influx rate and cumulative aquifer influx; and fieldwide WORand GOR values. These quantities are output as functions of time and timestepnumber.26.2.3 Well Report File - WTEMP.WELRates and cumulative production/injection data for each layer of each wellare summarized in the well report at times you specify. Field totals are alsoincluded.26.2.4 Distribution Arrays File - WTEMP.ROFYou may output the following arrays whenever desired: pressure,saturations, bubble point pressure, cumulative aquifer influx, eompressionalvelocity, acoustic impedance, and seismic reflection coefficient. Output arraysmay be used as input pressure and saturation distributions for restarting a run.It is usually unnecessary to print all of the arrays. To avoid excessiveoutput and correspondingly large output files, you should be judicious indeciding which arrays are printed. In addition to arrays, you may wish to outputdigital contour plots.Digital contour plots provide a simplified picture of the physical parameterdistribution. The plot subroutine finds the minimum (AMIN) and maximum(AMAX) values of the array APLOT. A new array AOUT is constructed usingthe normalized parameter values given byAV = (APLOT(I, J, K) - AMIN)/ADIFwhere ADIF = AMAX - AMIN > 0.001. The values of AOUT are defined asfollows:AOUT-12Meaning (±0.05)AV<0.05AV = 0.10AV = 0.20TEAM LinG - Live, Informative, Non-cost and Genuine!Part IV: WINB4D User's Manual 281AOUT3456789TMeaning (±0.05)AV = 0.30AV = 0.40AV = 0.50AV = 0.60AV = 0.70AV = 0.80AV = 0.90AV > 0.95Digital contour plots highlight changes in parameter values and let youvisually monitor such items as saturation fronts, movements of pressure pulses,and changes in acoustic impedance. The output array AOUT is printed so thatit can be used for drawing a rough contour plot.TEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!PartVTechnical SupplementsTEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 27Simulator FormulationWINB4D is an implicit pressure-explicit saturation finite differencesimulator. It can simulate isothermal Darcy flow in up to three dimensions.Reservoir fluids are described by up to three fluid phases (oil, gas, and water),whose physical properties are functions of pressure only. Solution gas may bepresent in both the oil and water phases.27.1 EquationsThe black oil simulator mass conservation equations for the oil-, water-and gas-phases are derived in Chapter 4. They can be succinctly written in vectornotation as follows:OilaB0 Pose dtWater4> -r-l (27.1)Bo285TEAM LinG - Live, Informative, Non-cost and Genuine!286 Principles of Applied Reservoir SimulationGasR-V •V j?2 SO -*„-„„,*?„ -4. .m._LJ_Li.L, ij1 K 'R 0 °-D 15^ ^a |' a, jg + ^o5g 50v-v^ (27.3)Letting the subscript / denote o (oil), w (water), and g (gas), the symbols in Eqs.(27.1) to (27.3) have the following definitions:Bt = formation volume factor of phase iqi = mass flow rate per unit reservoir volume of phase /Rso = solubility of gas in oilRsw = solubility of gas in water5, = saturation of phase iv, = Darcy's velocity of phase iPise ~ density of phase i at standard conditions<j) = porosityThree additional equations - called auxiliary equations - are employedwhen solving the preceding fluid flow equations. They are the saturationconstraintS0 + Sw + Sg=l (27.4)and the capillary pressure relationshipsPcOW(Sw) = P0 - Pw (27.5)Pcgo(Sg) = P8 ' P0 (27.6)where P, is the pressure of phase i, Pcow is the oil-water capillary pressure, andPcgo is the gas-oil capillary pressure.Darcy's velocity for phase / is(27.7)TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 287where K is a permeability tensor that is usually assumed to be diagonalized alongits principal axes, kriis relative permeability and |l, is viscosity of phase i. Thephase potentials ^t are given as functions of depth z byo z o z o zA =p _.L£__ $ = P -P --L2L. d> =p +p _£*_ (27810 ° 144' w ° cow 144' * ° cgo 144 l jPhase densities are calculated from input PVT data asP0 = -^-[P0,c + ^0PgJ' Pw = "-[P0,c + ^wPgJ' P^-f^ (27.9)5o 5w 5gExpressions for rock and phase compressibilities are1 84) 1 dBg£. — ~ f\ = 2.^ 4>5p * BdP(27.10)C*. ~ ~_L^- B*8R>»" dP B dpThese equations are discretized and solved numerically in WINB4D. Theprocedure for solving these equations is outlined in Chapter 32.27.2 Coordinate OrientationThe WINB4D reservoir model assumes a block-centered grid with theaxes aligned iising the right-handed coordinate system illustrated irK *y \I1zi Figure 27-1.Figure 27-1. Coordinate system.TEAM LinG - Live, Informative, Non-cost and Genuine!288 Principles of Applied Reservoir SimulationThe top layer (K = 1) is shown. The second layer (K = 2) is below the K = 1layer, and so on. The top of each gridblock may vary from one block to another.This allows the model to perform calculations using grid representations ofreservoirs ranging from flat layer cake models to dipping structures such asanticlines and domes. An anticlinal structure is shown in Figure 27-2.D -6200-60001D-6400-6200• -6600-6400•-6800-6600North - SouthFigure 27-2. Depth to top of anticlinal structure.27.3 Petrophysical ModelMonitoring changes in the seismic characteristics of a reservoir as thereservoir is produced is the basis of time-lapse (4D) seismic monitoring[Anderson, etal., 1995; He, etal., 1996; Fanchi, etal. 1999]. Changes in seismiccharacteristics are determined in WINB4D by calculating seismic attributes asa function of time. The seismic attributes calculated in WINB4D are definedbelow.( ompressional and Shear VelocitiesSeismic compressional velocity and shear velocity are calculated fromthe expressions [Scho'n, 1996; McQuillin, et al., 1984]:'* + ty*" 3__ (27.11)PBandTEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 289V -rs(27.12)whereVp — compressional velocityVs = shear velocityK* = effective bulk modulusfl* = effective shear moduluspB = effective bulk density = (!-<J))pMa +4^pma = density of grains (solid matrix material)p, = fluid density = p0S0 +pwSw +pgSg<j> = porosityGassman [1951] derived an expression for K* from the theory of elasticity ofporous media [Schon, 1996; McQuillin, et al., 1984]:KKB + i K (27whereKB - bulk modulus of empty reservoir, that is, dry rock orporous matrix materialKG = bulk modulus of grains (solid matrix material)KF = bulk modulus of fluid = IIcfcf = fluid compressibility = c0 S0 + cw Sw + cg SgThe grain modulus KG equals the bulk modulus KB when porosity equalszero. Figure 27-3 shows that bulk modulus and shear modulus are linearfunctions of porosity for quartz sandstone [Murphy, et al., 1993] for porosityless than 35%.The WINB4D user must enter data that cannot be calculated fromtraditional black oil simulator input data. In particular, the user must enter KB,KG,\i*, and pma. The references give values that may be used if the data are notTEAM LinG - Live, Informative, Non-cost and Genuine!290 Principles of Applied Reservoir Simulationavailable from well logs such as shear wave logging tools or laboratory measure-ments of parameters such as acoustic velocities or the dry frame Poisson's ratio.Bulk and Shear Moduli for Quartz SandCorrelation Valid for Porosity < 35%0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35PorosityBulk (psia) -»- Shear (psia)~* Bulk (GPa) -"- Shear (GPa)Figure 27-3. Correlation for bulk and shear moduli.Acoustic Impedance and Reflection CoefficientsAcoustic impedance Z is defined asZ = 9BVP (27.14)The reflection coefficient RC at the interface between two layers with acousticimpedances Z, and Z2 is given byZ2 - Z,RC = —— (27.15)Z,2 + Z,jThe transmission coefficient TC is2Z.TC = z + z (27J6)or TC = 1 - RC.TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 29127.4 Material BalanceMaterial balance is one measure of the numerical stability and accuracyof a simulator. The WINB4D material balance calculation at time t is given byFIPMaterial Balance = : (27 17)OFIP - Prod + Inj lwhereFIP = fluid in place at time /OFIP = original fluid in placeProd = cumulative fluid produced at time /Inj = cumulative fluid injected at time tBased on this definition, material balance should equal one in an idealizedcalculation. Actual simulator material balance may not equal one.Material balance error reported by WINB4D is calculated using theformula{FTP 1— 1 \ x 100% (27.18)OFIP - Prod + Inj JMaterial balance can be a sensitive indicator of error. Material balance error isgreatest in WINB4D when a gridblock undergoes a phase transition, for example,when a gridblock passes from single phase oil to two-phase oil and gas duringa timestep.Material balance errors can be corrected by adding or subtracting enoughfluid to reestablish an exact material balance [Nolen and Berry, 1973; Spillette,et al., 1986], This material balance correction technique is equivalent to addinga source/sink term to the mass conservation equations for every gridblock. Theseterms are not included in the WINB4D formulation. The exercises in Parts I andII show that the uncorrected formulation can be used with good accuracy inmany practical situations.