Computing Risk for Oil Prospects: Principles and Programs (Computer Methods in the Geosciences)

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Computing Risk for Oil Prospects: _ Principles and Programs

COMPUTER METHODS IN THE GEOSCIENCES Daniel F. Merriam, Series Editor Volumes in the series published by Elsevier Science Ltd Geological Problem Solving with Lotus 1-2-3 for Exploration and Mining Geology: G.S. Koch Jr. (with program on diskette) Exploration with a Computer: Geoscience Data Analysis Applications: W.R. Green Contouring: A Guide to the Analysis and Display of Spatial Data: D.F. Watson (with program on diskette) Management of Geological Data Bases: J. Frizado (Editor) Simulating Nearshore Environments: P.A. Martinez and J.W. Harbaugh Geographic Information Systems for Geoscientists: Modelling with GIS: G.F. Bonham-Carter + Structural Geology and Personal Computers: D. DePaor (Editor) ^Volumes published by Van Nostrand Reinhold Co. Inc.: Computer Applications in Petroleum Geology: J.E. Robinson Graphic Display of Two- and Three-Dimensional Markov Computer Models in Geology: C. Lin and J.W. Harbaugh Image Processing of Geological Data: A.G. Fabbri Contouring Geologic Surfaces with the Computer: T.A. Jones, D.E. Hamilton, and C.R. Johnson Exploration-Geochemical Data Analysis with the IBM PC: G.S. Koch, Jr. (with programs on diskettes) Geostatistics and Petroleum Geology: M.E. Hohn Simulating Clastic Sedimentation: D.M. Tetzlaff and J.W. Harbaugh •Orders to: Van Nostrand Reinhold Co. Inc., 7625 Empire Drive, Florence, KY 41042, USA. Related Elsevier Science Ltd Publications Books GAAL & MERRIAM (Editors): Computer Applications in Resource Estimation: Prediction and Assessment for Metals and Petroleum HANLEY & MERRIAM (Editors): Microcomputer Applications in Geology I and II MACEACHREN & TAYLOR (Editors): Visualization in Modern Cartography TAYLOR (Editor): Geographic Information Systems (The Microcomputer and Modern Cartography) Journals Computers & Geosciences Full details of all Elsevier publications available on request from your nearest Elsevier office. 4- In preparation

Computing Risk for Oil Prospects: _ Principles and Programs John W. Harbaugh Professor of Geological Sciences Petroleum Engineering Stanford University Stanford, California USA

and

John C. Davis Chief, Mathematical Geology, Kansas Professor of Petroleum Engineering University of Kansas Lawrence, Kansas USA

Geological

Survey

Johannes Wendebourg Research Scientist, Geology/Geochemistry Institut Frangais du Petrole Rueil-Malmaison FRANCE

PERGAMON

Division

U.K.

Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, U.K.

U.S.A.

Elsevier Science Inc., 660 White Plains Road, Tarrytown, New York 10591-5153, U.S.A.

JAPAN

Elsevier Science Japan, Tsunashima Building Annex, 3-20-12 Yushima, Bunkyo-ku, Tokyo 113, Japan

Copyright ©1995 J. W. Harbaugh, J. C. Davis and J. Wendebourg All Rights Reserved. No part of this publication may be reproduced, stored in a retreival system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. First edition 1995 Library of Congress Cataloging in Publication Data Harbaugh, John Warvelle, 1926- . Computing risk for oil prospects: principles and programs/John W. Harbaugh, John C. Davis, Johannes Wendebourg. p. cm. Includes bibliographical references and index. 1. Petroleum-Prospecting-Data processing. 2. Risk assessment—Data processing. I. Davis, John C. II. Wendebourg, Johannes. III. Title. TN271.P4H277 1995 622M828-dc20 95-43691 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library.

ISBN 0 08 041890 2

Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn

DISCLAIMER Neither Elsevier Science Ltd nor the authors nor their assigns nor any employer of the authors shall be liable for any special, indirect, consequential, incidental or other similar damages suffered by the user or any third party, including, without limitation, damages for loss of profits or business or damages resulting from use or performance of the software, the documentation, or any information supplied by the software or documentation, whether in contract or in tort, even if Elsevier Science Ltd or its authorized representative has been advised of the possibility of such damages; and Elsevier Science Ltd and the authors or their assigns, and any employer of the authors shall not be liable for any expenses, claims or suits arising out of or relating to any of the foregoing. See also pages 26 and 386.

USER ASSISTANCE AND INFORMATION For advice and information about the software accompanying this book, please contact SURFACE III Office, Kansas Geological Survey, 1930 Constant Avenue, Lawrence, KS 66047, U.S.A. Fax: (913) 864 5317. e-mail: [email protected]

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CONTENTS ........................................... ............................................ GETTING THINGS ROLLING ...................................... Acknowledgments ................................................

Foreword . . . . . . . . . . . . Preface . . . . . . . . . . . . . .

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1 The Challenge of Risk Assessment

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1

THE NATURE O F EXPLORATION .................................. 2 Non-Geological Aspects of Risk ..................................... 3 DEFINING PROBABILITIES . . . . ................................. 4 Schemes t h a t Yield Scores but not Probabilities ....................... 5 Assessments by Individuals versus Groups ............................ 6 Assessment Using Ill-Defined Geological Factors ....................... 7 Additive Schemes for Predicting Discoveries .......................... 8 Multiplicative Schemes for Predicting Discoveries ...................... 9 Multiplicative Schemes for Estimating Amount of Oil . . . . . . . . . . . . . . . . . 10 GEOLOGICAL UNCERTAINTY AND OIL OCCURRENCE . . . . . . . . . . . . . 12 REGIONAL HYDROCARBON ENDOWMENT ........................ 15 Total Endowment . . . . . . . . . . .................... 17 Endowment Initially in Place in Pools .............................. 17 Endowment of Potentially Producible Hydrocarbons . . . . . . . . . . . . . . . . . .18 Estimating Endowments of Producible Hydrocarbons . . . . . . . . . . . . . . . . . .18 Exploration Maturity ............................................ 20 Statistical Distribution of Field Sizes ............................... 21 PRINCIPAL PREMISES OF THIS BOOK ............................ 22 Computer Programs for Analyzing Risk ....................... . . 25 Examples and Data .............................................. 26

2 . Field Size Distributions

.......................................... 29 ESTIMATING “Q” ................................................ 29 OIL FIELD POPULATIONS .................................. STATISTICS OF FREQUENCY DISTRIBUTIONS . . . . . . . . . . . . . . . Measures of Centrality ........................................... 37 Measures of Dispersion ........................................... 38 Percentiles ..................................................... 39 CUMULATIVE PROBABILITY PLOTS .............................. 42 Probability Estimates from Frequency Distributions . . . . . . . . . . . . . . . . . . .46 CAUTION: FUTURE DISCOVERIES . . . . . ...................... 49 Occurrence Probabilities Versus Discovery abilities . . . . . . . . . . . . . . . .50

3 . Success, Sequence, and Gambler’s Ruin

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55

SUCCESS RATIOS AND DRY HOLE PROBABILITIES . . LONG -TERM LUCK AND BINOMIAL DISTRIBUTION . Graphs of Binomial Distribution ....................

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4 Estimating Discovery Size from Prospect Size . . . . . . . . . . . . . . . . . . 71

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STATISTICAL CORRELATIONS BETWEEN PROPERTIES . . . . . . . . . . . . 71 Fitting Lines . . . . . . . . . . . . . 74 ESTIMATING VOLUMES FROM SEISMIC MAPS ..................... 80 Characterizing Prospects ......................................... 81 Incorporating Prediction Error in Probability Estimates . . . . . . . . . . . . . . .84

5 Outcome Probabilities and Success Ratios

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89

Computing Risk for Oil Prospects GEOLOGY AND DRILLING RESULTS .............................. 89 Bayesian Conditional Probabilities and Success Ratios . . . . . . . . . . . . . . . . .93 Trend Surface Residuals and Conditional Success Ratios . . . . . . . . . . . . . . .97 BAYESIAN REVISION O F REGIONAL SUCCESS RATIOS ............. 99 Some Algebraic Background ...................................... 104 BAYESIAN REVISION O F DISTRIBUTIONS ........................ 107 Bayesian Revision of Expected Field Size ........................... 108

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............................................. 111 APPEAL O F THE SIMULATION APPROACH ....................... 111 STEPS IN MONTE CARL0 SIMULATION .......................... 113 Risked or Unrisked Distributions? ................................. 116 Simulating Field Size Distributions in Magyarstan . . . . . . . . . . . . . . . . . . .118 Simulating a Specific Prospect in Magyarstan ....................... 121 Incorporating Risk in Simulation .................................. 124 DISTRIBUTIONS AND PARAMETERS ............................. 128 ARE GEOLOGIC PROPERTIES INDEPENDENT? . . . . . . . . . . . . . . . . . . . 133

6 Modeling Prospects

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7 Mapping Properties and Uncertainties

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COMPUTER CONTOURING ...................................... HOW CONTOUR MAPS ARE MADE ............................... Conventional Contouring Programs ................................ Trends and Residuals ........................................... GEOSTATISTICS IN RISK ASSESSMENT ........................... The Semivariance ..............................................

.............................................. ..................................

d a w a y Out

137 137 139 139 142 149 150 159 165

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8 Discriminating Discoveries and Dry Holes ...................... 173 COMBINING GEOLOGICAL VA Misclassification of Drill Holes The Conditional Probability of ASSESSING A NEW AREA ........................................ 181 Combining Individual Geologic Maps .............................. 182 Mapping Combined Geological Properties .......................... 183 UPDATING ASSESSMENTS ....................................... 189 EXPLORING MAGYARSTAN TARGET AREA ...................... 191 THE IMMATURESTAGE ......................................... 192 Discriminant Analysis in Immature Stage ........................... 196 Including Target Area Information in Probability Function . . . . . . . . . . . . 199 Planning for Subsequent Drilling .................................. 199 THE INTERMEDIATE STAGE ..................................... 202 Comparing Results ............................................. 203 Completing the Move t o Target Area .............................. 207 THE MATURE STAGE ........................................... 209 THE FINAL STAGE . . ........................................ 211

9. Forecasting Cash Flow for a Prospect

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FINANCIAL OVERVIEW ......................................... DISCOUNTED NET CASH FLOW ANALYSIS ....................... Risk Analysis Tables ............................................

............................................. ............................................. overy’s Future Production ...................... USING CASHFLOW . . . . . . . . . . . . . . . ......................... Information Required by CASHFLOW ............................. OUTPUT FROM CASHFLOW .....................................

215 215 218 219 220 221 221 226 229 252

Contents EXAMPLE APPLICATIONS O F CASHFLOW . . . . . . . . . . . . . . 255 Example 9.1 ............................. Example 9.2 . . . . . . ......... ........................ 258 Example 9.3 . . . . . . ..................................... Example 9.4 ................................................

10. The Worth of Money

.......................................... 265 A DRY HOLE VERSUS A DISCOVERY ............................. 265 T h e Utility Function ............................................ 266 Obtaining a Utility Function ..................................... 268 Utility Function in Response to Prospect Proposals . . . . . . . . . . . . . . . . . .275 Graphing a Utility Function with RISKSTAT ....................... 278

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............................. 279 RATS LINK OUTCOMES W I T H RISK .............................. 279 H O W RATs T R E A T INFORMATION ............................... 285 Row-by-Row Description of Tables Generated by RAT . . . . . . . . . Incorporating Utility in RAT .............................. EXAMPLE RISK ANALYSIS TABLES . . . . . . . . . . . . . . . . . . . W h a t We Can Conclude About RATs . . . . . . . . . . . . . . . . . . . . . . . . . .298 DECISION AIDS F O R ACTIONS ........................ Sensitivity of EMVs to Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 CONSTRUCTING DECISION TABLES ............................. 312 DECISION T R E E S . . . . . . . . . . . . . . . ........................ 312 Expected Utility Tables and Trees ......................... 314 RAT Cages . . . . . . . . . . . . . . . . . . . ......................... 319 OVERVIEW AND A LOOK TO T H E F U T U R E ....................... 319

11 RATS, Decision Tables, and Trees

12. Bringing It Together ...........................................

323 RISKING T H E ROSKOFF P R O S P E C T .............................. 323 T H E PRIZE O F F E R E D . . . . . . . . . . . . . . .................... 324 TROYSKA AND ROSKOFF AREAS COMPARED .................... 327 ESTIMATING ROSKOFF’S DRY HOLE PROBABILITY . . . . . . . . . . . . . . . 328 Conditional Dry Hole Probability for Roskoff Well Location ........... 331 PROBABILITIES ATTACHED T O FIELD VOLUMES . . . . . . . . . . . . . . . . . 333 Estimating Probabilities from Area of Structural Closure . . . . . . . . . . . . . .335 Probabilities Conditional on Three Geological Variables . . . . . . . . . . . . . . . 337 ROSKOFF P R O S P E C T FINANCIAL ANALYSIS . . . . S O M E FINAL POINTS ...............................

Bibliography

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353

Appendix A- SOFTWARE INSTALLATION of RISK: Software to perform probabilistic assessments and financial calculations in Computing Risk for Oil Prospects .......................

Appendix B

- RISKSTAT MANUAL: Linked computer programs t h a t perform statistical analyses in RISK

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Appendix C - RISKMAP MANUAL: Linked computer programs t h a t perform t h e mapping function in RISK

Appendix D - RISKTAB MANUAL: Linked computer

387

. . . . . . . . . . . . . . . . . . 405

. . . . . . . . . . . . . . . . . . . 427 .............................................................. 443

programs t h a t perform financial calculations in RISK

Index

383

Computing Risk for Oil Prospects

SERIES EDITOR'S FOREWORD Computing Risk for Oil Prospects continues the tradition of the Series in bringing to the geological profession books that emphasize the quantitative side of the geosciences. The authors are recognized for their contributions to geomathematics, geostatistics, and computer simulation in the earth sciences. Here they have applied their skills to quantitative assessment of oil and gas prospects. Building upon their pioneering research in the application of computers to the exploration for oil and gas, they now extend their methodologies to include the financial aspects of prospect appraisal. The "risks" they discuss are financial, the potential losses or gains that occur when a geologist defines a prospect and recommends that it be drilled. The public is increasingly conscious of risk analysis as a scientific pursuit, usually in a dramatic context such as estimating the chances of a rare, catastrophic event: nuclear war, a terrorist attack, an airplane crash. It may be concerned with violent natural processes such as floods, tsunamis, or hurricanes. The attempts of geologists to predict long-term events— earthquakes, volcanic eruptions, meteorite impacts—seldom fail to catch the public's fancy. However, the less-spectacular side of risk analysis is not concerned with disasters (except the financial type), but it is, perhaps, more important in our everyday lives. Economic risk is concerned with monetary investment and the potential for gain versus the possibility of loss. This form of risk motivates every business activity from buying insurance to playing the stock market; it is the very lifeblood of our economy. In Computing Risk for Oil Prospects, the authors emphasize that the search for oil and gas is both a business activity and a scientific endeavor, and they apply the tools of modern risk analysis to all the aspects of petroleum exploration. As the authors note in the Preface, they have conducted research on risk analysis in one context or another for about 25 years. Their ideas, formulated, tested, presented, and revised throughout this period, have culminated in a book that off^ers the reader an in-depth view of the subject as applied to oil and gas exploration, including the most favorable choice of investment strategies. Risk analysis is presented not only in words and pictures, but in the form of "hands-on" computer software that allows the reader to practice what the authors preach. All that is asked of the reader is an interest in petroleum exploration, a modest background in geology, some familiarity with statistics, and the desire to make the most money in the oil business with the least amount of risk. This book should appeal not only to petroleum geologists, but also to avid explorationists and investors, regardless of background. Every point is clearly explained; each argument is accompanied by examples. This book is a worthy addition to the authors' earlier contributions to petroleum assessment and prospect evaluation. D.F. Merriam

mil

PREFACE.

.Getting Things Rolling We have written this book for explorationists, those who explore for oil and gas. Most explorationists are geologists or geophysicists, but some may be engineers or landmen. We presume that our readers will not have much background in statistics, probability, or computer mapping technology, and so our discussions of these topics begin at an introductory level. However, the risk assessment procedures developed in this book reach a level that is substantially advanced over procedures in common use today, in either large or small companies. To help the reader understand the methods, we have included a library of computer programs and numerous data sets that extend the examples in the text. This book has a long history. Its roots extend back to 1966 when John Harbaugh developed a course at Stanford University on mineral resource appraisal. The class included risk and uncertainty, and one exercise used decision tables in an oil-exploration context, employing early computer software to generate the tables. At that time the idea of a comprehensive computer-based system for oil-exploration decision analysis began to crystallize, but there were formidable problems that could not be overcome because of the relatively primitive state of the available computers and software. Computer contouring, for example, would be an essential component of the risk assessment system, but in 1966 contouring programs were still under development and seldom used outside research centers. Meanwhile, John Davis began work at the Kansas Geological Survey on the statistical and computer mapping procedures that led to some of the

Computing Risk for Oil Prospects Survey's pioneering publications in the field in the 1960s and 1970s. Convinced of the need for effective computer mapping, and with the backing of Dan Merriam, Davis encouraged Robert Sampson to develop the computer mapping program that came to be known as "SURFACE II." By the early 1970s, the Kansas Survey was established as a leader in geological computer applications, and advances at the Survey and elsewhere suggested that it was technically feasible to create an integrated oil-exploration decision system. The Survey had some of the key components in hand, such as SURFACE II, and was interested in such a system in hopes that it would stimulate exploration in the State. In 1971 John Harbaugh was invited to join the Survey as a Visiting Research Scientist. During a sabbatical leave for part of the 1971-72 academic year, Harbaugh devised a plan for the Kansas Oil-Exploration Decision System, an integrated system for exploration risk assessment which quickly acquired the acronym, the "KOX" system. Initially the system existed only as a concept that was presented before several geological societies. A description of the KOX system was published by the Kansas Geological Survey (Harbaugh, 1972). The best-laid plans often go astray. Before development work was seriously underway, it became clear that computing technology was not yet advanced enough to support the integrated KOX system concept. The early 1970s were still in the era of the mainframe computer, and virtually any computing task required the mastery of complicated job-control languages and file-handling procedures. Although in principle it was feasible to construct an integrated decision system, the practical obstacles were formidable. The user would be required to go back and forth between separate computer programs, creating, translating and transferring files at every step, an exercise that often taxed the skills of even computer systems wizards, let alone those of working geologists for whom the software was intended. By the end of 1973 the KOX project languished; although the concept was sound, it was a decade ahead of its time. However, the research work didn't stop totally. A petroleum exploration decision system needs statistical underpinnings in order to generate the exploration outcome probabilities. John Davis continued to work on statistical applications at the Survey, where he was joined by John Doveton and a succession of Visiting Research Scientists. Meanwhile, John Harbaugh's students worked on resource appraisal applications, often in conjunction with the Kansas Geological Survey. In 1974, a research conference on the topic of evaluating undiscovered oil and gas resources was held at Stanford University. Several papers on procedures for generating exploration outcome probabilities were presented, based on work at the Survey and at Stanford. The conference proceedings were published by the American Association of Petroleum Geologists (Haun, 1975). Industry participants at the conference suggested that increased focus on probability methods was warranted, with the result that

Preface a second conference was organized at Stanford in 1975 by Davis, Dove ton, and Harbaugh. This second conference stimulated the publication of a book, Probability Methods in Oil Exploration^ by Harbaugh, Doveton, and Davis, that appeared in 1977. At that time, the oil industry was booming because of OPEC's success in raising oil prices throughout the world. There was great interest in methods for appraising hydrocarbon resources and assessing exploration outcome probabilities. The authors were asked to conduct another research conference on probability methods, but instead they concluded that practical training in the topic was needed, and so devised a week-long short course that was given in the United States and abroad from 1977 through 1980. When the training course was initially developed, computing was still in the mainframe era, although the first primitive personal computers apxpeared in 1977. Concepts introduced in lectures were illustrated by a manual of problem sets, but of necessity these involved only hand computations aided by the use of pocket calculators. The exercises proved difficult for participants because of the excessive arithmetic labor required. The objective of the course was that participants could put the principles they were taught into practice after they returned to their corporate homes, but few did so because of the difficulties of acquiring and coordinating the necessary software in a mainframe environment. In 1989, Tim Coburn of Marathon Oil Company requested that we teach a new version of the short course, with problem sets to be analyzed by course participants using personal computers. Davis and Harbaugh agreed to conduct the course, and with the assistance of Geoff Bohling of the Kansas Geological Survey and Johannes Wendebourg of Stanford University, the necessary computer software was written or adapted from existing sources. The risk assessment short course was given at Marathon's Petroleum Technology Center in 1990 and 1991, featuring a new manual that included the problem sets for analysis by personal computer. In 1993 and 1994, we adapted the manual and computer programs of the short course into a self-contained document incorporating both theoretical material on risk assessment and new computer software. The result is this book and library of computer programs. Acknowledgments This book and the accompanying computer software reflect the contributions of many people. First, we thank the students, in both the academic and industrial worlds, who have taken our courses on risk analysis and probability methods in oil exploration. We have taught risk analysis courses at Stanford University for nearly three decades, and have offered training courses for the oil industry for almost two decades. Our aggregate number of students is in the hundreds, all of whom contributed feedback in the form of comments and discussions, research and term papers, and responses to xi

Computing Risk for Oil Prospects problems and exercises. This input has helped shape our ideas and the viewpoints we present here. Next, we must thank those who have directly helped in the preparation of this book. Two persons deserve special mention: Jo Anne DeGraffenreid of the Kansas Geological Survey played a critical production role as editor, copy editor, and preparer of the camera-ready text, not only for this volume but also for the many preceding generations of short-course manuals. Our geologist colleague John H. Doveton of the Kansas Geological Survey was a collaborator on the earlier book. Probability Methods in Oil Exploration (1977), and on several short courses. He critically read drafts of many of the chapters of this book and offered numerous suggestions for their improvement. We also gratefully acknowledge the many years of support of our research into petroleum risk assessment by the Kansas Geological Survey under the directorship of Lee Gerhard and his predecessor, William Hambleton. In addition, Marathon Oil Company's Petroleum Technology Center generously provided the use of its research facilities, library, and data files. Their sponsorship of the training courses on which this text is built was essential. We especially appreciate the opportunities to discuss our ideas and concepts with Marathon management and personnel, and offer a special thanks to Tim Coburn, who encouraged us to place our short course material and exercises in fully computerized form, leading directly to the software that accompanies this book. Our procedures were converted into computer code by staff and students at the Kansas Geological Survey and Stanford University, in particular by Geoff Bohling and Bob Sampson at Kansas, and former Stanford students Dan Tetzlaff and Chris Murray. Claudio Bettini, a long-term member of the exploration staff of Petrobras in Brazil and former student at Stanford, suggested some innovative procedures which we have incorporated in the software. Warren Kourt of Stanford University and of Mineral Acquisition Partners, helped test the cash flow program and critiqued the details of the cash flow analysis presented in Chapter 9. Finally, John Davis wishes to acknowledge financial support, in the form of a Senior Fulbright Fellowship, from the Austrian-American Educational Commission during the time of final preparation of this book. Facilities were provided by the Applied Mathematics Department of the Montanuniversitat-Leoben, Joanneum Research Institute, and Geo- und Umweltinformatik, Leoben, Austria.

T r a d e m a r k N o t i c e : RISKMAP, RISKSTAT, and RISKTAB are trademarks of Davis Consultants Inc. SURFACE III is a trademark of the Kansas Geological Survey. All other designations used by manufacturers and sellers to designate their products, brand names, product names, or tradenames, are trademarks or registered trademarks of their respective owners or companies.

Xtl

CHAPTER

1

The Challenge of Risk Assessment This book is about financial risk in oil exploration, particularly the risk associated with the drilling of prospects. "Risk" connotes the possibility of loss and the chance or probability of that loss. In oil exploration, risk deals with the potential for loss, such as the cost of drilling a dry exploratory hole, versus a compensating gain such as the discovery and production of commercial quantities of oil. Throughout this book we present the view that risk can be treated systematically e use of statistics and probability theory, and that geologists and oil explorationists can make effective use of the large amounts of relevant data that are potentially available. In this process, the magnitudes of risks may not be reduced but they can be made explicit, and better decisions will result if they are based on realistic assessments of the risks involved. In our view, risk should be expressed in the form of numerical probabilities estimated on the basis of frequencies as far as feasible. In oil exploration there are many aspects of risk, but we will concentrate on those elements of risk attached to the drilling of individual prospects. A "prospect" may be defined as a specific locality within an area where we possess or may acquire a lease or concession and which we interpret to have geological or economic characteristics that may warrant testing by drilling. The alternative outcomes to the drilling of a prospect can be expressed as a discrete probability distribution. There is a probability that the hole will be dry and a complementary probabihty that it will be a discovery

Computing Risk for Oil Prospects — Chapter 1 well. If it is a discovery, the distribution can be subdivided to express the probabilities attached to different volumes of oil that may be discovered. If we can estimate the form of this distribution, it can be linked with the financial gains or losses corresponding to each of the alternative outcomes in the distribution. This linkage provides a succinct summary of the alternative financial gains and losses that can result when the well is drilled and weights them according to their probabilities of occurrence. The challenge in assessing risk is to obtain and use the most appropriate probability distribution for a prospect, and to link the distribution with procedures for financial analysis. That is what this book is about. Risk and uncertainty are associated with drilling operations, with field development after discoveries have been made, and with production. These are important components of the complete spectrum of risk in the oil industry, but are beyond the scope of our book, which is concerned only with evaluating prospects.

THE NATURE OF EXPLORATION In a general way, oil exploration proceeds in a sequential manner. First, "plays" usually are generated within a region. Plays consist of areas or geographic trends that seem especially promising from an exploration standpoint. A play may be a concept that is conceived by a single individual or company, or may be widely perceived and of broad interest to the entire exploration industry. Plays often are defined by geological attributes that suggest the presence of producible hydrocarbons, or plays may be touched off* by an actual discovery. At one extreme, a play may be based on details of geology and geophysics, and at the other may simply represent the collective behavior of industry (the "herd instinct") in a competition for acreage touched off by one company embarking on a vigorous leasing campaign in a specific area. An appropriate land position is critical, for there can be little serious involvement in a play by a company unless it has access to potential drilling sites. A land position may consist of a concession or a leasehold, and can be acquired at the beginning or late in the evolution of a play. Leases may be acquired early as a speculative investment, or their acquisition may be delayed until specific prospects have been generated. To reach their ultimate value, however, leases must be tested by drilling and production obtained, which requires significant additional investments. Before drilling, prospects within plays should be perceived as having geological and financial merits that are commensurate with the risks involved. If exploratory holes are warranted, they should be located on leases or concessions in positions that coincide with drill able prospects. As in real estate ventures, the three most

The Challenge of Risk Assessment important factors are "location, location, and location." If a discovery warrants additional drilling, stepout wells can expand the field, with the locations and numbers of stepouts depending on the geographic expanses of leaseholds and the magnitude of the field that has been discovered. The degree of geographic coincidence between a newly discovered field and the leaseholds is critical. Hindsight provides scant comfort to an operator who discovers a field, which on subsequent stepout drilling proves to lie mostly under leases held by others! The orderly development of plays and prospects is an ideal that is seldom achieved in practice. Exploratory wells may not be part of established plays, and even may be drilled with little or no geological information at locations where there is no perceivable prospect. Pending expiration dates for leases and concessions may stimulate drilling regardless of geological considerations. With neither an established play nor prospect, an exploratory hole may be drilled to protect the perceived value of a land position that would be lost if a hole were not drilled. If the hole results in a discovery and establishes production, the lease on which it is drilled will be preserved as long as oil or gas are produced by virtue of "held-by-production" (HBP) clauses that are standard in most lease and concession agreements. However, issues of risk are present regardless of the rationale (or lack thereof) for initially acquiring leases.

Non-Geological Aspects of Risk Risk assumes many forms, including the risk of future adverse changes in prices and costs. Most explorationists are uneasy about assuming a future price for oil or gas, or the magnitude of future expenses such as taxes and operating costs. Nevertheless, such assumptions must be made in order to arrive at rational decisions about the drilling of a well or the making of any other investment. A common assumption is that neither prices or costs will change much in the foreseeable future, but we know from experience that this is unrealistic. Taxes on oil and gas production increased markedly in the 1970's, and oil and gas prices have fluctuated over a tenfold range since 1973. The severe oil price declines in 1986 were financially disastrous for many small oil and gas producers. Clearly, the potential for increases in taxes or declines in market prices are risks that cannot be ignored. Expropriation is an example of a risk that has extreme consequences for an oil company operating in a foreign land. American and European oil companies usually attach a substantial risk factor to investments in nations where there is a history of expropriation, or where the poUtical situation seems potentially hostile to private industry. Mexico expropriated foreign-owned oil and gas producing properties in 1938, and Venezuela and Peru later followed with similar sweeping expropriations. The memory of

Computing Risk for Oil Prospects — Chapter 1 these past actions still tempers the present investment attitudes of private industry, and influences the willingness of companies to explore in some areas of the world. Fortunately, the risk of expropriation is small or nonexistent in western Europe and the United States and Canada. However, even in politically and economically stable countries, adverse regulation and litigation may have effects that are almost as severe as expropriation. States and municipalities in the United States, and even the Federal government itself, have at times prohibited oil companies from producing from properties they had already explored and developed, citing the potential for environmental damage as justification. While not "expropriation," the consequences of such restrictive action may be even more severe because there may be Uttle or no financial compensation. Clearly, the risk of being shut down by administrative or judicial decree cannot be ignored. Physical disasters also pose a risk of economic loss. Oil spills from outof-control wells may cause severe environmental damage, and fires, floods, hurricanes, tornadoes, high waves, landslides, earthquakes, and other natural disasters may pose significant risks. For example. Hurricane Andrew heavily damaged off-shore producing platforms in the Gulf of Mexico in 1992, and violent storms have damaged and destroyed producing platforms in the North Sea. Such physical risks vary with location; in some places the risk is small, while elsewhere it may be substantial.

DEFINING PROBABILITIES What does the word "probability" mean? The general dictionary definition is "the quality or state of being probable, or likely to occur," which is annoyingly circular. Technical dictionaries concede that probability is, strictly speaking, undefinable, even though most people have an intuitive feel for the concept. We can consider probability to be the relative frequency with which a specific outcome is observed within a large number of similar circumstances (that is, the past history of an event), or the strength of our belief that some outcome will occur in the future; either concept leads to the same mathematical formalism of probability. Both approaches can be used for our purposes, provided we express probability on a continuous scale that ranges from 0.0 to 1.0, or as a percentage between 0 and 100%. Probability statements commonly are used to express uncertainty about future events, such as the likely state of tomorrow's weather or the magnitude of next year's wheat harvest. The uncertainty remains until the event actually happens, although the uncertainty may be reduced if we have effective forecasting procedures. When we drill an exploratory hole, we use a probability statement in a slightly diff^erent sense. Uncertainty exists because the outcome of the hole lies in the future, but uncertainty also exists

The Challenge of Risk Assessment because we lack information about oil and gas that may be present. The actual presence or absence of oil and gas was fixed by nature long ago—the uncertainty arises because our knowledge is incomplete. Once we drill the hole, we obtain new information and the uncertainty is reduced. We may estimate probabilities from many different sources of information. Frequencies are widely used; weather forecasts expressed as probabilities ("the chance of rain today is 30%") are based on conditional relationships in the extensive historical weather record. Forecasters relate previous weather conditions with observed frequencies of temperatures, wind velocities, and the motions of air masses. In a similar fashion, probabilities for the various outcomes of exploratory holes could be based on frequencies of drilling results related to geological factors observed in the historical records of the industry. Probabilities also can be based on theoretical relationships, as in wagers on roulette wheels or card games where the rules are fixed and the number of possibilities is prescribed. Probabilities can be based also on well-established empirical relationships, although this borders on an expression of belief. Sometimes, neither frequencies nor theory apply and a probability estimate consists simply of a person's subjective opinion or "degree of belief," as for example, a probability assigned to the number of casualties in a nuclear war, an event for which frequency data are scant and for which theory presumably does not exist. Many authors have discussed the use of probabilities in an oil exploration context. Relevant articles and books include those by Abry (1973, 1975); Agterberg (1971); Antia (1988); Behrenbruch, Azinger, and Foley (1989); Bruckner (1978); Capen (1976, 1979); Crovelh (1981, 1984, 1986, 1988); Davis (1981, 1988); Dowds (1968); Drew (1972); Grayson (1960, 1962); Harbaugh, Doveton, and Davis (1977); Hayward (1934); Jones and Smith (1983); Kaufman (1963); Kaufman, Balcer, and Kruyt (1975); Lee and Wang (1983a, 6); McCray (1975); Megill (1985); Meisner and Demirmen (1981); Newendorp (1975); Northern (1967); Pirson (1941); Prelat (1974); Reznik (1982); Rose (1987); Roy (1975); Roy, Proctor, and McCrossan (1975); Ryckborst (1980); Singer and Wickman (1969); Sluijk, Nederlof, and Parker (1986); Smith (1980); and Solow (1988).

Schemes that Yield Scores but not Probabilities Most oil companies have formalized procedures for appraising prospects, and over the past few decades some have been modified to yield probabilities. Basically, these methods involve computing scores or ranks that express the relative attractiveness of the prospects being considered. Some older schemes (Benelli, 1967; Gotautas, 1963; andSchwade, 1967) are highly formalized and consider many geological factors on a presence/absence or

Computing Risk for Oil Prospects — Chapter 1 ranked basis. Each factor is weighted numerically according to its presumed relative importance and combined with other factors, yielding a composite numerical score for the prospect. Such scoring schemes were popular in the 1960's and 70's and are still in use. They do provide consistency in appraising and comparing prospects, but there are problems in assigning the weights to geological properties. The weights are supposed to reflect the importance of the properties with respect to the accumulation of oil and gas, but this assessment is entirely subjective and intuitive. The weightings have only a vaguely defined relationship to the frequencies of occurrence that might actually be observed (in part because no effort has been made to tabulate such frequencies). As a consequence, the scores do not incorporate uncertainty and cannot yield probabilities directly. However, a record of prospects and their scores awarded under one of these methods could be combined with the results of drilling to yield probability estimates of well outcomes conditional upon the scores.

Assessments by Individuals versus Groups One way to estimate probabilities is simply to guess them; although it is seldom admitted, most probability estimates seem to be obtained in this way. The person who originates a prospect also can supply an estimate of the probability for success, although attempting to do this in a disinterested manner may be psychologically painful. Geologists who generate prospects usually are reluctant to estimate probabilities, preferring to simply rank a prospect as "Grade A" or as "having a ten-million barrel potential." Such descriptions may be useful for comparing a prospect with competing alternatives, but they cannot be used with the formalized riskanalysis procedures needed in financial forecasting. Often the originator of a prospect is pressed to express an appraisal in the form of a discrete probability distribution having a few entries, such as a dry hole probability and probabilities for a small, modest, or large discovery. It is human nature to view one's own creation in a rosy light, and prospect originators often will do whatever it takes to "sell" their prospects. Unfortunately, subjective probabilities supplied to fit the needs of the moment are likely to be biased. Potential investors, either inside or outside the company, are then faced with the additional problem of judging the prospect originator as well as the prospect itself. Subjective estimates of outcome probabilities are widely accepted in the petroleum industry. In fact, many companies utilize probabilities that are no more than subjective guesses, although they would be reluctant to characterize them in such unscientific terms. Perhaps the best guesses are those provided by persons whose long experience in a region has allowed

The Challenge of Risk Assessment them to observe the collective results of drilling a substantial number of prospects over the years. Such a background provides an informal statistical base from which the individuals can supply reasonable estimates of outcome probabilities. Group experience can substitute for individual experience. For example, the aggregate experiences of a group of explorationists will surpass that of any single member of the group, so collective insight should be superior to that of any individual. If the group as a whole examines a prospect in detail and the group leader can elicit a genuine consensus of opinion from the group, the resulting collective subjective estimate may be the best available. A formal consensus-forming procedure called the "Delphi method" may be used in which individual divergent opinions are re-examined by the group, or less structured methods of agreement may be used. With either approach, obtaining a true consensus is difficult. For example, a manager or other person of authority may dominate the group's discussion, so instead of a true expression of collective judgment, the response may be only an echo of the dominant person's views. Project generators with reputations as "proven oil finders" often assert that they have developed an intuitive or instinctive "feel" for prospects that permits them to distinguish truly good prospects from the mediocre. Nothing succeeds like success, but it is hard to document whether these individuals actually possess psychic qualities, or whether they simply make effective use of their experience.

Assessment Using Ill-Defined Geological Factors The components of a prospect are often difficult to define with any degree of exactitude. For example, because of the lack of well control and seismic information, the geologic structure may be poorly perceived and a structure contour map may be highly interpretive. In such a circumstance, imaginative structural interpretations may be regarded as justifiable even though they are not strongly supported by the data. Sometimes a highly imaginative (but not necessarily realistic) interpretation is taken as appropriate because the data do not preclude its possibility! In areas of sparse data, only the largest scale components of geological features can be defined; smaller features cannot be resolved. This leads to the well-known tendency for large seismic structures defined with reconnaissance data to split into smaller structures when detailed seismic lines are shot at closer spacings. The large size of structures defined with limited data is not merely a manifestation of wishful thinking, but an expression of the inherent limits of resolution of the data. A computer contouring program, which has no "desire" to create large prospects, is as likely to draw vast areas of closure in areas of poor control as is a human interpreter.

Computing Risk for Oil Prospects — Chapter 1 Other geological properties of prospects may be even more difficult to define than subsurface structural configuration. The presence of source beds and carrier beds (through which migration may have occurred), and the sequence or "timing" of the regional and local structural evolution and the generation, migration, and entrapment of hydrocarbons are poorly understood, especially in frontier regions. Other factors that are regarded as vital for the concentration of prospective quantities of oil and gas include the existence of suitable reservoir intervals and the presence of traps and seals. Many companies attempt to formally incorporate these geologic factors in procedures which assign weights or probabilities to each factor to obtain a composite score or probability estimate for a prospect. These methods have laudable objectives, but there are severe difficulties in obtaining valid estimates of the necessary weights or probabilities that must be attached to the geologic factors.

Additive Schemes for Predicting Discoveries To illustrate the difficulties of obtaining valid weights, consider a naive procedure in which numerical weights are assigned as a matter of expert opinion to each geologic factor. An assessment for the prospect is made as the sum of the scores of the geologic factors. For example, one company assigns weights (probability estimates?) on a scale of 0.0 to 1.0 to each of six "essential geologic factors." These are added together and the total is divided by six to obtain a prospect score that ranges from 0.0 to 1.0. That is, H = {S-\-C-\-R + Tp-}-Sl-\-Tm)/6 where H = S = C = R = Tp = SI = Tm =

presence of producible hydrocarbons presence of hydrocarbon source beds presence of carrier beds presence of reservoir rocks presence of hydrocarbon trap presence of seal appropriate timing of factors.

At first glance, such a procedure may seem reasonable, but there is a potentially fatal problem. Suppose that all of the factors except one are given high probabilities, say 1.0, but the one remaining factor is regarded as impossible and is assigned a value of 0.0. The resulting prospect score is F = l + l + l - M - f l + 0 - 5 / 6 = 0.83 The value of 5/6 is nonsensical in a probabilistic sense because all of the factors are considered to be essential, and if any one of them has a low or 8

The Challenge of Risk Assessment zero probability, the overall probability must also be low or zero. Clearly, the combination of factor weightings or probabilities by addition is not appropriate, and even if it were deemed useful as a ranking procedure, it suffers from other shortcomings. If we use such an additive scoring scheme, we must assign an appropriate scale to each geologic factor. In the example, each factor has been rated on a scale of 0.0 to 1.0, but this is arbitrary and there is little reason why all the factors should be assigned values along the same scale. Some factors may be more significant and deserve greater weight, but we cannot objectively set the appropriate scale values without frequency data. Overall, the greatest shortcoming of methods that attempt to incorporate fundamental factors is the almost total lack of relevant frequency data. Has anyone compiled frequencies on "timing," for example? Timing as a geological factor is difficult to define for an individual play or prospect, and it is harder still to place into categories so that frequencies can be tabulated for the instances of "good timing," "moderate timing," and "poor timing." To our knowledge, almost no frequency data that pertain to fundamental geological factors have been gathered an3^where.

Multiplicative Schemes for Predicting Discoveries If additive methods are inappropriate, we may consider a multiplicative procedure in which geological factors are combined by H = SxCxRx Tp xSlx Tm using the same definitions as above. If any factor has a weight or probability of 0.0, the overall product also will be 0.0, which accords with our concept that the fundamental geologic factors must all be present in order for producible hydrocarbons to occur. Of course, the probability or weight assigned to each factor has to be determined along a standard unit scale, but scaling problems are avoided when we multiply rather than add them. Unfortunately, the problem is more subtle than this simple procedure implies, because multiplying the probabilities together presumes that the factors are independent of one other. We may assume that a potential trap may evolve structurally in a manner that is quite independent of the development of a source of hydrocarbons, but such an assumption may be erroneous. For example, the structural development of traps in a basin may be closely related to the overall structural evolution of the basin, including its depth of burial, temperature gradients, and hydrocarbon generation. Most major geological factors are unlikely to be wholly independent of each other, but instead are interdependent to varying degrees. If the dependency is small, combining probabilities by multiplication is warranted, but if the dependency is large, the combinatorial procedures must be adjusted. Estimating the correct probabilities requires frequency data which generally are

Computing Risk for Oil Prospects — Chapter 1 lacking, so we are at an impasse when using most procedures that combine probabihties assigned to major geological factors. If two geological events are truly independent, then the probability that both will happen (their joint probability of occurrence) is the product of their individual probabilities; this is what is assumed in a multiplicative scoring procedure. If two events are not independent, then the joint probability that both will occur is the product of the probability that the first event occurs times the conditional probability that the second occurs, given that the first event has already occurred. To estimate a conditional probability, we must have frequency data about how often the second event occurs for different conditions of the first event. This takes even more information than is needed to simply estimate the individual probabilities of occurrence. This topic will be discussed in considerably more detail in Chapter 5, when we consider how outcome probabilities can be modified during progressive stages of exploration.

Multiplicative Schemes for Estimating Amount of Oil Most major oil companies have formalized procedures for estimating hydrocarbon volumes that involve multiplying the major constituents of reservoir volume. A general equation for the producible hydrocarbon volume (PHV) in a reservoir is PHV =

HxAxx{l-Su;)xRfxS

where H = Height or thickness ofreservoir A = Area of reservoir (j) = Average porosity = Average water saturation Rf = Recovery factor S = Scaling constants Such an equation represents the rock volume of the reservoir (height times area of closure), scaled by porosity to yield gross pore volume, which is then scaled by a recovery factor after accounting for the proportion of pore space occupied by water, and finally scaled to compensate for expansion and other factors, and to express the volume in units such as stock tank barrels or MCF under surface conditions. The procedure could not be simpler, and if reasonably certain point estimates are provided for each of the components, their multiplication yields a single estimate of the volume of producible hydrocarbons. Unfortunately, the procedure is not so simple in practice. 10

The Challenge of Risk Assessment If the components are not known, then each can be represented by a probabihty distribution and the multiphcation carried out by Monte Carlo procedures. Depending upon our state of knowledge, multiplication may involve a mixture of probability distributions and point estimates. Whenever probability distributions are included in an equation, the product obtained by multiplication is also a probability distribution. The main difficulty with this multiplicative procedure is that we usually don't know appropriate values for the components of reservoir volume before a prospect is drilled.Porosity, water saturation, and recovery factors may be surmised from regional data and are appropriately incorporated either as point estimates or probability distributions, but height and area of a specific prospect usually are not well known until the prospect is tested. The kernel of the problem lies in estimating the gross rock volume of a prospect. We can't use regional estimates because then all prospects would have the same estimated volumes, which would make the exercise pointless. Clearly, we need estimates of reservoir rock volume that are conditional on the perceived properties of a specific prospect. It is common practice to assume probability distributions for height and area of closure and insert them in the equation. Often these are in the form of three-point or triangular distributions, consisting of a low, an intermediate, and a high value. It is very difficult to provide objective estimates of these distributions in advance of drilling. It often happens that estimates are supplied initially as poorly constrained guesses which are revised to yield estimated volumes that seem suitable. By that point, any claim to scientific objectivity has been lost. The old issue of interdependence between geological properties again arises. First, none of the factors is likely to be independent of the others. It is likely that reservoir height and area are strongly correlated in most structures. Even porosity is likely to be partially dependent on gross rock volume; for example, in some regions thicker sandstone reservoirs tend to be better sorted and to have higher porosities than do thinner reservoirs. We could compensate for these interdependencies if we had adequate frequency data, but usually such information has not been collected. We will return to the question of independence between geological properties in Chapter 6 where we discuss Monte Carlo simulation of reservoir volumes as it is used by many oil companies, both small and large. Should major geological factors even be considered and included in formalized risk-analysis procedures? Many oil companies strongly defend the use of such basic concepts because they help promote consistency in risking prospects. However, because of the inherent problems in their numerical combination and their often imponderable nature, it might be best if such 11

Computing Risk for Oil Prospects — Chapter 1 factors were treated as aspects of the factual and interpretative background when prospects are graded and ranked.

GEOLOGICAL UNCERTAINTY AND OIL OCCURRENCE Geological and geophysical information used in oil exploration comes mostly from wells and seismic surveys, although observations from outcrops, aerial photographs, and satellite imagery may also be useful. Unfortunately, we have no direct way of viewing the earth's geology in the subsurface. No one has ever examined an oil field from the inside, and the best that we can do is infer the geological processes that lead to an oil accumulation from the limited information available to us. The amount of information is never adequate; we never have enough well data or seismic data to interpret the geology fully, although our uncertainty is reduced as we continue to drill more holes and shoot more seismic lines. Each hole or seismic line increases our knowledge of the subsurface, but the reduction in uncertainty is seldom proportional to the expense and effort of gathering new information. Doubling the number of wells usually will not halve the uncertainty. This disproportionate reduction is a manifestation of the redundancy between observations, because the additional information we gain from a new well or seismic line partially repeats what we already know from pre-existing wells or seismic surveys. In addition, the locations of drill holes and seismic lines affect the reduction in uncertainty. A stepout well, in general, supplies less new geological information than will be gained from a wildcat well located at a distance, although the stepout well may be more useful for confirming the magnitude of a discovery than the distant well. The detail in geological interpretation that can be made in an area usually depends on the density of holes and seismic lines as well as the complexity of the geological features being interpreted. For example, an apparently uniformly sloping homocline that forms an almost perfect plane can be inferred from only a few wells, but a complex structure such as a salt dome or nappe might require a vast number of wells for correct interpretation. Structure contour maps in Figures 1.1 to 1.3 show interpretations that reflect the amount of well information, as they show progressively increasing detail with increasing numbers of wells. The three maps were made using an identical computer contouring procedure, so that differences between them are due solely to differences in the amount of information. A few wells are sufficient to establish the regional structure in Figure 1.1, but details shown in Figures 1.2 and 1.3 appear only when many more wells are available. There may be no limit to the degree of detail that could 12

The Challenge of Risk Assessment

Figure 1.1. Structure contour map of top of Lansing-Kansas City Group (Pennsylvanian) in an area of Graham County, northwest Kansas, based on 13 wells drilled by end of 1935. Geographic coordinates in miles from an arbitrary origin. Contours in feet. be interpreted if we possessed information measured without error in an unlimited number of wells, but in practice errors in depth measurements, inconsistencies in picking formation tops, and the impossibility of obtaining precise stratigraphic correlations are inescapable. Which of the three maps is "correct" or "true"? None of them, of course, although Figure 1.3 is more nearly correct than Figure 1.2, which in turn is more correct than Figure 1.1. This progressive increase in "correctness" (or reduction in error and uncertainty) can be measured and explicitly considered when estimating the risk associated with a petroleum prospect. We shall return repeatedly to this topic in later sections of this book. Our focus here, however, is on the uncertainty that involves the degree to which interpreted geology controls hydrocarbon accumulation. Usually, we are only peripherally interested in the uncertainty about the geological features themselves. Geological features are sometimes regarded as a surrogate for the occurrence of oil. Since oil and gas often are trapped in anticlines, explorationists search for antichnes. Although the relationship between anticlines and oil occurrence is well established in many regions and firmly entrenched in the folklore of petroleum exploration, the relationship may be less sound 13

Computing Risk for Oil Prospects — Chapter 1

Figure 1.2. Structure contour map comparable to Figure 1.1 except that it is based on 44 wells drilled by the end of 1950.

Figure 1.3. Structure contour map comparable to Figures 1.1 and 1.2 except that it is based on data from 196 wells available at end of 1965.

14

The Challenge of Risk Assessment than commonly believed. Not all oil occurs in anticlines, and not all anticlines contain oil. Of course, we can incorporate other geological features in our quest to reduce uncertainty, including the presence of source and carrier beds, paleotemperature indicators, and the presence of traps other than anticlines. All these characteristics bear on the occurrence of hydrocarbons, but none are wholly reliable, individually or collectively. If they were, the industry would drill few dry holes! Thus, geological features are uncertain, but their relationships to hydrocarbon occurrence are even more uncertain. The challenge is to understand these relationships and to estimate their strength with the limited information available to us. Unfortunately, "geological uncertainty" is hard to separate from "hydrocarbon-occurrence uncertainty." Uncertainty in oil exploration is more pervasive than most laymen realize, or many geologists acknowledge. An exploratory hole may be drilled through an oil-bearing interval and the presence of oil not detected. Fortunately, technology for "sniffing" hydrocarbons has advanced significantly in recent decades, and the uncertainty of hydrocarbon detection has been drastically reduced but not totally eliminated. Most exploratory holes are serviced by mud logging units capable of detecting traces of oil or gas in the drilling-mud stream. A strong oil or gas show at depth is a "red flag" indicating that producible hydrocarbons may possibly be present. Drill cuttings, petrophysical logs, and drill-stem tests provide additional information about the potential for production, although when the decision is made to complete the well as a producer or to abandon the hole as dry, substantial uncertainty about its true nature may still remain. If a hole is completed as a producer, the interval or intervals for production must be selected. All such decisions involve uncertainty, and even after the well goes into production, the volume that it will ultimately produce remains uncertain. This is as true for a well that is far advanced in its decline as it is for a new discovery just going on stream, although the relative degree of uncertainty is less.

REGIONAL HYDROCARBON E N D O W M E N T Although this book is about risk, it is specifically focused on the risk attached to different outcomes that may result when petroleum prospects are drilled. The other types of risk briefly discussed above will not be considered further, although all elements of risk are important in analyzing the total risk, and hence economic potential, attached to a prospect, an individual well, or a field. In the forecast of outcomes of plays and prospects, the broadest issue that affects risk is the regional endowment of hydrocarbons. The literature dealing with hydrocarbon endowment and exploration potential is extensive. Salient papers include those by Arps 15

Computing Risk for Oil Prospects — Chapter 1 and Roberts (1958); Attanasi, Drew, and Schuenemeyer (1980); Attanasi and Haynes (1984); Barton (1992); Beebe, Murdy, and Rasinier (1975); Bettini (1987); Bird (1986); Bloomfield and others (1980); Charpentier and Wesley (1986); Chung, Fabbri, and Sinding-Larsen (1988); Crovelli (1981, 1984, 1985, 1986, 1988); Dolton (1984); Dolton and others (1979, 1981); Drew, Scheunemeyer, and Root (1980); Drew and others (1987); Forman and Hinde (1985); Gehman, Baker, and White (1975); GrossUng (1975); Harff, Davis, and Olea (1992); Harris (1977); Haun (1975); Hubbert (1967, 1982); Ivanhoe (1986); Kaufman (1979, 1986); Kent and Herrington (1986); Mast and others (1980); Masters (1984, 1985, 1986); Masters, Klemme, and Coury (1982); Masters and others (1987); Masters and Peterson (1981); Masters, Root, and Dietzman (1983); Mayer and others (1979); McKelvey (1975); Meyer (1977); Miller (1979, 1981, 1982, 1986); Miller and others (1975); Momper (1979); Nederlof (1980); Nehring (1978); Pitcher (1976); Pratt (1937); Proctor, Lee, and Taylor (1982); Reznik (1982, 1986); Root and Attanasi (1980); Root and Drew (1979); Root and Schuenemeyer (1980); Ryan (1973a, 6); Schroeder (1986); Schuenemeyer and Drew (1977, 1983); Sheldon (1976); Uhler and Bradley (1970); Uri (1979a, 6); Varnes, Dolton, and Charpentier (1981); Weeks (1965); White and Gehman (1979); Wiorkowski (1981); Zapp (1962). A region's hydrocarbon endowment governs its potential in terms of pools and fields, and for that matter, the potential of individual prospects and wells. A prospect consisting of a trap of a particular form and size has greater potential in a region of large endowment than a similar prospect in a region of small endowment. Estimating the regional endowment is important in all stages of assessing risk, but especially at the outset. Uncertainty about regional endowment is greatest before any wells have been drilled in a region; while uncertainty declines progressively as more and more wells are drilled, it is never totally eliminated because we never have complete knowledge of the oil and gas pools of a region, even after all drilling has ceased. We can distinguish three main categories of regional hydrocarbon endowment in which Category 2 is a subset of Category 1, and Category 3 is a subset of Category 2: Category 1. Total hydrocarbon endowment of a region over its entire extent. Category 2. Endowment of hydrocarbons initially in place in pools or fields within the region. Category 3. Endowment of producible hydrocarbons in pools or fields that have been discovered or are potentially discoverable.

16

The Challenge of Risk Assessment

Total Endowment As the name implies, the total hydrocarbon endowment consists of the total quantity of hydrocarbons contained in the rocks of the earth's crust in the region of interest. This is a sweeping definition, because it includes all oil and gas in the known fields and undiscovered pools, non-liquid hydrocarbons bound up in fine-grained rocks, tar sands and other deposits of viscous hydrocarbons, gas in coal seams, and gas dissolved in pore water. The total hydrocarbon endowment is virtually unmeasurable and unknowable, although it may be crudely estimated from volumetric calculations and organic carbon analyses. Unfortunately, only a fraction of the total hydrocarbon endowment occurs in concentrations of potentially economic size, and therefore much of the total hydrocarbon endowment is unlikely to be produced even if we had accurate information about it. Presumably, non-liquid hydrocarbons in fine-grained rocks cannot be produced and are not themselves a potential resource (oil shales are an exception). Gas in coal seams is producible in some regions, and gas dissolved in pore water may be potentially producible, although it is not usually regarded as a potential resource. Tar sands are special and may or may not be potentially exploitable. The total hydrocarbon endowment therefore remains elusive because we usually have only information from holes drilled in the search for conventional hydrocarbons, and the rest of the total endowment remains unsampled and unmeasured.

Endowment Initially in Place in Pools The endowment of hydrocarbons initially in place in pools is of more direct interest because it strongly influences exploration. Each drill hole, whether producing or dry, samples this endowment and is potentially useful for estimating its magnitude. Although simple in concept, there are complications in estimating the endowment of hydrocarbons in pools because their lower size limit is impossible to define. A "pool" is a subsurface accumulation of oil and/or gas, whether discovered or not. The term "reservoir" refers to the characteristics of the rock body that contains a pool. A "field" is a contiguous geographic area within which wells produce oil or gas. A field may contain multiple reservoirs or pools. The geographic expanses of fields generally are defined by the locations of producing wells, which may range in number from one to thousands within a single field. Although the lower limit for a field in terms of number of producing wells is a single well, the lower limit in terms of volume of oil cannot be defined because no perceptible physical lower limit for volume exists. Some single-well fields may produce as little as a few hundred barrels of^oil before being abandoned, suggesting that the pools encountered were extremely small. Some 17

Computing Risk for Oil Prospects — Chapter 1 exploratory holes that penetrate small accumulations of oil or gas may be completed as dry holes because their potential return would be uneconomic; these holes yield virtually no information about the lower limit of volumes in pools that have been encountered, because the resulting "shows" are not included in production statistics. In most oil-producing regions, most of the hydrocarbons are produced from a small number of large fields, and the lower limit of fields sizes is of no practical consequence. The lower limit remains undefined and unmeasured.

Endowment of Potentially Producible Hydrocarbons The endowment of hydrocarbons that are potentially producible consists of the producible fraction of oil and gas initially in place in pools that can be discovered and are of sufficient size to be economic. This fraction is the recovery factor and differs widely depending on viscosity and gravity of the oil, and on reservoir characteristics that include gross volume, porosity, permeability, and connectivity, on the type of drive mechanisms in the reservoir (water drive, gas drive, gravity drainage), on the economics and technology of enhanced recovery, and on the economics of production at low rates. The endowment of producible hydrocarbons is complicated to define and difficult to estimate. Because of the economic considerations that are implicit in its definition, estimates are strongly influenced by oil and gas prices, by taxes and operating costs, and by lease and concession agreements, particularly royalty terms. Although the endowment of oil and gas in pools is defined by nature, the amount of potentially producible hydrocarbons is influenced both by nature and by economics and production technology. Recovery factors strongly reflect the technology and economics of enhanced recovery procedures. If oil prices rise, recovery factors also rise, so that the volumes of producible hydrocarbons may be highly sensitive to price. For example, a program for enhanced recovery may not be economic if oil is $19 a barrel, but if the price rises to $26 the investment may be very attractive. Of course, the efficiency and cost of future recovery technology as well as future prices of oil are uncertain, so the endowment of producible hydrocarbons is doubly uncertain.

Estimating Endowments of Producible Hydrocarbons Explorationists are primarily concerned with the endowment of producible hydrocarbons. The broad questions are how much oil and gas is there to be 18

The Challenge of Risk Assessment produced, and how and where is it distributed? With complete knowledge, all uncertainty about occurrence would be eliminated, and we would not need to explore because we could simply exploit the known resources. Although we shall never achieve such sweeping understanding, we can reduce uncertainty by estimating the regional endowment of producible hydrocarbons even without specifying the locations of undiscovered fields. There are two principal approaches to estimating the endowment of producible hydrocarbons in a region. The first is based on an analysis of previous discoveries in the region and assumes that past exploration results provide a statistical key to future discoveries. Frequency distributions of the sizes of fields discovered to date can be analyzed and trends in field sizes through time noted. Of course, if few fields have been discovered, the history of exploration may not provide much information for forecasting either the sizes of fields to be discovered in the future or the regional endowment. The second approach consists of comparing the region of interest with other, more mature regions whose geology and modes of oil occurrence seem more or less similar. This presumes that enough is known about the geology so that realistic analogue areas can be chosen. The statistical pattern of discovery in the more mature analogue region is used to predict the future of exploration in the less well explored region. As in the first procedure, we depend on production records for basic information, although we are "borrowing" these records from another area. The best estimates of the endowment of producible hydrocarbons are obtained in maturely explored and exploited regions where some fields have produced for a sufficient time so that their ultimate cumulative productions can be accurately forecast. The ultimate cumulative production of younger fields then can be estimated by comparison with more mature fields. Depending on the degree of exploration maturity in the region, the aggregate of the older and younger fields may approach the total producible endowment, although the resulting figure will always be an underestimate to some degree because exploration is never complete. The current degree of exploration maturity is a key factor in the estimation process. Although estimates of ultimate volumes of oil and gas deal with producible hydrocarbons, they also can yield estimates of oil and gas originally in place. It is unfortunate that we seldom recover more than 50 or 60% of the crude oil originally in place. The recovery factor can be evaluated by engineering methods during production. By considering the volume of ultimately producible hydrocarbons and the recovery efficiency, the total endowment originally in place in fields can be estimated.

19

Computing Risk for Oil Prospects — Chapter 1

160-r-

100 Oil in Place, bblsxIO^ Figure 1.4. Plot of exploratory drilling in millions of feet versus oil discovered in billions of barrels in the Permian Basin. Adapted from U.S. Geological Survey (1980).

Exploration Maturity The maturity of exploration in a region can be estimated in several v^ays. One involves comparison of cumulative exploration effort with cumulative volume of hydrocarbons discovered. When effort is plotted against discoveries, a J-shaped curve usually results, such as that shown in Figure 1.4 for the Permian Basin of West Texas. If the curve is extrapolated until it becomes asymptotic, the cumulative effort approaches infinity and the cumulative discoveries approach complete maturity or total ultimate production. The method is marvelously simple, but does contain certain pitfalls. "Exploration effort" is an arbitrarily defined concept. One possible measure of effort is the number of exploratory wells, another is the total footage of exploratory holes drilled, and still another uses all wells, exploratory and production, expressed either as the numbers of wells or their aggregate footage. 20

The Challenge of Risk Assessment A J-shaped curve can be extrapolated so that when it becomes vertical, the exploration effort plotted along the vertical axis has essentially become infinite. This yields an estimate of the total producible hydrocarbon endowment, usually denoted "Q" (Harbaugh, Doveton, and Davis, 1977). This estimate may be realistic if three conditions are satisfied: (1) The volumes of progressively discovered hydrocarbons have been estimated assuming constant economic and technological conditions that will be valid in the future, (2) the region is at least in a moderately advanced state of maturity, and (3) exploration technology and geological concepts employed in the past will continue to be used without drastic change in the future. These conditions are difficult to satisfy. Estimates of volumes depend partly on future prices and recovery technology, and different assumptions may change estimates by two- or threefold. Furthermore, opinions about the degree of maturity of a region are subjective. A steep J-shaped curve may not signify relative maturity if parts of a region remain relatively unexplored, possibly because of the depth of drilling required or geographic remoteness. It is well known that shallow fields usually are more extensively explored and exploited before deep fields, so shallow fields generally are relatively over-represented and deep fields under-represented in distributions of discovered fields. Finally, advances in exploration and exploitation technology may have a revolutionary effect in old regions, such as the advent of horizontal drilling of fractured reservoirs in the Austin Chalk of Texas. Other ways of estimating exploration maturity involve simulation experiments in established regions, where a population of geographic locations and volumes of known fields is repeatedly sampled by "drilling" at random. The resulting simulated "history" of discovery is used to estimate the volumes of hydrocarbons that would be discovered by varying numbers of exploratory wells. In such an experiment. Drew (1974) repeatedly "drilled" a population based on known fields in the Powder River Basin of Wyoming and used the results to forecast discoveries in similar basins with comparable geology and mode of oil occurrence. These simulation experiments must specify minimum spacings between exploratory wells and include procedures for allocating volumes among the multiple exploratory wells that may "discover" the same fields. Similar experiments also can be used to estimate the volumes of hydrocarbons to be discovered in unexplored or under-explored parts of a region.

Statistical Distribution of Field Sizes The procedures above yield estimates of total endowments and degrees of maturity, but do not provide estimates of the sizes of fields that may be discovered. Consider two hypothetical regions with identical total endowments of ten billion barrels of oil. One region contains a single field holding 21

Computing Risk for Oil Prospects — Chapter 1 all ten billion barrels, while the other contains ten thousand fields each containing one million barrels. Although extreme, these examples are not absurd. Contrast the Arctic Coastal Plain of Alaska where only a few large fields have been discovered (including the Prudhoe Bay Field containing 12 billion barrels of oil) with the Central Kansas Uplift which has several thousand fields whose median size is less than a million barrels. The economics and strategy for exploring in such diverse regions are drastically different, and it is obvious that knowledge of field size frequency distributions is enormously important. The usual sources of information for estimating field size distributions are field production summaries. Most oil-producing states in the United States publish such summary statistics, usually by county, geographic, or producing region subdivisions. The summaries usually give production from individual fields on both an annual and a cumulative basis. Cumulative production statistics by fields may diff^er substantially from estimates of ultimate cumulative production, but cumulative production figures are valuable if used appropriately. Graphs of production data are easily generated and may be used directly for forecasting purposes, although hazards in such forecasting will be discussed later, particularly the bias toward early discovery of large fields. A principal difficulty in using cumulative production by fields within a region is that the data may represent a mixture of old and new fields. Although older fields may be close to their ultimate producible yield, newer fields may be far below their ultimate cumulative production because they have produced for only a short time.

PRINCIPAL PREMISES OF THIS BOOK (1) Use frequencies to estimate probabilities: Our first premise is that outcome probabilities assigned to oil and gas prospects should be based on frequencies of occurrence. If we do not adopt frequencies as a basis, directly or indirectly, there is no realistic way of constraining outcome probabilities. Even in frontier regions we can use data from analogous areas to provide frequencies. In regions in which fields have been discovered, a substantial body of information usually exists that can provide frequencies suitable for estimating outcome probabilities. (2) Use two-part probability distributions: Our second premise is that outcome probabilities should be expressed in two parts. A general form of a probability distribution for representing prospect outcomes is shown in Figure 1.5. One part consists of the dry hole probability (shown as a discrete "spike") and the other consists of a continuous curve representing probabilities for diff^erent volumes of producible hydrocarbons. The two parts are complementary. If the dry hole probability is 40%, the probability 22

The Challenge of Risk Assessment ••—>'

^ 1 5

5

SI

2

Q-

-

"Shows" and Dry Holes

r

V

i

1

\

\

\

\

..............^

-i

-

—

1 Economic 7 Discoveries

1 1 iiiiit

10^

1 iiiiiiq

10^

1 f iiiiiii

10^

10"^ 10^

1 I I mill

10^ 10"

1 MiiiTif

10^ 10

11 mill

10'

T Trillin—imTTlli

10^

10

Discovery Size, bbis Figure 1.5. Graph of two-part probability distribution for prospect outcome. Spike represents discrete probability of dry hole and continuous curve represents hydrocarbon volumes. of a producer of some magnitude must be 60%. Sometimes distributions with only two numbers are useful, the dry hole probability and the producer probability, with volumes unrepresented. In Figure 1.5, the continuous probability distribution of possible volumes is not convenient to work with. Although its continuous form is realistic because actual volumes of producible hydrocarbons extend over a continuous range, we need distributions that are discrete so they can be linked with financial analyses. Figure 1.6 corresponds to Figure 1.5, except that volumes are expressed as discrete classes. (3) Optimize estimation of producer-versus-dry probabilities: Our third premise is that estimates of discovery probabiHties (that is, the probability of a producer versus the probability of a dry hole) can be optimized by making effective use of available data. If we lack regional geological and geophysical data, outcome statistics based on exploratory holes in the region can provide background data for estimating discovery probabilities that can be applied to a prospect. If the prospect involves a deep test, outcome statistics for other deep exploratory holes in the region may be useful as a base. We need to know how many were producers and how 23

Computing Risk for Oil Prospects — Chapter 1 1

....

! ^dry

j

1

i

j

j

1

1

i

*

1

, = 0.40 ;

^

1

S

1

2 a.

1 1

1 1

10^

1

1 I

Economic D*SCOveries

1

j

1

1

i

i

1 j

• 1 Mlllll 1 1 iMiiii 10^ 10^

r^ 10"*

1 r1 m i l

rnrrm

10^

1 1Tmnf i 10^ 10^

1 mil

iiiin

10^

1 I'll Mill

10^

Discovery Size, bbis Figure 1.6. Two-part distribution with volumes represented as a histogram. many were dry, because their proportions provide an initial estimate of the chance of success. This initial estimate can be subsequently refined with other information. In general, oil and gas prospects are conditional upon geological relationships as they are interpreted before drilling. If the geological properties can be shown on maps, there are procedures that can help us identify and enhance their subtle conditional relationships and express them as outcome probabilities. (4) Optimize estimation of hydrocarbon-volume probabilities: Our fourth premise is that probability estimates for hydrocarbon volumes can be optimized with frequency data. If fields have been discovered in a region, the frequency distributions that they form are important. We need to know how many are large fields, how many are intermediate, and how many are small, and whether they form distributions similar to those in other regions. By generating frequency distributions of field volumes, we have a start toward estimating probabilities of hydrocarbon volumes that we can refine subsequently with information specific to a prospect. Since conditional relationships are important, the geological features of a prospect can be compared with those of other prospects to provide a statistical base of perceived relationships. For example, if relationships 24

The Challenge of Risk Assessment between sizes of subsurface structures and hydrocarbon volumes exist, statistical procedures may quantify them and help estimate conditional probabilities. These four premises are aspects of a general philosophy that explorationists should use available data in order to generate frequencies that are useful for evaluating prospects. This approach is not intended to discourage imagination, but instead seeks to improve prospect appraisal by clarifying and enhancing relationships that may otherwise be obscure, with the objective of placing prospect appraisals in forms most suitable for financial analysis. (5) Use outcome probabilities in financial analysis: Our fifth premise is that explorationists who generate prospects should also analyze them financially. A geologist who proposes a prospect needs to understand its economic implications. If it is to be a good prospect, it must have good economic potential. Accordingly, we link procedures for obtaining outcome probabilities with procedures for analyzing their financial consequence. (6) Incorporate aversion to risk: Our sixth premise is that no individual or firm is truly neutral toward risk; losses hurt more than the benefits of equivalent gains. Therefore, the option of representing aversion to risk is included in financial analyses.

Computer Programs for Analyzing Risk The procedures described here require computers in most situations, as the computations are intense and the data may be voluminous. Risk analyses can be made using a combination of different commercial programs for the statistical, financial, and map calculations. Unfortunately, it's not easy to move data between different applications and the process may prove to be exceedingly cumbersome. To avoid this problem, we have provided RISK, a collection of integrated programs for your use. The diskettes that accompany this book contain all the software needed to carry out the probabilistic assessments and financial calculations described in this book. In addition, the diskettes contain files of data that you can use to teach yourself how to operate the software. These same data are discussed as examples in the text. The RISK software is organized into three main packages: RISKSTAT provides statistical tools, RISKMAP performs contour mapping and map analysis, and RISKTAB provides tools for financial analysis. User's manuals for RISKSTAT, RISKMAP, and RISKTAB are given in Appendices B, C, and D. The programs are written in FORTRAN, but are in compiled form so they can be installed directly on IBM-compatible personal computers using the MS-DOS operating system (Version 3.0 or later). An Intel 386 25

Computing Risk for Oil Prospects — Chapter 1 or better processor with 640K or more memory, a hard disk, a VGA color graphics card, a color monitor, and either a PostScript or DOS-compatible line printer are required. For plotted copies of maps and other graphics, a PostScript-compatible graphics printer is necessary. Instructions for installing the software on your computer are given in Appendix A. RISKSTAT was Written by Geoff Bohling, based on software originally developed by Steve N. Yee of Terrasciences Inc., as modified by Chris Murray. RISKMAP was written by Johannes Wendebourg and is based on a PC version of the SURFACE III graphics package written by Robert J. Sampson, copyrighted and distributed by the Kansas Geological Survey RISKTAB was written by Johannes Wendebourg. All of the RISK software has been copyrighted by Davis Consultants Inc., under appropriate license from original copyright holders. The RISK software is provided on an "as is" basis; neither the authors nor the publisher make any warranties, express or implied, about its performance or suitability for a specific purpose. For a limited time the Kansas Geological Survey will provide technical support to registered users of RISK in response to inquiries submitted by post or electronic mail to the SURFACE III Office at the Kansas Geological Survey. Information will be provided about any future releases of the RISK software and related computer programs. Instructions for registering your copy of RISK are given in Appendix A.

Examples and Data A wide variety of examples has been selected for this book from petroleum provinces in the United States. Field size data are provided for the DenverJulesburg, Powder River, Big Horn, and Wind River basins in the Rocky Mountains, the Permian Basin of Texas and New Mexico, the Western Shelf area of Kansas, and offshore areas of Louisiana and Texas. The most extensively used data, however, are from a foreign petroleum province in central Eurasia within the "Republic of Magyarstan," a mythical country that we have defined for illustrative and pedagogical purposes. "Magyarstan" lies near two major oil-producing regions of the Middle East. Oil and gas have been produced in the republic since the 1920's, so parts of Magyarstan are in a mature stage of exploration. The data from Magyarstan are derived from an actual petroleum province so the geological relationships are realistic in form, but the names for geographic areas and geological formations are synthetic. Both the statistical distributions of fields and the spatial distributions of geological properties are based on real relationships that have been transformed appropriately for the scale of the problems presented in this book. Data are given for several petroleum provinces in Magyarstan, including a mature area on the Zhardzhou Shelf 26

The Challenge of Risk Assessment discussed in Chapters 2, 5, 6, 7, and 8, and another area in the Belaskova Region presented in Chapter 12. The mature area includes a 36 x 36 km tract on the Zhardzhou Shelf, which is a structurally positive feature near the southwest border of the republic. This tract, referred to as the "training area," contains 18 fields that were discovered by drilling 93 exploratory holes. Production comes from Middle and Upper Jurassic (Callovian and Oxfordian) limestones. Reservoirs occur in stratigraphic units called the XV and XVa horizons of J3 (Upper Jurassic) and the XVb horizon of J2 (Middle Jurassic). The reservoir rocks are described as "bar reefs, barrier reefs, and solitary buildups," but it is not clear that a rigid reef framework is present. The reservoir facies is perhaps best interpreted as carbonate buildups of mixed bioclastic and biotic origin. Porosities of 8-10% are typical, although higher values are noted in some cores. The limestone reservoir rocks are interbedded with marine shales and overlain by a thin sequence of limestone. The reservoirs are sealed by evaporites of Early Cretaceous age. These consist of a lower salt unit, a middle anhydrite unit, and an upper salt unit. There is no evidence of salt tectonics, although the thickness of the lower salt varies with the paleotopography on top of the uppermost Jurassic limestone. The Upper Cretaceous consists of alternating marine and nonmarine elastics, becoming entirely nonmarine in the upper part. Oil also has been produced from Lower Cretaceous sandstones elsewhere in the republic. The Cretaceous is overlain by Paleogene (probably Eocene) nonmarine elastics which are very thick in some regions. The upper part of the section consists of Neogene continental sediments. A second tract, called the "target area," is in the Bakant Basin, a structural depression lying approximately 200 km east of the training area. The XVa Limestone is known to be present in this area, and presumably it contains porous zones that would be suitable reservoir host rocks. The K l salt and anhydrite are known to be present also and should form suitable seals on any traps. In the training area, reservoirs are not controlled by structure alone, but instead are localized by the interplay of structural configuration and variations in lithofacies. The variables measured in wells of the training area include structural elevation (in meters) of the top of the XVa Limestone, thickness of the XVa Limestone (in meters), shale ratio (calculated from the average of the gamma-ray log reading), and an interbedding index (calculated from the standard deviation of the gamma-ray log measured over the XVa interval). The shale ratio expresses the relative "cleanUness" of the limestone, while the bedding index contrasts massive carbonates with carbonates interbedded with thin shales. When mapped and considered in combination, these variables have proved useful in defining new prospects 27

Computing Risk for Oil Prospects — Chapter 1 that have an exceptionally high probabihty of success. The same four variables can be calculated from the well logs that will be run in holes drilled in the target area as exploration progresses and may prove to be a useful guide to oil in this area as well.

28

CHAPTER

2

Field Size Distributions ESTIMATING ^'Q'' FROM THE HISTORY OF EXPLORATORY DRILLING As an initial step in evaluating the remaining petroleum potential of a region at a moderate or mature stage of exploration, we can extrapolate the recorded production history until it reaches an asymptotic limit. This limit is an estimate of the ultimate volume of hydrocarbon that will be produced. Estimating the amount of ultimately recoverable oil and gas in a region is of interest to explorationists as well as economists and politicians. The issue is economic because the amount of oil and gas present may be much greater than the amount that is economically recoverable, either in today's economic cUmate or in the future. The past is the key to the future, and in maturely explored areas, we can employ a simple technique to show the relationship between discoveries and exploration activity (Harbaugh, Doveton, and Davis, 1977). We can obtain a J-shaped curve by plotting exploratory drilling (either the cumulative number of holes drilled or the aggregate feet or meters of hole drilled) versus amount of oil discovered (or oil plus gas expressed as BOE, that is "Barrels of Oil Equivalent"—5700 cubic feet of gas at standard temperature and pressure equals one barrel of oil). An example curve for the Permian Basin is given in Figure 1.4. Determining the amount of oil discovered requires estimating remaining reserves, because fields are not yet exhausted. Continued exploitation of fields inevitably will result

Computing Risk for Oil Prospects — Chapter 2 in increases in cumulative producible hydrocarbons because of additional drilling or use of enhanced recovery procedures. With this precaution, a Jshaped curve is a way of projecting the cumulative discoveries to date to the ultimate end of exploration, when all economically recoverable oil and gas have been discovered. Of course, in early stages, extrapolating a J-shaped curve is perilous because of wide latitude in the possible projections. Although a plot of drilling versus discovered hydrocarbons usually is J-shaped, the curve for gas discovered in the Permian Basin has a different form. The curves are not J-shaped and suggest, somewhat paradoxically, that discovery efficiency has increased with increasing exploration activity. As exploration continues, however, we can be confident that the curves in Figure 2.1 will grow steeper and eventually become J-shaped. By comparing Figures 1.4 and 2.1, it appears that much of the oil in the Permian Basin had been discovered by 1980, but the proportion of gas that has been discovered is much less. This may reflect the fact that deep gas remains an important exploration target, because deeper horizons are principally gas-bearing and not oil-bearing. Q can be estimated using simple graphical procedures such as those contained in the RISKSTAT software. Table 2.1 gives the number of exploratory wells drilled annually and the total cumulative production of the fields that have been discovered in a district in the mythical Magyarstan republic. Total production is calculated through 1990. For example, fields discovered in 1959 produced 44 million bbls by the end of 1990. These data also are contained on the diskette in the data file MAGYAR.DAT. The bivariate scatter plot option of the statistical graphics package in RISKSTAT will produce a plot similar to that shown in Figure 2.2. From Figure 2.2, we can estimate Q by extrapolating the curve until it becomes vertical. What is Q? A reasonable estimate is about 12.4 billion bbls of oil. Among the factors in the data that might bias the estimate is the fact that we used cumulative production figures rather than estimates of ultimate production. Because some of the fields have a short producing history, their cumulative oil volumes are low and the value of Q estimated from them will be conservative. Although it is better to use estimated ultimate production figures as data, these may be difficult to obtain and because they are estimated by different operators, they may be inconsistent. In contrast, production data usually are a matter of public record and are available from conservation commissions and similar regulatory agencies, and from information service companies. Estimating the ultimate production from a well, lease, or field usually requires detailed analyses of production decline curves or engineering studies of pressure drops, water production, and response to stimulation. It's possible to make empirical adjustments to cumulative production volumes 30

Field Size Distributions 160 n

10

20 30 Gas in Place, cu ft x lO"'^

40

50

Figure 2.1. Exploratory drilling versus gas discovered in the Permian Basin. Line labeled Older Paleozoic pertains to discoveries in pre-Mississippian (Lower Carboniferous) rocks. Line labeled Total includes all discoveries. Adapted from U.S. Geological Survey (1980). that may approximate ultimate production volumes, provided the conditions of production are not seriously changed over time. One empirical adjustment consists of adding some multiple of the last year's annual production to the cumulative production of a field, on the assumption that field production will continue to decline at a constant rate. Multipliers of 5 to 8 seem to give reasonable results in many areas, but it is worthwhile to examine the records of production decline of the older fields in an area to get an idea of the most appropriate multiplier. Prom a plot of the data in Table 2.1, we can see that sUghtly over 12 billion bbls of oil have been produced in this district. The difference between this value and the extrapolated value of Q suggests that only about 300 miUion bbls remain to be produced. This suggests that most of the major oil fields in this district of the Republic of Magyarstan have been found.

31

Computing Risk for Oil Prospects — Chapter 2 Table 2.1. Number of exploratory wells drilled annually in a district in the Republic of Magyarstan and total production through 1990.

Year

of

discovery

Number of wells

1957* 21 3 1958 1959 2 1960 0 1961 9 1962 3 1963 15 1964 17 1965 14 1966 7 1967 14 1968 18 1969 7 1970 40 1971 83 83 1972 1973 39 1974 84 1975 142 1976 115 1977 119 1978 199 1979 120 1980 50 1981 47 1982 84 1983 43 1984 47 1985 58 1986 45 1987 113 1988 70 1989 99 1990 91 *and prior years.

32

Cumulative number of wells

21 24 26 26 35 38 53 70 84 91 105 123 130 170 253 336 375 459 601 716 835 1034 1154 1204 1251 1335 1378 1425 1483 1528 1641 1711 1810 1901

Total production (Mbbls)

C u m . total production (Mbbls)

0

0

3,881,200 44,000

3,881,200 3,925,200 3,925,200 3,950,200 3,950,200 3,950,200 3,950,200 3,986,500 3,986,500 3,986,500 3,986,500 5,720,300 5,834,500 6,725,000 7,030,500 7,724,600 8,776,100 9,943,800 10,222,000 11,000,700 11,244,400 11,422,500 11,431,900 11,581,600 11,616,200 11,675,500 11,680,700 11,694,000 11,858,200 11,941,700 11,962,200 12,118,100 12,187,600

0 25,000

0 0 0 36,300

0 0 0 1,733,800 114,200 890,500 305,500 694,100 1,051,500 1,167,700 278,200 778,700 243,700 178,100 9,400 149,700 34,600

593 5,200 13,300 164,200 83,500 20,500 155,900 69,500

Field Size Distributions Q

2000JO

1) 0

....

1500-

1

JQ

E z 0)

>

p j 1000-

E

3

o

A

500(

l^

U

1

1

1

1 i 8 10 12 Cumulative Production, bbis x 10

14

Figure 2.2. Cumulative numbers of exploratory wells versus cumulative barrels of oil discovered in the Republic of Magyarstan, based on data in Table 2.1. The manually fitted smooth curve has been extrapolated to an asymptotic Hmit of approximately 12.4 billion bbls.