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 28Rock and Fluid ModelsThe interaction between reservoir rock and in situ fluids is modeled withrelative permeability and capillary pressure data. This chapter defines the three-phase oil relative permeability model used in WINB4D and its use intransmissibility calculations. It then presents additional details of the fluidproperty model after reviewing a few commonly used thermodynamic terms.28.1 Three-Phase Relative PermeabilityRelative permeability curves are some of the most critical data in thesimulator because relative permeability curves can have a significant impact onsimulator performance. Relative permeability curves are an important part ofthe algorithm that is used to model the interaction between reservoir rock andfluids. Unfortunately, relative permeability curves are often among the missingor poorer quality data.Relative permeability data are affected significantly by alterations inwettability conditions in the core. Ideally, the relative permeability data shouldbe measured in the laboratory under the same conditions of wettability that existin the reservoir. One method of approaching this ideal is to use preserved,"native state" core samples."Native state" core samples are cores that are drilled using crude oil ora special coring fluid designed to minimize wettability alterations. The coresare sealed at the well site to minimize exposure to oxygen or drying and thenpreserved until ready to undergo flow testing in the laboratory. However, this292TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 293process is expensive and most relative permeability data are obtained on restoredstate cores in the laboratory.In principle, three-phase relative permeability should be used when oil,water, and gas are flowing simultaneously. As a practical matter, the difficultyof accurately measuring three-phase relative permeabilities often makes theiruse meaningless. It is often sufficient to work with the two-phase relativepermeability curves only.Despite their shortcomings, it may be of interest to perform a simulationusing a set of three-phase relative permeability curves. For this case, WINB4Dcontains an option for computing a three-phase oil relative permeability curveusing water-oil and gas-oil relative permeability curves. As with most calcula-tions of this type, we assume:a. The water relative permeability curve (k^ obtained for a water-oilsystem depends only on water saturation, andb. The gas relative permeability curve (krg) obtained for a gas-oilsystem depends only on gas saturation.The validity of these assumptions depends on such factors as wettabilityand degree of consolidation. Given the above assumptions, £w and krg for water-oil and gas-oil systems, respectively, are also valid for a water-gas-oil system.The three-phase oil relative permeability kro3 is calculated as, (^row + ^rw) (krog + *rp ,* _,_ L \k^ = - 1 £- - (krw + krg) (28J)*rowwherekrow - oil relative permeability for water-oil systemkmg = oil relative permeability for gas-oil systemk*row= oil relative permeability for water-oil system evaluatedat the oil saturation corresponding to irreducible watersaturationEquation (28.1) is based on the work by Stone [1973], and it corresponds toModel II of Dietrich and Bonder [1976]. For a discussion of alternative modelsof three-phase oil relative permeability, see Blunt [1999].When the three-phase calculation is activated, the user must be sure theinput water-oil and gas-oil relative permeability curves are realistic. For example,TEAM LinG - Live, Informative, Non-cost and Genuine!294 Principles of Applied Reservoir Simulationif we write irreducible water saturation as Swr, the relative permeability constraintkmw (1 - Swr) = krog (S0 + Sw = 1.0) must be satisfied since Sg = 0 in both cases.28.2 TransmissibilityThe simulator offers no-flow boundary conditions, which lets you stopflow between specified gridblocks in chosen directions. The no-flow conditionsare implemented by setting transmissibilities at boundary interfaces to zero. TheTransmissibility Modifications section in Chapter 24.3.3 describes the directionalconventions for transmissibility in the model,Flow between neighboring blocks is treated as a series application ofDarcy's law. A transmissibility term at the interface between two blocks isdefined using the product of average values of relative permeability k^ of phase0, absolute permeability K of each block at the interface, and cross-sectionalarea Ac of each block at the interface, divided by the product of viscosity jle andformation volume factor B% of the phase in each block. The transmissibility toeach phase is determined using a harmonic average calculation of the productof absolute permeability and cross-sectional area at the interface betweenneighboring blocks. An arithmetic average of phase viscosities and formationvolume factors is used. The average relative permeability is determined usingan upstream weighted averaging technique. The resulting Darcy transmissibilityisA'-"•Airrt( upstream)Bu2(KACand the finite difference transmissibility Atii.]/2 for phase <! between block / -1and block / used in the simulator is+ *-t ; I/O •*••• a ;.. iwhere they, k indices are suppressed and the spatial differences areAx — x(— xf_}, Ax = XM - xiTEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 295Similar definitions of transmissibility apply in all three coordinate directions.28.3 Terminology and General CommentsSome of the fluid property terms that are most frequently used in blackoil simulation are defined here.DensityDensity is defined as the mass of a substance divided by the volume itoccupies. The density of a fluid depends on the pressure, temperature, andcomposition of the fluid.CompositionThe composition of a fluid depends on whether the fluid consists of a purecomponent, such as water or methane, or is a mixture. For example, petroleumand in situ water are mixtures. Petroleum is a mixture of hydrocarbon com-pounds, and in situ water usually contains dissolved solids, such as salt, and maycontain dissolved gases such as methane and carbon dioxide. The compositionof a fluid is a list of the components contained in the fluid.The relative amount of each component in a mixture is defined as theconcentration of the component. Concentration may be expressed in a varietyof units, such as volume fraction, weight fraction, or molar fraction. It isimportant to know the units associated with the composition. If the concentrationunits are not clearly expressed in a fluid report, they should be determined beforeuse in calculations. It is common to find composition expressed in molefractions. The symbols {*„>>„ zt} are often used to denote the mole fraction ofcomponent i in the oil phase, gas phase, and wellstream respectively.The equilibrium K value is a measure of the amount of component i inthe gas phase relative to the oil phase. It is defined as the ratioK, = y./xtIf component i exists entirely in the oil phase, then^, is 0 and K{ is 0. Conversely,if component i exists entirely in the gas phase, then xt is 0 and Kf approachesinfinity. Thus, the equilibrium K value for component / may range from 0 toTEAM LinG - Live, Informative, Non-cost and Genuine!296 Principles of Applied Reservoir Simulationinfinity. It should be noted that these concepts apply to both hydrocarboncomponents and any other distinct molecular species, such as carbon dioxideand nitrogen.PressureThe average pressure on a surface is the total normal force applied to thesurface divided by the area of the surface. The normal force is the componentof the force that is acting perpendicularly to the surface.Consider a fluid in the pore space of a rock. The pressure at any point inthe fluid is equal in all directions. If the fluid is at rest in the pore space, thepressure is equal at all points in the fluid at the same depth. Pascal's law saysthat pressure applied to an enclosed fluid will be transmitted without a changein magnitude to every point of the fluid and to the walls of the container.TemperatureTemperature is a measure of the average kinetic energy of a system.Several temperature scales are in use. The most commonly used temperaturescales are the Fahrenheit and Celsius scales. The relationship between thesescales iswhere Tc and TF are temperatures in degrees Celsius and degrees Fahrenheitrespectively.Applications of equations of state require the use of absolute temperaturescales. Absolute temperature may be expressed in terms of degrees Kelvin ordegrees Rankine. The Kelvin scale is related to the Celsius scale byTK = Tc + 273where TK is temperature in degrees Kelvin. The Rankine scale is related to theFahrenheit scale byTR = TF + 460where TR is temperature in degrees Rankine.TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 297Intensive and Extensive PropertiesPressure, temperature, and density are examples of intensive properties.An intensive property is a fluid property which is independent of the amountof material. For example, if a cubic cell of gas in an equilibrium state is dividedinto two halves by a vertical partition, the gas in each half of the cell should havethe same pressure and temperature. By contrast, the mass and volume in eachhalf will be one half of the original mass and volume. Mass and volume areexamples of extensive properties. An extensive property is a property thatdepends on the amount of material.CompressibilityIf the surface of an object is subjected to an external force, the resultingpressure applied to the object can change the volume of the object. Compressibil-ity is a measure of the volume change resulting from the applied pressure. Thefractional volume change AF/Fof an object may be estimated fromVwhere c is the compressibility of the object, AP is the pressure applied, and theminus sign implies that an increase (decrease) in applied pressure results in adecrease (increase) in the volume of the object.Formation Volume FactorFormation volume factor is defined as the volume occupied by a fluidphase at reservoir conditions divided by the volume occupied by the fluid phaseat standard conditions. The fluid phase volume may change substantially aspressure and temperature change.Ordinarily the volume of a fluid with constant composition will increaseas the applied pressure and temperature decrease. The behavior of petroleumis made more complex because it is a mixture and can experience a change incomposition as temperature and pressure change. For example, a barrel of oilat reservoir conditions (relatively high pressure and temperature) will shrink asthe barrel is brought to the surface (relatively low pressure and temperature).The shrinkage is associated with the release of solution gas as the pressure andTEAM LinG - Live, Informative, Non-cost and Genuine!298 Principles of Applied Reservoir Simulationtemperature of the oil decline from reservoir to surface conditions. Consequently,measurements of the change in volume as a function of pressure are desirable,especially for the oil phase.The determination of gas formation volume factor provides an interestingcontrast to the determination of oil formation volume factor. Gas formationvolume factor is often determined with reasonable accuracy using the real gasequation of state PF= ZnRT where n is the number of moles of gas in volumeFat pressure P and temperature T. The gas compressibility factor Z equals oneif the gas is an ideal gas. For real gases, Z * 1 for most pressures and tempera-tures,Specific GravitySpecific gravity is defined as the density of a fluid divided by a referencedensity. Gas specific gravity is calculated at standard conditions using air densityas the reference density. The specific gravity of gas is defined byg Ma(air) 29where Ma is apparent molecular weight. Apparent molecular weight is calculatedas the mole fraction weighted averageM = Yy.M.a L-t J i ii=\where Nc is the number of components,^, is the mole fraction of component /,and M{ is the molecular weight of component i.Oil specific gravity is calculated at standard conditions using fresh waterdensity as the reference density. Oils are often characterized by specifying theirAPI gravity, which is related to oil specify gravity y0 at standard temperatureand pressure by the equationToHeavy oils are oils with a relatively large J0 and a relatively low API gravity.Heavy oils typically do not contain much gas in solution. By contrast, light oilsTEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 299have a relatively small yo and a correspondingly large API gravity. Light oilstypically contain a large amount of gas in solution.Gas-Liquid RatioThe gas-liquid ratio is defined as the volume of gas divided by the volumeof liquid, usually oil or water. The gas volume and liquid volume should beexpressed at the same temperature and pressure.ViscosityThe coefficient of viscosity is a measure of resistance to flow of the fluid.In general, gases have a lower viscosity than liquids. The inverse of viscosityis called fluidity [McCain, 1990]. Thus, a fluid with a large viscosity has a lowfluidity.The relationship between viscosity and shear rate defines the rheologyof the fluid. If fluid viscosity is independent of flow rate, the fluid is referredto as a Newtonian fluid. If fluid viscosity depends on flow rate, the fluid isconsidered a non-Newtonian fluid.Two types of viscosity may be specified: dynamic viscosity p, andkinematic viscosity V. They are related by the expression [I = p v where p is thedensity of the fluid. Dynamic viscosity |i is used in Darcy's law to calculate therate of fluid movement fluid flow in porous media. Typically, the unit ofdynamic viscosity Ji is centipoise. If fluid density p has the unit of g/cc, thenkinematic viscosity v has the unit of centistoke. Thus, 1 centistoke equals 1centipoise divided by 1 g/cc.Reservoir fluid properties (PVT data) include fluid viscosities, densities,formation volume factors, gas solubilities, etc. These data are usually obtainedby laboratory analyses applied to fluid samples taken from the reservoir. Theyare sketched in ChapterB.Differential to Flash ConversionLaboratory reservoir fluid analyses generally provide data from both adifferential liberation experiment and a flash experiment approximating fieldseparator conditions. The differential and flash liberation data can be signifi-TEAM LinG - Live, Informative, Non-cost and Genuine!300 Principles of Applied Reservoir Simulationcantly different for some oils. The actual behavior of the production process issome combination of the differential and flash processes. The assumptionnormally made in preparing PVT data for use in a black oil simulator is that thedifferential liberation data represent the process occurring in the reservoir andthe flash data represent production to stock tank conditions. Thus, for use in thesimulator, the differential liberation data should be corrected to flash values atfield separation conditions. This procedure is described in the literature [Arnyx,et al, 1960; Moses, 1986] and is summarized below.Physical property data obtained from a testing laboratory for a black oilsystem will generally be a differential liberation study coupled with a separatorstudy. Most reservoir simulators require that these data be converted to flashform so that the effects of the surface separation facility are included. Conversionof the data is restricted to oil formation volume factor and solution gas-oil ratiodata. If the separator B0 and Rso are known, the conversion equations are:"odbpandBofbp"odbpwhere subscripts are defined as:d = differential liberation data/ = flash databp - bubble point28.4 Extrapolating Saturated CurvesGuidelines for extrapolating PVT data to pressures above the measuredsaturation pressure are presented below.i. The Bg versus pressure curve is strongly non-linear and an extrapolationof this curve to small Bg values at high pressures can result in errors. ForTEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 301most natural gases, the relationship \!Bg versus pressure will be verynearly linear, especially at moderate to high pressures. Plotting l/Bgversus pressure and extrapolating to PMAX should provide more realisticvalues of Bg at higher pressures. Interpolating Bg using l/Bg versuspressure substantially improves material balance.2. Once the Bg versus P curve is fixed, Rso versus P and B0 versus P curvesmust be extrapolated so as to avoid a negative oil compressibility beingcalculated over any pressure increment. To ensure that negative oilcompressibilities will not be calculated by the program, the following testshould be used. For any pressure increment P, to P2, where P2 > P{, thefollowing relationship should hold:0 * - (B02 - Bol) +v ;.5.615where the units of B0, Bg, and Rso are RB/STB, RCF/SCF, and SCF/ STB,respectively. Note that this test applies only to the saturated oil PVT data.3 . The above concepts also apply to the water PVT data. However, for mostsimulations, it can be assumed that Rsw = 0.0, thus - ABW/BWAP approxi-mates water compressibility.28.5 Gas PVT Correlation OptionBasic Gas PropertiesFollowing Govier [1975], real gas Z-factors are computed using theDranchuk, et al. [1974] representation of the Standing-Katz Z-factor charts[1942]. This representation employs the Benedict- Webb-Rubin [1940] eight-parameter equation of state to express the Z-factor as a function of pseudo-critical temperature Tr and pseudo-critical pressure Pr, thusZ = Z(Pr, Tr) (28.2)TEAM LinG - Live, Informative, Non-cost and Genuine!302 Principles of Applied Reservoir SimulationOnce Z is known, the gas formation volume factor is easily determined for agiven temperature and pressure using the real gas law.The isothermal gas compressibility c is obtained from Eq. (28,2) asc =11 1BZ"a P.