FREQUENCY DISTRIBUTIONS OF OIL FIELD POPULATIONS Frequency distributions provide the foundation for a probabilistic approach to risk assessment. Frequency distributions are obtained by counting. If we count the oil fields in a region and arrange them into categories based on their quantities of ultimately producible oil or gas, for example, we obtain a frequency distribution of field volumes. The frequency distribution can be represented by a set of numbers, or more commonly by a bar graph (histogram) to which a curve can be fitted. Figure 2.3 shows a histogram of the volumes of ultimately producible oil and gas in fields of the Big Horn Basin in Wyoming that had been discovered through 1977. The areas of the bars of the histogram are proportional to the number of fields that occur 33

Computing Risk for Oil Prospects — Chapter 2 100 80•u

il

60-

E 2

40-

20-

0-

••4—-j—-i—• — - j — •

r 11 n 200 i

...4.....}....<....<.—j....;

i i i i800M n400 i i 600

i j j j i i 1000 1200

Volume, bblsxIO^ Figure 2.3. Histogram of volumes of oil and gasfieldsdiscovered through 1977 in the Big Horn Basin, Wyoming. Oil and gas volumes are merged together as barrels of oil equivalent (BOE) and are 1978 estimates of ultimately producible oil and gas per field. in each size category. The histogram is highly skewed; that is, most fields occur in the smallest size category and far fewer occur in the categories of successively larger sizes. This suggests that any pattern in the field size data could be better seen if we changed the field sizes into their logarithms before dividing them into size classes. Taking the logarithms of field volumes would compress the values on the right-hand side of the histogram (where the larger fields occur) and expand the values on the left-hand side, where all the smaller fields are concentrated. This has been done in Figure 2.4, which shows a histogram whose classes represent volumes in BOE expressed as powers of 10. These histograms show how the sizes of oil and gas fields in the Big Horn Basin are statistically distributed. It provides an effective expression of hindsight. It tells us that there are only a few very large fields, and only a few extremely small fields. Most fields are modest in size, but the range of sizes is essentially continuous from vanishingly small fields to the largest field (Elk Basin Field, forecast to yield about 585 million BOE). You'll note that the most frequent size of field occurs in the middle of the histogram in Figure 2.4, approximately 10^ bbls. The frequency of both 34

Field Size Distributions 30

2 20 0)

0

E

10 -\

i

r-

5 6 7 Volume, Log^Q bbis Figure 2.4. Histogram of estimated ultimate production volumes in BOE of oil and gas fields discovered through 1977 in the Big Horn Basin, Wyoming, in size classes whose limits are logarithms to the base 10. The fitted curve is a normal distribution having the same mean and standard deviation as the logarithms of field size. larger and smaller fields declines away from this central value (the "bump" in the smallest size category probably is an artifact caused by the reporting process which groups together marginal fields.) This symmetrical, bell-like shape with smoothly declining frequencies in the tails is often seen in measurement data. A mathematical model of this bell-shaped distribution is referred to as a "normal" curve, and is a key concept in statistics. In addition to the many natural phenomena that produce frequency distributions that closely resemble normal curves, statistical theory predicts that the averages of randomly chosen samples of almost any property should follow a normal distribution. Repeated measurements on the same entity often result in a collection of values that follows a normal distribution, a consequence of the addition of many small, random, partially offsetting sources of error. We will discuss normal distributions at greater length when we consider the expression of uncertainty and other topics. For now, we will content ourselves with the observation that the normal distribution can be used as a useful description of the form of many frequency distributions that we will encounter in petroleum risk assessment. Because the equation for the normal distribution contains only two parameters whose values can 35

Computing Risk for Oil Prospects — Chapter 2 be estimated from the data themselves, it's relatively easy to fit a normal distribution* to a set of observations and use it as a model. The form of the normal distribution can be calculated exactly from its equation, so we can determine the proportion of the area under the curve that lies between any specified limits. That is, we can predict the relative frequency of occurrence of any histogram class in a distribution that is modeled by a normal curve. Since in Figure 2.4 the scale of field volumes is logarithmic (in powers of ten), the histogram can be modeled by a "lognormal" curve; that is, a set of data whose logarithms follow a normal distribution. Like the normal distribution, the lognormal is defined by a mathematical equation that has only two parameters whose values can be estimated from the data. Then, the equation can be evaluated and plotted on the histogram, as has been done in Figure 2.4. The fact that the curve does not coincide with the histogram perfectly should not concern us greatly. Only 108 fields had been discovered when the field volumes were estimated at the end of 1977, and when we divide this number of fields into a small number of frequency classes (represented by the histogram bars), it is not surprising that the histogram is somewhat irregular. Had we chosen different class intervals, the histogram would be slightly diff'erent in shape. What sort of mechanism might lead to a lognormal rather than a normal distribution? Just as a process that adds together small random displacements about a central value leads to a normal distribution, a process that multiplies together small random displacements will lead to a lognormal distribution. We can imagine a physical analogue consisting of a dispersion in water of uniform sized droplets of oil. Random motion will cause some oil droplets to coalesce, forming larger droplets, and the larger droplets will combine again and again, forming smaller and smaller numbers of ever larger blobs of oil. The end result will have a size distribution with a very large number of smaller droplets, and decreasing numbers of larger blobs. A lognormal equation will closely describe such a distribution. We can represent many kinds of information as frequency distributions. In fact, unless we organize our information as frequency distributions, we can say very little about the statistical or probabilistic behavior of the properties with which we must deal. Most information relevant to oil and gas exploration can be placed in frequency distributions. * The equation for a normal distribution having a mean /x and standard deviation a is

f{x;^,a)

36

= ~===exp v27rc7^

(^ - ^f 2^2

Field Size Distributions Literature on field size distributions is abundant and includes the following papers and reports: Attanasi and Drew (1985), Bettini (1987), Bloomfield and others (1980), Constau (1979, 1980), Drew and Griffiths (1965), Drew and others (1987), Houghton (1986, 1988), Houghton and others (1989), Howarth, White, and Koch (1980), Klemme (1986), Mayer and others (1979), McCrossan (1969), Meyer and Fleming (1985), Meyer and others (1983), Roadifer (1986).

STATISTICS OF FREQUENCY DISTRIBUTIONS In addition to making histogram and cumulative plots of a set of observations such as the sizes of oil fields in a basin, we can calculate statistics, or summary measures, that describe the general shape of the distribution. These descriptive statistics express where the values are located along the measurement scale (measures of centrality) and how they are scattered (measures of dispersion). There are other statistics that describe various attributes of the distribution—whether it is symmetrical, whether it is flattened in the middle—and still others that describe more subtle expressions of shape. However, these measures are less widely used than those measuring centrality and dispersion, so we will omit them here. Statistical measures discussed in this and in the following chapters are contained in all introductory texts on statistics. Presentations tailored for geologists and petroleum engineers are given by Barnes (1994), Cheeney (1983), Davis (1986), Duckworth (1968), Griffiths (1967), Hogg and Ledolter (1992), Koch and Link (1980), Krumbein and Graybill (1965), Mezei (1990), Miller and Kahn (1962), Till (1974), and Walpole and Myers (1993).

Measures of Centrality Three statistics commonly are used to indicate the "average" or "typical" or "representative" value of a collection of measurements. All of these statistics express the location of the center of the distribution along the scale used to measure the observations. These are: Mode: The value that occurs most often. This is the class of the histogram that contains the largest number of observations. Some distributions may have more than one mode. Median: The value such that half the observations are larger, and half are smaller. The median splits the histogram into two equal-sized parts. Mean: The arithmetic average, found by adding together all of the values and dividing by the number of observations. In equation form,

37

Computing Risk for Oil Prospects — Chapter 2 where X is the statistical symbol used to indicate the mean, Yl indicates the mathematical operation of summation, Xi is the i-th value in the set, and n is the total number of observations. The mean is one of the parameters of the normal distribution. In addition, we must be familiar with another measure of centrality, the geometric mean. It is especially useful for describing the average of a set of lognormally distributed observations. Geometric mean: The n-th root of the product of n observations. This is equivalent to the antilog of the arithmetic average of the logarithms of n observations. In equation form, G M = ( n X , ) ' ' " = antilog

( S i ^ )

where GM is the statistical notation for geometric mean, n is a mathematical operator indicating all values are multiplied together, Xi is the i-th value in the set, and the exponent 1/n indicates that the n-th root of the product is to be taken. If values of Xi are replaced with their logarithms (log Xi), the same results are obtained by finding their arithmetic mean, then taking the antilog of the mean. If a histogram is symmetric, all of the measures of centrality will be approximately the same. If a histogram is asymmetric, the various measures of centrality will have a certain relationship to each other. If the distribution has a long tail extending to high values, the mean will be largest, the median will be smaller, and the mode will be the smallest value. The geometric mean will be close to the median.

Measures of Dispersion These statistics measure how scattered the values are about the central point, or how much of a histogram is contained in bars other than that representing the central class. Only two measures are widely used: Variance: The average squared deviation of each observation from the mean. (Deviations from the mean are both positive—larger than the mean, and negative—smaller than the mean. If they are summed to find their average, the positive and negative deviations cancel. To avoid this, the deviations are squared. A similar effect can be achieved using absolute deviations, but the variance has more useful statistical properties.) In equation form. S

38

=

Field Size Distributions where s^ is the statistical notation for variance and {Xi — X) indicates the difference between an individual value Xi and the mean X. (In practice, a slightly smaller denominator is used to compensate for the fact that each observation is also used to calculate the mean, X.) The variance is the second parameter of the normal distribution. Standard deviation: The square root of the variance. The variance is expressed in units that are the squares of the units of the original observations. Taking the square root returns the statistic to the original scale of measurement. In equation form.

v?=.s'*-^)' Percentiles If we rank or arrange a set of observations in order from smallest to largest, we can count the numbers of observations that are smaller than certain values. The count can be expressed in a relative form as percentiles. The median is the 50th percentile; that is, 50% of the observations are smaller than the median. Any other percentile can be calculated, as well. For example, 10% of the observations are smaller than the 10th percentile, and 10% of the observations are equal to or larger than the 90th percentile {i.e., 90% are smaller than the 90th percentile). We can demonstrate the calculation of a field size distribution and the associated statistics by analyzing data on fields discovered in the Powder River Basin of Wyoming through 1988. These data are contained in a file called PRBASIN.DAT on the accompanying diskettes. RISKSTAT will compute the mean, variance, geometric mean, and various percentiles of field sizes of whatever variable is chosen for analysis. For the Powder River Basin, the relevant statistics for field volume (x 1000 bbls) include: 6.63 Number of points 369 Coefficient of variation Mean 7113.3 Minimum 1 60 Sum 2,624,816 25th percentile Variance 2,223,312,400 Median 573 3792 Standard deviation 47,152.01 75th percentile 869,800 Geometric mean 421.2 Maximum In order to compare histograms prepared using diff^erent numbers of observations, they must be plotted as relative frequencies rather than as absolute frequencies. That is, the scale along the Y axis is expressed as 39

Computing Risk for Oil Prospects — Chapter 2

"'"[ '

lUU _

904 804

.... — .... .... .... — — —

—H .... ..... —-r— .... .............. .... .... . . . . . j . — .... .... .... ....[.... .... ..... .... .... .... , . . . . i . . . . | . . . .... — — ...4....^

j

w 704

1 60-1

.... — .... ..... ....

I 30i H— U

....[....

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—

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.—j.—

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• ; 400 600 800 Volume, bblsx 10^

m

— .....

—j-.-.j....

—

—— .... — . . . . j . . . . .... .... —— —

.........^

m

1000

....; 1200

Figure 2.5. Histogram of estimated ultimate production volumes of oil and gas fields in BOE discovered through 1988 in the Powder River Basin of Wyoming, plotted with a relative frequency (percentage) scale. a percentage of the total number of fields. The histogram of field sizes in the Powder River Basin is shown in Figure 2.5 with a relative frequency scale. Otherwise, the scales of the Y axes of different histograms will differ according to the number of observations in the data set. If we examine the histogram and compare the geometric mean, the mean, and the median, it is immediately apparent that the distribution is highly skewed and not at all normally distributed. If the data are transformed by taking their logarithms, the resulting distribution may be more nearly normal in shape (Fig. 2.6). Exploration in the Powder River Basin perhaps can be characterized as moderately advanced, with the possibility of large pools remaining to be discovered with additional drilling in the deeper parts of the basin. Known fields occur in a variety of trap types that are either structurally or stratigraphically controlled. Other Rocky Mountain basins are at different stages in their exploration history, and the nature of plays may differ from basin to basin. It may be instructive to examine the field size distributions for different basins in the Rocky Mountain Province to see how the distributions vary. This can be done by analyzing the data on field sizes for other basins contained in files on the diskettes. 40

Field Size Distributions 20 "1

15 4 CO

0

o ioi

c0

0 CL

3

4

5 6 7 Volume, Log-jobbls

8

9

Figure 2.6. Histogram of logarithms of estimated ultimate production volumes of oil and gas fields in BOE discovered through 1988 in the Powder River Basin, plotted with a relative frequency scale. The Denver-Julesburg Basin (data file DJBASIN.DAT) is extensively explored and is at a very mature stage of its development. Also, almost all the fields are stratigraphically controlled and occur in only a few (primarily Lower Cretaceous) sands. These differences affect the form of the histogram of the field size distribution (Fig. 2.7). In the Big Horn Basin (file BHBASIN.DAT) production comes from a variety of stratigraphic horizons ranging in age from Ordovician to Cretaceous. Most fields are structurally controlled. These factors may exert a significant influence on the shape of the field size distribution (Fig. 2.8). The data set contains all fields discovered in the basin through 1988; the infiuence of the additional (mostly small) fields discovered since 1977 can be seen by comparing Figures 2.8 and 2.4. Field size data from the Wind River Basin are given in the file WRBASIN.DAT. The Wind River is a much smaller basin with fewer discoveries than in most other Rocky Mountain basins. The influence of this on the form of the field size distribution is apparent in the histogram of the data, and in the computed statistics (Fig. 2.9).

41

Computing Risk for Oil Prospects — Chapter 2

25^ (fi •D Q) UL

20-

C

i

Q.

03

4

5 6 7 Volume, Log^o ^^^^

8

9

Figure 2.7. Histogram of logarithms of estimated ultimate production volumes of oil and gas fields in BOE discovered through 1988 in the DenverJulesburg Basin of Colorado and adjacent states, plotted with a relative frequency scale.

CUMULATIVE PROBABILITY PLOTS The frequency distribution in Figure 2.4 is a histogram plotted in the conventional way. The curve fitted to it is a normal curve, although the horizontal scale is logarithmic. An alternative is to plot the histogram in cumulative form. A normal curve fitted to a cumulative distribution generally has a "lazy-S" or sigmoidal shape. Figure 2.10 is produced using the same data that were used to produce Figure 2.4 (field sizes in the Big Horn Basin through 1977), but is plotted in cumulative form. The numbers of fields in each frequency class are progressively accumulated, so that the histogram bar at the extreme right equals 100% along the cumulative scale. It's also possible to plot the individual observations in a form that is very similar to a cumulative histogram. Simply sort the data from smallest to largest and assign each a rank from 1 to n. Divide each observation's rank by n, converting it into a fractile, then multiply by 100 to express it as a percentile. Plot each percentile value against the volume of the corresponding field, as has been done in Figure 2.11. Because the logarithms 42

1

20(0

12 .9^ LL

Field Size Distributions

i

U 14... i

i

15-

\

" 1

h

1

1 10-

->

1^5^

Q-

5-

^m.

0 " 0

3

I

\

V'olurrle,

i LI=^910

y

8

(

bbis

Figure 2.8. Histogram of logarithms of estimated ultimate production volumes of oil and gas fields in BOE discovered through 1988 in the Big Horn Basin of Wyoming, plotted with a relative frequency scale. of field volumes in our data set form an almost perfect normal distribution, as we saw in Figure 2.4, the cumulative plot forms the characteristic "lazy S" we previously noted in the cumulative histogram (Fig. 2.10). We could judge how closely the distribution of any set of field sizes approached log normality by comparing their cumulative plot to the sigmoidal curve that represents a perfect lognormal distribution. Computing and drawing an ideal normal curve and making the comparison might be a bit difficult, but statisticians have borrowed an old engineer's trick that makes the process much easier. Rather than drawing a sigmoidal-shaped normal curve on ordinary graph paper, the grid of the graph can be distorted so that a cumulative normal curve plots as a straight line. It's much easier to compare a scatter of points to a straight line than it is to compare them to a complicated curve. The distorted scale of the graph is produced by dividing a normal distribution into successive segments that contain equal areas under the curve, say 5%. A segment near the tails of the distribution that contains 5% of the total area will be very wide, while a segment that contains 5% near the center of the distribution will be quite narrow. The graph is constructed so the spaces between lines correspond 43

Computing Risk for Oil Prospects — Chapter 2 20-r

15-1 (0

g>

LL

^ 10c 0 0

a.

5-f

4

5 6 Volume, LoQiobbls

Figure 2.9. Histogram of logarithms of estimated ultimate production volumes of oil and gas fields in BOE discovered through 1988 in the Wind River Basin of Wyoming, plotted with a relative frequency scale. to the widths of the successive intervals in the distribution. The resulting lines will be very closely spaced near 50%, and become farther and farther apart toward either end. Both zero and 100% on the scale lie at infinite distances, so they do not appear. Figure 2.12 shows cumulative field size data for the Big Horn Basin plotted on this special graph, which is called a probability plot. Since the field sizes are given as logarithms of volume, it actually is a log-probability plot, and a lognormal distribution will graph as a straight line. In this example, we can see that the data follow a lognormal distribution almost perfectly, with minor deviations only in the largest and smallest field sizes. (Note that conventional log-probability graphs are drawn with volume along the Y axis and cumulative percent or probability along the X axis. It is common to arrange the cumulative percentage scale in ascending order, so that the lower end of the cumulative percentage corresponds with the low end of the sequence of field sizes. The resulting plot extends from lower left to upper right. We'll follow this convention in the remainder of this book. The cumulative percentage scale also can be plotted in reverse order, and some plots contain cumulative percentage scales in both ascending and descending order, which are complementary.) 44

Field Size Distributions l\J\J

•8

8 0 -1

i

[

\

LL

o c 0 y

60

§

40 -

0 CL

3

E ,^ 20 O r\

A.'.'.'.'.W.y.'

3

4

6 7 /olunle.L ogio bbis

8

9

Figure 2.10. Frequency distribution of volumes in BOE of 108 fields in the Big Horn Basin plotted in cumulative form. Compare with Figure 2.4 which represents distribution plotted in conventional form. Vertical scale differs in the two figures. A major advantage in plotting field size data in this manner is that we are able to project the plotted distribution toward its two ends where probabilities are very low. This may be important in attempting to estimate the probabilities of large fields that have very low probabilities of occurrence. A giant field is a rare event, and it is difficult to estimate intuitively the probability of its occurrence, although estimates are readily obtained by projecting a log-probability plot. Of course, a log-probability plot is a simple model or theoretical abstraction which may not actually represent the real world, so it must be used with caution. (We may sometimes find a cumulative field size distribution that is s-shaped, or has some other unexpected form. This may indicate that the fields have been drawn from two or more populations that have quite distinctly different means.) Uses of log-probability plots in oil exploration and hydrocarbon resource assessment have been described by Barouch and Kaufman (1976a, 6); Harbaugh, Doveton, and Davis (1977); Harbaugh and Ducastaing (1981); Kaufman (1963); and Kaufman, Balcer, and Kruyt (1975).

45

Computing Risk for Oil Prospects — Chapter 2 Table 2.2. Probabilities attached to occurrence of fields in specific size classes in an unexplored basin analogous to the Big Horn Basin. Size class in BOE

Probability attached to each size class

Less than 10^ 10^ to 10^ 10^ to 10^ 10'^ to 10^ Greater than 10^

19-0 50-19 80-50 96-80 100-96

= = = =

19% 31% 30% 16% 4% 100%

Probability Estimates from Frequency Distributions A major advantage in using probability plots is the ease with which probability estimates can be read directly from the graphs. If we were exploring in an immaturely explored basin believed to be similar in endowment of oil and gas fields to the Big Horn Basin, we could use the Big Horn Basin as a geological and statistical model for the unexplored basin. The Big Horn Basin has been more or less extensively explored, and we can presume that most of its large- and intermediate-sized fields have been found. We have considerable knowledge about the population of fields in the Big Horn Basin, but not total knowledge because exploration is incomplete. We are interested in forecasting the probability of discovering fields of specified size in the unexplored basin. If the Big Horn Basin is an appropriate analogue, we can obtain probability estimates for the different field size classes from the log-probability plot of the Big Horn fields. Table 2.2 provides probabilities for five field size classes, which represent all possible field sizes and totals 100%. We could choose different classes if we wished. The probabilities can be read from Figure 2.13, which is similar to Figure 2.12, but the axes have been switched and it has solid lines that project field size classes onto the cumulative percentage scale, from which the probabilities in Table 2.2 are obtained. A log-probability plot can be extrapolated indefinitely in principle, but it is questionable practice to extend it much beyond existing data points. Nevertheless, we can extrapolate to obtain estimates for fields larger than those discovered. This presumes that the lognormal distribution remains appHcable, but this is doubtful because the largest field in the Big Horn Basin is Elk Basin Field, which will yield an estimated 585 million BOE. The probability of a field larger than Elk Basin Field in the Big Horn Basin 46

Field Size Distributions 100 (0

c 2

(D Q0)

.>

E

o

5 6 7 Volume, Log^o ^^'^ Figure 2.11. Plot of logarithms of volumes of individual fields in the Big Horn Basin versus percentile rank. Note similarity in form with Figure 2.10. in Wyoming is very small and may be zero. However, here we are interested in estimating probabilities of very large fields that may occur in an immaturely explored basin for which the Big Horn Basin is an analogue. Extrapolation of the plot for the Big Horn Basin (Fig. 2.14) may be useful for this purpose. For example, the cumulative percentage scale yields probabilities of about (100% - 99.2%) -= 0.8% for fields of 1,000 million BOE or larger, and about (100% - 99.97%) = 0.03% for fields greater than 10,000 milUon BOE. These small probabiUties may be reasonable if the Big Horn Basin is an appropriate analogue. Although the volumes of fields discovered so far in the Big Horn Basin convincingly follow a lognormal distribution, extrapolations must be employed with caution. Options in the RISKSTAT program will allow you to create all of the necessary graphical displays useful for analysis of field size distributions, including histograms, cumulative plots, and log-probability plots. Data can easily be transformed by taking their logarithms, or the original data can be plotted along axes with logarithmic scales. Be sure not to both transform data to logarithms and plot the transformed data on logarithmic axes, or your graphs will show logarithms of logarithms! The diskettes contain data files containing estimated ultimate field sizes for discoveries 47

Computing Risk for Oil Prospects — Chapter 2

3

4

5 6 7 Volume, Log-jo bbis

Figure 2.12. Log-probability plot of field volumes in the Big Horn Basin. A perfect lognormal distribution would form a straight line.

5 10 20

50

80 9095

99 99.9 99.99

Cumulative Percent of Fields Figure 2.13. Plot shown in Fig. 2.12 with axes switched. Solid lines define specific size classes and corresponding cumulative probabihty intervals. 48

Field Size Distributions 10000 q 8000 d

JO

(0

c o E O

>

70

80

90 95 99 Cumulative Percent of Fields

99.9

99.99

Figure 2.14. Extrapolation of log-probability plot of field sizes in the Big Horn Basin to extremely large field volumes. This graph is an extension of the upper right part of Figure 2.13. in several Rocky Mountain basins, including the Big Horn Basin, Powder River Basin, Wind River Basin, and the Denver-Julesburg Basin. Graphs similar to Figure 2.13 can be constructed easily.

CAUTION: F U T U R E DISCOVERIES ARE LIKELY TO BE DRAWN FROM POPULATIONS OF SMALLER FIELDS We have stressed the value of systematized hindsight in the form of frequency distributions of knov^n oil and gas fields. However, the population of oil and gas fields discovered in a basin may not be an entirely appropriate guide to the characteristics of fields that remain to be discovered. As a generalization, large fields tend to be discovered early and progressively later fields tend to be smaller. The early discovery of large fields is a well-known phenomenon, and we can learn valuable lessons by statistically analyzing 49

Computing Risk for Oil Prospects — Chapter 2 changes in field size as discovery proceeds. If regular trends in the sizes of fields discovered are detectable, these trends can be incorporated in probabiUty estimates of the size of future discoveries. Harbaugh and Ducastaing (1981), Kaufman, Balcer, and Kruyt (1975), and Lee and Wang (1985) deal with changes in field size statistics.

Occurrence Probabilities Versus Discovery Probabilities At this point, it is appropriate to distinguish between occurrence probabilities and discovery probabilities. Occurrence probabilities depend on nature's endowment of oil and gas pools, regardless of whether they have been discovered or not. Discovery probabilities, on the other hand, depend upon the capability of discovering pools which exist and have not yet been found. Of course, the two forms of probability are closely interdependent. Our knowledge of oil and gas occurrence depends entirely upon past discoveries, but if discoveries have been made, the pools discovered must have occurred. When a region has been thoroughly explored, we have a good understanding of the pools with which it has been endowed. When it is only partially explored, the population of discovered fields is an imperfect approximation of the population of actual pools, and until exploration is complete there is a gap between the occurrence population and the discovered population. Opportunities for future discoveries exist because of this gap. The magnitude of the gap at any instant is influenced by the basin's geological complexity, by exploration technology, by the depth to the producing horizons, and by the economics of exploration and exploitation and the availability of leases or concessions (Kaufman, Balcer, and Kruyt, 1975). We will now switch our attention from the Rocky Mountains to exploration in the mythical country of Magyarstan where many of the demonstrations in this book are set. File MAGVOL.DAT in the diskettes contains data on 98 fields discovered in a petroleum district in the country since exploration resumed following the end of World War II. These provide a good approximation of the endowed population of large- and intermediate-sized oil pools in the district. The relative maturity of the area suggests that the discovery of additional large fields has a low probability. If we extract probabilities from the present field size distribution and use them to forecast occurrence probabilities in a geologically analogous but little explored area, these estimates should be most reliable for large- and intermediatesized fields, and less so for small fields. The distribution of known fields in the district in Magyarstan is inappropriate for estimating additional discoveries within the district itself because new fields will come from a population consisting mostly of smaller 50

Field Size Distributions pools. The problem is to adapt information from the 98 previously discovered fields to forecast the form of the population of the remaining undiscovered oil pools. Progressive changes have taken place as this part of Magyarstan has been explored. An appropriate way to study these changes is to segregate the distribution of fields into subsets according to their discovery sequence. Then, we can examine successive distributions of the subsets and determine if there are systematic changes in their character that can be extrapolated into the future. In order to sort the field size data into subsets for analysis using RISKSTAT, it is necessary to first rank the data by discovery date of the fields and to convert the ranks to cumulative fractions. This will express the fields in terms of their relative order of discovery. The data can then be divided into subsets corresponding to the 33rd! and ^Qth percentiles, splitting the discovery history into three equal parts. Log probability plots of the three subsets are shown in Figure 2.15. Note the change in slope of the cumulative curves with time, and especially the change in the median field size discovered. What will the distribution of field sizes be for the next 32 fields to be discovered (fields 99 through 130)? An estimate can be obtained by projecting trends observed in earlier discoveries. Plots of the subpopulations contrast with the overall population. They intertwine and overlap; these irregularities probably reflect shifts in exploration practices, economic influences, and the statistical vagaries that result from the small sizes of the subpopulations. By plotting the median field sizes for the successive subsets of the data on a semilog graph, we can extrapolate the curve connecting these medians to the next 33% interval and also to 33% beyond that. Extrapolation of the medians is somewhat uncertain, but the downward trend is both reasonable and expected. A plot of cumulative volumes (in BOE) of oil and gas discovered per each 33% interval may also be useful, as the cumulative volume discovered in each interval decreases more regularly than do the medians. The necessary cumulative volumes can be obtained using the Univariate Statistics Option of RISKSTAT. The predicted median sizes for the next two intervals can be plotted on the graph of cumulative field sizes for the three subsets (Figure 2.16). Through these medians we can draw estimated distributions that follow the trends in the earlier size distributions; these are generalized projections of the earlier distributions. The projections are simple visual approximations, and the projected subpopulations may differ from the lognormal ideal in that their upper and lower ends may be smaller than in perfect lognormal distributions with the same medians. Other projections could be shown, such as straight lines for perfect lognormal distributions. 51

Computing Risk for Oil Prospects • 109l::::::::5

108-i:

:::^:::::::::i:::::[:

o First Third A Second Third • Third Third

Chapter 2 :::i:::t::1:::: H--4-H"

1:?:::::::::^:: ...O-i—

::::+::::::2fp::::: .....

^-f-•4'SlSf*"! ..JI"

E 107i:

:q::;;:::

106^: ••-1—r--i—-

105-

.01

.1

S|S2^=

T-1—r 1—r 5 10 20 50 80 9095 99 99.9 99.99 Cumulative Percent of Fields

tr

Figure 2.15. Log-probability plot of field volumes in a Magyarstan district with first 33, second 33, and final 32 fields discovered plotted separately.

1

r—1

5 10 20 50 80 9 0 9 5 99 99.9 99.99 Cumulative Percent of Fields Figure 2.16. Empirical models of field size distributions for successive thirds of fields discovered in Magyarstan district along with potential size distributions for next 32 fields discovered and 32 fields after that. 52

Field Size Distributions Table 2.3. Field size statistics for a district in the Republic of Magyarstan.

Percentage ranges

Percentage of present (MBOE*) total 100% 1,706,670

Total for Median interval

Number of fields

(MBOE*)

Entire basin

0-100%

98

13,660

Progressive discoveries through present

0-33% 33-66% 66-100%

33 33 32

19,566 16,097 5,949

881,451 556,537 268,682

52% 33% 16%

Forecast 100-133% future 133-166% discoveries

32 32

2,100 600

86,000 23,000

5% 1%

*Thousands of barrels of oil equivalent.

Table 2.4. Discovery probabilities in percent associated with two populations of future fields in a district of the Republic of Magyarstan. Probabilities pertain to the size of a field, given that a discovery has been made. Dry hole probabilities are not considered. Field size classes in BOE

Next 32 fields Next 32 fields after that

Less than 100,000

100,000 to 1,000,000

1,000,000 to 10,000,000

Greater than 10,000,000

5%

26%

64%

5%

20%

43%

36%

Less than 1%

From the statistical analyses of the three subsets of the Magyarstan data and the log probability projections, forecasts of the next 33% of fields to be discovered, and the 33% after that, can be made (Table 2.3). These projections, although somber, do represent extrapolations of established trends. If confirmed, they indicate that most of the district's oil and gas has been discovered, and the district is in a mature stage of exploration. Probabilities of field size classes can be read from the two future log probability distributions. Table 2.4 is determined by interpolating values from the plots. The plots represent only the probable size of a field, given 53

Computing Risk for Oil Prospects — Chapter 2 that one is discovered, and are not conditional upon local geology. They may provide a foundation for estimating discovery probabilities attached to specific prospects remaining in the Magyarstan republic, and could be revised upward or downward for specific prospects depending upon merits of the prospects. Dry hole probabilities must also be estimated, but these are not derived from field size data.

54

CHAPTER

3

Success, Sequence, and Gambler's Ruin In Chapter 2 we dealt with estimates of probabilities of field discoveries in different size categories provided that a field is discovered. These estimates tell us nothing, however, about dry hole probabilities. The oil field population estimates are based solely on information about fields and do not include information about exploration wells. Thus we need information about outcomes of exploratory holes so that we obtain dry hole frequencies as well as frequencies of field sizes. Estimating the dry hole probability attached to a specific prospect poses the same general problem as estimating the spectrum of discovery probabilities. Each prospect may be unique, but it should be compared with the industry's overall current performance in the region to provide a frame for reference. The dry hole probability can be adjusted upward or downward, depending on the qualities of the prospect. Meanwhile the industry average, known as its siiccess ratio, provides an important reference statistic and is defined as the proportion of exploratory wells that are "successful," that is, that are completed and produce some oil or gas. A success ratio is an estimate of the probability of a producer, and the dry hole probability is its complement. (Note that by this definition, a "successful" well can be an economic failure in that it may produce little oil or gss before being abandoned, possibly entailing substantial financial loss.)

Computing Risk for Oil Prospects — Chapter 3 Table 3.1. Rescaling of probabilities for an exploratory well attached to different field size categories for future discoveries in the Big Horn Basin, assuming a dry hole probability of 80%. Probabilities before rescaling (%)

Outcome Dry hole 10^ BOE or less 10^ to 10^ BOE 10^ to 10^ BOE 10^ to 10^ BOE 10^ BOE or more Totals

Probabilities after rescaUng (%)

4 8 26 45 17 100

80 1 2 5 9 3 100

SUCCESS RATIOS AND DRY HOLE PROBABILITIES If we are appraising a prospect, we need the dry hole probability plus the probabilities attached to different field sizes if a field is discovered. Each of the outcomes is mutually exclusive of the others, and since all possible outcomes should be represented, the probabilities should sum to 100%. Graphing such a probability distribution poses a problem because we cannot represent a dry hole with zero barrels on a logarithmic scale. However, it is convenient and often essential to represent oil field size distributions on a logarithmic scale. Fortunately, we do not need to represent the dry hole probability graphically because it is a single number. If we have probabilities attached to different field size categories and an estimate of the dry hole probability, we can simply rescale the probabiUties attached to the field size categories to accommodate the dry hole probability. Table 3.1 provides an example in which the dry hole probabiUty is 80%. Rescaling is done simply by multiplying the probability of making a discovery of a specified size by the probability of a success (100 - 80 = 20%). The computation of a success ratio is strongly infiuenced by the definition of wells involved. Rank wildcat exploratory wells usually have much lower success ratios than development or step-out wells. A company that has drilled a number of exploratory holes in a region can calculate its own success ratio, providing a useful benchmark in appraising subsequent prospects and a measure of its success as compared to that of its competitors. 56

Success, Sequence, and Gambler^s Ruin

LONG-TERM LUCK AND THE BINOMIAL DISTRIBUTION Luck is obviously a major factor in exploration. Even the best geological understanding and the most sophisticated geophysical interpretations will not guarantee success. The success ratio is a measure of success that is generally valid on a regional basis and is useful when either individual wells or outcomes of sequences of wells are to be considered. The binomial distribution provides a convenient method for estimating probabilities attached to sequences of events when there are only two possible outcomes per event. Two such outcomes might be drilling a producing well versus drilling a dry hole. In spite of its limitation to two mutually exclusive outcomes, the binomial distribution is both useful and simple. An introduction to the binomial can be found in Megill (1984) and in most textbooks on probability theory. McCray (1975) discusses application of binomial distributions. In our context the binomial distribution can be written where p = probabihty of a well being a producer, estimated by the success ratio d = probability of the same well being dry n = number of trials or number of wildcat wells drilled. We can expand the binomial {p + d) to the appropriate power according to the number of wells to be drilled in a sequence. Expansions to the fifth power are provided in Table 3.2 and they can be generated readily for higher powers. The table illustrates that the expansion progresses in a regular manner. In fact, we can use "Pascal's triangle" in Table 3.3 to obtain the coefficients of the terms in the expansion to the tenth power. The probabilities associated with the expansions are obtained by inserting values for p and d. Each term in an expansion denotes a particular outcome. Consider the expansion to the third power, which could represent a three-well situation: {p + df = p^ + 3p2d -I- 3pd^ _^ d^ p3 = 3p'^d = 2 producers 3 producers and 0 dry holes and 1 dry hole

Spd'^ = 1 producer and 2 dry holes

0 producers and 3 dry holes

Here there are four outcomes, all of which are mutually exclusive, and collectively, all-inclusive. When p and d are inserted and the probabilities 57

Computing Risk for Oil Prospects — Chapter 3 Table 3.2. Expansion of the binomial to the fifth power. Power

Expansion

{p + df ip + d)^ {p + d)^

p^ + 3p'^d-\-Zp(f-\-d^ p^ + V d + 6p2d2 _!_ 4^^3 ^ ^4 p^ + ^p^d + lOp^^ + 10^2^3 ^ 5^^4 ^ ^5

p'^ + 2pd-\-(P

Table 3.3. Coefficients of terms in binomial expansions represented by Pascal's triangle. Each number in the triangle is the sum of the two numbers immediately to the right and left of it on the line above. Power 0 1 2 3 4 5 6 7 8 9 10

Coefficients of Term 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 220 252 220 120 45 10 1

calculated, they must sum to 1.0, or 100%, because there are no other possibiUties. If the regional success ratio is 20% and the dry hole ratio is therefore 80% {p = 0.2 and d = 0.8), the probabilities attached to the four outcomes are: (0.2 + 0.8f - 0.2^ + 3(0.2^ x 0.8) + 3(0.2 x 0.8^) + 0.8^ = 0.008 + 0.096 -h 0.384 + 0.512 0.8^ = 3(0.22 X 0.8) = 3(0.2 X 0.82) = 0.2^ = 3 producers 1 producer 0 producers 2 producers and 0 dry holes and 1 dry hole and 2 dry holes and 3 dry holes Two important points need emphasis. First, we do not consider the sequence of outcomes. For example, the second term of the binomial equation 58

Success, Sequence, and Gambler's Ruin Table 3.4. Probabilities associated with eight tosses of a coin. Outcomes No. of No. of heads (H) tails (T) 8 7 6 5 4 3 2 1 0

0 1 2 3 4 5 6 7 8

Terms in expansion of binomial to Sth power H^ 8H^Ti 28 H ^ T 2 56 H^T^ 70 H^T^ 56 H^T^ 28 H^T^ 8HiT^ rpS

Probabilities attached to Arithmetic each outcome (0.5)« 8(0.5)^(0.5)1 28(0.5)^(0.5)2 56(0.5)5(0.5)3 70(0.5)^(0.5)4 56(0.5)3(0.5)5 28(0.5)2(0.5)6 8(0.5)1(0.5)^ (0-5)«

0.0039 0.0313 0.1094 0.2187 0.2734 0.2187 0.1094 0.0313 0.0039

for three holes involves two producers and one dry hole. The dry hole could be first, second, or third in the sequence. Secondly, the outcomes should be independent of each other. That is, the outcome from the driUing of one hole will not affect what will occur when subsequent holes are drilled. This important assumption may or may not be satisfied. The outcomes of three wells drilled close to each other are unlikely to be independent, but if they are many miles apart, they are more likely to be independent. The binomial distribution is a probability distribution involving discrete events such as "producer or dry," or "heads or tails." Coin-tossing experiments commonly are used to illustrate binomial probability. A tossed coin has a 50-50 probability of turning up heads or tails in a single toss. Furthermore, each toss in a sequence is independent, because the results of previous tosses obviously can have no influence on subsequent tosses. To obtain the probabilities associated with eight coin tosses we insert 0.5 and 0.5 into the expansion to the eighth power (Table 3.3), yielding values for the nine possible outcomes as illustrated in Table 3.4. Calculating binomial probabilities by hand is tedious. Tables that cross-reference expansions to a progression of different values of jo or c^ are easily generated by computer. Table 3.5 provides tables labeled in an oil-exploration context for sequences from two to ten wells that are cross-tabulated for nine different success ratios or values of p. The decimal fractions in the table are the probabilities attached to alternative outcomes.

59

Computing Risk for Oil Prospects — Chapter 3 Table 3.5. Table of binomial probabilities labeled for an oil exploration context. A = number of wells drilled in sequence; B = number of successful wells drilled.

A B "OOi

CTOS

OlO

Regional success ratios OTS 0 2 0 025 030

2 0 1 2 3 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0.9801 0.9025 0.8100 0.7225 0.6400 0.0198 0.0950 0.1800 0.2550 0.3200 0.0001 0.0025 0.0100 0.0225 0.0400 0.9703 0.8574 0.7290 0.6141 0.5120 0.0294 0.1354 0.2430 0.3251 0.3840 0.0003 0.0071 0.0270 0.0574 0.0960 0.0000 0.0001 0.0010 0.0034 0.0080 0.9606 0.8145 0.6561 0.5220 0.4096 0.0388 0.1715 0.2916 0.3685 0.4096 0.0006 0.0135 0.0486 0.0975 0.1536 0.0000 0.0005 0.0036 0.0115 0.0256 0.0000 0.0000 0.0001 0.0005 0.0016 0.9510 0.7738 0.5905 0.4437 0.3277 0.0480 0.2036 0.3280 0.3915 0.4096 0.0010 0.0214 0.0729 0.1382 0.2048 0.0000 0.0011 0.0081 0.0244 0.0512 0.0000 0.0000 0.0004 0.0022 0.0064 0.0000 0.0000 0.0000 0.0001 0.0003 0.9415 0.7351 0.5314 0.3771 0.2621 0.0571 0.2321 0.3543 0.3993 0.3932 0.0014 0.0305 0.0984 0.1762 0.2456 0.0000 0.0021 0.0146 0.0415 0.0819 0.0000 0.0001 0.0012 0.0055 0.0154 0.0000 0.0000 0.0001 0.0004 0.0015 0.0000 0.0000 0.0000 0.0000 0.0001

7 0 1 2 3 4 5 6 7

0.9321 0.0659 0.0020 0.0000 0.0000 0.0000 0.0000 0.0000

60

0.6983 0.4783 0.2573 0.3720 0.0406 0.1240 0.0036 0.0230 0.0002 0.0026 0.0000 0.0002 0.0000 0.0000 0.0000 0.0000

0.3206 0.3960 0.2097 0.0617 0.0109 0.0012 0.0001 0.0000

Oo

0.50

0.5625 0.4900 0.3600 0.2500 0.3750 0.4200 0.4800 0.5000 0.0625 0.0900 0.1600 0.2500 0.4219 0.3430 0.2160 0.1250 0.4219 0.4410 0.4320 0.3750 0.1406 0.1890 0.2880 0.3750 0.0156 0.0270 0.0640 0.1250 0.3164 0.2401 0.1296 0.0625 0.4219 0.4116 0.3456 0.2500 0.2109 0.2646 0.3456 0.3750 0.0469 0.0756 0.1536 0.2500 0.0039 0.0081 0.0256 0.0625 0.2373 0.1681 0.0778 0.0312 0.3955 0.3602 0.2592 0.1562 0.2637 0.3087 0.3456 0.3125 0.0879 0.1323 0.2304 0.3125 0.0146 0.0284 0.0768 0.1562 0.0010 0.0024 0.0102 0.0312 0.1780 0.1176 0.0467 0.0156 0.3560 0.3025 0.1866 0.0938 0.2966 0.3241 0.3110 0.2344 0.1318 0.1852 0.2765 0.3125 0.0330 0.0595 0.1382 0.2344 0.0044 0.0102 0.0369 0.0938 0.0002 0.0007 0.0041 0.0156

0.2097 0.1335 0.3670 0.3115 0.2753 0.3115 0.1147 0.1730 0.0287 0.0577 0.0043 0.0115 0.0004 0.0013 0.0000 0.0001 (cont.)

0.0824 0.2471 0.3177 0.2269 0.0972 0.0250 0.0036 0.0002

0.0280 0.1306 0.2613 0.2903 0.1935 0.0774 0.0172 0.0016

0.0078 0.0547 0.1641 0.2734 0.2734 0.1641 0.0547 0.0078

Success, Sequence, and Gambler's Ruin Table 3.5 (concluded). Table of binomial probabilities labeled for an oil exploration context. A = number of wells drilled in sequence; B = number of successful wells drilled. A B "OOi 8

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10

005

Regional success ratios OlO 0 1 5 020 0 2 5 O30

0.9227 0.6634 0.4305 0.2725 0.1678 0.1001 0.0576 0.0746 0.2793 0.3826 0.3847 0.3355 0.2670 0.1977 0.0026 0.0515 0.1488 0.2376 0.2936 0.3115 0.2965 0.0001 0.0054 0.0331 0.0839 0.1468 0.2076 0.2541 0.0000 0.0004 0.0046 0.0185 0.0459 0.0865 0.1361 0.0000 0.0000 0.0004 0.0026 0.0092 0.0231 0.0467 0.0000 0.0000 0.0000 0.0002 0.0011 0.0038 0.0100 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 0.0012 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.9135 0.6302 0.3874 0.2316 0.1342 0.0751 0.0404 0.0830 0.2985 0.3874 0.3679 0.3020 0.2253 0.1556 0.0034 0.0629 0.1722 0.2597 0.3020 0.3003 0.2668 0.0001 0.0077 0.0446 0.1069 0.1762 0.2336 0.2668 0.0000 0.0006 0.0074 0.0283 0.0661 0.1168 0.1715 0.0000 0.0000 0.0008 0.0050 0.0165 0.0389 0.0735 0.0000 0.0000 0.0001 0.0006 0.0028 0.0087 0.0210 0.0000 0.0000 0.0000 0.0000 0.0003 0.0012 0.0039 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9044 0.5987 0.3487 0.1969 0.1074 0.0563 0.0282 0.0913 0.3151 0.3874 0.3474 0.2684 0.1877 0.1211 0.0042 0.0746 0.1937 0.2759 0.3020 0.2816 0.2335 0.0001 0.0105 0.0574 0.1298 0.2013 0.2503 0.2668 0.0000 0.0010 0.0112 0.0401 0.0881 0.1460 0.2001 0.0000 0.0001 0.0015 0.0085 0.0264 0.0584 0.1029 0.0000 0.0000 0.0001 0.0012 0.0055 0.0162 0.0368 0.0000 0.0000 0.0000 0.0001 0.0008 0.0031 0.0090 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 0.0014 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

O40

0.50

0.0168 0.0896 0.2090 0.2787 0.2322 0.1239 0.0413 0.0079 0.0007 0.0101 0.0605 0.1612 0.2508 0.2508 0.1672 0.0743 0.0212 0.0035 0.0003 0.0060 0.0403 0.1209 0.2150 0.2508 0.2007 0.1115 0.0425 0.0106 0.0016 0.0001

0.0039 0.0312 0.1094 0.2188 0.2734 0.2188 0.1094 0.0312 0.0039 0.0020 0.0176 0.0703 0.1641 0.2461 0.2461 0.1641 0.0703 0.0176 0.0020 0.0010 0.0098 0.0439 0.1172 0.2051 0.2461 0.2051 0.1172 0.0439 0.0098 0.0010

61

Computing Risk for Oil Prospects — Chapter 3

Graphs of the Binomial Distribution Tables and graphs of the binomial distribution are useful for simple assessments of various outcomes in exploration. Figure 3.1 gives histograms of the binomial distribution expanded to the eighth power (that is, for nine alternative outcomes) for five different probabilities. We can estimate p, the probability of a discovery, by the regional success ratio. Then, these histograms give the probabilities of different numbers of discoveries in a drilling program consisting of eight holes. The possible outcomes range from no discoveries up to discoveries on all eight holes. When the regional success ratio is equal to 50%, the binomial distribution approximates a normal distribution. The binomial distribution becomes increasingly skewed as p diverges from 50%. At 5%, the distribution is extremely skewed. (If p were 95%—a highly unlikely exploration success ratio but appropriate for other circumstances—the histogram would be a mirror image of the 5% histogram shown.) Graphs of the binomial distribution are useful for exploration forecasting. For example, if we have enough funds to drill five wildcat wells, we may wish to estimate the probabilities of no discovery, one or more discoveries, two or more discoveries, and so on. The probability of one or more discoveries is equal to the sum of the probability of making exactly one discovery, plus the probability of making exactly two discoveries— Obviously, such cumulative probabilities will be higher than the probability of making exactly one discovery, exactly two discoveries, or any other specific outcome. Probabilities of "one or more" or "two or more" outcomes generally are more informative than are the probabilities for exact outcomes. Cumulative probabilities can be calculated from Table 3.5 by combining the individual probabilities, but it is easier to read them from graphs. Figure 3.2 is a graph of the binomial distribution for six different success ratios and up to 15 "trials" (or 15 wildcat wells) and gives the probabilities of "gambler's ruin," the condition when absolutely no discoveries are made. Figures 3.3 to 3.5 are similar, but they show, respectively, one or more, two or more, and three or more discoveries. Figure 3.3 for one or more discoveries is the inverse of Figure 3.2. Thus, probabilities read from the two graphs are complementary since the alternative results (one or more discoveries versus no discovery) are mutually exclusive. For example, if we drill five wildcats and the success ratio is 15%, Figure 3.2 indicates a probability of no discovery of 44%, and Figure 3.3 indicates a probability of one or more discoveries of 56%. Success ratios that are not directly shown on the graph, for example 25%, can be obtained easily by interpolating a curve lying between the 20% and 30% curves. 62

0.6

0.5

.2

1: 0

Number of Successes

0.2 0.1 0

Number of Succ0.5

O

0

1

2

3

4

5

6

7

8

N h r o f successes

0.4

.a 0.3

I

1 II

0.2 a

0.1 '

E 0

1

2

9

4

5

6

Number of S m w

7

8

Number of Smcwses

Figure 3.1. Series of histograms of the binomial distribution expanded to the eighth power, representing outcomes of an eight-well drilling program for five different success ratios.

Computing Risk for Oil Prospects — Chapter 3

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Figure 3.2. Graph of the binomial distribution for "gambler's ruin" or no discovery for six success ratios.

64

Success, Sequence, and Gambler's

Ruin

100

1

2

3

4

5

6 7 8 9 10 11 Number of Wildcats

12 13

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Figure 3.3. Graph of the binomial distribution for one or more discoveries.

65

Computing Risk for Oil Prospects — Chapter 3

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66

Success, Sequence, and Gambler's Ruin

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67

Computing Risk for Oil Prospects — Chapter 3 Prom histograms such as those shown in Figure 3.1, we can pose "what if... ?" questions and evaluate different possible scenarios. Such exercises can be especially valuable in appraising the relative attractiveness of different regions before investing time and expense in detailed regional studies and geological reviews. The collective experience of the industry in the regions can provide initial estimates of success ratios, and rough budgetary calculations can estimate the possible magnitudes of exploratory programs that might be undertaken. The binomial distribution then provides a way of predicting the results that may occur. For example, we may wish to know the probability of drilling all dry holes in an exploration program consisting of five wildcat wells, if the regional success ratio is 10%. The graph in Figure 3.2 quickly shows us that the probability of such an unpleasant result is almost 60%. (Table 3.5 provides an exact probability of 59.05%.) The same graph and table also provide probabilities related to more desirable outcomes. Suppose the regional success ratio is 15%, and we want to know how many wildcat wells must be drilled if there is to be an 80% probability of making a discovery? That is, we want the complement of the probability that we will make no discoveries, or the number of wells that must be drilled before the probability of gambler's ruin drops to 20%. From Figure 3.2, we can determine that we must drill at least ten wells before the probability drops to this level, if the regional success ratio is 15%. (From Table 3.5, we can see that the exact probability of some success in a ten-well program is 100% — 19.69% = 80.31%.) We also can invert the questions, and ask what regional success ratio must prevail if we are to achieve certain outcomes with specified probabilities. We might consider launching an exploration program in a region if we felt the possibility of complete failure in the initial venture was below some threshold. For example, if four wildcats are to be drilled, what must the regional success ratio be if there is to be only a 25% chance that all four wells will be dry? We can determine this easily from the graph in Figure 3.2, at the intersection of four wells and 25% probability; the necessary success ratio is approximately 30%. Figures 3.3 through 3.5 can be used to investigate the odds associated with various possible successful outcomes. Suppose eight wildcat wells are to be drilled in a program, and the regional success ratio is believed to be 10%. From Figure 3.3, we see that the probability of one or more discoveries is about 56%. From Figure 3.4, the probability of two or more discoveries drops to about 19%, and from Figure 3.5, the probability of three or more discoveries is down to only about 4%. These same graphs can be read in the inverse manner to determine what the regional success ratio must be in order to achieve some desired 68

Success, Sequence, and Gambler's Ruin objective with a specified probability. If we need two or more discoveries and ten wildcat wells are to be drilled, what must the regional success ratio be if there is to be at least an even chance (50% probability) of their discovery? The answer is at the intersection of 50% probability and ten wildcats on Figure 3.4, and is a success ratio of about 17%. Of course, the binomial distribution can be applied to the evaluation of exploration ventures in mature regions as well as those in frontier areas. The regional success ratio in the Republic of Magyarstan has been historically about 11%. If this ratio applies to subsequent exploration, how many additional wildcats must be drilled in an exploration program if we wish to increase the probabiUty of a discovery from 50% to 80%? First, we must determine the number of wildcat wells needed to achieve a probability of one or more discoveries when the success ratio is 11%. Interpolating an 11% curve on Figure 3.3 provides an estimate of six holes required to achieve a 50% probability of one or more discoveries. Tracing our interpolated 11% curve upward gives an estimate of 14 holes needed to increase the probability of one or more discoveries to 80%. Therefore, we will need an additional eight holes to increase our likelihood of success by the specified amount. The binomial distribution is an example of probabilities that come about as a logical consequence of the "rules of the game." In petroleum risk assessment, this is both a strength and a weakness. For the binomial distribution to provide a valid estimate of drilling outcome probabihties, we must make four assumptions: (1) There can be only two mutually exclusive outcomes; that is, each hole must result in a discovery well or be dry. (2) Each hole that we drill must be independent of all others. Our success or failure on a prior hole will not change the probabilities or what we do on a subsequent hole. (3) The regional success ratio does not change. And finally, (4) We will drill a fixed number of holes. Assumptions (1) and (4) can reasonably be made. Assumption (2) is more difficult to make, because we constantly reassess the geology and modify our perceptions as we gain experience and learn of our competitor's results. A drilling program is dynamic and evolving, and the results of early successes or failures inevitably cause changes in later plans. Finally, assumption (3) also may be questionable. We do not know the "true" regional success ratio; indeed, it is only an abstract concept. What we believe the ratio to be may change drastically, especially if the industry experiences a string of bad luck! This is most apt to happen in a frontier area, where there is limited drilling and little historical experience on which to base our estimates of the success ratio. 69

Computing Risk for Oil Prospects — Chapter 3 However, the binomial distribution provides a way of predicting what may occur in driUing ventures without requiring any additional knowledge beyond an estimate of the regional success ratio and a presumption of the four rules of the binomial. Since an assessment does not depend upon specific information about individual prospects, and no geologic knowledge (beyond that incorporated in the success ratio), we may presume that the addition of geological knowledge and skills will improve the probabilities of success. Thus, we might interpret the probabilities derived from the binomial distribution as a "worst case" that would apply to naive explorationists drilling at random. We may expect that our drilling ventures will do better than predicted by the binomial distribution, but the binomial provides a measure against which our performance can be judged.

70

CHAPTER

4

Estimating Discovery Size From Prospect Size STATISTICAL CORRELATIONS B E T W E E N PROPERTIES We can imagine an analogy for oil exploration in the form of a carnival game in which we search for hidden prizes by sticking pins through a curtain that conceals the prizes from our view. The worth of a prize is determined by its volume—the amount of oil it contains. But the likelihood that our pin will hit a prize is determined by the areas of the prizes relative to the area of the curtain. Obviously, there should be a relationship between area and volume, but it may not be a simple one-to-one relationship. It is important that we determine this relationship as best we can. To determine the relationship between field volume and field area, or between any two variables, we must add to our collection of statistical measures. The variance and standard deviation express the variability of one variable around its mean, and we can calculate the two means and two standard deviations of measurements on field volumes and areas. However, these do not tell us anything about their joint variation, or how the two properties vary together around their common means. Figure 4.1 is a cross plot of the logarithms of field volumes versus the logarithms of field areas for the Denver-Julesburg Basin. The data set (contained in file DJSIZE.DAT on the diskettes) is smaller than that used previously (file DJBASIN.DAT) because information on field area is much harder to glean from public records than information on volume.

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Figure 4.1. Log-log plot of field area in acres versus field volume in barrels for 98fieldsin the Denver-Julesburg Basin. Correlation between log area and log volume is r == 0.31. The pattern of points indicates there is a definite relationship, as we would expect, between log field volume and log field area. We can express this relationship by computing the covariance, a statistic describing the joint variation in two variables. The equation for covariance is sx,Y

=

j:{X,-X)iY,-Y)

n where SX,Y symbolically represents the covariance between two variables X and Y whose means are X and Y. Xi \s the value of the first variable measured on the i-th observation and Yi is the value of the second variable measured on the same observation. In our example, Xi and Yi might be the log area and log volume of the Agate Field, one of the fields of the Denver-Julesburg Basin. (Again, a slightly smaller denominator is used in practice to compensate for the fact that the same observations are used to calculate the means and the deviations from the means.) For the data set shown in Figure 4.1 we can calculate a covariance of 0.70. Covariance has the units of the original variables, in this instance acres and barrels. This results in rather awkward units of measurement (acres x bbls). We can standardize covariances to have dimensionless form, which 72

Estimating Discovery Size from Prospect Size will make them easier to compare. The standardized covariance is called a correlation, and it ranges in value from —1 to -hi. If the variables have a perfect inverse relationship (that is, a unit of increase in one variable is accompanied by a precise unit of decrease in the other), their correlation will be —1.0. If the two variables have a perfect direct relationship, their correlation will be +1.0. Two variables that have absolutely no relationship between them will have a correlation of 0.0. If there is a relationship, but it is less than perfect, the correlation will be intermediate in magnitude. Correlation may be calculated in two equivalent ways. The original data may be standardized by subtracting the mean from each observation and dividing the difference by the standard deviation. The resulting standardized variable will have a mean of 0.0, a standard deviation of 1.0, and will be dimensionless (the units of measurement will be cancelled out). The covariances of standardized variables are correlations. Alternatively, we can calculate the covariance between two raw variables and then standardize the covariance by dividing by the product of the two standard deviations. In equation form,

rxY —

sx,Y

'^'^

~

—

^{Xi-X){Yi-Y)

—

yJziXi-x)'j:iY-Yy

(The n's in the denominators of covariance and the standard deviations cancel out. Again, the equation used in practice is slightly more complicated.) For the Denver-Julesburg data set 0.70 "^'^^(0.42)(0.53)=^'^^ which indicates that there is a weak linear trend between field area and field volume when both are scaled logarithmically. The RISKSTAT program can be used to confirm these figures and to calculate the relationship between volumes and areas of fields discovered in the Denver-Julesburg Basin. The program's graphic routines can make a cross plot with linear or logarithmic scales along either axis, so a log-log plot can easily be produced. However, to compute the correlation between the log of field size and the log of field area, it is necessary to transform the variables to their logarithms before calculating the correlation. Next, we will examine the fitting of lines and how these lines can be used for estimation. This topic is covered in greater detail in introductory statistics books such as Devore and Peck (1986), and in books on statistics for engineers (Mezei, 1990), managers (Newbold, 1988), and geologists (Davis, 1986; Till, 1974). 73

Computing Risk for Oil Prospects — Chapter 4

Fitting Lines In many situations we require more information about the relationship between two variables than simply its strength as measured by the correlation. Certainly fields that are larger in area tend to contain larger volumes of oil, but if the field area is doubled in size, does the oil volume double as well? Or does volume increase at a faster or slower rate than area? We can imagine drawing a line through the cloud of points on a bivariate scatter plot of field area versus field volume that would express this tendency. The equation that defines any line is simply Y = a-\-h X^ where Y is the variable plotted along the vertical axis and X is plotted along the horizontal axis. The a term in the equation is the intercept^ or value of Y at the origin, where X is zero. The term h is the slope^ or rate of change in Y for a unit of change in X. If we knew appropriate values for a and 6, we could use the equation to predict what Y should be for any specified value of X. Regression is a statistical procedure for estimating values for a and h that has certain properties that are very desirable for petroleum risk assessment. The line describing the relationship between Y and X is fitted by the method of least squares. This means that if we measure the deviation of each observation of Y from the fitted line, then square the deviations (so deviations above the line are not cancelled out by deviations below), their sum is the minimum possible (Fig. 4.2). No other line could be chosen that would have smaller squared deviations. Stated in another way, the coefficients a and h have been calculated so the variance or scatter in Y around the line is as small as possible. Finding a and h requires solving a pair of simultaneous equations,

Y,Xa,-Y:xH = Y.^Y where n is the number of points. RISKSTAT contains routines that solve these equations and plots the line defined by the two coefficients on a scatter plot. In addition, it also will list various statistics that are useful for evaluating how well the line describes the variation in y as ^ changes. Regression assumes that Y is to be predicted for some specified value of X. Therefore, all of the variability is measured in terms of the scatter of Y around the fitted regression line, or as F — y , where Y (called "Y-hat") is the predicted value on the line that corresponds to the specific value of X. The scatter about the line in the X-direction is not considered. Of course, we could simply swap the roles of X and Y and fit a line in which the deviations in what had been variable X would be minimized, but the coefficients we calculate would be diff^erent and the line would go through 74

Estimating Discovery Size from Prospect Size 4.5-1

4.0 4 3.5 4 3.0 4 2.5 4 2.0

1—rn—I—|—T—r—I—r-|—n—i—r-|—r—i—r—i—|—i—i—i—i—pT—m—i—]—i—m—r

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

Figure 4.2. Least squares regression line fitted by minimizing the sum of the squares of the deviations in the Y direction (vertical lines). the cloud of points at a different angle. (Both regression lines would cross at a location corresponding to the mean of X and the mean ofY.) At times it seems unrealistic to fit a line considering only the scatter in the direction of one of the variables. The joint variation in both variables is used to fit a reduced major axis, or RMA line. The a and b coefficients are chosen so the RMA line goes through the joint means of X and Y and the slope of the line is equal to the ratio of the standard deviation of Y over the standard deviation of X. Although an RMA line does not have the nice statistical properties of a regression line, it does correspond more closely to what we visually perceive as the "trend" between two variables in a scatter plot. An option in RISKSTAT will fit an RMA line to a bivariate scatter plot, although a confidence band cannot be computed for the line. Figure 4.3 shows the differences between an RMA line and lines fitted by minimizing the deviations in Y and in X. We can examine fines fitted by different methods using the RISKSTAT software and data on volumes and areas of fields discovered in the DenverJulesburg Basin. The Bivariate Scatter Plot option allows you to choose to fit a regression line (Fig. 4.4) which predicts log volume as a function of log area of a field. Since the probability of hitting a field with an exploratory hole is related to the area of the field, but the amount of oil discovered 75

Computing Risk for Oil Prospects — Chapter 4

Figure 4.3. Lines fitted by minimizing squared deviations in Y direction (solid line), squared deviations in X direction (dashed line), and the product of the deviations of X and Y, the reduced major axis or RMA line (dotted hne). is a function of the volume, this seems the most useful relationship to investigate. However, in later chapters you will need to estimate the number of development wells necessary to produce a field which contains different specified volumes. The number of wells required is related to the well spacing and the field area. Therefore, you will need the regression giving log area as a function of log volume. To find the inverse regression, simply interchange the roles of X and Y. You will note that the line of regression of log area on log volume, shown in Figure 4.5, is not the same as the regression of log volume on log area. We can use the coefficients of the regression line to predict what Y might be for any specified value of X by inserting X into the regression equation and solving. Predicted values of Y won't coincide with the actual value of Y unless there is a perfect correlation between X and Y. The amount that Y and Y will differ, on average, can be estimated by the predictive error (sometimes called the "standard error of prediction" or the "standard error of estimate") which is SY\X

5y V 1 — r^

where sy = standard deviation oiY, r = correlation between X and Y. 16

Estimating Discovery Size from Prospect

Size

8.0

7.0 H -

0

E ^

s>4L-.o. ^..

o §6 ^ ^

-• ^

6.0

o 5.0 H-

4.0

—T—I—I—I—I—I—I—I—I—I—r—T—I—I—I—rni—i—|—i—i—i—i—|—i—i—i—i—j

1.0

2.0

Log^QArea

3.0

4.0

Figure 4.4. Regression line fitted to log volume and log area of fields in the Denver-Julesburg Basin expressing field volume as a function of field area.

4.0 n-

I I I I I I I I I I I

6.0 Log^o Volume

8.0

Figure 4.5. Regression line as in Figure 4.4 fitted to log area and log volume. This regression expresses field area as a function of field volume.

77

Computing Risk for Oil Prospects — Chapter 4 We can see how this measure reflects the scatter of the estimates about their true values by considering two extreme situations. If the correlation between X and Y is perfect, r = \. Then ^Y\X = 5 y \ / l — 1^ = 5 y \ / l — 1 = Sy\/0 = 0 Because the regression predicts the exact value of Y at every point X, there is no predictive error! At the other extreme, if there is absolutely no correlation between X and y , r = 0 and ^Y\X = 5y V 1 — 0^ = 5yVl — 0 = SyVl = Sy The predictive error is equal to the standard deviation of Y\ that is, the regression line provides an estimate of Y that is no better than simply guessing Y to be equal to its mean, Y. We can use the predictive error to calculate probabilistic limits around the regression. These limits, called confidence hands or confidence intervals, are based on the characteristics of the normal distribution. The regression line (as determined by its two coefficients, a and 6) provides us with a statistical estimate of the expected or mean value of Y for a given X. The predictive error SY\X provides the standard deviation of the estimated values of Y. Recall that the normal distribution has only two parameters, its mean and its variance (or standard deviation). By assuming that the errors in prediction (that is, values of the diff^erence Y — Y) are normally distributed, we can evaluate the normal distribution and calculate intervals in which future observations will fall with specified probability. The confidence bands have the form Y ± Za/2SY\x^ where Zc,/2 indicates the upper and lower limits of a standardized normal distribution that contains a specified proportion of the area under the curve, say 95%. The probability is 95% that future observations of Y at that value of X will fall within the confidence band. Other bands can be chosen that will have different probabilities of containing future observations. If we observed a very large number of y ' s at this specific value of X, they should closely approximate a normal distribution centered around the regression line at that point. By calculating confidence intervals for successive values of X, we can plot confidence bands around a fitted regression that is shown as a line on a cross plot. RISKSTAT provides 95% confidence bands around a fitted regression line, although the procedure used is more complicated than the one described here. If we have a large number of points and they form a narrow cloud along the line, the confidence bands will be narrow. On the other hand, if we have only a few points, or they scatter widely, the confidence bands will be wide and fiaring (Fig. 4.6). 78

Estimating Discovery Size from Prospect Size

J

I

1

^- ^ T

I

n^-^^^-^^^'

i ^-^^r I *^

I

3

I

4

I

I

I

!

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I

i

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1

4- .... •

i

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5

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I

6

7

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c

s

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Figure 4.6. Regression lines and 95% confidence bands for points tliat (a) closely follow a straight line with r = 0.97; (b) have a moderate correlation of r = 0.86; and (c) scatter widely around a line with r = —0.02. The mean of Y is 3.36 and its standard deviation is 0.66. The mean of X is 6.68 and its standard deviation is 0.89.

79

Computing Risk for Oil Prospects — Chapter 4

ESTIMATING OIL FIELD VOLUMES FROM STRUCTURAL CLOSURE ON SEISMIC MAPS Regression provides a way of predicting one property from values of another. If we measure geological or geophysical characteristics of known fields and relate these to the volumes of hydrocarbons contained in these fields, the regression may give us a way of predicting what will be found in undrilled prospects. That is, provided the predictor variable is something that can be determined in advance of drilling. Such a statistical study was conducted in the "Pleistocene Trend" of the U.S. Gulf Coast OCS, or Outer Continental Shelf (Fig. 4.7). The original objective was to see if the government assignment of a "Minimum Acceptable Bid" for offishore tracts could be improved, but the results also indicated how regression could provide probabilistic forecasts of the results of drilling exploratory wells (Davis and Harbaugh, 1983). Exploration in the OCS is based almost exclusively on the drilling of seismically detected structural closure. It is possible that oil may be trapped by mechanisms other than structural closure (updip sand pinchouts, for example), but "selling" a prospect that does not include seismic indications of closure would be difficult, indeed. Most seismically detected structures in the Outer Continental Shelf of Louisiana and Texas are the result of diapiric salt or shale movement and many possess shapes that are exceedingly complex. There is no published systematic classification of Gulf Coast salt domes in a form suitable for subsequent statistical treatment. Therefore, a relatively simple classification was devised in which apparent area of structural closure is the principal geological parameter. In addition, engineering estimates of recoverable oil and gas provided by the former Conservation Division of the U.S. Geological Survey for individual lease tracts were combined to form estimates of the reserves per prospect and expressed as BOE. The structural properties were obtained principally from regional seismic reflection-time maps prepared at a scale of 1 in = 4000 ft. Most of the measurements were made using regional maps of "Horizon II," a Pleistocene reflecting horizon. For four structures it was necessary to use maps of the shallower Pleistocene "Horizon I" because of diflSculties in interpreting Horizon II. Within the area of study, all structures exhibiting closure on the reflection-time maps were tabulated; most structures are interpreted as complexly faulted (although some are not), and they may have piercement cores of salt. Individual fault blocks that form the components of a structure have been aggregated for statistical purposes.

80

Estimating Discovery Size from Prospect Size

Figure 4.7. Index map showing outline of the area within the "Pleistocene Trend" of the Louisiana-Texas OCS. From Davis and Harbaugh (1983).

Characterizing Prospects "Area of closure" was defined as the area of an anticlinal structure which is bounded by the lowest closing reflection-time contour. In a faulted anticlinal structure, one or more faults may also form boundaries to the area of closure, but the deepest part of the area of closure within a fault block is necessarily defined by the lowest closing reflection-time contour. The lowest closing contour value may diff^er from one fault block to another, as shown in Figure 4.8. The total area of closure of a faulted structure is the sum of the areas of closure of all closed fault blocks. The other principal form of information used in this study were estimates of recoverable oil and gas, expressed as barrels of oil equivalent (BOE). Gas was converted to BOE by dividing the estimated recoverable gas in MCF by 5.7, as this quantity of gas is approximately equal to one 42-gallon barrel of oil on an energy equivalence basis. Geophysical measurements and reserve estimates are given in Table 4.1 and in file OCS.DAT on the diskettes. 81

Computing Risk for Oil Prospects — Chapter 4

Figure 4.8. Hypothetical prospect consisting of a piercement dome within study area as indicated by seismic reflection-time map of "Horizon H." Each square is a 5000-acre offshore lease tract. Contour lines are 50-miUisecond seismic reflection times. Gray area represents approximately 11,900 acres of closure with a maximum height slightly greater than 700 milliseconds and a volume of 14 million acre-feet. Areas indicated by cross hatches represent salt piercements. Data gathering consisted of tabulating all seismically mapped structures within the area of the study which exhibit closure and which have been tested by drilling. The distribution of field sizes can be investigated by plotting size (in BOE) against cumulative percentage of fields, as log probability graphs. Area of closure of the producing structures also can be plotted as a log probability graph of area versus cumulative percent. Areas of closure of structures that have been tested but which are dry or non-commercial can be shown in the same form. It is important to note that the frequency distributions of producing structures and dry structures that exhibit closure (and have been tested by drilling) are diff'erent. The median area of structures that contain producible hydrocarbons can be calculated by computing their univariate statistics using RISKSTAT or can be read directly from the graph in 82

Estimating Discovery Size from Prospect Size Table 4.1. Mapped seismic structures in the Pleistocene Trend, LouisianaTexas OCS, that exhibit closure and have been tested by drilling. Area 1726

734 293 1359 2167 1689 3011 1322

954 1836 1395

550 6387 1933 6285 2975 3048 1175 4354 4113 3691 5285 2718 3864 3011 1101 6758 3195

Amount*

0.0 4.6 0.5 13.0 10.5

0.0 2.7 4.4 5.5 0.0 0.0 0.0 4.0 0.0 5.8 4.6 4.7 0.0 7.5 24.5

9.8 33.4

2.5 5.5 0.0 0.0 32.5 33.9

Area 1542 26,629 11,569 16,234 14,765 5823 4554 10,579 2092 21,874 9990 16,712 5142 7272 15,242 4040 3856 13,149 5549 7456 4554 2820 8264 2854 11,753 3856 16,895

Amount*

0.0 45.7 22.0 21.2 14.8 128.6 14.9 143.2 10.7 40.7 17.2 21.0

0.0 15.9 132.2 32.4 22.5

0.0 0.0 30.7 98.9

4.2 0.0 2.1 27.5

8.6 22.7

Area 8558 3673 1579 2681 25,711 13,416 2938 4223 19,466 3185 1908 5876 2020 3011 4334 15,242 7639 1138 2607 8447 26,996 11,569 4811 8998 10,100 3636 2671

Amount* 12.4

0.0 0.0 0.0 18.2 36.4 30.9 25.3 218.0

9.2 0.0 9.5 0.0 9.5 18.9

0.0 0.0 0.0 9.2 0.0 0.0 59.1 12.4 100.9 27.1 15.2 31.8

*Volume of recoverable hydrocarbons (MMBOE) Figure 4.9 and is about 4300 acres; the median size of i:ion-producing tested structures is about 2300 acres. The areas of producing and dry structures also differ in their log standard deviations, which is expressed as a difference in the slopes of the two plotted distributions. It is clear that the two variables (area of closure and field volume) are essentially lognormally distributed, but we do not know the relationship between the two variables. We have a strong incentive to find out what relationship exists, however, because structural closure can be interpreted 83

Computing Risk for Oil Prospects — Chapter 4 100000-3

-4

^

\

\—\."\—\—'<""\

\-—

tP

p

(0 10000c

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iCP...<.....i.^.^4..^....|...^.

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I A '.99 I I I : I I I :$::::„:::::Ux:::;":":i"":i":5:"£"5""4!!!!i:"":5:"";:":"":":": *...^..<...<....<....< \...i...i...i.".i- — i

-1—I—I—r I I I

-I

* {

{... {•••

1 — I — - T —

50 5 10 20 80 9095 Percent of Structures

99

99.999.99

Figure 4.9. Log-probability plot of areas of closure of producing structures (open circles) and dry structures (solid dots) in the Pleistocene Trend. from seismic information in advance of drilling and should be useful in predicting both the presence and volume of hydrocarbons associated with the structures. The fact that both variables are lognormally distributed suggests that the relationship between the two should be represented by plotting both on logarithmic rather than linear scales. Figure 4.10 shows a regression of log volume on log area and provides an equation that can be used to predict the volume of hydrocarbons that a prospect may contain. The regression is based only on structures that contain hydrocarbons, so it is a conditional relationship because the possibility that a prospect will be barren is not considered. It is necessary to base the regression on producing structures only for the very practical reason that we cannot take the logarithm of zero volume. Aside from this, it also is easier and more effective to treat the dry hole risk as a separate problem of estimation.

Incorporating Prediction Error in Probability Estimates We have related volume of hydrocarbons in discovered fields to the seismic closure about these same fields by using simple linear regression and the data in file OCS.DAT. This gives us a "best estimate" of the relationship 84

Estimating Discovery Size from Prospect Size

3.0 3.5 4.0 Area, Log^Q Acres

5.0

Figure 4.10. Regression line fitted to log field volume and log area of closure of producing structures in the Pleistocene Trend. Correlation is r = 0.65. Curved lines form a 95% confidence band. between closure and field volume in productive structures, and a relationship that can be used to predict the volume of hydrocarbons in BOE contained in seismic anomalies that have not yet been tested. The regression line is enclosed in a 95% confidence band. We expect that 95% of the field volumes and seismic closure areas of both the known fields and those that may be discovered in the future will lie within these limits. Similar bands can be constructed for other confidence levels. The regression can be used to estimate the form of the probability distribution of possible field sizes associated with a seismically defined structural prospect in the Pleistocene Trend. It is necessary to assume that the scatter in values of log volume around the regression line follows a normal distribution, but this is a reasonable assumption given the fact that both field sizes and areas of structural closure are lognormally distributed. As an example. Figure 4.11 shows the probability distribution for a structure having 1000 acres of closure on the seismic map of Horizon II. The center of the distribution represents the value of the regression line corresponding to a prospect area of 1000 acres (that is, X — 1000) and other points along the curve are taken from confidence bands. Although the best estimate of the volume of recoverable hydrocarbons in BOE contained in the structure is almost 5 million bbls, the probabiUty that the prospect contains less than 85

Computing Risk for Oil Prospects — Chapter 4

-4.0

"T

100,000

-3.0

-1.0 0.0 1.0 Standard Error ~l—I—I—I—I—T" T T 5.5 6.0 6.5 7.0 Log Volume, bbis 1

-2.0

1—I I 1 i I

1,000,000

1

1

1 I I I I I

10,000,000 Volume, bbIs

2.0

3.0

7.5

4.0 8.0

n—I—I

I i 111

100,000,000

Figure 4.11. Lognormal distribution of possible field sizes in BOE corresponding to a prospect having 1000 acres of closure. Shaded areas correspond to 5% of the distribution (upper and lower limits of the 95% confidence band). 2 million bbls is equal to 1/6, the proportion of the probability distribution which lies to the left of —1.0 standard error. Similarly, the probability is 1/6 that the prospect may contain over 11 million bbls, because this is the proportion of the distribution that exceeds one standard error. Other discovery sizes and their associated probabilities can be estimated, using the characteristics of the normal distribution. The field size associated with the upper 5% of the probability distribution is often considered critical, because it is taken as the upper limit on the size of field that may be discovered. In this example, this uppermost field size is 20 million bbls; the probability that a larger field will be associated with a structure having 1000 acres of closure is less than 5%. The predicted volume of hydrocarbons, in barrels, is subject to substantial error. For a given forecast, conditional upon a specific area of closure in the study area, 95% of the prospects having the same area of 86

Estimnting Discovery Size from Prospect Size 10S

9.0-n

8-4 QH-^JO SI

s^ E lO^J o > 10H

10^-' o

.01

.1

T H — I I I I I I I TT" 5 10 20 50 80 90 95 Probability, Percent

T 1 1 99 99.9 99.99

Figure 4.12. Cumulative probability distributions for field sizes associated with three different areas of seismic closure in the Pleistocene Trend study area.

closure should fall within the 95% confidence band. Assuming that the logarithms of the deviations from the forecast of field size are normally distributed (which seems reasonable), 95 out of 100 prospects with a given area of closure can be predicted to lie inside the envelope. The fitted regression line lies halfway between the two confidence lines, and is an estimate of the mean of the range of volumes for a given area of closure. The means and the medians of the log-transformed volume should be nearly coincident, so estimates of field volumes at 2.5%, 50%, and 97.5% cumulative probabilities can be inferred for a specified area of closure. These three points are sufficient to fit a straight line representing the field size distribution for a given area of closure. Figure 4.12 shows lines that represent field volume distributions for structures having 5000, 10,000, and 20,000 acres of closure. The probable volumes of hydrocarbons contained in diff'erent areas of closure in the Pleistocene Trend can be derived from these fitted distributions, given that a discovery is made. Table 4.2 contains a tabulation of such probabilities. The probabilities estimated in Table 4.2 do not consider the dry hole risk, or probability of failing to find commercial production. We can estimate the dry hole probability from the proportion of dry structures listed in Table 4.1. The probabilities in Table 4.2 can be adjusted by incorporating 87

Computing Risk for Oil Prospects — Chapter 4 Table 4.2. Conditional probabilities of volumes of hydrocarbons contained in structures of specified size in the Pleistocene Trend study area, given a discovery. Volume (MMBOE)

Area of closure (acres) 5000 10,000 20,000

Less than 1 0.8 1 to 2.5 4.2 10 2.5 to 5 5 to 10 20 10 to 25 35 25 to 50 10 50 to 100 16 Greater than 100 4 Totals 100%

0.2 3.3 4.5 12 30 12 28 10 100%

0.03 0.8 2.2 5.0 22 10 32 28 100%

the complement of the dry hole risk, which is the probability that a seismically defined area with closure will yield commercial production. This "success ratio," measured on a per-structure basis, is 70%. (Note that success or failure pertains to a tested structure, and not to an individual exploratory hole.) Since we consider the probability that a structure might prove barren regardless of its closure to be independent of the conditional probability of discovery size given that a discovery is made, the two are combined simply by multiplication, as in Table 4.3. Table 4.3. Outcome probabilities for testing structures of specified size in the Pleistocene Trend study area. Size of field (MMBOE)

Area of closure (acres) 5000 10,000 20,000

Barren 30.0 Less than 2.5 0.6 2.5 to 5 2.9 5 to 10 7.0 10 to 25 14.0 25 to 50 24.5 50 to 100 7.0 100 to 250 11.2 Greater than 250 2.8 Totals 100%

88

30.0 0.1 2.3 3.2 8.4 21.0 8.4 19.6 7.0 100%

30.0 0.0 0.6 1.5 3.5 15.4 7.0 22.4 19.6 100%

CHAPTER

5 Revising Outcome Probabilities and — Success Ratios

PROBABILITIES CONDITIONAL U P O N GEOLOGY AND DRILLING RESULTS Most prospects are based upon geological and geophysical interpretations. Understanding conditional relationships between hydrocarbon accumulations and geological properties forms the essence of petroleum geology, whether or not mathematical tools are employed. If we are to express these relationships as conditional probabilities, we need to extract frequencies of occurrence from the relevant background information. For example, stratigraphic wedgeouts preferentially trap oil and gas, and while the presence of a wedgeout does not guarantee that commercial quantities of hydrocarbons are present, drilling wedgeouts usually is a better strategy than drilling at random. The challenge is to express relationships between the presence of wedgeouts and oil accumulations to obtain estimates of well outcomes that are conditional upon wedgeouts. It is here that Bayes' theorem provides the key. There is a voluminous literature on so-called Bayesian statistics, but most of this is concerned with problems and concepts in statistics itself (a recent survey of the field by Press (1989) includes a reprint of Rev. Bayes' original article) or with applications in business and management decisionmaking. The relatively few articles in the petroleum exploration literature include those by Demirmen (1975); Grayson (1962); Grender, Rapoport, and Segers (1974); King (1974); Lee and Wang (1983, 1985); Meisner and

Computing Risk for Oil Prospects — Chapter 5 Demirmen (1981); Newendorp (1972); Prelat (1974); Sluijk, Nederlof, and Parker (1986); Solow (1988); and Stone (1990). Bayes' theorem is also discussed in books by Harbaugh, Doveton, and Davis (1977) and Newendorp (1975). Reverend Thomas Bayes was a 19^/i-century Presbyterian minister who stated algebraic relationships that permit probabilistic forecasts of future events to be based on knowledge of previous events. Bayes' theorem can take various forms and is equally useful for making probabilistic statements about tomorrow's weather or the outcome of next year's wildcat well. It is often stated in words and widely used intuitively. Many individuals use Bayes' theorem routinely without realizing they are "Bayesians." In fact most explorationists are informal Bayesians because unless prospects are drilled at random or on a grid, conditional relationships are employed that are inherently Bayesian in their nature. One form of Bayes' theorem may be written in words as / ,. . , 1 1 .1. N (joint probability) (conditional probability) = (marginal probability) Subsequent derivations and embellishments of Bayes' theorem stem from this equation. Because words are cumbersome, there is an advantage to stating Bayes' theorem in algebraic notation:

where p{A\B) represents the conditional probability that event A will occur given knowledge that event B has occurred (the vertical bar signifies that A is conditional upon JB), p{Ay B) represents the joint probability that both events A and B will occur, and p{B) represents the marginal probability that event B will occur. Conditional probability is a probability that is conditional upon knowledge of an event. It thus depends upon a specific condition being satisfied. For example, if you were to estimate the probability of rain later today (event A), knowledge of whether it is presently cloudy (event B) or clear (event C) would affect your estimate. The conditional probability of rain would be higher given knowledge that it is cloudy rather than clear. In this situation, {A\B) can be read "the probability of rain later today given knowledge that it is presently cloudy." Joint probability pertains to the probability of two (or more) events occurring together, or jointly. We could obtain an estimate of the joint probability of both rain and it being a cloudy day from frequencies provided 90

Revising Outcome Probabilities and Success Ratios by weather records, or we could intuitively guess the probability from our experience. If A represents rain and B represents cloudy day, the joint probability of rain and a cloudy day is written {A, B). Marginal prohability is the probability that an event will occur. In our scenario, (B) represents the probability of a cloudy day, and might be estimated from weather records that provide information on the proportion of cloudy days over a month or year. Since Bayes' theorem is solely concerned with probabilities or frequencies, it is common practice to drop the p 's (which signify probabilities) and write the equation in simpler notation as

Bayes' theorem can be rearranged so that (^4, B) = {A\B) (B) or in words, the joint probability is equal to the conditional probability times the marginal probability. Another rearrangement of Bayes' theorem solves for the marginal probability

which states that the marginal probability is equal to the joint probability divided by the conditional probability. Oil exploration is inherently a Bayesian activity. A forecast of the outcome of drilling a prospect is conditional upon geological knowledge about the prospect. If anticlines are believed to be favorable structures for the occurrence of oil, we can use Bayes' theorem to estimate the probability of a discovery conditional upon knowledge that an anticline is present:

where {P\A) is the conditional probability that the hole will result in a producer, P , given knowledge that an anticline. A, is present. (P, A) is the joint probability that both a producer and an anticline coexist at a locality (perhaps estimated as the proportion of all prospects that have been drilled on anticlines and proved to be discoveries). (A) is the marginal probability of an anticline (perhaps estimated as the proportion of prospects involving anticlines relative to the total population of prospects). Before plunging into adaptations of Bayes' theorem, let's do some simple experiments by drawing colored balls at random from a proverbial 91

Computing Risk for Oil Prospects — CJtapter 5

10Black (B)

White (W)

Column Totals

Red (R)

Green (G)

Row Totals

(B.R)

(B.G)

(B)

5

50

55

(W,R)

(W,G)

(W)

20

25

45

(R)

(G)

Total

25

75

100

Figure 5.1. Contents of a statistician's urn containing balls that are jointly black and red {B,R), white and red {W,R)^ black and green {B,G), and white and green (VF, G). Shaded areas are marginal frequencies. "statistician's urn" whose contents are statistically described in Figure 5.1. Each ball has two colors which occur with the specified frequencies. If we know the contents of the urn, we can respond effectively to questions that involve conditional relationships. For example, if we draw a ball that is black on one side, what is the probability that the other side is red? A glance at the figure will tell us, but it's instructive to work through the algebra. The relationship is {R\B)

{B,R) {R)

_5 — = 0.20 25

where {R\B) represents the probability of drawing a red ball, given that it is black on one side, {B,R) is the probability (proportion) of balls in the urn that are both black and red, and {R) is the probability (proportion) of balls that are red plus any other color. This equation can also be written as

(mm =

(-^--^)

=_ i _

^ ' ^ {B,R)-\-{W,R) 5 + 20 since the number of balls that are partially red (the marginal sum for red) is equal to the number of balls that are jointly black and red, plus those that are jointly white and red: {R) = {B,R) + {W,R) 92

Revising Outcome Probabilities and Success Ratios We have defined Bayes' theorem in terms of probabilities, although Figure 5.1 contains frequencies. The frequencies represent probabilities perfectly because they describe the entire population of balls and it makes no difference whether the numbers in the table are actual counts, relative frequencies, or probabilities. In real situations, frequencies that are based on samples generally provide the basis for estimates of probabilities that we ascribe to the underlying populations, but the Bayesian relationships are the same. Bayes' theorem is widely used regardless of the source of the probabilities. Probabilities can be derived from frequencies such as the balls in the urn, or they can be subjective guesses about future events, such as nuclear attacks, for which frequency data fortunately do not exist. Bayes' theorem does not distinguish between the sources of the probabilities, although there is a widely mistaken notion that "Bayesian" operations are confined to probabilities based on subjective estimates; this is a totally unwarranted restriction.

Bayesian Conditional Probabilities and Success Ratios Stratigraphic thicknesses and lithologic compositions are readily interpreted from well logs, and can be expressed numerically as thickness in feet or percent of shale or other constituents. These same properties can be expressed as mutually exclusive qualities such as "thick" versus "thin" and "clean" versus "shaly." The outcome of drilling an exploratory well can be classified as a "producer" or "dry hole." Such observations can be assembled in tables of frequencies from which contingent relationships and conditional probabilities are readily extracted in the form of success ratios and producer probabilities. Throughout this book we often use the term "success ratio" as a surrogate for "producer probability" or "probability of discovery." We commonly calculate success ratios from frequency data; they are simply the proportion of those wells that are producers in the total population of drill holes being considered, and are estimates of the probability of discovery. As far as Bayesian algebra is concerned, these terms can be used interchangeably. Multiple geological properties often are useful in defining prospects, as in the time-honored practice of overlaying subsurface maps on a light table and looking for places where key geological features coincide. Rather than looking at maps, we can examine the well records on which the maps are based. At every well location, we can categorize the geologic properties of interest and tabulate their relationships. For example. Tables 5.1 to 5.3 give drilling results cross-tabulated against categories of thickness and shale 93

Computing Risk for Oil Prospects — Chapter 5 Table 5 . 1 . Two X two contingency table of well status (producing or dry hole), versus thickness of the XVa Limestone ("thick" is defined as greater than 37.5 m, and "thin" as less than 37.5 m).

FREQUENCY

Marginal totals

Producing

Dry

Thick

11

32

43

Thin

7

33

40

18

65

83

Producing

Dry

Marginal probs.

Marginal totals

PROBABILITY

Thick

0.13

0.39

0.52

Thin

0.08

0.40

0.48

Marginal probs.

0.21

0.79

1.00

Grand total

Grand total

ratio of t h e XVa Limestone, an Oxfordian (Upper Jurassic) oil-producing interval in t h e Magyarstan Republic. ( T h e geological background of this region is described in Chapters 1 a n d 7.) Table 5.2. Two x two contingency table of well status (producing or dry hole), versus shale ratio in the XVa Limestone ("shaly" is defined as a shale ratio greater than 0.44 and "clean" as less than 0.44).

FREQUENCY

Dry

Shaly

0

42

42

Clean

18

23

41

Marginal totals

18

65

83

PROBABILITY

94

Marginal totals

Producing

Producing

Dry

Marginal probs.

Shaly

0.00

0.51

0.51

Clean

0.22

0.27

0.49

Marginal probs.

0.22

0.78

1.00

Grand total

Grand total

Revising Outcome Probabilities and Success Ratios Table 5.3. Three-way (two X two X two) contingency table combining information about thickness and shale content of the XVa Limestone. FREQUENCY THICK

THIN

Prod.

Dry

23 1

0

19

19

9

20

7

14

21

32

43

7

33

40

Prod.

Dry

1 Shaly

0

23

1 Clean

11 11

PROBABILITY THIN

THICK

Prod.

Dry

Prod.

Dry

Shaly

.00

.28

.28

.00

.23

.23

Clean

.13

.11

.24

.08

.17

.25

.13

.39

.52

.08

.40

.48

These tables are immediately useful. Table 5.1 shows that where holes have penetrated the XVa Limestone and it is relatively thick (more than 37.5 m), the success ratio {SR) is higher than where the hmestone is thin. Success ratios extracted from Table 5.1 are: Thick intervals: SR = 11/43 = 26% Thin intervals: SR = 7/40 = 18% Useful as thickness is, the shale ratio has an even more pronounced conditional relationship with discoveries (limestone with a low shale ratio is relatively "clean" or free of clay). The success ratios extracted from Table 5.2 are: 0/42 0% Shaly: SR 18/41 44% Clean: SR Since both the shale ratio and thickness are important, we should consider both of these properties at the same time. This has been done in Table 5.3, yielding joint conditional success ratios: Shaly Shaly Clean Clean

and and and and

thick: thin: thick: thin:

SR SR SR SR

= = = =

0/23 0/19 11/20 7/21

= = = -

0% 0% 55% 33% 95

Computing Risk for Oil Prospects — Chapter 5 The conditional success ratios associated with clean intervals are sharply greater than the unconditional success ratio of 18/83 = 22%, which is the proportion of producers in the total population of exploratory holes, or 18 out of 83. Clearly the joint conditional success ratios are superior guides in exploration and we should seek prospects where the XVa Limestone is both clean and thick and avoid places where the unit is shaly, regardless of whether it is thick or thin. Although these conditional relationships are simple, they are a manifestation of Bayes' theorem and can be represented formally. Both joint frequencies and joint probabilities are contained in Tables 5.1 to 5.3, as are the marginal frequencies and marginal probabilities. The conditional probabilities must be determined by dividing the joint probabilities by the appropriate marginal probabilities. For example, to estimate the probability of a discovery conditional upon knowledge that the XVa Limestone is thick, we can write (P|T) =

^

where P is the probability of a producer, and T is the probability that a thick interval will be encountered in the XVa Limestone. We can use either the frequencies or the probabilities in Table 5.1, since both will yield the same results. There are 11 wells in the collection that are producers and in which the XVa Limestone is thick, and the total number of wells (both producers and dry holes) where the XVa Limestone is thick is 43. Thus, {P\T) = 11/43 = 0.256 or, equivalently using probabilities from Table 5.1, ( P J r ) = 0.133/0.516 = 0.256. By contrast, the unconditional probability of a producer is the marginal probability (P), or proportion of all wells in the population that are producers: (P) = 18/83 = 0.217. The conditional probability of a producer given knowledge that the XVa Limestone is both thick and clean at a location may be written as ( P | T . C ) = < ^ where (P|T, C) is the conditional probability of a producer (P) in the XVa Limestone, given knowledge that interval at the prospect locality is both thick (T) and clean (C). (P^T.C) is the joint probability that a well is a producer and the XVa Limestone in the well is both thick and clean. (T, C) is the marginal probability that the XVa Limestone is both thick and clean, regardless of the well status. We can estimate the desired conditional probabilities from frequency data in Table 5.3, giving ( P | r , C) = 11/20 = 0.55, or we can obtain the same results if we use probabilities (P|T, C) = 0.133/0.241 = 0.55. 96

Revising Outcome Probabilities and Success Ratios

Trend Surface Residuals as a Source of Conditional Success Ratios These simple examples show that geological data and information on well status are readily manipulated to yield conditional probabilities. This form of conditional analysis can employ different kinds of information, including trend surface structural residuals, which may be useful in regions where geologic structures strongly influence hydrocarbon accumulation. Calculation of trend surfaces and trend surface residuals is discussed in Chapter 7. One difficulty with using conventional contour maps of subsurface structure is that the features represented may be difficult to define quantitatively. For example, we may want to express the height of closure of a structural dome in meters or feet, but this requires that we compare its elevation with that of its immediate surroundings. This may be difficult to do because the boundaries of a specific feature, such as a dome, may be ill-defined. It is a simple matter, however, to fit a trend surface and subtract it from the structural contour map to yield a trend surface residual map. Structures such as anticlines, domes, and synclines will be isolated and explicitly defined. When expressed as trend surface residuals, categories of ranges of residuals, measured in meters or feet, can be compared with the presence or absence of hydrocarbons. Such relationships can be determined in regions that have been explored to some substantial extent, providing estimates of producer probabilities that are conditional upon the magnitudes of trend surface residuals. Table 5.4. Proportion of the area in southwest Magyarstan underlain by different classes of fourth-degree polynomial trend surface residuals. Residual class underlain by Nonknown Producing producing Total production Percentage of area

Trend surface residual classes (m) Strong positive (> 40) Med. positive (20 to 40) Weak positive (0 to 20) Weak negative (0 to —20) Med. negative (-20 to -40) Strong negative (< —40)

3.82 1.96 1.50 0.71 0.36 0.16

Totals

8.51

13.78 16.87 29.49 20.04 7.99 3.32 91.49

17.60 18.83 30.99 20.75 8.35 3.48 100.00

21.70 10.41 4.84 3.42 4.31 4.62

In southwest Magyarstan, there is a close concordance between structural "highs" and oil fields that occur in the XVa Limestone of Late Jurassic 97

Computing Risk for Oil Prospects — Chapter 5 age. Table 5.4 documents the strong relationship between the positive or negative classes of trend surface residuals and hydrocarbon accumulations in this maturely explored region. In Table 5.4, fourth-degree polynomial trend surface residuals are cross-tabulated for the known oil fields, revealing that residual highs greater than 40 m are about five times more frequently underlain by oil fields than are areas of negative or weakly positive residuals. Obviously, such a strong conditional relationship could be highly useful in exploration. Suppose we had no knowledge of the magnitude of the trend surface residual at a prospect location where an exploratory hole was to be drilled and we wish to estimate the unconditional probability of a producer. Since 8.51% of the area is underlain by known production, the unconditional probability of a producer is at least 8.51%, but may be higher depending on the extent to which the non-producing complement (100 — 8.51 = 91.49%) has been explored. If it is incompletely explored, parts of the area may be underlain by undiscovered hydrocarbons. Alternatively, if an exploratory hole is drilled on a strong positive trend surface residual, the probability of a discovery is 21.7%, which is markedly better than the unconditional probability. In passing, we should note that geologically interesting inverse relationships may be extracted from tables such as Table 5.4. Suppose that an exploratory hole is completed as a producer, and although the magnitude of the trend surface residual at the well site is not known, we would like to estimate the probability that there is a pronounced positive trend surface residual at the location. Clearly, this is a conditional relationship that can be expressed by Bayes' theorem, because the presence of a strong positive residual is conditional upon the knowledge that a producing well has been drilled. Table 5.4 shows that the unconditional probability of a strong positive residual in the region is only 0.18, but the conditional probability is more than twice as great:

Here, (5, P) is the percentage of the total area that is underlain both by strongly positive residuals and by production, and (P) is the percentage of the total area that is underlain by all known production.

98

Revising Outcome Probabilities and Success Ratios

BAYESIAN REVISION OF REGIONAL SUCCESS RATIOS We will now pursue some Bayesian applications that are more algebraically challenging. In a frontier region, results of early exploratory holes may strongly influence the industry's outlook toward the region as a whole. For example, disappointing results in early exploratory drilling on the U.S. Atlantic Outer Continental Shelf dampened enthusiasm for the Shelf's overall potential; a major discovery would have produced a very diff'erent outlook. The results from early exploratory holes can be used to adjust previously assigned regional success ratios, using Bayesian relationships. Suppose extensive drilling in an offshore region has allowed us to calculate industry-wide success ratios for six different areas, and that a new area has been opened to leasing in the same offshore region. What success ratio should be assumed for the new area prior to drilling? Clearly, the success ratios seen in the six established areas should influence our assignment, but how should this information be applied? Table 5.5. Success ratios (SR) for six offshore areas and the probability that each is the appropriate success ratio to assign to a new area. Offshore area I II III IV V VI

Success ratio (%) 0 10 20 30 40 50

Probability of correct assignment 0.40 0.30 0.15 0.08 0.05 0.02

Since each of the six success ratios is a possible candidate to be used as the initial success ratio for the new area, each should be assigned some weight or probability. It seems reasonable that these probabilities should depend on the geological similarity and geographic proximity of each of the older areas to the new area. Considering these factors, subjective probabilities for each success ratio were assigned in Table 5.5. The probabilities express the "degree of belief that each success ratio is the correct one to be applied to the new area. (Because success ratios are themselves probabilities, the term "success ratio" or "^^i^" has been used to avoid confusion, 99

Computing Risk for Oil Prospects — Chapter 5 as we are dealing with probabilities attached to probabilities.) Prom this information, the expected success ratio for the new exploration area before a well is drilled can be calculated by multiplying each success ratio by its assigned probability and summing the products: E (SR) - (0 X 0.4) + (0.1 X 0.3) + (0.2 x 0.15) + (0.3 x 0.08) + + (0.4 X 0.05) + (0.5 X 0.02) = 0.114 = 11.4% It is obvious that the results of the first hole drilled in the new area will have a major influence on the industry's perception, even though a single hole won't provide enough information to establish a success ratio. Intuition tells us that if the hole is successful, the expected success ratio should increase, but a dry hole should cause it to decrease. Unfortunately, intuition does not tell us how much the success ratio should be changed. To determine this we'll need the formal algebra of Bayes' theorem, first defining these events: Ai = SR of new area = SR of area z, z = 1,2,..., 6 Bj = outcome of first well, S i = dry, JB2 = discovery Next, let's define the following probabilities if the first well is dry: (Ai) is the a priori probability that the new area has the same SR as area i. {Bi\Ai) is the conditional probability of a dry hole, given that the SR of the new area is the same as the SR of area i. This dry hole probability is equal to the likelihood of the observed outcome for each SR. We need to adjust the probability {Ai\Bi), which is the probability that the new area has the same SR as area i, given that the first well drilled has proven dry. This is given by Bayes' theorem, with calculations laid out in Table 5.6:

Let's examine what's been done. Before drilling, a priori probabilities of 0.40, 0.30, 0.15, 0.08, 0.05, and 0.02 were assigned to success ratios of 0%, 10%, 20%, 30%, 40%, and 50%. By using a suite of possible success ratios, we are expressing our uncertainty about which SR is really applicable. As Table 5.6 shows, the revised probabilities for 0% and 10% success ratios have increased, and the probabilities attached to all of the higher success ratios have decreased. The gi'eatest decreases have occurred in the probabilities assigned to the highest success ratios, all of which accords with our intuition. 100

Revising Outcome Probabilities and Success

Ratios

Table 5.6. Adjustment of probabilities attached to different success ratios if initial well in new area is a dry hole. Column A = candidate success ratios, B = probabilities initially attached to each SR, C = probability of one initial dry hole for given SR^ D = product of columns B and C, E = column D rescaled to sum to 1.0. A

B

Ai

(Ai)

0% 10% 20% 30% 40% 50%

0.40 0.30 0.15 0.08 0.05 0.02

Sums

1.00

C

(BMi) 1.00 0.90 0.80 0.70 0.60 0.50

D {Bt\Ai)x{Ai)

E 0.886

0.400 0.270 0.120 0.056 0.030 0.010

0.452 0.305 0.135 0.063 0.034 0.011

0.886

1.000

Now, let's see how t h e probabilities would be revised if t h e first hole discovered oil. Applying t h e same principles, we obtain

which is reflected in Table 5.7. Table 5.7. Adjustment of probabilities attached to different success ratios if initial well in new area is a producer. Column A = candidate success ratios, B — probabilities initially attached to each SR^ C = probability of one initial success for given SR^ D = product of columns B and C, E = column D rescaled to sum to 1.0. A

B

Ai

(Ai)

0% 10% 20% 30% 40% 50%

0.40 0.30 0.15 0.08 0.05 0.02

Sums

1.00

C {B2\Ai) 0.00 0.10 0.20 0.30 0.40 0.50

D {B2\Ai) X {Ai)

E (B2|Ai)x(A,) 0.114

0.000 0.030 0.030 0.024 0.020 0.010

0.000 0.263 0.263 0.211 0.175 0.088

0.114

1.000

101

Computing Risk for Oil Prospects — Chapter 5 If the first exploratory well in the new area is successful, there are two immediate effects. First, we can set the probability attached to the 0% success ratio to zero, because a success ratio of 0% is impossible if there has been a discovery. The second effect is to cause the values assigned to the remaining probabilities to be adjusted. Since only a single well has been drilled, we can multiply our initial estimates of probabilities (column B in Table 5.7) by the probabilities that a single exploratory hole would have resulted in a discovery (given in column C). We have done this for each of the iSiJ's and recorded the results in column D. This represents our revised estimate, and needs only to be rescaled so that it sums to 1.0 (column E). The examples in Tables 5.6 and 5.7 show the revisions that must be made to reflect outcome of only a single exploratory hole. We could readily extend the tables to include other outcomes, such as the probabilities that would be attached to the results from two or three successive holes. For example, revised probabilities stemming from two discoveries made in succession can be obtained by expanding the binomial equation, as illustrated in Table 5.8 which is based on the same set of initial success ratios as Tables 5.6 and 5.7. Table 5.9 shows the revisions that would be required if one discovery and one dry hole resulted from the first two initial drilling activities. These examples demonstrate that revision of the initial probability estimates of regional success ratios follow from Bayes' theorem, but there are weaknesses in this approach. First, there are only two classes of outcome (producing or dry) and second, the successive well outcomes must be presumed to be independent of each other. In reality, there are many possible outcomes. For example, while an initial exploratory well might not be a producer, it might yield such favorable geological information that other prospect locations in the vicinity would become more attractive. Under these circumstances the outcomes of successive wells cannot be presumed to be independent of each other.

102

Revising Outcome Probabilities and Success

Ratios

Table 5.8. Adjustment of probabilities attached to different success ratios if initial two exploratory holes in new area are producers. Column A = candidate success ratios, B ~ probabilities initially attached to each SR^ C = probability of two successes for given SR^ D = product of columns B and C, E = column D rescaled to sum to 1.0. A

B

Ai

{Ai)

0% 10% 20% 30% 40% 50%

0.40 0.30 0.15 0.08 0.05 0.02 1.00

Sums

C [B2 .B2\Ai) 0.00 0.01 0.04 0.09 0.16 0.25

D , 5 2 | ^ i ) x {Ai) (52:

E (B2,B2|Ai)x(Ai) 0.0292

0.0000 0.0030 0.0060 0.0072 0.0080 0.0050

0.0000 0.1027 0.2055 0.2466 0.2740 0.1712

0.0292

1.0000

Table 5.9. Adjustment of probabilities attached to different success ratios if initial two exploratory holes in new area result in a discovery and a dry hole. Column A = candidate success ratios, B = probabilities initially attached to each SR, C = probability of one success and one dry hole for given SR, D = product of columns B and C, E = column D rescaled to sum to 1.0. A

B

Ai

{Ai)

0% 10% 20% 30% 40% 50%

0.40 0.30 0.15 0.08 0.05 0.02

Sums

1.00

C {B2. ,Bx\Ai) 0.00 0.18 0.32 0.42 0.48 0.50

D ( B 2 , B i | ^ i ) x {Ai)

E {B2,Bi\Ai)^{Ai) 0.1696

0.0000 0.0540 0.0480 0.0336 0.0240 0.0100

0.0000 0.3184 0.2830 0.1981 0.1415 0.0590

0.1696

1.0000

103

Computing Risk for Oil Prospects — Chapter 5

Some Algebraic Background What we've done is simple enough, but how is it a manifestation of Bayes' theorem? We can see, by using the following notation for marginal, conditional, and joint probabilities associated with regional success ratios: (SRi) (H) {SRi\H)

= marginal probability of the i-th success ratio = marginal probability of drilling outcome H = conditional probability of i-th success ratio given knowledge of drilling outcome H {H\SRi) = conditional probability of drilling outcome H given knowledge of the i-th success ratio {SRi, H) = {H, SRi) = joint probability of the i-th success ratio and outcome H.

Bayes' theorem relates conditional, joint, and marginal probabilities. In this notation

and

{SRm = ^ ^ ^ These equations can be rearranged so that the joint probabilities are expressed as the product of the conditional and marginal probabilities: {H,SRi)^{H\SRi){SRi) and {SRi,H)

=

{SRi\H){H)

The expressions for the joint probability are equivalent; that is, {SRi,H)

= {H,SRi)

Therefore, it follows that {SRi,H)

=

{H\SRi){SRi)

Substituting this relationship into the equation for {SRi\H) above, we obtain

(5ft|g)='"iy By this means, we have adapted Bayes' theorem to obtain the probability of the i-th success ratio, given knowledge of drilling outcome H. 104

Revising Outcome Probabilities and Success Ratios The set of success ratios that we define must include all possible success ratios, and the individual success ratios must be mutually exclusive. If there are n possible success ratios, the probabilities attached to each of them must be such that they sum to 1.00. That is, (SRi) + {SR2) + . . . + (SRn) = 1.00 The probability of a specific drilling outcome {H) is the sum of the joint probabilities of the specific outcome and the various success ratios: {H) - ( F , SR,) + ( F , SR2) + . . . + ( F , SRn) But we can substitute the product of the conditional and marginal probabiUty for each of the joint probabilities in this equation, yielding (H) = {H\SR,){SR^)

-f {H\SR2){SR2)

+ ... +

{H\SRn){SRn)

The equation can be rewritten in a more compact form: n

k=l

This expression for {H) can now be substituted into the denominator of the equation for {SRi\H), above:

E

{H\SRk){SRk)

k=i

This is the Bayesian equation which we use in this section. Figure 5.2 is a Venn diagram that expresses the relationship graphically. Let the total area within the rectangle of Figure 5.2 be equal to 1.0. The circle has an area equal to the probability of event H, and is some quantity less than 1.0. The four shaded regions represent the probabilities attached to four different success ratios, labeled SRi to SR4. The sum of these four individual areas is equal to 1.0, indicating that the four success ratios represent all the possibilities. There are four smaller areas labeled a, 6, c, and d. Each of these smaller areas is formed by the intersection of a particular SR and H. The term "intersection" means those parts that are common to both areas where they overlap. For example, subarea c is the intersection of H and SR^. 105

Computing Risk for Oil Prospects — Chapter 5

SR^

^^^ ^k

/

L/

b

a

^-'^c'T? SR4

jS*

1

SF^ •

1

6

.«S^^M

Figure 5.2. Venn diagram showing intersection of the probability of driUing outcome H with the probabihties attached to four alternative success ratios. Joint probabilities are indicated by a, b, c, and d.

Now we can obtain the probability of SRs conditional upon outcome H. This can be defined, using the diagram, as {SRs\H)

=

area of intersection of SRs and H total area of H

or {SRs\H) = - = H (a + 6 + c + d) This is equivalent to the more general form of the equation for {SRi\H) given on the previous page. If we express the specific conditional probability (SR^IH) in the general form, then

iSRs\H) = iH\SRi){SRr)

106

(i/|5i?3){5i?3) + {H\SR2){SR2) + {H\SR3){SR3)

+

{H\SR4){SR4)

Revising Outcome Probabilities and Success Ratios

BAYESIAN REVISION OF FIELD SIZE PROBABILITY DISTRIBUTIONS Now consider problems associated with assigning probabilities attached to discrete field size distributions before a new concession is drilled. Assume that distributions obtained from three extensively explored areas are thought to be reasonable analogues and that each is represented by three field size classes: small (20 MMbbls), medium (60 MMbbls), and large (150 MMbbls). The difference between the three areas lies in the frequencies with which the three classes of field sizes occur, as represented by probabiUty estimates in Table 5.10. Table 5.10. Probabilities attached to different field sizes in three areas. Column A = probability of small field, B = probability of medium field, C = probability of large field, D = sum, E — a priori probability that field size distribution is correct choice for new concession.

Area I Area II Area III

A 20*

B 60*

C 150*

D

E

0.40 0.60 0.75

0.40 0.25 0.15

0.20 0.15 0.10

1.00 1.00 1.00

0.333 0.333 0.333

* MMbbls

The difficulty lies in deciding which field size distribution is most appropriate. Since this is uncertain, Table 5.10 contains probabilities that have been assigned to each of the three explored areas; these are subjective estimates of the likelihood that the new concession will be characterized by a similar field size distribution. In the absence of any evidence that might favor one or the other of these candidate distributions, each has been subjectively assigned an equal probability that it correctly represents the new area, or in other words, a probability of one-third. Suppose that a large field is now discovered in the new concession; we must ask the question, "what effect does this discovery have on the probabilities assigned to the three different field size classes for the concession?" Intuitively we would realize that if the new discovery is large, a field size distribution similar to that of Area I is more likely and the probability that the field size distribution is like that of Area III will decrease. Unfortunately, intuition will not tell us how much the probabilities will change, nor does it tell us how the probability associated with a field size 107

Computing Risk for Oil Prospects — Chapter 5 distribution like that of Area II will be affected. We need Bayes' theorem to calculate answers to these questions about field size distribution. First, we must define the events: Ai = field size distribution of new concession = field size distribution of previously explored area i Bi = size of first field (J5i = small field; B2 = medium field; Bs= large field). Hence, the probabilities: {Ai) = a priori probability of field size distribution Ai {B^\Ai) = conditional probability of obtaining large field S3, given that the applicable field size distribution is Ai {Ai\Bs) = conditional probability of field size distribution Ai, given that a large (Ss) field is discovered {AilBs)

=

(B3|Ai)(Ai)+(B3|A2)(A2)+(B3|A3)(A3) •

As Table 5.11 shows, the probability that the field size distribution in Area I correctly represents the field size distribution for the new concession has increased to 0.4443, the probability that Area II is correct has increased very slightly to 0.3336, and the probability that Area III is correct has been reduced to 0.2221. Table 5.11. Calculation of probabilities attached to different field size classes for the new concession after a large field has been discovered.

A, A2 ^3

Sums

0.3333 0.3333 0.3333

0.20 0.15 0.10

0.0666 0.0500 0.0333 0.1499

0.4443 0.3336 0.2221 1.0000

Bayesian Revision of Expected Field Size Now we will consider a slightly more complex situation that extends this example. Let's assume that a 150-million-barrel field has been discovered in the new concession, and that the other facts are the same as presented in Table 5.10. We want to know how the expected value of the size of 108

Revising Outcome Probabilities and Success Ratios Table 5.12. Expected size of field in new concession prior to discovery of initial field. Column A = probability initially assigned to each field size distribution, B = individual probabilities attached to field sizes within each distribution, C = field sizes, D = expected value of field size for each distribution, E = column A x column D. A I

II

III

B

C (MMbbls)

0.3333 0.40 0.40 0.20

20 60 150

0.60 0.25 0.15

20 60 150

0.75 0.15 0.10

20 60 150

0.3333

0.3333

D (MMbbls)

E (MMbbls)

62.0

20.664

49.5

16.498

39.0

12.999

Sum = 50.161* * Initial expected value offield size for new concession

the next field to be discovered in the new concession has been affected by the news of the discovery of a 150-million-barrel field. Assume that the geology of the next prospect to be drilled in the new concession is similar to the prospect that yielded the initial discovery. Intuition tells us that the expected volume of oil should increase, but again we'll need the algebra of Bayes' theorem to determine how much. The details are laid out in Table 5.12, which repeats information from Table 5.10 and also provides an expected value (column D) for each of the field size distributions for Areas I, II, and III, and an a priori expected field size for the new concession (the sum of column E). Table 5.13 demonstrates the necessary computations for revising the field size forecast following the discovery of a 150-million-barrel field. The probabilities initially attached (0.3333) to each alternative field size distribution have been changed (column H) to accord with those obtained in Table 5.11. The expected value in column D of Table 5.12 for each field size distribution has been reentered in column D of Table 5.13 and multiplied by the revised probability for that field size distribution. The product of 109

Computing Risk for Oil Prospects — Chapter 5 Table 5.13. Expected size of next field to be discovered in a concession after discovery of a 150-million-barrel field. Column A = probability initially assigned to each field size distribution, F = probability of discovering a 150-million-barrel field, G = column A x column F, H = column G rescaled to sum to 1.0, D = expected value of each distribution, I = column H X column D.

I II III

A

F

G

H

D (MMbbls)

I (MMbbls)

0.3333 0.3333 0.3333

0.20 0.15 0.10

0.0667 0.0500 0.0333

0.4443 0.3336 0.2221 1.0000

62.0 49.5 39.0

27.547 16.513 8.662

Sum =

52.722*

* Revised expected value of field size for new concession

this multiplication is given in column I. The sum of products in column I is the revised expected field size for the new concession. Before revision, the expected field size value v^as 50.2 MMbbls; after revision it has been increased to 52.7 MMbbls. In summary, Bayes' theorem is widely applicable in exploration when conditional relationships are to be expressed as probabilities. Most regions provide a wealth of geological and production information from which useful conditional relationships can be extracted. Bayesian operations can provide consistent procedures for obtaining and revising discovery probabilities (and complementary dry hole probabilities) and field size distributions. Often these relationships are so obvious that formal Bayesian algebra is not required and intuition can guide the arithmetic. In other circumstances the conditional relationships may be obscure and formal use of Bayes' theorem is essential.

110

CHAPTER

6 ^h

Modeling Prospects APPEAL OF T H E SIMULATION A P P R O A C H Simulation is a popular method of performing risk analysis in petroleum exploration, and is widely used by the major oil companies. Indeed, some multinational companies have become so enamored of this particular riskassessment technique that a standardized simulation program must be run on each and every prospect that company geologists generate throughout the world. The results of the standardized assessments are used to rank the prospects in order of their desirability. It's easy to understand why simulation is so popular within the oil industry. Simulation has been widely and successfully used by engineers to model reservoir performance, to design offshore platforms, and to evaluate alternatives for production facilities. Corporate economists have used simulation to study the possible effects of changes in the future price of oil, in discount rates, or in investment policies. However, perhaps the most important reason why simulation is so appealing is because it allows geologists and engineers to address a strongly probabilistic problem in a manner that is almost deterministic. We can imagine how we might compute the amount of oil in a prospect if we had definite information about certain specific reservoir properties. If, for example, we knew the thickness, areal extent, and porosity of the reservoir unit, we could calculate the volume of the reservoir rock that is filled with fluids. If in addition we knew the oil saturation, we could

Computing Risk for Oil Prospects — Chapter 6 calculate how much of the pore volume contains oil. Indeed, this is exactly the volumetric calculation method that petroleum engineers use to estimate the initial oil in place in reservoirs that are being developed. However, we do not know any of these constituents of volume prior to the drilling of a prospect and the discovery of oil. We may have some indirect evidence, perhaps about the thickness of a potential reservoir interval, or the height of closure and area of closure on a reflecting horizon from seismic surveys, but most of the critical variables cannot be known in advance of drilling. We may have good ideas about some potential ranges, such as the possible porosities that sandstone reservoir rock might possess, but no definite information about the specific prospect being modeled. In such a circumstance, we cannot calculate a single, specific estimate of volume, but instead we might imagine calculating a very large number of combinations of possible values of the various constituents of volume. In eff^ect, we would be posing a number of "what if..." scenarios, each representing a possible state of the reservoir. The collection of possible outcomes, if sufficiently large, forms a distribution of volumes of oil that the prospect might contain. If the inputs are given in the form of probability distributions which describe the likelihood that the input properties have certain values, the output distribution is also a probability distribution of the likely volume of oil. The numerical technique named for the famous Monte Carlo casino in Monaco is the basis for most petroleum risk-assessment simulation procedures, and indeed, for simulation methods in many fields of application. Monte Carlo techniques operate on pure chance, drawing samples at random from distributions and combining these to obtain a possible output or realization. By repeating the process of random selection and combination, a distribution of outcomes is eventually created. Monte Carlo simulation procedures were developed during World War II in connection with research on the atomic bomb (Metropolis and Ulam, 1949). It is widely used by mathematicians and statisticians to solve numerical problems that otherwise would be intractable. In the 1960's, simulation techniques were applied to the analysis of business decisions (Hertz, 1964) and soon found their way into the petroleum industry (Stoian, 1965). Simulation is commonly used by reservoir engineers and facilities designers (Smith, 1968; Walstrom, Mueller and McFarlane, 1967) and plays an important role in resource assessments made by government agencies (Crovelli, 1984; Dolton and others, 1981; Lee and Wang, 1983). Bernard and Akers (1977) describe the computer program used for the United States government's Monte Carlo assessment of offshore prospects. Newendorp's (1975) textbook contains a lengthy and especially readable discussion of Monte Carlo procedures applied to financial evaluation of petroleum prospects. 112

Modeling Prospects Textbooks devoted to Monte Carlo procedures include those by Hammersley and Handscomb (1964), Law and Kelton (1982), and Rubinstein (1981). Initially, programs for Monte Carlo analysis had to be custom created for each application and the iterative procedure strained the capabilities of the available computers. As a consequence, the method was adopted primarily by the major oil companies (McCray, 1969), but the development of very powerful and inexpensive personal computers and general-purpose Monte Carlo software have placed the technique within the reach of any interested explorationist (Davis, 1992).

STEPS IN MONTE CARLO SIMULATION The process that leads to the formation of an oil reservoir can be regarded as a succession of steps, each of which has an associated uncertainty. For some of these steps, the uncertainty relates to whether a critical event has occurred or not, and hence to the likelihood that a prospect might contain oil. That is, these steps relate to the dry hole risk. For others, the uncertainty relates to the magnitudes of the variables involved, and hence to the amount of oil that may be contained in a prospect. We can consider only a few components in the process of formation, or we can attempt to model nature in great detail. Many assessment schemes attempt to simulate the possible status of petroleum formation and accumulation at five critical steps. These steps are designated generation (or source)^ migration, timing, trap, and seal (Fig. 6.1). Generation relates to the actual presence of possible source rocks and their characteristics: volume (usually broken down into areas and thickness of source rocks), organic content, and degree of maturation. Simulation of this phase yields a distribution of the possible volumes of oil that might have been generated from rocks comprising the source beds. Migration is a nebulous concept that deals with movement of oil from the source(s) to the trap. It considers the probability that the source beds and the reservoir unit are connected by carrier beds, and whether intervening faults or other geometric barriers may be present. Simulation usually results in a distribution of the likelihood that migration has successfully occurred. Generation and migration are collectively related to charge, the quantity of oil that might be created and expelled from source beds. Some companies consider it important to model both the volume of charge and the capacity, or volume of a prospect. A comparison of the two indicates whether a prospect may be only partially filled, filled to capacity, or filled to overflowing with the potential for excess oil to have migrated farther updip into additional traps or to be lost. Timing assesses the probability that oil was generated and migrated to the location of the prospect at a time after creation of the trapping 113

Computing Risk for Oil Prospects — Chapter 6

TOC, Source Area, Bed Thickness

Source

I — I — I — r — I — I — I — I — I — I — I — I

Generated Volume

Efficiency, Timing

Timing, {Migration

% Uncertainty that Migration Occurred Area, Net Pay, Porosity, So, Sg, Free Gas

I

I

I—I

I

I

I—I

I

Trap

I—I—I

Reservoir Volume

Seal

Seal Integrity

% Volume Loss Various Conversion Factors

Oil Volume I—r—1—1—I—I—I—I—I—I—I—I

Recoverable Oil Figure 6.1. Schematic diagram of Monte Carlo simulation of an oil prospect. Source represents generation of a charge of hydrocarbons; migration and timing act to decrease the volume of the charge; trap determines the potential capacity of a prospect; seal reduces the volume retained in the trap. TOC = Total Organic Carbon. 114

Modeling Prospects mechanism. This step attempts to take into account the possibility that a prospect is barren because all of the oil potentially in place moved updip prior to formation of a trap. Trap is the most complicated and definitive of the steps in a simulation, because it relates to the geometry of a specific prospect, and usually the most detailed information has been gathered about its location and configuration. The trap ordinarily is broken down into various constituents of oil-bearing pore volume, such as reservoir thickness, area, percent fill, porosity, and oil saturation. The product of these variables is the volume of oil contained within the reservoir. In some simulation procedures, percent fill is not a specified variable, but is determined by the interplay of source, migration, and timing. Specific schemes may include somewhat different components, such as net-to-gross ratio, but basically these are the same variables that an engineer would use to estimate the amount of oil in a reservoir by the volumetric calculation method. Seal describes the integrity of a trap, or the extent to which it can contain oil over time. The seal may be perfect, enabling the trap to retain all of the oil that migrates into it until the prospect fills to the spillpoint. Conversely, the seal may be leaky, perhaps fractured or semipermeable, causing oil to be lost from the trap at an uncertain rate. In some of the more elaborate Monte Carlo simulation schemes, variables that relate to the possible volumes of gas and condensate are incorporated. These include additional properties such as the gas-oil ratio, thickness of the gas cap, and the proportion of condensate. Gas, condensate, and oil require quite different volumetric calculations and have very diff'erent economic implications, but all constituents must fit inside the same gross volume of reservoir. This means that multiphase Monte Carlo modeling schemes must (or should) include negative constraints between the constituents of the reservoir. In a Monte Carlo simulation, each of the constituents is represented by a probability distribution that describes how likely any range of values is believed to be. The probability distributions may be of standard form, such as normal or lognormal distributions, or they may have empirical forms such as rectangular, triangular, or more complicated shapes. The parameters of the distributions may be chosen based on studies of the geological properties in the area or in analogous areas, or on personal or collective experience. No matter what their source, the distributions represent a codification of the expectations of the appraiser; the probabilities distinguish between values that are considered likely, and those that are considered unlikely or impossible. The result of a Monte Carlo run is a joint probability: This amount of oil is what the prospect will contain if this is the amount generated and this is the amount that migrates and the trap is so large and The joint 115

Computing Risk for Oil Prospects — Chapter 6 probability of several events is the product of their individual probabilities, so the Monte Carlo technique involves multiplying all of the probabilities attached to the individual outcomes at each step. A value is drawn randomly from the probability distribution that describes a property and is multiplied by a value randomly drawn from the probability distribution that describes the second property, and so on. Values are drawn more frequently from portions of the distributions that have high associated probabilities. (In fact, in the limit, the proportion of observations drawn from any specific part of the distribution is exactly equal to the probability.) Successive iterations or repetitions are performed, and each time new values are drawn randomly and multiplied together. After several hundred or several thousand iterations, the collection of outcomes (each of which is a possible volume of oil) is analyzed to determine the proportions of outcomes that fall within specific ranges. When assembled into a distribution, either in the form of a histogram or a cumulative curve, the result is a probability distribution of the volume of oil.

Risked or Unrisked Distributions? In prospect evaluation, the input probability distributions can be specified in either of two ways, as risked or unrisked distributions. The difference is illustrated in Figure 6.2, which shows the possible distribution of a reservoir property such as thickness of net pay. The risked distribution has two parts, a spike at zero thickness and a skewed part that begins at 1 m, rises to a maximum at 5 m, and then tails ofT at greater values. The spike represents the probability that the prospect is barren and hence has zero thickness of oil-saturated interval; this is equivalent to the dry hole risk. The remainder of the distribution describes the possible thickness of pay in a producing prospect. Since this is a probability distribution, the total area under the curve must be 1.00 or 100%. The area under the spike (or dry hole risk) is 50%, so the area under the remaining part of the distribution also must be 50%. In the second form of the distribution, the dry hole risk is not included. Instead, the probability distribution specifies the likely nature of a successful prospect. This is a conditional probability distribution, because it describes the probabilities attached to different thicknesses of net pay, given that the prospect contains oil. The dry hole risk and the uncertainty in the amount of oil contained in the prospect are treated separately. Since an unrisked distribution is also a probability distribution, the area under the curve must also be 1.00 or 100%, but because the spike of the dry hole risk is absent, the distribution is much simpler in form and can be modeled using an equation such as the normal or lognormal. 116

Modeling

Prospects

50 45 40 35 '^^ Dry Hole Risk

® 30

I

^ 25 !5

I 20 a. 15

0\sV*^^°''

10

0

2 4

6

201

8 10 12 1416 18 2022 24 26 2 8 3 0 32 34 Net Pay, Meters

(D

iJ 1 5 a>

tou^:^o^

^^.^^^'''

Q.

•t 10 (0

0

2 4

6

8 10 12 1416 18 2022 24 26 2 8 3 0 32 34 Net Pay, Meters

Figure 6.2. (a) Risked probability distribution with a spike which corresponds to the probability of a dry hole, (b) Unrisked probability distribution that is conditional upon a discovery having been made. 117

Computing Risk for Oil Prospects — Chapter 6 Monte Carlo simulation can involve either form of distribution, but it is simpler to specify the form of an unrisked curve. For this reason, most risk-simulation procedures treat the dry hole risk separately; they model the conditional probability related to the volume of oil in a prospect, given that the prospect will be a discovery. The final result is then converted into an unconditional probability by multiplying the probabilities of the output distribution by the complement of the dry hole probability. This yields a result identical to that obtained when a simulation is based on a risked distribution.

Simulating Field Size Distributions in a District of Magyarstan We can demonstrate a simple application of the Monte Carlo simulation technique using a prospect in southwestern Magyarstan. Exploration began in earnest in this area in the early 1930's, and numerous fields have been discovered in Jurassic rocks since that time. The XV and XVa intervals of the J3 and the XVb interval of the J2 consist of alternating limestones and shales deposited in a shallow epeiric sea. Seven or more major limestones are included, each a component within a transgressive-regressive cycle. The uppermost, regressive limestone of each cycle commonly has a porous, grain-rich reservoir interval near its top. In this area, 141 fields have been discovered, each producing from one or more of these intervals. Producing zones may be 2 to 10 m thick, and several producing zones usually occur in a field. The reservoirs seem to be combination structural and stratigraphic traps, localized by a complex interplay of limited structural closure (typically 10 m or less) and the local development of porous facies. Structural and stratigraphic components may be interrelated, with thicker and more porous lithologies having originated as exposed and leached marine banks whose topographic expression was further emphasized by compaction of enclosing shales. As a consequence of their mode of formation, the fields have limited areal extent; the largest field in the area covers less than 5 km^. Information on the areal size, location, discovery date, producing interval, and production history of fields in the area can be extracted from the records of the Magyarstan Scientific Research Ministry of Economics of Mineral Resources and Hydrocarbons. Prom these data, estimates can be made of the ultimate production that will be obtained from fields still in production. These data provide information on the characteristics of fields that have been discovered in the region, and by extension, characteristics of pools that are undiscovered. Unfortunately, nothing has been systematically collected on average net pay or producing interval thicknesses, porosities, or 118

Modeling Prospects oil saturations. Distributions of these properties must be based on general characteristics of carbonate reservoirs (compiled for North America and elsewhere in the world), detailed descriptions of typical fields in the region, and the personal knowledge of local experts. These sources of information can be used to calibrate the simulation of a "typical" carbonate reservoir in this area of Magyarstan. The parameters of the input distributions can be adjusted until the output distribution closely matches the known field size distribution for the region. This provides reassurance that the simulation is reasonable, but is subject to two caveats: First, simulations are not unique; if their diff^erences are mutually compensating, many different combinations of inputs may yield similar outputs. Second, the input distributions required to simulate the population distribution of field sizes in a region are different from the input distributions needed to simulate a probability distribution expressing the uncertainty in size of an individual prospect. Since this area is a mature petroleum province, we can presume that adequate hydrocarbons were generated in source beds, and that migration occurred with appropriate timing. The only uncertainty is associated with the characteristics of the prospects themselves, and the proportion of oil in the prospects that will be recovered (the latter uncertainty need not be considered if we content ourselves with simulating oil-in-place; however, for comparison with historical records, we must simulate the amount of oil that might be produced). A five-component Monte Carlo simulation as shown in Figure 6.3 is adequate to model a "typical" field. During each iteration of the Monte Carlo process, a value is drawn at random from each of the distributions shown in Figure 6.3. These randomly selected values are combined to yield a value of ultimate oil production in barrels according to the formula: Ultimate Production (bbls) = Area (ha) x Thickness (m) x Porosity (%) x Oil Saturation (%) x Recovery Factor (%) x 62,900. The numerical factor of 62,900 converts hectare-meters (10,000 m^) to barrels, assuming that porosity, oil saturation, and recovery factor are given as decimal fractions and that stock-tank and reservoir barrels are equivalent. All input distributions were modeled as truncated continuous probability functions whose parameters are given in Table 6.1. The areas of oil fields in the region are known to follow a highly skewed distribution, and the distribution of thicknesses is thought to be highly skewed as well; both area and thickness were modeled as truncated lognormal distributions. Porosities, oil saturations, and recovery factors may be presumed to vary more or less symmetrically about central values, so these were modeled as truncated normal distributions. 119

Computing Risk for Oil Prospects — Chapter 6 30-1 251 20 15H

100

200 300 400 Area, Hectares

10 15 20 25 Porosity, Percent

500

30

30i

25

25

20

20

15

15

10

10-1

5

5-^

0

5

10 15 20 25 Thickness, Meters

30

30 40 50 60 70 80 90 100 Oil Saturation, Percent

30-1

25J 20-^

15J 10-1 5 0

10 20 30 40 50 60 Recovery Factor, Percent

Figure 6.3. Input probability distributions used in Monte Carlo simulation of ultimate production from Jurassic fields in southern Magyarstan. Parameters of distributions are given in Table 6.1. The output distribution is shown in Figure 6.4a and may be compared to Figure 6.4b which shows the observed distribution of estimated ultimate production from the 141 Jurassic fields in the area. The simulation is 120

Modeling Prospects Table 6.1. Monte Carlo input parameters derived from characteristics of fields in a district in Magyarstan. Property Area (ha) Thickness (m) Porosity (%) Oil Saturation (%) Recovery Factor (%)

Mean

Standard deviation

Lower limit

Upper hmit

80 12 10 70 30

96 4 5 10 7.5

20 4 2 0 10

1200 36 30 100 50

very similar to the actual distribution, indicating that reasonable choices of distributions and parameters have been made.

Simulating a Specific Prospect in Southern Magyarstan This exercise in "history matching" reassures us that the simulation model produces acceptable results. The next step is to substitute distributions that describe the characteristics of a particular prospect that we wish to evaluate into the model. We will apply the model to a prospect, shown in Figure 6.5, which was proposed by the international partner in a joint exploration venture. Inputs to the Monte Carlo risk-assessment model are taken from the company's prospect folio. When applying Monte Carlo methods to the analysis of an individual prospect, it is very important to keep in mind that the input probability distributions describe the range of likely values that properties might assume in that specific prospect. These distributions have nothing directly to do with the distributions we have used to describe the variation in average properties of the collection of fields from the southwestern Magyarstan area, except that hopefully the properties of the specific prospect will become a new single set of values in these distributions. For example, perhaps the prospect can have at most a trap area of 160 ha, and cannot conceivably have an area of less than 120 ha. These values should define the upper and lower extremes of the distribution of area, even though the resulting distribution is quite different than the distribution of field areas in the region. Obviously, if the regional distributions pertained to the individual prospects, then all prospects would be expected to have exactly the same probability distribution! Another caveat is that the distributions of thickness, porosity, and oil saturation refer to field-wide average values and the uncertainties about the exact magnitudes of these field-wide averages. The uncertainty about 121

Computing Risk for Oil Prospects — Chapter 6 20-

15o Q.

CO

n 2

5H ^^^rrrru

1

2 3 Ultimate Production, Millions of bbis

1

2 3 4 Ultimate Production, Millions of bbIs

20-

15(0 0)

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I

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(H

l::::::::A:::::::::fc:::::x^^^^^^^^^

Figure 6.4. (a) Simulated ultimate production in barrels for 500 iterations of a Monte Carlo simulation of Jurassic fields in southern Magyarstan. (b) Actual distribution of estimated ultimate productions of 141 fields in southern Magyarstan. 122

Modeling Prospects

Figure 6.5. Prospect map of an area in southern Magyarstan, as developed during a joint venture. Inputs to Monte Carlo simulation are based on characteristics of this prospect. Grid has 1-km spacing. Contours in meters below sea level. average porosity or oil saturation is not the same as the variation in porosity or oil saturation that may occur between one depth and another in a well, or even the variation in average values that occurs between wells. Similarly, the uncertainty about the possible average thickness of net payis not necessarily the same as the possible variation in net pay thickness that occurs across the field. The model parameters selected, after some experimentation, are given in Table 6.2. Area and thickness were modeled using truncated normal distributions, rather than skewed lognormal distributions. As can be seen from 123

Computing Risk for Oil Prospects — Chapter 6 Table 6.2. Parameters of distributions used to model a specific prospect in southern Magyarstan. Property Area (ha) Thickness (m) Porosity {%) Oil Saturation (%) Recovery Factor (%)

Mean 164 6 10 70 30

Standard Minimum Deviation 40 3 5 10 7.5

30 3 2 0 10

Maximum 208 9 30 100 50

the prospect map in Figure 6.5, a field could not be larger than about 200 ha, or it would already have been encountered by one of the nearby drill holes. Explorationists who developed the prospect saw no reason to presume that the distribution should be asymmetrical, so a normal distribution of area was used. Nearby productive wells produce from one, two, or at most three zones in the XV and XVa limestones of J3, as several of the transgressive-regressive cycles are missing and no carbonates are present in the J2. Production comes from leached porous zones that typically are about 3 m thick at the tops of the limestones. Therefore, if one productive limestone is encountered, the net pay will be about 3 m; if two are encountered, the net pay will be about 6 m; and if three are encountered, the net pay will be about 9 m. The thickness of the net pay was represented by a normal distribution with a relatively large standard deviation and close cutoffs. Use of a uniform or rectangular distribution would yield similar results. Since no specific information was available on porosity, oil saturation, or recovery factor, these were modeled using the same parameters as used for the regional simulation. A Monte Carlo simulation of 500 iterations yields the distribution of the ultimate oil production (in bbls) from the prospect, if a discovery is made (Fig. Qt.^). The expected (mean) amount is 1,300,000 bbls.

Incorporating Risk in the Simulation A critical missing component in the simulation to this point stems from the fact that we have modeled a probability distribution that is conditional upon oil being discovered. That is, the distribution we have produced is unrisked. To produce the more useful risked, or unconditional form of the distribution, we must include the dry hole probability in our considerations. In this area, the dry hole probability can be estimated at about 47%, based 124

Modeling Prospects

15

c 0)

p

0

1

2 3 Ultimate Production, Millions of bbis

4

5

Figure 6.6. Simulated probability distribution of ultimate production in barrels for a prospect in southern Magyarstan based on 500 iterations of a Monte Carlo simulation. Expected value is 1,300,000 bbls. on the proportion of wildcats drilled that have been abandoned as dry. The probability of a discovery of some magnitude is then 100% — 47% — 53%. The probabilities of the unrisked distribution are simply multiplied by the probability that a discovery will be made. The area under the probability distribution will not sum to 100%, but rather will sum to 53%; the remaining 47% of the distribution is the probability that the prospect will be dry. Perhaps the most convenient way to show the probability distributions is not as histograms in which the bars represent the probabilities associated with equal intervals of oil volume or monetary worth, but as cumulative probability curves. These are made by graphing the successive percentiles of the probability distribution against barrels of oil produced (Fig. 6.7), producing a plot of the probability that a specified volume or less of oil will be discovered. Sometimes it is more convenient to use the complement of this probability, so the graph expresses the probability of the discovery of a specified volume or more (Fig. 6.8). Because the highest probabilities may be associated with the smallest amounts of oil, it may be useful to plot volume on a logarithmic scale (Fig. 6.9). Plots of the risked probability of discovery have the same form as unrisked plots, but the curves do not begin at 100%, but rather begin at the complement of the dry hole risk. Figure 6.10 shows a risked distribution on 125

Computing Risk for Oil Prospects — Chapter 6

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Figure 6.7. Probability that a discovery will contain a specified volume of oil or less, given that a discovery is made. a logarithmic scale; note that the distribution begins at 53%, the probability of any discovery, regardless of magnitude. RISKSTAT contains a simple Monte Carlo simulator that allows you to experiment with the procedure for assessing an individual prospect. The Monte Carlo routine in RISKSTAT uses four input variables: area, thickness, porosity, and oil saturation. (The program assumes that area is in acres and thickness is in feet, so the results of your simulation will be off by a constant if you use hectares and meters. Output can be scaled to metric units by multiplying the number of barrels of oil by 8.1, or roughly increasing them by an order of magnitude.) For each input variable, you will be asked to choose a form for the distribution—normal, lognormal, exponential, or uniform. Depending upon your choices, you will then be asked for the appropriate parameters. RISKSTAT provides the option of simulating "oil-in-place" or "recoverable oil." If you choose to model recoverable oil, you must also specify the form of distribution and parameters for the recovery factor. RISKSTAT will also request that you specify the number of iterations in the 126

Modeling Prospects

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Figure 6.8. Probability that the specified volume of oil, or more, will be discovered in the prospect, given that a discovery is made. simulation. Depending upon the speed of your computer, time required for Monte Carlo simulations can be lengthy. It may be prudent to try an initial run using only 50 iterations or so to confirm that the proper input parameters have been chosen. When you are satisfied that appropriate forms of distributions and their parameters have been specified, the actual simulations should be run for at least 500 iterations. It is not unusual to specify several thousand iterations when using fast mainframe computers or workstations. The results of simulations can be displayed as histograms using the graphics options of RISKSTAT.

127

Computing Risk for Oil Prospects — Chapter 6 100

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SELECTING DISTRIBUTIONS AND SETTING PARAMETERS There are many numerical distributions, both discrete and continuous, that could be used as models in Monte Carlo simulation. Usually, workers choose a normal distribution to model properties that they believe are more or less symmetrical, and lognormal distributions to model properties that may be skewed with a tail extending to large values. However, no one knows what forms really describe the populations of geological variables, and the normal and lognormal models are chosen primarily for convenience and conformity with accepted usage. For example, some researchers have argued that field sizes should follow a Pareto distribution (Schuenemeyer and Drew, 1983), while others have advocated a log-gamma model (Davis and Chang, 1989). @RISK, a popular Monte Carlo simulation program for personal computers, contains a library of almost 30 different distributions that could be used in modeling a prospect. 128

Modeling Prospects

Ultimate Production, Millions of bbis

Figure 6.10. Probability that the specified volume of oil, or more, will be discovered in the prospect, adjusted for the dry hole risk. Volume plotted on a logarithmic scale. Faced with such a plethora of choices and very little guidance, many users settle for the simplest approach, which is to specify distributions as triangular in form, defined by a "lowest possible" limit, a "most likely" peak value, and a "highest possible" upper limit. Figure 6.11 shows a comparison between a triangular distribution and a normal distribution that have the same means and standard deviations {X = 20, s = 4.08). At first glance, there is little difference between the two, but a triangular distribution may be seriously misleading when used in the evaluation of petroleum prospects. The triangular distribution in Figure 6.11a has a minimum lower limit of 10 and a maximum upper limit of 30; no values can be drawn from the distribution that are smaller or larger than these limits. In contrast, the normal distribution shown in Figure 6.11b is theoretically limitless. In 100 random draws from the distribution, the smallest value was —1.82 and the largest was 41.83; even more extreme values would be drawn if the simulation were run for several hundred, a thousand, or more iterations. Of course, these extreme values are very rare; only 5% of the draws exceeded 129

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Thickness, Meters Figure 6.11. Comparison between triangular distribution (a) and normal distribution (b) used to describe the thickness of a reservoir interval. Both have means of 20 m and standard deviations of 4.08 m. 26.71 in the simulation having 100 iterations. But these very large outcomes that are associated with small probabilities of occurrence play a critical role in risk assessment. It is the chance of discovering a bonanza that often drives exploration. Even if the probabilities of finding giant fields are very small, the potential rewards may be so vast that they strongly influence the worth of prospects. If triangular or other bounded distributions (such as the uniform distribution) are used, we run the risk of unknowingly truncating the size of field that a prospect might contain. We will not be able to estimate the probability associated with a giant field because our simulation will be 130

Modeling Prospects incapable of creating such a field, regardless of the number of iterations that are run. In effect, we have decreed that the occurrence of a very large field is not unlikely but, rather, impossible because of our choice of input distributions. If this is truly our intent, well and good. The danger lies in inadvertently creating a simulation which cannot represent the range of possibilities, and failing to realize that the potential economic range of the model is thereby constrained. In the absence of specific knowledge (or strong belief) about the form of various geological distributions, it seems reasonable to model most of them using normal and lognormal distributions. Some properties must be bounded; for example, it is not possible to have negative thicknesses or percentage variables such as porosities or oil saturations outside the range of 0% to 100%. Experience in a specific area or play may lead us to believe that the bounds are more restricted. In the simulation of a Magyarstan prospect, information from other fields leads us to believe that recovery factors can be no less than 10% and no greater than 50%. Having settled on the appropriate form of distribution for a geological property, we are then faced with the problem of specifying its parameters. Most distributions require a measure of the center (the mean, median, or mode) and one or more measures of spread (the standard deviation, range, or upper and lower limits). Geologists seem to have reasonably consistent and reliable ideas about the average or "most typical" values for many geological variables, but a poor grasp of the possible extremes. Psychological experiments have shown that people (including geologists) consistently underestimate the magnitudes associated with rare occurrences (z.e., those in the extreme tails of distributions). As a consequence, there is a tendency to be conservative in estimating the spread of distributions in Monte Carlo simulation. The result is an unwarranted reduction in probabilities associated with the most extreme outcomes, including the probability of making a very large discovery. The problem of specifying appropriate parameters is complicated if the normal or lognormal distributions are used, because the standard deviation is one of the required parameters for these distributions. Most geologists have only a vague "feel" for the meaning of the standard deviation, and little or no experience that might guide them in selecting appropriate values. The most appropriate estimates of these parameters are statistics calculated from data collected in the same province or play as the prospect being modeled. This was done, for example, in the simulation of a prospect in Magyarstan. Sample means and standard deviations can be calculated on the properties measured in known fields and used to guide the specification of parameters in a simulation. 131

Computing Risk for Oil Prospects — Chapter 6 Sample means and standard deviations may not be entirely reliable because both measures are sensitive to the occurrence of unusual values, especially if calculated from small data sets. In such circumstances, it may be better to estimate the center of a distribution by the median of the regional data, and to approximate the standard deviation by ranking the known observations, determining their 15th and S5th percentiles, and dividing the difference between them by two. This approximation is based on the fact that the interval within one standard deviation of either side of the mean of a normal distribution contains almost 70% of the area under the curve. Unfortunately, deriving modeling parameters from regional data is only feasible in relatively mature areas where abundant observations are available. In virgin areas, or when modeling a prospect based on a totally new geological concept, these data do not yet exist. And, of course, for many geological variables that are included in some of the more complicated Monte Carlo procedures, direct knowledge is almost never available. Who can really say what a distribution might be like that describes a property such as "adequacy of seal?" For some geological properties, there are national or worldwide compilations that can provide guidance for specifying realistic parameters. The American Petroleum Institute, for example, issues statistical summaries of the characteristics of oil and gas fields in the United States (American Petroleum Institute, 1967, 1984), and numerous authors have published studies of specific properties {e.g., Maxwell, 1964; Nehring and Van Driest, 1981; Schmoker, Krystinik, and Halley, 1985; Sluijk and Nederlof, 1984). In Monte Carlo schemes that encompass the complete sequence of oil generation, migration, entrapment, and recovery shown in Figure 6.1, some variables are not actual geological properties. An example is "trap timing," which is supposed to express the chance that a potential trap was formed prior to the migration of hydrocarbons through the location of the trap. The likelihood of a fortuitous coincidence of events is a probability, and is given in percent; its sole eff^ect is to reduce the quantity of oil or gas that is available to be included in the prospect. Usually, "timing" is not given as a single value but as a distribution having a lower limit, a most likely value, and an upper limit. In other words, it is a probability distribution of probabilities! It is extremely difficult to imagine how the parameters of this distribution might rationally be specified, even though they may have a significant effect on the final volume of oil contained in the prospect. Similar comments apply to variables such as "migration efficiency" and "seal quaUty," which also are expressed as percentages {i.e., probabilities of occurrence or failure). The foregoing comments might be taken as lightly veiled criticisms of some widely used Monte Carlo simulation procedures. Our remarks are 132

Modeling Prospects based on the belief that specifying in great detail an extremely uncertain series of events does not make the outcome any less uncertain. If geologists have difficulty assessing the volume of oil that might be contained in a prospect, their task is not made any easier (or the results more precise) if the prospect is broken down into a large number of components whose characteristics are even less well understood. In many areas, the bestknown property associated with oil fields is how much oil they contain. Geologists might well do a better job of estimating the distribution of field sizes directly, rather than estimating a large number of secondary attributes that are poorly known, and then multiplying these together to produce a distribution of field sizes.

ARE GEOLOGIC PROPERTIES INDEPENDENT? In Monte Carlo simulation of a petroleum prospect, values of geological variables are selected at random from the specified distributions and multiplied together to obtain the distribution of their products. Some of the variables are percentages and some are areas or thicknesses; the end result is a distribution of volumes. When we draw a value of one variable, the number we obtain has no effect on the value we will draw for another variable. That is, the variables are completely independent of one another. Is this a reasonable assumption, and what are the consequences if it is not? In Chapter 4, we discussed the relationship between field area and field volume. There are similar positive relationships between field areas and reservoir thicknesses (bigger fields tend to have thicker oil columns as well as greater areal extent) and sometimes between other geological variables (in some sandstone reservoirs, thicker intervals tend to be cleaner, and hence have higher porosities; in turn, oil saturations tend to be higher in reservoirs with higher porosities). Productivity may be correlated with geological characteristics (the recovery may be low for tight formations). Typically, these correlations are not especially pronounced, but if they are not considered in simulation, the results may be biased. Figure 6.12a shows a lognormal distribution that we will use to model field area; the distribution has a mean of 120 ha and a standard deviation of 20 ha. (Note that the distribution is skewed to the right when plotted on an arithmetic scale as shown here; if the distribution were plotted on a log scale it would be symmetrical.) Figure 6.12b is a plot of a normal distribution representing reservoir thickness; it has a mean of 10 m and a standard deviation of 2 m. We can sample randomly from each of these distributions and obtain their products, which will express the gross rock volume of the prospect. 133

Computing Risk for Oil Prospects — Chapter 6

0 6

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Figure 6.12. Input distributions used in a simulation of reservoir volume, (a) Area in hectares (lognormal with mean = 120 ha, standard deviation = 20 ha), (b) Thickness in meters (normal with mean = 10 m, standard deviation = 2 m). The distribution shown in Figure 6.13a is the result of 1000 iterations in an ordinary Monte Carlo procedure. The output distribution has a mean of 1198 hectare-meters (or 11.98 million m^). The upper ^bth percentile is 1733 hectare-meters and the maximum value calculated in 1000 iterations is 2571 hectare-meters. A simulation based on the same input parameters is shown in Figure 6.13b, but the thickness and area are specified as having a positive correlation of r = 0.80. (This is much higher than the correlations usually seen between reservoir properties.) The mean of the output distribution is somewhat higher, being equal to 1234 hectare-meters. However, the upper 95^/i percentile is 1985 hectare-meters, or 252 hectare-meters greater than for the ordinary simulation. The maximum value calculated in 1000 iterations was 5215 hectare-meters, more than twice the maximum calculated when the properties were assumed to be independent. (Correlations between variables in Monte Carlo simulation can be induced by a two-stage sampling procedure that orders the observations by their rank. Technical details are given by Iman and Conover (1980). Newendorp (1975) also discusses the problems that may arise in simulation with dependent variables and gives several ad hoc procedures for introducing dependence into a simulation.) The differences between results from the simulation that assumes the input variables are independent (Fig. 6.13a) and one that does not (Fig. 6.13b) might seem to be minor, but note where the bigger discrepancies appear. They occur in the tails, and particularly the upper tail if one or more of the input distributions are positively skewed {i.e., lognormal). As we have noted, this low probability but high payoff part of the field size 134

Modeling Prospects

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distribution plays a critical role in the worth of a prospect. Our simple experiment has shown that ignoring correlations between variables will not dramatically affect the resulting simulation, but it does have the potential to cause critical parts of the final distribution to be underestimated. Unfortunately, almost nothing has been published on possible interdependencies between geological variables (except for the interrelationship between porosity and water saturation), so it is difficult to determine if this might be a significant problem in Monte Carlo simulation of prospects. As another cautionary note, remember that our experiment involved multiplying 135

Computing Risk for Oil Prospects — Chapter 6 together only two variables, and in a full-scale simulation we might multiply a dozen or more distributions representing different geological properties. If several of these variables are dependent and are mistakenly treated as if they were independent, the combined effect may be much more severe than what is seen here. A prudent course of action for anyone who relies on Monte Carlo simulation of prospects would be to experiment and determine the effects, if any, of assuming independence versus nonindependence between the input variables they use. If possible, data should be collected from fields within the same play as the prospects being appraised, and statistical correlations calculated. Considering the enormous investment that many oil companies have made in Monte Carlo prospect evaluation software, it's surprising that so little has been done to verify the assumptions built into the technique.

136

CHAPTER

7 Mapping Geological Properties and Uncertainties

C O M P U T E R CONTOURING Some of the geological properties that we use to define prospects are continuous surfaces, such as the contact between two stratigraphic units (expressed as either the top or bottom of an interval) and the contacts between oil and water and gas and oil in a reservoir. We may know the elevations of formation tops in wells that penetrate the formations, or we may have estimates of their depths at locations along a seismic survey. However, we obviously do not know the true values of such surfaces at a prospect site because this would mean that the prospect has already been drilled! We must, in some manner, calculate estimates of the elevations of the surfaces at the prospect locality. More generally, we would like to estimate values of the surfaces of interest at all locations within the region, because we could then represent the surfaces as contour maps. The construction of contour maps was one of the first major tasks to be delegated to computers by exploration geologists; primitive contouring programs appeared in the early 1960's (c/. IBM, 1965). Initially, programmers were happy if their code could generate an acceptable-appearing contour map in a tolerable length of time. By present standards, the mainframe computers available in the 1960's and 70's were enormously expensive, had minuscule memories, and were agonizingly slow. Only the larger oil companies, universities, and government research laboratories could aff^ord to experiment with computer contouring. Because of the limited capabilities

Computing Risk for Oil Prospects — Chapter 7 of the available machines, great emphasis was placed on computational efficiency and the use of programming "tricks" that would allow larger data sets to be processed than would otherwise be possible. The major oil companies realized that by automating one of the geologist's most time-consuming tasks, computer contouring could greatly speed the work of their exploration groups. In a competitive business like oil exploration, being first is often critical. In addition, these same companies had already invested heavily in computers for geophysical processing, record-keeping, and business purposes, and it seemed natural that their explorationists should make use of this investment as well. The geologists, however, often took a dim view of computer contouring and considerable effort was devoted to convincing the skeptics that computer-generated maps were acceptable (Dahlberg, 1975; Walters, 1969). Commercial software houses developed contouring programs to appeal to the large oil companies with their enormous investments in computing equipment. Like the computers themselves, the programs were large, complex, and extremely expensive. Because of their complexity, the programs were not operated by geologists, but by computer specialists. Since the programs were so expensive to purchase and costly to maintain and run, it was essential that the visual quality of their output reflect the magnitude of the investment. Consequently, great emphasis was placed on graphical perfection and cartographic embellishments. Also, because the programs were used in conjunction with large corporate and commercial well data bases, they included sophisticated data management capabilities. The one critical aspect of contouring that was overlooked during this period of software development was an assessment of the reliability of the finished computer-drawn contour maps. Maps produced by computer were judged on the basis of their visual appearance, which meant how closely their features resembled those on manually contoured maps. About the only quantitative measure of acceptability was the degree to which a map "honored the data points." Arguments that this was not a valid criterion for judging a mapping procedure whose primary function was estimation and not reproduction were roundly ignored (Davis, 1976). There are extensive discussions about how computer contouring programs operate, and exhaustive comparisons of the characteristics of different algorithms. We will not pursue these details, but instead refer the reader to discussions in Davis (1986), Hamilton and Jones (1992), Jones, Hamilton, and Johnson (1986), Robinson (1982), and especially Watson (1992), who has a complete bibliography on contouring algorithms. Here we will briefly outline the underlying principles of computer contouring and place them in the broader scope of geostatistics, which provides a means of assessing the reliability of contour maps in probabilistic terms. 138

Mapping Properties and Uncertainties In risk assessment, it is essential that we regard contouring as more than a quick and convenient way of creating a pretty picture of, say, the subsurface structural form of a prospect. We must be able to use contouring as a forecasting tool, producing an estimate of the likely shape of a structure that is only dimly perceived from scattered observations. Furthermore, we must be able to assess the uncertainty in our estimate, and to define the structure's likely upper and lower limits. Only when we do this can we include the potential range of the prospect in our appraisal.

HOW CONTOUR MAPS ARE MADE All computer contouring algorithms make several assumptions about the surface being mapped. The surface is presumed to be single-valued at each point or geographic location, to be continuous everywhere within the limits of the map, and to be autocorrelated over a distance that is greater than the typical distance between the available data points. ("Autocorrelation" is a statistical concept borrowed from time series analysis; it indicates the degree of similarity between a signal and itself after it has been shifted.) If these assumptions are valid, the known values of the surface at the control points can be combined to produce estimates of the surface between the control points. A geological surface such as the contact between two stratigraphic intervals obviously is single-valued and continuous unless it has been thrust faulted or recumbently folded. Other geological properties of interest, such as porosity, are not so obviously single-valued since more than one measurement of porosity may be available for a reservoir interval in an individual well. However, we can think of such multiple observations as samples that can be represented by a statistical value such as their mean. We may assume that the mean porosity of an interval is a single-valued property that is continuous, and hence mappable. Although geophysicists routinely utilize the concept of autocorrelation in seismic processing, it is less familiar to geologists. However, the concept is essential for an understanding of spatial phenomena. "Spatial autocorrelation" refers to the degree to which values at specific locations are related to values at other locations which are separated by a constant distance in a fixed direction. It provides a formalization of the commonsense concept that points on a surface are very much like nearby points, and less like more distant points.

Conventional Contouring Programs Computer contouring programs perform a number of distinct operations. First, the program must sort through the data and organize them so data 139

Computing Risk for Oil Prospects — Chapter 7 points can be selected quickly according to their location. Next, the program establishes an imaginary regular gridwork of locations across the map area where the estimates of the surface will to be made in succeeding operations. Ordinarily, the user must specify the dimensions of this grid, either as the number of rows and columns or the distances between the rows and columns. Other information may be necessary if the map is to be either larger or smaller than the extremes of the locations of the data points. Once this gridwork is defined, the program is ready to estimate the values of the surface at the nodes of the grid. To estimate an individual grid node, the nearest control points—usually well locations or seismic shot points—are combined in the form of a weighted average. At this time, the user may be faced with a number of decisions. How is "nearest" to be defined? How many "nearby" points should be used in each estimate? And what weights should be applied to each point? Diff^erent programs incorporate different choices in their coding, and the vendors of some commercial software packages tout their particular choices as superior. Other programs leave these decisions up to the user, and provide a variety of diff'erent alternatives. The fact is, such choices are arbitrary; the combination of data selection criterion, number of points, and weighting function that proves eff^ective with one data set may not be best for a diff^erent variable or a diff'erent set of data points. The algorithms are sensitive to the characteristics of the property being mapped and the geometric arrangement of the observations. Experience has shown that some algorithms tend to work well with the type of data that commonly are mapped in petroleum exploration, such as the structural configuration of a subsurface horizon or the thickness of an interval. Typically, the program will select 8 or 16 control points around a grid node that is to be estimated. The area around the grid node is divided into eight wedge-shaped sectors and the nearest one or two points are found in each sector. This ensures some degree of radial control of the estimate even if the data points are closely spaced along widely separated lines (as is typical of seismic and airborne geophysical measurements). Finally, the selected points are combined as an average in which each observation is weighted by the inverse of the square of the distance from the observation to the grid node. The weights are scaled so the most distant point has a weight near zero and the sum of the weights is equal to 1.0. There are many other factors that must be considered in practice. A maximum search radius and a minimum acceptable number of control points for each estimate must be specified, or otherwise the algorithm may incorporate far distant control points in areas where there are few observations, or fail to estimate grid nodes near the edges of the map. The literature on contour mapping (which is extensive but widely scattered) is 140

Mapping Properties and Uncertainties Table 7.1. Summary statistics for dry holes and producing wells in the Magyarstan training area. Data are contained in the file TRAINWEL.DAT on the diskette. Elevation Thickness Shale Bedding ratio index (m) (m) Producing wells (n = 18) Maximum -1404.0 90% -1409.4 -1431.2 75% 50% -1554.5 25% -1676.0 10% -1682.7 -1689.0 Minimum Mean -1550.9 Standard deviation 119.2

46.73 46.52 44.77 38.97 36.57 35.28 35.09 40.73 4.29

0.408 0.390 0.379 0.371 0.342 0.315 0.293 0.361 0.029

0.270 0.260 0.250 0.243 0.235 0.225 0.215 0.243 0.013

Dry holes (n == 65) Maximum 90% 75% 50% 25% 10% Minimum Mean Standard deviation

46.77 42.52 40.69 37.72 34.28 32.17 27.61 37.26 4.19

0.650 0.567 0.514 0.476 0.445 0.407 0.392 0.483 0.056

0.338 0.319 0.295 0.275 0.243 0.231 0.196 0.274 0.033

-1190.0 -1355.4 -1468.5 -1557.0 -1618.5 -1659.0 -1705.0 -1529.6 114.4

filled with discussions of the effects that these and other design decisions have on the resulting maps. Table 7.1 gives statistical summaries of measurements on four geological properties of the XVa Limestone as measured in 83 wells drilled in a small area of Magyarstan on the Zhardzhou Shelf that we refer to in this book as the "training" area. The data are included in the diskette file TRAINWEL.DAT. The geological setting and significance of these particular variables are described in Chapter 1, and Bayesian estimates of conditional probabilities based on the magnitudes of some of these variables are given in Chapter 5. Here, we will use these properties to illustrate computer contouring and subsurface analysis performed using the RISKMAP software, which is included on the accompanying diskettes. 141

Computing Risk for Oil Prospects — Chapter 7 The area to be mapped measures 35 x 35 km. Each geological property will be represented by a grid of values that contains 36 rows and 36 columns, so the contouring program must estimate values of a surface at grid nodes spaced at 1-km intervals. The origin of the grids is in the southwest corner and has coordinates of 1740 (east-west or Xi-direction) and 6455 (north-south or X2-direction). These are transformations of realworld UTM coordinates. We will use a simple contouring algorithm that estimates each grid node from the eight nearest wells, weighting the value at each individual well by the inverse of the square of the distance between the grid node and the well. Figure 7.1 is a contour map of the subsurface structural configuration of the top of the XVa Limestone unit; the contours represent meters below sea level. Contour lines have been drawn at intervals of 25 m. In addition to local irregularities in form, there is a general tendency for the surface to dip to the southeast. Figure 7.2 is an isopach map showing the thickness of the XVa Limestone unit, measured in meters. Thicker intervals are interpreted as being carbonate buildups or reefs, particularly if the carbonate interval is massive (see Chapter 1). Figure 7.3 is a contour map of the shale ratio, based on the average response of the gamma-ray log. Lower values represent relatively pure limestone that is free of included shale. Figure 7.4 is a contour map of the bedding index, a measure in which low values represent massively bedded limestones and thick shales at one extreme and high values represent finely interbedded limestones and shales at the other extreme. The bedding index is derived from the standard deviation of the gamma-ray log, which is sensitive to the shale content of the rocks.

Trends and Residuals Sometimes it is helpful to transform a contour map in different ways, to clarify features or to separate the map into components in order to see relationships that might otherwise be hidden. A widely used procedure is trend surface analysis, in which a dipping plane or simple curved surface is used to approximate the observations. This "trend" is fitted by a least squares criterion, in which the coefficients of the surface are determined in a manner that minimizes the sum of the squared deviations of the actual observations from the trend surface. This is exactly the same procedure that is used to fit a line to data in a cross plot, as described in Chapter 4. The fitted trend is an approximation of the data, and is a smooth representation of the large-scale form of the surface. If the observations are structural elevations, the trend often is interpreted as a regional component of the structure. Davis (1986) provides an extensive discussion of trend surface analysis and its alternative interpretations. 142

Mapping Properties and Uncertainties

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Figure 7.1. Structure contour map of top of XVa Limestone in the Magyarstan training area. Coordinates given in kilometers. Contours given in meters below sea level. Contour interval is 25 m. Solid dots are producing wells; other symbols indicate dry holes. Summary statistics for the 83 wells are listed in Table 7.1.

The trend can be subtracted from the actual surface, leaving residuals where the two do not coincide. These residuals also can be displayed in the form of contour maps, consisting of positive "highs" where the actual surface lies above the trend, and negative "lows" where it is below the trend. In effect, the residuals represent small-scale variations in the shape of the mapped property that are not contained in the large-scale trend. If we regard the trend as representing aregional component of structure, the residuals represent local features. 143

Computing Risk for Oil Prospects — Chapter 7 6490.

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Figure 7.2. Isopach map of the thickness of the XVa Limestone in the Magyarstan training area. Contour interval is 2 m.