(28.3)where Pc is the critical pressure (psia).Real gas viscosities are computed using the method described in Govier[1975]. This method is a computerized version of the Carr, Kobayashi, andBurrows [1954] hydrocarbon gas viscosity determination procedure.Pseudo-Pressure CalculationsPseudo-pressures are defined byiK/0 = 2/ ^-dP' (28.4)Pe »gZwhereP1 - dummy integration variable with pressure units (psia)P0 = reference pressure = 14.7 psiaP = specified pressure (psia)|lg = gas viscosity (cp)Z = gas compressibility factorThe pseudo-pressure t|f (P) is often written as m(P). Since \lg and Z dependon P', evaluation of Eq. (28.4) is accomplished by numerical integration usingthe trapezoidal rule and a user-specified pressure increment AP' ~ dP'.Gas Property DescriptionFour different gas property descriptions may be specified. Their descrip-tions and control parameter (KODEA) values follow:KODEA]GAS DESCRIPTIONSweet gas:0.0.0. 1.0input 12 component mole fractions as0. 0. 0. 0. 0. 0. 0.TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 303KODEA234GAS DESCRIPTIONSour gas: input 12 component mole fractions in they\ y-i y3 y* 0- o. o. o. o. o. o. o.where j/j = mole fraction of H2S,y2 = mole fraction of CO2y3 = mole fraction of N2, andSweet or sour gas with the following 12 componenttions read in the following order:ordermole frac-H2S, CO2, N2, Cj, C2, C3, iC4, nC4, iC5, nC5, C6, C7+.The sum of the mole fractions should equal one.Same as KODEA = 3 but also read critical pressure,temperature, and molecular weight of C7+.criticalCorrelation Range LimitsThe following range limits apply to correlations used in calculating gasZ-factors, compressibilities and viscosities:1.05 < — < 3.0Tc0.01 < — < 15.0PC0.55 < SPG < 1.540 < T < 400whereTc = pseudo-critical temperature (°R)Pc = pseudo-critical pressure (psia)T = temperature (°R)P = pressure (psia)SPG = gas specific gravityNo values of r, P, or SPG should be used that exceed the above correlationranges. If the range limit is exceeded, a fatal error will occur.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 29InitializationIt is important when making cross-section or 3D runs that the pressuresin the model are correctly initialized. If not, phase potential differences due togravity terms could cause fluid migration even though no wells are active.Consequently, a simple pressure initialization algorithm is used in WINB4D.It is reviewed below along with an option to correct pressures to a user-specifieddatum and an option to initialize saturations using gravity segregation.29.1 Pressure InitializationConsider a gridblock that may have a gas-oil contact and a water-oilcontact as in Figure 29-1.DatumXy\> „GOC,)EL(+)trI woe*w\r\fjL.\^ 'fFigure 29-1. Depths for pressure initialization algorithm.We assume the pressure in the gridblock at model location (/, j, k) isdominated by the density of the phase at the block midpoint and that there are304TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 305no transition zones between different phases initially. The pressure and depthat the gas-oil contact are PGOC and GOC, respectively. Similarly, for the water-oil contact we have PWOC and WOC.The initial pressure assigned to the gridblock in Figure 29-1 is determinedby the depth of the node (midpoint) relative to the respective contact elevations.Let us define the depth of the block midpoint from datum as ELiJk. Withthis definition, the pressure in the block is given by the following algorithm;a. IfEL,yA<GOCthenpg = pgJBgandP..k = PGOC + pg (EL,/t - GOC)/144b. IfEL,7fc>WOCthenPw = (Pw*c + R™ ' Pgsc)/5w andP.jk = PWOC + Pw (EL,.., - GOC)/144c. If GOC <; ELljk < WOC thenPO = (Post + Rso ' P«c)^o andPijk = PWOC + p0 (ELiJk - GOC)/144The above algorithm should be reasonable for systems with initial transitionzones that are small relative to the total thickness of the formation.Pressure Corrected to DatumPressure P(I, J, K) of gridblock I, J, K with mid-point elevation EL(I, J,K) may be corrected to a datum depth PDATUM by specifying a pressuregradient GRAD. The pressure at datum is given by PDAT(I, J, K) = P(I, J, K)+ (PDATUM - EL(I, J, K))*GRAD.29.2 Gravity-Segregated Saturation InitializationA simple model of a gravity-segregated saturation distribution iscalculated when KSI = 1. For depths increasing downward, we calculateelevations and thicknesses using the geometry shown in Figure 29-1 as follows:Block BOT = EL + 0.5 *DZBlock THICK = DZTEAM LinG - Live, Informative, Non-cost and Genuine!306 Principles of Applied Reservoir SimulationBlock TOP = BOT - THICKWater zone thicknessWTHICK = BOT - WOCGas zone thicknessGTHICK = GOC - TOPThe user must specify the initial oil saturation (SOI) for an oil-water system andthe initial gas saturation (SGI) for a water-gas system. Given the initialsaturations SOI and SGI, the following algorithm is applied [Fanchi, 1986].Case1Case2Case3Case4GOCTOPBOTWOCTOP ,GOC }ftWOC ,BOT } ATOPGOCBOT J-'WOCGOCTOP[/WOC ;BOTS, = 0S0 = SOISw=l-SOIf GTHICKg THICKf WTHICKTHICKC* IOC*ow - 1 - 50 - bg, l GTHICKTHICKS0=\- SOI*fSg = (l-f)*SGIC = 1 C C^w L - *->o " ^gf_ l WTHICKTHICK5=0gSw = 1 - SOI*fS0 = SOI*fIf ^^^^ then50-°0sg = fg*SGIUS0 < Sor, thenS — 05W = 1 - 5G/Sg - SGIlfS0 < Sor, thenS0 = 05W = 1TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 307Case5Case6TOPEOTGOCwoeGOCwoeTOPBOTS0 = 0Sw=l-SGISg = SGIS0 = Sg = QSw=lWater saturation is calculated as Sw = 1 - S0 - Sg in all cases. Cases 2 through 4require the user to enter residual oil saturation Sor.29.3 Aquifer ModelsA reservoir-aquifer system can be modeled using small gridblocks todefine the reservoir and increasingly larger gridblocks to define the aquifer. Thisapproach has the advantage of providing a numerically uniform analysis of thereservoir-aquifer system, but it has the disadvantage of requiring more computerstorage and computing time because additional gridblocks are used to model theaquifer. A more time- and cost-effective means of representing an aquifer is torepresent aquifer influx with an analytic model. Three models are available asoptions in WINB4D.Pot AquiferAquifer influx is calculated assuming the aquifer is both small andbounded. The pot aquifer influx rate qwp is dependent on the pressure changeover a timestep for a specified gridblock:POT(Pn -POT * 0 (29.1)where P", Af" are gridblock pressure and timestep at the present time level n\Pn + \ Ar"+' are gridblock pressure and timestep at the future time level n + I;TEAM LinG - Live, Informative, Non-cost and Genuine!308 Principles of Applied Reservoir Simulationand POT is the pot aquifer coefficient. The minus sign preceding the bracketedterm indicates water is entering the block when Pn> Pn + l.Steady-State AquiferThe steady-state aquifer model is based on Schilthuis's assumption thatthe water influx rate qwss is proportional to the pressure difference between theaquifer and the hydrocarbon reservoir. It is further assumed that the aquifer issufficiently large that it experiences no net pressure change throughout theproducing life of the reservoir. With these assumptions, WINB4D computessteady-state aquifer influx into a specified gridblock asqwss = -[SSAQ (/>° - Pw + 1)]; SSAQ * 0 (29.2)where P"+l is the gridblock pressure at the future time level n + 1 ; P° is the initialgridblock pressure; and SSAQ is the proportionality constant. The minus signpreceding the bracketed term indicates water is entering the block when we havethe inequality p°>pn + \Carter-Tracy AquiferThe Carter-Tracy [ 1 960] modification of the Hurst- van Everdingen [ 1 949]unsteady-state aquifer influx calculation is available in WINB4D. The Carter-Tracy aquifer influx rate qwa for a specified gridblock is*Wr = -M - B(Pn+l - P")] (29.3)where P",P"+l are gridblock pressures at time levels n and n + 1, respectively,The coefficients A and B are given byA = KtP(p° -/>")-DENOMwith(29.4)= Kt H (29 5)' DENOM l JDENOM = P"D+l - t£p't£+l (29.6)TEAM LinG - Live, Informative, Non-cost and Genuine!P.'n+1tDPart V: Technical Supplements 309dP.tDdtD(29.7)Kt = 0.00633 ^-- = AQPAR1p = 2n$hcrs = AQPAR2 (29.9)andc = cr + cw (29.10)The quantities ffl and PtD are dimensionless time and pressure, respectively, withtD = Kttand PtD