A trend surface is calculated from the original observations using a statistical procedure that is a direct extension of the regression technique discussed in Chapter 4 and incorporated in RISKSTAT. The mapped variable, perhaps elevations of a subsurface horizon as measured in a number of wells, is the dependent Y variable and the geographic coordinates of the well locations are two independent X variables. A linear trend is defined by the equation Y = a -\- b^Xi -\-b2X2, where Y might be elevation, Xi the east-west coordinate, and X^ the north-south coordinate. The coefficient 61 gives the slope of the trend in the east-west direction, and 62 gives its slope in the north-south direction. The fitted surface has the form of a uniformly dipping plane. Coefficient a is the intercept, or value of the 144

Mapping Properties and Uncertainties

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Figure 7.3. Contour map of shale ratio, calculated as the average gamma-ray response in the XVa Limestone in the Magyarstan training area. Contour interval is 0.02 units. Low values indicate intervals with little clay content, high values indicate intervals with relatively high clay content. trend at the origin, and reflects the average value of y . The coefficients are found by solving a set of simultaneous equations that are direct extensions of those described in the section on fitting lines in Chapter 4. If a dipping plane seems too simple a description of the regional trend, we can expand the trend surface equations to fit more complicated surfaces. This is done by adding new variables to the trend surface equation and solving for the additional coefficients by expanding the set of simultaneous equations. The new variables are powers and cross products of the geographic coordinates. For example, a second-degree trend surface has an equation: Y = a-[- hiXi + 62X2 + h^Xl -f- 64XI + 65X1X2. The additional 145

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Figure 7.4. Contour map of bedding index calculated as standard deviation of gamma-ray response in the XVa Limestone in the Magyarstan training area. Contour interval is 0.02 units. Low values indicate massive intervals, high values indicate thin-bedded intervals. squared terms allow the surface to bend or change slope in each of the two coordinate directions. So, a second-degree trend surface might have the form of a dome, basin, or saddle. More complicated forms are possible by using additional terms of higher powers, such as Xf, X^, or combinations of powers of the two coordinates, such as XJX2' Once the trend surface coefficients have been estimated from the data, the trend equation can be quickly evaluated for any set of geographic coordinates. Therefore, it is a simple matter to evaluate the equation at all the grid node locations where our contouring program has made an estimate of the surface. We will then have two grids of values, one containing estimates 146

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6455.

1740.

1745.

1750.

1755.

1760.

1765.

1770.

1775.

Figure 7.5. Second-degree trend surface fitted to elevation of top of the XVa Limestone in 83 wells in the Magyarstan training area. Contours given in meters below sea level. Contour interval is 25 m.

of the surface itself, and the other containing the trend. By subtracting the two grids, we produce a grid of residuals. All three of these grids can be displayed as contour maps. Figure 7.5 is a second-degree trend surface of the subsea elevation of the top of the XVa unit, the surface shown in Figure 7.1. The pronounced structural dip to the southeast is readily apparent. Figure 7.6 is a map of residuals produced by subtracting the grid of Figure 7.5 from that of Figure 7.1. The regional dip has been removed, leaving local areas that are relatively "high" with respect to the trend distinguished from areas that are relatively "low." 147

Computing Risk for Oil Prospects — Chapter 7 6490.

6485.

6480.

6475.

6470.

6465.

6460.

6455. 1740.

1745.

1750.

1755.

1760.

1765.

1770.

1775.

Figure 7.6. Residuals from second-degree trend surface (Fig. 7.5) fitted to elevation of top of the XVa Limestone in the Magyarstan training area. Residuals are found by subtracting trend surface grid from structural grid used to construct Figure 7.1. Contour interval is 10 m. Often a structural contour map, such as Figure 7.1, will show no traces of structural closure, although "noses" and other plunging features may be evident as deflections in the contour lines. If the regional trend is removed, these deflections may be transformed into closed features. These closed contours are easier to discern than are the deflections from which they are derived, and in some circumstances they even can be interpreted as originally closed features that have been tilted by regional movement. However, it's not necessary to attach a genetic significance to a trend surface and its residuals; it is sufficient to regard them as arbitrarily transformed versions of the original map that may be useful in the search for oil and gas. 148

Mapping Properties and Uncertainties Trend and residual maps can be prepared for any geological variable, not just elevations of subsurface stratigraphic units. Residuals from an isopach map show locations where the mapped unit is unusually thick or thin, and trend maps of porosity and other highly erratic properties may be easier to interpret than contour maps of the raw data. Just as there are many geologic properties that could be mapped, there are any number of trend surface and residual maps that could be made. The critical issue is whether such maps are useful in delineating prospective areas.

GEOSTATISTICS IN RISK ASSESSMENT Conventional contouring programs may produce acceptable maps of a geological surface, in the sense that the appearance of the maps meets our expectations. Unfortunately, there is no way to test the overall "goodness" of a conventionally produced contour map, nor is there any way to delineate areas on a map that may be more uncertain or less reliable than other areas. However, it seems reasonable that predictions about a subsurface unit should be better where the unit is penetrated by many wells than where it is relatively undrilled. Because conventional contouring algorithms are empirically based, there is no underlying theory that can be used to produce measures of reliability or uncertainty. Geostatistics is a branch of applied statistics that treats the variation of properties through space, in both two and three dimensions. As you might expect, the concept of autocorrelation plays a key role in this field. Geostatisticians have developed methods for estimating the spatial autocorrelation of a surface from scattered observations such as exploratory drill holes, and then using models of the autocorrelation to construct optimal estimates of the surface. These estimates can be displayed in the form of contour maps. Conventional contouring programs calculate estimates of a surface as distance-weighted averages of nearby points. The weights are assigned arbitrarily, usually as the inverse of the squared distances between the location where the estimate is being made and the control points being used. In contrast, geostatistical estimators are "custom-made" for the degree of autocorrelation in the actual surface, and the specific arrangement of points around every location being estimated. The geostatistical literature is vast and often couched in nomenclature and mathematics of fearsome complexity; Isaaks and Srivastava (1989) provide a modern, comparatively lucid introduction to the topic, while Cressie (1991) examines the field from the viewpoint of a statistician. A classical treatment with applications to mining is given by Journel and Huijbregts (1978). Deutsch and Journel (1992) provide software. The seminal works by Matheron (1962, 1965) are reserved for the mathematically masochistic. 149

Computing Risk for Oil Prospects — Chapter 7

T h e Semivariance The spatial autocorrelation of a surface is measured by a special statistic called the semivariance, which is simply the average squared difference between pairs of points that are separated by a constant distance. This statistic can be calculated for various distances and the results plotted as a graph of distance versus semivariance. Such a graph is called a semivariogram. There should be no difference between a measurement made on a surface at a point and a second measurement made at the same point (that is, between two measurements that are separated by a distance of zero), so the semivariance for a surface over a distance of zero should be zero. In other words, the semivariogram plot should go through the origin. Since values at locations that are separated by small distances should be similar, their average difference will be small and the semivariance of a surface over short distances should be low. If more distant locations on the surface are compared, there will be greater differences between values on the surface, and the semivariance over large distances will be greater. However, the differences between pairs of points will not continually increase with distance, but instead will become a more or less constant value. This upper limit will be numerically equal to the variance of all measurements made on the surface, without considering their spatial locations. A simple equation for semivariance is

^^^

2^

where xi is the value on a surface at some point i and xi^h is the value at another point located a distance h away. There are n such points, so there are 2n possible pairs of comparisons. The semivariance corresponding to a distance h between the pairs of points is indicated by 7/1. However, it is obvious that calculating numerical values for 7/^ requires that we have a set of points on the surface that are separated by a constant distance. If we have such a traverse of equally spaced points (such as a seismic hne or row of development wells in a large field), we can easily calculate the semivariance for multiples of the spacing between observations (Fig. 7.7). The multiples of spacing are referred to as lags, a terminology borrowed from time series analysis. Often, the X axis of the semivariogram is given in terms of lags, or multiples of the basic distance between pairs of points, rather than in actual distances. Figure 7.8 is a plot of average porosities measured in a series of production wells drilled into a shallow sandstone reservoir in eastern Kansas. The field is drilled with a 10-acre spacing, so the 20-well traverse extends for 2.5 mi and the unit distance between wells is about 660 ft. The first step 150

Mapping Properties and Uncertainties

h= 1

+ + + + + + + + + + + + + + + + + +

\J\JKJKJ\JKJ\J\J h=2

+ + + + + + + + + + + + + + + + + +

Figure 7.7. Straight traverse along a row of equally spaced wells showing pairwise comparisons between wells. For h = 1, every well is compared with its neighbors; for h = 2, every well is compared to every other well; and for h = 3, comparisons are between wells separated by two intervening wells. in calculating the semivariance is to find the squared differences between all possible pairs of points for successive lags. These differences are shown in Figure 7.9, plotted against lags. The actual distances can be found by multiplying the lag number by 660 ft, the basic distance between wells. (Although there appear to be fewer measurements at small lags, this is because the plotted points overlap.) Squared differences are shown for lags from 1 through 10, a distance of 1 mi. Next, averages are found for each of the lag distances. In Figure 7.10, the averages are shown by heavy lines and "box-and-whisker" plots show the distributions of the individual squared differences for each lag. It's apparent that both the averages and the scatter in the squared differences increase up to about lag 6 and then remain more or less constant. Figure 7.11 shows the completed semivariogram, without the intermediate values used in its calculation. This is referred to as an experimental semivariogram because it is based on a sample of observations rather than on theoretical considerations. The semivariogram will be used for calculation of weights in the geostatistical contouring algorithm, which goes by the name of kriging. (The name honors a prominent South African mining engineer, Danie Krige, who introduced statistical methods into mine evaluation.) In kriging, the semivariance may be required for any distance, not just those corresponding to the discrete lags of the experimental semivariogram, so we must create a continuous model of the semivariogram. This model semivariogram is an 151

Computing Risk for Oil Prospects — Chapter 7 25-

I

I

I

I

10 Observation

I

I

I

1

15

1

I

20

Figure 7.8. Profile showing clianges in average porosity along a traverse through 20 wells in an eastern Kansas oil field.

150-

§100-

i •8 (0 CO

-

J! 1

L '2

'3

' 4

' 5

' 6 Lag

' 7

' 8

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Figure 7.9. Squared differences between pairs of wells separated by 1 to 10 lags along traverse shown in Figure 7.8. Lag interval is 660 ft. 152

Mopping Properties and Uncertainties IOU~ •

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Figure 7.10. Distributions of squared differences between pairs of wells shown by "box-and-whisker" plots. For each lag, "boxes" enclose central 75% of squared differences, and "whiskers" extend to cover 90% of squared differences. Line connects average squared differences, which are equivalent to the semivariance. Lag interval is 660 ft. idealized representation of the spatial continuity of the surface. Because the model has the form of an equation, it can be evaluated for any required distance. Geostatisticians have developed a library of model equations that are especially useful in representing the experimental semivariograms that are found commonly in petroleum exploration. Some of these are shown in Figure 7.12; all begin at the origin and rise smoothly to an upper limit called the sill. They reach this limit at a distance called the range. The values at points on the surface which are closer to each other than the range are related; if the points are separated by distances greater than the range they are statistically independent of one another. The experimental semivariogram may not be so well-behaved as the one shown in Figure 7.11, especially if the number of wells is hmited. The semivariance may change erratically at successive lags, making the task of fitting a model seem hopeless. In such circumstances, the best that can 153

Computing Risk for Oil Prospects — Chapter 7

Figure 7.11. Experimental semivariogram for average porosity along traverse through eastern Kansas oil field. Lag interval is 660 ft.

be done is to define a model that approximately forms a convex hull or envelope around most of the semivariances, and which reflects as closely as possible those nearest the origin. Semivariogram models cannot be fitted readily by least squares or similar techniques, because the objective is not to find a model that goes through the middle of the scatter of experimental values, but rather one that encloses most of the values. This will result in a conservative estimate of spatial continuity and conservative estimates of the amount of uncertainty about the surface. The simplest model for the semivariogram is a straight line, called a linear model (Fig. 7.12a). It begins at the origin and rises at a constant rate until it reaches the sill; from that point it is a constant. In equation form, 7/i = ah

forh
Ih = ^ 0

{ox h> a

where CTQ is the variance of all of the points measured on the surface. The model is fitted by trial and error, adjusting the slope, a, of a straight line 154

Mapping Properties and Uncertainties

\ - < - '

u

O

o

o

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Distance, h Figure 7.12. Popular models for semivariance (theoretical semivariograms). Range is indicated by a, value of sill by G\. (a) Linear semivariogram with a constant slope up the sill, (b) Spherical semivariogram. (c) Exponential semivariogram. (d) Gaussian semivariogram. through the origin until it just encloses the majority of the semivariances that are near the origin. The most widely used model for the semivariogram is the spherical model (Fig. 7.12b) which is very straight near the origin but bends and merges smoothly with the horizontal line that represents the sill. The equation for the spherical model is 2/3/1

h^\

7/i = ^0 loxh> a To fit a spherical model, all that is necessary is to define a range, o, for the semivariogram, and then to calculate 7/^ for various values of h. As a first 155

Computing Risk for Oil Prospects — Chapter 7 approximation, the sill may be set equal to the variance of the observations. It may be necessary to experiment with different levels for the sill and values of a in order to find a semivariogram that is acceptable. Sometimes the experimental semivariogram rises faster at the origin than does the spherical model. In such instances, an exponential model (Fig. 7.12c) may provide a better fit. The equation for this model is 7;, =al{l-

e"^/^)

7/1 = o^l

for / i < a for h > a

Again, only the range, a, and the sill, aQ, are needed to fit this model. The range has a slightly different meaning in this model, because the exponential curve approaches the value of the sill asymptotically and so never quite becomes equal. To avoid this problem, the range is defined based on a nominal distance at which the semivariance "approaches closely" to the sill. A practical rule is to set the range as y/Sa, where a is the nominal distance at which the semivariance reaches 95% of the value of the sill. Experimentation with different sills and values of a may be needed to fit the model to the data (Olea, 1991). Geologic properties that are exceptionally smooth and continuous will produce an experimental semivariogram that rises slowly at low lags, then more rapidly with increasing distance. Such behavior can be represented by a Gaussian model (Fig. 7.12d), which is parabolic in form near the origin. The Gaussian model equation is 7/1

= al{l-

7/i =

(JQ

e-^'^'/a^)

for / i < a for h>

a

Like the exponential semivariogram, the Gaussian model approaches the sill asymptotically. The same procedure can be used to select an appropriate value for the range. Other more complicated models are found in the geostatistical literature, as well as models formed by combining simpler forms; these we leave to the geostatistical sophisticate. We will only mention that sometimes the experimental semivariogram does not seem to go through the origin, either because of fluctuations in the surface over distances shorter than the spacing between wells, or because repeated measurements at a location may result in different values (properties such as porosity may exhibit this behavior). This is referred to as a nugget effect, and simply means that the semivariogram starts at a value greater than zero. The nugget effect can be included in any of the semivariogram models by adding a constant. 156

Mapping Properties and Uncertainties

50-

jz

40-

0

o

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Sill = 26

o

on ^"

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O

/o

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0-c

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' 2

' 3

' 4

' 5

' €;

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' 7 ' 8 ' 9

'10 '

Figure 7.13. Spherical semivariogram model with a sill of 26 and a range of 6 fitted to experimental semivariogram of Figure 7.11. Figure 7.13 shows a spherical model fitted to the experimental semivariogram of Figure 7.11. The model has a range of 6 lags, equal to a distance of almost 4000 ft. The sill has been set to 26, which is slightly greater than the variance of the 20 observations along the line of wells. Although calculation of the experimental semivariogram ideally involves use of observations equally spaced along a straight traverse across the surface, conditions are seldom ideal. In some circumstances it may be possible to devise a more or less straight path from well to well across a densely drilled area, constructing a traverse along which the observations are approximately equally spaced, and so compute a semivariogram in the classical manner. However, most people would prefer a computerized procedure for creating the semivariogram automatically, regardless of the locations and spacings between the wells. Such a procedure can be developed using the directional search capabilities of a contouring program, and is included in the RISKMAP software. An automated semivariogram calculation procedure requires that a number of approximations be made. First, since we are unlikely to find 157

Computing Risk for Oil Prospects — Chapter 7 a single, straight traverse of points in the data set, the program uses the octant search procedure to locate pairs of points that lie within the same broad, general orientation. For example, the program might find all pairs of points that are generally aligned north-by-northwest; the compass orientations of the lines connecting such pairs could vary from N to N45°E (or S to S45*^W). For the purpose of calculating a semivariogram, all of these pairs of points are considered to reflect the variation in the surface along a specific, average orientation. Similarly, the pairs of points that are found will not all be separated by the same distance; indeed, except for the effect that may be caused by a minimum allowable well spacing, in a typical set of wells we expect to see a continuous distribution of the distances between pairs of wells. This continuous range of distances is divided into discrete intervals, and all pairs of wells which fall within an interval are considered to be separated by a constant distance. It is as though we have divided the space around each well into a pattern of bins, and we pretend that any other well that falls into a bin is located at the exact center of the bin. In this way, we achieve a set of equidistant pairs of points whose squared differences can be averaged to yield an estimate of the semi variance. Of course, we've introduced inaccuracies into the estimation because the pairs of points are not really located along straight lines at equal intervals, but we hope that this uncertainty is overwhelmed by the very large number of observations that can be included if we use these approximations. Figure 7.14 shows the experimental semivariogram for thickness of the XVa Limestone as calculated by the automated procedure in RISKMAP. It is necessary to specify the radius of the bin; in this example the radius of a bin has been set so the effective distance for one lag is 1 km. The program automatically sets the radial segments at successive 45° intervals. Semivariances can be calculated for any of these preset orientations; or if there are no significant differences with orientation, they can be averaged to yield an omnidirectional semivariogram. Thickness does not show a pronounced grain, so an omnidirectional semivariogram has been calculated and is plotted in the figure. The program also will fit a semivariogram model based upon specified parameters. In Figure 7.14, a spherical model has been fitted, after some experimentation, using a sill of 21 m^ and a range of 13 km. This model appears to give a good approximation to the experimental semivariogram over a distance of about 10 km or so. Since the influence of the nearest wells is most critical, and at most locations the nearest wells will be within 10 km of the grid node being estimated, this model seems adequate.

158

Mapping Properties and Uncertainties 3025Sill = 21 o

o

o o

"~T—

15 Distance, km

—}—

20

25

Figure 7.14. Spherical semivariogram model fitted to thickness of XVa Limestone in 83 wells in the Magyarstan training area using modeling procedure in RISKMAP. The sill is 21 m^ and the range is 13 km.

Kriging The purpose of determining a model for the semivariogram is to provide information on the spatial rate of change to a procedure for estimating the form of the surface. The estimation procedure is kriging, which is a form of weighted averaging just like the procedures used in ordinary contour mapping. We can use kriging to estimate values of a surface at the nodes of a regular grid across a map, and then use contour-drawing procedures identical to those included in ordinary mapping packages to construct a contour map. The difference is that the map produced by kriging will be based on a weighting function that is tailor-made for every location in the map, and which varies with the distances and arrangements of the wells that are used in each estimate. The weights are found from the semivariances that correspond to the distances between each of the wells used in the estimate, and the distances between these wells and the point where the estimate is required. These semivariances are entered into a set of simultaneous equations, one for each well used in the estimation process. The set of equations is solved to yield the weights that are applied to each of the wells when their values are averaged to form the estimate. Actually, it is 159

Computing Risk for Oil Prospects — Chapter 7 necessary to include another equation in the set, which acts to constrain the weights so that they sum to 1.0; otherwise, the surface may be consistently biased. The kriging algorithm implemented in RISKMAP searches for the nearest well within each of the eight octants around the grid node being estimated. Therefore, a set of nine simultaneous equations must be solved and evaluated for every node in the surface grid; as you can imagine, this is computationally more demanding than ordinary contour mapping. Davis (1986) provides a simple numerical example of kriging that shows how the values from the semivariogram model are inserted into the kriging equations, how the set of equations are solved, and how the resulting weights are used to estimate the value at the location being evaluated. We will not repeat this development here, but simply show the basic equations. The simplest set of kriging equations that can be solved consists of four simultaneous equations and is based on the semivariances corresponding to the distances between three wells. The expansion of this equation set to eight wells is straightforward. ^ i 7 i i + ^2712 + ^^3713 + A = 7ip Wi^n

-h ^t^2722 -h ^/^3723 + A = 72p

w;i7i3 H- t(;2723 + ^^3733 + A = 73p

The unknown wis are the weighting coefficients to be determined, and the 7ij's are the semivariances taken from the semivariogram model. For example, 713 is the semivariance corresponding to the distance between well 1 and well 3, and 72^ is the semivariance corresponding to the distance between well 2 and the point being evaluated. The set of equations can be rewritten and set in matrix form for solution: 711

712

713

1"

712

722

723

1

7l3

1

723

1

1

733

0

1

.

'Wl' tV2 Ws

. A .

^

•71P 72p 73p

. 1

Once we've found the weights, they are used to estimate the value of the surface at locations p. We simply multiply the value Yi at each of the wells by its corresponding weight and add the products: Yp = wiYi -f ^V2Y2 + ivsYs However, we've also estimated a fourth coefficient. A, which can be used, along with the semivariances, to estimate the uncertainty in the kriged 160

Mapping Properties and Uncertainties value. This is done by weighting each of the semi variances for the distances to the point, and adding the products to A:

This value is called the error variance; conventionally, we take its square root so its units are the same as the units of measurement of the variable being mapped. Then, it is the standard deviation of the error (or simply, standard error) in our kriged estimate. For every point on the grid, we have calculated two numbers: the estimated value of the surface itself, and the standard error of that estimate. Both numerical grids can be displayed as contour maps. Figure 7.15 is a contour map of the kriged thickness of the XVa Limestone measured in the 83 wells of the Magyarstan training area data set. The mapping conventions (scale, contour interval, symbols, etc.) are the same as used in the conventionally produced contour map of thickness shown in Figure 7.2. Careful comparison of the two maps will show that near the data points the contours have the same values, but the maps convey distinctly different general appearances. Which is best for exploration purposes? Such a question can be answered only by drilling, but the map produced by kriging does have one distinct advantage that is especially significant for risk analysis. Figure 7.16 is the second map produced by kriging the thickness data; this map shows the standard error of the estimated surface. The standard error is given in the same units as the original map, or in this example, in meters of thickness. The values on the contour map give the width of one standard deviation about the estimated value of the surface. Near wells, the uncertainty about the thickness of the XVa Limestone is very low (at the exact locations of the wells we know the true thickness, so the standard error is 0.0). At a great distance from the nearest wells, such as a location in the center of the map, the uncertainty is relatively large and the standard error exceeds 3 m. If we assume that the kriging errors of estimation follow a normal distribution, we can make a probabilistic interpretation of the standard error map. Imagine that it would be possible to drill another set of 83 wells in this area, with a somewhat different configuration of locations. If we calculated a semivariogram from these new wells, it should be very similar to our original, and if we drew a new map using kriging, it should also be very similar. In fact, if we were to repeat this imaginary exercise many times, we would expect that at any location, 68 out of 100 times we would estimate a value by kriging that would lie within plus or minus one standard error of the original kriging estimate. Ninety-six times out 161

Computing Risk for Oil Prospects — Chapter 7

6490.

6485.

6480.

6475.

6470.

6465.

6460.

6455. 1740.

1745.

1750.

1755.

1760.

1765.

1770.

1775.

Figure 7.15. Isopach map produced by kriging thickness of the XVa Limestone in the Magyarstan training area. Contour interval is 2 m. Compare map to conventionally produced contour map in Figure 7.2. of 100, we would expect the estimates at a location to fall within plus or minus twice the standard error of the original kriging estimate. These expectations are derived from the properties of the normal distribution, as can be seen in Figure 7.17. A common (but arbitrary) choice for probability limits is 90%; if we set hmits of (±1.63 x standard error) around the kriging estimate, we have defined an interval that should capture the true value of the mapped surface 90% of the time. That is, the probability is only 10% that the true value at that location lies outside the specified limits. There is a 5% probability that the true value is below the lower limit, and a 5% probability that it is above the upper limit. 162

Mapping Properties and Uncertainties 6490.

6485.

6480.

6475. h

6470. r

6465.

6460.

6455. 1740.

1745.

1750.

1755.

1760.

1765.

1770.

1775.

Figure 7.16. Contour map showing magnitude in meters of one standard error in the isopach map of thickness of XVa Limestone in the Magyarstan training area. Contour interval is 0.5 m. Since the kriging operation has produced estimates of the thickness of the XVa Limestone at a regular array of locations across the map area, and estimates of the standard error at each of these locations, we can combine the two grids to create maps of the greatest likely and smallest likely thicknesses for the unit. Figure 7.18 shows a map obtained by multiplying the values in the grid of standard errors by 1.63 and then adding the result to the grid of kriged estimates of thickness. The map conventions are the same as in Figure 7.2. At the well locations, the greatest possible values of thickness are equal to the measured thicknesses in the wells, since the standard error is zero. Away from the wells, the possible thickness could be greater, because of the reduced amount of control. Figure 7.19 shows 163

Computing Risk for Oil Prospects — Chapter 7 Lower Limit

Standard Error

Kriging Estimate

Upper Limit

n

- 3 - 2 - 1 0 1 2 3 I I I I I I I I I I I I I I I I I I I I I Thickness, m 20 25 30 35 40 Figure 7.17. Distribution of error around a kriged estimate of thickness of XVa Limestone in the Magyarstan training area, where the estimated thickness is 30 m and the standard error is 3 m. Upper and lower Hmits are defined as (±1.63 X standard error), forming an interval from 25 m to 35 m which contains the true thickness with 90% probability.

the corresponding lower limits on thickness, produced by subtracting (1.63 X standard error) from the kriged thickness. Perhaps the easiest way to visualize the concept of the kriging estimate and its standard error is to examine a profile across the map. Figure 7.20 shows a plot of thickness of the XVa Limestone as estimated along row 10 of the map grid. This is an east-west line at the X2-axis coordinate 6464, which passes through two dry holes in the eastern part of the area. On the profile, the kriged estimates along this row are shown by circles, each one representing a column in the grid. Near the two wells on the profile, the upper and lower limits pinch together and become equal to the kriged estimate, which in turn is equal to the measured thickness in the wells. Between the wells and elsewhere along the profile, the upper and lower limits flare out to an extent determined by the semivariance of thickness and the configuration of wells near the profile line. We expect that the true thickness hes within this envelope. The probability is only 10% that the true thickness lies outside the envelope at any location. 164

Mapping Properties and Uncertainties

6490.

6485.

6480.

6475.

6470.

6465.

6460.

6455. 1740.

1745. 1750. 1755.

1760. 1765.

1770. 1775.

Figure 7.18. Upper limit of thickness of XVa Limestone in Magyarstan training area found by adding (1.63 X standard error) to thickness estimated by kriging. The probability is 5% that the true thickness exceeds this limit.

A Complication and a Way Out Occasionally, we will encounter a geological variable which perversely refuses to yield a well-behaved semivariogram such as that shown in Figure 7.14 for the thickness of the XVa Limestone. Instead, the semivariance will increase with increasing lag, showing every indication of continuing upward forever. No sill will be apparent in the plotted semivariogram. As an example, we can analyze the data on structural elevation of the XVa Limestone in the file TRAINWEL.DAT, and we will see that its semivariogram has these unfortunate properties (Fig. 7.21). This presents serious problems for kriging, because we cannot define either a sill or a range on 165

Computing Risk for Oil Prospects — Chapter 7 6490.

6485.

6480.

6475.

6470.

6465. h

6460. h

6455. 1740.

1745.

1750.

1755.

1760.

1765.

1770.

1775.

Figure 7.19. Lower limit of thickness of XVa Limestone in Magyarstan training area found by subtracting (1.63 X standard error) from thickness estimated by kriging. The probabiHty is 5% that the true thickness is less than this limit. such a semivariogram and we will be unable to apply any of the standard semivariogram models. The behavior implies that the influence of a point extends without limits, and that the uncertainty in estimation can increase without bounds. It also creates severe mathematical problems in the solution of the kriging equations. What is causing this condition? A basic assumption in ordinary kriging is that the geologic property is stationary; that is, its average value is approximately the same everywhere. Although the mapped surface fluctuates and may be rich in features, there is no persistent tendency for it to rise or fall in value. This is not the case with structural elevation in the Magyarstan area, as we can see in 166

Mapping Properties and

~1 5

.

1 ..

10

..J, . . . . 15

.|.,,. 20

•• 1

25

•

Uncertainties

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30

1

35

Grid Column Figure 7.20. Profile along row 10 of the grids used to draw Figures 7.15, 7.18, and 7.19. The line of section runs east-west at coordinate 6464 of the X2 (north-south) axis and passes through the location of two dry holes. Kriging estimates at grid nodes are indicated by open circles.

20000-

15000H CM

E o

§ 10000H

•c

E

0) CO

o o

5000 o o 0-^>-o-ii. o o

5

—I—

~T—

10

15

20

25

Distance, km Figure 7.21. Experimental semivariogram for structural elevation of the top of the XVa Limestone in the Magyarstan training area. 167

Computing Risk for Oil Prospects — Chapter 7 1000-

7504

Sill = 750

o

Range = 10; -tT— 10 15 Distance, km

o

20

25

Figure 7.22. Experimental semivariogram for second-degree trend residuals from the top of the XVa Limestone in the Magyarstan training area. A spherical semivariogram model with a sill of 750 m^ and a range of 10 km has been fitted. Figure 7.1. This tendency for a persistent change in structural elevation was emphatically shown by trend surface analysis in Figure 7.5. A geological property that exhibits a pronounced trend is said to be nonstationary, and it cannot be correctly mapped by ordinary kriging techniques. (In fact, there will be subtle distortions of the mapped surface regardless of what mapping procedure is used; positive features will tend to be shifted up-slope and negative features shifted down-slope.) If the structural surface is leveled by removing the trend (in effect, tilting the surface back to a horizontal aspect) as was done in Figure 7.6, we will find that the semivariogram of the trend residuals exhibits more acceptable behavior (Fig. 7.22). This suggests that we could solve the problem of nonstationarity by removing a trend from the data and computing the semivariogram of the stationary residuals. Kriging could be used to map the residuals, and the resulting residual map added to the trend surface map to recreate a map of the original surface. This effectively splits the mapping problem into two parts: the fitting of a deterministic nonstationary trend and the estimation 168

Mapping Properties and Uncertainties of the stationary residuals by kriging. Although such a procedure will work in many instances, it is considered inelegant by geostatistical practitioners who have devised a number of alternatives for estimating nonstationary properties and assessing the estimation errors. Perhaps the simplest of these is called "universal kriging," which combines the removal of nonstationarity and the estimation of the residuals into a single step. RISKMAP contains options that will perform universal kriging of nonstationary geologic variables. The set of kriging equations is expanded by adding two or more coefficients that must be estimated along with the kriging weights. The equations necessary to determine these additional coefficients contain powers and cross products of the geographic coordinates of the data points in the neighborhood around the location being estimated. That is, we incorporate the equation of a trend surface directly into the kriging equation and solve both sets of coefficients simultaneously. However, what we are finding is not really a trend surface in the sense that we have used the term earlier, because it does not extend throughout the map area but rather is confined to the neighborhood of the location being estimated. In fact, the trend surface is not actually computed at all; by calculating the kriging weights at the same time as the trend coefficients, the kriging weights are automatically adjusted for the presence of the trend. The universal kriging equations have the form 711

721

731

741

751

712

722

732

742

752

713

723

733

743

753

714

724

734

744

754

715

725

735

745

755

1

1

1

1

1

^11

-^12

-^13

Xi4

^15

^21

-^22

-^23

^24

^25

1 1 1 1 1 0 0 0

^11

-^21 '

'Wi'

"7ip

^12

-^22

W2

72p

-^13

-^23

ws

73p

Xi4

^24

W4

74p

-^15

-^25

W^

75p

0 0 0

0 0 0 .

A

1

^1

^Ip

L62J

-^2p

which is fearsome in appearance but is a simple extension of the ordinary kriging equations given earlier. Davis (1986) provides a relatively simple numerical example of universal kriging. Geostatisticians refer to the fitted polynomial function in universal kriging as a "drift" rather than a "trend" to distinguish it from the global function of trend surface analysis. After determining all of the coefficients, the surface is estimated by Yp = wiYi + W2Y2 + tv^Ys + W4Y4 4- wsYs The error variance (and from it, the kriging standard error) are found as before: si = wijip

+ ^t;272p + ^373p + ^474p + ^575p + A

169

Computing Risk for Oil Prospects — Chapter 7 The drift itself could be estimated and shown in the form of a contour map, but this is seldom done as it has no readily interpretable meaning. It is simply a mathematical construct useful to create local stationarity in the mapped property. Since the kriging weights and the drift coefficients are determined simultaneously, the kriging weights reflect the influence of the drift calculation. Because values of the surface at specific locations are estimated by an equation whose kriging weights reflect the drift, the effect of the drift is included in the final map grid. Figure 7.23 is a kriged map of structural elevation in the Magyarstan training area produced by universal kriging using RISKMAP. A first-order drift has been specified (RISKMAP allows choice of a first- or second-order polynomial drift). The accompanying map of the standard error of the estimated structural elevation is shown in Figure 7.24. To produce the kriged map, the user must provide the parameters of an appropriate model of the semivariogram that describes the spatial structure of the drift residuals, and this brings up an annoying problem. In order to apply universal kriging, we must provide the kriging program with several parameters, including the order of the drift and the parameters of the semivariogram of the residuals. However, the semivariogram of the residuals can be modeled only from the residuals, which must be determined by removing the drift from the variable to be mapped. But to remove the drift we must specify the neighborhood size (the range of the semivariogram) and order of the drift, which are part of the very semivariogram model that we seek! Breaking out of this circular impasse requires experimentation, first choosing an arbitrary order for the drift, arbitrarily selecting a model (linear, spherical, etc.) for the semivariogram, and setting a "convenient" neighborhood size. Using these assumed parameters, we then estimate the drift, find the residuals, and calculate the semivariogram of the residuals. If we've chosen appropriate parameters, the experimental semivariogram and the model semivariogram will coincide. If they don't, we adjust one or more of the parameters (neighborhood size, model, or order of drift) and try again. Since the drift is an arbitrary construct, we may expect that there will be several combinations that produce an acceptable fit. In general, it is desirable to have a large neighborhood (we are less apt to encounter locations where we cannot estimate the surface because there are too few points inside the neighborhood), so it is often better to specify a first-order drift and a large neighborhood rather than a second-order drift and a small neighborhood that gives equally good results. In our example, we can obtain very good initial estimates for the semivariogram parameters from the semivariogram of trend residuals shown in Figure 7.22. This suggests that a spherical model with a range of 10 km may be appropriate, and these are the parameters used in calculating the 170

Mapping Properties and Uncertainties

6490.

6485.

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6475.

6470.

6465.

6460.

6455. 1740.

1745.

1750.

1755.

1760.

1765.

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1775.

Figure 7.23. Structure contour map fitted by universal kriging to the top of the XVa Limestone in the Magyarstan training area. Contour interval is 25 m. Compare to conventional contour map in Figure 7.1. universal kriging map of Figure 7.23. Although the experimentation necessary to estimate the parameters of the semivariogram model is a nuisance, it is a necessary step if we wish to incorporate nonstationary geological properties into our analysis. As in other aspects of petroleum risk assessment, the time and effort spent in carefully establishing appropriate parameters and extracting relationships from data will be repaid in improved analyses and more reliable interpretations. Kriging can provide an important component in a petroleum riskassessment system. Suppose that in a certain play, thickness is believed to play a critical role in the localization of petroleum. Kriging can be used to map thickness, just as we have done using thickness data on the XVa 171

Computing Risk for Oil Prospects — Chapter 7 6490.

6485.

6480.

6475.

6470. b

6465.

6460.

6455. 1740.

1745.

1750.

1755.

1760.

1765.

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1775.

Figure 7.24. Contour map showing magnitude in meters of one standard error in the structural map of XVa Limestone in the Magyarstan training area. Contour interval is 5 m. Limestone, and to predict locations where the thickness seems to be sufficient for oil accumulation. By assuming that the uncertainty follows a normal distribution, the kriging standard error allows us to express our estimate of thickness in probabilistic terms, and to specify likely upper and lower limits on its value. It provides a quantitative measure of the reliabihty of our perception of the mapped property, and directly furnishes a probability distribution in a form suitable for input into subsequent risk analyses.

172

CHAPTER

8

Discriminating Between Discoveries and Dry Holes In earlier days of oil exploration, it was sometimes sufficient to map a single geologic property, such as structural elevation, in order to define a drillable prospect. In the intermountain basins of the Rockies or the coastal basins of California, for example, closed structures could be defined by planetable mapping of surface outcrops. Some of these structures, when drilled, yielded the enormous returns that fueled the growth of the petroleum industry in its formative years. Unfortunately, these large, conspicuous targets are long gone; delineating prospects today requires the consideration of many factors, both local and regional, that contribute to the entrapment of petroleum. The traditional approach to searching for subtle prospects has been to prepare separate maps of those properties that were thought to be important. These might include closure as revealed on a structural elevation map, variations in thickness as shown on isopach maps of key intervals, maps of trends in porosity, and maps of lithofacies or other lithologic measures. Mentally integrating the complex patterns shown on these maps is not an easy task, and geologists spent hours armed with colored pencils and tracing paper in the attempt to discern meaningful confluences. A large light table where several maps could be overlaid and considered simultaneously was an essential fixture in all exploration offices. With the development of computer contouring algorithms, the number of geological properties that could be considered in the delineation of a prospect or play was greatly increased, and was ultimately limited only by

Computing Risk for Oil Prospects — Chapter 8 the availability of data and not by the time required to prepare the maps. Unfortunately, this has made the problem of simultaneously considering the patterns on a thick stack of maps even more acute. The problem is complicated by the recognition that not all of the possible maps are equally important; some properties may be critical in the formation of a trap, and others may be of lesser importance or even not a factor at all. What is required is a way of weighting each of the geological properties in the order of its importance in defining a successful prospect, and then combining the weighted properties into a single map.

COMBINING GEOLOGICAL VARIABLES We can imagine an equation in which each geologic property is weighted by a coefficient and the products summed to yield a single value. An equation of this form is called a linear combination and appears, for example, in the computation of trend surfaces. The trend surface equation is a linear combination of powers and cross products of the geographic coordinates. In trend surface analysis, the coefficients are chosen so the sum of the squared deviations between the fitted trend and the observations is a minimum. This is called the least squares criterion and is especially convenient because weights that satisfy this criterion can be found by solving a simple set of normal equations. We can extend this principle to the situation where the variables in the linear combination are diff'erent geologic properties rather than geographic coordinates, and we can develop an equation that will combine these properties into a single new variable. However, in order to develop a suitable set of equations, we must specify some criterion that we wish to minimize. In trend surface analysis, we minimized the squared diff^erences between the values estimated by the linear combination (the trend equation) and the geologic property being mapped. This assured us that the fitted trend surface was the best possible in the sense that no other surface defined by the same number of coefficients in a linear equation would have a smaller variance in the residuals. That is, the scatter about the fitted surface would be as small as possible. In the present situation, however, we are not specifically concerned with an individual geologic property. Instead, we want to combine a number of geologic properties in a way that emphasizes the difference between areas containing oil and areas that are barren. That suggests we could define a criterion that involves minimizing the squared differences between a linear equation of the differences in average properties of producing and dry areas. Just as we used the least squares criterion to estimate trend surface coefficients, we can also use it to estimate the coefficients of an equation that predicts the difference between areas where oil has been discovered and 174

Discriminating Discoveries and Dry Holes areas that have been drilled but proved barren. Such an equation is called a discriminant function, and it might have the form Si — /3iXu + y^2^2i + /^3^3i + • • ' + l3mXmi +^i j wherc Si is a score or composite value at location i, Pi is the coefficient or weight assigned to geologic variable Xu, and €i is an error or residual. In order to estimate the unknown coefficients, we must divide our data into two sets; that from producing wells, and that from dry holes. For each property, we must then calculate the average in each of the two groups, and find their differences (subtracting dry holes from producers gives the same answer as the reverse). The discriminant function will weight most heavily those properties which provide the greatest distinction between the producing and dry groups. There are, however, two additional complications that must be considered. First, the units of measurement of the geological variables must be reflected in the magnitudes of the coefficients. Second, the amount of dispersion in the variables must also play a role, because if a property exhibits a large difference in average value between producing and dry areas but also exhibits an enormous scatter in values, the difference may well be meaningless. If we incorporate the variances of the different variables in calculating the discriminant function, we produce a function which simultaneously maximizes the differences between the producing wells and dry holes, while minimizing the scatter of the individual observations in the two groups around their means. Option 3 of RISKSTAT will compute a linear discriminant function between two sets of observations, one of which can be a collection of producing wells and the other a collection of dry holes. The program uses many of the mathematical routines that are used to fit trend surfaces, but different intermediate values must be calculated. A vector of differences between the means of the two groups is found after determining the mean of each variable in each group. The variances and covariances of the variables are calculated for the producing and dry groups as matrices, and these are then pooled, or combined in a manner that weights each according to the number of wells in each group. Finally, these quantities are entered into a matrix equation of the form

which is solved to yield the discriminant coefficients. Davis (1986) provides a demonstration of the calculations for a problem involving only two variables, so the operations can be performed by hand and all results can be shown graphically. Discriminant functions are discussed at length in most texts on multivariate statistics, as it is a popular methodology in many fields. Among the numerous general texts are those by Dillon 175

Computing Risk for Oil Prospects — Chapter 8 and Goldstein (1984), Gnanadesikan (1977), Harris (1975), Jackson (1991), Kendall (1980), and Rao (1952). The training area on the Zhardzhou Shelf of Magyarstan contains 83 drill holes, of which 18 are producers and the remainder dry exploratory holes. Summary statistics for the data are listed in Table 7.1, and the data themselves are given in the file TRAINWEL.DAT on the diskette. Figure 8.1 is a cross plot of thickness of the XVa Limestone versus the bedding index for the interval. Data from producing wells and dry holes are indicated on the diagram by symbols. It is obvious that the producing wells tend to be clustered in one part of the diagram, and that by considering both variables simultaneously we can more completely distinguish producing from dry holes than is possible considering the two variables individually. The distinction between producing wells and dry holes might be more pronounced if we considered additional variables, such as the shale ratio and trend surface residuals. It is not possible to visualize directly the fourdimensional space that would be defined if these properties were added to the two shown in Figure 8.1. We can, however, compute the discriminant function between producing wells and dry holes from these four variables and use the function to calculate scores for each observation. In effect, we are projecting each well onto the discriminant function. Figure 8.2 shows the results of discriminant analysis applied to the 83 holes in the training area. Intermediate values required to find the discriminant function are given in Table 8.1. The discriminant function is Discriminant score = 58.4 Shale ratio -{-49.1 Bedding index — 0.452 Thickness — 0.01 Trend residual. Figure 8.2 also shows several special points projected onto the discriminant function. The projection of the centroid of the producing wells gives a score of 14.3; projection of the dry hole centroid gives a score of 24.8. The discriminant index is a point exactly halfway between these two values and is the projection of the average of the two group centroids; it often is used as a threshold for assigning new observations to one or the other of the two groups. In this instance, the discriminant index is 19.6; if a prospect had a discriminant score less than this value, it would be assigned to the producing category because the score of the centroid of the producing group also is less than the index score. Conversely, if a prospect scored higher than 19.6, the new location would be classified with the dry holes.

Misclassification of Drill Holes Although a discriminant function might be the best possible linear combination of variables for the purpose of distinguishing two categories, the "best" may not be too good if the variables chosen do not reflect genuine diff^erences between the groups. We can determine the effectiveness of the discriminant function by examining the misclassification ratios, which are 176

Discriminating Discoveries and Dry Holes

g 0.28 T| o) 0.26-j c

+

V ^iVt>

++

"S 0.24 H m -«-+

i •

^

•

^

*

V

0.22 H 0.20 H 0.18

-f •~T"

30

"~1 ' T" 35 40 Thickness, m

45

Figure 8.1. Thickness versus bedding index of XVa Limestone in Magyarstan training area. Producing status of individual drill holes is indicated by well symbols. Large well symbols designate centroids (means) of the two groups. Dark shade in histograms indicates proportion of producing wells; light shade indicates proportion of dry holes. the proportions of the scores of the original observations that fall on the wrong side of the discriminant index, and hence are classified as belonging to an incorrect group. Table 8.2 gives the numbers of drill holes predicted to fall into the producing class and dry class among the 83 wells in the training set. In this example, the classification by the discriminant function is quite good; only 10% of the holes are misclassified. Statistical tests of the significance of a discriminant function are discussed in many texts on multivariate statistics; a good summary with comments on diff'erent procedures used in popular computer programs is given by Marascuilo and Levin (1983). Anderson (1984) provides an exhaustive discussion of the subject. Determining relative contributions of individual 177

Computing Risk for Oil Prospects — Chapter 8 10

15

35

20 25 Discriminant score

Figure 8.2. Histogram of discriminant scores of exploratory holes in Magyarstan training area. Group centroids are indicated by well symbols, and d.i. is the discriminant index. Dark shade in histogram indicates producing wells and light shade indicates dry holes. Table 8 . 1 . Statistical terms needed to calculate discriminant function based on four geological variables measured in 83 holes in Magyarstan training area.

Thickness Shale ratio Bedding index Trend residual

Dry hole mean

Producing mean

Difference

37.26 0.483 0.274 —5.48

40.73 0.361 0.243 19.78

-3.47 0.122 0.031 -25.26

Variance-Covariance Matrix / 17.7 1 0.0637 1 0.0217 \ 25.6

178

0.0637 0.0026 -0.000044 -0.0617

0.0217 -0.000044 0.00087 -0.0811

25.6 -0.0617 -0.081 611

\ j 1 /

Discriminating Discoveries and Dry Holes Table 8.2. Misclassification table for four-variable discriminant function applied to 83 holes in Magyarstan training area. Upper tabulation gives frequencies, lower tabulation gives proportions. Predicted Dry holes Producers Actual totals Predicted Dry holes Producers Actual totals

Actual Dry holes Producers

Predicted totals

1 17 18

63 20 83

Actual Dry holes Producers

Predicted totals

62 3 65

95.4% 4.6% 100.0%

5.6% 94.4% 100.0%

75.9% 24.1% 100.0%

variables to the discrimination is somewhat complicated for reasons discussed by Davis (1986). Although most programs for multivariate statistics have procedures for selecting the most effective variables for discrimination, it often is simplest to run repeated analyses using different combinations of variables and see which does best. For our purposes in risk assessment, we have limited interest in tests of significance or in seeking a statistically optimal combination of geologic variables. What we require is a set of geologic properties that we believe (or hope!) will provide a distinction between producing and dry localities and whose values can be estimated at undrilled prospect sites by quantitative procedures such as kriging. It also should be pointed out that geological variables differ in one critical respect from ordinary variables that often are analyzed by discriminant functions, such as sociological or anthropological measurements. Geological properties have spatial characteristics that other variables lack, and this may have a profound effect on any probabilistic interpretation made from them. The estimation of probabilities in the misclassification table (Table 8.2) assumes that all observations are independent, or at least dependent to an equal degree. This assumption is reasonable if producing wells are not significantly more closely spaced than are dry holes. Then, the proportion of producing wells can be interpreted as an estimate of the area occupied by production, and for a spatially distributed variable, the probability of occurrence is based on the proportion of area occupied by the variable. In the Magyarstan training area, producing wells are not so closely spaced that this is troublesome, and our probability estimates can be based simply on counts of wells. 179

Computing Risk for Oil Prospects — Chapter 8

ioon

10

I

I

12 14 16 18 20 22 24 Discriminant score

I

I

I

I

I

26 28 30 32 34

Figure 8.3. Empirical distribution having seven overlapping intervals of probability versus discriminant score for Magyarstan training area. Curve is drawn through center of each overlapping interval (see Table 8.3).

The Conditional Probability of Success The misclassification ratio can be thought of as a two-step function of the discriminant scores; one step includes all values lower than the discriminant index and the second includes all values greater than the index. However, it seems reasonable that producing wells whose discriminant scores are close to the centroid of the producing group should have a higher probability of being correctly classified than wells whose scores are closer to the discriminant index. That is, the probability of correct classification should vary with the magnitude of the discriminant score. The probability mapping options of RISKMAP incorporate this commonsense idea and calculate an empirical function that relates probabilities of correct classification and the discriminant scores for wells that are mapped. The empirical probability function is determined by dividing the distance between the centroids of the dry and producing groups into an arbitrary number of intervals and counting the number of correctly classified and misclassified producing wells whose scores fall into each interval. The segments are chosen so they overlap in order to increase the total number of observations that fall into each interval. The resulting distribution will be smoother and more nearly approximate a continuous function (Fig. 8.3). The distribution provides the conditional probability that an exploratory hole will be a producer, given a specific discriminant function score. The complement is the probability that the hole will be dry. 180

Discriminating Discoveries and Dry Holes Table 8.3. Empirical distribution of the probability of a producing well given a discriminant score for Magyarstan training area. Score interval Start End 13.6 7.1 10.3 16.8 13.6 20.0 16.8 23.3 20.0 26.5 23.3 29.7 26.5 32.9 Unconditional probabiUty

Counts Producers Dry holes

Conditional probability Producers Dry holes

7 16 10 1 1 0 0

0 0 6 23 42 36 17

1.000 1.000 0.625 0.042 0.023 0.000 0.000

0.000 0.000 0.375 0.958 0.977 1.000 1.000

18

65

0.217

0.783

In this example from the Magyarstan training area, the minimum discriminant score is 7.11 and the maximum is 32.94, giving a difference of 25.83. If we divide this difference by {k + 1), we will get the width of half of each of k overlapping intervals, or 4.3 for A; = 7 intervals. Each of the seven intervals will have a width of 8.6 and will overlap its neighboring intervals by one-half, as shown in Figure 8.3. The number of producers and their proportions, and the resulting conditional probabilities are given in Table 8.3.

ASSESSING A N E W AREA Of course we have no interest in classifying holes that have been drilled and whose status we already know. We are concerned with these historical results merely as a way to estimate both the coefficients of the discriminant function and the discovery probabilities associated with discriminant scores. Once these are in hand, our attention can turn to our real objective, which is to estimate an appropriate discriminant score for an undrilled prospect, and from this score to estimate the prospect's probability of success. The prospects we wish to evaluate may be part of the same play as the drill holes used to develop the discriminant function, in which case we can be confident that the function is appropriate for assessment purposes. In other circumstances, because we must evaluate prospects in a relatively virgin play, we may need to transfer a discriminant function from a more mature area or interval where data are available for estimating probabilities. This introduces an additional element of uncertainty that we cannot initially 181

Computing Risk for Oil Prospects — Chapter 8 assess, but as we shall see later, our initial estimates of the probability of success can be updated as our knowledge of the new play increases. Since no exploratory hole has yet been drilled on a prospect, we do not have direct knowledge of the geologic properties needed to calculate the prospect's discriminant score. However, we can estimate values of these properties using geostatistical techniques described in the last chapter. All of the geological variables used in the discriminant function are mappable {i.e., they are spatially autocorrelated, single-valued, and continuous), and since the discriminant function is a linear combination of mappable variables, the discriminant score also is a mappable variable.

Combining Individual Geologic Maps There are two alternative approaches to estimating the discriminant score at an undrilled location. The most obvious is to produce maps of each geological variable used in the discriminant function, then take estimates from these maps and enter them into the function and calculate the discriminant score. This necessitates modeling the semivariogram and solving the kriging equations for all of the geological variables. Because kriging would be used to map each geological variable throughout the area of interest, not just at the prospect locality, a series of structural, isopach, and other geological maps would be produced. Each map would be accompanied by a kriging error map expressing the uncertainty in the estimate of the variable throughout the map area. The difficulties with this approach arise when we attempt to assess the uncertainty or error distribution of the calculated discriminant function. The expected, or average, value of the discriminant score at a specific prospect location will be produced if we simply enter values taken from the individual kriged maps of each geological property. Although the uncertainty associated with this calculated score must be related in some way to the standard errors of the individual kriged variables, combining these into a single error distribution for the discriminant score is not straightforward. If we were certain that the geological variables were statistically independent of one another, we could estimate the standard error of their linear combination by propagation of error methods, or even by Monte Carlo simulation. However, we are quite confident that the geological variables chosen are not independent, because they were all selected with the thought that they would contribute to the discrimination between producing wells and dry holes, and hence they must have some degree of relationship. In order to produce the most pronounced discrimination possible between producing and barren locations, it is essential that all of the variables contain a degree of contributory information. For this reason, any error bands for 182

Discriminating Discoveries and Dry Holes

6

8 10 Distance, km

r 14

16

Figure 8.4. Experimental semivariogram and fitted spherical model for discriminant scores of wells in Magyarstan training area. the discriminant score that we might calculate by assuming each geological variable to be independent of all others would be unrealistically wide.

Mapping Combined Geological Properties The alternative approach is to calculate the discriminant score at every producing well and dry hole and treat the scores as another geologic variable (which it is, in fact, although of a complicated and unfamiliar kind). We can estimate a semivariogram based on the discriminant scores just as we determined semivariograms for structural elevation or thickness. The experimental semivariogram produced by RISKMAP for discriminant scores of the 83 holes in the Magyarstan training area is shown in Figure 8.4. Prom this semivariogram, we can estimate a value for the discriminant score at every location in the area, and can display these estimates as a contour map made by kriging. Figure 8.5 shows such a contour map; this represents a weighted combination of the four original variables of thickness, shale ratio, bedding index, and residuals from a second-degree trend surface fitted to structural elevation. Each original geological variable has contributed to this map in proportion to its effectiveness in distinguishing producing from dry locaUties. 183

Computing Risk for Oil Prospects — Chapter 8 6490.

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Figure 8.5. Contour map of discriminant scores of exploratory holes in Magyarstan training area, made by kriging. Contour interval is 2.0 units. Prom the map in Figure 8.5 we can select a location of interest and read off an estimate of the discriminant score at that spot. Using this estimated discriminant score and the function relating probability of a producer to score magnitude (Fig. 8.3), we could produce a measure of the probability that we will discover oil at that location. However, our estimate of the discriminant score is uncertain because it is based on interpolations from distant locations, and we must take this uncertainty into account in calculating the discovery probability. The kriging procedure provides us with the standard error of estimate at all locations (Fig. S.6), and the error for our specific location of interest can be read from this map and used to adjust the final probability. We must assume that the uncertainty in our kriging estimates is normally distributed; such an assumption is quite reasonable for discriminant 184

Discriminating Discoveries and Dry Holes 6490.

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Figure 8.6. Contour map of standard error in estimated discriminant scores of Magyarstan training area, made by kriging. Contour interval is 1.0 units. scores, since they are combinations of many variables, and the Central Limits Theorem assures us they should approach normality as the number of variables increases. At any location there is a normal distribution of possible discriminant scores. The center of this distribution is the kriging estimate itself, and the spread in the distribution about this central value is given by the kriging standard error. The form of the normal distribution is known explicitly, so for any given mean and standard error, the area of any desired segment under the curve can be determined. These areas are, of course, proportional to probabilities. The empirical conditional function relating discriminant scores and producing probabilities has been estimated for specified intervals of the discriminant score. The normal error distribution can be divided into matching intervals. The two distributions can then be combined as shown in 185

Computing Risk for Oil Prospects — Chapter 8 Figure 8.7 to yield a single estimate of the probability of production, given a discriminant score and its associated standard error. The example is for a location where kriging has estimated the discriminant score to be 14.0, with a standard error of 2.0. Calculations for the seven individual intervals of the distributions are given in Table 8.4; each of the products is the joint probability that a producer will occur and the score will lie in a specific interval. The probability of success at any point reflects the spectrum of possible discriminant scores that might occur at that point, weighted by multiplying each possible score by its probability of occurrence. Summing the resulting products yields a probability that is "unconditional" in the sense that it does not depend on what discriminant score actually appears on the map. However, perhaps a term such as the "error-weighted conditional probability" is more appropriate because it expresses the concept that the discriminant scores (and hence the corresponding probabilities) are conditional (or dependent) upon geological variables, but also incorporate the uncertainty in mapping the discriminant score at that location. Unless the point being estimated coincides with the location of a drill hole (where the map error is zero), the probability estimated for that point will not correspond directly with the discriminant score mapped at that point. Table 8.4. Calculation of probability of production at a location with an estimated discriminant score of 14.0 and standard error of 2.0. (A) Lower and (B) upper limit of interval in units of standard deviation; (C) lower and (D) upper limit of interval in discriminant score units; (E) probability (proportional area) of interval; (F) probability from corresponding interval of conditional distribution shown in Figure 8.3. (G) Product of columns (E) and (F). B

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We can perform the same transformation for every estimated discriminant score in our kriged map of Figure 8.5 and its accompanying error map in Figure 8.6. This will produce a grid of probability estimates that can also be shown as a contour map. Figure 8.8 is such a map for the 186

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Figure 8.8. Map of probability of discovery, based on discriminant scores in Magyarstan training area. Contour interval is 0.1, with an additional contour of 0.05.

Magyarstan training area, based on a discriminant analysis of the 83 holes in the area. It is unfortunate that the probabilities shown are not particularly encouraging, as the locations that have relatively high probabilities of success are step-outs of known production.The northwest-southeast trend in low discriminant scores that is especially apparent in the southeast part of the score map is well reproduced as an area of non-zero probabilities. Additional production other than step-outs of the two known fields is only moderately likely along this trend. The discriminant score trend reflects patterns in the original geological variables, particularly the structural trend residuals (Fig. 7.6). The area along this trend to the northwest of the field in the southeast corner seems to be a candidate for closer inspection. 188

Discriminating Discoveries and Dry Holes It is surprising that procedures for combining geological variables for assessment of prospects have been largely neglected in the literature of petroleum geology, given their potential importance. Of course, Monte Carlo schemes for assessing the probability of production in a prospect are currently popular, but these involve assigning individual probabilities to different geological factors such as the presence of trap, seal, carrier bed, etc., and merging these probabilities by Monte Carlo multiplication. As noted in Chapter 5, the procedures are suspect because of the subjective nature of the probability estimates and possible interdependencies between the geological factors. An alternative is to generate conditional probabilities that incorporate map error and are based on combinatorial procedures such as discriminant function analysis that have been specifically adapted to the objective of distinguishing producing from dry locations.

UPDATING ASSESSMENTS It may be that we are not particularly concerned about prospects in the Magyarstan training area, which has been drilled relatively intensely. Instead, the analysis may have been performed to gain quantitative "experience" in the form of knowledge about useful geological properties and estimates of discriminant coefficients and semivariogram parameters for kriging. This quantitative experience can be transferred to another area considered to be geologically similar and used for an initial analysis of discovery probabiUties. This will allow us to estimate the likelihood of success in the second area even though there may be too few (or no) discoveries from which to calculate reliable estimates of discovery probabilities. Under such a scenario, the first area is a "training area" and the second is a "target area" where the lessons learned in the training area will be applied. This approach embodies the same exploration philosophy as the use of geological analogues for the interpretation of new areas, except that the concept is extended to include the transfer of quantitative and probabilistic functions from the analogue for the initial appraisal. However, the process of "learning" does not cease with analysis of the training area, but continues as prospects are developed and tested. Each new hole that is drilled provides information on success or failure that should be used to update and improve the probability estimates, just as the hole provides geologic information that should lead to revision of the structural, isopach, and other geological maps. Exploration should be thought of as an evolutionary process; the drilling of a prospect provides data that result in the revision of the initial geological interpretation, including the discriminant score map. Subsequent error maps are modified because of the presence of the additional control point. The success or failure of the hole 189

Computing Risk for Oil Prospects — Chapter 8 alters the ratio of producers to dry holes, changing both our estimates of the dry hole probability and the conditional probabilities between geological properties and the discovery of oil. Initially, these events result in a relatively modest and progressive adaptation of distributions and parameters transferred from the training area, but as they accumulate with continued exploration they become more significant and at some point completely supplant the transferred knowledge. At that point, the discriminant analysis and all related analyses must be redone, using only information from the target area itself. In effect, the region has matured to the point that it can serve as its own training area. RISKMAP provides mechanisms for updating the unconditional success ratio by Bayesian methods, as described in Chapter 5. The conditional probability distribution is updated by changing the numbers of successes and failures in the intervals of the original distribution and recalculating the probabilities. The kriged map and standard error map of the discriminant score are automatically updated (although the semivariogram and the kriging parameters are not changed) whenever the maps are redrawn, which should be done whenever data are available from a new hole. Consequently, the probability map changes dynamically to reflect the results of continued exploration in the region. At the user's option, RISKMAP will abandon the training area parameters and distributions and recompute the discriminant function and conditional probability distribution using only data from the area being analyzed. Of course, sufficient exploratory holes must have been drilled prior to the recomputation to support estimation of the new conditional probabilities. It also will be necessary to recalculate the semivariogram of the discriminant scores, since the new scores will diff'er from the scores based on the training area function. Although the potential exists for dramatic changes, we should not anticipate that any will occur if our initial assumption that the training area is a suitable analogue for the target area is correct. The discriminant functions for the two areas should be similar, and the spatial relationships also should be comparable. It is difficult to specify an exact point at which analysis should shift entirely to the target area and the training area parameters be abandoned. This depends upon a complicated interplay of geometrical and geological factors and the degree of exploratory success. Obviously, probability estimates based on the area itself should be more reliable than those transferred from an analogue training area. However, if exploration of the new target region has proceeded in an erratic manner and large areas are unexplored while other small areas are intensely drilled (a statistically unfortunate but typical circumstance), the conditional estimates relating discriminant scores and probabilities may be unreliable, as may be the semivariogram of 190

Discriminating Discoveries and Dry Holes the discriminant scores. A prudent course is to conduct parallel analyses, one based on an analogue training area and the other based on the target area itself, until it is apparent that the various probabilities and statistical parameters are not changing erratically with continued exploration. Using these techniques, it is possible to evaluate an entire map area and estimate the probability of a discovery at every point, with little more effort than that required to produce the geological maps that ordinarily are made in the course of an exploration program. The virtues of this approach lie in its greater consistency and its explicit expression of discovery probabilities in a form that can be used in financial analysis and decision making. Those areas delineated as especially promising can be analyzed further, using techniques discussed in Chapters 4 and 6, to estimate the conditional magnitude of discovery. However, the volume of oil contained in a prospect is best estimated from the characteristics of a detailed model (either statistical or deterministic) of the prospect rather than from a regional analysis. In principle, it would be possible to expand the analytical procedure described here to estimate the probabilities of discovering fields of different sizes, but a very large amount of data would be required to produce stable probability estimates. It seems better to divide the assessment process into two stages; estimation of the probability of success, followed by conditional assessment of the size of discovery, given that a discovery is made.

EXPLORING T H E MAGYARSTAN TARGET AREA IN T H E BAKANT BASIN We can draw upon our experience in the Magyarstan training area and apply the knowledge to exploration of a region in the Bakant Basin that we will refer to as the "target" area. Although this second area now also has been extensively drilled, we are able to examine its exploration history and simulate how our prospect evaluation techniques might have been applied at successive stages. For example, we can "explore" the target area at a time when drilling had just begun and data were limited and then compare our results with those actually achieved. The target area can be explored progressively, much as any new area is opened up and developed, and we will see how exploration strategies and evaluation procedures must be modified as increasing amounts of information become available. Initially, we will assume that relationships between oil occurrence and geology in the target area are similar to those we have observed in the training area, although we shouldn't expect the two areas to be identical. In other words, the training area will serve as a geological analogue of the target area. The target area has the same geogi-aphic dimensions as the training area, oil occurs in the same reservoir formation (the XVa Limestone), and 191

Computing Risk for Oil Prospects — Chapter 8 the main geological variables influencing the occurrence of oil and gas are the same. As drilling proceeds in the target area, we will gain more and more information, allowing us to judge how realistic our original assumptions about the similarity between the two areas have been. To assess how closely the two regions resemble each other at any stage in the simulation, we can compare maps of the geological variables using only data available at that stage. It will be especially instructive to compare probability functions relating oil occurrence to discriminant scores and semivariograms that express the map error of the discriminant scores for the two areas. For the purpose of this simulation, drilling results for the target area have been segregated into four stages: an "immature stage" in which only 17 initial dry holes have been drilled, an "intermediate stage" in which one field has been discovered, a "mature stage" in which three fields have been discovered, and a "final stage" in which all known exploratory holes and producing wells are represented. For convenience, results are presented for only these stages, but keep in mind that in actual practice, drilling results should be reanalyzed after each well has been completed.

THE IMMATURE STAGE In the immature stage, 17 exploratory holes have been drilled in the target area; their results are given in the data file TARGET1.DAT. Although all of these holes were unfortunately dry, they provide valuable information that allows us to update our geological interpretations. Even though 17 dry holes is a discouraging outcome, an area the size of the target area with seemingly favorable geologic conditions should not be condemned. There are many regions where a long sequence of dry holes preceded a significant discovery. Using data from the 17 holes, we can generate contour maps of the five geological variables: structural elevation (Fig. 8.9), structural trend surface residuals (Fig. 8.10), thickness (Fig. 8.11), shale ratio (Fig. 8.12), and bedding index (Fig. 8.13). These are the same properties that we used to describe the geology of the training area, and we can map them using the same contouring procedures. Drill-stem test (DST) results from the XVa Limestone are indicated by special well symbols on the contour maps. The symbol conventions are given in the caption of Figure 8.9. Four DST results are recorded: mud, salt water, oil-cut salt water, and oil. Recovery of only mud indicates low permeability, whereas recovery of salt water, oil-cut salt water, or oil indicates suitable permeabilities. The structural elevation map of the target area (Fig. 8.9) shows a nearly featureless homocline dipping towards the southeast that is similar in gross form to its counterpart structure shown on the structural map 192

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Figure 8.9. Structure contour map of top of XVa Limestone in the Magyarstan target area at immature exploration stage. Coordinates given in kilometers. Contours given in meters below sea level. Contour interval is 10 m. Map based on 17 exploratory holes. Well symbols indicate drill-stem test results: • = oil "^ = oil-cut salt water -fy = salt water -^ = mud. of the training area (Fig. 7.1). However, the second-degree trend surface structural residual map of the target area (Fig. 8.10) is different from that of the training area (Fig. 7.6), particularly because there are several notable residual highs and lows in the target area. The northeast-southwest strike in the training area also can be seen in the target area, although it is not as clear. Keep in mind that much of the differences in details of the two maps are related to the difference in well density, as there are 83 holes in the training area and only 17 in the target area. The isopach map of the target area (Fig. 8.11) also differs markedly from its counterpart in the training area (Fig. 7.2). Figure 8.11 reveals an 193

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Figure 8.10. Residuals from second-degree trend surface fitted to elevations of top of XVa Limestone in the Magyarstan target area at immature exploration stage. Contour interval is 10 m. increase in thickness towards the west-southwest, where only mud has been recovered in drill-stem tests of the four holes that have been drilled there, a distinctly unfavorable sign. In the training area, by contrast, increased thickness vaguely accords with the presence of producing wells. On the map of shale ratio in the target area (Fig. 8.12) there are increasing shale ratios toward the southwest, where high shale ratios are measured in four dry holes whose drill-stem tests recovered mud. Furthermore, we note in the training area (Fig. 7.3) that producing wells generally are associated with low shale ratios. These relationships suggest that subsequent exploratory holes should be located where low shale ratios are projected to occur in the eastern and northern parts of the target area. High shale 194

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Figure 8.11. Isopax^h map of thickness of the XVa Limestone in the Magyarstan target area at immature exploration stage. Contour interval is 1 m.

ratios and greater thickness seem to be correlated with dry holes in the target area, a combination that should be avoided in subsequent drilling. This correlation is not so obvious in the training area. However, in both the training and target areas, low shale ratios are necessary for low discriminant scores and high producer probabilities. In the target area, the bedding index tends to be low in the northern part and high in the southern. Therefore, it is only partially correlated with the shale ratio. In contrast, the bedding index and shale ratios are correlated to a greater extent in the training area. The map of bedding index in the target area (Fig. 8.13) shows only modest variations. The map differs in detail from its counterpart in the 195

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Figure 8.12. Contour map of shale ratio of the XVa Limestone in the Magyarstan target area at immature exploration stage. Contour interval is 0.01 units. training area (Fig. 7.4). However, there is a vague inverse concordance with the shale ratio map of the target area (Fig. 8.12).

Discriminant Analysis in the Immature Stage A discriminant function that would combine all the geological variables into a single map might help in assessment, but to generate a discriminant function we must have samples from populations of both producers and dry holes; and in the immature stage, our target area contains only dry holes. An interim solution is to borrow a discriminant function, as well as a probability function, from the training area. This is the discriminant function used to create the score map shown in Figure 8.5. Using this same 196

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Figure 8.13. Bedding index of the XVa Limestone in the Magyarstan target area at immature exploration stage. Contour interval is 0.01 units. discriminant function, we can transform the geological variables measured in the target area's 17 dry holes into a discriminant score for each locality. The contouring program can interpolate these values over most of the target area (Fig. 8.14). To estimate the map error in the discriminant score map, we can utilize the semivariogram calculated for the training area scores (Fig. 8.4) and use it to generate a grid of standard errors of estimated discriminant scores in the target area. The pattern of contour lines in Figure 8.14 reflects the locations of the dry holes and the geological variables measured in them. However, the mapped scores also are influenced by the discriminant function and the semivariogram of discriminant scores, both of which have been borrowed from the geologically analogous training area. The discriminant score map 197

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Figure 8.14. Contour map of discriminant scores in the Magyarstan target area at immature exploration stage. Discriminant function and semivariogram are based on training area. Contour interval is 1.0 units. thus is a hybrid of influences from the target area and the training area. Interrelationships between the geological properties and their spatial continuities are derived from the training area, but the specific geological measurements and the well locations are provided by the target area. If we compare the discriminant score maps of the target and training areas (Figures 8.14 and 8.5), we note that the training area exhibits a greater range of scores. We also note that relatively high scores in the target area occur in the southwest, where mud was recovered in drill-stem tests, a distinctly unfavorable response. High scores also coincide with dry holes over much of the training area. This all suggests that high scores in the target area will be unfavorable for oil occurrence, just as they were 198

Discriminating Discoveries and Dry Holes in the training area. This conclusion suggests that the use of the training area as an analogue for the target area is appropriate even though some geological variables have a slightly different statistical behavior when viewed individually.

Including Target Area Information in the Probability Function Since the Bakant Basin target area is in the process of being explored, the challenge is to locate prospects that have not yet been drilled and which have relatively high probabilities of success. To obtain such probabilities, we must transform the discriminant scores into probabilities, using a function in which probabilities are conditional on scores. Here we face a dilemma. We could borrow the probability function from the training area (Fig. 8.3) and apply it directly to the target area, but this would ignore information from the 17 dry holes already in the target area that are not represented in the function. As an alternative, we could incorporate information from these dry holes to generate a new function. Should we use the training area function, or should we use a new function based on a mixed population of wells that includes the 17 dry holes from the target area and the 83 holes from the training area? How can we decide? First, we can compare probability functions based on different collections of exploratory holes. Figure 8.15a is based only on training area holes, whereas Figure 8.15b combines the 83 training area holes with the 17 target area dry holes. The two are very similar because the preponderance of the data come from the training area; nevertheless, we have elected to use the second function because it does incorporate information from the target area. Having chosen a probability function, the next step is to generate a probability map for the target area, based on the 17 dry holes. The map of discriminant scores provides an estimated score at each grid node of the map. Using the probability function, we transform each score estimate into a probability estimate at each grid node through the steps outlined in Figure 8.7. Contour lines are then fitted to the new grid of estimated probabilities of success to create the probability map shown in Figure 8.16.

Planning for Subsequent Drilling There are several localities shown on Figure 8.16 where the estimated probabilities exceed 25%. In general, the southeastern and northern parts of the target area seem to be most promising; these areas are characterized by low shale ratios and low discriminant scores. Recall from the discriminant function that the producing group is characterized by low discriminant scores 199

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Figure 8.15. Empirical distributions relating discriminant scores and discovery probabilities, (a) Based on 18 producing wells and 65 dry holes in training area, (b) Based on 18 producing wells and 82 dry holes from the training and target areas combined, (c) Based on 43 producing wells and 106 dry holes from the training and target areas combined, (d) Based on 25 producing wells and 41 dry holes in the target area at intermediate exploration stage, (e) Based on 58 producing wells and 60 dry holes in the target area at mature exploration stage. while the dry hole group is characterized by relatively high scores. This is reflected in the discriminant score maps and the probability maps for both the training and target areas. However, discriminant scores in the target area are not as low as those in the training area and, consequently, probabilities of success are not as high in the target area as they are in the training area. Note that an exploratory well at coordinates (4980E, 7333N) in the target area is not enclosed by high probability contours, even though this well recovered oil-cut salt water during a drill-stem test. Although recovery of oil-cut salt water is usually regarded as an encouraging sign, the probability map is relatively discouraging for the possibility of production in the immediate vicinity. At this point, weVe extracted virtually all useful information from the 17 dry holes in the target area, and this information has been linked in an optimum manner with experience previously gained in the training area. 200

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Figure 8.16. Map of probability of discovery in the Magyarstan target area at the immature exploration stage. Discriminant function and semivariogram are based on training area. Empirical probability distribution (Fig. 8.15b) is based on training area and target area combined. Contour interval is 0.05. While it's possible that other areas would be better analogues of the target area and allow improved predictions, we have made the best use of the information that we've selected to use in creating a probability map of the target area. But how do we know if the probability map is effective? The old oil patch saying decrees that "the proof of the prospect is in the drilling." We must drill more exploratory holes to determine if our procedures are successful, following the strategy of drilling where probabilities are highest. On Figure 8.16, the highest probabilities of success are in the eastern and northern parts of the target area, with the best defined high probabilities in the southeast where the contours range above 25%. During the next 201

Computing Risk for Oil Prospects — Cliapter 8 round of drilling, some prospects should be developed in this area, particularly where the probabilities of success exceed 20 to 25%. We note that those parts of the area with higher probabilities have scant well control, but the geological signs are favorable and support the need for more drilling.

THE INTERMEDIATE STAGE Now we consider an intermediate stage of exploration when 49 additional holes have been drilled in the target area, for a total of 66 holes consisting of 25 producing wells and 41 dry holes. Information from these drill holes is contained in data file TARGET2.DAT. Figure 8.17 is a map of discriminant scores in the target area, based on data from the 66 holes that have been drilled there, but using the discriminant function from the training area. The contour map of scores has been made by kriging, with a semivariogram model that also has been derived from the training area. In Figure 8.18, the discriminant scores and their kriging estimation variances have been transformed into a map of probabilities of discovery. The transformation function (Fig. 8.15c) has been calculated from the 83 exploratory holes in the training area in combination with the 66 holes in the target area. These maps show that all producers are located in a field in the southeastern part of the target area that partly coincides with the closed contour on Figure 8.16, based on the 17 dry holes. The probability closure, however, has been strongly influenced by the presence of the dry hole at coordinates (4997E, 7322N) which subsequently proved to lie close to the rim of the oil field. The reservoir interval encountered in that particular dry hole had characteristics similar to those in producers within the oil field; in hindsight, we can see that this was a favorable sign. Although the center of the oil field does not coincide exactly with the probability high (the field is off^set to the west by about 2 km and the highest probabilities coincide with the field's eastern margin), the discriminant score and probability maps were helpful for selecting drilling locations that led to the discovery of the field. In Figure 8.18, note that some poorly defined probability highs occur along the northern and eastern edges of the target area. We wish to test these areas in subsequent exploratory drilling, but again we face a dilemma. Now that there are both producers and dry holes in the target area, should the discriminant function and semivariogram continue to be based on data from the training area, or should the training area be abandoned and only target area wells used to generate these two functions? This is a difficult decision, but a comparison of maps prepared using the two alternatives should help decide. 202

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Figure 8.17. Contour map of discriminant scores in the Magyarstan target area at intermediate exploration stage. Discriminant function and semivariogram are based on training area only. Contour interval is 1.0 units.

Comparing Results Although the discriminant score and probability maps are drawn using the 66 holes available in the target area at the intermediate stage, the maps retain the discriminant function and semivariogram derived from the training area. The probability map utilizes an updated probability function based on a mixture of the training area's 83 holes and the ^^ holes of the target area. Although we haven't broken the dependence on the training area, this could be done because there are now 25 producers in the target area that would permit development of a new discriminant function and probability function derived entirely from target area wells. In addition. 203

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Figure 8.18. Map of probability of discovery in the Magyarstan target area at the intermediate exploration stage. Discriminant function and semivariogram are based on training area. Empirical probability distribution (Fig. 8.15c) is based on training area and target area combined. Contour interval is 0.1, with an additional contour of 0.05. a new semivariogram could be based entirely on target area wells. The question is whether this should be done at this time. How can we decide? We can compare the discriminant function we have used with a discriminant function based solely on data from the target area by examining their relative efficacy in discriminating between producing and dry locations in the target area. A cross plot such as Figure 8.19 compares scores calculated for the 66 target area wells using the new (target area) discriminant function versus those calculated using the old (training area) discriminant function. If the two functions were identical, the scores would plot on a 45° line. Although there is a modest amount of scatter in the plot, it 204

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Figure 8.19. Scores for target area holes calculated from training area discriminant function versus scores for target area holes calculated from target area discriminant function. See Figure 8.9 for key to well symbols. confirms that scores calculated with both functions vary linearly and the two functions are similar in their ability to segregate producing wells from dry holes. Note that scores calculated using the discriminant function from the target area have a smaller range than those calculated using the discriminant function based on the training area. The lowest scores based on the target area discriminant function are about four score units higher than equivalent scores calculated using the training area discriminant function. Figure 8.19 also emphasizes that low scores are associated with producers and high scores with dry holes, relationships observed when score maps are compared with probability maps. The question of which discriminant function is best can be resolved by plotting histograms of scores for producing wells and dry holes. Figure 8.20 shows frequencies of discriminant scores for the 66 target area holes, based 205

Computing Risk for Oil Prospects — Chapter 8

20 25 Discriminant score Figure 8.20. Distributions of discriminant scores for wells in Magyarstan target area at the intermediate exploration stage, (a) Based on training area discriminant function, (b) Based on target area discriminant function. Dark shade in histograms indicates producing wells, light is dry holes. on the two discriminant functions. It is apparent that the discriminant function derived from target area wells more effectively segregates producers from dry holes than does the discriminant function based on the training area. Therefore, it is appropriate to abandon the training area discriminant function and utilize a discriminant function based solely on target area wells. As more wells are drilled in the target area, the discriminant function can be progressively recalculated—each time taking advantage of the new well information that becomes available. 206

Discriminating Discoveries and Dry Holes 25-

Sill = 19.6

Distance, km Figure 8.21. Experimental semivariogram of discriminant scores in Magyarstan target area at the intermediate exploration stage. A Gaussian semivariogram model with a nugget of 0, sill of 19.6, and a range of 12.9 km has been fitted.

Completing the Move to the Target Area It is now time to move completely to the target area and replace all the functions that were transferred from the training area. The succeeding analyses will use a new discriminant function based only on the target area drill holes. The semivariogram used in kriging the discriminant scores also will use only the target area holes, and is represented with a Gaussian model having a nugget of 0, a sill of 19.6, and a range of 12.9 km (Fig. 8.21). A new probability transformation (Fig. 8.15d) will also be based exclusively on relationships in the target area. Figure 8.22 is a discriminant score map that incorporates a new grid of the standard error of the estimated scores. At each grid node, we carried out the operations illustrated schematically in Figure 8.7 to produce the new probability map in Figure 8.23. We must now ask the question whether this new probability map is better than the old one (Fig. 8.18). There are several important differences. First, probability highs in the northern part are much more sharply defined and have higher values. The high value of over 50% in the northwest corner is suspect because it is based on 207

Computing Risk for Oil Prospects — Chapter 8

7350. k

7345.

7340.

7335.

7330. r

7325.

7320. 4970.

4975.

4980.

4985.

4990.

4995.

5000.

5005.

Figure 8.22. Contour map of discriminant scores in Magyarstan target area at the intermediate exploration stage. Discriminant ftinction and semivariogram are based on target area only. Contour interval is 2 units. extrapolations toward the map boundary. In contrast, the elongated high of more than 40% in the northeast quarter lies between several dry holes and therefore is based on local conditions. Note that this seems to be the same NW-SE trend seen in the training area. These observations lead us to the conclusion that the probability map based on the target area discriminant function better represents the characteristics of the target area, and we will use it as the base for future exploration. As always, however, the proof is in the drilling.

208

Discriminating Discoveries and Dry Holes —1

1

1 —

7350.

7345.

7340.

7335.

7330.

7325. h

7320.

4970.

4975.

4980.

4985.

4990.

4995.

5000.

5005.

Figure 8.23. Map of probability of discovery in the Magyarstan target area at the intermediate exploration stage. Discriminant function, semivariogram, and empirical probability distribution (Fig. 8.15d) are based on target area. Contour interval is 0.1, with an additional contour of 0.05.

THE MATURE STAGE We can now analyze results of continued exploration in the target area, which has reached a mature stage after the drilling of 52 more holes, making a total of 118 drill holes that includes 61 dry holes, plus 57 producers in two oil fields. The well data are in the file TARGET3.DAT. Using these additional wells, we can calculate another new discriminant function, a new semivariogram (Fig. 8.24), and a new probability transformation (Fig. 8.15e). The new semivariogram of discriminant scores is shown fitted with a Gaussian model that has a nugget of 0, a sill of 6.6, and a range of 15.5 km. These functions will allow us to prepare new discriminant 209

Computing Risk for Oil Prospects — Chapter 8 10-

Sill = 6.6

10 Distance, km

20

Figure 8.24. Experimental semivariogram of discriminant scores in Magyarstan target area at the mature exploration stage. A Gaussian semivariogram model with a nugget of 0, sill of 6.6, and a range of 15.5 km has been fitted. score and probability maps (Figs. 8.25 and 8.26). Of course, in practice we would have recalculated these functions and maps after each new hole was drilled and its outcome determined, thus creating an evolving succession of probability maps. In Figure 8.25 the range of scores differs from those of previous maps. The range of scores in the new probability function (Fig. 8.15e) also differs from the range of scores in earlier probability functions. These differences stem from the updated set of target area drill holes that now includes 24 new producing wells drilled in a new field discovered in the north-central part of the target area. The change in scores illustrates that specific values are of little significance, but that the values of scores relative to each other are important. Having progressed from CS to 118 exploratory holes in the target area, it is worth looking back to see which of the two probability maps based on 66 drill holes provided the better forecast of the succeeding 52 prospects. The choice is between a map based on the discriminant function and semivariogram of the training area (Fig. 8.18) and a map based on the discriminant function and semivariogram of the target area (Fig. 8.23). Subsequent 210

Discriminating Discoveries and Dry Holes

7350.