is the Carter-Tracy influence function for the constant terminal rate case.The functions PlD andP',D are numerically represented by regression equations[Fanchi, 1985]. All remaining parameters are defined as follows:cr - rock compressibility (psi'1)cw = water compressibility (psi"1)h ~ aquifer net thickness (ft)k ~ aquifer permeability (md)re = external aquifer radius (ft)rw = external reservoir radius (ft)s = 0/360° where 6 is the angle of aquifer/reservoir interfaceWe = cumulative water influx at time level«, SCF[i = aquifer water viscosity (cp)(f) = aquifer porosityTEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 30Well ModelsThe well models contained in WINB4D are described in this chapter.User-specified parameters for controlling these well models are defined inChapter 25.30.1 Rate Constraint RepresentationCase 1: Oil Production Rate Q0 SpecifiedIn this representation, rates may be specified for injectors or producers.We assume the well may be completed in a total of K connections, and theproduction rates for each connection k for a specified oil rate are:Oil(30.1)WaterO ± - O L i fin 9\»w* *£ok i /T, PU./JK '(PID)— -°J*K£A=lA "(PH))-±o310TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 31!GasaI.JBg gQok + (Rso\Qak + (RSW\QWk (30,3)0 ° > kwhere A.c is the fluid mobility of phase Q. and PID is the well productivity index.For a more detailed discussion of PID, see Chapter 31. Notice that a PID maybe specified for each connection. This capability lets the WINB4D user take intoaccount permeability contrast.Case 2: Water Production Rate Qw SpecifiedAssuming the well may be completed in K connections, the productionrates of connection k for a specified water rate are:WaterQwk = Qw K (30.4)S/"DTT\\ 1 / D( "LU i At I Jo,_.L w WJ kOile°*= ^w* T^/if (30-5)Gas•^^ ? P ^-J /T» \ /-k . /•»> \ /-V/ , + (K )L{S , + (R )k\s . /3Q 5)? gMB.Case 3: Gas Production Rate Qg SpecifiedAssuming the well may be completed in K connections, the productionrates of connection k for a specified gas rate are:GasQ = Q*k *TEAM LinG - Live, Informative, Non-cost and Genuine!312 Principles of Applied Reservoir SimulationOil/""I /"^ I ® O \y0k= ygk\ . ,„ (30.8)Watera, = O J —— — flf) Q\wk X'gkl \ I n \J\J.y)Solution gas in both oil and water is neglected when a gas production rate isspecified. This is a reasonable assumption for wells producing primarily freegas.Case 4: Total Production Rate SpecifiedWhen the total reservoir voidage rate QT is specified, we first computethe phase mobility ratio for all connections:Oil Mobility Ratio*( Ko }aoT = SI ^ + ^ + ^ (30.10)Water Mobility RatioK ( A... 1Gas Mobility RatioK ( * }agT ~ ^ 7 T T~ (30.12)*=1\ ^o + ^w + ^gj kWe now compute the total oil rate(\ x-vT^^. \^~ (30.13)whereTEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 313*0 = 7 £ (B0\ (30.14)A Jt=lis the average oil formation volume factor for all connections in which the wellis completed. Given Eq. (30.13), we simply proceed as in Eqs. (30.1) through(30.3) above.Case 5: Injection Rate SpecifiedIf the well is a water or gas injector, the user must specify the total wateror gas injection rates Qw or Qg, respectively, and a well injectivity index (WI)for each connection. The injection rate for each connection is then allocated asfollows:Water Injection Rate(30.15)Gas Injection Rate[WI(A,0 + Xw + A,p]tgk ~ g~K~ ~ (30.16)k~ 1It is important to note that allocation of injection fluids is based on totalmobilities, and not just injected fluid mobility. This is necessary for thefollowing reason: If an injector is placed in a block where the relative permeabil-ity to the injection fluid is zero, then the simulator using injection fluid mobilityonly would prohibit fluid injection even though a real well would allow fluidinjection. A common example would be water injection into a block containingoil and irreducible water. To avoid the unrealistic result of no fluid injection,we assume the total mobility of the block should be used. For most cases, theerror of this method will only persist for a few timesteps because, in time, themobile fluid saturation in the block will be dominated by the injected fluid.TEAM LinG - Live, Informative, Non-cost and Genuine!314 Principles of Applied Reservoir Simulation30.2 Explicit Pressure Constraint RepresentationCase \: Oil and/or Water Production WellsWe assume that flowing bottomhole pressures (PWF) and well PIDs arespecified for a pressure-constrained well. The oil and water rates in STB/D forconnection k are given byVQnk =PID(Pn - PWF), (30.17)andA,(Pn - PWF), (30.18)B kwhere the explicit pressure P" is used. If P" < PWF, the well is shut in. WhenP" > PWF, Qok and Qwk are calculated and then substituted into Eq. (30.3) toCase 2: Gas Production WellThe laminar-inertial-turbulent (LIT) method may be used to represent agas production well. The LIT method entails fitting gas well test data to theequationaQ + bQ = i|r - t|f (30.19)wheretyR = pseudo-pressure corresponding to shut-in pressurePR (psia2/ cp)t|V = pseudo-pressure corresponding to a specified well flowingpressure /^(psiaVcp)agg = laminar flow^(?g2 ~ inertial and turbulent flowWINB4D employs user specified values of a, b, Pwf, and a table of pseudo-pressure versus pressure values to compute total gas well production rate asTEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 315(3020)where tyR is the pseudo-pressure corresponding to the nodal pressure P". Ratesfor each phase in connection k are computed by mobility allocation as shownin Eqs, (30.7) through (30.9).Case 3: Injection WellsThe injection rate for a water or gas injection well is computed from( A. + A, + VI B,where the subscript/? denotes water or gas, and PID=WI. Fluid injection occurswhen Pn < PWF. If P" > PWF, the injection well is shut in. Also note that totalmobility is used for the injection well rate calculation. The reason for this wasdiscussed in the first section of this chapter.30.3 GOR/WOR ConstraintsMaximum gas-oil and water-oil ratios (GORMAX, WORMAX respec-tively) are input by the user and apply to every oil production well. GOR for awell is defined as total gas production divided by total oil production for allactive well completion intervals. If GOR for the well exceeds GORMAX, thenthe completion interval (connection) with the highest GOR will be shut in. Ifmore than one connection has the same maximum GOR, the shallowestconnection will be shut in first. The procedure is repeated until GOR is less thanGORMAX or until the well is shut in.The ratio WOR is defined as total water production divided by total oilproduction for all active well completion intervals. If WOR for the well exceedsWORMAX, then the completion interval (connection) with the highest WORwill be shut in. If more than one connection has the same maximum WOR, thedeepest connection will be shut in first. The procedure is repeated until WORis less than WORMAX or until the well is shut in.TEAM LinG - Live, Informative, Non-cost and Genuine!316 Principles of Applied Reservoir Simulation30.4 Fluid Withdrawal ConstraintsFluid withdrawal from explicit pressure controlled production wells canbe constrained as follows:a. A minimum oil production rate can be specified;b. A maximum oil production rate can be specified; andc. A maximum liquid (water plus oil) withdrawal rate can be speci-fied.A positive value of QO for a pressure controlled production well is usedas the minimum allowed oil production rate. If the calculated oil production ratedrops below the minimum allowed value, the well is shut in.A positive value of QW for a pressure controlled production well is usedas the maximum allowed oil production rate. If the calculated oil production rateexceeds the maximum allowed value, calculated production will be reduced tothe allowed value. Production from each connection is proportionally reducedby the ratio of allowed to calculated oil production rates.A positive value of QT for a pressure controlled production well is usedas the maximum allowed liquid withdrawal rate. If the sum of oil and waterproduction exceeds the maximum allowed value, calculated production isreduced to the allowed value. The reduction is made by multiplying productionfrom each connection by the ratio of allowed-to-calculated liquid withdrawalrates. IMPORTANT: When used to control total liquid withdrawal, the unitsof QT are STB/Day.30.5 Fluid Injection ConstraintsFluid injection into explicit pressure controlled injection wells can beconstrained as follows:a. A maximum water injection rate can be specified; andb. A maximum gas injection rate can be specified.A negative value of QW for a pressure controlled water injection well isused as the maximum allowed water injection rate. If the calculated waterinjection rate exceeds the allowed value, calculated water injection will beTEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 317reduced to the allowed value. Water injection into each connection is proportion-ally reduced by the ratio of allowed to calculated water injection rates.A negative value of QG for a pressure controlled gas injection well is usedas the maximum allowed gas injection rate in direct analogy to the waterinjection rate constraint described previously.