7345.

7340.

7335. h

7330.

7325. 7320. 4970.

4975.

4980.

4985.

4990.

4995.

5000.

5005.

Figure 8.25. Contour map of discriminant scores map in Magyarstan target area at the mature exploration stage. Discriminant function and semivariogram are based on target area only. Contour interval is 2 units. comparison of results of drilling will show that neither map was perfect, but the map based exclusively on the target area more sharply defines localities with higher probabilities and has proven to be the better choice based on results in the mature stage of exploration. In particular, the pattern of high probabilities in the northern part of Figure 8.23 is much more indicative of the presence of oil than the diffuse pattern shown on Figure 8.18.

T H E FINAL STAGE In the final stage of exploration we add 60 producers that have been drilled in the target area since the mature stage. The boundaries of the two previously discovered oil fields have been extended and two new fields have been 211

Computing Risk for Oil Prospects — Chapter 8

7350.

7345. r

7340. h

7335.

7330.

7325.

7320. 4970.

4975.

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4995.

5000.

5005.

Figure 8.26. Map of probability of discovery in the Magyarstan target area at the mature exploration stage. Discriminant function, semivariogram, and empirical probability distribution (Fig. 8.15e) are based on target area. Contour interval is 0.1, with an additional contour of 0.05. established. Locations of these new wells were influenced by the probability map (Fig. 8.26) that was based on the 118 wells known at the mature stage of exploration. The map outlines two strongly favorable areas, one broadly north of the previously discovered oil field in the north-central part of the target area and the other in the east-central part of the target area. It is instructive to make a predrill/postdrill comparison. This can be done readily by superimposing the 60 new wells on a probability map whose contours are based only on information known in the mature stage, as is done in Figure 8.27. The new producers are located in areas where probabilities had been forecast that ranged from about 0.05 to 0.60. Although the predictions are far from perfect, the probability contours are useful 212

Discriminating Discoveries and Dry Holes

7350. h

7345. r

7340.

7335. K

7330. r

7325.

7320. 4970.

4975.

4980.

4985.

4990.

4995.

5000.

5005.

Figure 8.27. Map of probability of discovery in the Magyarstan target area at the mature exploration stage with the 60 producing wells drilled afterwards superimposed, facilitating comparison between forecasts (contoured probabilities) and outcomes of wells drilled after contours were generated. Contour interval is 0.1, with an additional contour of 0.05. guides to the results that will be achieved upon drilling. For example, one of the two new fields was discovered where probabilities ranging from 0.40 to about 0.50 occur along the northern edge of the target area. The other new field coincides with an area whose probabilities ranged from about 0.20 to 0.40 along the eastern edge of the area. The procedures illustrated in this chapter can be used routinely to update the statistical relations derived from predrill/postdrill comparisons whenever a new well is completed, whether the well is a stepout or a rank wildcat. Not only can contour maps of conventional geological variables be updated, but discriminant score maps and probability maps also can 213

Computing Risk for Oil Prospects — Chapter 8 be updated, as well as the discriminant function, the semivariogram, and the empirical probability function on which they are based. Such updating would be virtually impossible manually, but can be carried out easily and quickly with appropriate computer software. In Chapters 7 and 8 we have outlined procedures for combining geological variables by using discriminant functions and have emphasized the usefulness of maps of discriminant scores. We have also emphasized the pervasive error and the need to take this source of uncertainty into consideration when making probability assessments. Finally, we have emphasized the creation and use of probability maps as direct guides for exploratory drilling. Chapter 8 stresses the systematic collection of information gained by experience in a training area and the application of this knowledge to exploration in a geologically analogous target area. Explorationists have informally employed "training" or analogue areas since the beginning of the quest for petroleum, but the application of statistical procedures for systematically extracting information from training areas and applying it to a "target" area is a recent advance. Statistical procedures not only quantify the degree to which training areas and target areas are similar, they also promise greater scientific objectivity. Exploration of the Magyarstan target area has been described during four stages of maturity. In actual practice, maps would be updated and reevaluated much more often, usually after the drilling of each exploratory hole. Such frequent updating and reevaluation would be almost impossible with manual contouring and conventional evaluation procedures. Using computers and appropriate software, however, maps of optimum usefulness can be generated and updated easily. An old axiom declares that opportunities for discovering oil are greatest when information is most meager. This adage may or may not be true, but it is nonetheless important to make optimum use of all available information at all stages of exploration so we can recognize good prospects while they're available. Although our immediate goal is to delineate promising prospects, our ultimate goal is to exploit the prospects we identify to achieve financial gain; this requires that the risk of monetary loss be carefully considered. The probability maps that we have created yield estimates of the probabilities of success, quantifying the dry hole risk associated with prospects within the map area. The next three chapters focus on analyzing the potential profitability of prospects we have identified and describe procedures for balancing the risk of losses against the potential for gains.

214

CHAPTER

9 .v>:<'W'^*-'''f<
Forecasting Cash Flow -for a Prospect FINANCIAL OVERVIEW Estimates of outcome probabilities for a prospect are not much use by themselves because the estimates need to be tied to decisions about the prospect, and in the end these decisions are primarily financial and not geological. The overriding issue is whether the prospect has an acceptable chance of becoming a financial success. If it doesn't it is not a good prospect, regardless of its geological qualities. The real central objective in the oil business (as in all businesses) is to make money and not to generate prospects, so the geological qualities of prospects are irrelevant if the outlook for profits is not reasonable. Risk analysis links the geological qualities of prospects with financial considerations and focuses on the objective of making optimum decisions. Probability estimates are the form of the necessary linkages that permit potential gains or losses to be analyzed in terms of the risk involved. In the oil exploration business, prospects are generated and risks taken with the expectation that some prospects will yield handsome rewards and others will be bitter disappointments. In the oil business, we know that we will have to take the bad with the good, and that no extension of technology is capable of eliminating all risk. However, the specific actions we take will profoundly aff^ect our progress over the long run, and we should strive to make the best decisions by weighing alternative investment opportunities as they become available and take those actions that are consistent and

Computing Risk for Oil Prospects — Chapter 9 optimal for our financial resources and risk outlook. This is the essence of oil exploration risk analysis. The literature on decision analysis in oil exploration and exploitation is large. Several books provide overviews, including the classic introduction to the topic by Grayson (1960), and readable, thorough tomes by McCray (1975), Megill (1985, 1988), and Newendorp (1975). There are many other books, as well as theses and articles dispersed throughout geological, engineering, and economics journals. The following works provide background in decision analysis in the exploration side of the oil business: Arps (1965), Arps and Arps (1974), Behrenbruch, Azinger, and Foley (1989), BennelU (1967), Brown (1962, 1966), Bruckner (1978), Bunn (1984), Caldwell (1986), Campbell (1962), Campbell and Schuh (1961), Capen (1979), Capen, Clapp and Campbell (1971), Charreton and Bourdaire (1983), Cozzolino (1977), Cozzolino and Falconer (1977), DeGeoffroy and Wignall (1985), Fuda (1971), Garb (1985, 1988a, b), Gotautas (1963), Halter and Dean (1971), HoUoway (1979), Hosseini (1986), Kaufman (1963), King (1974), Kirkwood (1985), Marsh (1980), Martin (1988), McCray (1969, 1975), Newendorp (1967), Newendorp and Root (1986), Newendorp and Quick (1987), Northern (1967), Pirson (1941), Raiffa (1968), Rose (1987), Rosenberg (1985), Schlaifer (1969), Schuyler (1990), Schwade (1967), Sluijk, Nederlof, and Parker (1986), Walls (1990), Wignall and DeGeoffroy (1987), and WilUams (1987). Many other articles could be added to this Ust, most of which are included in the Bibliography at the back of this book. There is no shortage of material to be read on the topic of risk analysis in oil exploration, but the discussions are weak on methods of estimating outcome probabilities and they fail to systematically link geological factors with financial considerations. Most risk-analysis procedures require the use of computer programs. Otherwise, the arithmetic and data handling that are necessary are almost unmanageable. Spreadsheet programs typically are used for this purpose, but they cannot easily be linked to statistical and mapping programs that are essential for estimating outcome probabilities based on geological relationships. General-purpose risk-analysis and financial programs appropriate for business forecasting in manufacturing or sales are inadequate for risk assessment in petroleum exploration. For this reason, we have provided a software package called RISKTAB that contains modules linking the tools for financially analyzing prospects with the tools contained in RISKSTAT and RISKMAP for estimating outcome probabilities. The translation of risk to dollars can be made using program modules whose linkages are shown schematically in Figure 9.1. The names of the modules connote their roles: CASHFLOW generates discounted net cash flows and in turn is linked with RAT, which generates risk analysis tables, including an extension involving 216

Forecasting Cash Flow for a Prospect

Financial information

CASHFLOW Discounted net cash flow (DNCF) analyses

Probability distributions

RAT Risk-analysis tables

Utility function

DECISION Expected monetary value (EMV) decision tables

DECISION Expected utility value (EUV) decision tables

EMV decision tree

EUV decision tree

Figure 9 . 1 . Interconnections between financial modules and sources of information in financial analyses. Financial information supplied to CASHFLOW generates discounted net cash flows (DNCFs) and net present values (NPVS). Outcome probability distributions supplied to RAT generate risk analysis tables and utility tables. RATs can be aggregated by DECISION to create EMV or EUV decision tables which can be linked to form decision trees. Multiple DNCFs are required for a RAT, multiple RATs for a decision table, and multiple decision tables for a decision tree. 217

Computing Risk for Oil Prospects — Chapter 9 aversion to risk. Files containing probability distributions based on geological properties, production data, and well outcomes also link with RAT. DECISION links with RAT to produce decision tables. Finally, decision trees can be created by manually linking sequences of decision tables.

DISCOUNTED NET CASH FLOW ANALYSIS The first step in financial analysis involves computing discounted net cash flows (DNCFS) associated with each possible outcome for a prospect. Before a prospect is drilled, we face a spectrum of possible results: the prospect could be a dry hole, a real bonanza, or something in between. We won't know how the prospect will turn out until it is drilled; but, prior to drilling, we need forecasts of how much a big discovery would be worth, what a dry hole would cost, and what intermediate outcomes would be worth. DNCF analysis gives us these forecasts. As its name implies, a DNCF consists of calculating the cash flows for an investment and discounting them according to their time of occurrence in the future so that all cash inflows and outflows are expressed on an equivalent, net present-value (NPV) basis. Although cash flow analysis may seem complex, most people have a basic grasp of its principles even if they have never formally studied finance. Consider an ordinary bank checking account; the money in the account at any given moment is the net (or algebraic sum) of all cash infiows and all cash outflows since the account was established. The inflows are the deposits and interest earned; the outflows are the checks written on the account, the cash withdrawals, and any bank charges. If we tabulate all the inflows and outflows from the beginning of the account until the present, we would have a complete cash flow summary for the account. A cash flow analysis for a prospect diff'ers from our analogy only because it considers possible inflows and outflows that may occur in the future, rather than the historical records of past events. Because they are projections, these future transactions are uncertain. We don't know what the price of oil will be in the future, although we can make projections that may be reasonable. Likewise, we may be uncertain about the rate of taxation on oil production in the future, particularly in light of lawmakers' perennial desire to change the tax laws. We must make assumptions about these costs in spite of their uncertainties in order to compute a cash flow for a prospect, even though changes in prices and costs could have a major influence on the projected cash flows. Megill (1988) provides an excellent introduction to cash flow analysis in an oil-exploration context. There is a another major difference between the cash flow history of a checking account and a cash flow forecast for an oil prospect. We must discount all future cash flows if we want to analyze their effect on the present 218

Forecasting Cash Flow for a Prospect value of the prospect. Suppose we are faced with making a decision at the present time and wish to compare alternative exploration investments. We need to compare these alternatives as though they occur at the same moment in time, and this can be done most simply (and most understandably) if we compare their values at the present time. To do this, we must put future inflows and outflows on the same basis even though they may occur at quite different times. A dollar received in the future does not have the same worth as a dollar received at the present. Quite apart from inflation (which also must be considered), a future dollar is worth less than a dollar at present because of the time value of money. For example, if we can earn 10% a year on an investment, the value of $100 to be received one year in the future is only $91 at the present time. This is because we can invest only $91 now and it will grow to $100 in one year at a 10% rate of interest. In preparing a DNCF, we must systematically discount all future inflows and outflows. The more distant these financial transactions are in the future, the more they must be discounted. Outflows are discounted in the same manner as inflows; the present worth of a dollar to be paid out in the future is less than a dollar paid out today. The literature on cash flow analysis and incorporation of discounting is voluminous, but much of it is only indirectly relevant to oil exploration. Salient articles and books in our context include those by Adelman (1986) and Megill (1988).

Risk Analysis Tables While DNCF forecasts are useful, they don't tell the whole story because they do not provide a weighting by which good results can be compared with intermediate or poor results. Diff'erent DNCFs can represent forecasts of the present value of a 100-MMbbls discovery, the present value of a 5-MMbbls discovery, or the present-value cost of a dry hole. Any of these results may be possible, but what are the probabilities associated with the alternatives? We cannot analyze risks with DNCFs unless we consider the likelihood that the alternative outcomes will come to pass. In other words, we need to "risk" the different forecasts by weighting each of them by the probability of their occurrence. To weight by risk, we must multiply the net present value of each speciflc outcome by the probability of that outcome, and then sum the products to obtain an overall value that incorporates both favorable and unfavorable outcomes. The arithmetic is conveniently organized in a risk analysis table (RAT), and the single risk-weighted figure that results is an expected monetary value (EMV), an estimate expressed in dollars or other monetary units. (Alternatively, the analysis may involve imaginary units called utiles that are derived from a utility function expressing the degree of risk aversion of an individual or corporation. This topic, which extends 219

Computing Risk for Oil Prospects — Chapter 9 a risk analysis table to yield an expected utility value or EUV expressed in utiles, will be discussed in Chapter 10.) Preparing a risk analysis table or RAT requires calculating a series of DNCFs. A main purpose for organizing information in a risk analysis table is to put the information in a convenient form for generating a succession of DNCFs. For example, we might envision a spectrum of outcomes for a prospect, ranging from the discovery of a small field to the discovery of a giant. However, we can't conveniently deal with a continuous spectrum of outcomes, so we must represent them as a discrete distribution having a manageable number of field size classes. The discovery of a field of a certain size would require drilling a number of development wells, and hence the expenditure of an amount of money that would depend on the size of the field discovered. This means that we must analyze each possible class of field size separately. We will have to forecast the number of producing wells and their average cost, the number of development wells that will be dry and their average cost, the average amount of oil and gas that each producing well will yield, the gross income the production will provide, and the capital costs, operating costs, and taxes that we will have to pay. Armed with this information, we can generate a DNCF for each well in a field of this specific size, and from the DNCF we can estimate the net present value of each producing well. If we then multiply this NPV per well by the number of producing wells in a field and subtract the cost of the dry development wells, we obtain an overall monetary value of a field in this specific size class. We can repeat these steps for the other possible field size classes. We must also treat the cost of a dry exploratory hole as an outcome class because the initial hole drilled on the prospect may be a failure. When we have calculated the NPVs for all the diflferent outcome classes that we've defined, we can risk each of them by multiplying their present values by their probability of occurrence and then summing the products. This sum is the "bottom line" of the RAT and is the expected monetary value or EMV for the prospect. While an EMV is simple to obtain in principle, the volume of arithmetic computations can be formidable. Module RAT systematizes the steps in generating a risk analysis table, reducing confusion and minimizing arithmetic.

Decision Tables A decision table provides foresight and not hindsight, which was the basis for the frequency distributions discussed earlier. While hindsight may be comforting because uncertainty has been removed, analyzing a prospect requires foresight so that we can make the best choice among alternative 220

Forecasting Cash Flow for a Prospect actions that may be available to us and which are commensurate with the risk and uncertainty involved. Module RAT in RISKTAB analyzes the consequences of a single action, such as the drilling of a prospect by a sole investor. If we wish to consider alternatives such as farming out the prospect or organizing a joint venture with other operators to share costs (and rewards), a RAT must be prepared for each alternative strategy and the results compared. For comparison purposes, we can group different RATs into a decision table where different outcomes are tabulated in columns and different actions in rows. Each row represents a specific action, and the EMVs or EUVs at the end of the rows show which choice is best. If we are risk neutral and adhere to an expected monetary value criterion, we can use the DECISION module to generate an EMV decision table. If we are averse to risk and can provide the necessary utility function, we can use DECISION to generate an EUV decision table.

Decision Trees In oil exploration as in life we generally are faced with sequences of events, because one thing tends to lead to another. If we drill a successful exploratory well, we will probably drill an offset well. If the offset is successful, then we will most likely drill a third ofTset, and so on. Thus, the present potential of a prospect is affected by the outcomes of a succession of events that may take place in the future. The pattern of future events can be expressed with decision trees, whose branches represent pathways along which possible future decisions alternate with possible outcomes. While we cannot make future decisions now, we can decide how we will respond in future circumstances as opportunities for decisions arise and outcomes take place. A decision tree shows how alternative future decisions and outcomes affect us now, and most importantly, what is the best path to choose at this point in time.

T h e Worth of a Discovery's Future Production DNCF analysis is simple in principle, but the arithmetic is unbearably tedious if the computations are done by hand. It is small wonder that DNCF analysis was not commonplace until computers became widely available. The CASHFLOV^ module of the RISKTAB program is designed to perform these computations in a form specifically tailored for oil ventures. In the following discussion, the examples illustrate the type of analyses that can be done using this software. Why discount? First, the income from an oil-drilling venture (assuming a discovery is made and oil is produced and sold) will be received at some time in the future rather than immediately. If we are to compare alternative 221

Computing Risk for Oil Prospects — Chapter 9 choices in a drilling venture, we must calculate financial consequences at a specific point in time, and the most suitable time is the present because we need to make the initial decision now. In other words, we require present values to assess these alternatives, and as we've pointed out, present values require discounting. Since decision analysis involves choosing among alternatives, we must analyze the alternatives in similar fashion. Megill's (1988) introduction to cash flow analysis in an oil-exploration context provides a good discussion of discounting. Given that we drill a successful well on a prospect, we must wait to recoup our investment, but we also may be able to delay many of the expenses that accompany the operation of a well. Therefore, we'll need to merge future income and future expenses to obtain a consolidated stream of cash, all of which must be discounted to put the well in terms of its present worth. Discounting can have powerful eff^ects, so that the present value of future income may be much less than if received at the present time. Interest rates and discount rates are opposite sides of the same coin, and the rate that is assumed may have a large influence on the value of a property or action. For example, if you earned 10% per year on money invested and the funds compounded annually, you would have to invest only $3860 now to have $10,000 in 10 years (ignoring taxes). To put it another way, the net present value (NPV) of $10,000, received as a lump sum in 10 years, is only $3860 discounted annually at 10%. Discounted at 20%, the present value of $10,000 received in 10 years is only $1620. Table 9.1 provides factors for four alternative discount rates for diff'erent numbers of years. Prom the table, we can see that $10,000 to be received after 20 years and discounted at 20% is worth only $260 today. (Note that the discount factors in Table 9.1 pertain to the end of each year, while module CASHFLOW uses discount factors that pertain to the middle of each year.) To develop an appreciation for the influence of discount rates, imagine what eff^ect interest rates on credit cards would have if you regarded them as discount rates. What would you amass in 20 years if you invested $1000 with a credit card company and it paid you interest at the same rate it charges when it loans you money? At a typical annual rate of 19.8% compounded monthly (ignoring taxes), you would accumulate $50,789! Clearly, the time value of money has major influence on the present worth of an investment. Suppose you were faced with two alternative investments that both paid off* 5 to 1. One investment, however, requires 20 years for payout, and the other requires only 10 years. Obviously the 10-year payout is better, but how much better? DNCF analysis will tell us, depending, of course, on the discount rate and the timing of inflows and outflows. 222

Forecasting Cash Flow for a Prospect Table 9.1. Present value of 1.000 discounted annually for up to 20 years. Discount factor =

1

(i+i)-

i = 0.05 i = 0.10 5% 10%

i = 0.15 15%

i = 0.20 20%

0 1 2 3 4 5 6 7 8 9 10

1.000 0.952 0.907 0.864 0.823 0.784 0.746 0.711 0.677 0.645 0.614

1.000 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 0.386

1.000 0.870 0.756 0.658 0.572 0.497 0.432 0.376 0.327 0.284 0.247

1.000 0.833 0.694 0.579 0.482 0.402 0.335 0.279 0.233 0.194 0.162

15

0.481

0.239

0.123

0.065

20

0.377

0.149

0.061

0.026

Years (n) hence

Table 9.2. Simplified discounted net cash flow for a hypothetical oil well, in $1000 units. Discount rate is 10% compounded annually at the end of each year. (1)

(2)

0 1 2 3 4 5 6 7 8 9 10

0 500 600 700 720 680 540 440 380 280 240

(3)

(5) (6) (7) (4) Net Cum. DisCash Cash cash Discount counted undisinflow outflow flow counted net flow factor Year cash flow $ for year $ $ $ $ % -1500 --1500 -250 250 -190 410 -170 530 -190 530 -210 470 -220 320 -220 220 -230 150 -230 50 -240 0 1430

-1500 -1250 -840 -310 220 690 1010 1230 1380 1430

1.000 0.909 0.826 0.751 0.683 0.621 0.564 0.513 0.467 0.424 0.386

-1500 227 339 398 362 292 180 113 70 21 0 502

(8) Cum. discounted cash flow $ -1500 -1273 -934 -536 -174 118 298 411 481 502

223

Computing Risk for Oil Prospects — Chapter 9 Table 9.3. Schematic outline of principal streams in cash flow analysis for an exploratory well. Nonflows are for income tax computation. I.T.O. represents income tax outflows. Bottom-line figures include (A) undiscounted value and (B) discounted present value.

Time

Years or Quarters Lease Cost

c o © 5

Drilling and Wellhead Equipment Costs

Tangible intangible

Production Income c c

Credits Against Taxable Income Royalties Paid

0)
uj O

Severance Taxes Operating Expenses Overhead (If Charged)

o

1o

I.T.O.

Depletion Depreciation Income Taxes Net Cash Flows Cumulative Net Cash Flows

15 2:

Discounted Net Cash Flows Cumulative Discounted Net Cash Flows

224

•*© •®

Forecasting Cash Flow for a Prospect Table 9.2 provides a simplified discounted cash flow analysis on a yearby-year basis for a hypothetical oil well. Each year, cash flows in from oil produced and sold (Column 2), and cash flows out (Column 3) for the initial drilling costs, subsequent operating costs, and taxes and royalties paid. The initial well cost in Year 0 is $1,500,000. Subsequent cash outflows in Column 3 represent operating costs, severance taxes, income taxes, and royalty payments by year. Cash inflows in Column 2 are sales of oil beginning in Year 1. The well reaches its economic limit and is abandoned during Year 10 when the net cash flow reaches zero. These inflows and outflows provide the raw material for the analysis. The net difference between the inflows and outflows for a year is the yearly net cash flow (Column 4). The net cash flows can be accumulated (Column 5), which tells us when we recover our "front end" investment (the accumulated net flow passes from negative to positive in Year 4). We can insert the discount factor for each year (Column 6) and multiply to get the discounted net flow for each year (Column 7). The yearly discounted net flows in turn can be accumulated to obtain the NPV (Column 8). The NPV for this drilling venture, when discounted at 10%, is $502,000. The undiscounted value is $1,430,000, an enormous difference! What are the NPVs discounted at 5, 15, and 20%? You can easily calculate them from information in Table 9.2 using discount factors taken from Table 9.1. Table 9.3 illustrates the main components of a cash flow analysis for a proposed exploratory well. The increments of time used in a cash flow analysis, such as years, quarters, or months, can be selected arbitrarily. There are five kinds of cash flows: inflows, outflows, nonflows, net flows, and cumulative flows. As their names imply, inflows and outflows represent money received and money spent. Nonflows do not involve cash flowing in or out, but rather represent depletion and depreciation. Although these are not cash items, they are essential for computing income taxes, which ordinarily are outflows (Burke and Starcher, 1993). Net flows represent the algebraic sums of inflows and outflows. Cumulative flows are the progressive sums of flows (usually net flows) through time. Thompson and Wright (1985) provide detailed examples of cash flow analysis in an oil-production context.

225

Computing Risk for Oil Prospects — Chapter 9

USING CASHFLOW Cash flows can be generated for a variety of needs. Engineers, for example, often are interested in projecting cash flows for specific wells whose production rates and decline functions are predicted by engineering analyses. By contrast, in this book we are concerned with evaluating and computing the risk for an entire leasehold, concession, or prospect, and in general do not have (nor require) the detailed production forecasts that engineers use. We are concerned with an entire field or leasehold before it is drilled and seek to place a series of values on it that correspond to different possible outcomes. An evaluation engineer usually deals with wells that have already been drilled and attempts to project their performance into the future. However, if we are to place a value on a prospect, we must analyze the alternative outcomes if a field is discovered in terms of production and income streams that may be derived from its wells, and this requires that we generate successions of cash flows that incorporate production rates and decline functions for wells in different possible classes of production size. Cash flows are so tedious to calculate by hand that even simple applications require the use of computers. The program module CASHFLOW is used here for generating cash flow forecasts on a well-by-well basis, CASHFLOW is flexible enough to apply in many actual situations, and yet is relatively easy to use (as cash flow programs go). Some of the components of CASHFLOW are described in terms that will be most familiar to oil operators in the United States, but the procedures are broadly applicable around the world. For example, oil prices can be specified in American dollars, British pounds, Saudi Arabian riyals or in any unit of currency desired. The user provides CASHFLOW with estimates of initial production rates, producing life span, production decline rates, ultimate cumulative production, oil and gas prices, changes in future prices, income and severance tax rates, royalty rates, discount rates, percentages of revenue interests and working interests, and other information relevant to a prospect. CASHFLOW then generates a detailed projection of cash infiows and outfiows over the life of the well, and in the end provides a single number that represents the undiscounted net present value of the well in whatever monetary unit is used, and also a NPV for every discount rate that has been specified. The tables generated by CASHFLOW are commonly expressed in multiple units, such as thousands of barrels or thousands of dollars. Although computations in CASHFLOW are done using high-precision floating-point arithmetic, the numbers in the output tables are rounded for easier assimilation. As a consequence, values printed in a specific row may not appear to sum exactly to the printed total. Users can be assured, however, that there is no compounding or propagation of arithmetic errors due to 226

Forecasting Cash Flow for a Prospect rounding because all rounding is performed after other calculations have been performed. CASHFLOW is designed to compute cash flows from the standpoint of an investor. For example, suppose we generate a prospect on a leasehold that we already own and wish to calculate an NPV if we drill a well with a 100% working interest. We also may wish to contrast this result with the NPV if we were to farm out the lease and receive a 10% overriding royalty if the well produces, or we may wish to analyze the investment opportunity from the point of view of the owner of the mineral rights. CASHFLOW^ can generate such cash flow projections and NPVs for a variety of alternative situations. Bear in mind, however, that if we farm the prospect out and receive income in the form of an overriding royalty, the cash flow projection and NPV will be very different than the projection and NPV that we would calculate if we were the person or company taking the prospect as a farmin. In other words, CASHFLOW analyzes investments from "our side of the fence" and we must supply input data that are appropriate for this purpose. CASHFLOW analyzes situations from only one viewpoint at a time. If we are the operator, the royalty owners and other working interest owners are on the other side of the fence. Of course, we can exchange roles and view the alternatives from the opposite side. There are circumstances where analyses from "both sides of the fence" are needed. For example, consider a concession in which we simultaneously hold a royalty interest (which lacks a working interest), plus another revenue interest with an associated working interest, CASHFLOW can analyze such a situation, but it requires making two runs and combining the bottom-line results. The initial run could be for the royalty interest, and the second for the other revenue interest with its associated working interest. The NPVs for the two runs (either discounted or undiscounted, or both, as the user chooses) then can be added together. Another "both sides of the fence" situation may arise when an overriding royalty interest changes to a revenue interest accompanied by a working interest after payout. This situation can be analyzed without the need for successive runs. Before payout, the overriding royalty interest may be entered as an ordinary revenue interest, and the accompanying working interest percentage set to zero. After payout, the revenue interest percentage can be adjusted and the appropriate working interest percentage specified. Although CASHFLOW is flexible and adequate for many applications, the program was designed to serve as an introduction to cash flow analysis in an oil exploration context and is simpler than the integrated cash flow programs that may be used in larger oil companies. Users with more complicated financial requirements may find it inadequate for specific needs. A major function of CASHFLOW is to serve as an integral component of module 227

Computing Risk for Oil Prospects — Chapter 9 RAT for generating risk analysis tables when a spectrum of outcomes must be analyzed, the usual situation in prospect evaluation. When CASHFLOW is operated by itself, the analysis is confined to generating a forecast for a specific outcome for an individual well. In addition, neither CASHFLOW nor RAT consider specialized aspects of oil exploration such as evaluating the worth of seismic surveys or alternative lease bidding strategies. CASHFLOW analyzes the simultaneous production of both oil and gas from a well. If only oil is produced, the production rates for gas can be entered as zero, and vice versa if only gas is produced. If both gas and natural gas liquids are produced, production rates and prices for natural gas liquids can be substituted for oil in CASHFLOW. Experienced users will note that CASHFLOW treats taxes in a simplified manner. No direct provision is made for ad valorem taxes (taxes assessed upon the value of a property), which vary widely and may require a complicated calculation of the tax base. Provision is made, however, for computing severance or production taxes (which are widely imposed and generally simple in form) and for income taxes (which are also widely imposed but not quite so simple). If ad valorem taxes are important for a cash flow application, they could be approximately represented indirectly in CASHFLOW by adding an increment to the percentage specified for severance taxes, or alternatively by adding an increment to the operating expenses. CASHFLOW makes calculations on a year-by-year basis as a simplification. In practice, cash flow projections may be made over shorter intervals, such as months or quarter-years. Bottom-line results from analyses, including internal rates of return and net present values, differ only slightly if intervals of less than a year are used. Data may be directly entered into CASHFLOW if it is used in stand-alone form. However, CASHFLOW is repeatedly called by module RAT, because production rates for representative wells will vary from size class to size class in the spectrum of field sizes represented by a probability distribution. This requires that an NPV be generated for a representative well in each class. Mercifully, information required for input to CASHFLOW need be entered only once when RAT is employed, even though separate NPVs are required for each size class. RAT interfaces efficiently with CASHFLOW to generate the succession of NPVs without repeated intervention by the user. When we compute a cash flow for a prospect, there is uncertainty in the resulting forecasts because there is uncertainty in the information on which the forecasts are based. A projected income stream for an oil well requires assumptions about production rates and oil prices, both of which may be highly uncertain. Most explorationists are uneasy when forecasting future prices because they have little background for making such estimates. However, they may take some comfort by generating alternative cash flow 228

Forecasting Cash Flow for a Prospect projections that contrast different assumptions for prices and producing rates. It is a wise practice to examine the effects of different discount rates, oil and gas prices, production rates, decHne functions, taxes, and operating expenses over ranges that are reasonable. Making such comparisons may require many different cash flow projections, but they will show how the assumptions affect the projections, particularly since each projection is reduced to a single number representing the present value in dollars or other monetary units.

Information Required by C A S H F L O W We can think of the inputs to CASHFLOW as forming a multidimensional matrix which collectively provides about 5000 variations in the manner that CASHFLOW can be used. These variations fall into eight principal categories: (I) Treatment of a well's revenue from the viewpoint of whether we are (1) working interest owners or (2) royalty owners. (II) Treatment of a well's expenses from the viewpoint of whether we (1) do hold a working interest or (2) do not hold such an interest. (III) Treatment of whether we have (1) incurred a cost in acquiring a leasehold, a mineral rights interest, or an overriding royalty interest or (2) have not incurred such a cost. (IV) Treatment of the proportions of revenue and working interests with (1) changes after payout, or (2) no changes after payout. (V) Depreciation of tangible capital costs which may involve (1) straightline depreciation, (2) unit-of-production depreciation, or (3) an empirical depreciation table supplied in an external file. (VI) Depreciation of the cost of a leasehold, royalty interest, or mineral rights, which may involve (1) straight-line depreciation, (2) depreciation (termed cost depletion) in the form of unit-of-production depreciation, or (3) an empirical depreciation table supplied in an external file. (VII) Projection of a well's production stream, which may involve (1) an empirical stream supplied in an external file, or (2) a stream based on a parametric function that involves (a) an exponential decline or (b) a hyperbolic decline. These two parametric decline functions can be generated using inputs in the form of {i) ultimate cumulative production, initial production, and the well's life span in years, or {ii) initial production, a production rate for a subsequent year before the economic limit is reached, and the production rate at the economic limit. If a hyperbolic decline curve is selected, a decline exponent must also be supplied. 229

Computing Risk for Oil Prospects — Chapter 9 (VIII) Selection of up to four different discount rates (an undiscounted cash flow as well as a cash flow discounted at the internal rate of return is provided automatically). The user has complete freedom in selecting units for hydrocarbon volumes as well as the monetary units. Examples in this and subsequent chapters utilize barrels (bbls), thousands of cubic feet (MCF), acres, and U.S. dollars; other units such as metric tons, cubic meters, hectares, Japanese yen, German marks, and Piench francs are equally acceptable. Of course the user must ensure that units are in accord with each other. If oil volumes are expressed in metric tons, then oil prices also must be expressed on a metric-ton basis, and so on. The user also should take care that entries expressed in units per day are not confused with units per year. The specific types of input are discussed below in the order in which they must be supplied in an input file. The inputs can be provided interactively using the computer's screen and keyboard by supplying information as requested by the program. After an initial input file has been created in this manner, the user may bring the file back to the screen for editing to create a revised input file. For example, the user may wish to change only a single parameter and rerun the cash fiow, perhaps changing an income tax rate, the escalation rate for oil prices, the number of years of the well's producing life span, or an initial production rate. Revemie interest (RI) consists of the proportion of revenue derived from the well's production stream that is to be received. The overall revenue from the well is shared between the mineral rights owners (who receive a royalty) and the investors in the well. The investors include the operator and perhaps others with fractional working interests, as well as investors who may have purchased or earned royalty interests in the form of overriding royalties. In the United States, it is common for some wells to have many revenue interest owners. A revenue interest is sometimes referred to as a net revemie interest (NRI), but this term is somewhat confusing, so we use the simpler expression "revenue interest" here. At the outset we must make a distinction between revenue interests and working interests. Working interests pertain to a well's costs, including both capital and operating costs. These costs are borne by the operator and by other investors (if there are other investors), with costs allocated in proportion to their respective working interests. The distinction between revenue interests and working interests is vitally important. For example, an investor's working interest in a well is usually higher than the investor's revenue interest. In CASHFLOW, capital costs are shared in proportion to the working interest percentages, including initial capital costs, and subsequent downstream capital costs. If the proportions of working interests change 230

Forecasting Cash Flow for a Prospect Table 9.4. Percentages assigned to revenue interests and working interests before and after payout in Examples 9.3 and 9.4. —Before payout— Revenue Working interest interest

—After payout— Revenue interest

Working interest

Operator

54.0

58.0

47.5

55.0

Other investor

27.5

42.0

32.0

45.0

Mineral rights owner's royalty

12.5

—

12.5

—

6.0

—

8.0

—

Overriding royalty owner

100.0

100.0

100.0

100.0

after payout, the allocation of operating costs and downstream capital costs change accordingly. Consider Examples 9.1 and 9.2 in the section entitled "Example Applications of CASHFLOW." In a simple but common situation, the operator of a well holds an 87.5% revenue interest and a 100% working interest, with the mineral rights owner holding the remaining 12.5% revenue interest by virtue of a royalty interest. In a more complicated situation presented in Examples 9.3 and 9.4 and summarized in Table 9.4, there are two royalty owners, the mineral rights owner with a 12.5% royalty interest and an overriding royalty owner who initially receives a 6% royalty interest. If we were the well's operator, we would receive a 54% revenue interest at the outset, and the other working interest owner (the "other investor," who invested after the prospect was generated) initially receives a 27.5% revenue interest but has agreed to accept a working interest that is substantially higher in proportion. Table 9.4 exactly summarizes the division of revenue and working interests. Of course, the collective revenue interests of all who have a claim to the well's production stream, including royalty owners and working interest owners, must sum to 100%. Provision often is made for shifts in proportions of revenue interests and working interests after a well reaches payout^ which is defined in CASHFLOW as that point where the capital and other costs of the well (except lease costs and income taxes, which are excluded) have been repaid (without discounting) to the investor or investors who hold working interests. Alternatively, an overriding royalty interest before payout may change to 231

Computing Risk for Oil Prospects — Chapter 9 a revenue interest accompanied by a working interest after payout. Many variations are possible. In Table 9.4, payout for the operator will occur before the payout for the other investor, requiring a consistent definition of payout for purposes of defining when changes in revenue and working interests are to occur. In CASHFLOV^, "payout" for purposes of specifying when changes in percentages are to occur is defined in terms of the aggregate of all investors who hold working interests, and is calculated as: payout = total cumulative net operational income before payout — total capital costs (excluding leasehold costs) incurred before payout = 0. In other words, at the payout point, the cumulative sum of all funds that have been previously paid out by working interest owners, including initial and subsequent capital costs, operating expenses, production taxes, and royalties (with income taxes excepted), are balanced by the cumulative sum of the income that they have received. Leasehold costs are generally excluded from the definition of payout. We should also note that the well's initial capital costs are shared in proportion to working interest percentages before payout. Downstream capital costs incurred before payout also are in proportion to working interest percentages held before payout. When working interest percentages change after payout, any subsequent capital costs are shared in proportion to the changed working interest percentages. Payout for an investment in a royalty interest, for example, is diff'erent than payout as defined above. Payment for a specific investment can be ascertained by examining the cash fiow tables. Astute observers will notice that the percentages for revenue and working interests for the operator and investor in Table 9.4 are not in proportion to each other, either before or after payout. One consequence is that the well's operating costs and royalty burden are disproportionally borne by the other investor. While such disparities may be uncommon (and perhaps unrealistic or unjust), the examples in Table 9.4 were chosen to illustrate the flexibility of CASHFLOW. Users can provide any assumptions they wish in allocating the proportions of revenue and working interests, subject to the constraint that aggregate revenue interests and working interests must both sum to 100%. Multiple investors may participate in funding a well's capital costs. In CASHFLOW, the contribution by an investor to capital costs is assumed to be in proportion to the working interest held by the investor. However, this proportion may not be the same as the proportionate revenue interest held by the same investor, so that payout may be attained at different times for different investors with working interests in the well. Overriding royalty rates (ORR) often change after payout, but this requires that the aggregate revenue percentages held by others also change 232

Forecasting Cash Flow for a Prospect so that the overall sum remains at 100%. Table 9.4 shows, for Examples 9.3 and 9.4, that the aggregate percentages assigned to revenue interests after payout must change to accommodate the change in the ORR percentage. Royalty owners who do not participate in a well's costs will be affected by payout if their proportionate revenue interests are subject to change following payout (just as payout affects the working interest owners as defined above). Keep in mind that payout with respect to a royalty owner's investment in a property also can be calculated, but generally will be very different than the payout for the working interest owners. For example, if an investor purchased a royalty interest in a property for $60,000, payout in terms of that investment would be reached when the cumulative royalty income equals $60,000, regardless of when payout is reached for the aggregate of working interest owners. Of course, payout may never be reached by either royalty owners or working interest owners, depending on the circumstances. Working interests (WI) pertain to the allocation of the well's costs, including capital and operating costs (although the capital costs of leases often are excepted in the allocation of costs). As stated above, working interests commonly differ in proportion from revenue interests, and except for royalty owners (who generally lack working interests by definition), the percentages of working interests for operators and other investors are usually greater than their corresponding revenue interests. Consider Examples 9.1 and 9.2. Since there is only a single operator, the operator has a 100% working interest, but the operator's revenue interest is 87.5% because the mineral rights owner receives the remaining 12.5% as a royalty. In other words, the operator receives 87.5% of the well's revenue (before taxes), but pays 100% of the well's capital costs and operating expenses. Meanwhile, the mineral rights owner receives the royalty, but pays none of the well's cost and operating expenses. Next, consider Examples 9.3 and 9.4 in which two revenue interest owners share capital costs and operating expenses between them in proportion to their working interest percentages, but these are different in proportion to their revenue interest percentages, as summarized in Table 9.4. Before payout, as the operator we might hold a 54% revenue interest and a 58% working interest, while the other investor holds a 27.5% revenue interest and a 42% working interest. In other words, before payout (from our perspective) we will receive 54% of the well's revenue and pay 58% of its capital costs and operating expenses. In contrast, the other investor will receive 27.5% of the well's revenue but pays 42% of its capital costs and operating expenses. Before payout, the two royalty owners receive 6% and 12.5% of the well's revenues, respectively, but pay none of the well's capital costs or operating expenses. 233

Computing Risk for Oil Prospects — Chapter 9 After payout in Examples 9.3 and 9.4, both revenue interest and working interest proportions change. Our revenue interest as the operator decreases to 47.5%, the other investor's revenue interest increases to 32%, the mineral rights owner continues to receive 12.5%, and the overriding royalty interest increases to 8%. Simultaneously, the working interest proportions also change; our working interest as operator decreases to 55% and the other investor's working interest increases to 45%. A situation may arise where an overriding royalty interest changes to a revenue interest accompanied by a working interest after payout. Such a situation can be readily handled by CASHFLOW. Before payout, the overriding royalty interest may be entered as a revenue interest at the specified percentage, and the corresponding working interest percentage set to zero. After payout, the revised revenue interest percentage can be entered, accompanied by the appropriate working interest percentage. Royalties may be either paid or received. There are three categories: (1) Royalties paid to mineral rights owners^ (2) overriding royalties paid to other persons^ and (3) royalties received. The royalty rates for the three categories are assumed to be the same for both oil and gas. Royalties paid to mineral rights owners and royalties paid to other persons are exempt from operating expenses. While severance or production taxes usually are imposed on royalties, such taxes are the responsibility of the recipients and therefore do not enter into cash flow calculations made from the viewpoint of the well's operator. On the other hand, royalties that are received are subject to severance and other production taxes, and if we are the recipients as royalty owners, we should include these taxes in our cash flow calculations. Royalties received, in common with royalties in general, are not generally subject to operating expenses or capital costs, although under some circumstances royalty interests may bear certain costs required to bring the oil and gas produced to a marketable state. As noted above, provision can be made for varying royalty rates after payout. The income tax rate consists of the combined tax rate for all income tax obligations (federal, state, and local) lumped together and is expressed in percent. The oil severance ti\x rate pertains to the tax levied on the value of the oil produced or "severed from the ground," and is expressed in percent. Severance taxes are also termed "production taxes." In the United States, the individual states commonly impose severance taxes of several percent or more. Note that severance taxes on royalties paid out are the responsibility of the recipients and do not enter into cash flow calculations. No provision is made in CASHFLOW for representing ad valorem taxes directly, but they can be accommodated if they are represented as a proportion of the value 234

Forecasting Cash Flow for a Prospect of the oil produced. This may be done by adjusting the severance tax rate, or by adding an incremental component to the operating costs. The gas severance tax rate pertains to the tax on the value of gas produced and is expressed in percent. Its computation is the same as the severance tax for oil, except that a different rate can be specified. Discount rates are expressed in percent per year; the user may specify from none to four diff^erent discount rates, CASHFLOW will calculate discounted cash flows for each specified rate, plus an undiscounted cash flow and a cash flow discounted at the internal rate of return^ which is deflned as the discount rate for which the present value is zero. Note that CASHFLOW employs mid-year discounting, represented as 1

(l + if-^-^ where: i = annual discount rate expressed as a decimal fraction (not percent) n = number of years If CASHFLOW is operated in conjunction with RAT, only a single discount rate is supplied (which may be zero if desired) because RAT analyzes a spectrum of outcomes for a prospect and all of them must be calculated at the same discount rate. If different discount rates are desired, RAT must be run repeatedly for each discount rate. Depreciation options apply to tangible capital costs. CASHFLOW provides for treating the depreciation of capital assets that pertain directly to the well, such as the well casing and production equipment and other hardware separately from the depreciation applied to the cost of the leasehold, to mineral rights, or to any overriding royalty. The user has the option of depreciating the entire tangible costs of hardware, or alternatively, depreciating the difference between its cost and salvage value. For both categories, three depreciation options are available: (i) "straight-Une" depreciation, (ii) "unit-of-production" depreciation, and (m) "empirical" depreciation. (a) If straight-line depreciation is selected, years over which depreciation occurs, and the proportion depreciated each year is the is specified, 20% of the capital investment

the user specifies the number of the rate per year is set so that same. If a five-year depreciation is depreciated each year.

(b) \i unit-of'production depreciation is selected, the proportion allocated changes each year. A unit of production for any specific year is defined as the production in that year divided by the reserves forecast to be produced over the remaining life of the well, including the current year's production. 235

Computing Risk for Oil Prospects — Chapter 9 The amount depreciated each year may be expressed as (Cost — Accumulated depreciation) x

Production for year Remaining reserve

Since CASHFLOW provides for two production streams, oil and gas, either oil or gas may be specified as the commodity that provides the basis for unit-of-production depreciation calculations. The predominant commodity should be selected by the user. (c) If empirical depreciation is selected, the user can supply a depreciation schedule of any form, including such widely used schemes as sum-of-thedigits depreciation and declining-balance depreciation. Empirical depreciation thus allows total flexibility. If the user specifies the unit-of-production option for the cost of the leasehold (or concession), mineral rights, and overriding royalty, CASHFLOV^ will automatically treat its cost from a depletion standpoint (discussed below). However, if the user specifies either the straight-line or empirical depreciation options, the cost of the leasehold, mineral rights, and overriding royalty will be treated from a depreciation standpoint. In the United States, leasehold, mineral rights, and overriding royalty costs generally are subject to depletion instead of depreciation, but in other countries the cost of a concession, for example, may be subject to depreciation rather than depletion. Depreciation and depletion are similar in concept and purpose. Percentage depletion is expressed as the percent of the gross income from production that is exempt from income tax. For a well operator, percentage depletion applies to gross income less royalties, but for a royalty owner percentage depletion is applied to gross income. The percentage depletion rate for Federal income tax in the United States for smaller operators is 15%, although operators of marginal wells may claim higher rates. Some U.S. states also provide for percentage depletion in calculating state income tax, usually at the same rate as applied to Federal income tax. The rate is set at zero for operators not entitled to percentage depletion. Cost depletion is applicable to operators in the United States, and the user should specify whether cost depletion is to be taken. Depletion applies to income from production, in recognition that the stream of production is a wasting asset that eventually will be exhausted. In the United States there are two general forms, cost depletion and percentage depletion. Both are based on income from production, but cost depletion is applied to the cost of a leasehold, mineral rights, and any overriding royalty interest, and provides a mechanism for depreciating their costs directly. Percentage depletion represents a fraction of the value of oil or gas that is produced and sold, and permits that fraction to be exempt from income tax. Cost depletion is 236

Forecasting Cash Flow for a Prospect calculated in a manner identical to that for unit-of-production depreciation, and may be expressed as (Cost of lease — Accumulated depletion) x — T-. Remainmg reserve The term "remaining reserve" includes the year for which the calculation is made. Since CASHFLOW provides for two production streams, oil and gas, either oil or gas may be specified as the commodity that provides the basis for unit-of-production depreciation and cost depletion calculations. The predominant commodity should be selected by the user. Cost depletion often is applicable early in the life of the lease when the production rate is low. The general rule is to take whichever form of depletion is greater (percentage depletion or cost depletion), which is referred to as the allowable depletion. The allowable depletion, however, may be subject to a provision that the percentage depletion cannot exceed a specified percent (which the user specifies) of the net for depletion purposes (NFDP) from a property. For an operator, the NFDP is defined as gross income less royalties, less operating expenses and other intangible costs, less severance taxes, and less depreciation. If the user specifies that cost depletion is to be taken, CASHFLOW^ will then calculate and compare both cost depletion and percentage depletion, and take the maximum as the allowable depletion, subject to the constraint that percentage depletion not exceed the NFDP. If cost depletion is specified and percentage depletion is set at zero, the calculations will be carried out but only cost depletion will be taken. Leasehold cost is the cost incurred when a lease is acquired. Any lease delay rentals (which are usually paid annually) can be prepaid and lumped with the overall lease cost (common practice in the United States), or treated as downstream expenses. Once a well goes into production, the lease is held by production ("HBP") and delay rentals are no longer due. While leases generally are considered to be tangible assets, lease costs usually are not depreciated in the same manner as other tangible assets. However, in the United States, cost depletion provides a substitute for depreciation of a lease. A lease has no salvage value. If a lease is obtained and the property fails to produce, the lease cost is taken as an expense in the year in which the lease is abandoned. In countries where the government grants concessions, the cost of a concession can be entered into CASHFLOW^ in place of the leasehold cost. Mineral rights cost refers to the cost for the right to exploit "minerals." In the United States, mineral rights generally are privately owned except on public lands, and the owner of mineral rights is entitled to royalties if oil or gas are produced from the owner's property. 237

Computing Risk for Oil Prospects — Chapter 9 It is common practice in oil- and gas-producing regions of the United States to separate the ownership of mineral rights from the ownership of the land's surface, so that some or all of the mineral rights attached to a parcel of land may be sold separately from the land itself. There is an active market for mineral rights, and one way of participating in oil production is to buy the mineral rights for areas that produce or are potentially productive. Mineral rights are wasting assets when oil and gas are produced, so depletion provides a substitute for depreciation, at least in the United States. The costs of mineral rights are treated in the same manner as leasehold costs within CASHFLOW. Overriding royalty costs also are treated identically to leasehold costs. Initial tangible costs consist of those initial capital costs that are to be depreciated rather than expensed in the initial year. They include the tangible costs of drilling, casing, and completing the well, production equipment, and tangible costs incurred in providing physical access to the drill site. In CASHFLOW, the initial capital costs (both tangible and intangible) are allocated in proportion to the working interest percentages held initially. As an option, downstream tangible capital costs can be supplied for subsequent years. In CASHFLOW, such downstream costs (both tangible and intangible) are scaled to be proportional to the working interest percentages held at the time the costs are incurred, whether before or after payout. Downstream intangible capital costs also can be supplied for subsequent years, and likewise are scaled to the working interest proportions. Salvage value is expressed as a percentage of the aggregate costs of all capital tangible assets over the life of the well, with the exception of leasehold, royalty, and concession costs. The salvage value is a cash inflow when the well is abandoned. Although leases also are tangible assets, they have no salvage value when abandoned or surrendered. Abandonment costs are expressed as a percentage of the well's initial capital costs, both tangible and intangible, and are a cash outflow when the well is abandoned. The year in which the well goes on production must be supplied. Normally this is in Year 1, but the user may specify a later time. Operating costs can be specified on a year-by-year basis, for any number of years. The user must at least specify the operating cost for the first year. After the last cost is provided, the operating costs change at a fixed ratio given by the ratio of change in annual operating costs. This specifies the relative change from one year to the next after the last year in which specific estimates are provided. If overhead (general administrative expenses) is to be charged, it can be incorporated as a component of the operating costs. The life span of the well's production is given in years. The production forecast for the life of the well can be specified either as (a) an empirical 238

Forecasting Cash Flow for a Prospect production stream on a year-by-year basis for the life of the well, (b) as an exponential (constant rate) decline, or (c) as a hyperbolic decline (p. 243). While the user can choose different units for the oil or gas produced, the following examples use barrels (bbls) and MCF. Users should make sure that units are compatible, so that barrels per day, for example, are not confused with barrels per year. If the option empirical production stream is selected, the user must supply the amount of oil and of gas to be produced each year over the producing life of the well. The decline in oil production may differ from the decline for gas. If oil is produced and gas is not, the user must supply a value of zero for each year for gas. Values of zero must be supplied for oil if only gas is produced. The file can be edited using RISKSTAT, providing total flexibility because the well's production stream can be of any form and can include production constraints such as proration allowables and pipeline capacities. However, when CASHFLOW is used with module RAT, the production-stream option is not available because providing individual year-by-year production forecasts for wells in each field size class would be excessively complicated. RAT requires either an exponential or hyperbolic decline function. If the exponential decline option is selected, the proportional decline in production each year remains constant. Oil and gas declines are treated separately. For oil, the user specifies (1) initial producing (IP) rate in bbls (or other units) per day, (2) ultimate cumulative production in bbls, and (3) life span of the well in years. For gas, the user specifies initial production per day and cumulative production in MCF (or other units), but only one life span is specified, so it is necessarily the same for gas and oil. If only oil is produced, the IP and ultimate cumulative production figures for gas are set to zero, and vice versa if only gas is produced. CASHFLOW automatically computes decline rates for oil and for gas, each based on the three values provided. Clearly, the initial production, decline rate, producing Hfe span, and ultimate cumulative production must be consistent because they collectively form a closed system. Figure 9.2 shows cumulative production versus time at a constant decline rate; the plot forms a straight line on semi-log paper. Note that in the equations that follow we specify the initial decline rate, even though the decline rate in exponential decline remains constant. We use this notation to be consistent with our subsequent treatment of hyperbolic decline, where the decline rate is not constant. Any year's production can be calculated for an exponential decline with the following equation: Qt — Qi^

where: 239

Computing Risk for Oil Prospects — Chapter 9 1 00000 •!

;

,

CO 0

-°— •

SI

Exponential decline Hyperbolic decline

^ lOOOOzl: c .o o 3

100010

Year

20

30

Figure 9.2. Exponential and hyperbolic production-decline curves for oil well with initial production of 55 bbls per day and cumulative production of 200,000 bbls. Life span of well is 30 years. Decline exponent b for hyperbolic decline function is 0.5. Qi = initial production rate Qt = production rate at year t Di = initial decline rate. Cumulative production at any year can be calculated by: Qt =

Di

v^here: Qt = cumulative production at year t. CASHFLOW calculates the production stream as follows: Using the exponential decline equation, the production rate during the last year of the well's life is given by qe = qie'^'^^ where: qe = production rate at economic limit in the final year, tg. 240

Forecasting Cash Flow for a Prospect We can also calculate the rate in this final year by rearranging the equation for cumulative production, yielding: qe = Qi-

QeDi

where: Qe = cumulative production at the economic limit. Combining the final two equations above, we can set up a nonlinear equation for the initial decline rate and then calculate each year's production:

This equation can be solved numerically for the initial decline rate by Newton's method:

^' ^ r{D,) where: n = iteration index f{Di) = function determined from the initial decline rate. The production rate beyond the specified life span may still be substantial because the ultimate production and life span may not be consistent with the specified decline function. For example, in Figure 9.2, the production during the last year is greater than 2000 bbls per year for hyperbolic decline. The user can compensate for this by specifying a longer life span that will include most of the tail of the decline curve. CASHFLOW^ provides an alternative to exponential decline that does not require estimating the ultimate cumulative production nor the life span of the well. If we provide the initial production rate (in bbls per day) and a production rate (also in bbls per day) for any subsequent year excepting the final year, the decline rate and each year's production can be calculated by rearranging the equation for cumulative production:

A = iln(^ Figure 9.3 shows annual production for a well having an initial production of 55 bbls per day. The upper curve is based on an ultimate cumulative production of 200,000 bbls and a life span of 30 years. The lower curve is based on a production rate in the middle of Year 5 of 20 bbls per day (7300 bbls per year), and a production rate of 1 bbl per day at the economic 241

Computing Risk for Oil Prospects — Chapter 9 20000 Based on cumulative production Based on production rate 0)

t

0)

§• 10000 c .o o 3

Year Figure 9.3. Exponential production decline curves based on cumulative production (top curve) and production rate (bottom curve). limit. This results in a cumulative production of only 87,000 bbls. If the production rate at Year 5 for the lower curve had been set at 35 bbls per day, the two curves would be nearly coincident. The lifetime of a well can then be derived from this equation, knowing the production rate at the economic limit: t. =

^ln

(I)

These two alternative forms of input for specifying exponential decline provide added flexibility in the use of CASHFLOW. Hyperbolic decline is more complicated than exponential decline because it involves one more parameter, the hyperbolic decline exponent h which can range between zero and one. A hyperbolic decline plots as a curved line on semi-log paper (Fig. 9.2), where the sharpness of curvature is specified by the decline exponent (Fig. 9.4). Higher values of h indicate lower production in early years, which is compensated by higher production in later years. The exponential decline curve is a special case of the hyperbolic decline curve when the decline exponent is zero. Tinker and 242

Forecasting Cash Flow for a Prospect 100000 inHyperbolic decline, b = 0.8 Hyperbolic decline, b = 0.5 Hyperbolic decline, b = 0.2 Exponential decline, b = 0.0

t SI

B "(5 cc 10000^: c o

1000Year Figure 9.4. Hyperbolic production decline curves showing effect of different decline exponents, b. Initial production is 55 bbls of oil per day, cumulative production is 200,000 bbls, and life span of well is 30 years. Skov (1993) provide an extensive discussion of the use of hyperbolic decline functions. Any year's production rate can be calculated for hyperbolic decline according to the following equation:

where: b = decline exponent. Cumulative production at any year can then be calculated from: Qt =


(7f-^^-.ro

In CASHFLOW, the first year's production, ultimate cumulative production, and the life span of the well are provided by the user. The initial decline rate can then be calculated from the equation below, which is derived in a manner analogous to the derivation of the exponential decline 243

Computing Risk for Oil Prospects — Chapter 9 function: 1/(1-6)

qi (1 + hDite)

hjrrh)

^ ^

An alternative form of input is also provided. Given an additional production rate, the initial decline rate can be calculated by rearranging the hyperbolic decline curve equation:

""W,

)

b

-1

For the initial decline rate, we can calculate each year's production by solving the hyperbolic decline curve equation, which also can be rewritten to yield the lifetime of the well if the production rate at the economic limit is specified: U =

bDi

(I)'

1

Oil and gas price forecasts are required to compute cash flows. The prices during the first year of production must be estimated. Following the first year, the user has the option of supplying price estimates for succeeding years on a year-by-year basis for any number of years. As an alternative, the user can specify changes in subsequent years as a ratio representing the change from one year to the next. The ratio for annual change in the price of oil may differ from that for the price of gas. Each ratio can represent an increase, decrease, or no change. Output options include several alternatives, including printing or not printing the output. If the output is printed, the user again has two options: (1) Cash flow projections are printed out until the economic limit is reached. The economic limit is deflned as that point when operating expenses equal or exceed net revenue after severance taxes have been subtracted. (2) Cash flow projections are printed out until either the economic limit is reached or a specified number of years has elapsed, whichever comes first. However, cash flow calculations are carried out until the economic limit is reached, regardless of the number of years represented in the printed output. 244

Forecasting Cash Flow for a Prospect Table 9.5. Input to CASHFLOW for Example 9.1.

***REVENUE*AND*WORKING*INTERESTS*IN*PERCENT****************** Before After 1 Royalties paid to mineral rights payout 12.50 12.50 owners 0.00 0.00 2 Overriding royalties paid 0.00 0.00 3 Royalties we receive 0.00 0.00 4 Revenue paid to other working interests 5

6 7

Revenue we receive as working interest owner Sum Our working interest

87.50 100.00

87.50 100.00

100.00

100.00 0.00 100.00

0.0

Working interest of others Sum

100.00

3(c 3|c * 3|e 3|e 3|c 3|c * 3|c ]«£ H e * * 3 K 9fc 9|c 3|c 3|c 3|e 3|c * 3|c 3ic 3|c 3ic ale * )|c J k * * * * * * >te * * * * * * * * * 3ic * * > k

***TAX*AND*DISCOUNT*RATES************************************ 1 2 3

Income tax rate CD Oil and gas severance tax rate (%) Discount rates (none up to 4) CO

6.00

5.00 9.00

28.00 5.00 12.00

******3|c*********a)c3|e:jcj|c*****************************************

***TANGIBLE*AND*INTANGIBLE*COSTS***************************** 1 Tangible costs at year 0 ($) 120000.00 2 Intangible costs at year 0 ($) 135000.00 3

Number of years for which a schedule of tangible & intangible capital costs is to be entered

4

Capital costs for subsequent years ($) Intangible Year Tangible

1 2 3 4 5 6 7

20

0. 0. 0 0. 0.

0. 0. 0. 0. 0.

15000.

17000.

0. Cont.

0.

245

Computing Risk for Oil Prospects — Chapter 9 8 9 10 11 12 13 14 15 16 17 18 19 20

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

18000.

16500.

***DEPLETION************************************************* 1 Percentage depletion 15.00 2 Limit for depletion C/,) 65.00 Jtc 3tC *

*

3tC 3|C % 3tC :4c 3(C 3|C 3iC :)C 3(C 3|C 3|C 3|C 3|C SK 3(C *

3K 3|C ](C 3|C 3|C 3|C 3tC 9|C 3K *

3|C *

*

*

3|C >K *

*

*

*

*

3K }*( *

>^

***DEPRECIATION OF TANGIBLE COST***************************** 1 Depreciation Function Straight line (1) Unit of production (2) Empirical depreciation (3) 2 ***ADDITIONAL*COSTS****************************************** 1 Leasehold cost ($) 32000.00 2 Mineral rights or ORR cost ($) 0. 3 Abandonment cost as proportion of well's aggregate capital cost (X) 5.00 4 Salvage value as proportion of well's tangible cost (X) 8.00 ***DEPRECIATIGN*OF*LEASEHOLD,*ORR,*OR*MINERAL*RIGHTS*COST**** 1 Depreciation Function Straight line (1) Unit of production (2) Empirical depreciation (3) 2 ***PRINT*QPTIONS********************************************* 1 Print cashflow tables (y/n) y 2 Print results up to economic limit (1)

Cont. 246

Forecasting Cash Flow for a Prospect or up to specified year or economic limit whichever comes first (2)

1

***OPERATING*COSTS******************************************* 1 Year in which well starts operating 1 2 Number of years for which specific operating costs are entered 3 Operating costs for these years 3 Year / Operating costs ($) 1 4000.00 2 4200.00 3 4400.00 4 Ratio of change in operating costs (costs of current yeair/previous year) 1.04 ***WELL*PRODUCTION*FORECAST********************************** 1 Production stream from file (f) or by pairametric decline function (p) p 2 Decline based on cumulative production (1) or based on a second production rate (2) 1 3 Exponential(1) or hyperbolic(2) decline 1 Production of first operation year 4 Average oil production (bbls/day) 55 5 Average gas production (MCF/day) 0 Ultimate cumulative production 6 Ultimate cums. for oil (bbls) 200000 7 Ultimate cums. for gas (MCF) 0 8 Lifespan of well (years) 30 ***HYDRGCARBON*PRICE*FORECAST******************************** 1 Number of years for which specific price forecasts are entered (>=1) 2 Price forecast for these years 2 Year Gil price ($/bbls) Gas price ($/MCF) 1 19.00 1.50 2 19.50 1.55 Change of price in subsequent years (price of current year/previous year) 3 Change in oil price 1.03 4 Change in gas price 1.04

247

Computing Risk for Oil Prospects — Chapter 9

Table 9.6. CASHFLOW output for operator with 100% working interest DISCOUNTED NET CASH FLOW FOR A SINGLE PRODUCING WELL

NET PRESENT VALUES ( i n $ X 1000) :

2715. Ex 9- 1, looy, working interest Undiscoimted[ 07. 1564. at 6.0y. ECONOMIC LIMIT REACHED AT YEAR 30 Discounted 1260. at 9.07. PAYOUT DURING YEAR Discounted 1 1044. at 12.07. Discounted INTERNAL RATE OF RETURN « 135.88 PERCENT •••••••••••••••••*********NET*OPERATING*INCOME*SCHEDULE* 4c4i4c4>4<4>4(4e4c4c4'4(4'4c)«c4c4c4c4c4(«* 10 7 8 9 2 5 6 3 1 4 YEARLY YEARLY OIL NET SEVRNC OPERAT NET GROSS R0YLT+ GAS TAX COSTS OPERAT GAS PRICE PRICE INCOME OTHER REVENU OIL YEAR PRODUC PRODUC (O+G) INCOME REVINT X 1000 X 1000 X 1000 X 1000 X 1000 X 1000 X 1000 X 1000 BBL/YR MCF/YR $/BBL $/MCF $ $ $ $ $ $ %4t**4(*4c«« • • • • • • • • • • • « • • • • • • • • • • • • • • • • • • • « « • • • • • • • • • • • • • • • • • • « * • • • • • • • • • • • • • •

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0. 19.2 17.4 15.9 14.4 13.1 11.9 10.9 9.9 9.0 8.2 7.4 6.8 6.2 5.6 5.1 4.6 4.2 3.8 3.5 3.2 2.9 2.6 2.4 2.2 2.0 1.8 1.6 1.5 1.4 1.2

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

TOTAL

200.

0.

248

19.00 19.50 20.08 20.69 21.31 21.95 22.61 23.28 23.98 24.70 25.44 26.21 26.99 27.80 28.64 29.50 30.38 31.29 32.23 33.20 34.19 35.22 36.28 37.36 38.48 39.64 40.83 42.05 43.32 44.61

1.50 1.55 1.61 1.68 1.74 1.81 1.89 1.96 2.04 2.12 2.21 2.29 2.39 2.48 2.58 2.68 2.79 2.90 3.02 3.14 3.27 3.40 3.53 3.67 3.82 3.97 4.13 4.30 4.47 4.65

0. 364. 340. 318. 298. 280. 262. 246. 230. 216. 202. 189. 178. 166. 156. 146. 137. 128. 120. 113. 106. 99. 93. 87. 81. 76. 72. 67. 63. 59. 55.

0. 45. 42. 40. 37. 35. 33. 31. 29. 27. 25. 24. 22. 21. 19. 18. 17. 16. 15. 14. 13. 12. 12. 11. 10. 10. 9. 8. 8. 7. 7.

0. 318. 297. 279. 261. 245. 229. 215. 201. 189. 177. 166. 155. 146. 136. 128. 120. 112. 105. 99. 92. 87. 81. 76. 71. 67. 63. 59. 55. 52. 48.

0. 16. 15. 14. 13. 12. 11. 11. 10. 9. 9. 8. 8. 7. 7. 6. 6. 6. 5. 5. 5. 4. 4. 4. 4. 3. 3. 3. 3. 3. 2.

0. 4. 4. 4. 5. 5. 5. 5. 5. 6. 6. 6. 6. 7. 7. 7. 7. 8. 8. 8. 9. 9. 9. 10. 10. 10. 11. 11. 12. 12. 13.

0. 299. 278. 260. 244. 228. 213. 199. 186. 174. 162. 151. 141. 132. 123. 114. 107. 99. 92. 85. 79. 73. 68. 63. 58. 53. 49. 44. 41. 37. 33.

4947.

618.

4329.

216

228.

3885.

Forecasting Cash Flow for a Prospect and 87.5% revenue interest based on input file presented in Table 9.5.

•••••iNVESTMENT*SCHEDULE**** •••••••••••***INCOME*TAX*SCHEDULE***********•*•* 11 12 13 14 15 16 17 18 19 20 21 LEASE TANGIB INTANG ABDCST DEPREC COST PERCNT PERCNT ALLOW NT TAX INCOME ORR OR COST COST - SLVG DEPLET DEPLET DEPLET DEPLET INCOME TAX7. MR CST COST LIMIT 28.000 X 1000 X 1000 X 1000 X 1000 X 1000 X 1000 X 1000 X 1000 X 1000 X 1000 X 1000

$

$

$

$

$

$

$

$

$

$

$

>|i «4(« • • • • • • • • • • • ••4c4c4i«««4c«^4'4c^>('4c%>f>(c^««%4c%«%% • • • • • • • • • • • ' • ( • • • • • • • • ^ ( • • • • • • • «

32. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 32.

120. 0. 0. 0. 0. 0. 15. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 18. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

135. 0. 0. 0. 0. 0. 17. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 17. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.

0. 11. 10. 10. 9. 8. 9. 8. 7. 7. 6. 5. 5. 4. 4. 4. 3. 3. 3. 3. 5. 4. 4. 4. 3. 3. 3. 2. 2. 2. 2.

0 3 3 3 2. 2. 2. 2 2

153.

169.

1.

153.

32.

0 0 0 0 0 0 0 0 0 0

0. 48. 45. 42. 39. 37. 34. 32. 30. 28. 27. 25. 23. 22. 20. 19. 18. 17. 16. 15. 14. 13. 12. 11. 11. 10. 9. 9. 8. 8. 7. 652.

0. 187. 174. 163. 153. 143. 122. 124. 116. 109. 102. 95. 89. 83. 77. 72. 67. 62. 58. 53. 37. 44. 41. 38. 35. 32. 29. 27. 24. 22. 20.

0. 48. 45. 42. 39. 37. 34. 32. 30. 28. 27. 25. 23. 22. 20. 19. 18. 17. 16. 15. 14. 13. 12. 11. 11. 10. 9. 9. 8. 8. 7. 649.

-135. 239. 223. 209. 196. 183. 153. 159. 149. 139. 130. 121. 113. 105. 98. 92. 85. 79. 73. 68. 44. 56. 52. 48. 44. 40. 37. 33. 30. 27. 24. 2914.

-38 67 63 59 55 51 43 44 42 39 36 34 32 30 28 26 24 22 21 19 12 16 14 13 12 11 10 9 8 8 7 816.

249

Computing Risk for Oil Prospects — Chapter 9

Table 9.6.

Concluded.

• ••••••••»»»»»»»»»»»»»»»»r:ASH4cFT.nW4cSnffl?nTTT.F.»*»*»»»»»»»»»»»»»* 22 23 24 25 26 27 28 29 NET LEASE TANGBL INTANG INCOME NET CUM NT CUM NT YEAR OPERAT ORR OR COST +ABCST TAX CASH CF W/0 CF W/ INCOME MR CST -SLVGE FLOW INCTAX INCTAX X 1000 X 1000 X 1000 X 1000 X 1000 X 1000 X 1000 X 1000 $ $ $ $ $ $ $ $ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 TOTAL

250

0. -32. -120. 299. 0. 0. 278. 0. 0. 260. 0. 0. 244. 0. 0. 228. 0. 0. 213. 0. -15. 199. 0. 0. 186. 0. 0. 174. 0. 0. 162. 0. 0. 151. 0. 0. 141. 0. 0. 132. 0. 0. 123. 0. 0. 114. 0. 0. 107. 0. 0. 99. 0. 0. 92. 0. 0. 85. 0. 0. 79. 0. -18. 73. 0. 0. 68. 0. 0. 63. 0. 0. 58. 0. 0. 53. 0. 0. 49. 0. 0. 44. 0. 0. 41. 0. 0. 37. 0. 0. 33. 0. 0.

-135. 0. 0. 0. 0. 0. -17. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -17. 0. 0. 0. 0. 0. 0. 0. 0. 0. -1.

38. -249. -67. 232. -63. 216. -59. 202. -55. 189. -51. 176. -43. 138. -44. 155. -42. 144. -39. 135. -36. 126. -34. 118. -32. 110. -30. 102. 95. -28. -26. 89. 83. -24. 77. -22. 71. -21. 66. -19. -12. 32. -16. 58. 53. -14. 49. -13. 45. -12. -11. 42. -10. 38. 35. -9. 32. -8. -8. 29. -7. 26.

3885. -32. -153. -169. -816. 2715.

-287. 12. 290. 550. 794. 1021. 1202. 1401. 1587. 1761. 1923. 2075. 2216. 2348. 2474. 2585. 2692. 2791. 2883. 2968. 3013. 3086. 3154. 3217. 3274. 3327. 3376. 3421. 3461. 3498. 3531.

-249. -18. 198. 400. 589. 765. 903. 1058. 1202. 1337. 1463. 1580. 1690. 1792. 1888. 1976. 2099. 2136. 2207. 2274. 2306. 2364. 2417. 2467. 2512. 2554. 2592. 2627. 2660. 2689. 2715.

•DNCF*SCHEDULE*Q*6y, 30 31 32 DISCT DISCTD CUMU FACTR NCF DNCF X 1000 X 1000 67. $ $

1.00 0.97 0.92 0.86 0.82 0.77 0.73 0.68 0.65 0.61 0.57 0.54 0.51 0.48 0.46 0.43 0.41 0.38 0.36 0.34 0.32 0.30 0.29 0.27 0.25 0.24 0.23 0.21 0.20 0.19 0.18

-249. 225. 198. 174. 154. 136. 100. 106. 93. 82. 72. 64. 56. 49. 43. 38. 34. 29. 26. 23. 10. 17. 15. 13. 12. 10. 9. 8. 6. 6. 5. 1564.

-249. -24. 173. 348. 502. 637. 738. 844. 937. 1019. 1091. 1155. 1211. 1261. 1304. 1342. 1376. 1405. 1431. 1453. 1464. 1481. 1497. 1510. 1521. 1531. 1540. 1548. 1554. 1560. 1564.

Forecasting Cash Flow for a Prospect

••DNCF*SCHEDULE*Q*97.*** **DNCF*SCHEDULE*Q* 12*/.***•*•** ***DNCF*SCHEDULE*Q*135.887.

33

34

35

36

37

38

(39-

42

44

43

41) DISCNT FACTOR

97.

DISCTD

NCF X 1000

$

CUMULT

DISCNT

DISCTD

DNCF

FACTOR

NCF

X 1000

$

X 1000 127.

$

CUMULT

DISCNT

DISCTD

DNCF

FACTOR

NCF X 1000

X 1000

$

1367.

$

CUMULT DNCF X 1000

$

«4e 4(4c4i««« ••4'••• •••>tc4c 4c ••••^(•••••••••••••••••••••••••••4e« • • • • • • • • • • • • • • •

1.00

-249.

-249.

1.00

-249.

-249.

1.00

-249.

-249.

0.96

222.

-27.

0.94

219.

-30.

0.65

151.

-98.

0.88

190.

162.

0.84

182.

152.

0.28

-39.

0.81

163.

325.

0.75

152.

304.

0.12

0.74

140.

464.

0.67

127.

431.

0.05

0.68

120.

584.

0.60

106.

536.

0.02

0.62

0.09

86. 88. 76. 65. 56. 48. 41. 35. 30. 25. 22. 19. 16. 13. 6. 10. 8. 7. 6. 5. 4. 4. 3. 3.

0.08

2.

0.57 0.52 0.48 0.44 0.40 0.37 0.34 0.31 0.29 0.26 0.24 0.22 0.20 0.19 0.17 0.16 0.14 0.13 0.12 0.11 0.10 0.09

1260.

670.

0.54

758.

0.48

834.

0.43

899.

0.38

954.

0.34

1002.

0.30

1043.

0.27

1077.

0.24

1107.

0.22

1133.

0.19

1154.

0.17

1173.

0.15

1189.

0.14

1202.

0.12

1208.

0.11

1218.

0.10

1227.

0.09

1234.

0.08

1240.

0.07

1245.

0.06

1249.

0.06

1253.

0.05

1256.

0.04

1258.

0.04

74. 74. 62. 51. 43. 36. 30. 25. 21. 17. 14. 12. 10. 8. 4. 6. 5. 4. 3. 3. 2. 2. 1. 1.

1260.

0.04

1. 1044.

1013.

0.00

1016.

0.00

1022.

0.00

1027.

0.00

1031.

0.00

1034.

0.00

1036.

0.00

1039.

0.00

1040.

0.00

1042.

0.00

1043.

0.00

60. 24. 9. 4. 1. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

1044.

0.00

0.

611.

0.01

685.

0.00

746.

0.00

798.

0.00

841.

0.00

876.

0.00

906.

0.00

931.

0.00

952.

0.00

969.

0.00

983.

0.00

995.

0.00

1005.

0.00

-15.

-6. -2. -1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0.

251

Computing Risk for Oil Prospects — Chapter 9

OUTPUT FROM CASHFLOW The output from CASHFLOW can be sent to an output file for printing. The example input file shown in Table 9.5 produces the output file given in Table 9.6 (these tables pertain to Example 9.1, described later). CASHFLOW prints both input and output files, providing the user with a complete record of each cash flow analysis. The economic limit in years is defined as that time at which operating costs equal or exceed net revenue after severance taxes have been subtracted, and the well is no longer economic. The economic limit may be reached before the last year specified in cash fiow projections has been reached, but calculations and printing of the cash flow streams cease once the economic limit has been reached. In all circumstances, CASHFLOW carries out the calculations until the economic limit is reached. If the last year specified to be printed is reached before the economic limit, the final row of each table gives a summary of the years between the last year specified and the year when the economic limit is reached. The payout year is when the cumulative undiscounted cash flow reaches zero, or the year when the accumulated inflows equal the accumulated outflows and the capital and other costs incurred by the working interest owners considered in the aggregate have been recouped from production income (disregarding income taxes and leasehold costs). The internal rate of return (IRR) is the specific discount rate at which the cumulative discounted net cash flow is zero. Usually there is only a single IRR, although under certain circumstances more than one may be defined. Internal rate of return represents the rate that the investment actually provides, and since it is usually represented by a single number for each investment, it is convenient for comparing alternative investments. The operating income schedule is provided as a table containing numbered columns in which each year is represented by rows. The user may select the units that are used. In the following examples, barrels (bbls), MCF, and U.S. dollars are used. The table contains the following columns: Col. Col. Col. Col. Col.

Oil production in thousands of bbls. Gas production in thousands of MCF. Oil price in dollars per barrel Gas price in dollars per MCF. Gross income in dollars [oil production (1) x oil price (3) + gas production (2) x gas price (4)] Col. 6: Royalties paid [royalty rate x gross income (5)]. Col. 7: Net revenue [gross income (5) — royalties (6)]. Col. 8: Severance tax [severance tax rate x net revenue (7)]. 252

1: 2: 3: 4: 5:

Forecasting Cash Flow for a Prospect Col. 9: Operating costs for the working interest, which can include overhead costs if taken. Col. 10: Net operating income [net revenue (7) — severance tax (8) — operating costs (9)]. The investment schedule is presented in a second table which has the following columns: Col. 11: Leasehold, concession, mineral rights, or overriding royalty cost in thousands of dollars. Col. 12: Tangible capital costs in thousands of dollars. Col. 13: Intangible capital costs in thousands of dollars. Col. 14: Abandonment costs less salvage value in thousands of dollars are entered in the year in which the economic limit is reached. The income tax schedule forms the third output table with these columns: Col. 15: Depreciation in thousands of dollars of tangible costs, including that for both the well's tangible assets or "hardware," plus depreciation for the cost of the leasehold (or concession), overriding royalty, or mineral rights, provided that the unit-of-production option has not been specified. For example, the capital cost of a concession in a foreign country may be subject to straight-line depreciation, while in the United States the cost of a leasehold is generally subject to depletion (which involves unit-of-production calculations if cost depletion is considered). Col. 16: Cost depletion in thousands of dollars with respect to the cost of the leasehold, ORR, or mineral rights. This column, as well as columns (17), (18), and (19) will be supplied with information only if the unit-of-production option is selected for depreciation of the leasehold, ORR, or mineral rights cost. Otherwise, depreciation of the leasehold, ORR, or mineral rights will be merged with depreciation of the well's tangible costs and printed in column (15). Col. 17: Percentage depletion in thousands of dollars [percentage depletion rate x net revenue (7)]. Col. 18: Percentage depletion limit in thousands of dollars. In the United States, the amount that may be taken for percentage depletion may be hmited to a specified percentage of the net for depletion purposes, defined as [net operating income (10) — depreciation of the well's tangible costs (15) — intangible capital costs (13)]. Col. 19: Allowable depletion in thousands of dollars is equal to the maximum of either cost depletion (16) or percentage depletion (17); 253

Computing Risk for Oil Prospects — Chapter 9 in the United States, percentage depletion may be subject to the percentage depletion Umit (18). Col. 20: Net taxable income [net operating income (10) — intangible capital costs(13) — depreciation (15) — allowable depletion (19)]. Col. 21: Income tax [income tax rate x net taxable income (20)]. If there is net loss for the year, a negative sign denotes a credit against other taxable income, assuming the investor has other taxable income against which the credit may be applied. The undiscounted cash flow schedule forms the next table, in which the algebraic signs, positive or negative, denote inflows or outflows: Col. 22: Net operating income in thousands of dollars [repeated from column (10) for clarity]. Col. 23: Leasehold, concession, mineral rights, or overriding royalty capital cost [same as (11), with negative sign to indicate an outflow]. Col. 24: Tangible capital costs [same as (12), with negative sign to indicate an outflow]. Col. 25: Intangible capital costs, abandonment cost, and salvage value grouped in the same column, with negative signs to indicate outflows, except that the algebraic sum of salvage value and abandonment cost may be positive if the salvage value exceeds the abandonment cost. Col. 26: Income tax, which is the same as (21), except that negative signs indicate outflows. If positive, a tax credit is implied because there is a net loss for the year, and the loss creates an inflow (assuming taxes on other income subject to tax are reduced accordingly). Col. 27: Undiscounted net cash flow [net operating income (10) — leasehold cost (11) — tangible capital costs (12) — intangible capital costs (13) - income tax (21)]. The net cash flow for each year may be positive or negative. Col. 28: Cumulative undiscounted net cash flow without income tax considered [sum for that year plus all previous years]. Col. 29: Cumulative undiscounted net cash flow with income tax considered [sum for that year plus all previous years]. Finally, discounted net cash flow schedules for each discount rate specified are provided for none to four different discount rates. Groups of three columns numbered consecutively are provided for each specified discount rate. Schedules for the first discount rate are provided in columns (30) through (32): 254

Forecasting Cash Flow for a Prospect Col. 30: Discount factor for year at the given rate, calculated for middle of year. Col. 31: Discounted net cash flow for year [undiscounted net cash flow with income tax considered (29) x discount factor (30)]. Col. 32: Cumulative discounted net cash flow [sum of current year (31) and all previous years]. Subsequent sets of three columns provide cash flow schedules for additional discount rates specifled by the user, up to Col. 41. A net cash flow schedule discounted at the internal rate of return (IRR) is provided in the last group of columns labeled (42), (43), and (44). Note that these column numbers are used even if fewer than four discount rates have been calculated.