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 3131.1 Productivity IndexProductivity index (PI) is defined as the ratio of rate Q to pressure dropAP, or PI = £?/AP, where AP ~ Pe- Pw, Pe = average reservoir pressure, andPw = wellbore bottomhole pressure BHP. From Darcy's Law for radial oil flowwe can write PI asPI =_AP " Mo[H'A,) + S\(31.1)The meaning and units of all terms are given as follows:\i0 = oil viscosity (cp)B0 = oil FVF (RB/STB)re = drainage radius (ft)rw = wellbore radius (ft)S = skinKe = effective permeability (md) = kro Kabskro = relative permeability to oilKahs = absolute permeability (md)hnet = net thickness (ft)Q0 = oil rate (STB/D)318TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 319Some of the terms in Eq. (31.1) depend on time-varying pressure andsaturation, while other factors change relatively slowly or are constant withrespect to time. We separate these terms to obtainPI =where the quasi-stationary factors are collected in the PID term, that is,0.00708 J: . AThe WINB4D user is expected to provide a PID for each well connection. Aconnection is a gridblock with a well perforation.31.2 Vertical WellsA value of the connection flow index PID for a vertical well can beestimated from a formula derived by Peaceman [1978]:0.007080PID, =(31.2)wherere - ro = 0.14(A*2 + Ay2)'7'for an isotropic system. With respect to permeability, an isotropic system is asystem in which x direction and y direction permeabilities are equal, (Kx = Ky).For a square well block in an isotropic system, AJC = Ay and r0 « 0.2 AJC. Thesubscript k in Eq. (31.2) denotes the &th connection. For a well in a rectangulargridblock and an anisotropic system (that is, Kx * Ky), well PID is estimatedusing an effective permeabilityK = ]/KXKyand an equivalent well block radiusTEAM LinG - Live, Informative, Non-cost and Genuine!320 Principles of Applied Reservoir Simulationr = 0.28(Ky/Kxf< + (KxIK/<The remaining parameters are defined as:K = horizontal permeability of connection k (md)h - thickness of connection k (ft)rw = wellbore radius (ft)S = dimensionless skin factorIn principle, the well flow index can be related to measured values. In practice,however, the terms re, S, and kro /\10B0 are seldom well known, especially fora multiphase flowing well. As a matter of expediency, therefore, Eq. (31.2) isoften used to compute an initial estimate of PID. This value can then beimproved by adjusting it until the well rates computed by the simulator matchthe initial observed well rates.31.3 Horizontal WellsThere are many ways to estimate connection flow index PID for ahorizontal well [Joshi, 1991 ]. A PID value can be estimated for horizontal wellsin a manner similar to that for vertical wells by using the Joshi formulapro,QMlOSKhIna+ .\2 _ I L II 2j( LI2 )(\h +5J ^^ O2rwj(31.3)where£0.5N0.25TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 321The subscript k in Eq. (31.3) denotes the &* connection. The remainingparameters are defined as follows:K = horizontal permeability of connection k (md)h = thickness of connection & (ft)L = horizontal well length (ft)rw = wellbore radius (ft)reh = drainage radius of horizontal well (ft)S = dimensionless skin factorThe drainage radius reh of the horizontal well needs to account for elliptical flowinto the wellbore. Babu and Odeh [ 1989] present a procedure for estimating reh.TEAM LinG - Live, Informative, Non-cost and Genuine!Chapter 32The IMPES FormulationThe following section from Fanchi, et al. [1982] shows how the flowequations for a black oil simulator can be recast in a form that is suitable forsolution by a numerical technique. The numerical technique is based on theformulation originally presented by Sawyer and Mercer [1978].32.1 Flow Equations and Phase PotentialsThe form of the Darcy velocities (Eqs. (4.10) through (4.12)) may besimplified by defining the potential €>p of phase p as$ = p - _Z (32>i)P P 144and we have used the assumption that g=gc. In this notation, including x, y, andz directional permeabilities and unit vectors i, j, k, the Darcy velocities maybe written asv = -Kdx y By z dz(32.2)K • A...VO... = -A.322(32.3)TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 323a$° " - --- '• "-- (32,4)iK —- + jK — + kKBy z dzhave used the dyadic notation K to signify that permeability is a tensor of ranktwo. The expanded form of Eqs. (32.2) through (32.4) employs the commonassumption that the coordinate axes of our reference system are aligned alongthe principal axes of K. As discussed in Chapters 7 and 8, and associatedreferences, this assumption impacts the ability of the simulator to accuratelymodel fluid flow.Combining Eqs. (4.27) through (4.29) with Eqs. (32.2) through (32.4)gives(32.5)St B(32.6)and(32.7)Equations (32.5) through (32.7) are equivalent to Peaceman's [1977] Eqs. (1-105) through (1-107) for a three-dimensional system, except we have alsoallowed gas to dissolve in the water phase. Our rate and coordinate system signconventions also differ. If these differences are taken into consideration, theformulations are seen to be equivalent.32.2 Introduction of the Capillary Pressure ConceptThe presence of oil-, water-, and gas-phase pressures in Eqs. (32.5)through (32.7) complicates the problem. We simplify the handling of the phaseTEAM LinG - Live, Informative, Non-cost and Genuine!324 Principles of Applied Reservoir Simulationpressures and potentials in the flow equations by using the capillary pressureconcept. Let us define the difference in phase pressures aspcow = Po - Pw (32.8)andP = P ~ P - (32 9)ego g o \~>^.J/The differences Pcow and Pcgo are the capillary pressures for oil-water and gas-water systems, respectively. Experimentally Pcow and Pcgo have been observedto be principally functions of water and gas saturations, respectively. Using Eqs.(32.8) and (32.9) lets us write the water and gas phase potentials asPw*$ = P ~ P - -2— (32 10)*w Jo cow ~AA \J±.iv)144and®8 = po + pcgo - ~7 (32.11)* * 144Combining Eqs. (32.5) through (32.7) with Eqs. (32.10) and (32.11) andrearranging yieldsOil— ~. t %•„} _ _ ?„ d . S }~ (32-12)« I A, 1 aV • K • — VP + CGL - 'v^J Po,c 5rWater^L = ~~ U~ (32.13)f —\ w iTEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 325Gas(32.14)d_'dtB^ B,The gravity and capillary contributions to the phase pressures have beencollected in the terms CG0, CGW, and CGg:CG„ = -V • K•Z V= -V • K(32.15)(32-16)and,CG„ = V • K - -*-, , 144- VegoJLtt(32.17)>V \P.1 A i \ n I COW t A A144 / S I 144Essentially our task is to solve Eqs. (32.12) through (32.14) and saturationconstraint Eq. (4.20) for the four unknowns P0,S0,SW, and Sg. All other physicalproperties in the equations are known, in principle, as functions of the fourunknowns, or from field and laboratory data.32,3 The Pressure EquationThe procedure used in WINB4D to solve the flow equations requires thatwe first combine Eqs. (4.20) and (32.12) through (32.14) such that we have onlyTEAM LinG - Live, Informative, Non-cost and Genuine!326 Principles of Applied Reservoir Simulationone equation remaining for the unknown pressure P0. We proceed by using thefollowing shorthand for Eqs. (32.12) through (32.14):OilWaterr d f . OL = — 4>-Jt (32.19)a* I *JGasI. =aSL R.-S^ RS"g "so~o(32.20)whereL=V-K- — VP+CG- —, (32 21)/) __ rt /) ? f *»! JM' * JU J. I« Ar = v • K • -andL0 - V •gK • g + so ° +R RL \ "g "°*«A )w /CG - -^- (32.23)Recognizing that formation volume factors, gas solubilities, and porosityare functions of pressure, we use the chain rule to expand the accumulation terms(time derivatives) of Eqs. (32.18) through (32.20):OilQ\ O i3 n O CD /i T ^ "*"^TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 327WaterL - A ^"GasL. =B_ dtB. a/> R2 a p.o odP,dB^(32.26)dtB... dtB.o w odtThe saturation constraintSo + S* + Sg = 1 (32.27)is now used to remove dS^/d/ from Eq. (32.26). Differentiation of Eq. (32.27)by t and rearranging givesas as as—I = _—°- - —in. (32.28)a? a/ arSubstituting Eq. (32.28) into Eq. (32.26) and simplifying yieldsTEAM LinG - Live, Informative, Non-cost and Genuine!328 Principles of Applied Reservoir SimulationJLt ^ "°~ABsl4>dS..dt \ B. Bl dtas2 ^o ^ 3P0 a/*(32.29)Equations (32.24), (32.25), and (32.29) are three equations for the threeunknowns P0, S0, Sw. Multiplying Eq. (32.24) by (B0 - Rso Bg), Eq. (32.25) by(Bw - Rsw Bg\ Eq. (32.29) by Bg, and adding the results givesTEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 329B dt?0 *J dtBPno odt-Edt!4. s»* a^ , J« S°J?"' a* I 8P°8P0 Bg dP, B0 dP0 j dtdPdta*,(32.30)*„ a/>. B, dp0 B* dwhere some simplification has been performed. This mess can be greatlysimplified by multiplying the bracketed terms and then combining with appro-priate terms in the curly brackets. We also notice the terms involving timederivatives of S0 and Sw vanish identically. The result is+ s S\d^ -o) dpg °.dt (32.31)£,BTEAM LinG - Live, Informative, Non-cost and Genuine!330 Principles of Applied Reservoir SimulationOil and water compressibilities include the effect of gas:B0 dPo(32,32)W ^Gas compressibility isB4- £(32.34)while rock compressibility has the form1 ad:r* (32.35)0Total compressibility for the system is the sumct = cr + coso + cws» + cgsg (32.36)Employing these definitions, Eqs. (32.21) through (32.23) and (32.27) in Eq.