EXAMPLE APPLICATIONS OF CASHFLOW Input files for four DNCF analyses involving Examples 9.1 to 9.4 made with CASHFLOW^ are shown in Tables 9.5, 9.7, 9.9 and 9.11. Results of the analyses are shown in Tables 9.6, 9.8, 9.10, and 9.12. All four examples involve the same well and incorporate identical assumptions about well costs, oil prices, and the well's production stream, but each is different from an investment viewpoint. The percentages assigned to all revenue interests and working interests are given in each input file. Example 9.1 involves drilling and operating a well with an 87.5% revenue interest and 100% working interest. Example 9.2 concerns the income stream derived from a 12.5% royalty interest by the owner of the mineral rights. Examples 9.3 and 9.4 are more complicated because both involve changes in percentages of revenue interests and working interests after payout, as summarized in Table 9.4. Example 9.3 is from the standpoint of an operator who shares the well's revenue stream with another investor (who bears part of the costs of the working interest), and also with an overriding royalty owner and the mineral rights owner. Example 9.4 involves farming out the lease and retaining the overriding royalty. These four analyses of the same well show how CASHFLOW^ can be used to investigate widely differing investment actions.

255

Computing Risk for Oil Prospects — Chapter 9

Example 9.1 In this example, we as the operator hold all of the working interest in a prospective oil well whose CASHFLOW inputs are Usted in Table 9.5. The mineral rights owners are to be paid a 12.5% royalty interest and we hold the remaining 87.5% revenue interest. No changes in the proportions of revenue or working interests occur after payout. The income tax rate is 28%, the severance tax rate is 5%, cash flows are to be generated at discount rates of 6%, 9%, and 12% (a fourth discount rate could have been specified but was not, and an undiscounted cash flow and a cash flow discounted at the internal rate of return are generated automatically). The initial well capital costs are $120,000 for tangibles and $135,000 for intangibles, and the downstream capital costs will be incurred in Year 6 ($15,000 tangible and $17,000 intangible) and in Year 20 ($18,000 tangible and $16,500 intangible). In Example 9.1, percentage depletion at 15% is taken, the Umit for percentage depletion is set at 65%, and unit-of-production depreciation of tangibles (not including the leasehold cost) is specified. The leasehold cost is $32,000, abandonment costs are 5% of the well's aggregate capital costs, and the salvage value upon abandonment is 8% of the well's aggregate tangible capital costs. The lease will be depreciated indirectly through depletion, but since percentage depletion has been specified, percentage depletion will be taken (up to the specified limit) instead of cost depletion whenever percentage depletion exceeds cost depletion. Cash fiow tables will be calculated and printed up to the economic limit. The well will begin producing in Year 1. Operating costs for the first three years are provided ($4000, $4200, and $4500, respectively), and they increase in subsequent years at the rate of 4% a year (a factor of 1.04). Exponential decline is specified, with the decline rate calculated by CASHFLOW to be in accord with an average production rate during the first year of 55 bbls of oil per day (gas will not be produced), and an ultimate cumulative production of 200,000 bbls over a life span of 30 years. Oil prices for the first and second years of production are $19.00 and $19.50 per barrel respectively, followed by an increase of 3% a year thereafter (a factor of 1.03). Although prices and subsequent annual changes are also provided for gas, they have no effect because gas is not produced. This information is the raw material for the analysis. Table 9.6 presents the output, including the cash fiows themselves, with salient bottom-line figures presented first. The economic limit is reached in Year 30, payout occurs in Year 1, the internal rate of return is 136% per year, and the prospect has the following net present values (with income tax considered): Undiscounted it is $2,715,000, discounted at 6% it is $1,564,000, discounted 256

Forecasting Cash Flow for a Prospect 3000

X 2000 o CO 1000 O %

z .>

Undiscounted Discount Rate Discount Rate Discount Rate

E o -1000-

-110

Year

—r20

0% 6% 9% 12% 30

Figure 9.5. Undiscounted and discounted cumulative net cash flows (Ex. 9.1). 20000•

CO JO

c 10000 •5

o

Year Figure 9.6. Plot of yearly production versus time (Ex. 9.1). 257

Computing Risk for Oil Prospects — Chapter 9 at 9% it is $1,260,000, and discounted at 12% it is $1,044,000. The cumulative net cash flows are shown as graphs in Figure 9.5. A plot of production versus time is given in Figure 9.6. From this analysis, we would conclude that the well is an extremely attractive investment from our standpoint as operator, but risk has not been considered and the NPVs generated by CASHFLOW pertain only to the outcome specified by the input file in Table 9.5. Clearly, we must consider the risk of loss, as well as gains that might stem from other alternative outcomes; this is provided in Chapter 11, where Example 11.1 examines the same prospect from a probabilistic point of view.

Example 9.2 Example 9.2 involves the same well as Example 9.1 except that it is treated from the standpoint that we are the mineral rights owner and have purchased the mineral interest for $20,000 and will receive a 12.5% royalty. Table 9.7 represents the input (entries unchanged from Table 9.5 are not shown) and Table 9.8 presents the summary lines of the output, with cash flow tables omitted. In Table 9.7, our interest is listed under "royalties we receive," and the analysis is now from the "other side of the fence" with respect to the operator. As in Example 9.1, no changes in the proportions of revenue or working interests occur after payout. Figure 9.7 shows cumulative cash flows through time at the specified discount rates. Table 9.7. Abbreviated input for CASHFLOW for Example 9.2. ***REVENUE*AND*WORKING*INTERESTS*IN*PERCENT******************

0.00 0.00 12.50 87.50

After payout 0.00 0.00 12.50 87.50

0.00 100.00 0.00 100.00 100.00

0.00 100.00 0.00 100.00 100.00

Before

1 Royalties paid to mineral rights 2 3 4 5

owners Overriding royalties paid Royalties we receive Revenue paid to other working interests Revenue we receive as working interest owner

Sum 6 Our working interest 7 Working interest of others Sum

258

Forecasting Cash Flow for a Prospect Table 9.8. Summary output from CASHFLOW for Example 9.2 corresponding to input in Table 9.7. DISCOUNTED NET CASH FLOW FOR A SINGLE PRODUCING WELL Ex 9-2, 12.57o royalty interest ECONOMIC LIMIT REACHED AT YEAR 30 PAYOUT DURING YEAR 1 340.00 PERCENT INTERNAL RATE OF RETURN NET PRESENT VALUES (in $ X 1000) : 429. Undis counted 05i 249. Discounted at e.O'/o 202. Discounted at 9.0% 170. Discounted at 12.07o

450

Undiscounted Discount Rate Discount Rate Discount Rate Year

0% 6% 9% 12% 30

Figure 9.7. Undiscounted and discounted cumulative net cash flows (Ex. 9.2). With the exception of shifting to the other side of the fence and incurring a capital cost for purchase of the mineral rights for $20,000 instead of a leasehold for $32,000, other factors in the input are unchanged from 259

Computing Risk for Oil Prospects — Chapter 9 Example 9.1. The analysis, however, is quite different because costs, except for severance and income taxes, have no effect on the royalty calculations, as a royalty interest owner generally bears none of the well's costs. The oil price and production rate forecasts remain directly relevant, however, as do the tax and depletion percentages. The income stream is now much less because it involves a 12.5% revenue interest instead of an 87.5% revenue interest, although the tax rate is unchanged. Results from our standpoint as owners of the royalty interest are presented in Table 9.8. The relevant numbers (with income taxes considered) include payout during Year 1, an IRR of 340%, and an undiscounted NPV of $429,000. Discounted at 6%, the NPV is $249,000, at 9% it is $202,000, and at 12% it is $170,000. The economic limit is reached in Year 30, and the well will cease production at that time. Keep in mind, though, that the economic limit applies directly only to the operator (who holds all the working interest and pays the well's capital and operating costs) and not to us as royalty owners. We are indirectly affected because the owner's income stream ends when the well is abandoned, but because operating expenses are not incurred by royalty interests, an economic limit from our standpoint as royalty owners will never be reached. The investment represented by this single cash flow analysis is also very attractive, but as in Example 9.1, it has not been risked—an exercise left to Chapter 11.

Example 9.3 This is a more complex situation than Examples 9.1 and 9.2 because there are two investors with working interests, an overriding royalty owner and a mineral rights owner. The situation is further complicated because there are changes in percentages of both revenue interests and working interests after payout, as summarized in Table 9.4. Only the revenue interest held by the mineral rights owner remains unchanged as a percentage. The analysis is done from the standpoint of ourselves as the operator. Before payout, we hold a 54% revenue interest and a 58% working interest (Table 9.9). After payout, we hold a 47.5% interest and a 55% working interest. All other input factors remain unchanged from Example 9.1. The revenue and working interests presented in Table 9.9 include all that are involved in the well. Calculations of the well's capital and operating costs are scaled to our respective working interests before and after payout, and the income stream also is scaled to our respective revenue interests before and after payout. Results for us are presented in Table 9.10, which reveals that the economic limit is reached in Year 30, that payout occurs in the first year, that our IRR is 120.7%, and that the undiscounted NPV is $1,468,000. Discounted at 6%, the NPV is $843,000, discounted at 9% it is $678,000, and 260

Forecasting Cash Flow for a Prospect Table 9.9. Input for CASHFLOW for Example 9.3. Only those parts of the input that are changed from Table 9.5 are shown. *******R,EVENUE*AND*WORKING*INTERESTS*IN*PERCENT************** Before After payout 1 Royalties paid to mineral rights 12.50 12.50 owners 8.00 6.00 2 Overriding royalties paid 0.00 0.00 3 Royalties we receive 32.00 27.50 4 Revenue paid to other working interests 47.50 54.00 5 Revenue we receive as a working interest owner Sum 100.00 100.00 55.00 58.00 6 Our working interest 42.00 45.00 7 Working interest of others Sum 100.00 100.00

Table 9.10. Output from CASHFLOW corresponding to input in Table 9.9. Only summary lines are shown. DISCOUNTED NET CASH FLOW FOR A SINGLE PRODUCING WELL Ex 9-3, change of Revenue Interest and WI after payout ECONOMIC LIMIT REACHED AT YEAR 30 PAYOUT DURING YEAR 1 INTERNAL RATE OF RETURN = 120.75 PERCfeNT NET PRESENT VALUES (in $ x 1000) : Undiscounted 0% 1468. Discounted at 6.OX 843. Discounted at 9.0% 678. Discounted at 12.070 560.

discounted at 12% it is $560,000. Figure 9.8 shows cumulative cash flow through time at the specified discount rates. In Chapter 11, Example 11.4 treats the same investment from a risk standpoint.

261

Computing Risk for Oil Prospects — Chapter 9 2000-

X

10004 LL CO

o

CD

>

04 Undiscounted Discount Rate Discount Rate Discount Rate

E

o

0% 6% 9% 12%

-100010

Year

20

30

Figure 9.8. Undiscounted and discounted cumulative net cash flows (Ex. 9.3).

Example 9.4 The same property as in the previous examples is analyzed in this example, but now it is from the standpoint of ourselves as overriding royalty owners who paid $32,000 for the leasehold and then farmed it out to an operator who bears all other costs. The ORR we are to receive is 6% before payout and 8% afterwards, as listed under "royalties we receive" in Table 9.11. Note that the "revenue paid to other working interests" changes from 81.5% to 79.5% after payout to accommodate the increase in our ORR from 6% to 8%. Other facts are summarized in Table 9.11. Oil prices and production rate forecasts and capital and operating costs are unchanged from Table 9.5. While these data are essential for calculation of payout, the costs have no direct effect on our ORR interest, although the oil price and production rate forecasts remain strongly relevant, as do the tax and depletion percentages. Results from our standpoint are presented in Table 9.12. The salient figures (with an investment in the leasehold of $32,000) are an internal rate of return of 70.0%, and an NPV undiscounted of $250,000. Discounted at 6%, the NPV is $135,000, at 9% it is $105,000, and at 12% it is $84,000. Table 9.12 indicates that payout for the working interest owners occurs in 262

Forecasting Cash Flow for a Prospect Table 9.11. Input for CASHFLOW for Example 9.4. *******p£VENUE AND WORKING INTERESTS IN PERCENT************** Before After 1 R o y a l t i e s paid t o minerail r i g h t s payout owners 12.50 12.50 0.00 0.00 2 Overriding r o y a l t i e s paid 8.00 6.00 3 R o y a l t i e s we r e c e i v e 79.50 81.50 4 Revenue paid t o o t h e r working i n t e r e s t s 5 Revenue we r e c e i v e as a working 0.00 0.00 i n t e r e s t owner 100.00 100.00 Sum 0.00 0.00 6 Our working i n t e r e s t 100.00 100.00 7 Working i n t e r e s t of o t h e r s 100.00 100.00 Sum

Year 1, but the cash flow tables (not included) show that payout for us as royalty owners is in Year 2. Figure 9.9, which shows cumulative cash flows through time at the specified discount rates, also shows our payout occurring in Year 2. Example 11.5 in Chapter 11 treats the same investment from a risk standpoint.

Table 9.12. Output from CASHFLOW corresponding to input in Table 9.11. DISCOUNTED NET CASH FLOW FOR A SINGLE PRODUCING WELL Ex 9-4, ORR r e c e i v e d , change a f t e r payout ECONOMIC LIMIT REACHED AT YEAR 30 PAYOUT DURING YEAR 1 INTERNAL RATE OF RETURN = 70.00 PERCENT NET PRESENT VALUES ( i n $ x 1000) : Undiscounted 0% 250. Discounted a t 6.0% 135. Discounted a t 9.07o 105. Discounted a t 12.070 84.

263

Computing Risk for Oil Prospects — Chapter 9

300

Year Figure 9.9. Undiscounted and discounted cumulative net cash flows (Ex. 9.4).

264

CHAPTER

10

The Worth of Money AVERSION TO A DRY HOLE VERSUS T H E DESIRE FOR A DISCOVERY In Chapter 9 we mentioned the use of a utiUty function to represent the outlook of a person toward risk. When people make decisions that involve risk, they realize they will either win or lose, and so they weigh the desire for gain and contrast it with their aversion to loss. Here we're dealing with an inner psychological issue that is called utility. Utility represents the outlook in which the desire for gain is weighed against the aversion to loss. Probabilities are integral components of utility. In a risk venture, we're not only cognizant of the financial gains and losses attached to the venture, but are also aware of the probabilities attached to the losses and gains. We may be willing to subject ourselves to a potentially large loss in a venture if the probability of incurring it is small. The literature on utility is extensive, but most of it has been written by mathematical decision and business theorists, e.g.^ Swalm (1966). Grayson (1960) provides the pioneer treatment of utility in an oil exploration context and his readable book provides insight into early reactions to the concept of utility by persons in the petroleum industry. Newendorp extended Grayson's work by analyzing reactions of managers at Pan American Petroleum Corporation (now Amoco) to hypothetical drilling decisions, as described at length in his Ph.D. thesis (Newendorp, 1967), and went on to provide a very comprehensive treatment (Newendorp, 1975) in which

Computing Risk for Oil Prospects — Chapter 10 he substituted the term "risk preference" for "utility." In general, though, literature on the application of utility to oil exploration drilling decisions is scant and formalized treatment of risk aversion has received much less attention than it deserves. Some recent articles include those by Schuyler (1990) and Walls (1990).

T h e Utility Function For financial analysis, we need to express utility in the form of a utility function that can be incorporated into risk analysis tables and decision tables. The use of utility functions in these tables is described in Chapter 11. What's your own utility? If you lost $100 on a wager, would that have greater negative consequences for you than the positive consequences of winning $100? What if the stakes were higher, and you were faced with the alternatives of winning $1000 versus losing $1000? What if the stakes were $10,000? For most people, losses have greater negative consequences than the positive consequences of equivalent gains. This is risk aversion. Most of us are risk averse, but the degree to which we're risk averse varies greatly. College students, with their limited resources, typically are much more risk averse than are venture capitalists. Some people are actually "risk seeking" rather than risk averse in their outlook—at least over a range. You can well imagine that a person who enjoys financial risk might be found in Atlantic City or Las Vegas! Analyzing a person's risk outlook is not easy, but it can be done. Let's analyze your responses. Suppose I'm a "financial masochist" who enjoys losing money! I propose that you and I have a series of wagers, each involving a single toss of a coin. Heads you win, tails you lose. My first wager is that you win $2.00 if it's heads, but you only lose $1.00 if it's tails. This is a "good deal" for you because the odds are in your favor, as shown by the EMV: Product Outcome Probability Gain or Loss $1.00 Heads (you win) 0.5 x $2.00 = Tails (you lose) 0.5 x --$1.00 = -$0.50 $0.50 Sum of Products = EMV = Now let's escalate the wagers, each one involving a single coin toss. Say you win $20 if it's heads and lose $10 if it's tails: EMV = $5. Would you take the wager? Yes? Suppose it is heads you win $200/tails you lose $100. Would you still take it? How about you win $2000 if it's heads and lose $1000 if it's tails? Better yet, say you win $20,000 if it's heads and lose only $10,000 if it's tails. Will you still take the wager? Sooner or later you'll 266

The Worth of Money drop out of this contest because the potential damage of losing will exceed the gain of winning, even though the odds are always in your favor. By analyzing your responses to this hypothetical situation, we could generate a curve that represents your utility function and expresses your desire for gains versus your willingness to suffer losses in the pursuit of those gains. Remember, for every risk investment, an investor has the potential both for gain and for loss. Otherwise it's not a risk investment. An example utility function is shown in Figure 10.1. The horizontal axis represents dollars gained or lost and the vertical axis represents hypothetical units called utiles, which are arbitrary psychological units. Positive utiles correspond to gains and negative utiles to losses. Plotted as shown, a consistently risk-averse individual has a function that is always convex upward. Since different individuals have different attitudes towards risk, we can envision a series of utility functions (Figure 10.2) ranging from strongly risk averse, through risk neutral, to risk seeking. A risk-neutral outlook is identical to a pure EMV criterion, where a financial loss is identical to a gain of an equal amount of money in its psychological effect, except that a loss has negative consequences and a gain has positive consequences, regardless of the amount of money at stake. However, no one is truly neutral toward risk and a strictly EMV approach isn't realistic. Large gains don't offset equivalent large losses. Even the largest oil companies are averse to risk at some point. For example, several major oil companies may join together in bidding on an expensive offshore lease, because each company considers the risk too great to go it alone. Many individuals and companies exhibit aberrant behavior toward risk. Some may be risk seeking when lesser amounts of money are involved, but become strongly risk averse when larger amounts are at stake (Figure 10.3). Most people who gamble at casinos are risk seeking over only a limited range. They drop out at some critical point when the exposure to loss becomes excessive. While virtually no individual or company is truly risk neutral, few individuals or companies formalize their risk policies in the form of a specific utility function. Instead they operate with informally understood, generalized policies with respect to risk. Within companies, however, these policies may not be effectively understood by managers because of the interplay between probabilities and potential gains and losses.

267

Computing Risk for Oil Prospects — Chapter 10 Positive utiles 200

-$200,000 -$100,000 I Financial losses

$100,000 i

$200,000 $300,000 \ I Financial gains

-300 Negative utiles Figure 10.1. Idealized risk-averse utility function. Dotted lines show correspondence between utiles and dollars for point pairs.

Obtaining a Utility Function Clearly a utility function can be graphed or expressed in numerical form, but how can it be obtained? An individual's utility function is a very personal thing that cannot be provided by someone else. The presumption is that each of us has a utility function, even though we may never have thought about it formally. Presumably corporations also have utility functions, but these, too, are rarely formalized. However, they exist and the lack of formalization doesn't mean that risk aversion is absent. The fact that each of us is risk averse to some degree signals that there is a personal utility function inside us that strongly influences our decisions. Finding the utility function of an individual presents psychological as well as scientific challenges for the investigator. Determining the utility function of a 268

The Worth of Money

Positive utiles

Increasing risk aversion

Increasing risk seeking

Negative utiles

Figure 10.2. Family of utility functions that range progressively from strongly risk averse (convex upward), through risk neutral (sloping straight line), to strongly risk seeking (concave upward).

corporation is less fraught with psychological problems, but may be difficult because corporate decisions are often made by diverse individuals, each with his or her own characteristic response to risk. In either situation, an attempt to deduce the utility function is worthwhile, if only for the insight that it provides into likely reactions to future risk investments. One way to determine utility is to analyze decisions that have been made in the past. Another way is to obtain responses to a series of hypothetical investment alternatives in which losses and gains and attached probabilities are specified for alternative outcomes, which assumes that 269

Computing Risk for Oil Prospects — Chapter 10

Positive utiles

Negative utiles Figure 10.3. Utility function of a person who is risk seeking over smaller range, but risk averse over larger range.

the responses exemplify those that would be obtained in actual situations. Sometimes the responses are so erratic that it is difficult to graph them. A utility function is relevant only in a risk-and-reward situation in which the outcome is uncertain or probabilistic. A utility function can be represented by a utility equation that can be solved repeatedly to represent the function. We can express the equation in words: (Probability of gain x Utility of gain) + (Probability of loss X Utility of loss) = Net utility This equation cannot be solved as written. It can be solved if we are given the two probability values and one of the utility values for the point at which 270

The Worth of Money a person is ambivalent or neutral (the null point) and the net utility is zero. In other words, at the null point the positive utility of a gain weighted by its probability is exactly equal to the negative utility of the loss weighted by its probability, and the sum of the two products is zero. The challenge is to find the null point. Ideally, the null point can be found by analyzing an individual's responses to actual risk-investment opportunities. This is difficult, however, because the terms of actual opportunities rarely coincide with an individual's null point. More often, the individual is not ambivalent and is either strongly positive or negative about an opportunity. What is needed is a finely graduated series of risk-investment opportunities that permit us to bracket the individual's null point over a small range that spans the difference between being acceptable and unacceptable. Let's return to the wager involving the toss of a single coin and a twoto-one payoff in case of a win, such as heads you win $2/tails you lose $1. We can write the utility equation for this wager as follows: (0.5 X Utility of gain of $2) + (0.5 x Utility of loss of $1) = Net utility If you accept this offer, it has positive net utility and the equation can't be solved, because we don't know what the net utility is. But we can escalate the wager by small increments until you reach the point where you decline any further offers. Then we've found your null point and can solve the utility equation. Let's say the last wager you are willing to accept involves winning $20,000 versus losing $10,000. If the wager increases to win $20,010 or lose $10,005, you balk. Thus your null point lies between these two; but, we're close enough to solve the equation by setting the right-hand or net utility term to zero: (0.5 X Utility of gain of $20,000) + (0.5 x Utility of loss of $10,000) = 0 Before proceeding, we must assign a utility value to either the gain or the loss. Let's pick the loss. What is the utility value of a loss of $10,000? We can assign it an arbitrary number of utiles (since utiles are purely arbitrary units). Let's set —10 utiles equivalent to $10,000 lost. Inserting —10 in the equation, we get (0.5 X Utility of gain of $20,000) + (0.5 x (-10)) = 0 0.5 X UtiUty of gain of $20,000 - 5.0 - 0 UtiHty of gain of $20,000 - 10.0 The utility of a gain of $20,000 is 10 utiles, exactly the same as for a loss of $10,000, except that the sign is positive. This isn't surprising because the probabilities of a gain and a loss are both 0.5, and since we're at the 271

Computing Risk for Oil Prospects — Chapter 10 null point, the respective positive and negative utilities exactly balance. Now we have two pairs of points that we can plot. Actually we have three, because at the origin of a utility function, both dollars and utiles are zero. So our table of point pairs has three entries: Dollars •10,000 0 20,000

Utiles -10 0 10

These three points are plotted in Figure 10.4, and two dashed straight lines connect them, one in the lower left quadrant where dollars and utiles are negative, and the other in the upper right quadrant. However, straight lines do not provide a good representation of an actual utility function—a curved line is needed. We know this because of your responses. When we were within the win $20,000/lose $10,000 range, you accepted the wagers and therefore solutions to your utility equation had positive net utility (although we don't know how positive.) Likewise, when you rejected wagers beyond the win $20,000/lose $10,000 threshold, your net utility was negative (although, again, we did not know the magnitude). We must add a few more points to better establish the curve. We can't get additional points by continuing our series of wagers involving a single coin toss and a two-to-one payoff. Furthermore, we're stuck for the moment with our arbitrary assignment of —10 utiles to a loss of $10,000. But we can change the rules for the wagers to obtain different probabilities and payoff ratios. Let's go to a double coin toss, where either two heads yield a win for you, or a head and a tail (in either order) win, but two tails lose. Thus, the win/lose probabiUties are now 75-25, a major improvement in odds from your standpoint; but the payoff can be lowered because of the changed odds. Now let's find out what minimum amount you're willing to accept if you win, keeping your loss at $10,000 if two tails turn up. Let's say that you accept all wagers involving winning $4000 or more, but that you balk below $4000. In other words, you accept $4000 and decline $3995, thus bracketing your null point and permitting us to solve your utility equation again to obtain another point pair: (0.75 X Utility of $4000) -f (0.25 x Utility of -$10,000) - 0 0.75 X Utihty of $4000 = 2.5 Utihty of $4000 = 3.33 Let's obtain one more point pair for the positive quadrant, this time fixing the payoff at win $36,000/lose $10,000, and change the win/lose probabilities. The question now is: What is the minimum probability for a win 272

The Worth of Money Positive utiles 15 / 10 /

1-5 1-5 -$30,000 -$20,000 -$10,000 I

I

I

Financial losses

/

t

/

/

/

/

/

/

/

/

/ /

/

$10,000

$20,000

$30,000

Financial gains

/ / /

h-5

/ / h-10 / / "-.-15 Negative utiles Figure 10.4. Plot of three point pairs representing initial responses to wagers. Connecting dashed straight lines inadequately represent the utility function. that you will accept? Suppose we probe and find that you accept all wagers where the probability of winning is 0.45 or more, but at 0.44 or below, you balk. Again, we've bracketed your null point and can solve the utility equation: (0.45 X Utility of $36,000) + (0.55 x Utihty of -$10,000) = 0 (0.45 X Utility of $36,000) + (0.55 x -10) = 0 0.45 X Utihty of $36,000 = 5.5 Utihty of $36,000 = 12.2 Given these two additional points, we can plot a curve representing your utility function (Figure 10.5). The differences between the initial straight 273

Computing Risk for Oil Prospects — Chapter 10 Positive utiles r15

-$30,000 -$20,000 -$10,000 I

I

Financial losses

$10,000 $20,000 $30,000

I

I

I

Financial gains

1

L-IS Negative utiles Figure 10.5. Addition of two more point pairs to plot in Figure 10.4 enabling plotting of curve that represents utility function.

lines and the curve accords with your responses to the wagers. We could add more points using these same general procedures, particularly for the lower quadrant where one or two more points are desirable. But we've illustrated the concept of the utility equation and how it can be solved to yield a utility function. Unfortunately, most respondents don't respond as consistently as above, so solving their utility equations and graphing their utiUty functions can be a challenge!

274

The Worth of Money Table 10.1. Data for prospects for which decisions were made by exploration manager. Forecasts for dry hole outcome Probability

Cost (1000$)

Utiles*

Forecasts for producer outcome Probability

DNCF value (1000 $)

Utiles*

Prospect A B C D E F G H

.25 .60 .50 .55 .40 .30 .45 .60

I J K L M

.35 .40 .80 .55 .40

Establish Positive Side of Utility Function -100 -100 .75 50 -100 -100 .40 400 -100 -100 .50 250 -100 -100 .45 350 -100 -100 .60 60 -100 .70 160 -100 -100 -100 .55 140 -100 -100 .40 300 Establish Negative Side of Utility Function -150 -204 .65 250 -200 -225 .60 350 -40 .20 200 -22 -100 200 -74 .45 -160 .60 260 -173

33 150 100 122 67 43 82 150 110 130 90 90 115

* Values obtained after manager's responses, either supplied arbitrarily before other utile values were obtamed, determined by solving the utility equation, or read from the fitted line of the utility function.

Obtaining a Utility Function in Response to Prospect Proposals Now let's deal with a more advanced situation in which an oil company exploration manager is faced with drilling decisions involving 13 different prospects. For each prospect, an estimate has been made of the probabilities involved and the monetary consequences of failure and of success. For the first eight prospects (prospects A through H in Table 10.1), the losses are consistently estimated at $100,000, this being the dry hole cost at all eight locations. For each prospect, the manager has specified the minimum estimate of the value of a success, in dollars, that would have to be present before committing the company to drill the prospect. We now have enough information to calculate the positive utiles associated with each decision for prospects A through H. We can arbitrarily set 275

Computing Risk for Oil Prospects — Chapter 10 a value of -100 utiles to correspond with the loss of $100,000. Since there is no set scale of utiles, we can choose to peg a specific number of utiles to a specific gain or loss in dollars (or other monetary units). Once we do that, the scale of utiles to dollars is established by the utility function. Then, by solving the utility equation, we can calculate the positive utiles that correspond to producers if they are obtained as outcomes for prospects A through H:

where: Pd = probability of dry hole Pp = probability of producer Ud = utility of dry hole (loss) Up = utility of producer (gain). Since both probability values are given, as well as the utility of a dry hole (—100 utiles corresponding to —$100,000), the equation can be solved for the utility of a producer. Each solution yields two point pairs (dollars versus utiles)—one pair for the producer outcome, and the other pair (already established as —100 utiles and —$100,000) for the dry hole outcome. This provides some of the points we need to plot the utility function. We plot the positive utile values and their corresponding monetary values in the positive quadrant of the graph and can then fit a smooth curve to the points extending from the origin, with the line passing above some points and below others. We can also plot the single point in the negative quadrant, but we should delay extending the curve to this point until we have more points in the negative quadrant. We can use the same utility equation to get additional points for the negative quadrant, but now we need solutions involving the manager's response to maximum acceptable losses for proposed prospects in which gains have already been established, as provided in proposals I though M in Table 10.1. Then we can use the fitted line in the positive quadrant to estimate the positive utile value associated with each positive outcome, insert the positive utile value into the equation, and solve the equation to obtain the negative utile value that corresponds to the maximum acceptable loss. We can obtain solutions for each of these additional prospects because for each one we have found the manager's null point where the net utility is zero. Now we can extend the curve from the origin into the negative quadrant, thus providing a graph of the manager's utility function with respect to decisions made on the company's behalf over the range for which we have obtained responses. 276

The Worth of Money

-400

-300

r

r

L......

t

-200

-100

1 TheDusands of Do iars

-vlso -,6-100 -150 I

"05'

-200 -250 -300

Figure 10.6. Plot of utility function for exploration manager based on responses in Table 10.1. Continuous curved line in positive quadrant was fitted first, and dashed line representing extension to negative quadrant was fitted subsequently.

Examination of the plotted utility function (Fig. 10.6) shows that the manager has responded rationally and consistently. Since the fitted hne forms a smooth curve, it is probably reasonable to extend it beyond the range of the responses. An extension in the negative quadrant suggests that the manager is almost infinitely risk averse when exposed to losses of more than about $210,000, regardless of whether highly favorable outcomes and probabilities accompany prospective investment opportunities. 277

Computing Risk for Oil Prospects — Chapter 10

Graphing a Utility Function with RISKSTAT The user can create graphs of utihty functions using RiSKSTAT. The user must supply a suitable number of point pairs (dollars versus utiles). Five point pairs are a minimum, and at least one pair should be in the negative quadrant, although two are better. The first step is to enter the pairs of values into a new data file using System Control and Management Option 3: Create a New Data File. Then use Statistical Graphics Package Option 4: Bivariate Scatter Plot to plot them, and finally select the suboption to fit a cubic spline through the plotted points.

278

CHAPTER

11

RATs, Decision Tables, and Trees RATs LINK ALTERNATIVE P R O S P E C T OUTCOMES W I T H RISK Chapter 9 deals with cash flow projections and net present values for individual wells. An NPV for a well is useful by itself, but it doesn't consider the spectrum of possible outcomes for a well. The next step is to collectively analyze the alternative outcomes, including a dry hole and discoveries of various sizes. To do this we need NPVs for aggregates of wells having different production histories scaled to different sizes of discoveries. Then we must create a risk analysis table (or RAT) by weighting each outcome by its associated probability of occurrence. Module RAT does this by systematically gathering and treating information from various sources, including module CASHFLOW. For example, input tables prepared for analysis with CASHFLOW can be directly entered into RAT. Even a risk analysis table isn't an end product, because it represents only a single action to be taken with respect to a prospect. If we're to consider alternative actions before we make a decision, each action will need its own risk analysis table. Then the results of the risk analysis tables can be combined to form a decision table, requiring yet another step which is taken up later in this chapter. The role of module RAT is schematically shown in Figure 11.1, which extends the outhne of Figure 9.1 to a comparison of two prospects, A and B. Assumptions for oil and gas price forecasts, as well as for the capital

Computing Risk for Oil Prospects — Chapter 11 Production well capital and operating costs Oil and gas prices

Prospect B

Prospect A

Joint distribution of field areas and volumes

Dry development well ratio

CASHFLOW generates DNCFs for wells in each size class

Dry development well ratio

Exploratory dry hole probability and cost

Exploratory dry hole probability and cost RAT generates risk-analysis table for Prospect A

Physical access costs

Leasehold costs

Joint distribution of field areas and volumes

RAT generates risk-analysis table for Prospect 6

Utility function

/

/

\

Physical access costs

Leasehold costs

[Decision table contrasts different prospects RATs from other prospects and alternative investments

Decisbn tree

Figure 11.1. Flow chart showing relationships between RAT and CASHFLOW in which two prospects, A and B, are analyzed and compared. Producing well capital and operating costs are the same for both prospects. and operating costs for production wells, are supplied to CASHFLOW and are the same for both prospects. Other data specific for each prospect are supplied directly to RAT, but a single utility function serves both. Bottomline results for each RAT are supplied to a decision table where they are contrasted with each other and with RATS for other prospects. Figure 11.1 emphasizes that a two-way relationship exists between RAT and CASHFLOW, with information passing back and forth between them. Furthermore, RAT can be used to analyze multiple prospects in which the same cash flow projections are used. The generalized form of a risk analysis table is shown schematically in Figure 11.2. Each such RAT analyzes a single action for an individual 280

Risk Tables and Trees exploratory well. The exploratory well may lead to discovery from a spectrum of field size classes, and multiple development wells are provided for each size class. Except for the two columns on the right in Figure 11.2, each column in the table represents a specific field size class. Only three field size classes are shown there, but RAT can accommodate from two to seven field size classes. Construction of a RAT requires that fields be segregated into discrete size classes. While field sizes range continuously in nature, we cannot readily analyze a continuous distribution of sizes and must base our analysis on a relatively small number of discrete classes. The terms "risked" and "unrisked" indicate whether the risk of a dry hole for the exploratory well has been incorporated. The summary of a risk analysis table may be transferred to a decision table for comparison with summaries of other RATS.

The use of module RAT is illustrated by Examples 11.1 through 11.5, as summarized in Table 11.1. All five examples represent the same prospect, but have been analyzed as investments from different points of view. A constraint imposed by the size of the leasehold (640 acres) is incorporated in Example 11.2, which otherwise is identical to Example 11.1. All applications incur leasehold costs of $32,000 except Example 11.3, which incurs a cost of $20,000 to purchase mineral interests, RATS are not shown for Examples 11.3-11.5, but are contained in files EXll-3 and EX11-3.RAT through E X l l - 5 and EX11-5.RAT. Each example in Table 11.1 is from the point of view of the investor and pertains to the same prospect, all of which involve a 10% discount rate. Table 11.1. Variations in revenue and working interests in Examples 11.1 through 11.5 that illustrate module RAT's use. Revenue and working interests are given in percent.

Example number 11.1 11.2 (const.) 11.3 (royalty int.) 11.4 11.5 (ORRint.)

Revenue interest Before After payout payout 87.5 87.5 12.5 54.0 6.0

100.0 100.0 12.5 47.5 8.0

Working interest Before After payout payout 87.5 87.5

100.0 100.0

58.0

55.0

Table numbers 11.3 to 11.5 11.6 & 11.7

Depending on the areal extent of a leasehold (or concession), an exploratory well might discover a field whose areal extent is larger than the area of the leasehold. The area of the leasehold therefore establishes an 281

Computing Risk for Oil Prospects — Chapter 11 Small

Medium

Large

a. Unrisked probabilit es

1.000

b. Field volumes c. Field areas d. No. producing wells e. Ultimate cumulative production per well f. IPper well, bbls/day g. PV of indiv. prod, well h. No. dry devel. wells i. Cost of dry development wells j. PVs of field-size classes

2. —

Q. (D X ^ LU ^

k. Risked probabilities 1. Risked PV in each size class m. Exploratory dry hole c DSt

-D

n. Lease, mineral, or ORR cost

.^ c "^

0. Net physical access cost p. Expected monetary va ue

iiiiiiii (0

ijiiii

q. Unrisked PV in utiles r. Risked PV in utiles

"8

s. Neg. utility of sum (m) thru (o) left col. t. Expected utility (EUV)

EUV

^.

Figure 11.2. Schematic diagram of risk analysis table as represented in module RAT. Rows (a) through (p), supplied to or generated by RAT, yield an expected monetary value (EMV) in dollars or other currency units. Optional extension in rows (q) through (t) generates an expected utility value (EUV) in utiles. upper limit on the magnitude of a discovery whose value will benefit the investor, and in turn requires that the probability distribution attached to the prospect's hydrocarbon volume size classes be constrained to represent 282

Risk Tables and Trees

in xf CM

CM in CO

52 in

9

CO

0> r^

CO 00

CM h* CM r-

Unconstrained Field Area, Acres

in

'^ CM

CM in CO

CO T—

in

'^J-

CM

o

^ CD

Constrained Field Area, Acres

CO O T"

O)

in

'^

CD CM y— T-

o> in in CM

C3> xt o '^

o

s CD

Unconstrained Volume, bbis x 1000

' ^ CM

CO O T—

o> in Tf

CD CM T— T—

o 00 o CM

CM CO o> CM

00 O in 00

Constrained Volume, bblsx 1000

Figure 11.3. Hypothetical probability distributions showing how area of leasehold constrains oil volumes in RAT. (a) Unconstrained distribution of field areas, (b) Corresponding unconstrained distribution of field volumes, (c) Revised distribution of area classes constrained by 640-acre limit, (d) Corresponding constrained distribution of volume classes.

the area of the investor's leasehold. Figure 11.3 shows how such a constraint can be represented by a distribution. If a field that has a large area is discovered, it is likely that the thickness of the producing interval or intervals in the reservoir will be greater than if the field is small in area. (The relationship between field areas and volumes in producing regions was discussed in Chapter 4, and is employed 283

Computing Risk for Oil Prospects — Chapter 11 in Example 11.2 in which a constraint is imposed by a 640-acre leasehold.) As a consequence, production rates and ultimate cumulative production volumes tend to be larger on a per-acre or a per-well basis in fields that are large in area. This tendency should be represented by a probability distribution such as that shown in Figure 11.4 and attached to a leasehold whose areal dimensions are constrained. In Figure 11.3c, the probabilities attached to field areas for the three uppermost classes (which exceed the 640-acre limitation) have been incorporated into the largest class that falls under this limit. The three uppermost volume classes in Figure 11.3d apply to a leasehold of 640 acres.

00 0>

CO 0> CM

in 0> 00

TT iP i^ ^

in CO CM CO

o CD in ^

CO CM CO

Barrels per Acre Figure 11.4. Hypothetical probability distribution showing yield in barrels per acre. Module RAT is simplified in that it does not consider the span of time that may be required to develop a field. For example, drilling of development wells in a field may span a number of years, but RAT does not make such a distinction. If we compute undiscounted net values, this omission has no effect, but when discounting is applied, the result is sensitive to the discount rate. As the discount rate increases, the span required for drilling of development wells has a greater effect on a prospect's net present value (NPV).

While module RAT interfaces closely with CASHFLOW, there are important differences in the use of CASHFLOW in conjunction with RAT as compared with CASHFLOW'S use on a stand-alone basis. When used by itself, CASHFLOW provides for incorporation of the leasehold, royalty, or mineral rights cost as part of the input. The user also may incorporate special costs related to physical access to the overall prospect as part of the initial costs, both tangible and intangible. When we deal with multiple 284

Risk Tables and Trees wells, these costs are not repeated. The lease cost, for example, is incurred only once, as are physical access costs to reach the leasehold. The term "physical access costs" designates all initial costs related to access such as constructing a drilling platform or building an access road or bridge. The adjective "physical" distinguishes costs of access by equipment and personnel, in contrast to "legal" access provided by lease agreements or concession agreements. Physical access costs presumably are incurred only once because, for example, only one platform is necessary for testing and developing a prospect. These one-time costs are considered by RAT, and while they are not repeated for individual wells if a discovery is made, the tangible component of physical access costs is subject to depreciation and is allocated among the producing wells when each size class is considered. Similarly, the lease, royalty, or mineral rights cost may be subject to depletion (or another form of depreciation), which also is allocated among the producing wells in each size class. Many different types of financial arrangements between operators and other investors can be devised when wells are drilled. For example, the operator may have a "carried" interest when part of the initial capital costs are borne by other investors. An "interest carried to casing point" is common in the United States. In such an arrangement, all or part of the capital costs of the well before it is completed as a producer and before casing is set may be borne by other investors. If the well is dry, the well's entire costs may be borne by other investors, but if casing is set and the well is completed as a producer, the operator shares in the completion costs. Module RAT can be used to analyze such situations if appropriate entries are provided as input.

HOW RATs TREAT INFORMATION The risk analysis tables generated by module RAT organize information in rows (a) through (o), as shown in Figure 11.2. RAT then generates an expected monetary value (EMV) as a bottom-line result in row (p). The user has the option of providing values from specific data sources (such as frequency distributions, regressions, or functions), or alternatively entering values based on subjective judgment or derived by other means. The user also has the option of extending the table to rows (q) through (s) to obtain an expected utility value (EUV) as an additional bottom-line result in row (t). A summary description of the types of tables and information that RAT generates is given in Table 11.2. Much of the input to RAT is the same as for CASHFLOW, but there are important differences. The similarities stem from the fact that RAT 285

Computing Risk for Oil Prospects — Chapter 11 serves as a "shell" that passes information back and forth between itself and CASHFLOW. The differences include the fact that only a single type of hydrocarbon and a single discount rate can be analyzed, either an exponential decline or hyperbolic decline must be assumed (empirical production schedules cannot be accommodated), and the entry of a utility function is optional, RAT also requires a discrete probability distribution of field volumes and corresponding field areas. The midpoint of each size class in the distributions represents a specific number of barrels, MCF, or other units of volume for which there is a corresponding specific number of acres or other units of area. For each size class there must be an estimate of volume, an estimate of area, and an estimate of the attached probability. RAT, therefore, is "probabilistic," whereas CASHFLOW is deterministic. The number of field size classes in RAT can vary. RAT can accommodate from two to seven size classes. If an empirical field size distribution is employed, the user must supply the probability and the corresponding volume and area for each class. Since the distribution is empirical, no presumption is made about its form. The user also has complete flexibility in defining the midpoints of the size classes. Alternatively, a data set can be supplied to RAT in the form of an external file containing field volumes and areas observed in a region. For example, a file of oil volumes and areas from the Denver-Julesburg Basin has been used as input for Examples 11.1 through 11.5; the data are graphed in Figure 4.5. From such a file, a field size distribution may be extracted in a form suitable for RAT. Operations on this file involve taking the logarithms of both field volumes and areas, and then regressing log areas on log volumes. A frequency distribution of volumes is then calculated using procedures described in Chapters 2 and 4, and which the user can employ independently using modules in RISKSTAT. If the user elects to provide a data set containing volumes and areas, between two and seven field size classes can be specified; RAT will then automatically select midpoints for each of the field size classes when generating the distribution. The field area that corresponds to the field volume at each class midpoint is obtained from the regression. The term "net physical access cost" refers to both intangible and tangible costs, which are presumed to be incurred in year zero. There is no provision in RAT for entering additional physical access costs in subsequent years. The intangible physical access cost is expensed in year zero, but the tangible physical access cost is depreciated if a discovery is made, using a depreciation method specified by the user. Through its interface with CASHFLOW, a depreciation schedule for tangible physical access cost is generated on a pro rata basis for an individual producing well in each field size class, which in turn is multiplied by the number of producing wells in the 286

Risk Tables and Trees size class to represent the tangible physical access cost for the size class as a whole. The intangible physical access cost, while expensed in year zero, is also allocated on a pro rata basis to an individual well in each field size class, and then multiplied by the number of producing wells in the class to calculate its overall effect on the size class. The salvage value of the tangible physical access cost is supplied by the user and is also incorporated in calculating the net physical access cost. In addition, information must be provided that enables RAT to estimate the number of producing wells and dry field development wells in each field size class. Information is organized by RAT in rows as outlined in Figure 11.2 and Table 11.2, and described below in greater detail on a rowby-row basis. Note that the user may substitute units other than barrels, acres, or dollars by editing file UNIT.DEF. Module RAT treats either oil or gas, but not both. However, oil and gas can be merged for analysis by expressing them in barrels-of-oil-equivalent (BOE) units, using a conversion factor that equates gas to oil based on price or energy equivalence.

Row-by-Row Description of Tables Generated by Module RAT (a) Unrisked probabilities for field volume size classes are obtained by generating a probability distribution of field volumes representing ultimate cumulative production and segregating it into specific field size classes. Three classes are shown in Figure 11.2, whereas Tables 11.3 to 11.7, which pertain to Examples 11.1 and 11.2, have seven classes. Probability distributions of field volumes can be created from observed frequency distributions or obtained from other sources. When derived from other sources, the probabilities are expressed on a unit scale and must sum to 1.0. In this form the probabilities represent all possible outcomes for the prospect, if a discovery is made. The adjective "unrisked" refiects the fact that the probability for a dry hole is not included at this point, although it is incorporated later. Class intervals of probability distributions are assigned by RAT in the following way: When the distribution is to be based on a file of field volumes, the number of discrete class intervals into which the distribution is to be divided, which may range from 2 to 7, is specified. The maximum and minimum values of the logarithms of the volumes are determined to obtain the range, which is divided by the number of class intervals to yield the span of each size class. The number of fields in each size class is then counted and that number is divided by the overall total to yield the proportion of fields in that size class. These proportions, expressed on the unit scale, are 287

Computing Risk for Oil Prospects — Chapter 11 Table 11.2. Description of entries in risk analysis tables organized by module RAT. Letters (a) through (t) identify rows in Figure 11.2. Entry in table

Source

Units

(a) Unrisked probabilities attached to each field volume size class (dry hole probability is not represented)

Frequency distribution of field sizes

Probabilities on unit scale

(b) Ultimate cumulative production for each volume size class

Midpoints of size classes in frequency distribution of field sizes

Barrels or MCF

(c) Field area size classes that correspond to each volume size class

Regress field areas on field volumes

Acres

(d) Number of producing wells in each volume size class

Divide (c) by well spacing to yield nearest whole number

Number of producing wells

(e) Average cumulative production per well in each volume size class

Divide (b) by (d)

Barrels or MCF

(f)

Use function to obtain from (e)

Barrels or MCF

CASHFLOW com-

Dollars

Average daily production in initial year per well in each volume size class

(g) Present value of average individual producing well in each size class, without physical access costs

(h) Number of dry development wells in each size class

putes net present value per well in each size class, with (e) and (f) as part of input Use function to obtain from (d)

(Cont.) 288

Number of dry development wells

Risk Tables and Trees Table 11.2.

Continued.

Entry in table

Source

Units

(i)

Cost of dry development wells in each size class

Multiply (h) by cost per dry development well (supplied by user)

Dollars

(j)

Present value of aggregate producing wells in each size class

Multiply (g) by (d) and subtract (i)

Dollars

(k)

Adjust probabilities by incorporating dry hole probability and rescale to complete spectrum of outcome probabilities

Add dry hole probability to field size probabilities in (a) and rescale by dividing by sum

Probabilities on unit scale

(1)

Risked present values for size classes

For each size class, multiply (j) by (k) and sum the products

Dollars

(m) Exploratory dry hole cost (left) followed by risked dry hole cost (right)

User supplies dry hole cost which is then multiplied by dry hole probability

Dollars

(n)

User supplies cost which is then multiplied by dry hole probability

Dollars

Leasehold, mineral rights, or ORR cost (left) followed by risked cost (right)

{Cont) printed as row (a) in RAT, and represent the probabilities attached to each field size class. (b) Ultimate cumulative production for each field volume size class in barrels or MCF (or other units) is obtained from the probability distribution 289

Computing Risk for Oil Prospects — Chapter 11 Table 11.2. Concluded. Rows (q) through (t) incorporate optional extension involving transformation of dollars to utiles with utility function. Entry in table

Source

Units

(o) Net physical access cost (left) followed by risked net physical access cost (right)

User supplies net physical access cost which is then multiplied by dry hole probability

Dollars

(p) Expected monetary value (EMV) for prospect as a whole

Algebraic sum of right column of rows (1), (m), (n), and (o)

Dollars

(q) Present values of field size classes expressed in utiles

Transform PVs for size classes in (j) with utility function supplied by user

Utiles

(r) Risked PVs for field size classes in utiles

Multiply (q) by risked probabilities for field size classes in (k) and sum the products

Utiles

(s) Negative utility of loss in event that hole is dry and lease abandoned

Obtain algebraic sum of unrisked exploratory dry hole cost (m), unrisked lease cost (n), and unrisked net physical access cost (o). Transform sum to its utility value and place in left column; multiply by dry hole probability to obtain risked utility and place in right column

Utiles

(t) Expected utility value (EUV) for prospect as a whole

Algebraically add right columns of (r) and (s) to obtain (t)

Utiles

290

Risk Tables and Trees of field volume sizes classes used for (a). The entry for each size class should be the midpoint for that size class. (c) Field areas that correspond to each field volume size class in acres (or other units) are required and can be obtained by regressing field areas on field volumes. The entries in Examples 11.1 through 11.5 have been obtained by regressing observations of field areas on field volumes, using the relationship graphed in Figure 4.5. If desired, the user can employ modules within RISKSTAT for both calculation and graphic display of regression relationships. Such a regression provides an estimate of the area that corresponds to the hydrocarbon volume in each size class. Alternatively, the user can supply estimates from other sources, including subjective estimates. Estimates of field areas are needed because the number of development wells that will be required is a function of both field area and well spacing. The midpoints of size classes of field areas should correspond to the midpoints of the field volume classes entered in (b). (d) Number of producing wells for each field volume size class is obtained by dividing the estimated field area (c) for that size class by the unit spacing per well (which is supplied by the user), and rounding to the nearest whole number. For example, if the unit well spacing is 40 acres and the estimated field area obtained by regression for a specific volume size class is 155 acres, the estimated number of wells for that size class is four. Alternatively, the user may supply an estimate of the number of wells in each size class from another source, including subjective estimates. If an areal constraint is imposed by the size of the leasehold, then the number of wells also will be constrained. Such a constraint is calculated internally in RAT.

(e) Average ultimate cumulative production per well for each field volume size class in barrels or MCF is obtained by dividing the ultimate cumulative production for that size class (b) by the number of producing wells for that size class (d). The estimated ultimate cumulative production for an average producing well in each size class is one of the key inputs to CASHFLOW for computing an NPV for a representative well in a size class. (f) Average production in barrels or MCF per day during initial year for producing well in each size class can be obtained with a function that relates initial production to ultimate cumulative production. We need initial production as a component in generating the production stream for a representative well in the size class. RAT requires that the stream's decline function be either exponential or hyperbolic in form. A function relating the initial year's average daily production to ultimate cumulative production has been plotted in Figure 11.5. (g) Net present value of representative producing well in each size class in dollars or other units is obtained through RAT'S interface with 291

Computing Risk for Oil Prospects — Chapter 11 200

^

150

SI

c

1 100

? O •c

50

100

200 300 400 500 Ultimate Cumulative Oil Production (bblsxIOOO)

600

700

Figure 11.5. Initial year's average daily production versus ultimate cumulative production for prospect analyzed in Tables 11.3 through 11.7. Information based on this function has been printed in row (f) of Tables 11.4 and 11.7. CASHFLOW, yielding an NPV for a representative well. Only a single discount rate, including zero, can be assumed. Leasehold (or mineral rights or overriding royalty) costs and physical access costs also have been calculated on a per-well basis at this point and are incorporated in the net present value for an individual producing well in each size class, with leasehold and tangible physical access costs depreciated as specified. (h) Number of dry development wells in each size class is assumed to be related to the number of wells in that size class. The number of dry wells is taken from a fitted relationship based on user-supplied data. The line may be fitted as a second-degree polynomial (the procedure used in Figure 11.6), or by piecewise linear interpolation. In general, the proportion of dry holes will decline as the number of producing wells in a field increases. Because the ratio of perimeter to area decreases as field areas increase, the ratio of dry and producing wells decreases because fields are often bounded by 292

Risk Tables and Trees

5 3 E

Q. jO

•

>

-c

>

<

}.

I^ Q

4-

-i^i/;.

0

I 1 2

10

20 Total Number of Wells

30

40

Figure 11.6. Function relating number of dry development wells to total number of wells, based on three sets of observations. An estimate of the number of dry development wells in each size class based on this function has been supplied in row (h) of Tables 11.4 and 11.7.

dry holes. Furthermore, fields that are large in area may have thicker and more continuous producing intervals, which also affect the ratio of dry and producing wells. The number of dry development wells is limited if an areal constraint has been imposed. Such a limit is calculated internally in RAT. (i) Cost of dry development wells in dollars in each field size class is obtained by multiplying the dry hole cost for a development well, which the user suppUes, by the number of dry development wells in that class (h). (j) Present value of producing wells in each class size in dollars is obtained by multiplying the PV for a representative producing well in each size class (g) by the number of producing wells in that size class (d) and subtracting the costs of dry development wells (i) in that size class. The leasehold (or mineral rights or overriding royalty) costs and physical access costs previously calculated on a per-well basis are incorporated in the net 293

Computing Risk for Oil Prospects — Chapter 11 present value for each size class, and incorporate depreciation and depletion as specified by the user. (k) Incorporate dry hole probability and rescale probabilities for field size classes by adding the dry hole probability to other probabilities in (a) and dividing by the total so that the rescaled probabilities sum to 1.0. The set of probabilities is now "risked" because it incorporates the probability of a dry hole. (1) Risked present values for all classes of producing well outcomes are obtained by multiplying the present value of each producing outcome (j) by the probability associated with that outcome (k). The products are then summed and the total entered in the column at the right edge of the table. (m) Incorporate exploratory well dry hole cost and obtain risked dry hole cost. The exploratory dry hole cost is supplied by the user (in left column) and is then multiplied by the dry hole probability in (k) to yield the risked exploratory dry hole cost in the right column. (n) Incorporate leasehold cost and obtain risked leasehold cost. The leasehold (or mineral rights or overriding royalty) cost is entered in the left column and multiplied by the dry hole probability in (k) to yield the risked leasehold cost in the right column. (o) Incorporate net physical access to leasehold and calculate risked net physical access cost. The net physical access cost (left column) is multiplied by the dry hole probability in (k) to yield the risked net physical access cost (right column). The net physical access cost is defined as the sum of both the tangible and intangible physical access costs, less the salvage value of the tangible physical access assets (for example, the salvage value of a bridge or platform.) (p) Expected monetary value for the prospect as a whole in dollars is obtained by algebraically adding the four entries in the right column of rows (1) through (o). Entries in the right column of three rows (m) through (o) have been "risked" by multiplying the unrisked entries in the left column of these rows by the dry hole probability. The EMV obtained as a bottom-line result in row (p) is the sum of the risked present values for each of the field size classes (which have previously been summed) in row (1), plus the sum of the risked dry hole cost (m), risked leasehold cost (n), and the risked net physical access cost (o). Collectively, the EMV incorporates the spectrum of gains if a field is discovered (with each field size outcome tempered by its probability) plus the risk of loss if the exploratory well is dry, merging all outcomes into a single number. The EMV is neutral with respect to risk; if risk aversion is important, the analysis should be extended to obtain an expected utility value or EUV. 294

Risk Tables and Trees 600

400

200 (0

J

0I

',

'.

'.

'. •

Jj^

'•

'

'•

I

'

!

'>

'

I

!

-200

-400

-600 -5000

0 5000 Monetary Value ($ x 1000)

10000

Figure 11.7. Utility function for input to RAT in rows (q) and (s) of Tables 11.4 and 11.7, based on user-supplied data. Straight lines are piecewise linear interpolation, curved line is second-degree polynomial. Note unrealistic decrease in polynomial with high monetary values. This behavior can be avoided by using more data points.

I n c o r p o r a t i n g U t i l i t y in t h e R A T At the option of the user, utility can be incorporated in the RAT by the inclusion of rows (q) through (t) in Table 11.2. These are described below: (q) A utility value for each field size class is obtained by transforming the present values in dollars in (1) into utiles by a utility function supplied by the user. Tables 11.3 through 11.7 use the utility function shown in Figure 11.7. (r) A risked utility value for each field size class is obtained by multiplying the utility of its present value in row (q) by its probability in (k) and summing the products. (s) Negative utility of the combined losses if the hole is dry and leasehold abandoned is obtained by summing the dollar consequences in the left column of rows (m) through (o), transforming this sum to its utility value 295

Computing Risk for Oil Prospects — Chapter 11 and placing it in the left column of (s). This utility is multiplied by the dry hole probability from (k) and the product placed in the right column. (t) The expected utility value (EUV) for the prospect as a whole is obtained by algebraically adding the right column in rows (r) and (s), which represent the risked gains if a field is discovered and the risked loss if the exploratory well is dry. The bottom-line EUV combines the effect of all outcomes, each reflecting the degree of aversion to risk expressed by the utility function, and each tempered by its attached probability.

EXAMPLE RISK ANALYSIS TABLES GENERATED WITH MODULE RAT The construction of risk analysis tables by module RAT is illustrated by Examples 11.1 through 11.5. Examples 11.3 through 11.5 are not given as printed tables, but are contained in files EXll-3 and EX11-3.RAT, E X l l 4 and EX11-4.RAT, and E X l l - 5 and EX11-5.RAT. All examples pertain to the same prospect and to the same investment scenarios whose cash flows were analyzed in Examples 9.1 through 9.4 in Chapter 9. The big difference, of course, is that the examples now involve risk and incorporate a spectrum of possible outcomes consisting of seven field size classes and a dry hole. However, the basic financial assumptions are the same. Production outcomes on a per-well basis for the seven field size classes are shown in Figure 11.8. Figure 11.8 is an unusual but effective way to display a probability distribution of annual production volumes. The probability, in percent, for a representative well in each field size class is tabulated in the box within the figure. Each decline curve represents the annual production for a well in that field size class over the life span of the field. Alternatively, production through time can be plotted in cumulative form. Examples 11.1 through 11.5 incorporate identical assumptions about well production forecast data, forecasting procedure, well spacing, exploratory dry hole probability and cost, cost of each dry development well, relationship between the number of dry development wells and producing wells, physical access costs, percentages assigned to abandonment costs and salvage value, form and rate for depreciation of physical access tangible costs, and utility function. With the exception of Example 11.2, the differences are entirely in the investment scenarios. The input tables for Examples 11.1 through 11.5 involving RAT (Tables 11.3 and 11.6) are the same as those used in Examples 9.1 through 9.4 involving CASHFLOW (Tables 9.5, 9.7, 9.9, and 9.11); these CASHFLOW input tables were supplied essentially without change to the latter part of RAT'S input files. The investment scenarios in that part of RAT entitled 296

Risk Tables and Trees 60000 —D—

3% Probability

—#—

4% Probability

—D—

9% Probability >1% Probability [5% Probability 16% Probability 1% Probability

Year Figure 11.8. Annual production per well versus time for seven field size classes for Examples 11.1 through 11.5. Total annual production for the field in each size class is obtained by multiplying the per-well annual production by the number of producing wells in the size class. " R E V E N U E AND W O R K I N G I N T E R E S T S IN P E R C E N T " vary between examples, as outlined in Table 11.1. There is also a major difference in the production forecasts. In CASHFLOW the production forecast pertains to an individual well, whereas in RAT the production forecast involves multiple wells in each field size class, RAT'S abihty to incorporate input files generated earlier for CASHFLOW is an important advantage, and provides a major link between the two programs. Details of the input and output files for RAT and CASHFLOW are given in Appendix D. As outlined in Table 11.1, Examples 11.1 and 11.2 involve a 100% working interest and an 87.5% revenue interest. These correspond to Example 9.1 in Chapter 9. Input for Example 11.1 is summarized in Table 11.3. Part of the table has been omitted because it is identical to pages 2 and 3 of Table 9.5. The corresponding output is presented as a risk analysis table in Table 11.4, followed by cash flow summaries for individual field size classes in Table 11.5. The cash flow tables for representative wells in each size class (as generated by CASHFLOW) are not shown in Table 11.5, although 297

Computing Risk for Oil Prospects — Chapter 11 users have complete flexibility in specifying tables to be printed. Graphs of cumulative net cash flows discounted at 10% for each of the seven field size classes are provided for Example 11.1 (Fig. 11.9) and Example 11.2 (Fig. 11.10). The only difference between Examples 11.1 and 11.2 involves a constraint imposed by the size of the leasehold. In Example 11.2, a constraint of 640 acres has been imposed, but there is no constraint in Example 11.1. The effect of the constraint is shown by the differences between Tables 11.4 and 11.7. In row (d) of Table 11.4, the number of producing wells increases in each increasing field size class, whereas in Table 11.7, the number of producing wells remains at 14 for three of the larger size classes. The well spacing of 40 acres provides 16 well locations on 640 acres, but two of the locations for each of the larger size classes are presumed to be occupied by dry development wells, as based on the relationship between producing and dry development wells, for which data were supplied as part of the input. The effect of the areal constraint of 640 acres also is revealed in the present values for each size class in row (j). In Table 11.7, the present values continue to increase with increasing field sizes, even though the number of producing wells remains at 14. The progressive increase in present value of individual producing wells in row (g) is due to the greater producing rates and larger cumulative volumes for each individual well in the larger field size classes. The present value of each size class is weighted by multiplying by the attached probability to calculate the EMV. Table 11.7 shows that the EMV for this investment scenario (an 87.5% revenue interest and 100% working interest) in the prospect is $1,558,000 with the areal constraint. Without an areal constraint (Table 11.4), the EMV is $2,091,000. Expected utility values (EUVS) also are affected, although not in proportion to the EMVs because of the nonlinear form of the utility function. Analyses produced by RAT for the remaining examples (Examples 11.3 through 11.5) are not shown, but are given in files as noted previously. Comparison is facilitated by grouping the salient results from the analyses, which consist of the EMVs and EUVs, together in a decision table. Module DECISION has done this for us in Tables 11.13 and 11.14, which permit ready comparison of the different investment scenarios.

W h a t We Can Conclude About RATs Risk analysis tables are powerful tools and permit the ready examination of alternative investment scenarios. The user must be continually aware, however, that the validity of the tables depends on the assumptions that have been supplied as input. 298

Risk Tables and Trees

Table 11.3. Input to module RAT for Example 11.1 with an 87.5% revenue interest and a 100% working interest in the prospect, and no areal constraint. Compare with Tables 9.5 and 9.6 for an analysis of a production stream for a single well with identical investor's terms. RAT output based on Table 11.3 is given in Tables 11.4 and 11.5. Ex 1 1 - 1 , 1007o working i n t e r e s t , 87.SX revenue i n t e r e s t ***RAT*DATA************************************************** 1 F i e l d s i z e d i s t r i b u t i o n (1) or d a t a ( 2 ) 2 2 Name of f i e l d s i z e f i l e djsize.dat 3 Name of CASHFLOW d a t a f i l e ex9-l 4 D i s c o u n t r a t e CD 10.00 Field Size Distribution Probability Field Size F i e l d Area (•/.) (bblsxlOOO) (acres) 3. 28. 255. 4. 79. 329. 9. 216. 424. 22. 591. 547. 45. 1617. 705. 16. 4422. 908. 1. 12094. 1170. ***RAT*PRODUCTION*FORECAST*********************************** 1 Type of hydrocarbons ( o i l = 1, g a s = 2 ) 1 Cumulative/Initial o i l production function 2 L i n e a r (1) or p o l y n o m i a l (2) f i t 1 3 Number of d a t a p a i r s 3 4 Num (bbls) (bbls/day) 1 17000. 20. 2 200000. 80. 3 600000. 175. 5 L i f e s p a n of f i e l d (years) 30 6 E x p o n e n t i a l ( 1 ) or h y p e r b o l i c ( 2 ) d e c l i n e 1 ***RAT*FIELD*PARAMETERS************************************** 1 Well s p a c i n g ( a c r e s ) 40. 2 Area l i m i t f o r p o t e n t i a l l y p r o d u c t i v e l e a s e h o l d (0 = no l i m i t ) ( a c r e s ) 0. 3 Dry h o l e p r o b a b i l i t y CD 80.000 4 Dry h o l e c o s t ( $ ) 80000. 5 Cost of dry development h o l e ( $ ) 65000. iCont) 299

Computing Risk for Oil Prospects — Chapter 11 Table 11.3. 6 7 8

Concluded,

Dry development holes function Linear (1) or polynomial (2) fit Number of function data pairs Num Total Dry 1 6 1

2

16

1 3

2

3 30 3 ***PHYSICAL*ACCESS*COST************************************** 1 Tangible physical access cost ($) 10000. 2 Intangible physical access cost ($) 5000. 3 Abandonment cost as proportion of aggregate physical access cost (7,) 5.00 4 Salvage value as proportion of physical access tangible cost (X) 6.00 ***DEPRECIATION*OF*PHYSICAL*ACCESS*COST********************** 1 Depreciation Function Straight line (1) Unit of production (2) Empirical depreciation (3) 1 2 Number of years to depreciate 10 ***UTILITY*FUNCTION****************************************** 1 EUV analysis (y/n) y 2 Linear (1) or polynomial (2) fit 1 3 Name of utility function file utile.dat ***TITLE*OF*CASHFLOW***************************************** Ex Appl 9-1, lOO'/o working interest, 87.57, revenue interest

300

Risk Tables and Trees

Table 11.4. RAT output for Example 11.1 in the form of a risk analysis table corresponding to input in Table 11.3. Operator holds an 87.5% revenue interest and a 100% working interest in the prospect. Discount rate is 10%. Cash flow summaries for each field size class are given in Table 11.5. Compare Table 11.4 with Table 9.4 for an individual well. Graphs of cumulative net cash flows discounted at 10% for each of seven size classes in Figure 11.9.

Ex 11-1, 1007o working interest, ST.B'/o revenue interest *********5ic***********p,iSK*ANALYSIS*TABLE********************* Dry 1 2 3 4 5 6 7 hole Sum (a) Field size probability (b) Field size, (bblsxlOOO)

.031 .041 .092 .214 .449 .163 .010

28 79 216

(c) Field size, (acres) 255 329 424

591

1.000

1617 4422 12094

547 705

908 1170

(d) Producing wells

5

7

10

12

16

21

(e) Cum. prod. (bblsxlOOO)

5

11

21

49

101

210 465

(f) Init. prod, (bbls/day)

10

15

22

36

54

83

(g) PV of well ($ X 1000) - 149 -97

-9

227 648

(h) Dry devel. wells (DDW)

—

26

145

1466 3305

1

1

1

2

2

2

3

(i) Cost of DDW ($ X 1000) 65

65

65

130

130

130

195

CCont.) 301

Computing Risk for Oil Prospects — Chapter 11 Table 11.4. Concluded. **************>ic******RI3K*ANALYSIS*TABLE********************* Dry 1 2 3 4 5 6 7 hole Sum (j) PV of class ($ X 1000) -810 -744 -163 2594 10241 30660 85736 (k) Risked probability .006 .008 .018 .043 .090 .033 (1) Risked PV ($ X 1000)

-4

—

2192

(m) Dry hole cost ($ X 1000)

-79

-63

(n) Lease-MR cost ($ X 1000)

-32

-25

(o) Net P-A cost ($ X 1000)

-14

-11

-6

-3

111 919

1001

.002 .800 1.000

174

(p) EMV ($ X 1000)

2091

(q) PV utility (utiles) -81

-74

-16

180 457

(r) Risked util. (utiles) -.5

-.6

-.3

7.7

(s) Cost util. (utiles) (t) EUV (utiles)

302

41.1

1070 2722

35.4

5.6

—

87.8

-12.7 -10.1

77.7

Risk Tables and Trees

Table 11.5. Summaries of net cash flows for seven field size classes of Example 11.1 as analyzed by RAT involving prospect with an 87.5% revenue interest, 100% working interest, and discount rate of 10%.

WELL FOR FIELD SIZE CLASS NUMBER

1

DISCOUNTED NET CASH FLOW FOR A SINGLE PRODUCING WELL Ex 9-1, 100% working interest, 87.5X revenue interest ECONOMIC LIMIT REACHED AT YEAR 4 PAYOUT NEVER REACHED INTERNAL RATE OF RETURN CANNOT BE CALCULATED NET PRESENT VALUES (in $ x 1000) : Undiscounted OX : -141. Discounted at lO-O'/e : -149. % 3|c * * :|e sic j|c 3|c : k * * 3ic 3ic 3|c :|c:((>|e 9|c 3|e 3ic 3)c 3|c 3|e :|c ^ 3|e 3|e 3)e 3|c 3)c 3|e 3ic :|c 3|c 3jc 9)c 3K 3|c 3 k * ^

WELL FOR FIELD SIZE CLASS NUMBER

2

DISCOUNTED NET CASH FLOW FOR A SINGLE PRODUCING WELL Ex 9-1, lOO'/o working interest, 87.57o revenue interest ECONOMIC LIMIT REACHED AT YEAR 5 PAYOUT NEVER REACHED INTERNAL RATE OF RETURN CANNOT BE CALCULATED NET PRESENT VALUES (in $ x 1000) : Undiscounted O'/o : -79. Discounted at 10.0% : -97. 3 ^ 3|e 3|c 3ic 3|c 3|c ^ c ^ e sic ^ c 3(c 3|c )ie ^ 3|e )ie * 3(c * 3|c 3(c 3|c * 3|c ^ c 3)c :tc jfc 3tc * * * sic * 3|e * 9|c 9(c * >(c 3|e 3ic 4 c * *

WELL FOR FIELD SIZE CLASS NUMBER 3|C 3|C sic sic 9)C sic S K *

*

*

*

>lC *

*

3|C sic sic 3|C S|C sic *

sic *

*

sic 3|C %

3|C 3iC *

3

sic sic sic 3|e sic 3|C S|C 3iC sic *

3k

DISCOUNTED NET CASH FLOW FOR A SINGLE PRODUCING WELL Ex 9-1, lOO'/o working interest, 87.57o revenue interest ECONOMIC LIMIT REACHED AT YEAR 5 PAYOUT DURING YEAR 4 INTERNAL RATE OF RETURN = 6.97 PERCENT NET PRESENT VALUES (in $ x 1000) : Undiscounted OX : 27. Discounted at 10.0% : -10. sicsics|cs|csicsies{eslcsicsicsicsicsicsicsics{cs|cs|cs|csics|cs|c3ics|csicsicsic:ics|csicsicsicsicsics(cs)csics{cs|csic

WELL FOR FIELD SIZE CLASS NUMBER

4

s|c s{c sic sic s|c s|c sic sic sic sic sic sic s(c sic sic sic s|c s{c sic s|c sic sic : k * * * * * 3|c s{c sic s|c s|c sic sic s k sic s k sic s|c ^

DISCOUNTED NET CASH FLOW FOR A SINGLE PRODUCING WELL Ex 9-1, 1007o working interest, 87.570 revenue interest ECONOMIC LIMIT REACHED AT YEAR 14

Computing Risk for Oil Prospects — Chapter 11 Table 11.5. Concluded. PAYOUT DURING YEAR 2 INTERNAL RATE OF RETURN = 60.50 PERCENT NET PRESENT VALUES (in $ x 1000) : Undiscounted 0*/, : 369. Discounted at 10.OX : 227. s|c 9(c j|c 3|c :|c :(c 3|c 3ie :|e 3|c 3|e j|c 3|c :(c 3|c 3)c 9|e 9k ^ e * 3|c 3tc * * 3 k * * * * 3|c 3|e 3k s k * a k 9|c 3|e )|c 9(e 9|c 9|c 9|c *

WELL FOR FIELD SIZE CLASS NUMBER

5

3k9k*******3k>|c**3k*ate*3<e3k3k*****9(c9k9(e**3|c**3ie***3te*9|e:k******a|e***********

DISCOUNTED NET CASH FLOW FOR A SINGLE PRODUCING WELL Ex 9-1, 1007o working interest, 87.5*/, revenue interest ECONOMIC LIMIT REACHED AT YEAR 19 PAYOUT DURING YEAR 1 INTERNAL RATE OF RETURN = 139.50 PERCENT NET PRESENT VALUES (in $ x 1000) : Undiscounted 07, : 1065. Discounted at 10.0*/, : 648. 9k>k***3k3k3k*3|c*3k3k**t*3|e*3|c>k******3k3k:|c*******3te**3te3tt3k*3|cjk3|c3k**3|c*3k****

WELL FOR FIELD SIZE CLASS NUMBER

6

DISCOUNTED NET CASH FLOW FOR A SINGLE PRODUCING WELL Ex 9-1, lOO'/o working interest, 87.5'/, revenue interest ECONOMIC LIMIT REACHED AT YEAR 30 PAYOUT DURING YEAR 1 INTERNAL RATE OF RETURN = 302.00 PERCENT NET PRESENT VALUES (in $ x 1000) : Undiscounted OX : 2691. Discounted at 10.07, : 1466. :k9k3k3(c^^>|c9k^^^3k9k^>K^^^^3('^^^^^3k9k3k^3|(>k3|(^^9k9k>K>l'>l'>k>l'%^^'tc^^^^'|(%^^^^'l<^^^^'K

WELL FOR FIELD SIZE CLASS NUMBER

7

DISCOUNTED NET CASH FLOW FOR A SINGLE PRODUCING WELL Ex 9-1, 1007o working interest, 87.57o revenue interest ECONOMIC LIMIT REACHED AT YEAR 30 PAYOUT DURING YEAR 1 INTERNAL RATE OF RETURN = 850.00 PERCENT NET PRESENT VALUES (in $ x 1000) : Undiscounted 07o : 6727. Discounted at 10.07, : 3305.

304

Risk Tables and Trees

100000 il

80000

CO

•g;

"S

c ^ O X

60000

40000 1

0 >

/

P JA

X

. i i A A A A i ^ ^ ^ ^ ^ ^ • • " "^

A-X-A-i

A

20000

E

hlJi'2-Ct^^^-o-o-^-^-o-o-^-o :

o

fill -20000

1

1

1

1

1

10

15 Year

20

- D — 3% Probability -^ 4% Probability - D — 9% Probability - 0 — 2 1 % Probability

25

30

45% Probability 16% Probability 1% Probability

Figure 11.9. Cumulative net cash flow versus time discounted at 10% for seven different field size classes corresponding to Tables 11.3 to 11.5, for Example 11.1. Graph is an alternative display of a probability distribution of cumulative cash flows.

DECISION AIDS FOR ACTIONS Decision analysis involves weighing the financial consequences of alternative choices that are tempered by quantitative estimates of risk. As we have stressed, the alternatives must be convolved with, or multiplied by, their probabilities. This means that good decisions depend on good estimates of probabilities. Although good estimates of probabilities don't eliminate risk, they do help make the best choices given the information at hand. Essential to this decision-making process are procedures that provide probabilities 305

Computing Risk for Oil Prospects — Chapter 11 Table 11.6. RAT input for Example 11.2 for an 87.5% revenue interest and a 100% working interest in a prospect with an areal constraint of 640 acres. Discount rate is 10%. Compare with Tables 9.5 and 9.6 for analysis of a single well with the same investor's terms. Remainder of input table (not shown) is identical with Table 11.3. Output from RAT based on Table 11.6 is given in Table 11.7 and shown in Figure 11.10. Ex 11-2, 1007, working i n t e r e s t , 87.B'/, revenue i n t e r e s t ***RAT*DATA************************************************** 1 F i e l d s i z e d i s t r i b u t i o n (1) or d a t a (2) 2 2 Name of f i e l d s i z e f i l e djsize.dat 3 Name of CASHFLOW d a t a f i l e ex9-l 4 Discount r a t e (X) 10.00 ***RAT*FIELD*PARAMETERS************************************** 1 Well spacing (acres) 40. 2 Area l i m i t for p o t e n t i a l l y productive l e a s e h o l d (0 = no l i m i t ) ( a c r e s ) 640. 3 Dry hole p r o b a b i l i t y CD 80.000 4 Dry hole cost ($) 80000. 5 Cost of dry development hole ($) 65000. Dry development holes function 6 Linear (1) or polynomial (2) f i t 1 7 Number of function d a t a p a i r s 3 8 Num T o t a l Dry 1 6 1 2 16 2 3 30 3

and allow us to rank alternatives according to their favorability. These procedures include decision tables and decision trees. A decision table provides a systematic way of comparing a variety of alternative actions against a spectrum of possible outcomes for a prospect. A risk analysis table (RAT) is generated for a spectrum of outcomes, but it involves only a single action, such as drilling a prospect with a 100% working interest. A decision table, in contrast, considers multiple actions that are cross tabulated against different possible outcomes that are weighted by their probabilities of occurrence.

306

Risk Tables and Trees Table 11.7. Output from RAT for Example 11.2 for a prospect with 87.5% revenue interest, a 100% working interest, and an incorporated areal constraint of 640 acres. The output corresponds to input in Table 11.6. Compare with Table 9.5 and Table 9.6 for analysis of a specific production stream for a single well with identical investor's terms. Ex 1 1 - 2 , lOO'/o w o r k i n g i n t e r e s t , 87.57, revenue i n t e r e s t ****************** j|c**p^I3j(*ANALYSIS*TABLE********************* Dry 1 2 3 4 5 6 7 h o l e Sum (a) Field size probability (b) Field size, (bblsxlOOO)

.031 .041 .092 .214 .449 .163 .010

28

(c) Field size, (acres) 255 (d) Producing wells

5

(e) Cum. prod. (bblsxlOOO)

5

(f) Init. prod, (bbls/day)

10

(g) PV of well ($ X 1000) - 149 (h) Dry devel. wells (DDW)

591

216

329

424

547

705

10

12

14

14

14

465

11

15

-97

1.000

1617 4422 12094

79

7

—-

908

1170

21

49

101

210

22

36

54

83

-9

227

647

1465

3303

2

2

1

1

1

2

2

(i) Cost of DDW ($ X 1000) 65

65

65

130

130

130

145

130

(j) PV of class ($ X 1000) - 810 -744 -163 2594 8939 20382 46120

iCont) 307

Computing Risk for Oil Prospects — Chapter 11 Table 11.7. Concluded, **************** j|e***>ieRiSK*ANALYSIS*TABLE********************* Dry 1 2 3 4 5 6 7 h o l e Sum

(k) Risked probability .006 .008 .018 .043 .090 .033 ..002 .800 1.000 (1) Risked PV ($ X 1000)

—

1659

(m) Dry hole cost ($ X 1000)

-79

-63

(n) Lease-MR cost ($ X 1000)

-32

-25

(o) Net P-A cost ($ X 1000)

-14

-11

-4

-6

-3

111 802 665

94

(p) EMV ($ X 1000)

1558

(q) PV utility (utiles) -81

-74

-16

(r) Risked util. (utiles) -.5

-.6

-.3 7.7

180

418

761

37.6 24.9

(s) Cost util. (utiles) (t) EUV (utiles)

1534

3.1

—

71.8

-12.7-10.1

61.7

Sensitivity of E M V s to Probabilities First we should examine the degree to which expected monetary values are sensitive to probabilities. As an example, we will consider estimates of the dry hole probability for a prospect. Suppose that the regional 308

Risk Tables and Trees 50000

-10000 — D — 3% Probability —^ 4% Probability — D — 9% Probability —0 2 1 % Probability

45% Probability 16% Probability 1% Probability

Figure 11.10. Cumulative discounted net cash flows versus time for seven field size classes in Example 11.2. Investment involves an 87.5% revenue interest and a 100% working interest, is subject to an areal constraint of 640 acres, and corresponds to Tables 11.6 and 11.7. success ratio of 10% has provided us a starting point, but our estimate is to be conditioned upon the apparently superior geological features of the prospect. Whatever assumptions we make about the dry hole probability will have large influence on the prospect's EMV. We might be interested in the EMVs that would result if the dry hole probability is set at 80, 85, 90, and 95%. Assume a dry hole cost of $1,000,000 and a $10,000,000 present value producer. Table 11.8 shows the consequences, revealing that the prospect has a negative EMV when the dry hole probabihty is a little over 90%. We can calculate the exact value that coincides with an EMV of zero: Let p = probability of a producer and d = probability of a dry hole. 309

Computing Risk for Oil Prospects — Chapter 11 Table 11.8. Different EMVs for a prospect in which dry hole probabilities range from 80 to 95%.