(32.31) gives(*o - Rs« A a+ CG - —V • K - —VP + CG+ CG -*(32.37)a/>^ t -%dtEquation (32.37) is called the pressure equation. The pressure equation does notcontain any time derivatives of saturations. WINB4D is coded to solve the three-TEAM LinG - Live, Informative, Non-cost and Genuine!Part V: Technical Supplements 331dimensional, three-phase flow equations by first numerically solving the pressureequation for P0, then using the results in Eqs. (32.18), (32.19) and (32.27) to findthe phase saturations. This procedure is an example of a numerical methodknown as the IMplicit Pressure/Explicit Saturation (IMPES) procedure.TEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!ReferencesAbramowitz, M.A. and LA. Stegun (1972): Handbook of MathematicalFunctions, ninth printing, New York: Dover.Aguilera, R. 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(1984): "A Study of Spatial Approximations for Simulating FluidDisplacements in Petroleum Reservoirs," Computer Methods inApplied Mechanics and Engineering, New York: Elsevier, pp. 3-46.TEAM LinG - Live, Informative, Non-cost and Genuine!This page intentionally left blankTEAM LinG - Live, Informative, Non-cost and Genuine!INDEXA 145,153, 154,204,229,absolute permeability 25,26, 28, 285, 289, 300, 32246,274,294,318 blockaccumulation 32, 142-144,231, centered 160, 161326 pressure 148, 170, 171, 220,acoustic 107-109, 112, 115, 229, 221230, 272,279-281,290 bottomhole pressure 192, 218,acoustic impedance 107-109,115, 219,273,318229, 230, 272, 279-281,290 boundaries 64, 113, 114, 164, 165,algorithm 45, 90, 116,230,231, 169, 185,215266, 292, 304-306 boundary conditions 45,49, 50,allocation 1, 6, 146, 313, 315 52, 165, 294analytic aquifer 217,267,279 bubble point 57,121, 122, 127,aquifer 14, 60, 70, 102, 116-119, 128, 130, 148, 154, 155, 200,128, 158,178, 183, 189, 198, 204,206,207, 230,231, 258,209-212,216, 217, 223,234, 259, 271, 280, 300237, 267-269, 271, 279, 280, Buckley-Leverett 39,41,44, 49,307-309 90, 141,150,155,234aquifer model 212, 216, 217, 237, buildup 113, 170,220267, 279, 308 bulkareal model 154, 177, 192 density 107,108,289average43,46,62, 76, 85, 99,103, modulus 118, 119,245-248,104,107,108,114, 131, 178, 289183, 185,202,208,209, 217, volume 166,208, 209220,265,280, 294,296, 298,313,318 CC-D equation 44,45B calibration 168, 187barrier 185 capillary pressure 198,22-24, 26-base case 92, 139, 188, 223-225 30, 39,44,49, 52,131, 135-baseline 85, 92 139,145,154,159,169,212,bioremediation 82 257, 278, 286, 292, 323, 324black oil 36,37, 71, 121-127,143, Carter-Tracy 308, 309145, 152-154, 174, 175, 176, Cartesian 158, 160, 163, 164, 166,177, 204, 229, 260, 285, 289, 170, 173295, 300, 322 cash flow 1, 2, 77, 78, 81, 186black oil chemical 21, 68, 69, 72, 73, 121,model 126 132, 187simulator 36, 37, 71,125,127, coalbed methane 71347TEAM LinG - Live, Informative, Non-cost and Genuine!348 Principles of Applied Reservoir Simulationcompletion 7, 169, 177, 180, 207, corporate 2, 81,152, 191315 correction 83, 127, 139, 170-172,compositional 175, 178, 204, 205, 220,291model 125, 126 correlation 20, 90, 102, 108, 110,simulator 125, 153, 173 115, 132, 134, 139,212,213,compressibility 15, 17, 18, 82, 83, 260,290, 301, 30385, 98, 118, 124, 128, 154, 169, cricondentherm 122171, 181, 182,202,209,221, cross-section 112, 117,119, 177,259-262, 269, 289, 297,298, 197, 199, 215-217, 304301, 302, 309, 330 cross-section model 119,177, 217compressional velocity 107, 108, cubic 73, 124, 125,297117, 119, 229, 230, 272, 280, cylindrical 158, 160, 164288, 289computer Dmapping 158,166 Darcy's law 48, 131,133, 142,program 7,161 144,294, 299, 318concentration 32,44,45, 150, 182, data295 acquisition 116, 176conceptual model 165, 181 preparation 168condensate 121, 121,124, 127, quality 180145, 1.53, 173 datum 99, 170, 244, 262, 263, 304,configuration 231,232 3 05conformance 153 decision making 6, 81coning 160, 173 decline curve 11,16, 17, 80,90,consensus modeling 2 192conservation density 20,22-24, 28,33,37, 61,laws 91, 142, 145 68, 72, 107-109, 112, 138,144,of mass 12, 31,40, 91,142 160,220,235,245-247,249,constraint 35, 275, 286,294, 310, 262,286, 289, 295, 297-299,314,317,325,327 304contact angle 19,21-23,139 density gradient 22,23continuity equation 33,48 depletion 17,65, 67, 72, 73, 82,contour42,110, 111, 157,271, 83,85,94,117, 119, 122,128,272, 280,281 129,154, 173,179, 188, 192,convection 39,44,45, 143, 144 209,210,230, 234, 235core 12,21, 63, 96, 97, 102, 131, depth 32, 56, 98, 99, 109, 121,132, 134, 136, 158, 169, 180, 139, 165, 199, 204, 214,219,212, 292 223, 235, 244, 258, 262, 263,core 287,288,296, 305analysis 12, 97, 132 deterministic 101, 102permeability 102 dew point 122, 127comer-point 160, 161 diagonal grid 162TEAM LinG - Live, Informative, Non-cost and Genuine!differential 41,45, 52, 53,127, equivalent radius 221146,176,204, 205, 299, 300 expenses 1, 78,186, 190differential explicit pressure 314,316equations 52, 146liberation 204, 299, 300 Fdigitize 156 facilities 91, 125, 127, 128, 145,dimensionality 153,172 146,190dipping 28, 51, 52,65,116,117, falloff 114158, 161, 197, 288 fault 185, 188, 198, 216direct solution 266 files 7, 8, 149, 232, 271, 278-280directory 7 financial 82, 191disciplines 2-4, 11,93,100,119, fine grid 141,156, 157168 fingering 139-141discount rate 77-79, 81 finitediscretize 147 difference 147,162-164,285,dispersion 39,44, 143, 144,150, 294151,155,266 element 164drawdown 113, 178,183 five-point 163drill stem test 201 five-spot 59, 60,62,162drive 7,13,15,17,59-61, 65-67, flash 127,176, 204, 205, 299,30070, 73, 89, 168, 172, 173, 232 flowdry gas 122-124, 153, 173 chart 149dyadic 323 equations 31,33, 36, 37,45, 91,Dykstra-Parsons 62, 63 99,143,145-149, 162, 173,dynamic viscosity 299 286, 322, 324, 325, 331unit 97-99, 154E fluideconomic contacts 136, 139,156,169,forecast 2, 77 177,180,214recovery 1, 2, 186 modeling 124economics 61, 75, 84 movement 158, 299elevation 136, 305 properties 120,125,127,128,endpoint 233 154, 184,203,204,233, 299energy balance 145 sampling 128Enhanced Oil Recovery 57, 64, 68, type 56, 60, 112, 122, 123, 125,69 153,172environment 75, 81, 82, 95,120 fluidity 299EOR 69-71, 74 flux 31-33, 37,165, 185equation of state 124,298, 301 forecasting 92, 191equilibration 143, 159, 177 format 101, 176, 212, 232, 234,equilibrium 14,22,126,143,159, 278237, 263,295,297 formation volume factor 12,18,TEAM LinG - Live, Informative, Non-cost and Genuine!350 Principles of Applied Reservoir Simulation57, 126-128, 171, 204, 209, grain density 245-247, 249230, 259, 260, 286, 294, gravity297, 298, 300, 302, 313 drainage 65, 70, 134fractional flow 19, 26-29, 39-42, segregated 26344, 46, 49 segregation 15, 65, 129, 304fracture 71, 113, 114, 136, 169 grid 143, 147, 150, 156, 157-165,frontal advance 40-42, 48, 51-53, 167, 170, 173, 174, 176, 215,150, 162 217, 220, 223, 235, 239, 240,full field model 101, 164, 168 242, 287, 288fully implicit 148-151, 160 gridorientation 150, 158, 159, 161-G 163gamma ray 111,112 preparation 158, 164, 174, 215gas size 164, 240, 242, 243cap 8, 14, 17, 18, 65, 66, 129, gross thickness 100, 102, 156, 166,137, 173,181 198,241-243hydrate 72, 73gas-oil 12, 18, 20, 24, 29, 60, 70, H122, 127-130, 135, 137, 153, hand-drawn 102, 103160, 177, 178, 180, 200, 204, hard drive 7, 232213, 231, 263,265, 286, 293, heavy oil 122, 173300, 304, 305, 315 heterogeneity 62, 63, 103, 104,gas-oil 115, 141, 182contact 24, 60, 137, 180, 263, hierarchy of uncertainty 180304, 305 historical data 92, 151, 209ratio 12, 18, 122, 127-130, 153, history178,200, 204, 265, 300 match limitations 184gas-water 18, 72, 73, 118, 135, matching 92, 100, 104, 132,137, 145, 148, 160, 176, 264, 136, 153, 174, 176-182,324 184,188,218geologic model 172, 181, 199 Honarpour 136, 212, 213geology 173 horizontalgeometry 40, 51, 60, 160, 161, permeability 114, 134, 141,164, 