(1) (2) (3)

(4) (5) (6)

(7) (8) (9)

(10) (11) (12)

Producer

Dry hole

Row sums

.20 $10,000,000

.80 -$1,000,000

1.00

$2,000,000

-$800,000

$1,200,000

Probability Present value Product of (4) X (5)

.15 $10,000,000

.85 -$1,000,000

1.00

$1,500,000

-$850,000

$650,000

Probability Present value Product of (7) X (8)

.10 $10,000,000

.90 -$1,000,000

1.00

$1,000,000

-$900,000

$100,000

Probability Present value Product of (10) X (11)

.05 $10,000,000

.95 -$1,000,000

1.00

$500,000

-$950,000

-$450,000

Probability Present value Product of (1) X (2)

At break-even, the weighted cost of a dry hole plus the weighted reward of a producer must cancel out, d{-\, 000,000) + p(10,000,000) = 0. However, the probability of a producer and the probability of a dry hole are complementary, p=l-d Substituting into the break-even equation, we get d(-l,000,000) + (1 - d)(10,000,000) = 0 Multiplying out and taking negative terms to the opposite side gives 11,000,000^-10,000,000;

d = 0.9091, or 90.91%

If the dry hole probability exceeds about 91%, we should not drill the prospect. 310

Risk Tables and Trees Table 11.9. Simplified financial consequence table for hypothetical prospect involving specific outcomes (columns) versus specific actions (rows). Outcomes (thousands of dollars) Actions

Small producer

Medium producer

Large producer

Very large producer 0

0

0

0

Drill with 100% working interest

-600

3000

10,000

Drill with 50% working interest

-300

1500

5000

11,000

90

360

1200

2700

Do nothing

Farmout with 1/16 overriding royalty

Dry hole 0

22,000 -1000 -500

Table 11.10. Decision table incorporating information in Table 11.9 plus outcome probabilities. Totals for rows (3), (5), (7), and (9) are EMVs.

Outcomes thousands of dollars' Small prod.

Med. Large Very Ig. prod. prod. prod.

Row totals Dry hole

0.06

0.08

0.04

0.02

0.80

(2) Do nothing

0

0

0

0

0

(3) Prod, of (l)x(2)

0

0

0

0

0

(4) Drill with 100% working interest

-600

3000 10,000

22,000

- 1000

(5) Prod, of (l)x(4)

-36

240

400

440

-800

(6) Drill with 50% working interest

-300

1500

5000

11,000

-500

(7) Prod, of (l)x(6)

-18

120

200

220

-400

(8) Farmout with 1/16 ORR

90

360

1200

2700

0

(9) Prod, of (l)x(8)

5.4

28.8

48

54

0

(1) Probability (%)

1.00 0

244

122

136.2

311

Computing Risk for Oil Prospects — Chapter 11

CONSTRUCTING DECISION TABLES The situation illustrated in Table 11.8 is unrealistically simple. First, there may be a wide range of present values depending on the well's production rate. We should consider a spectrum of different possible production rates and their corresponding present values. Second, we've considered only one action, the drilling of a hole with 100% working interest. We could also consider farming out the prospect (assuming we hold the lease), or bringing in partners and retaining a fractional working interest and a lower revenue interest. These alternatives have different financial consequences, which also depend on outcomes. To examine the range of alternative actions and the possible outcomes, we need a table that cross-compares the different financial consequences of each combination. If the table includes probabilities, it is a decision table because it will show us which action is optimal by ranking the alternative actions according to their desirability. First, however, consider a financial consequences table that does not include probabilities (Table 11.9); this example is for a hypothetical exploratory hole in which the financial consequences of five outcomes are cross-tabulated against three possible actions. The table shows the most desirable outcome for each action, but does not show which action is most appropriate, because the lack of probabilities means that expected monetary values cannot be computed. Table 11.9 has been expanded in Table 11.10 to include probabilities and EMV's, thus providing a ranking of the four possible actions. If we adhere to an EMV criterion in decision-making, the optimum decision is to drill with a 100% working interest, and the second best alternative is to farm out the prospect, although drilling with a 50% working interest is only slightly less attractive. Doing nothing (which almost always is a possible alternative) is the worst course of action in this scenario.

DECISION TREES Decision tables can be extended to form decision trees for analyzing sequences of decisions and outcomes—when an outcome affects the next decision, which in turn involves an action that leads to the next outcome, and so on. A decision tree has a trunk, forks, and branches, and depending on its complexity, may branch repeatedly. There are two kinds of forks. At a chance fork there are multiple alternative outcomes, each of which is a matter of chance. The outcome of drilling an exploratory hole is a matter of chance, and the possible alternatives form a chance fork in a decision tree. In contrast, a decision fork involves the selection of a specific alternative action. 312

Risk Tables and Trees Decision trees and decision tables are similar. The rows of a decision table such as Table 11.10 (except for the row containing probabilities) represent the alternative actions that can be taken at a decision fork, whereas the columns represent the outcomes of a chance fork, each pertaining to the financial consequences of a specific action. The alternative outcomes (and actions) in a decision tree are mutually exclusive, just as they are in a decision table. This exclusivity is essential in evaluating a decision tree. At each decision fork, we should select the best alternative action from our viewpoint and cross off the others. Having done that, then we can consider the spectrum of outcomes that may result, of which only one will actually occur. Of course we don't know what specific outcome will occur when we choose the action to be taken, but to evaluate the tree, we must decide in advance which action we will select depending on what has taken place at the preceding chance fork. Table 11.10 can be graphed, yielding the simple tree shown in Figure 11.11, which contains four decision forks and five chance forks. If we adhere to an EMV criterion in our decisions, we should select the decision fork branch with the algebraically highest EM v. To do this, we double back from the tips of the branches, obtain the EMV for each branch, and cross off the other branches with lower EMVs. As the tree (and Table 11.10) show, the highest EMV is $244,000, which involves drilling with a 100% working interest. We didn't need a decision tree to find the optimum route because Table 11.10 contains the same information. But, when a succession of decisions and outcomes alternate, a decision tree is helpful because it represents a series of decision tables linked together. Figure 11.12 is a slightly more complicated hypothetical EMV decision tree involving two alternations of decision and chance forks—similar trees appear in most textbooks on decision analysis. The first decision fork involves selecting one of three alternatives: (a) sell the prospect (for $45,000), (b) shoot seismic, or (c) drill without seismic. Chance forks follow both the "shoot seismic" branch and the "drill" branch. The shoot-seismic branch leads to four alternative outcomes, namely the discovery of a dome, nose, homocline, or syncline. Each seismic outcome is in turn followed by a decision fork with "drill" and "don't drill" branches. Beyond the drill branches is the "well-outcome" chance fork, where a large discovery is worth $430,000, a small one is worth $130,000, and a dry hole costs -$50,000. To evaluate the tree, we go to the tips of the branches, calculate the EMV's, select the optimum (algebraically highest) branch at each decision fork (crossing off the rest), and work back to the trunk, which yields an overall EMV. Thus, a decision tree leads to a single bottom-line expected monetary value based on the optimum action selected at each decision fork. 313

Computing Risk for Oil Prospects — Chapter 11

Dry -$1,000,000 _ Small -$600,000 ^^Med

$3,000,000

Large $10,000,000 Vy. Large $22,000,000

EMV= $244,000

Dry -$500,000 Small -$300,000

o iy$3

2L%Med

$1,500,000

Large $5,000,000

Decision forl(

Vy. Large

$11,000,000

Dry $0

Cliance fori<

• Terminal event

Small $90,000 Med $360,000 Large $1,200,000 Vy. Large $2,700,000

Figure 11.11. Decision tree equivalent to Table 11.10. As the tree shows, the optimum path involves shooting seismic and drilling a hole, provided that the seismic survey doesn't reveal the presence of a syncline. Since we've decided in advance how we will act based on the seismic results, we assign an EMV of $66,800 to the prospect. If we took the "drill" branch, the EMV is $34,000, which is far from optimum and should be avoided. The "sell the prospect" branch has an intermediate EMV of $45,000, but is still far from the optimum. The reason why the EMV of the "shoot seismic" branch is best is because it avoids drilling if a syncline is found, which should not be drilled because synclines have an associated dry hole probability of 90%.

Expected Utility Tables and Trees An EMV decision table is fine for those who are neutral toward risk, but who is really neutral? Virtually no one is, so EMV tables and trees are not realistic decision-making tools. Table 11.10 shows that driUing with a 100% working interest involves an 80% probability of losing a million dollars. For most individuals that's too much risk and we need an expected utility value (EUV) decision table that incorporates our personal aversion to risk or the corporate risk philosophy of our company. 314

Risk Tables and Trees :!^-$50 •

Dry hole

$130 Small $430 Large

< ^ Cost of seismic

"^-^$45

r~^

Decision fork

^ ^

Chance fork

m

Terminal event

Figure 11.12. Decision tree involving two alterations of decision forks and chance forks. Dollar figures are in thousands. "Financial gate" pertains to cost of seismic ($10,000 in this example). The concept of expected value applies to utiles just as it does to money, so we can substitute utility values for dollars in decision tables and trees. To do this, we need a utility function that is appropriate for our philosophical outlook and can use it to transform monetary gains and losses into utiles. Outcomes expressed in utiles can be weighted by their probabilities of occurrence and the products summed to yield an EUV. We can transform Table 11.10 into an EUV table using the utility function in Figure 11.13 and its enlargement (Fig. 11.14). The range of the utiUty function is large enough to span the extremes of Table 11.10, which extends from a loss of $1,000,000 to a gain of $22,000,000. The transformation is facilitated if we first prepare a table of dollar-to-utile conversions in Table 11.11 and then insert them in the EUV decision table (Table 11.12). Comparison of Table 11.12 with Table 11.10 reveals that the optimum actions are drastically different. An investor whose utility function is represented by Figure 11.13 is sufficiently risk averse that the only suitable action is to farm out the prospect. By contrast, if an EMV criterion is employed (Table 11.10), the optimum choice is to drill with 100% working 315

Computing Risk for Oil Prospects — Chapter 11

Positive utiles 300+ 200+ 100+ Losses

Enlarged area -—, shown in Figure 11.14 H 5

-5

1 h— 10 15 Millions of dollars

Gains

— F — 20

+ -100 + '200 -f-SOO Negative utiles Figure 11.13. Utility function of consistently risk-averse investor used to transform financial outcomes in Table 11.10 to utiles in Table 11.11. Figure 11.14 enlarges part of the curve.

Table 11.11. Correspondence between dollars and utiles provided by utility function in Figures 11.13 and 11.14.

Financial gains and losses from Table 11.10 (million dollars) -1.00 -0.60 -0.50 -0.30 0.00 0.09 0.36 1.20

316

Corresponding utility values -130.0 -58.0 -44.0 -22.0 0.0 4.5 14.6 40.0

Financial gains and losses from Table 11.10 (million dollars) 1.50 2.70 3.00 5.00 10.00 11.00 22.00

Corresponding utility values 50.0 80.0 90.0 128.0 207.0 220.0 333.0

Risk Tables and Trees Positive utiles -40 [30 -20 -10/ Losses I

-1.0

1 -0.5

1 1

0.5 --10

1 1

Gains 1 1

1.0 1.5 Millions of dollars

--20 --30 --40

Negative utiles Figure 11.14. Enlarged segment of the utility function shown in Figure 11.13.

interest. The degree of risk aversion represented by the utility function in Figure 11.13 may be too extreme for many oil companies, but a decision table that is less risk averse could be prepared readily with a more appropriate utility function. Expected utility value or EUV decision trees are formulated in exactly the same manner as EMV decision trees, except that the financial consequences of outcomes are specified in utiles instead of dollars.

317

Computing Risk for Oil Prospects — Chapter 11

Table 11.12. Expected utility decision table in which financial outcomes of Table 11.10 have been transformed to utiles with the utility function in Figures 11.13 and 11.14. Table 11.11 provides correspondence between dollars and utiles in Tables 11.10 and 11.12. Totals for rows (3), (5), (7) and (9) are EUVs for alternative actions.

Outcomes (in utiles) Small Med. 1Large Very Ig. prod. prod. prod. prod.

Row totals Dry hole

0.06 0

0.08 0

0.04 0

0.02 0

0.80 0

1.00

(3) Prod. o f ( l ) x ( 2 )

0

0

0

0

0

0

(4) Drill with 100% working interest

-58

90.0

207.0

333.0 -130.0

(5) Prod, of (l)x(4)

-3.5

7.2

8.3

6.7 -104.0

(6) Drill with 50% working interest

-22

50.0

128.0

220.0

-44.0

(7) Prod, of (l)x(6)

-1.3

4

5.1

4.4

-35.2

(8) Farmout with 1/16 ORR

4.5

14.6

40.0

80.0

0

(9) Prod, of (l)x(8)

0.3

1.2

1.6

1.6

0

(1) Probabihty (%) (2) Do nothing

318

-85.3

-23.0

4.7

Risk Tables and Trees

RAT Cages A decision table can enclose summary lines from a series of RATs for specific actions, and some have dubbed such a construction a "RAT cage" because of its enclosing nature. An arrangement in which outcomes are represented by columns and actions by rows is convenient because RATs in this book are arranged in similar fashion. Tables 11.13 and 11.14 are summaries derived from the RATs produced in Examples 11.1 to 11.5 which describe four different alternative actions for the same prospect that is outlined in Table 11.1. For the degree of risk aversion represented in the analysis, the optimum action is to drill the prospect with an 87.5% revenue interest and 100% working interest, and with no change after payout. If the aversion to loss were greater, the optimum action would be to avoid the potential loss from drilling a dry hole and instead acquire a royalty interest.

OVERVIEW AND A LOOK TO THE FUTURE Decision tables that express the contrasts between alternative investment opportunities in the form of expected monetary values or utility values, as do Tables 11.13 and 11.14, are near the culmination of a long series of steps in which information is transformed, distilled, and expressed in a form best suited for making decisions about oil and gas prospects. Decision trees provide the ultimate expression of these decision-making tools because they represent a succession of decision tables linked together. In looking back, we can appreciate that the information on which decision tables and trees are based includes the knowledge that is relevant in the exploration side of the oil business. Geology and geophysics provide perceptions about potential traps that form individual prospects; production histories provide the background statistics about populations of producing wells and fields; and Bayesian relationships link geological and production data to yield conditional probabilities of discovery. Cash flow analyses link the costs of leasing and drilling with income that will be gained from production if discoveries are made. The possible outcomes for prospects are represented by probability distributions in risk analysis tables, where the potential gains and losses are weighted according to their probabilities of occurrence and the operator's aversion to risk. Finally, decision tables and decision trees systematically contrast the alternative exploration investment opportunities. The analytical sequence is complete, or nearly so. But as in all human endeavors, there is more that could be done. Some especially appealing lines for the future include: 319

Computing Risk for Oil Prospects — Chapter 11 Table 11.13. Decision table summarizing EMVs for the five investment scenarios outlined in Table 11.1 for Examples 11.1 through 11.5. EUVs are summarized in Table 11.14. EMVs are in thousands of dollars. )|c9|c3|c9|e3|e3|e******>|c****>(c9(c3|c*3icEMV

FSize Class

1

Field Size (bblsxlOOO)

28

DECISION

2

79

TABLE*********************

3

4

5

6

7

216

591 1617 4422 12094

Dry

Sum

0

Field Size

Probability .006 .008 .018 .043 .090 .033 .002 .800 1.000 EMV: ($ X 1000) Action 1 Ex 11-1, 1007. WI, 87.57o RI

2091

Action 2 Ex 11-2, 1007o WI, 87.57, RI, airea constrained

1558

Action 3 Ex 11-3, 12.57« RI Action 4 Ex 11-4, change of RI and WI after payout Action 5 Ex 11-5, ORR received, change after payout

399 1147

220

(1) Development of a better understanding of the degree to which geological properties of reservoirs are statistically interdependent, including such basic characteristics as reservoir areas and thicknesses, porosity and permeability, water saturation, and recovery factors. (2) Development of a better understanding of shifts in field size populations with discovery sequence, including the eff^ects of field area-volume relationships, reservoir depths, type of trap, and seismic, well-logging, and drilling technology. (3) Development of better statistical characterizations of the geometrical properties of fields, including area-volume ratios, directional orientation, areal shape, volumetric shape, and length-width ratios. (4) Improvement of procedures for quantifying the financial worth of geological and geophysical information. 320

Risk Tables and Trees Table 11.14. Decision table summarizing EUVs for five investment scenarios outlined in Table 11.1. :|e>|c3|c9|c3|e3|e)(e3|e3|e*9|c3|e**3|c3|c3|e3|c3|e3|e3|c3|e£UY

FSize Class

1

Field Size (bblsxlOOO)

28

2

DECISION

3

79

216

4

TABLE*********************

5

6

7

Dry

591 1617 4422 12094

Sum

0

Field Size

Probability EUV: ( U t i l e s )

.006 .008 .018 .043 .090 .033 .002 .800 1.000

Action 1 Ex 11-1, lOO'/o WI, 87.57. RI

77

Action 2 Ex 11-2, 1007o WI, 87.570 RI, area constrained

61

Action 3 Ex 11-3, 12.57o RI

26

Action 4 Ex 11-4, change of RI and WI after payout

51

Action 5 Ex 11-5, ORR received, change after payout

16

(5) Expansion of the incorporation of error measures for geological and geophysical data. At present, error measures are largely confined to treatment of contour maps. (6) Improvement of procedures for generating probability maps. (7) Improvement of methods for eliciting probabilities based on expert judgment. (8) Improvement of methods for eliciting responses that are useful for generating personal utility functions, and analytical methods for generating an optimum utility function for a corporation or other collective entity. (9) Incorporation of uncertainty in oil and gas price forecasts, perhaps by use of Monte Carlo procedures in computing cash flow forecasts. (10) Development of software for the flexible organization of decision trees. 321

Computing Risk for Oil Prospects — Chapter 11 (11) Development of procedures for the geographic expression of expected monetary values and expected utility values, perhaps in the form of contour maps. (12) Development of procedures for devising optimum exploration drilling strategies that can be linked with EMV maps and EUV maps and which can be updated when new information is received.

322

CHAPTER

12

^>V3x-:

Bringing it Together RISKING T H E ROSKOFF P R O S P E C T To this point, we have demonstrated individual steps in risk assesment; this final chapter integrates all of these procedures for the analysis of a prospect from start to finish. We will use the computer tools in the RISK package to analyze a prospect in northern Magyarstan known as the Roskoff prospect. The Magyarstan Scientific Research Ministry of Economics of Mineral Resources and Hydrocarbons has responsibility for the oil and gas resources of the country, including their exploration and production, and has offered a concession of 2500 ha (25 sq km) that encompasses the prospect. The Roskoff prospect lies in the Belaskova region of northern Magyarstan and is the only concession currently offered in the region, although there is substantial production in another part of Belaskova known as the Troyska area. The Roskoff prospect has attracted wide interest because of its potential, which is regarded as substantial in view of the large oil fields in the Troyska area, about 80 km from the Roskoff prospect. Fields in the Troyska area produce from the XVIIb Limestone of Cretaceous age, which is also the target interval at the Roskoff prospect. The oil and gas resources of Magyarstan are being progressively privatized. While the Ministry formerly drilled the prospects it generated, its new policy is to invite foreign operators to drill them. The Ministry's large base of information is useful for analyzing prospects in the region and the Ministry can provide seismic data, well logs, drilling and completion

Computing Risk for Oil Prospects — Chapter 12 reports, and production records for individual wells and fields. While the Ministry charges for this information, its accessibility is vital for decisions about Magyarstan's oil lands, including Roskoff. Our task is to decide whether the Roskoff prospect is of interest to us, and if so, whether to bid on it now or later. At this stage, we need an analysis of the prospect based on the information that the Ministry has provided us. Our first task is to obtain an outcome probability distribution for the prospect, for it will form a key input in our financial analysis of the prospect and our response to the Ministry's request for bids. The distribution must include an estimate of the dry hole risk and estimates of the probabilities attached to different volumes of producible oil if a discovery is made. The dry hole component is particularly important in view of our limited ability to sustain large losses.

THE PRIZE OFFERED The Roskoff prospect is based on a domelike seismic structure that spans about 2000 ha of structural closure (Fig. 12.1). Vertical relief within the structure is roughly 20 milliseconds two-way travel time. While the structure is attractive, we note that four dry holes resulted when four earlier prospects based on nearby structural closures were tested with exploratory holes. Interest remains strong, however, because of the presence of the XVIIb Limestone and because the Roskoff's structure is similar to producing structures in the Troyska area. Information provided by the four dry holes is summarized in Table 12.1 and file ROSKOFF.WEL in the diskettes. Tables 12.2 and 12.3 summarize information about fields and exploratory dry holes in the Troyska area. The Ministry will award the concession containing the Roskoff prospect to the highest bidder. Applicants must submit sealed bids stating the bonus to be ofTered (in United States dollars), accompanied by a cashier's check for 10% of the amount. The winner's check will be retained and the losers' checks returned. The winner has 30 days to pay the remainder of the bid. Other terms are fixed. The winner must drill an exploratory hole to test the XVIIb Limestone of Cretaceous age. If a discovery is made, a new stepout well must be begun at least every three months (on a 10-ha spacing with wells spaced about 330 m apart on a square grid). Any well locations that remain after the operator ceases drilling will revert to the Ministry. Since the concession area spans a total of 2500 ha (Fig. 12.1), 250 production well locations are available at a 10-ha spacing, although it would be unlikely that all would warrant drilling. The royalty rate is 48%; this is high by U.S. domestic standards but is moderate by international 324

Bringing It Together 45850

45845

45840

45835

45830

45825

790 ha 45820 12770

-L

12775

12780

12785

J-

12790

_L

12795

12800

Figure 12.1. Map of Roskoff area (30 x 32 km) showing seismic structure of reflecting horizon equated to top of Creta<;eous XVIIb Limestone. Contours in miUiseconds. Concession shown as dashed outhne; arrow indicates proposed exploratory well location. Locations of previous holes shown with dry hole symbols. Dashed lines represent seismic traces. Bold line represents fault trace. Coordinates given in kilometers from arbitrary origin.

standards, and is partially offset since Magyarstan levies neither production nor income taxes. The Ministry handles all oil and gas matters for the Magyarstan government, including collection of bonuses and royalty payments. The Ministry also purchases all oil and gas produced at prevailing prices in international markets. As a prospective operator, our task would be simplified because we would deal with a single agency and a complex division of production payments to different royalty owners would not be required. We may, however, engage in joint ventures with other operators, so we can 325

Computing Risk for Oil Prospects — Chapter 12 Table 12.1. Information from four exploratory dry holes in the Roskoff area of northern Magyarstan (Fig. 12.1). Areas of structural closure based on seismic surveys; other information from well logs. Easting

(m)

12783 12779 12800 12794

Northing

(m)

Area of ThickShale ness (m) ratio (%) closure (ha)

45823 45845 45824 45842

91 93 117 119

42 46 29 32

980 685 790 1250

Table 12.2. Information from 16 oil fields in Troyska area of Belaskova region in northern Magyarstan. Thicknesses and shale ratios are field averages for XVIlb Limestone based on logs of all producing and dry wells. Areas of structural closure on top of XVIIb Limestone are based on seismic surveys conducted prior to drilling.

Ultimate cum. prod. Field of field area (MMbbls) (ha)

1 1 2 3 3 7 14 19 26 29 32 39 41 62 98 146

326

120 145 180 270 305 450 920 870 1020 1400 1290 1720 1980 2100 2350 3250

Aver. Aver. cum. initial No. prod. prod. prod. per well per well wells (Mbbls) (BOPD)

21 19 24 23 41 52 66 72 61 93 98 141 182 173 196 292

47.6 52.6 83.3

130 73.1

134 212 243 426 311 326 276 225 358 501 499

28 30 41 60 34 54 82 110 150 112 132 96 88 115 171 187

No. dry

Area of 'Thick- Shale ratio dev. closure ness holes (ha) (m) (%)

9 7 11 9 14 6 14 16 15 18 23 22 29 30 24 32

460 780 600 580 640 780 1100 1030 1400 1630 1450 2900 2200 3250 3800 4100

121 116 123 107 116 110 123 136 141 140 136 129 114 98 120 146

30 32 28 38 38 34 28 32 30 21 28 31 24 26 24 21

Bringing It Together Table 12.3. Thickness and shale ratio of the XVIIb Limestone in 20 exploratory dry holes drilled in the Troyska area. Areas of structural closure on top of the XVIIb Limestone are based on seismic surveys before drilling. Thickness

Shale ratio

(m)

(%)

96 84 102 108 102 90 75 90 110 108

42 36 40 35 32 26 28 40 43 32

Area of closure (ha)

Thickness

Shale ratio

(m)

(%)

Area of closure (ha)

260 270 300 340 360 400 720 980

96 46 72 96 82 89 76 107 112 92

38 48 46 41 39 29 41 37 37 36

1600 1720 1780 2200 2600 2740 3100 3200 3200 3560

1020 1380

consider various joint ventures as alternatives to solo acquisition of the concession. The immediate question is whether to bid on the prospect and what bonus to offer? We expect competition from other operators, but our bid must be reasonable so that the prospect's economics remain attractive. Our analysis will be strongly affected by forecasts of oil prices over the next decade. The analysis also will reflect the probability distribution of expected volume, which in turn must be strongly influenced by the Ministry's previous results in the Troyska area where all discoveries have been made.

TROYSKA AND ROSKOFF AREAS COMPARED Although the Roskoff and Troyska areas are separated by 80 km, comparisons between them are appropriate because the Troyska area is in an advanced stage of exploration and has geological attributes similar to those of the Roskoff area. Unless we go entirely outside the Belaskova region, we have no other choice for an analogue or statistical training area. If we were to go outside, we could use the training and target areas described in Chapters 7 and 8, but they are 300 km to the south and produce from the Jurassic XVa Limestone instead of the Cretaceous XVIIb Limestone. Furthermore, in these areas geologic structure has only a modest influence 327

Computing Risk for Oil Prospects — Chapter 12 on oil accumulation, but in the Troyska area and presumably also at the RoskofT prospect, structure is strongly influential. The two previous areas and the Troyska area are similar in that limestone thicknesses and shale ratios influence reservoir properties. Although the areas studied in Chapters 7 and 8 could provide a training set for the RoskofF prospect, the Troyska area seems better, and by using information from both Tables 12.2 (file TROYSKA.DAT) and 12.3 (TROYSKA.WEL), we can obtain an initial training set for the RoskofF prospect. Table 12.2 summarizes information about oil fields in the Troyska area. Each field is given an estimated ultimate cumulative production in barrels of oil, area in hectares, number of producing wells, average ultimate cumulative production per well, average initial production for producing wells in barrels per day, number of dry development wells, area of structural closure on top of the XVIIb Limestone (determined seismically), and averages of thickness and shale ratio of the XVIIb Limestone encountered in all wells. The production decline is exponential in form throughout and is approximately 10% a year for all fields. Oil is uniformly low in sulfur and its gravity ranges from about 34° to 36° API. There is no oil price diff^erential based on differences in API gravity. Table 12.3 contains information from 20 exploratory dry holes that tested prospects in the Troyska area, including the area of structural closure determined seismically before the prospects were drilled and the thickness and shale ratio of the XVIIb Limestone estimated from logs of each exploratory well. The subsurface geology of the Roskoflf area is represented by a seismic reflection time map of the top of the XVIIb Limestone (Fig. 12.1), a contour map of the shale ratio (Fig. 12.2), and a contour map of the thickness of the interval (Fig. 12.3). Equivalent maps of the Troyska area are not available, but Tables 12.1 through 12.3 can be used for comparing the Roskoff and Troyska areas.

ESTIMATING THE ROSKOFF'S DRY HOLE PROBABILITY The Roskoff's producer and dry hole probabilities are critical. There is no doubt that the prospect is attractive because of its large structural closure, and Table 12.2 documents the strong relationship between structural closure and production volumes in the Troyska area. However, Table 12.3 documents the fact that many other structures in the Troyska area are dry, as are four other structures near the Roskoff prospect whose exploratory holes were dry. Although none of these previous dry holes condemns the Roskoff prospect, it is clear that the dry hole risk is substantial. 328

Bringing It Together 45850

45845

45840

45835

45830

45825

45820 12770

12775

12780

12785

12790

12795

12800

Figure 12.2. Contour map of shale ratio in XVIIb Limestone in RoskofF target area based on logs of four exploratory dry holes. Contour estimates made by inverse-distance weighting procedure. Scale and dimension of map same as in Figure 12.1. The Rostock site is indicated by the open circle. As a first step, we can obtain unconditional probabilities from the outcomes of exploratory wells in the region. Tables 12.2 and 12.3 show that the 36 exploratory wells in the Troyska area yielded 20 dry holes and 16 discoveries, for a dry hole probability of 20/36 = 56% and a producer probability of 16/36 = 44%. An unconditional exploratory dry hole probability of 56% isn't bad, but applying the dry hole probability from the Troyska area to the Roskoff prospect is suspect because the four exploratory wells drilled near the Roskoff area were all dry (Table 12.1), which would yield a dry hole frequency of 100%. If we truly believe that a dry hole probability of 100% is appropriate for the Roskoff prospect, we need not proceed any further. 329

Computing Risk for Oil Prospects — Chapter 12 45850

45845

45840

45835

45830 *%:

"^ 'S^'^ % \

45825 91 45820 12770

12775

12780

117

12785

12790

12795

12800

Figure 12.3. Thickness in meters of XVIIb Limestone in RoskofF area based on information from four exploratory dry holes. Contour estimates made by inverse-distance weighting procedure. Scale and dimension of map same as in Figure 12.1. The Rostock site is indicated by the open circle.

What is the proper alternative? Should we use 56% or 100% for the dry hole probability, take an average of 56% and 100% (which is 78%), or use some other percentage? The problem is that a dry hole probability based on exploratory drilling outcomes from either the Troyska area or the Roskoff area is not wholly applicable. Another alternative is to combine wells from both areas, giving a total of 40 exploratory wells of which 24 are dry, for a dry hole ratio of 24/40 = 60%. Although a dry hole probability of 60% isn't as good as one of 56%, it is close. The problem is that we don't know whether 56% or 60% is the better estimate, suggesting that unconditional probabilities 330

Bringing It Together based on outcome frequencies aren't entirely satisfactory, although they're better than naive guesses. What is needed is a conditional probability that more closely reflects the geology of the Roskoff prospect. Tables 12.2 and 12.3 suggest that dry hole frequencies in the Troyska area are related to areas of structural closure and to variations in shale ratio and thickness of the XVIIb Limestone. We need a dry hole probability that is conditional on the three geological variables and is specific for the location of the well that would test the Roskoff prospect.

Conditional Dry Hole Probability for the Roskoff Well Location Obtaining a conditional dry hole probability that is specific for the Roskoff location poses a problem because we have direct information about only one of the variables at the location, which is the seismic structural closure that has been inferred from seismic lines in the vicinity of the location. Estimates of thickness and shale ratio must be interpolated from the four nearby dry holes, as has been done to produce the contour maps shown in Figures 12.2 and 12.3. We need all three, but combining this "apples and oranges" mixture of seismic and well information poses a challenge. We can use a composite variable to represent the three variables in the form of discriminant scores that can be calculated from a discriminant function. Discriminant functions are described in Chapter 8. The coefficients of the discriminant function must be found whose specific purpose is to discriminate between producing locations and dry locations. For this purpose we will have to use the data from the Troyska area, because the Roskoff area does not yet have any producers. Of course we do not expect the discriminant function to give us exact answers. However, we can adapt the discriminant function to obtain the probability that the Roskoff well location will prove productive. We'll use the discriminant function to calculate a score for the well site, and then we'll convert the score to a probability estimate with another function (Fig. 12.4) that relates scores to producing probabilities, as also described in Chapter 8. The estimates for shale ratio and thickness of the XVIIb Limestone involve interpolation, as shown on the contour maps in Figures 12.2 and 12.3. We realize that there is an uncertainty involved in the interpolation, but we have no way of dealing with it because we have only four wells in the Roskoff area and cannot compute interpolation error functions with such limited information. Furthermore, we can't obtain error functions for shale ratio and thickness from the Troyska area because we do not have logs from the individual producing wells and also lack information about any of the 331

Computing Risk for Oil Prospects — Chapter 12 100 -T

Or

80 -4

fe 60 4 £L

40 - I p

20 -\

T -8 -6 -4 Discriminant score Figure 12.4. Empirical function relating probability of production to discriminant scores. While applied to the Roskoff area, it is based on information from the Troyska training area. Function has been obtained by plotting points for overlapping intervals in which specific probabilities correspond to specific scores. A probability of about 32% corresponds to a discriminant score of —6,6, as indicated by the dashed straight lines. well locations, whether producing or dry. Although error functions would improve our probability estimates, our philosophy is to make the best use of the information that is available, realizing that we can never have all the information that we might like for appraising a prospect. Both the discriminant function and the function relating scores to probabilities must be based on data from the Troyska area contained in Tables 12.2 and 12.3. However, the discriminant score we need requires point estimates for each of the three geological variables at the proposed Roskoff well site. The discriminant function has been calculated using RISKSTAT, and may be written: Score = 0.0001 Xi - 0.1203X2 + 16.22 X3 where Xi is area of closure in hectares, X2 is thickness of the XVIIb Limestone in meters, and X3 is shale ratio of the XVIIb Limestone. The score calculated for the Roskoff prospect's well location is —6.6. This is obtained by inserting the three point estimates, Xi — 2000 (hectares 332

Bringing It Together of closure estimated from Fig. 12.1), X2 = 105 (meters of thickness estimated from Fig. 12.3), and X3 = 0.36 (shale ratio estimated from Fig. 12.2), into the discriminant equation. When a score of —6.6 is transformed by use of Figure 12.4, a producer probability of about 32% is obtained, corresponding to the complementary dry hole probability of 68%. Although this conditional dry hole probability is less optimistic than an unconditional probability, it reflects specific geological conditions at the well site and hence we will use it as the dry hole probability.

PROBABILITIES ATTACHED TO A S P E C T R U M OF FIELD VOLUMES Now that we have a conditional dry hole probability, we need probabilities attached to the spectrum of possible field volumes to complete our conditional probability distribution. Again we turn to the Troyska area for background information, making use of information in Table 12.2. This gives field-by-field information on the three geological variables, the areas of fields, and the volumes of estimated ultimate cumulative production. Information about dry exploratory holes given in Table 12.3 is no longer relevant because it does not bear on field volumes. We can plot the statistical distributions of these variables, and a simple inspection will show that none is normally distributed. As we might expect, ultimate cumulative production, field areas, and areas of structural closure are more or less lognormal in form. Accordingly, we can plot them as logprobability graphs as has been done in Figure 12.5, which shows ultimate cumulative volumes of producible oil for the 16 fields in the Troyska area. Figure 12.6 is a log-probability plot of the areas of structural closure. The two plots confirm that field volumes and areas of closure are approximately lognormally distributed. While the log-probability plots in Figures 12.5 and 12.6 are reveaUng, they do not provide the conditional relationships that we need. Before proceeding, however, we should determine if useful relationships exist between oil volume and variations in the thickness and shale ratio of the XVIIb Limestone. For example, inspection of Table 12.2 shows that larger field volumes are associated with lower shale ratios, although the relationship between field volumes and thickness is less obvious. To investigate, we can prepare cross plots of oil volume and shale ratio and thickness and area of closure. Before we can make such plots, however, we need to know how the averages of shale ratio and thickness in the 16 fields of the Troyska area are statistically distributed. Examination of Table 12.2 suggests they are approximately normally (not lognormally) distributed, and the probability plots in Figures 12.7 and 12.8 confirm this. 333

Computing Risk for Oil Prospects — Chapter 12 1000 -zr-

100-4

c o o

13

Q. 0)
•-»

10^

CO

E

.01

.1

5 10 30 50 70 90 95 Cumulative probability

99 99.9 99.99

Figure 12.5. Log-probability plot of ultimate cumulative volumes of producible oil for 16 fields in the Troyska area. 10000 -^

5 10 30 50 70 90 95 Cumulative probability

99 99.9 99.99

Figure 12.6. Log-probability plot of structural closure for 16fieldsin the Troyska area. 334

Bringing It Together In preparing cross plots, it is desirable that both variables follow the same general form of distribution. Therefore we can plot normally distributed variables such as the shale ratio and thickness on a linear scale, and use a log scale for lognormal variables such as oil volume and area of closure. The cross plot of oil volume versus shale ratio reveals a pronounced relationship (Fig. 12.9), with a correlation coefficient of —0.654, whereas the relationship of oil volume to thickness is less pronounced (Fig. 12.10) and has a correlation coefficient of 0.351. The correlation between oil volume and area of closure is very pronounced (Fig. 12.11), with a correlation coefficient of 0.932.

Estimating Probabilities of Oil Volumes from Area of Structural Closure Before discussing relationships between oil volumes and all three geological variables considered simultaneously, we should examine the relationship between oil volume and area of structural closure, since this is very strong in the Troyska area. Both oil volume and area of closure are approximately lognormal, so their joint or bivariate distribution is probably also lognormal, although 16 fields are not enough to confirm this. From Figure 12.11 we can estimate oil volumes associated with specific areas of closure. Since the Roskoff prospect consists of about 2000 ha of seismic closure, we need a probability distribution for a spectrum of oil volumes conditional upon this specific area of closure. Because we have assumed that the bivariate distribution is essentially lognormal, we can construct a line that corresponds to the logarithm of 2000 ha that intersects the regression line and the confidence bands that represent ± 2 standard deviations. The points of intersection provide three estimates of oil volume on the cumulative probability scale: 10 million bbls corresponds to 2.5% cumulative probability or —2 standard deviations, 31 million bbls corresponds to 50% cumulative probability or the regression line estimate, and 100 million bbls corresponds to 97.5% cumulative probability or +2 standard deviations. The distribution has a log mean of 1.50 and a log standard deviation of 0.25. When entered into RISKSTAT, a spectrum of discrete probabilities and corresponding volumes is obtained (Table 12.4). Although this distribution in Table 12.4 could serve as one of the inputs to module RAT in the financial analysis of the Roskoff prospect, it would be better if we used a probability distribution for oil volumes that is jointly conditional upon all three geological variables, because Figures 12.9 and 12.10 show that field volumes are related to both shale ratio and thickness of the XVIIb Limestone. 335

Computing Risk for Oil Prospects — Chapter 12 40-t-

1—n—I—TT—rr 5 10 30 50 70 9095 99 99.9 99.99 Cumulative probability Figure 12.7. Probability plot of average shale ratio of the XVIIb Limestone per field for 16 fields in the Troyska area. .01

.1

1

150T

1404 1304

I 120o

1101004

90 .01

—1

1

r-i—n—I—r-i—r-i

.1

1

5 10

30 50 70

90 95

Cumulative probability

1

1—i

99 99.9 99.99

Figure 12.8. Probability plot of average thickness of the XVIIb Limestone per field for 16 fields in the Troyska area.

336

Bringing It Together 1000 -q

c .g •-» o

100 -4

Q.

0)

c5

E

30 Shale ratio, percent

40

Figure 12.9. Oil volume in barrels (log scale) versus shale ratio (linear scale) of the XVIIb Limestone for 16 fields in the Troyska area. Shale ratios are averages from logs of all wells in each field; r ~ —0.65. Dashed lines bound 95% confidence interval.

Oil Volume Probabilities Conditional on Three Geological Variables While the probability distribution for oil volumes in Table 12.4 is conditional only on area of closure, we can get a more reliable distribution by regressing oil volume on all three geological variables simultaneously. That is, we can regress the log of field volume simultaneously on the log of area of closure and on untransformed values of shale ratio and thickness. We cannot easily represent such a regression graphically because there are four variables, but its algebraic representation is straightforward and yields a probability distribution for oil volume that is jointly conditional upon 2000 ha of structural closure, a shale ratio of 0.36, and a thickness of 105 m. The multiple linear regression calculated by RISKSTAT yields the following equation: log(volume) = -6.61 + 0.008 x thickness + 0.814 x shale ratio -f 2.088 X logio(area of closure) When we insert the values for the three geological variables at the Roskoff location, we obtain 337

Computing Risk for Oil Prospects — Chapter 12 1000 ^

— -! ]—I—i—T—I—j—I—r-O-T—jOi—n—I—f—T—I

90

100

110

120 130 Thickness, m

r T j T—I—I—r

140

150

Figure 12.10. Oil volumes versus thickness of the XVIIb Limestone in the Troyska area. Thicknesses are field-wide averages; r = 0 . 3 5 .

1000-zr

100

1000 Area of closure, ha

10000

Figure 12.11. Area of closure versus cumulative oil production volumes in the Troyska area. Line is oil volume regressed on area of closure; r = 0.93. 338

Bringing It Together logio(volume) = -6.61+0.008 x 105 + 0.814 x 0.36 + 2.088 X 3.3 = 1.4134 volume = antilog (1.4134) = 25.9 million barrels RISKSTAT also calculates a correlation coefficient of 0.943, a log mean of 1.41, and a log standard deviation of 0.23, yielding the discrete probability distribution given in Table 12.5 (file ROSFSIZ.DAT). This distribution is in principle identical to that obtained using the regression in Figure 12.11, except that we cannot readily produce a graph of the regression because we're dealing with three predictor variables simultaneously rather than a single predictor.

Table 12.4. Probabilities of field volume classes of ultimately producible oil for the Roskoff prospect conditional on 2000 ha of structural closure. Log mean of 1.50 and log standard deviation of 0.25 estimated from Figure 12.9. Dry hole probability not incorporated. Probabilities (%)

Ultimately producible oil (MMbbls)

1.5 8.4 23.5 33.2 23.5 8.4 1.5 100

7.2 11.8 19.3 31.6 51.8 84.8 139.0

If we compare Tables 12.4 and 12.5, we see that the probability distribution of oil volumes that is jointly conditional on area of closure, shale ratio, and thickness is less optimistic than the distribution based solely on area of closure. The difference stems principally from the estimated shale ratio of 0.34 at the Roskoff location, which is moderately high by comparison with average shale ratios in the larger fields of the Troyska area. We decide that Table 12.5 provides better estimates.

339

Computing Risk for Oil Prospects — Chapter 12 Table 12.5. Probabilities of field volume classes conditional on 2000 ha of structural closure, shale ratio of 0.36, and thickness of 105 m. Log mean of 1.41 and log standard deviation of 0.23 estimated from multiple regression analysis. Dry hole probability not incorporated. Probabilities (%)

Ultimately producible oil (MMbbls)

1.5 8.4 23.5 33.2 23.5 8.4 1.5 100

6.5 10.3 16.3 25.7 40.4 63.7 100.0

ROSKOFF PROSPECT FINANCIAL ANALYSIS The final step is to analyze the financial potential of the Roskoff prospect, using RISKTAB'S module RAT. The distribution of field volumes in Table 12.5 forms part of the input, as does a dry hole probability of 68%. Inputs derived from graphical relationships include the ratio of dry development wells to all wells per field (Fig. 12.12), the relationship between average ultimate cumulative production per well and average initial production in barrels per day (Fig. 12.13), the relationship between areas of fields and ultimate cumulative production (Fig. 12.14), and the utility function that represents our outlook with respect to losses and gains (Figs. 12.15 and 12.16, file ROSUTL.DAT). Other inputs, provided as point estimates, are itemized in Table 12.6 and given in files ROSCASH and ROSRAT. An exponential decline of production has been specified. Annual decline rate varies, depending on field volume size, and is calculated internally to be commensurate with ultimate cumulative production, initial production rate, and maximum production life span. We have entered three different prices that might be paid for the concession ($1,000,000, $1,500,000, or $2,000,000), coupled with a 100% working interest, in order to find the maximum bid commensurate with our attitude toward risk as embodied in the utility function. The output from RAT for a concession price of $1,500,000 assuming a 100% working interest and 52% revenue interest is given in Table 12.7. Table 12.8 compares the bottom-line figures from RAT for the three alternative 340

Bringing It Together

I

T—rr

0

50

1 I

I I

I I

r r [ I I

t I

I 1 I 1 r I r r I

100 150 200 250 Total development wells

r [ 1 r 1 I

300

350

Figure 12.12. Dry development wells in each field in the Troyska area versus total number of wells in each field. 600-r

50

100 150 Initial production, bbls/day

200

Figure 12.13. Average ultimate cumulative production per well versus average initial production per well for fields in the Troyska area. 341

Computing Risk for Oil Prospects — Chapter 12 10000 Tr-

io 100 1000 Ultimate production, MMbbIs Figure 12.14. Log-log cross plot of field areas versus ultimate cumulative oil volumes for fields in the Troyska area. prices for the concession bonus. A series of cumulative net cash flows versus time for the seven field size categories is shown in Figure 12.17. The Roskoff prospect has a very attractive EMV for our involvement at 100% working interest (Table 12.8), but given our risk outlook, the amount of the bonus that we propose to bid is critical. The bonus cannot be much more than $1,500,000 before the expected utility becomes negative. We could offer a bid of as much as $1,600,000 and not incur a negative EUV, but we expect to be outbid substantially by competitors unless we offer a significantly higher bid. As an alternative, we could enter into a joint venture with a partner, each sharing equally all costs and all revenue after royalties are paid (thus our revenue interest would be 26%). Under these circumstances, we might propose to enter a bid of $3,000,000 for the prospect. Incorporating these assumptions in RAT yields an EMV of $15,770,000 and an EUV of 2.9 utiles. Given this attractive EMV and the fact that the EUV is positive, our decision is to enter into a joint venture on a 50-50 basis and submit a bid of $3,000,500 on the presumption that a bid should be slightly more than an even amount. We submit a bid of $3,000,500 accompanied by a cashier's check for $300,050. When the winner is announced, we learn that we have lost, for the 342

Bringing It Together

400-r

-100"~f~i—II I 'I I—n—r-|—n—i—r—|—rn—m—pi—i—n—[

-50

0

50

100

150

i i i—r-|—i—rn—r-|

200

250

300

Millions of dollars Figure 12.15. Utility function expressing our attitude toward losses and gains. 80-T-

n — I — I — I — I — I — I — I — I — I — I — I — I — I — p — I — I — I — I — I

0

5 10 Millions of dollars

15

20

Figure 12.16. Enlargement of part of utility function shown in Fig. 12.15.

343

Computing Risk for Oil Prospects — Chapter 12 $500 $400 $300 = g

$200

i

$100 ///

$0

/lx<^^ 1-1 n-n-D-D-D'Ci-n-D-D-n-n-D-n-n-n

. . . i?/g^g:gcn;a?B=D.-QrQrOra-Or.DrDrDr.DrDrDrnrDrn.

$-1001 $-200

10

—D—6.5MMbbls • 10.3 MMbbIs — D — 1 6 . 3 MMbbIs 0 25.7 MMbbIs

Year

15

20

25

40.4 MMbbIs •63.7 MMbbIs 100.0 MMbbIs

Figure 12.17. Cumulative net cash flows for seven field volume sizes for the RoskofF prospect defined in Table 12.5 with a 100% working interest and a concession bonus of $1,500,000.

winning bidder paid a bonus of $4,800,750. We realize that the prospect would still have an attractive EMV with a bonus of that amount, but it would have been too much for us to pay given our risk philosophy. The exploratory well that was subsequently drilled to test the RoskofF prospect was dry. We can conclude, in hindsight, that our analysis and subsequent actions were prudent and in accord with our risk-investment strategy.

344

Bringing It Together Table 12.6. Values supplied as point estimates to Module RAT in the analysis of the Roskoff prospect. Parameter Well spacing: Area of concession: Maximum producing life span of wells: Type of decline: Dry hole exploratory well cost: Dry development well cost: Producing well intangible cost: Producing well tangible cost: Tangible concession physical access cost: Intangible concession physical access cost: Royalty rate: Working interest: Income tax rate: Abandonment cost as % of well's capital cost: Salvage value as % of well's tangible cost: Operating costs initial year per well: Ratio of annual change in operating cost relative to previous year: Oil price in initial year: Ratio of annual change in oil price relative to previous year: Bonus paid for concession (three different values were supplied for three different analyses):

Discount Rate: Depreciation and depletion:

Value 10 ha 2500 ha 30 years Exponential $450,000 $400,000 $350,000 $400,000 $230,000 $175,000 48% 100% 0% 5% 3% $35,000 1.03 $20 1.04

$1,000,000 $1,500,000 $2,000,000 7.5% Unit of production

345

Computing Risk for Oil Prospects — Chapter 12 Table 12.7. Output summary table produced by RAT for the Roskoff prospect assuming a 100% working interest and a concession bonus of $1,500,000. •****j|csie**************RlSK*ANALYSIS*TABLE********************* Dry 1 2 3 4 5 6 7 h o l e Sum (a)

Field size .015 .084 . 2 3 5 . 3 3 2 .235 .084 .015 1.000 prob. (b) Field size 6500 10300 16300 25700 40400 63700 100300 (bblsxlOOO) (c) Field size 450 600 830 1100 1400 2020 2800 (ha) (d) No. of Prod. 34 48 69 93 120 176 221 wells (e) Cum. prod. 191 214 236 276 336 361 404 (bblsxlOOO) (f) Init. prod. 79 85 91 102 123 131 146 (bbls/day) (g) PV of well 312 498 668 972 1454 1660 2001 ($ X 1000) (h) Dry devel 11 12 14 17 20 26 29 wells (DDW) (i) Cost of DDW 4400 4800 5600 6800 8000 10400 11600 ($ X 1000) (j) PV of class 6216 19122 40500 83648 166514 281862 430827 ($ X 1000) (k) Risked prob- .005 .027 .075 .106 .075 .027 .005 .680 1.000 ability (1) Risked PV 29 514 3045 8886 12521 7516 2067 34642 ($ X 1000) (m) Dry hole cost -449 -305 ($ X 1000) (n) Lease-MR cost -1500-1019 ($ X 1000) (o) Net P-A cost -404 -275 ($ X 1000) {Cont.) 346

Bringing It Together Table 12.7. Conchided. 1

2

3

(p) Exp. mon'y. value (EMV) ($ X 1000) (q) PV util. 39 70 (utiles) ( r ) Risked util. .2 1.9 (utiles) (s) Cost util. (utiles) ( t ) Exp. u t i l . value (EUV) (utiles)

4

5

6

7

Dry hole Sum 33041

100

142

220

330

472

7.6

15.0

16.6

8.9

2.3

52.4 -67.8-46.1

•

6.3

Table 12.8. Comparison of bottom-line EMVs and EUVs for the Roskoff prospect for bonuses of $1,000,000, $1,500,000, and $2,000,000; a working interest of 100%; and a revenue interest of 52%. Bonus paid for concession

EMV

(utiles)

$1,000,000 $1,500,000 $2,000,000

$33,541,000 $33,041,000 $32,541,000

23.5 6.3 -10.9

EUV

SOME FINAL POINTS (1) Throughout this book, we have stressed the use of frequency data to estimate probabihties that are relevant for appraising petroleum prospects. Without a background based on frequencies of occurrence, there is virtually no way to constrain our estimates of the probabilities that are attached to different possible outcomes of drilling. Even for the rankest wildcat prospect, frequency data can be derived from analogue areas, nearby wells, and outcrops that are relevant for estimating probabilities of success and magnitude of discovery attached to the wildcat. 347

Computing Risk for Oil Prospects — Chapter 12 (2) We hg^ve seen how many forms of geological and production information can be represented as frequency distributions, including the outcomes of drilling exploratory holes in a region and development wells in a field; the records of ultimate cumulative production from individual wells; the volumes and areas of oil and gas fields; the areas, volumes, and shapes of geologic structures; regional geological properties such as organic carbon content and vitrinite reflectance indices; and characteristics of reservoir units, including their thickness, bedding indices, shale ratios, porosities, and permeabilities. These properties can be expressed either qualitatively or quantitatively, although quantitative measures generally are preferred. Qualitative expressions, however, can be tabulated as frequencies and also placed in numerical form. By representing geological and geophysical information in the form of relative frequency distributions, the uncertainty associated with the geological properties can be expressed quantitatively, which facilitates our goal of minimizing uncertainty. (3) Prospects commonly are evaluated from geological interpretations that are based on subsurface information from well logs and seismic surveys, plus the records of production from previously discovered wells and fields. An explorationist's objective is to uncover conditional relationships between the perceived geology of prospects and the presence and quantities of producible hydrocarbons that may be associated with the prospects. We have seen how this objective can be achieved using statistical procedures that will help reduce, or at least explicitly quantify, the uncertainty in these relationships between geological and geophysical properties and the occurrence of oil and gas. (4) Any action in the analysis of a prospect that increases our uncertainty about the chance of success and the probable magnitude of any discovery that might be made is inappropriate and obviously should be avoided. However, we have seen that some currently popular procedures for appraising prospects do appear to increase, rather than reduce, the uncertainty. Some oil companies multiply together probabilities attached to a succession of essential but difficult-to-assess "geological factors" for a petroleum prospect, such as the presence of source beds, carrier beds, reservoir beds, presence of traps and seals, and the appropriate timing of events such as oil generation, migration, and trap formation. These factors are closely linked with modern genetic concepts in petroleum geology and are attractive from that standpoint. Unfortunately, probabilities assigned to individual factors tend to be unconstrained guesses that have substantial uncertainty unless they are based on observations of their relative frequencies of occurrence. The problem is that frequency data relevant to these basic genetic factors are scarce or nonexistent. Because the probabilities that are assigned 348

Bringing It Together to the factors are combined by multiplication, if any one of the factors is pronouncedly inappropriate, the product obtained by multiplication will be highly misleading even if the estimates of other probabilities are realistic. Increasing the number of factors considered does not improve the final result, and may even produce an unintended decrease in the final assessment of the probability of the "presence of producible hydrocarbons" (or an increase in its complement, the dry hole probability). These prospect assessment procedures, although apparently rigorous and quantitative, may result in greater uncertainty in prospect appraisal, with the added defect that the amount of uncertainty remains unmeasured and unappreciated. (5) Similar problems arise if multiplicative procedures are used for estimating volumes of producible hydrocarbons that will be discovered, prior to the drilling of a prospect. One popular strategy has been to multiply estimates of the trap's area by its vertical dimension, yielding gross reservoir rock volume. This is multiplied by porosity, oil saturation, recovery factor, and volumetric change to yield the volume of oil potentially contained in the prospect. These physical properties of the prospect may be represented either as point estimates or as probability distributions. Because some of these volumetric factors often are poorly known prior to drilling, their assessment may be nothing more than an unconstrained guess, resulting in large increases in uncertainty in the final product. (6) We have noted that there is an additional concern with use of multiplicative procedures for obtaining either the dry hole probabiUty or a distribution of field volumes if the essential geological factors are strongly interdependent. Ironically, corrections for interdependence could be made if frequency data were available to assess the strength of the interdependencies, but since these data generally have not been collected, the issue of possible interdependence between properties is likely to be as unmeasured and as uncertain as the estimates of the factors themselves. (7) Although multiplicative procedures are widely used by major oil companies for prospect appraisal and decision making, the methods usually yield rankings instead of valid estimates of the probable results of drilling. These and similar formalized procedures have been regarded as useful for ranking prospects for decades, but because they generally do not yield true outcome probabilities, they cannot be used validly in financial procedures to obtain expected monetary or expected utility values. (8) We have stressed the estimation of outcome probabilities that are conditional upon geological, geophysical, and production information relevant to prospect appraisal and which can be measured or estimated statistically and objectively at a prospect location before drilling. We have cautioned 349

Computing Risk for Oil Prospects — Chapter 12 that it is unwise to use unconstrained guesses for any of the geological or geophysical aspects of the prospect. To do so will increase the uncertainty and be contrary to our objective of reducing uncertainty in the appraisal. Our goal is to utilize information available before a prospect is drilled in a manner that reveals conditional relationships and minimizes uncertainty. (9) An enormous amount of information useful for statistical appraisal of prospects has accumulated in most petroleum-producing regions, although most of this has been underutilized, at least for extracting conditional probabilities. The first eight chapters of our book describe procedures that can be helpful in extracting conditional probabilities between geological properties and the occurrence of oil and gas in an objective manner. These procedures are conceptually simple and employ widely available software (including that provided with this book), and include procedures for plotting frequency distributions, Bayesian extraction of conditional probabilities, regression, contour mapping and kriging, Monte Carlo simulation, and discriminant function analysis. None are unique to the evaluation of petroleum prospects, and all of the methods are widely applied in other fields. Our contribution has been to link these procedures in a logical sequence for our specific application, and to illustrate how they can be applied to the measurement of uncertainty in petroleum prospect evaluation. (10) Statistical techniques such as discriminant function analysis can be especially useful for exploiting conditional relationships in the analysis of prospects. Although conditional relationships between geological features and the presence and volumes of hydrocarbons often are obscure, particularly if multiple geological factors are responsible for the localization of oil in traps, the degree to which each factor is relevant may be difficult to determine. Discriminant function analysis weights the geological factors according to the degree to which they can distinguish between known producing and barren localities. The discriminant function merges the relevant geological variables into a composite variable in the form of discriminant scores. Their geographic variations can be represented in a contour map, just as individual geological properties can be shown as maps. (11) Contour maps are among the geologist's most widely used tools for estimating the value of a geological property at a prospect location. Variations in lithologic composition in a stratigraphic interval that may serve as a reservoir commonly are represented by contour maps of the proportions or ratios of specific lithologic types. For example, a map of the percent of clay in an interval, based on well log data, can provide an estimate of the cleanliness of a reservoir unit at a prospect location. The uncertainty of the estimate may vary widely, depending on the distances from the prospect to existing wells, the spatial patterns of the wells, and the lateral variability 350

Bringing It Together of the reservoir rock. If the proportion of clay in the reservoir rock is a key factor for the prospect, it is important that the uncertainty associated with the estimate based on the contour map be measured and incorporated into the probabihty estimate for the outcome of the prospect. For this reason, we have emphasized procedures such as kriging for estimating uncertainties associated with contour maps prepared by computer. Composite variables such as discriminant scores can be represented conveniently on contour maps, and their associated uncertainties also can be assessed using these same techniques. (12) Our readers should keep in mind that all of the procedures we have discussed have been directed toward the single goal of appraisal of prospects before they are drilled. Some of the techniques also may be useful for evaluation of producing properties and assessing alternative development schemes for fields, but we have focused on the analysis of prospects prior to an initial discovery and not on subsequent developmental drilling. It is essential, however, to consider the possible outcomes for every exploratory prospect in terms of the spectrum of field sizes that may be discovered. This requires that the financial consequences of different possible numbers of development wells be considered in analyzing a prospect. (13) The statistical, geostatistical, and financial procedures we have presented in this book have been generalized and simplified, partly for illustrative purposes and partly so that risk analyses for a spectrum of field sizes that might be discovered when a prospect is drilled could be linked with financial calculations, and yet have all of this fit into a single book and a computer program for a small personal computer. To examine all of the topics we have touched upon in depth would require a library of books and computer software. Some readers may feel that our RISK software is inadequate for their applications; if so, more powerful commercial packages for statistical analyses, more sophisticated mapping programs, and specialized financial software for analyzing production streams are widely available and will provide all of the functionality required. We strongly encourage those who wish to progress beyond our introductory software, but offer two notes of caution. First, transferring the necessary data files from one program to another requires considerable eff'ort. Making independent programs run harmoniously is not a simple task, especially if operation in an easy and convenient manner suitable for a nonspecialist user is desired. Second, the limitations plaguing risk assessment tend to refiect the inadequacies of the data on which probabilities are based, rather than the sophistication of the methods used to manipulate the probabilities. Gathering additional information to refine estimates of probabilities and conditional relationships may be more fruitful than constructing more refined computer programs. 351

Computing Risk for Oil Prospects — Chapter 12 (14) Because financial analysis is an integral part of the analysis of a prospect, we strongly recommend that explorationists who develop new prospects based on geological and geophysical factors also analyze these prospects from production and financial viewpoints. There are three reasons for such an approach. First, those who search for prospects should be cognizant of the financial aspects at an early stage, because this will help them focus on those prospects that are most likely to be financially attractive and in accord with their companies' attitudes and policies with respect to risk. Second, the procedures used to analyze prospects should be closely linked, leading continuously to the end product: risked financial projections that permit comparisons between prospects on a consistent basis of risked financial worth. Third, each explorationist who generates a prospect should view the prospect as a whole, which requires an awareness of the many factors that bear on its worth. These include not only the geologic and geophysical aspects, but also the costs of any additional geological and geophysical information that might be needed, the spectrum of potential field volumes that might be discovered, the number of development wells that might be required, possible future production rates and decline functions, possible changes in future oil and gas prices, possible changes in operating costs and taxes, and alternatives in royalty rates, other revenue interests, and working interests. Of course, reservoir engineers, financial analysts, managers, and investors will continue to review and analyze prospects generated by others, but they will also find it helpful to use consistent procedures such as those described in this book. Then the consequences of their diff^ering assumptions and premises can be appropriately contrasted and consistently assessed. (15) This book represents a beginning more than an end, for it is obvious that much remains to be done in the art and science of prospect appraisal. We believe that explorationists can use geological, geophysical, production, and other information to develop a consistent, probabilistic framework for prospect appraisal. The increasingly scientific aspects of prospect evaluation can be accomplished at the expense of the artistic component, with a corresponding reduction in the uncertainty associated with petroleum exploration. Some may see this approach as taking the romance out of the search for oil and gas, but the regrets should be off"set by the rewards that will appear in the bottom line. Good prospecting!

352

BIBLIOGRAPHY This bibliography includes sources cited in the text, plus a compilation of literature on the general topics of petroleum resource appraisal and geological risk assessment. It also contains references on the geological, geophysical, and engineering aspects of oil and gas exploration and production, and on a variety of technical subjects such as basic probability and statistics, operations research, risk analysis, utility theory, and applied economics. Various aspects of theory, methodology, and applications are represented, along with a number of specific case studies, many of which are taken from the international arena. Special emphasis has been placed on materials related to the oil and gas industry in the former Soviet Union. Although this compilation is extensive, it is by no means exhaustive; many additional articles have appeared in technical reports, symposium volumes, and other forms of the "gray" literature that have limited circulation and which often are not cited or indexed. The authors will appreciate being informed of errors or omissions. Abramovich, M. V., 1960, Estimate of reserves of prospective areas in folded oil and gas regions: Pet. Geology, v. 4, no. 6-A, p. 315-318. Abry, C.G., 1973, Quantitative Estimation of Oil-Exploration Outcome Probabilities in the Tatum Basin, New Mexico: Unpub. PhD Thesis, Stanford Univ., Dept. of Applied Earth Sci., 151 pp. Abry, C.G., 1975, Geostatistical model for predicting oil, Tatum Basin, New Mexico: Am. Assoc. Pet. Geologists Bull., v. 59, p. 2111-2122. Abry. C.G., 1975, Geostatistics can indicate highest potential acreage: World Oil, v. 181 (August), p. 38-44. Adelman, M.A., 1970, Economics of exploration for petroleum and other minerals: Geoexploration, v. 8, p. 131-150. Adelman, M.A., 1986, Oil-producing countries discount rates: Resources and Energy, v. 8, p. 309-329. Adelman, M.A., 1993, The Economics of Petroleum Supply—Papers, 1962-1993: MIT Press, Cambridge, MA, 556 pp. Adelman, M.A., Houghton, J.C., Kaufman, G.M., and Zimmerman, M.B., 1983, Energy Resources in an Uncertain Future: Ballinger Pub. Co., Cambridge, MA, 434 pp. Agterberg, P.P., 1971, A probability index for detecting favourable geological environments, in McGerrigle, J.I., ed.. Decision Making in the Mineral Industry: Canadian Inst, of Mining and Metallurgy, Special Vol. 12, p. 82-91. Agterberg, P.P., 1974, Geomathematics: Elsevier Scientific Pub. Co., Amsterdam, 596 pp. Agterberg, P.P., 1980, Mineral resource estimation and statistical exploration, in Miall, A.D., ed., Facts and Principles of World Petroleum Occurrence: Canadian Soc. Pet. Geologists Memoir 6, p. 301-318. Aguilera, Roberto, 1978, The uncertainty of evaluating original oil-in-place in naturally fractured reservoirs: SPWLA 19th Ann. Symp. Transactions, p. A1-A17. Ahrens, L.H., 1954, The lognormal distribution of the elements (a fundamental law of geochemistry and its subsidiary): Geochim,ica et Cosmochimica Acta, v. 5, p. 49-73. Aitchison, L.H., and Brown, J.A.C., 1969, The Lognormal Distribution: Cambridge University Press, Cambridge, 176 pp. Allais, M., 1957, Method of appraising economic prospects over large territories: Management Science, v. 3, p. 285-347. Allen, P.A., and Allen, J.R., 1990, Basin Analysis—Principles and Applications: Blackwell Sci. Pub., Oxford, 451 pp. American Petroleum Institute, 1967, A Statistical Study Of Recovery Efficiency, 1st ed.; Am. Pet. Inst. Bull. D-14, 27 pp. American Petroleum Institute, 1984, Statistical Analysis of Crude Oil Recovery and Recovery Efficiency, 2nd ed.; Am. Pet. Inst. Bull. D-14, 47 pp. Anderson, T.W., 1984, An Introduction to Multivariate Analysis, 2nd ed.; John Wiley & Sons, New York, 675 pp. Andreatta, G., Kaufman, G.M., McCrossan, R.G., and Procter, R.M., 1988, The shape of Lloydminster oil and gas deposit attribute data, in Chung, C.F., Fabbri, A.G., and Sinding-Larsen, R., eds., Quantitative Analysis of Mineral And Energy Resources: D. Reidel Pub. Co., Dordrecht, p. 411-431. Ang, A.H.S., and Tang, W.H., 1984, Probability Concepts in Engineering Planning and Design, v. 2; John Wiley & Sons, New York, 562 pp. Anonymous, 1983, Industry wrestles with geological and economic uncertainty in oil and gas project evaluation: Jour. Pet. Tech., v. 35, p. 84-87.

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Tissot, B.P., and Espitalie, J., 1975, L'evolution thermique de la matiere organique des sediments: Applications d'une simulation mathematique: Revue de VInstitut Frangais du P6trole, V. 30, p. 743-777. Tissot, B.P., and Welte, D.H., 1978, Petroleum. Formation and Occurrence: Springer-Verlag, Berlin, 538 pp. Townes, H.L., 1966, Using Economics in Exploration Decisions: Symposium on Economics and the Petroleum Geologist: West Texas Geol. Soc, Publ. 66-53, Midland, TX, p. 108-117. Trask, P.D., 1936, Proportion of organic matter converted into oil in Santa Fe Springs Field, Califomia: Am. Assoc. Pet. Geologists Bull., v. 20, no. 3, p. 245-257. Uhler, R.S., and Bradley, R.G., 1970, A stochastic model for determining the economic prospects of petroleum exploration over large regions: Jour. Am. Statistical Assoc, V. 65, no. 330, p. 623-630. 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380

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APPENDIX A - SOFTWARE INSTALLATION RISK: Software t o perform probabilistic assessments and financial calculations in Computing Risk for Oil Prospects

Hardware Requirements The source code of the RISK software is written in FORTRAN. The programs provided on the accompanying diskettes have been compiled into executable files. They require an IBM-compatible personal computer (model 386 or later) equipped with an MS-DOS operating system, a hard drive, and VGA color graphics card and monitor. To produce printed maps, a PostScript-compatible printer and an appropriate downloading utility are required.

Contents of Diskettes The RISK software includes executable programs and example data files. All of the necessary executable programs are contained in the directory \ R I S K \ E X E . Data files are in directory \RISK\DATA, which includes four subdirectories: DATA\RSTAT, DATA\RISKMAP, DATA\RISKTAB, and DATA\CHAP12. These subdirectories contain all the data files referred to in the text. A complete listing of all files on the two diskettes is given in Table A.l. The RISK software is distributed on two diskettes, RISKl and RISK2. RISKl contains data files and executables; RISK2 contains only executables.

Installing the RISK Software Installing the RISK software consists primarily of copying files from the diskettes provided onto your computer's hard drive. In addition, two system files on your computer, AUTOEXEC.BAT and CONFIG.SYS, must be modified before rimning the software. In the following description, it is assumed that your hard drive is drive C and the RISK distribution diskettes are read on drive A. If the drive names differ on your system (for example, the hard drive is drive E or you are reading the diskettes on drive B), then replace C and A in the following instructions with the appropriate drive designations. The RISK software is distributed on two diskettes. Copy all the files from both diskettes to your hard drive. The software and data files are contained in a number of subdirectories underneath a parent directory called RISK. For example, all the executable programs are contained in a subdirectory named RISK\EXE. When copying these files from the diskettes to the hard drive, you should be careful to preserve the directory structure on the diskettes. The simplest way to ensure that this is done is to use the DOS command XCOPY, which will copy entire directory trees. Begin the process by inserting the first RISK diskette into drive A. At the DOS prompt, type the following command: XCOPY A : *. * C : \

/S

[Upper or lowercase is acceptable, since DOS is not case-sensitive.] This tells DOS to copy all files (*.*) from the diskette in drive A to C:\, the root directory on the hard drive. The / S switch causes XCOPY to maintain the directory structure during the copy. Thus, DOS will create the appropriate subdirectories under C:\

Computing Risk for Oil Prospects — Appendix

A

T a b l e A . l . Content of diskettes comprising RISK software. RISKl \RISK\DATA\RSTAT\ BHBASIN.DAT

PRBASIN.DAT

MAGYAR.DAT

WRBASIN.DAT

DJBASIN.DAT

DJSIZE.DAT

OCS.DAT

MAGVOL.DAT

TRAINWEL.DAT

TRAINWEL.GRD

PROB.DAT

SVARIO.DAT

KRIGE.DEF

DST.DEF

GRID.DEF

DISCR.DEF

TARGET1.DAT

TARGET2.DAT

TARGET3.DAT

TARGETl.GRD

TARGET2.GRD

TARGET3.GRD

PROB.DEF

\RISK\DATA\RISKMAP\

\RISK\DATA\RISKTAB\ CUMINP

FSIZE.DAT

DJSIZE.DAT

UTILE.DAT

DECISION.DAT

UNIT.DEF

EX9-1

EX9-1.INP

EX9-1.TAB

EX9-2

EX9-2.INP

EX9-2.TAB

EX9-3

EX9-3.INP

EX9-3.TAB

EX9-4

EX9-4.INP

EX9-4.TAB

EXll-1

EXll-l.INP

EXll-l.RAT

EXll-2

EX11-2.INP

EX11-2.RAT

EXll-3

EX11-3.INP

EX11-3.RAT

EXll-4

EX11-4.INP

EX11-4.RAT

EXll-5

EX11-5.INP

EX11-5.RAT \RISK\DATA\CHAPI2\ TROYSKA.DAT

TROYSKA.WEL

ROSKOFF.WEL

PROSPECT.WEL

PROSPECT.GRD

UNIT.DEF

GRID.DEF

DISCR.DEF

PROB.DEF

KRIGE.DEF

DST.DEF

ROSCASH

ROSRAT.INP

ROSRAT.RAT

ROSUTL.DAT

ROSFSIZ.DAT

ROSRAT

PROB.DAT MAKEPS.EXE

\RISK\EXE\ RMAP.EXE

RTAB.EXE

DATCONV.EXE

TREND.EXE

SVARIO.EXE

SCORE.EXE

PROB.EXE

WELGRID.EXE

DECISION.EXE

CUM.EXE

RISK.MAN

RISK.BAT

MAP.BAT

QUIT.BAT

STAT.BAT

TAB.BAT

NORMAL.FNT

DPRINT.PRT

RISK2 \RISK\EXE\ RAT.EXE

CASHFLOW.EXE

RISKMAP.EXE

RSTAT.EXE

as it copies. The top-level directory will be C:\RISK, directly under the root directory. XCOPY will list the names of the files as they are being copied. Once the XCOPY command is finished, insert the second diskette into the floppy drive and 384

RISK Software

Installation

enter the same XCOPY command again. The additional files will be put into the appropriate subdirectories as they are transferred to the hard drive. When the files have been copied, you must modify files AUTOEXEC.BAT and CONFIG.SYS, both located in the root directory of your hard drive. Add the entry C : \RISK\EXE to the list of entries in the SET PATH= hne in AUTOEXEC.BAT. This will add the directory C:\RISK\EXE to the execution path, allowing the software to be accessed from any subdirectory. In addition, a line: SET DEVICE-ANSI.SYS should be added to the CONFIG.SYS file. The line should include the complete path for the file ANSI.SYS. For example, if ANSI.SYS is in the C:\DOS subdirectory on your machine, then this fine should read: SET DEVICE=C : \DOS\ANSLSYS The changes to AUTOEXEC.BAT and CONFIG.SYS will not take effect until your computer is restarted.

Running the RISK Software Once C:\RISK\EXE has been added to the execution path, the RISK software can be run from any subdirectory. The simplest way to work with RISK is to first change directories so you are in the subdirectory containing the data files you wish to analyze. For example, if you wish to use files in the directory \RISK\DATA\RISKMAP, make this the working directory by entering CD C : \RISK\DATA\RISKMAP Then start RISK by typing the word RISK at the DOS prompt. If you wish to work with data files that are in a number of different subdirectories, you should copy them all to a common subdirectory and then use the CD command to place you in that subdirectory before starting the RISK software.

Printing Graphics Files RISKMAP creates printable graphics as PostScript files that can be printed directly on a PostScript-compatible printer. However, the graphics files produced by RISKSTAT are metafiles and must be converted to PostScript files using the conversion program MAKERS. For example, if you use RISKSTAT to create a graphics file named MYGRAPH, it can be converted to PostScript with the following command: MAKERS MYGRAPH MYGRAPH.PS The PostScript file MYGRAPH.PS can now be printed on a PostScript printer using the download utility appropriate for the printer. 385

Computing Risk for Oil Prospects — Appendix A

D a t a Files RISK data files are in matrix form, with rows corresponding to data points and colmnns corresponding to variables measured at these data points. RISK data files are in readable text (ASCII) format. RISK includes a data file conversion program called DATCONV which will convert RISK data files into input files that are compatible with many spreadsheet programs. For example, a RISK data file named BIGBEND.DAT can be converted to a form that can be read by Excel, Quatro-PRO, Lotus 1-2-3 or other spreadsheet programs by typing the command DATCONV BIGBEND.DAT BIGBEND.CNV at the DOS prompt. BIGBEND.CNV will be the name of the new file, in spreadsheet format.

Disclaimer The accompanying RISK computer programs and data sets are provided only for educational use by purchasers of this book, Computing Risk for Oil Prospects: Principles and Programs. The RISK programs are designed for the sole and exclusive purpose of demonstrating calculations and procedures discussed in the book. Although the authors have made reasonable efforts to determine that the RISK programs perform as described, the programs cannot be warranted to be free of errors or defects. The RISK programs and accompanying data sets are provided on an "as is" basis, and the authors and publisher do not warrant the operation and applicability of the RISK software for any purpose. Any use of the RISK software and data sets is strictly at the user's risk. The data sets are provided solely to demonstrate the use of procedures described in the book and are not necessarily complete nor accurate; the data cannot be considered to represent actual features in any specific geographic area. The RISK software is copyright by Davis Consultants Inc. Some components of the RISKSTAT program are copyright by TerraSciences Inc., with all rights assigned to Davis Consultants Inc. Some components of RISKMAP are copyright by Robert J. Sampson and the Kansas Geological Survey and are used by permission.

Registration You may register your ownership of the RISK computer programs by contacting the SURFACE III Office of the Kansas Geological Survey. Your name and address will be kept on file and you will be informed of any future updates or new releases of the software, and information on related software products. Please contact the SURFACE III office by letter, fax or e-mail only; telephone inquiries about RISK cannot be accepted. SURFACE III Office Kansas Geological Survey 1930 Constant Avenue Lawrence, KS 66047 USA fax: (913) 864-5317

e-mail:

386

danaQmsmail.kgs.ukans.edu

APPENDIX B - RISKSTAT MANUAL RISKSTAT: Linked c o m p u t e r programs that perform statistical analyses in t h e R I S K software

Running RISKSTAT RISKSTAT is a computer program that performs many of the statistical procedures used in RISK. The name of the executable file for the RISKSTAT program is RSTAT.EXE. Once the RISKSTAT software is loaded according to the installation procedure, RISKSTAT can be accessed through the RISK shell or can be invoked separately by typing R S T A T at the system prompt. When analyzing large data sets, it is very easy for the computer screen to "run away" from you, displaying information faster than you can absorb it. RISKSTAT seeks to avoid this problem by incorporating numerous pauses in its operation. When the program pauses, it will display the instruction Press return to continue : When you see this prompt on the screen, press J [the E N T E R or R E T U R N key]. RISKSTAT automatically places you in the data file module when the program is first started, where you tell the program what data file you are going to analyze. The prompt on the screen says Enter name of input data f i l e : If you have already created a data file, enter its name now. If you have not yet built a data file or you wish to create a new file for analysis, enter the name you wish to assign to this new data set. RISKSTAT will search its directory to find a data file having the name you have entered. If it finds such a file, it will load it into memory and then give you the prompt Press return t o go to the main menu of options : Prom the main menu you will be able to select the type of analysis you wish to perform. If the file name you type cannot be found in the current directory, RISKSTAT will assume that you wish to enter a new data file having this name. Suppose we enter the name of a file—perhaps TESTX—which RISKSTAT cannot find in the current directory. RISKSTAT will respond with the message File TESTX does not exist. Are you creating a new file? y/n (yes/no) If you respond n (no), RISKSTAT will assume that you made a mistake and wish to retype the correct file name. It will respond Enter the name of input data f i l e : If you respond with the name of a data file that RISKSTAT has in its directory, it will load it into memory. You will then go to the main menu of options. Suppose, however, that T E S T X does not exist in the RISKSTAT directory and you respond to the query above by typing y (yes)—you wish to create a new file. RISKSTAT will automatically place you in the Data Entry Module. The first menu is: Enter method of data entry : 1 « Keyboard

Computing Risk for Oil Prospects — Appendix B 2 « ASCII file with user-specified format 3 « ASCII file with list-directed format You must select one of the three options. Option 1 will allow you to create a RISKSTAT data file by entering numerical values through the keyboard. Option 2 allows you to create a RISKSTAT data file from a "foreign" data file. The format of the "foreign" file must be provided as a standard FORTRAN format statement with enclosing parentheses. Option 3 allows you to do the same as Option 2, except the "foreign" data have been recorded in list-directed or free format. That is, successive data values are separated by commas, by spaces, or by returns. We will assume that you wish to enter a simple data set through the keyboard. Enter Option 1 by typing 1 J . RISKSTAT will now ask a series of questions that define the size and nature of the data set. The first instruction is Enter number of variables or columns : RISKSTAT will allow you to enter fi:om one to 12 variables, each as a separate column of values. Each column must have a title, so RISKSTAT will request a "variable description" for each successive column. Usually this is the name of the variable or some unique identifier consisting of up to 15 alphanumeric characters. The request for a variable description will be repeated for as many times as there are columns. Next you will be told to Enter number of groups : N o t e : The only acceptable response is 1. RISKSTAT is preset to use one group only! Multiple groups are not required for the linear discriminant analysis module. You will be asked to enter the number of observations (rows), up to a maximum of 1200. Now that the size of the data arrays has been established, you will be asked to enter the numerical values corresponding to each element in the array. The first request will be for the value in Row 1, Column 1. If a particular observation is missing, enter the letter H. In successive queries, all of the observations for Row 1 are entered. When the first row of data is complete, RISKSTAT will request the observations in the second row and so on, until all elements have been entered. Values may be entered as integers (without a decimal point) or as real values containing a decimal point. Leading and trailing zeroes are optional. When all elements have been entered, the data are automatically stored in the TESTX file. If a mistake is made and an invalid character is entered, the computer will beep and the message Error in number field. Invalid entry. Try again : will appear. This will occur if a number is entered using a special character other than a decimal point or an alphanumeric character other than the letter H is entered to indicate a missing value. Simply type in the correct entry and continue. If an incorrect number was typed in the previous entry, you can go bax:k to that previous value by typing U. You can then enter the correct value at that time. To go to the main menu of options when all entries have been completed, press J .

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R I S K S T A T File N a m e s Data and other information used by RISKSTAT are stored in files identified by unique names. A RISKSTAT data file may be given any valid DOS file name. A file name may contain up to eight characters, plus a three-character extension separated from the main part of the name by a period (.). DOS file names cannot contain special characters or embedded blanks; only upper- and lowercase letters and numbers may be used. [DOS does not distinguish between upper and lower case; so, for example, TESTX, TestX, Testx, and testx are all the same name.] If you attempt to specify a file name that does not conform to these conventions, RISKSTAT will beep a warning and print the message Invalid f i l e name. Try again : If you enter a new file name that is identical to a name on a file that is already in the RISKSTAT directory, the program will beep a warning and ask if you wish to select a different new file name. If you respond n (no), the contents of the old file will be destroyed as the new file having the same name is created. Note that RISKSTAT will not allow you to assign a name to an output file that is the same as the currently active data file, since this would result in changing the file contents while an analysis was being performed. RISKSTAT Variable N a m e s In RISKSTAT, data are contained in files which consist of information records and matrices or data arrays. In the matrices, columns represent variables and rows represent observations. Each column is identified by a column head or variable name, which may be a description up to 15 characters in length. The descriptors may contain any printable characters, including upper- and lowercase letters, numbers, and special characters. The description may contain embedded spaces or blank characters. Variable names need not be unique, but obviously the same descriptors should not be used more than once in a single data set.

Going Back to Previous Menus RISKSTAT is menu-driven. You may make a wrong choice from a menu, and find yourself in a part of the program where you do not want to be. You can easily go back to the previous menu by typing Q ^ . You can then select the correct option. If you are at a submenu within an option, typing Q will return you to the option menu. Typing Q at the option menu will return you to the main menu.