173,239, 305 160, 185, 221, 320, 321geophysics 101, 106, 111,115, well 188, 230, 274, 320, 321116 hysteresis 136, 145geostatistics 101, 102, 104, 115,181 Igiga scale 96, 106, 111 immiscible 19, 20, 22-24, 29, 39,gradient 22, 23, 65, 71, 72, 121, 40, 44,49, 68-70, 136, 138144, 165, 220, 258, 262, 263, IMPES 147-151, 160, 222, 229,305 231,279,322,331TEAM LinG - Live, Informative, Non-cost and Genuine!Index 351implicit 148-151, 160,229, 285, linear 27,29, 39,43,45,48, 52,331 53, 58, 143, 147, 150, 158, 206,Improved Oil Recovery 68 233, 289, 300, 301incompressible 39, 40,49, 61, 210 LIT gas well analysis 276, 277infill 61,68 LIT method 314inflation rate 79 local grid refinement (LGR) 163,influx 14, 18, 65, 70, 116, 118, 174119, 183, 209-212, 216, 267, logging 97, 111, 132, 290269,271,279,280,307-309initialization 138, 176, 177, 208, M230, 233, 239, 262, 263, 278, Macleod-Sugden 20, 139304, 305 macro scale 97, 131integration 18, 54, 95, 97, 118, mapping 103, 156-158, 166119,302 massinterfacial tension 19, 20, 22, 44, balance 14268, 69, 139 conservation 32, 33, 35, 142,interference testing 169 285, 291inverse problem 92,184 materialinvestment 77-79, 81, 85, 188, 214 balance equation 13-15, 17, 66,irreducible 42, 117, 118, 135-137, 209-211212, 258, 264,293,294, 313 balance error 148, 155, 224,isothermal 14, 91, 128, 143, 145, 231, 291153, 229, 285, 302 matrix 4, 71, 72, 136, 148, 245,isotropic 161, 171, 221, 319 266,289iterations 266 matrix material 245, 289mega scale 96, 111, 113, 114, 131J micro scale 97, 131, 134Jacobian 148 microbial 69-71Joshi formula 320 miscible 39, 44,45, 57, 69, 143,173K miscibility 69, 70k value 295 mobility 19,24-26, 34,46,49, 50,kinematic viscosity 299 55, 61-63, 67, 69, 70, 114, 146,311-313,315L mobility ratio 25, 26,46, 50, 55,laboratory 21, 22, 71, 93, 127, 128, 61-63, 69, 70, 312131, 132, 134, 136, 139, 152, model initialization 176, 177, 208153, 169, 203, 204, 209,212, molar conservation 142, 144290, 292, 293, 299, 300, 325 mole fraction 20, 144, 261, 295,laboratory measurements 21,128, 298, 303131, 132, 139,204 molecular weight 20, 123, 261,layer cake 158, 159, 244, 288 298, 303TEAM LinG - Live, Informative, Non-cost and Genuine!352 Principles of Applied Reservoir Simulationmomentum 91, 142, 143 petrophysical model 288multidisciplinary 104 phasebehavior 121, 125N envelope 121, 123naturally fractured 173 plot file 279near wellbore 173, 181 Poisson's ratio 83, 86,290net present value 77-81 poreneutron 112 radius 23, 30Newton-Raphson 147, 148 volume 14,40,44, 56, 151,nine-point 163 166, 181,182,185, 208,nonlinear 41, 53,133,145,146, 209,217, 221-223, 265,280206 porous 6, 34, 39,48,49, 89,131-normal distribution 63, 76, 77, 85 133,140,143,245,289,299numerical pot aquifer 267,268, 307, 308dispersion 150,151, 155, 266 prediction 2, 77, 92, 93, 119, 168,174, 186-188,207,223-225O prediction process 187,224objectives 5, 85, 90, 153,154, 160, pressure165, 168, 169, 172-174, 176, correction 170, 220183, 197 equation 325, 331Ockham's Razor 3, 154, 160, 172- initialization 262,263, 304174 maintenance 67, 68, 83, 173oil productive capacity 110, 115 price forecast 1, 92, 93oil-water 21,24,26,28,46,49, 60, primary production 6470, 135-138, 148,264,286, pseudoization 159,160306,324 pulse 114oscillations 279 PVT 121, 127, 128, 145, 154,155,output 6, 8, 17,47, 63, 149,165, 169,175, 176,204,207,232,232, 270, 271,278-281 234,235,239,240,255-259,261,265,278,299-301P PVT region 175, 232,255-258,P-T diagram 121,123 278, 287parallel grid 162parachor 20 Rpartial differential equations 146 radialpattern 56, 58-62, 69, 141, 162, coordinates 173192 flow 171, 221payout 78, 81 radius 22,23, 30, 61, 95,170, 171,PEBI164 175,201-203,219, 221, 269,performance 274, 309, 318-321data 3 ratepredictions 89, 186,192,218 constraint 275, 310, 317TEAM LinG - Live, Informative, Non-cost and Genuine!Index 353of return 77, 80, 81 robustness 151,152,231realizations 3,104,116,176 rockreasonableness 3,184 compressibility 181,182, 260-recovery efficiency 13, 56-58,61, 262,269, 309, 33063, 65,68,122 quality 110, 115,134recurrent data 233,237,270,271, region 154,232,255-257, 263-279 265reflection coefficient 108,119, rock-fluid interaction 131,145230, 272, 280,290 ran summary 279regression 20,126,189, 309reliability 96, 180,183 Srepeat formation test 139 saturated 121,125,127,152, 153,representation 3,97,125,134, 234,259,300, 301137, 146, 153,156,158,190, saturation constraint 286, 325, 327208, 230,276, 301, 310, 314 scale 21,95-97,99,103,106, 111,representative elementary volume 113, 114,131, 134, 136,164,142 187,217,296reserves 73, 75-77, 80, 81, 85,190 secondary production 67reservoir seismicarchitecture 96,111,143,156, history matching 179, 180161,172,223 line 197, 198characterization 82, 89,96, trace 106, 107, 109101, 109,116,176 velocity 245,272description 92,115, 173,233 waves 106, 107, 110engineering 6, 80, 89,121,142, semi-variance 102192, 229,230 sensitivity 2, 76, 80, 82,116,131,geophysics 101,106, 111, 115, 139,157, 185, 186,188-190,116 192,224management 1,2, 65,68, 75, sensitivity82, 83, 90-93, 101, 106, analyses 188,192152,160,162,186, 187, study 190, 224190,192,197,207,229 separator 91,122,128, 146,204,simulation 6,11, 80, 89, 92, 93, 205, 299, 300119, 136,142,143,151, shear158, 192,197 modulus 245-248,289structure 7, 8, 60, 106, 145, velocity 117, 119, 230, 272,197,214 288,289resistivity 112 simulator selection 153,172,173restart 231, 263 slanted well 230restored state 293 slope 16,17,42-44,99,259revenue 78, 79, 190 solubilities 33, 34,48, 49, 53, 54,risk 93, 95, 103, 188, 189 299, 326TEAM LinG - Live, Informative, Non-cost and Genuine!354 Principles of Applied Reservoir Simulationsolution method 265,266 relative permeability 258, 292,sonic 109, 112 293source/sink 32, 143, 144, 291 throughput 151, 160, 163,221,spacing 58, 60, 61, 68, 169, 234 222SPE/WPC 75, 76 time-lapse (4D) seismic 106,227,specific gravity 29, 30, 261,298, 288303 timestep 38, 147-151, 155, 221,spontaneous potential 112 222, 224,233,264,266 270,stability 86, 120, 149, 151, 161, 272, 278-280, 291, 307224,279,291 timestepstandard deviation 76, 85 summary 38,278,279Standing-Katz 301 tomography 118steady-state aquifer 223, 308 tracer 103, 169, 177, 182stochastic image 102, 181 transient tests 113, 114, 169subsidence 82, 83, 85, 86 transition zone 23,24, 30,49, 136-surface model 90, 91, 146 139,165sweep 39, 57, 58, 62 transmissibility 103, 130, 141,sweep efficiency 39, 58, 62 185,223, 230, 235,249, 253-symnietry 141, 161 255, 292, 294, 295transmission coefficient 108,290T two-point upstream 267tank model 216,230Taylor 150 Uteam 4-6, 76, 89, 90, 93,114, 132, uncertainty 93, 104, 132, 157, 180;134, 135, 168, 170, 174, 184, 183, 188,192189,190 unconformity 198,199,223temperature 65,91, 92, 121 -124, undersaturated 94,127, 129, 154,127-129,139, 143, 145, 153, 155, 192, 197, 200,204,206,177, 261, 295-299, 301-303 207,209, 210, 234,259tensor 134, 144, 163, 287, 323 uniqueness problem 184terminal 309tertiary production 68 Vthermal 68-70, 91,112, 145, 173, validity 92, 116, 128, 139, 179,177, 187 180,186, 191, 192,293thickness 12, 58, 61, 82, 83, 85, vector 36, 37, 102, 148, 285100-102, 110, 137, 139, 146, velocity 33, 37, 42, 44, 49, 50, 52,156, 166, 182, 198,208,219, 53, 107-110, 117, 119, 144,223, 241-244, 269, 274, 305, 150, 229, 230, 245, 272, 280,306, 309, 318, 320 , 321 286, 288, 289three-dimensional model 222 verticalthree-phase conformance 153flow 33, 331 equilibrium (VE) 159TEAM LinG - Live, Informative, Non-cost and Genuine!Index 355well 273, 319 wellviscosity 25, 28-30, 34,46, 61, 63, density 61, 6869, 70, 126, 128, 133, 140, 144, log 12, 95, 96, 111, 112, 114,171,202, 206, 207, 259, 260, 131, 132, 139, 182, 198,269,287, 294,299, 302, 309, 199, 214, 216, 219, 223318 model 90, 91, 143, 145, 146,viscous fingering 139 181,218,219voidage 274, 312 pattern 56, 60volatile oil 122, 124, 153 productivity 311volume element 40,99 report 280volumetric 12, 18, 26,29,40, 56- spacing 60, 61, 6858, 62, 89, 103, 166, 177, 208, testing 113211,212, 216, 222 wellbore 61, 86, 90,91, 95, 123,127-129, 132, 143, 145, 146,W 160,169-171, 173,180,181,water 201, 219-221, 274, 318, 320,drive 65, 67, 173 321saturation 24,26,40-44,46,47, wellbore50, 117,119,136, 137, 148, model 90, 91, 145198,202,212,213,257, storage 171, 221258,263,264,271,293, wet gas 122294, 307 wettability 19-22, 68, 135, 139,water-oil 292,293contact 138, 139, 214,223, window 164, 165, 168, 185, 188263, 304, 305 workover 169ratio 141, 155, 177, 178,207,264 Ywaterflood 26, 42,43, 57, 63, 68, yield 11, 69, 76, 90, 121, 124, 126,70, 73, 90, 134, 135, 141, 150, 132, 145, 148, 165, 185, 190,155,188,234 192wave 86, 107-110,290Weinaug-Katz 20, 139 ZWelge42 Z-factor301TEAM LinG - Live, Informative, Non-cost and Genuine!