Main Menu of Options This menu describes options that are available in RISKSTAT. The five options are: 1 2 3 4 0

SYSTEM CONTROL AND DATA MANAGEMENT STATISTICAL ANALYSIS STATISTICAL GRAPHICS PACKAGE FIELD SIZE SIMULATION EXIT RISKSTAT

We will examine these options in order.

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Computing Risk for Oil Prospects — Appendix B Option 1 - S Y S T E M C O N T R O L A N D D A T A M A N A G E M E N T The SYSTEM CONTROL AND DATA MANAGEMENT menu of options contains 14 options: 1 Change output options 2 Change system data f i l e 3 Create new data f i l e 4 List data 5 Edit data 6 Merge data 7 Sort data 8 Rank data 9 Find cumulate of a variable 10 Find decimal fraction of a variable 11 Find cumulative fraction of a variable 12 Standardize a variable 13 Logarithmic transform of a variable 0 Return to main menu of options The first option of this menu allows us to change the way RISKSTAT presents its output. Most of the other options allow us to edit or change a preexisting file in some manner. In the Running RISKSTAT section we have been using the third option of this menu to create a new data file. Each option can be selected simply by entering the number that corresponds to the function we wish to perform. 1 Change o u t p u t o p t i o n s : The menu title "Module to Change Output Option" appears on the screen. It fists four options. Each of these consists of a statement followed by y (yes) or n (no). Entering the number corresponding to an option causes the "yes" or "no" response to change. RISKSTAT automatically presets these options so that results of analyses are displayed on the computer's monitor but are not printed nor saved in data files. The fist of options are: 1 2 3 0

Output results to monitor : y Output results to printer file : n Store calculated values in data file : n Exit

If you want your results sent to a printer file, select 2. RISKSTAT will then rewrite the menu, but you will note Option 2 now is set to yes (y). To change back to n, simply select 2 again. Select 0 to exit. If you elect to store the results of calculations, RISKSTAT will ask you to describe the calculated variables whenever you perform an analysis. Some options store calculated results in the input data file. Other options store results in a separate data file. In the latter case, the name of the separate output data file will be requested. It then wiU ask you to name the output variables or columns of the data file where the computed values are to be placed. The exact sequence of queries depends upon the specific computation performed. Generally, this information will be requested immediately before RISKSTAT displays a description of the currently active file. Important w a r n i n g : If 3 (—Store calculated values in data file) is set to y (yes), you should back up your input data files beforehand by copying them onto a diskette. If an error occurs during calculations, or an incorrect operation is accidentally specified, your currently active file may be destroyed because results are written onto the file as they are calculated.

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2 Change system d a t a f i l e : This option of the SYSTEM CONTROL AND DATA MANAGEMENT menu allows you to change the currently active data file. After selecting 2, RISKSTAT will ask you to enter the name of the input data file that you wish to replace the current data file. If you enter the name of a file that exists in the RISKSTAT directory, that file will be placed in active memory. If RISKSTAT cannot find a file with the name you have entered, it will ask if you wish to build a new data file. A "yes" response (y) will automatically take you to the data entry module. If the file cannot be found because you mistyped the file name, enter n. RISKSTAT will repeat its request for the new file name. 3 Create new d a t a f i l e : This option allows you to create a separate new RISKSTAT data file while retaining the currently active data file as your system data file. The use of this option has been described in the Running RISKSTAT section. RISKSTAT will first ask for the name of the file where the data are to be stored. If you provide a name which is the same as a file that already exists, RISKSTAT will beep a warning and ask if you wish to use another file name. If you respond y, you will be asked to provide the new name. If you respond n, RISKSTAT will write the new data onto the preexisting file, erasing the information already stored in it. RISKSTAT will then ask you to select the method of data entry. There are three choices 1 Keyboard 2 ASCII file with user-specified format 3 ASCII file with list-directed format

1—Keyboard. RISKSTAT will first ask for the number of variables or columns. Up to 12 columns may be specified. You will next be asked for a variable description or column heading for each column. The request will be repeated until all columns have been described. RISKSTAT will ask for the number of groups in the data set. Only one group should be specified! RISKSTAT will now begin to create the first group in the new data file. You will be asked to enter the number of observations, with a maximum of 1200. Each observation constitutes a row in the data matrix. You will then be asked to enter the elements in the data matrix for each combination of row and column. Indicate missing data by entering H. You can go back to a previous entry by typing U. When all elements of the data matrix have been entered, RISKSTAT will store the information on the disk and return to the SYSTEM CONTROL AND DATA MANAGEMENT menu of options. 2—ASCII f i l e with user-specified format. RISKSTAT will request information on the number of variables (or columns) and groups in the data file, as described under Option 1 Keyboard. It will then ask the name of the input ASCII file. RISKSTAT will read the ASCII file from the currently active disk drive. If the named file cannot be found, a warning message will appear. RISKSTAT will then request information about the ASCII file, including the number of header records, the code used to indicate missing values, and the format of the data. The format should be given in standard FORTRAN notation, with enclosing parentheses. 3—ASCII f i l e with l i s t - d i r e c t e d format. You will be requested to enter the number of variables or columns, the variable description or column headings for each column, and the number of groups, as described under Option 1 Keyboard. RISKSTAT will request the name of the input ASCII file. It will then look for the ASCII file in the currently active disk drive. It will next request the number of header records in the ASCII file and the value used to represent missing data. If missing value representation is not used, enter - . l E - f 3 1 . RISKSTAT will then compute the number of observations or rows in the ASCII file by counting line

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Computing Risk for Oil Prospects — Appendix B returns. RISKSTAT assumes that each variable (column) is separated by a space or comma, and each observation (row) is separated by a line return. In Options 1, 2, and 3, a new RISKSTAT data file is created which can be used with the RISKSTAT program in the future. O p t i o n 4 L i s t d a t a : . This procedure allows you to produce listings of data files. There are three options. The first produces a short summary of the data, while the second produces a complete listing of all entries in the data file. 1 List status summary of data, 2 List data values, 0 Exit.

1—List status summary of data. RISKSTAT will present a series of screens giving the number of variables or columns, the number of groups, the names of the variables and of the groups, the number of records in the file, the number of records which contain missing values, and the maximum, minimum, and average values for each variable. During listing the screen display will pause several times. You may press J to continue the display or type Q to quit. This feature allows you to cancel what may otherwise be excessively long fistings of data files. Summaries of the contents of each group in a file are listed successively. 2—List data values. RISKSTAT will produce a listing of the individual data values. Data are listed by variable (column), so you will be requested to enter numbers corresponding to variables that you wish to display. The variables can be selected firom a list of variable descriptions that will appear on the screen. Selecting 0 will cause the variable descriptions to be relisted, a valuable option if the number of variables is large and the listing exceeds the number of lines that can be displayed on the computer screen without scrolling. Note that RISKSTAT allows you to terminate the listing process at numerous points by typing Q (quit). The data will be displayed in columnar tables. A maximum of five columns or variables can be displayed at any one time. 0 - E x i t . This will return you to the SYSTEM CONTROL AND DATA MANAGEMENT menu of options. O p t i o n 5 E d i t d a t a : Under this option, RISKSTAT will permit you to change individual values in the currently active data file. The editing option menu lists nine choices: 1 2 3 4 5 6 7 8 0

Modify variable/group descriptions Change single data value Insert data records Delete data records Delete entire variables or columns Delete entire groups Permute order of variables in file Permute order of groups in file Exit

The operation of the edit commands is self-explanatory and will not be described in detail. In all options, variables are listed and are selected by entering the number which corresponds to their column in the data file. Within any option, entering 0 will cause the variable or group descriptions to be relisted. Entering -1 signals that editing is complete for that particular edit option and RISKSTAT will return to the Option 5 Edit Data menu. To change individual data items, the entry to be modified is selected by specifying variable (column) and group. You are then asked for the number of 392

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the row that contains the element. RISKSTAT will display the current value of the element. Simply type in the new data value. Type H to indicate a missing value. When inserting or deleting records (entire observations or rows in a data set), you will be asked to select a reference record number. This is a number between 1 and n, where n is the nimiber of rows (observations) in the group being edited. You will be asked if you wish to insert a new record before or after the reference record. You also will be asked for the number of records to be inserted at that point. During the insertion of new values, you can move back to the previously entered value by typing U. When deleting records, you will be asked to enter record numbers of the first and last records (rows or observations) that are to be deleted. The numbers should be separated by a space, comma, or return. After the designated records or rows are deleted from the data file, the remaining records will be automatically renumbered. When you exit the Edit Data option by selecting 0 (Exit), you will be asked if you wish to save the changes that have been made. Selecting y (yes) will permanently change the edited file on your disk. If you select no (n), the copy of the file on disk will not be modified. Option 6 Merge d a t a : This instruction causes a second data file to be merged into the currently active data file. This convenient feature permits you to analyze a subset of data, appraise the results, and then combine the data with a larger data set. When Option 6 is selected you will be given the choice of merging variables (columns) or merging observations (rows). You will next be asked for the name of the second data file to be merged with the active file. Note that the two files to be combined must have the same number of columns if they have common variables, or the same number of rows if they have common observations. If these conditions are not met, an error message will appear and RISKSTAT will return to the menu of options. If you are merging observations, the variables in the two files must be in the same sequential order. If you need to change the order of variables in a file before merging, you can do this by selecting 7 (Permute order of variables) or 8 (Permute order of groups) under Option 5 Edit data. Option 7 S o r t d a t a : This procedure allows you to sort a data file, based on the values of one of the variables in the file. Sorting is done in ascending order. When you select Option 7, you will be presented with a list of variables, and asked which variable you wish to sort on. After picking the variable, the entire file is sorted, and the results written directly to the data file. You will then be returned to the SYSTEM CONTROL AND DATA MANAGEMENT menu. Option 8 Rank d a t a : This option produces a new variable that contains the ranks of the observations of one of the variables in the system data file. You will be asked to choose the variable that you wish to rank. Select the variable by entering the number of that variable. Next you will be asked to enter the name of the new variable that will be created. A new variable, equal to the rank of each observation in the file, will then be written to the data file. Option 9 Find cumulate of a v a r i a b l e : This procedure produces a new variable equal to the cumulative sum of an old variable. You will be asked to select the variable for accumulation from the menu. You will then be asked to select a variable which will determine the order of accumulation. For example, you may wish to accumulate field volume versus discovery date, to generate a new 393

Computing Risk for Oil Prospects — Appendix B variable which contains cumulative volume produced versus time. In this case, tell RISKSTAT to accumulate volume based on the order of discovery date. To accumulate a variable from its smallest to largest value, simply accumulate that variable based on the order of the variable itself (select the same variable at both prompts). Then you will be asked to provide the name of the new variable that will contain the cumulative sum. The cumulative sum of each observation of the old variable will be entered in the data file xmder that variable name. Option 10 Find decimal f r a c t i o n of a v a r i a b l e : This option creates a new variable that is equal to the fraction that each observation represents of the sum of a selected variable. You will be asked to select one of the variables in the data file from the menu. Next, you will be prompted for the name of the new variable. The program then calculates the decimal fraction that each observation of the old variable represents of the sum of that variable. The output is then written to the data file, and you will be returned to the SYSTEM CONTROL AND DATA MANAGEMENT menu. Option 11 Find c u m u l a t i v e f r a c t i o n of a v a r i a b l e : This option calculates the cumulative fraction of a variable. This option behaves exactly like Option 9, except that the individual accimiulated values are divided by the overall sum of the variable. First, select from the menu the variable for which you wish to find the cumulative fraction. As with Option 9, you will then be prompted to specify which variable should determine the order of accumulation. Next you will be asked to enter the name of the new variable that will be created. The cumulative fraction represented by each observation is calculated. The new variable containing the cumulative fraction is written to the data file, and you will be returned to the SYSTEM CONTROL AND DATA MANAGEMENT menu. Option 12 S t a n d a r d i z e a v a r i a b l e : This option creates a standardized variable from a preexisting variable. The standardized variable will have a mean of zero and a standard deviation of 1. After selecting one of the variables in the file, you will be asked to provide the name of the standardized variable that will be created. This procedure first calculates the mean and standard deviation of the variable that is to be standardized. The standardized variable is then calculated by subtracting the mean from each observation and dividing the result by the standard deviation. The standardized variable is written directly to the data file, and you will be returned to the SYSTEM CONTROL AND DATA MANAGEMENT menu. Option 13 Logarithmic transform of a v a r i a b l e : This procedure creates a new variable by taking the natural logarithm, common logarithm (base 10), or base 2 logarithm of a preexisting variable. Select the variable whose logarithm is to be taken from the menu. Then select the logarithm base to use. Next, you will be prompted for the name of the new variable that will be created. The logarithm of the selected variable will be calculated and written to the data file. Option 0 Return t o main menu of o p t i o n s : Self-explanatory.

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Option 2 - S T A T I S T I C A L A N A L Y S I S The STATISTICAL ANALYSIS module of RISKSTAT allows you to perform simple univariate and bivariate statistical analysis. In addition, discrete normal distribution, multiple regression, and discriminant function analysis (DFA) are included in the STATISTICAL ANALYSIS module, which contains six options: 1 2 3 4 5 0

Univariate statistics Bivariate statistics Normal distribution Multiple regression Discriminant Function Analysis Return to the main menu

1 U n i v a r i a t e s t a t i s t i c s : The univariate statistics module is able to run statistics on a subset of the data set, such as all oil fields discovered from 1945 to 1965. If you do not want to analyze a subset of the data, answer n (no) at the prompt. If you do want a subset of the data, answer yes (y). You will be presented with a menu listing the variables in the data file, and asked to select the variable to be used to subset the data. Enter the number of the variable. In this example, that would be the number of the column containing the discovery dates. You will then be asked for the lower limit and upper limit of the interval to be analyzed. If you want to analyze all fields discovered from 1945 to 1965, inclusive, enter 1945 for the lower limit, and 1965 for the upper limit. You can also choose a subset that is everything else but the selected subset. When RISKSTAT asks you: Do you want the s p e c i f i e d range as subset (1) or i t s complement (2) ?

you enter 2. This will plot all fields discovered before 1945 and after 1965. If you want to analyze all fields discovered after 1965, enter 1966 for the lower limit, and any number greater than or equal to the latest date {e.g., 2000). After you have chosen the interval that you want to study, you will be presented with a menu of the variables in the data file, and asked to select the variable to be analyzed. (The same variable that was used to subset the data also can be analyzed.) RISKSTAT will then list the number of observations used in the calculations, followed by the univariate statistics for the chosen variable. The statistics generated include the mean, sum, variance, standard deviation, geometric mean, coefficient of variation, minimum value, 25th percentile, median value, 7bth percentile, and maximum value. A value of —999 for the geometric mean indicates that the variable had negative values, so the geometric mean does not exist. 2 B i v c u r i a t e s t a t i s t i c s : This option produces an analysis of the joint variation of two variables. If you select this option, the data set must contain at least two columns or variables. If it does not, an error message will appear. RISKSTAT will fist all variable names for the currently active file. You will be asked to select one of these to be variable X and another to be variable Y. RISKSTAT will then list the number of observations, the means, variances, and standard deviations for variables X and F, and the covariance and correlation between the two variables. If you want to view a cross plot of the two variables, use STATISTICAL GRAPHICS PACKAGE, Option 3 on the main menu. 3 Normal d i s t r i b u t i o n : This option produces a discrete normal distribution. You will be asked to enter a mean and a standard deviation of a continuous normal distribution and the number of discrete probability classes. RISKSTAT will then 395

Computing Risk for Oil Prospects — Appendix B list the probabilities for each class and their midpoint values. If you specify the distribution sis lognormal, RISKSTAT will automatically express the midpoints as their antilog values. 4 M u l t i p l e r e g r e s s i o n : This option produces the weights of a linear multiple regression equation. If you select this option, the data set must contain at least two columns or variables. If it does not, an error message will appear. RISKSTAT will list all variable names for the currently active file. You will be asked to select one of these to be the dependent variable and at least one other to be the independent variable(s). RISKSTAT will then calculate the regression equation and list all observations, the regressed values, and the deviations. Then, RISKSTAT will display the regression coefficients for all independent variables plus a constant term. In addition, the standard error of the regression and the correlation coefficient are shown. 5 D i s c r i m i n a n t F u n c t i o n A n a l y s i s f o r two g r o u p s : RISKSTAT will compute a linear discriminant function using a procedure related to linear regression. The procedure will find the most efficient linear combination of variables for distinguishing between two groups. The identity of members in the two groups must be known beforehand. RISKSTAT will ask for the variable you want to use to identify members of the two groups, and will present a menu of the variables in the data file. Select a variable by entering the number of the variable from the menu. Next, you will need to enter a cutoff level or threshold value for splitting the data set into two parts. For example, you might want to differentiate between producers and dry holes, using a variable that contains the coded results of drill-stem test data. If dry holes are coded with a 2, 12, or 14, and a producing well has a code of 13, then enter the number 13 that equals the cutoff level. All wells having a drill-stem test code value 13 (i.e., producers) will be placed in one group, and the non-producing wells will be placed in the second group. RISKSTAT will ask for the number of variables to be used in computing the linear discriminant function. It will then list the variable names or column descriptors in the currently active data file. You will be asked to select the number corresponding to each successive variable to be used in the equation. Your choices will be confirmed. You will also be asked for the name of the variable for the discriminant scores, which are automatically stored in the system data file. RISKSTAT will report on the number of observations in each group. The vector mean of group 1 will be calculated, followed by the vector mean of group 2. These two vectors will be subtracted to produce the vector of mean differences. The pooled variance-covariance matrix for the two groups is calculated and reported. RISKSTAT will now solve the discriminant function equation and present a table of statistics. The first of these is an F-value, with degrees of fireedom listed, for testing the significance of the difference between the two groups. The F-value is based upon Mahalanobis' Distance. RISKSTAT now presents the coefficients of the linear discriminant function The standardized differences listed by RISKSTAT provide an approximate assessment of the effectiveness of the variables as discriminators. The sign of a standardized difference is not important, as it simply reflects which of the two groups was selected to be group 1 and which was selected to be group 2. RISKSTAT will now compute discriminant scores for the two groups, and store them in the system data file. RISKSTAT will also write the discriminant

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function to a file called DISCR.DEF that may be used in RISKMAP. You will then be returned to the STATISTICAL ANALYSIS menu. 0 R e t u r n t o t h e main menu: Self-explanatory. Option 3 - STATISTICAL G R A P H I C S P A C K A G E The STATISTICAL GRAPHICS PACKAGE allows you to create high-resolution graphic plots from your data. The five options that are available are: 1 2 3 4 0

Histogram Cumulative histogram Normal probability plot Bivariate scatter plot Return to the main menu

Generation of Graphic Files Any of the plots produced by the STATISTICAL GRAPHICS PACKAGE can be printed. When you are satisfied with the appearance of the plot on the screen, type P J . You will be asked to provide a name for the printer file that you want to create. Note that this file is a print metafile and cannot be printed directly. Use program MAKEPS at the DOS level to convert your print metafiles into PostScript files which can then be printed on PostScript-compatible printers (see Appendix A - Software Installation for instructions). The first three statistical graphics options produce univariate plots. Each of the univariate plots provides an option to plot a subset of the data, for example, all oil fields discovered from 1945 to 1965. If you do not want to plot a subset of the data, answer n (no) at the prompt. If you do want a subset of the data, answer y (yes), and you will be presented with a menu of the variables in the data file and asked to select the variable to be used to subset the data. Enter the number of the variable. You will then be asked for the lower limit and upper limit of the interval to be plotted. If you want to plot all fields discovered from 1945 to 1965, inclusive, enter 1945 for the lower limit, and 1965 for the upper limit. You also can choose a subset that contains everything but the selected subset. When RISKSTAT asks you: Do you want the s p e c i f i e d range as subset (1) or i t s complement (2) ?

you enter 2. This will plot all fields discovered before 1945 and after 1965. If you want to analyze all fields discovered after 1965, enter 1966 for the lower limit, and any number greater than or equal to the latest date (e.^., 2000). Each of the plotting options will now be discussed separately. 1 Histogram: The first prompt will ask if you want to plot a subset of the data. If you do, refer to the subset option described above. After you have responded to the prompts for preparing subsets, RISKSTAT will present you with a menu listing all of the variables in the data file. Enter the number of the variable to be plotted as a histogram. RISKSTAT will display the number of observations, then plot the histogram. The variable used to subset the data, as well as the lower and upper cutoflTs, will be displayed in the upper right-hand margin of the graph. If you want to print the histogram, type P J . You will be asked to name the print file that will be created. If you do not want to print the histogram, simply press U to clear the screen. You will be 397

Computing Risk for Oil Prospects — Appendix B asked if you wish to change any of the plot parameters. An answer of n (no) will return you to the STATISTICAL GRAPHICS PACKAGE menu. If you want to change the plot parameters, enter y (yes) and you will see the histogram option menu. RISKSTAT returns repeatedly to the histogram option menu until you select option 0, so several plotting parameters can be changed before the plot is regenerated. The histogram plotting options that are available are: 1 Change the histogram class limits and/or the number of classes

2 CHiange Y-axis parameters 3 Choose between absolute (on) and relative (off) frequencies 4 Overlay a plot of the normal frequency distribution on the histogram (no) 5 Change plot t i t l e —> HISTOGRAM 0 Return without making any (more) changes 1—Change the histogram class limits and/or the number of classes. RISKSTAT displays a message giving the current number of classes in the histogram, the width of each class, the starting point of the histogram, and the number of decimal places in the X axis labels. If you want to change any of these parameters, enter the new parameters that you wish to use, and press J . 2—Change Y-axis parameters. This option is useful for ensuring that several histograms have a constant vertical scale so they can be compared easily. RISKSTAT prints the current number of tick marks on the Y axis, the width between tick marks, and the maximum value for the Y axis. Enter the new parameters that you wish to use, and press J . 3—Choose between absolute (on) and relative frequency (off). This option toggles between the use of absolute and relative frequencies on the Y axis. Absolute frequency expresses the number of observations falling in each class. Relative frequency is expressed in decimal fractions; the number of observations in each class is divided by the total number of observations, so the sum of the relative frequencies is equal to 1. 4—Overlay a plot of normal frequency distribution on the histogram (no). This option can be used to qualitatively assess the degree of normality in the data. A normal curve is plotted that has the same mean and standard deviation as the observations. Selecting this option toggles the normal plot on and off. 5—Change plot t i t l e —> HISTOGRAM. This option gives you the current title and will prompt for a new title, which can be any alphanumeric string, up to 40 characters long. The new title will replace the previous title at the top of the plot. 0—Return without making any (more) changes. Choose this option when you are finished altering the plotting parameters for the histogram. The histogram will then be plotted using the new parameters that were chosen. 2 C u m u l a t i v e histograLin: This option includes the ability to plot a cmnulative histogram of a subset of the data. See the instructions contained in the introductory paragraphs of the STATISTICAL GRAPHICS PACKAGE option for producing subsets of data. After you have responded to the prompts to select a subset of the data file, you will be presented with a menu listing all of the variables in the data file. Choose the variable to be plotted. RISKSTAT will print out the nimiber of observations and plot the cumulative histogram. If you want to print the cumulative histogram, type P J . You will be asked to provide the name of the print metafile to be created. If you do not want to print the cumulative histogram, enter a return ( J ) to clear the screen. Next, you will be asked if you

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want to change any of the plot parameters. If you answer n (no), you will be returned to the STATISTICAL GRAPHICS PACKAGE menu. If you answer y (yes), you will be presented with the cumulative histogram option menu. The options on that menu include: 1 Change the histogram class limits and/or the number of classes

2 Change Y-axis parameters 3 Choose between absolute (on) eind relative (off) frequencies 4 Choose between a cumulative bar chart (on) and a curve (off) 5 Change plot t i t l e —> CUMULATIVE HISTOGRAM 0 Return without making any (more) changes 1—Change the histogram class limits and or the number of classes. RISKSTAT prints a message giving the current number of classes in the histogram, the width of each class, the starting point of the histogram, and the number of decimal places used on the X axis. To change any of these parameters, enter the new parameters that you wish to use, and press return. 2—Change Y-axis parameters. This option is useful for ensuring that several histograms have a constant vertical scale so they can be compared more easily. RISKSTAT prints out the current number of tick marks on the Y axis, the width between tick marks, and the maximum value for the Y axis. Enter the new parameters that you wish to use, and press return. 3—Choose between absolute (on) and relative (off) frequencies. This option toggles between the use of absolute and relative frequencies on the Y axis. The absolute frequency expresses the number of observations falling in each class. The relative frequency scale reads in decimal fractions; the number of observations in each class is divided by the total number of observations, so the sum of the relative frequencies is equal to 1. 4—Choose between a cumulative bar chart (on) or a curve (off). This option toggles between plotting the cumulative histogram using bars to represent the frequencies, or as a "curve" that connects the upper limit of each frequency class with straight line segments. 5—Change plot t i t l e —> CUMULATIVE HISTOGRAM. This option will display the current plot title and prompt for a new title, which can be any string, up to 40 characters long. The new title will replace the previous title at the top of the plot. 0—Return without making any (more) changes. Choose this option when you are finished altering the plotting parameters for the cumulative histogram. The cumulative histogram will then be plotted using the new parameters that were chosen. 3 Normal P r o b a b i l i t y P l o t : This option plots the observations on a normal probability scale. If the data are normally distributed, they will lie along a straight line. If the data are lognormally distributed, then logarithms of the data values will be plotted along a straight line on a normal probability scale. The normal probability plot can be generated for subsets of the data; the first prompt under this option will determine if you want to subset the data. Instructions for creating a subset of the data are contained in the introductory material for the STATISTICAL GRAPHICS PACKAGE. Because of the limited size of the graphics screen, if a subset of the data is chosen, the name of the variable used to define the subset and the specified limits of the chosen interval will be displayed only on printed copies of the normal probability plot. After you have responded to the prompts for producing a subset of the data, RISKSTAT will provide a menu

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Computing Risk for Oil Prospects — Appendix B listing all of the variables in the data file. Select the variable to be plotted. RISKSTAT will print the number of observations and then generate the normal probability plot. If you want to print the normal probability plot, type P J . You will be asked for the name of the print metafile that will be created. If you do not want to create a print file, press J to clear the screen. After the screen clears, you will be asked if you want to change the plot parameters. If you respond n (no), you will be returned to the STATISTICAL GRAPHICS PACKAGE menu. If you want to change the plot parameters, respond y (yes). The normal probability plot option menu will be displayed. There are three entries in the menu: 1 Choose between linear (on) or logarithmic (off) plot 2 Change plot t i t l e —> NORMAL PROBABILITY PLOT 0 Return without making any (more) changes 1—Choose between linear (on) or logarithmic (off) plot. This option toggles between a linear and logarithmic plot. If the logarithmic option is chosen, the logarithms (base 10) of the data are taken, and a logarithmic scale is used for the Y axis. 2—Change plot t i t l e —> NORMAL PROBABILITY PLOT. This option will display the current plot title and prompt for a new title, which can be any alphanumeric string up to 40 characters long. The new title will replace the previous title at the top of the plot. 0—Return without making any (more) changes. This option regenerates the normal probability plot. 4 B i v a r i a t e S c a t t e r P l o t : This option of the STATISTICAL GRAPHICS PACKAGE produces a cross plot of two variables. You will be asked if you want to discriminate between two subsets of the data by plotting the two sets using different symbols. If you respond n (no), you will be asked to select two variables for the cross plot. If you respond y (yes), you will be asked which variable you want to use to determine the two subsets. RISKSTAT will present a menu listing all of the variables in the data file. Select a variable by entering the number of the variable from the menu. Next, you must enter a cutoff value or threshold level that will be used to split the data set into two parts. For example, you might want to differentiate between producers and dry holes, using a variable that contains the coded results of drill-stem test data. If dry holes are coded 2, 12, or 14 and producing wells have a code value of 13, then enter the number 13 as the upper cutoff level and also as the lower cutoff value. All wells with a drill-stem test code less than 13 or greater than 13 will be plotted using one symbol, and producing wells will be plotted using another symbol. After you have responded to the prompts to select subsets of the data file, you will be asked to select two variables for the cross plot. The first variable that you select (Variable 1) is the X variable (plotted along the horizontal axis) and the second variable is the K variable (plotted along the vertical axis). After you have selected the plot variables, RISKSTAT will display the number of observations and then plot the data. If a variable has been used to divide the data into subsets, the name of the variable and the threshold level will be displayed in the upper right margin of the plot. If you want to print the plot, enter P J . You will be asked for a name for the print metafile that will be created. If you do not wish to print the cross plot, enter a return ( J ) to clear the screen when you have finished viewing the plot. Next, you will be asked if you want to change any of the plot parameters.

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If you answer n (no), you will be returned to the STATISTICAL GRAPHICS PACKAGE menu. If you want to change any of the plot parameters, answer y (yes) and you will be presented with the Bivariate Scatter Plot Option menu. That menu includes seven options: 1 Change the plotting parameters of the X axis 2 Change the plotting parameters of the Y axis 3 Change the X axis to a logarithmic axis (no) 4 Change the Y axis to a logarithmic axis (no) 5 Add a regression line or polynomial or cubic spline or semivariogram 6 Change plot t i t l e —> SCATTERPLOT 0 Return without making any (more) changes) 1—Change the plotting parameters of the X axis. This procedure allows you to change the number of tick marks, the width between tick marks, the origin for the X axis, and the number of decimal places used in labels of the tick marks. RISKSTAT will print the current value for these parameters. You will then be asked to enter new values for the parameters on a single line, leaving a space between each of the parameters. 2—Change the plotting parameters of the Y axis. This option functions in the same manner as Option 1, but changes the parameters of the Y axis. See Option 1 for a more detailed description. 3—Change the X axis to a logarithmic axis (no). This option toggles between a linear and a logarithmic X axis. 4—Change the Y axis to a logarithmic axis (no). This option toggles between a linear and a logarithmic Y axis. 5—Add a regression line or polynomial or cubic spline or semivariogram. This option gives you a submenu which contains six choices: (1) Plot the ordinary linear regression of Y on X. (2) Plot the ordinary linear regression of Y on X, with a 95% confidence band around the regression. (3) Plot the reduced major axis (RMA) regression between X and Y. The reduced major axis regression is useful in situations where it is not clear which variable should be considered the dependent variable. If a bivariate scatter plot with a regression line is plotted, the slope and intercept of the regression line and the linear correlation coefficient will be listed, as well as the type of regression. (4) Add a fitted polynomial to the plot. If you choose Option 4, you will be prompted for the degree of the equation to be fitted, which can be any integer between 0 and 10. You will then be asked if you want an estimated intercept. If you respond to this prompt with n (no), the intercept term will be set to zero, forcing the polynomial curve through the origin. (5) Connect the data points with a piecewise cubic spline function. If this option is chosen, then a spline curve will be added to the plot. If either a polynomial curve or a spline curve appears on the current plot, you will be presented with the following prompt after you clear the screen and before you are asked if you want to make any further changes: Enter f i l e for storing regression coefficients (RETURN for none) : [The word "regression" is replaced by "spline" if a spline curve has been drawn.] If you want to save the regression or spline coefficients, then enter the name of a new file at this prompt. Otherwise, simply press J . A file containing 401

Computing Risk for Oil Prospects — Appendix B polynomial coefficients for an nth-degree polynomial will contain n + 1 lines, each consisting of a coefficient label followed by the value of the coefficient. The first line contains the intercept (Oth-OTdei) term. An example of a coefficient file for a cubic polynomial would be: J5(0) = - 1 . 5 7 3 3 B ( l ) = 5.4654 B(2) = - 1 . 2 4 8 3 B{3) = .10064 In this example, the predicted Y value at any X location is given by Y = B{0) + B{1) * X + B{2) * X^ + B(3) * X^ A spline coefficient file contains coefficients for cubic interpolation between successive data points (ordered on X) in the data file. Each line, i, in the coefficient file contains five values, X(i),C(i,0),C(i,l),C(i,2),C(i,3) Each X{i) corresponds to an X value in the data file, unless there are duplicate (or near-duplicate) X values. If there are duplicate X values, then only that point with the median Y value for each set of duplicates is retained for the spline coefficient calculation and only this point is included in the coefficient file. Predicted Y values as a function of X are given by Y{X)

= C{i, 0) + C{i, 1) * ( X - X{i))

+ (C(i, 2)/2) * ( X - X ( 0 ) ^ +

+(C(i,3)/6)*(X-X(0)'

forX(i) <X

<X{i-\-l)

(6) Add a model semivariogram. You must specify the desired type of semivariogram model, which can be either linear, spherical, exponential, or Gaussian. The nugget, the sill, and the range must also be specified. 6—Change plot t i t l e —> SCATTERPLOT. This option will display the current title and prompt for a new title, which can be any string, up to 40 characters long. The new title will replace the previous title at the top of the plot. 0—Return without making any (more) changes. Choose this option when you have finished changing the plot parameters. The cross plot will then be regenerated. 0 Return t o main menu of o p t i o n s : Self-explanatory.

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Option 4 - FIELD SIZE S I M U L A T I O N The FIELD SIZE SIMULATION option uses Monte Carlo simulation techniques to simulate a distribution of field sizes by sampling from parametric distributions of area, thickness, porosity, and hydrocarbon saturation that are truncated by user-defined minimum and maximum values. Note: Although the Monte Carlo procedure is general and can be used to create combinations of any distributions, in RISKSTAT it has been preset for four specific variables relating to reservoir volume. All the necessary constants for converting acres, feet, and percent into barrel volumes are incorporated. The Monte Carlo simulation can be performed to produce distributions of recoverable reserves or original resources in place. If original resources are simulated, then the field size in barrels is calculated from the formula: field size = ((1 + / ) / 2 ) * (area) * (thickness) * (porosity/100) * (saturation/100) * (7.758) where / is a dimensionless geometry factor, area is given in acres, thickness in feet, and porosity and saturation in percent. If recoverable reserves are simulated, the field size is multiplied by (recovery factor/100), where the recovery factor is also specified as a percentage value. The value for area used in this simulation is the area of the bottom of the field. The dimensionless geometry factor / is based on the assumption that the field is a truncated cone whose top area is equal to / times the bottom area, and whose cross-sectional area varies linearly between the bottom and the top. RISKSTAT will first ask for the name of the file where the results of the simulation are to be stored. For each of the four variables, 1) area in acres, 2) thickness in feet, 3) porosity in percent, and 4) hydrocarbon saturation in percent, you will be asked for the type of distribution to be assigned to the variable. The available distributions include normal, lognormal, exponential, and uniform. The choice of distributions should not be made arbitrarily, but should be related to the properties of the variables that are being simulated. For example, areas of producing fields usually have a positively skewed distribution that can be modeled by a lognormal distribution. If you select a normal or lognormal distribution, you will be asked for the mean and standard deviation of the distribution. If you choose an exponential distribution, you will be asked for the mean of the distribution. For a uniform distribution and all others you will be asked to specify the upper and lower limits. The next question posed by RISKSTAT will be whether you want to simulate original oil in place or recoverable oil. If you want to simulate recoverable oil, you will be asked for the type of distribution to be used for the recovery factor, which is expressed in percent. The distributions available, and the parameters required for each distribution, are the same as those discussed in the preceding paragraph. If you want to specify a constant recovery factor, choose a normal distribution with a mean equal to the desired recovery factor and a standard deviation of zero. RISKSTAT will then list the maximum number of simulation cycles, and ask how many reahzations you want. For each realization, a single size of field will be

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Computing Risk for Oil Prospects — Appendix B simulated. You will also be asked to supply a seed—an initial number—for the random number generator. After you have provided those parameters, RISKSTAT will run the simulation. A file will be created in standard RISKSTAT format that can be analyzed using the STATISTICAL ANALYSIS or STATISTICAL GRAPHICS PACKAGE options. The variables in the file will include the field size (in thousands of barrels of oil) and its log transform, and the corresponding area, thickness, porosity, and hydrocarbon saturation. If recoverable reserves have been simulated, the recovery factor for each simulated reservoir will also be included in the file. When the simulation is complete, you will be returned to the main menu of options.

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APPENDIX C - RISKMAP MANUAL R I S K M A P : Linked c o m p u t e r programs that perform t h e mapping function in t h e R I S K software

RISKMAP requires well data files that are written in RISKSTAT format. For example, TRAINWEL.DAT contains 83 rows and 7 columns. Each row corresponds to a well and each column to a variable. Well data files may contain up to 1200 wells (rows) and up to 12 variables or properties measured on each well (columns). The variables are explained in the header of the file. The variables must include an X geographic coordinate (usually an east-west or easting) and a Y geographic coordinate (usually a north-south or northing). Coordinates must be expressed in decimal units, not latitudes and longitudes or State Plane Coordinates. They can be UTM or similar map projection coordinates, or may be Cartesian coordinates measured from an arbitrary origin. To avoid problems that may arise from rounding errors, it may be necessary to scale very large coordinate values (such as UTM coordinates) by subtracting constants in order to reduce the number of significant digits. RISKMAP well data files also must contain DST results expressed as codes as defined in the file DST.DEF. The beginning of the file TRAINWEL.DAT is shown below. The first line consists of a one-line title or file name, and can consist of any alphanumeric information. The second line defines the number of rows and columns in the data file. The succeeding seven lines give the names of the variables corresponding to each of the seven columns in the file. The data then appear in the following 83 rows. 1 1 83 7 X-Co ordinate Y-Coordinate

DST Result Structure (m) Thickness (m) Shale Ratio Bedding Index 1741.0 6491.0 1747.0 6488.0 1752.0 6489.0

12.0 -1190.0 35.56 0.522 0.238 12.0 -1244.4 33.48 0.553 0.239 12.0 -1249.1 29.75 0.472 0.239

The variable DST (Drill-Stem Test result) is expressed as a DST code number. DST codes are defined in file DST.DEF and can be changed by the user. The first fine of file DST.DEF contains the number of DST codes that are used. Following fines give the specific code numbers and the meaning of each number. The DST codes are the same as the symbol codes used to control the plotting of symbols at well locations in CONTOUR. An example of a DST.DEF file is shown below. (The program actually reads only the initial numeric values from each line; the remaining comments are reminders.) 4 t o t a l number of DST codes 2 mud 12 s a l t water 14 o i l show 13 o i l producer (must be last DST)

Computing Risk for Oil Prospects — Appendix C RISKMAP creates "grids" that consist of rows and columns of values estimated from well data. The rows and columns correspond to geographic coordinates spaced regularly across the mapped area, and are used to produce contour maps. Grid files can contain several grids that have been created from different variables in a well data file. Individual grids can have a maximum of 36 rows and 36 columns. A grid file can contain a maximum of 15 grids.

Running RISKMAP In this manual, each option of RISKMAP will be explained by guiding the user through the individual modules. Input to programs is demonstrated using data from file TRAINWEL.DAT. You can always escape from anywhere in the RISKMAP program by depressing the CRTL key and typing c which will return you to the RISKMAP main menu. Once the RISK software is loaded according to the installation procedure, RISKMAP can be accessed through the RISK main menu. A menu labeled RISKMAP will appear that offers 7 options: 1 TREND 2 SCORE 3 SVARIO 4 GRID 5 PROB 6 CONTOUR 0 EXIT Enter your

Create trend surface Calculate discriminant scores Calculate semivariogram Create grid from well data Create producing probability grid Draw contour plot of grid choice :

Each option is chosen by entering its associated number. For example, if you want to draw a contour map, you type 6 and press J [the E N T E R or R E T U R N key]. Option 0 - E X I T This will return you to the RISK main menu. Option 1 - T R E N D TREND reads a well data file and calculates a trend surface of degree one up to five for a user-chosen variable. TREND adds the resulting trend residuals at the well locations to the well data file as an additional column. TREND may also add the calculated trend surface to a grid file. First, TREND asks for the well data file name: Enter well d a t a f i l e name (0 t o q u i t ) :

TRAINWEL.DAT

TREND will then display a report listing all variables in the specified well data file: These are the variables in the well data file : 1 X-Coordinate 2 Y-Coordinate 3 DST R e s u l t 4 S t r u c t u r e (m) 5 Thickness (m)

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In a dialog, you will be asked for a number corresponding to the variable for which you want the trend surface. Next, you will be asked for the desired degree of the trend. For example, if you want to calculate a second-degree trend surface of structural elevation, enter: Which variable do you want to calculate the trend residuals for (0 to exit) ? 4 Enter degree of trend surface (max. 5) ? 2 TREND will then calculate the trend surface coefficients, calculate the value of the trend surface at each well location, and subtract the trend value at each well location to produce trend residuals. An additional column will be appended to the well data file that will contain the trend residuals for each well. The warning message Column will be appended will appear. If the well data file already contains a trend residual column, TREND will ask if you wish to replace the previous trend residual column with the new residuals. If you enter n*J, TREND will append an additional column of residuals to the file. There Is already a column Structure (m)(TRes2). Do you want to overwrite i t ? n (no) Column will be appended In addition to the residuals, you can also save the trend surface as a grid file: Do you want to add trend surface to grid f i l e ? y (yes) TREND will then add the trend surface grid to an already existing grid file. Be sure that the grid file already exists because TREND can only add trend surface grids to previously existing grid files. For example, if you want to append the trend surface grid to the grid file TRAINWEL.GRD you enter: Enter grid f i l e name (0 to quit) : TRAINWEL.GRD Grid will be appended Trend surface calculations finished Option 2 - S C O R E SCORE reads a well data file and a discriminant function and calculates discriminant scores for all wells, appending these scores to the well data file as an additional column. Alternatively, SCORE will read a grid file and a discriminant function and calculate discriminant scores for the rows and columns of the grid, writing these to an additional grid in the grid file. The discriminant function must have been created previously by performing a discriminant function analysis using the statistical analysis option of RISKSTAT. Be sure that you have calculated a discriminant function with RISKSTAT and have created the file DISCR.DEF in your directory. Do you want scores for (1) well data or (2) a grid : 1 If you want scores calculated for well locations, SCORE asks you for the well data file name: Enter well data f i l e name (0 to quit) : TRAINWEL.DAT

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Computing Risk for Oil Prospects — Appendix C SCORE lists the variables used for the discriminant function, calculates the discriminant scores, and appends them to the well data file as a new column. Variables used for scores : Thickness (m) Shale Ratio Bedding Index Structure (m)(TRes2) Discriminant scores appended If a column containing discriminant scores already exists in the data file, SCORE will ask if you wish to overwrite this column with new discriminant scores: Discriminant scores already exist on t h i s f i l e . Do you want to create new scores ? y If SCORE discovers that not all variables necessary to calculate discriminant scores are present in the well data file, it will print the following message and exit to the RISKMAP menu. Variable(s) missing for score calculations You must update your well data file by including the missing variables in order to use the SCORE option. If you want discriminant scores calculated for a grid, SCORE will ask you for the grid file name: Enter grid f i l e name : TRAINWEL.GRD Variables used for scores : Thickness (m) Shale Ratio Bedding Index Structure (m)(TRes2) If grids for all the specified variables are present in the grid file, SCORE will calculate the score grid and include it in the grid file: Discriminant scores appended to TRAINWEL.GRD Option 3 - S V A R I O SVARIO reads a well data file and creates an experimental semivariogram for a selected variable and writes the semivariogram to a file in RISKSTAT format. Note that SVARIO assumes that the variogram is omnidirectional. First, SVARIO asks for the name of the well data file: Enter well data f i l e name (0 to quit) : TRAINWEL.DAT SVARIO will then display all variables in the well data file: Columns in the data f i l e :

1 2 3 4 5 6 7 8

X-Coordinate Y-Coordinate DST Result Structure (m) Thickness (m) Shale Ratio Bedding Index Structure (m)(TRes2)

9 Disciminant score If you wish to calculate a semivariogram of the discriminant scores, enter: 408

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Enter column for which variogram will be calculated : 9 SVARIO then asks for the lag or interval of distance over which semivariances will be calculated: Enter h or size of lag : 1 SVARIO next asks for the maximum distance (in terms of the maximum number of lags) over which the semivariance will be calculated: Enter maximum distance for variogram : 15 Semivariogram was written to f i l e SVARIO.DAT SVARIO calculates successive values of the experimental semivariogram and writes them as a column in the file SVARIO.DAT. This file is in RISKSTAT format and can be graphed and printed using RISKSTAT. Also, a model semivariogram can be superimposed on the experimental semivariogram in RISKSTAT and can be used subsequently in kriging operations. The file SVARIO.DAT consists of three columns; the first column specifies the lag distance, the second column contains the experimental semivariance for each lag, and the third column contains the number of wells used to calculate the semivariance for each lag. An example SVARIO.DAT file is given below. The first line of the file contains an alphanumeric title or identification information. The second line specifies the number of rows and columns, the next three lines identify the column variables, and the following 15 lines contain successive values for the lag, semivariance, and number of wells used in the calculations. 1 1 15 3 Distance Semivariance

Frequency 1.0 6.26 2.0 16.43 3.0 23.25

36.0 47.0 43.0

Option 4 - G R I D GRID creates a numerical matrix or grid in which the rows and columns represent geographic coordinates spaced uniformly across an area to be mapped. The values in the grid are interpolated from data contained in a well data file. GRID interpolates using either an inverse distance weighting algorithm or universal kriging. GRID first asks for a well data file name and then produces a report listing the variables in the specified well data file: Enter well data f i l e name (0 to quit) : TRAINWEL.DAT Columns in the data f i l e : 1 X-Coordinate 2 Y-Coordinate 3 DST Result 4 Structure (m) 5 Thickness (m) 6 Shale Ratio 7 Bedding Index Enter 0 to continue

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Computing Risk for Oil Prospects — Appendix C If you wish to create a grid of estimated thickness values, for example, enter the number 5. An asterisk will be placed beside the selected variable in the list of variables. Which coliimn do you want to grid (-1 for a l l ) ? 5 Columns in the data f i l e : 1 2 3 4 • 5 6

X-Coordinate Y-Coordinate DST Result Structure (m) Thickness (m) Shale Ratio

7 Bedding Index Enter 0 to continue Continue to select columns imtil all the variables are marked that you want to grid. If all the variables in the well data file are to be gridded, they can be selected at once by entering - 1 . When you are finished, enter 0 and GRID will ask for the name of the grid file where the grids are to be stored: Which column do you want to grid (-1 for a l l ) ? 0 Enter name of grid data f i l e (0 to quit) : TRAINWEL.GRD If the grid file is a new file, GRID will open a file and request that you specify the dimensions of the grids. Open a new grid f i l e What i s the grid size in x and y : 36 36 The grid size is given as the number of columns (relative size in the X direction) and the number of rows (relative size in the Y direction). The example specifies that the grid will consist of 36 rows and 36 columns. GRID will then request the distance between rows and columns, in the units of measurement used for the well coordinates. In the file TRAINWEL.DAT, well coordinates are given in kilometers firom an arbitrary origin, so to produce a grid having a 1-km spacing between rows and columns, we would enter: What i s the grid spacing in x and y : 1.0 1.0 We can specify an origin for the grid, in terms of the coordinate system used to designate the well locations, so our map will consist of the desired geographic area. We do this by giving the coordinates of the lower left corner of the grid. To produce a grid that will encompass the well locations in the TRAINWEL.DAT data set, we must set the easting origin to 1740 and the northing origin to 6455. The right edge of the grid will then represent an easting coordinate of 1776 and the top of the grid will represent a northing coordinate of 6491. What i s the g r i d ' s origin : 1740 6455 If the grid file already exists, GRID will notify you: File exists already Grids already in the data f i l e :

1 Structure (m) 2 Thickness (m) 3 Shale Ratio 4 Bedding Index Enter 0 to continue

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You can select those grids that you want to retain in the file by entering the number of the grid. An asterisk will appear beside the grid number. Continue to select grids until all that you wish to retain have been marked. If you enter - 1 all the grids will be kept. If you initially enter 0, none of the grids will be kept: Which colimin do you want to keep (-1 for a l l ) ? 0 If a grid has been marked and the corresponding column variable also has been marked in the well data file, you must decide if you want to replace the old grid or to create a new grid from the well data and add it to the grid file while retaining the old grid. The following dialog is appropriate if you want to replace a pre-existing grid: There are columns to be gridded that you also want to keep as grids : Do you want to (1) keep grid Thickness (m) or (2) regrid from well data ? 2 You will now be asked, for each grid specified, whether the grid is to be created by inverse distance estimation or by kriging. Inverse distance estimation creates a single grid of surface estimates while kriging creates two grids, one containing estimates of the surface and a second grid containing the standard errors of the estimates. If you choose inverse distance, enter: Do you want gridding of Thickness (m) by inverse distance (1) or kriging (2) ? 1 GRID reads the inverse distance control parameters from file GRID.DEF, which contains two parameters, the number of wells to be used in making each estimate and the exponent of the inverse distance weighting fimction. Be sure this file is present in your directory. GRID will display the following table confirming the parameters that will be used in gridding: Inverse distance gridding parameters for Thickness (m) Number of columns and rows in grid 36 36 Grid spacing dx and dy 1.0 1.0 Grid origin in x and y directions 1740.0 6455.0 1 Number of Influence wells 6 2 Inverse distance exponent 2 0 Continue The grid specifications are taken from the GRID.DEF file. These can be modified at this point if desired. 1 Number of influence wells: This specifies the maximum number of wells that will be used to calculate the estimate at each grid node location. 2 Inverse distance exponent: The inverse distance exponent determines how weights assigned to each well depend on an inverse power function of the distance between a grid node and the well. If you want to change a parameter, enter the number of the option and the desired value. For example, if you want to change the inverse distance exponent to 1.5, enter: Enter choice and value : 2 1.5 When you are satisfied with your choice of parameters, enter 0 to continue: Enter choice and value : 0 GRID will then update file GRID.DEF, inserting the new parameters, and perform the inverse distance gridding operation:

411

Computing Risk for Oil Prospects — Appendix C Now gridding Thickness (m) If you wish to estimate the grid by kriging, ate: Do you want gridding of Thickness (m) by inverse distance (1) or kriging (2) ? GRID will read ten parameters necessary from the file KRIGE.DEF. Be sure this file is parameters in the presently active KRIGE.DEF the screen:

1 2 3 4 5 6 7 8 9 10 0

the following dialog is appropri-

2 to define the kriging operation present in your directory. The file will be listed in a report on

Kriging Parameters for Thickness (m) Number of columns and rows in grid : 36 36 Grid spacing dx and dy 1.0 1.0 Grid origin in x and y directions 1740.0 6455.0 Variogram model (1 « linear, 2 • spherical, 1 3 • exponential, 4 « gaussian) Degree of drift polynomial 0 Nugget 0.0 Sill 30.0 Range 8.0 Max. number of search points per octant 2 Min. total number of search points 3 Max. search radius in dx and dy units 15.0 Min. number of octants with >• 1 point 1 Angle of octant search (-22.5 - 22.5) 0.0 Continue

Information in the first three lines of the report is taken from the dialog in which the characteristics of the grid are specified. These include the number of rows and columns in the grid, the spacing between rows and columns in map coordinate units (kilometers, in this example), and the geographic origin of the map grid. The remaining information in the report is read from the file KRIGE.DEF. 1 Variogram model: Four alternative variogram models can be used in the kriging algorithm. These are (1) linear, (2) spherical , (3) exponential, and (4) Gaussian models. The appropriate model can be determined by plotting experimental semivariograms from file SVARIO.DAT using RISKSTAT's graphics scatter plot option. 2 Degree of drift polynomial: Kriging may be performed assuming local stationarity (no drift) or may be performed in the presence of local nonstationarity or drift. Choices include 0 no drift, 1 linear drift, or 2 quadratic drift. 3 Nugget: If a semi variogram has a nugget, this implies there is an irreducible error or variance at the sample points. The nugget can be estimated experimentally by plotting semivariograms from file SVARIO.DAT using RISKSTAT and superimposing model semivariograms. If a nugget is specified, the kriged surface may not honor all data points. 4 S i l l : The sill is the highest value of the semi variogram model, beyond which the variance is assumed to be a constant. The sill can be estimated experimentally by plotting semivariograms from file SVARIO.DAT using RISKSTAT and superimposing model semivariograms. 5 Range: The range is the distance at which the semi variogram reaches the sill and defines the neighborhood around a location. Data points separated by distances greater than the range are statistically independent. The range can be 412

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estimated experimentally by plotting semivariograms from file SVARIO.DAT using RISKSTAT and superimposing model semivariograms. 6 Maximum number of search points per octant: The kriging algorithm in GRID uses an octant search to find neighboring wells around each grid node. Every octant is searched over progressively greater distances until the specified maximum number of data points is found or the maximum search radius is exceeded. 7 Minimum t o t a l number of search points: The sum of all data points in all octants must be at least the specified number before an estimate will be made. 8 Maximum search radius in dx and dy units: The search within each octant is limited by this radial distance, expressed in numbers of grid intervals searched. 9 Minimum number of octants with at least 1 data point: A search is successful only if data points are found in at least the specified number of octants. 10 Angle of octant search: The search octants can be rotated so they are oblique to the grid axes. The rotation may be specified firom —22.5 to -f22.5 degrees deviation from the direction of the Y (north-south) axis. If you want to change a parameter, enter the number of the option followed by the desired parameter value. For example, if you want to change the variogram model from linear to exponential, enter: Enter choice and value :

1

3 J

Continue to specify parameters in this manner until all desired changes have been made. When you are satisfied with your choice of parameters, enter 0 to continue: Enter choice and value :

0 J

The parameters entered during this dialog are written to file KRIGE.DEF and replace any previous parameters. The new parameters will remain in eff'ect until they are changed. GRID will now perform the gridding for the specified column variables in the well data file: Now gridding Thickness (m) After finishing the last grid, GRID will write the complete grids to the grid file: Grids are written to TRAINWEL.GRD

Option 5 - P R O B PROB calculates a producing probability grid firom specified grids of estimates and errors. PROB either creates a new producing probability distribution from values in the well data file, or updates a probability distribution that had been created earlier and written to file PROB.DEF. PROB first reads the specified well data file: Enter name of well data f i l e (0 to quit) : TRAINWEL.DAT You can create only a new probabihty distribution or create both a probability distribution and a probability map: Do you want to calculate a probability map (m) or just a probability distribution (d) ? m If you wish to create both a probability distribution and a probability map, PROB must read a grid file:

413

Computing Risk for Oil Prospects — Appendix C Enter name of grid f i l e (0 to quit) :

TRAINWEL.GRD

PROB then displays the variables (columns) in the specified well data file: Columns in the data f i l e : 1 X-Coordinate 2 Y-Coordinate 3 4

DST Result Structure (m)

5 Tliickness (m) 6 Shale Ratio 7 8

Bedding Index Structure (m)(TRes2)

9 Discriminant score You may select any variable in the well data file as the basis for calculating a probability map. However, the grid file must contain both a grid of estimates and an error grid for the specified variable (that is, the variable must have been gridded using the KRIGE option or an equivalent process). Unless both a grid of estimates and an error grid are present, a probability map cannot be created. If a grid of estimates for the specified variable is not present in the grid file, PROB prints the following error message: Grid f i l e does not contain estimate grid of variable If an error grid for the specified variable is not present in the grid file, PROB prints the following error message: Grid f i l e does not contain error grid of variable If PROB cannot find either an estimate grid or an error grid, the procedure will terminate. You must create these grids (for example, by option KRIGE) before option P R O B can be used. If grids of estimates and errors are present, you must specify the variable (column) in the well data file to be used in calculating the probability map. For example, if you wish to base the probabilities on discriminant scores, enter 9: Which column variable do you want to use for probability distribution ? 9 PROB will then present a statistical summary of the selected variable: Kriging Results of Discriminant Score Number of data points Min. value in data Max. value in data Number of data points in grid Number of invalid data points Min. estimated value in grid Max. estimated value in grid Min. error of estimated value Max. error of estimated value

83 7.11 32.94 1296 3 7.11 32.94 0.00 4.21

This table provides a quick check of the specified variable by comparing calculated and estimated values. Note that invalid estimates are produced by the kriging algorithm wherever the specified search rules preclude calculation of a kriging estimate. PROB will then present four options: 414

RISKMAP Manual 1 2 3 0

New probability distribution from file Update probability distribution from file Choose different variable Quit

Option 1: Reads the discriminant function from file DISCR.DEF and calculates a producing probability distribution from the original data. This distribution will be written automatically to file PROB.DEF which stores information about the current probabihty distribution. Note that calculating a producing probabihty distribution is only possible if there are producing wells in the well data file. Option 2: Reads the discriminant function from file DISCR.DEF and reads a previously calculated probability distribution from file PROB.DEF. The previously calculated producing probability distribution will be updated, using information from the current well data file. Note that under this option, the well data file may not necessarily contain producing wells. However, a previously calculated probability distribution must be available in file PROB.DEF. The previous probability distribution may have been calculated using a well data file for another (analogous or training) area, or may have been calculated from an earlier, smaller well data file for the same area. Option 3: Lists the well data so another variable may be selected. If you wish to create a new producing probability distribution of the discriminant scores, for example, enter: Enter your choice : 9 PROB will ask for the number of probabihty classes in the distribution: How many classes for probability distribution ? 5 PROB will then calculate and list the new producing probabihty distribution with 5 probability classes: Variable used for probability distribution : Discriminant score Number of wells in data f i l e 83 Number of producers in data f i l e 18 Min. value in data f i l e 7.1 Max. value in data f i l e 32.9 New Producing Probability Distribution Producers Non-Producers Unconditional Probability 0.217 0.783 Interval Conditional Probability 7.1 15.7 1.000 0.000 11.4 20.0 0.727 0.273 15.7 24.3 0.156 0.844 20.0 28.6 0.020 0.980 24.3 32.9 0.000 1.000 The first part of this display gives characteristics of the selected variable in the well data file. The second part lists the producing probability distribution. The success ratio or unconditional probability is given, followed by columns containing the class limits and the producing probabilities for each class conditional upon values within these classes . Note that the classes are overlapping in order to calculate smoother probability curves with limited data. PROB then offers three options: 1 Calculate probability grid 2 Recalculate probability distribution 0 Quit

415

Computing Risk for Oil Prospects — Appendix C Under the first option, a producing probability grid will be calculated from the probability distribution and from grid estimates that are weighted by the estimation errors taken from the error grid. The resulting probability grid wiU be appended to the grid file. Under the second option, the probability distribution will simply be recalculated, using a different number of probability classes. The third option returns you to the RISK MAP menu. If you wish to create a probability grid, enter: Enter your choice : 1 First, the probability distribution is written to a file: Writing probability distribution to PROB.DAT The file PROB.DAT contains five columns, which are the midpoints of the classes of the variable, the frequency of producing wells in each class, the frequency of dry holes in each class, and the producing probabilities and dry hole probabilities of each class. File PROB.DAT is written in RISKSTAT format and a probability distribution can be plotted using RISKSTAT. An example of a PROB.DAT file is given below: 1 1 5 5 Discriminant Score Frequency producers Frequency dry wells Producing probability Dry hole probability 11.41 13.0 0.0 1.000 15.72 16.0 6.0 7.27E-01 20.02 5.0 27.0 1.56E-01

0.000 2.73E-01 8.44E-01

Next the probability grid is written to a file. The probability grid can be expressed either as the probability of discovering a producing well or as the complementary probability of drilling a dry hole (dry hole probability = 1 — producing probability). If you wish to save the producing probability grid, enter: Do you want producing probability (1) or dry hole probability (0) ? 1 The grid will automatically be added to the grid file. Probability grid appended to TRAINWEL.GRD If a probability grid is already present in the grid file, PROB will pla^e a warning on the screen. You may choose to replace the previously calculated probability grid, or you may add the new grid to the grid file. If you wish to overwrite the earlier probability grid, enter: Grid f i l e already contains probability grid. Do you want to overwrite i t (1) or append (2) or quit (0) ? 1

Option 6 - CONTOUR This option draws a contour map using a specified well data file and a corresponding grid file. Maps can be saved as PostScript files which can be printed on PostScript printers using the appropriate utility programs. The CONTOUR option of RISKMAP is a DOS-compatible subset of the SURFACE III graphics 416

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package. Because there are many parameters that must be specified to tailor a map t o your exact needs, the list of commands and options is lengthy. T h e s e are listed in a dialog that appears when C O N T O U R (Option 6) is selected: CONTOUR MAIN MENU AD — ABOUT X-Y-Z DATA SET > AG — ABOUT GRID MATRIX > XY — LOAD X-Y-Z DATA > GM — LOAD GRID MATRIX > SM — SMOOTH GRID MATRIX > PO — POST X-Y-Z DATA POINTS > CO — CONTOUR MAP OF GRID MATRIX > SH — SHADED CONTOUR MAP OF GRID MATRIX > TR — TRANSECT PLOT OF GRID MATRIX > PS — CONVERT PLOT TO POSTSCRIPT FORMAT > ST — STOP - RETURN TO RISKMAP MAIN MENU PLEASE ENTER YOUR SELECTION : Select the desired option and enter the two-char ax: ter code, followed by J . A submenu will appear for all options except S T , which will return y o u t o the RISKMAP main menu. Some options, such as A D and AG, require that files be loaded before the options can be exercised. For example, if you select A D and have not previously loaded a well d a t a set, the warning message •***•• X-Y-Z DATA DOES NOT EXIST will appear. Simply type J and the C O N T O U R main menu will be restored and

a data set can be loaded. Each of the CONTOUR options and their submenus are described below in the order that the options appear on CONTOUR MAIN MENU. AD — ABOUT X-Y-Z DATA SET: A report will appear on the screen listing the number of data points in the currently active well data file, the X^ Y, and Z minimum and maximum values in the data, and whether the data have ID numbers and symbol codes. Press J to return to the CONTOUR main menu. AG — ABOUT GRID MATRIX: A report will appear on the screen listing the number of rows and columns in the currently active grid file, the real-world distances between rows and columns in the file, the X coordinate of the left and right edge of the grid, the Y coordinate of the bottom and top of the grid, and the maximum and minimum lvalues in the grid. The number of missing values will be reported. Press •J to restore the CONTOUR main menu. XY — LOAD X-Y-Z DATA SET: A menu will appear hsting the columns and the variable names in the currently active well data set. The listing will be followed by the LOAD DATASET submenu, which contains eight options. FN X Y Z MS SD RE GO

-

FILENAME OF RISKMAP X-Y-Z DATASET VARIABLE TO USE AS X COORDINATE VARIABLE TO USE AS Y COORDINATE VARIABLE TO USE AS Z VARIABLE TO USE AS MAP SYMBOL CODE SET ALL PARAMETERS TO DEFAULT VALUES RETURN TO MAIN MENU LOAD RISKMAP X-Y-Z DATA

REQUIRED REQUIRED REQUIRED REQUIRED 0

An option is selected by typing the option code followed by a space and the desired value of the parameter. Columns in the well data set are specified by entering their column number. Specified parameters appear on the right as they are entered. After the instruction, type: 417

Computing Risk for Oil Prospects — Appendix C PLEASE ENTER YOUR SELECTION :

fn

TRAINWEL.DAT J

T h e file n a m e T R A I N W E L . D A T will now replace t h e word REQUIRED t o t h e right of t h e F N c o m m a n d a n d a listing of colimms in t h e T R A I N W E L . D A T file will a p p e a r above t h e menu. Choose t h e next c o m m a n d in t h e m e n u by entering PLEASE ENTER YOUR SELECTION :

X

1 J

which will specify that the variable in column 1 of file TRAINWEL.DAT will be used as the X coordinate of the map. This will now appear to the right of the corresponding command in the menu. Next, enter Y 2 J to select the second column as the Y coordinate of the map, enter Z 5 J to specify the fifth column (thickness) as the mapped variable, then enter M S 3 J to specify that column 3 (DST result) is to be used as the map symbol code. SD - resets the LOAD DATA SET menu to its initial state. RE - returns to CONTOUR MAIN MENU without altering or initiating the LOAD DATA SET selections. GO - implements the selections you have made and loads the selected data into memory. The ABOUT X-Y-Z DATA report will appear. Press J to return to CONTOUR MAIN MENU. GM — LOAD GRID MATRIX: T h i s option functions in a m a n n e r similar to option X Y , except t h a t only t h e first two p a r a m e t e r s are required. T h e following L O A D GRID menu appears: FN GN SD RE GO

-

FILENAME OF RISKMAP GRIDS GRID NUMBER TO LOAD SET ALL PARAMETERS TO DEFAULT VALUES RETURN TO MAIN MENU LOAD RISKMAP GRID

REQUIRED REQUIRED

After entering the name of the desired grid file and typing J , a directory of the grids in the specified file will appear. Select the grid to be contoured by entering its number. Option SD resets the menu to its initial, empty state. Option RE returns to the CONTOUR main menu, leaving the LOAD GRID menu imchanged. Option GO implements the selections you have made and loads the specified grid into memory. The ABOUT GRID MATRIX report will appear. Press J to return to CONTOUR MAIN MENU. SM — SMOOTH GRID MATRIX: This option averages adjacent values in the grid matrix to produce a smoother representation of the surface being mapped. The degree of smoothing can be adjusted by changing one or more of five different parameters. The following SMOOTH GRID menu appears: NS - NUMBER OF TIMES TO REPEAT SMOOTHING FN - TYPE OF SMOOTHING (0) - GRID VALUES ARE NOT WEIGHTED (1) - GRID VALUES ARE WEIGHTED BY 1/D (2) - GRID VALUES ARE WEIGHTED BY 1/D**2 WF - WEIGHT FACTOR OF CENTER POINT NC - NUMBER OF COLUMNS IN SMOOTHING SPAN NR - NUMBER OF ROWS IN SMOOTHING SPAN SD - SET ALL PARAMETERS TO DEFAULT VALUES RE - RETURN TO MAIN MENU GO - SMOOTH GRID MATRIX

1 0

1.0000 2 2

NS - is an integer specifying the number of times the smoothing operation will be repeated. A larger number results in a smoother surface.

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RISKMAP Manual FN - specifies the type of smoothing operation. If 0 is specified, grid values are averaged. If 1, grid values are weighted by the inverse of the distance from the grid node to the center of the weighting span. If 2, grid values are weighted by the square of the inverse of the distance. A value of 0 will cause the greatest smoothing effect while a value of 2 will cause the smallest amount of smoothing. WF - specifies a multiplier for the central point in the weighting span. Specifying 5, for example, means the central value will have five times the influence on the final average. The larger the value, the less the smoothing effect. NC and NR - specify the number of columns to the right and left of the center point and the number of rows above and below the center point of the weighting span. Larger values will incorporate more values into the average and yield a smoother surface. SD and RE - have the same function as in other commands. After changing one or more of the smoothing parameters, enter GO*J to smooth the grid matrix. The report, ABOUT GRID MATRIX, will appear showing the range of the new, smoothed l v a l u e s . PO — POST X-Y-Z DATA POINTS: This creates a posting of the wells in the currently active well data file. A well data file must be loaded using the command XY on the contour main menu prior to selecting this option. The following submenu will appear: PO - POINTS AND LABEL OPTION 0 « SYMBOL I S DRAWN WITH NO LABEL 1 « Z VALUE I S WRITTEN AS LABEL 2 « SAMPLE IDENTIFICATION IS WRITTEN AS LABEL SC - SOURCE OF SYMBOL CODE PS - SET SYMBOL USED ON POSTING PL - SET DATA POINT LABEL OV - OVERLAY THIS MAP ON PREVIOUS MAP MS - SET MAP SIZE

BX BO BL NE SD RE GO

-

SET BOX EXTREMES DRAW INDEX MARKS AROUND BOX LABEL INDEX MARKS DRAW NEAT LINE AROUND MAP SET ALL PARAMETERS TO DEFAULT VALUES RETURN TO MAIN MENU DRAW POSTING

0

1 > > NO >

> > > > >

Using the default values will produce a 6 X 6-inch map on the screen, in which well locations are shown as crosses. There will be no labels on points or tick marks. PO - specifies the type of labels, if any, to be placed to the right of each posted point on the map. SC - if set to 0, will read symbol codes from the column specified by the XY command to contain well symbols. If SC is set to an integer greater than zero, the integer specifies the specific symbol to be used. There are 14 different symbols available in RISKMAP. The symbol code numbers and the symbols that will be plotted are:

419

Computing Risk for Oil Prospects — Appendix C

1+

sQ

110 14 +

2X

6%

12<> I s O

30

7j^ '^^ ''^

PS - brings up a submenu that allows the characteristics of the selected symbols to be changed. HS PS SC SS CI C2

-

HEIGHT OF SYMBOL PEN NUMBER FOR DRAWING SYMBOLS CODE FOR SYMBOL TO BE POSTED SYMBOL SELECTION CODE CODE 1 CODE 2

C8 SD RE GO

-

CODE 8 SET ALL PARAMETERS TO DEFAULT VALUES RETURN TO PREVIOUS MENU SET POSTING SYMBOL PARAMETERS

.100 1 1 0 0 0

0

Options HS and PS specify the size and color of the symbol. SC specifies the symbol to be used, chosen from the illustration above. SS specifies how a symbol is to be selected. If 0, symbol codes in the well data file can be changed to the specified symbol by entering the code values as parameters CI, C2, C3, etc. If SS is set to 1, the symbol specified will be assigned to wells whose code values fall within specified intervals. The intervals are defined by setting CI to an interval's lower limit and C2 to its upper limit, etc. Options SD, RE, and GO have the same functions as in other commands. PL - SET DATA POINT LABEL allows the size and placement of labels to be specified. The DATA POINT LABELS menu is self-explanatory. For most applications, the default options are appropriate. OV - OVERLAY THIS MAP ON PREVIOUS MAP allows a posting to be superimposed on top of a previously drawn contour map or shaded contour map. The responses are y (yes) or n (no). MS - SET MAP SIZE specifies the physical dimensions of the finished map. Typing M S J will bring up the following SET MAP SIZE submenu: MP - METHOD OF SPECIFING MAP DIMENSIONS ( 0 ) SCALE FACTOR ( 1 ) MAP SIZE IN INCHES XS - X DIMENSION OF MAP IN INCHES YS - Y DIMENSION OF MAP IN INCHES SD - SET ALL PARAMETERS TO DEFAULT VALUES RE - RETURN TO PREVIOUS MENU GO - SET MAP SIZE

1

12.0000 12.0000

In most instances, the size of a map will be specified directly in inches (option MP set equal to 1), which are entered by options XS and YS. If MP is set to 0, the XS and YS options will change to

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XS - SCALE FACTOR FOR X DIMENSION YS - SCALE FACTOR FOR Y DIMENSION Options S D , R E , and G O have the same functions as in other commands. BX - SET BOX EXTREMES permits specification of the coverage of the m a p {i.e., the extent of the m a p p e d region) in terms of t h e geographic coordinates. T h e S E T B O X E X T R E M E S menu is preceeded by a hsting of the minimum and m a x i m u m X and Y coordinates of wells in the currently active well d a t a file. T h e following submenu appears: XL XR YB YT SD RE GO

-

X VALUE AT LEFT EDGE OF MAP X VALUE AT RIGHT EDGE OF MAP Y VALUE AT BOTTOM EDGE OF MAP Y VALUE AT TOP EDGE OF MAP SET ALL PARAMETERS TO DEFAULT VALUES RETURN TO PREVIOUS MENU SET BOX EXTREMES

T h e default values for the first four parameters are set equal t o the X and Y extremes of the data set. Changing these values allows the map to extend beyond the data, or t o b e confined to an area smaller than the data. Options S D , R E , and G O have t h e same functions as in other commands. BO - DRAW INDEX MARKS AROUND BOX controls the characteristics of the geographic coordinate marks placed around the edge of the m a p . T h e options specify how far apart tick marks should appear, given in units of the geographic coordinates (every 5 km, 1 mile, or 1000 feet, for example), and the origins for the scales in b o t h the X and Y directions. Other parameters set the physical size of the ticks and usually can be left at the default choices, except for RG. Entering B O J will cause the following submenu to appear: XT YT LT PT RX RY RG PG SD RE GO -

UNITS BETWEEN TICK MARKS IN X DIRECTION UNITS BETWEEN TICK MARKS IN Y DIRECTION LENGTH OF TICK MARKS PEN NUMBER FOR DRAWING TICK MARKS REFERENCE VALUE FOR TICK MARKS ON X-AXIS REFERENCE VALUE FOR TICK MARKS ON Y-AXIS REFERENCE GRID OPTION PEN NUMBER FOR DRAWING REFERENCE GRID SET ALL PARAMETERS TO DEFAULT VALUES RETURN TO PREVIOUS MENU SET BOX PARAMETERS

1.0000 1.0000 .0500 1 .0000 .0000 0 1

Option RG has five choices: If 0, no tick marks and no coordinate grid will be drawn; if 1, all tick marks and coordinate grids will be drawn at every specified interval; if 2 , all ticks will be drawn and coordinate grids will b e drawn for every other specified interval; if 3 , all ticks and all coordinate grid intersections will be drawn; if 4, all ticks and every other specified coordinate grid intersection will be drawn. Options S D , R E , and G O have the same functions as in other commands. BL - LABEL INDEX MARKS provides control over the spax^ing and characteristics of the geographic labels around the margin of the map. T h e options on the S E T B O X LABELS menu are self-explanatory. In most circumstances, the default values are appropriate. T h e submenu is

421

Computing Risk for Oil Prospects — Appendix C ED - EDGE LABELING OPTION (1) BOTTOM AND LEFT EDGE ONLY (2) ALL EDGES (3) BOTTOM AND LEFT EDGES, X LABELS ROTATED (4) ALL EDGES, X LABELS ROTATED FX - FREQUENCY OF LABELING TICK MARKS ALONG X-AXIS FY - FREQUENCY OF LABELING TICK MARKS ALONG Y-AXIS HL - HEIGHT OF NUMBERS IN THE LABELS PL - PEN NUMBER FOR DRAWING LABELS FL - FONT NUMBER OF LABELS ND - NUMBER OF DECIMAL PLACES IN LABELS SD - SET ALL PARAMETERS TO DEFAULT VALUES RE - RETURN TO PREVIOUS MENU GO - SET BOX LABEL PARAMETERS

1

5 5 .1000 1 0 0

NE - DRAW NEAT LINE AROUND MAP specifies the physical characteristics of an optional neat line around the outside of a map. T h e options are self-explanatory and in most instances the default values will be satisfactory if a neat line is specified. T h e S E T N E A T LINE submenu has the following appearance: LN RN BN TN PN SD RE GO

-

WIDTH BETWEEN LEFT BORDER AND LEFT NEAT LINE WIDTH BETWEEN RIGHT BORDER AND RIGHT NEAT LINE WIDTH BETWEEN BOTTOM BORDER AND BOTTOM NEAT LINE WIDTH BETWEEN TOP BORDER AND TOP NEAT LINE PEN NUMBER FOR DRAWING NEAT LINE SET ALL PARAMETERS TO DEFAULT VALUES RETURN TO PREVIOUS MENU SET BOX LABEL PARAMETERS

.5 .5 .5 .5 1

T h e three final options on the P O S T DATA P O I N T S menu SD - SET ALL PARAMETERS TO DEFAULT VALUES RE - RETURN TO MAIN MENU GO - DRAW POSTING have the same functions as in previously described menus. After entering G O J , the screen will clear and a map showing the locations and status of wells in the currently active well data file will be drawn, having the characteristics specified in the options and suboptions of the P O S T DATA P O I N T S menu. CO - CONTOUR MAP OF GRID DATA produces a contour m a p of the currently active grid. Typing C O J will bring up the following D R A W C O N T O U R M A P menu: SL - CONTOUR LINE SMOOTHING 2 SC - SUPPRESS CLOSELY SPACED CONTOUR LINES .0500 DE - ANNOTATE CLOSED DEPRESSIONS 0 TK - SIZE OF TICK MARKS ON HATCHURED LINES . 1000 OV - OVERLAY THIS MAP ON PREVIOUS MAP NO CI - SET CONTOUR INTERVAL > BD - SET BOLD CONTOUR PARAMETERS > MS - SET MAP SIZE > BX - SET BOX EXTREMES > BO - DRAW INDEX MARKS AROUND BOX > BL - LABEL INDEX MARKS > NE - DRAW NEAT LINE AROUND MAP > SD - SET ALL PARAMETERS TO DEFAULT VALUES RE - RETURN TO MAIN MENU GO - DRAW CONTOUR MAP Many of the options on this menu will bring u p submenus that are identical t o those already described under PO - POST X-Y-Z DATA POINTS. See the descriptions under the P O command for instructions on setting these parameters. If parameters have been set on a submenu under a previous command, they remain

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in force and do not have to be re-entered under another command unless y o u wish to change their values. SL - CONTOUR LINE SMOOTHING. Contour lines may have a jagged appearance if the underlying grid is coarse. T h e appearance can be improved by interpolating to a finer interval along the contour lines. If option SL is set t o 0, the contours are not smoothed. If set to a non-zero integer, each grid cell is subdivided into smaller cells. SL specifies the number of intermediate points between each row or column in the grid. SC - SUPPRESS CLOSELY SPACED CONTOUR LINES. In areas of steep slope, contour lines may be t o o closely spaced. Contours closer together than the specified value (in inches) will be blanked out. DE - ANNOTATE CLOSED DEPRESSIONS. If 0, no hatchures will be drawn. If 1, all contours in closed depressions will be hatchured. If 2 , only bold contour lines in closed depressions will be hatchured. TK - SIZE OF TICK MARKS ON HATCHURED LINES. T h e length of hatch marks on contour lines in closed depressions can be specified in inches. OV - OVERLAY THIS MAP ON PREVIOUS MAP. See under P O command, above. CI - SET CONTOUR INTERVAL. CI specifies the contours that will appear on the map. Typing C l J will bring up the following S E T C O N T O U R INTERVAL menu: BL - BASE CONTOUR LEVEL CI - CONTOUR INTERVAL MI - MAXIMUM NUMBER OF CONTOUR INTERVALS IF 0, INTERVALS WILL BE GENERATED TO COVER RANGE FL - FREQUENCY OF LABELING OF CONTOUR LINES HL - HEIGHT OF NUMBERS IN LABELS PL - PEN NUMBER FOR DRAWING LABELS DL - NUMBER OF CHARACTERS TO THE RIGHT OF DECIMAL PLACE MD - MINIMUM DISTANCE BETWEEN LABELS SD - SET ALL PARAMETERS TO DEFAULT VALUES RE - RETURN TO PREVIOUS MENU GO - SET CONTOUR INTERVAL PARAMETERS

.0000 10.0000 0 5 .1000 1 0 .4000

Most of the options are self-explanatory and the default values are appropriate in most instances. To help select appropriate values, the minimum and m a x i m u m Z values in the currently active grid matrix are displayed above the menu. T h e first three options are most critical. Option BL specifies the value from which contour intervals will be calculated. N o contour lines will be drawn which are lower than this value. Option CI specifies the difference between successive contour lines. Option MI specifies the m a x i m u m number of intervals. If this is set t o 0, enough contour lines will be generated to completely cover the range of Z values in the grid matrix. Otherwise, it specifies the number of contours that will be drawn above the base level. B y adjusting these three parameters, it is possible to contour only a selected range of Z values (such as probabilities greater than 0.5, or only positive trend surface residuals, for example). Options S D , R E , and G O have the same functions as in other commands. BD - SET BOLD CONTOUR PARAMETERS. B D specifies which contours will appear as bold lines on the map. Typing B D J will bring up the following S E T B O L D C O N T O U R menu:

423

Computing Risk for Oil Prospects — Appendix C HL - PLOT EVERY NTH CONTOUR AS A HEAVY LINE ME - CONTROLS THE METHOD OF DRAWING BOLD LINES 0 - DRAW BY RETRACING IN REVERSE DIRECTION 1 - DRAW BY RETURNING TO ORIGIN, THEN RETRACE

4 0

2 - DRAW BY USING DIFFERENT COLOR PEN PEN NUMBER FOR DRAWING BOLD CONTOUR LINES 1 SEPARATION BETWEEN THE LINES USED TO PRODUCE BOLD LINES .01200 SET ALL PARAMETERS TO DEFAULT VALUES RETURN TO PREVIOUS MENU SET BOLD CONTOUR ANNOTATION PARAMETERS Option HL is used to specify the frequency of bold contour lines. Most of the other options are not relevent for screen displays and the default values are appropriate. Options SD, RE, and GO have the same functions as in other commands. The following options imder menu CO-DRAW CONTOUR MAP are explained under the P O command, above.

PB SE SD RE GO -

MS - SET MAP SIZE BX - SET BOX EXTREMES BO - DRAW INDEX MARKS AROUND BOX

BL - LABEL INDEX MARKS NE - DRAW NEAT LINE AROUND MAP

The three final options on the DRAW CONTOUR MAP menu have the same functions as in previously described menus. After entering GO*J, the screen will clear and a contour map of the specified grid in the currently active grid file will be drawn, having the characteristics specified in the options and suboptions of the DRAW CONTOUR MAP menu. SH — SHADED CONTOUR MAP OF GRID MATRIX: This option will produce a hypsometric map of the currently active grid in which the contour intervals are represented by bands of color. Typing S H J will bring up the following DRAW SHADED CONTOUR menu. Many of the options on this menu will bring up submenus that are identical to those already described under PO - POST X-Y-Z DATA POINTS. See the descriptions under the PO command for instructions on setting these parameters. If parameters have been set on a submenu under a previous command, they remain in force and do not have to be re-entered under another command unless you wish to change their values. SL LB HL WL HN PL DP OV SI MS

-

BX BO BL NE SD RE GO -

CONTOUR LINE SMOOTHING LOCATION OF LEGEND BAR HEIGHT OF LEGEND BAR WIDTH OF LEGEND BAR HEIGHT OF NUMBERS ON LEGEND PEN NUMBER FOR DRAWING LEGEND NUMBER OF CHARACTERS TO RIGHT OF DECIMAL OVERLAY THIS MAP ON PREVIOUS MAP SET SHADE INTERVAL SET MAP SIZE

SET BOX EXTREMES DRAW INDEX MARKS AROUND BOX LABEL INDEX MARKS DRAW NEAT LINE AROUND MAP SET ALL PARAMETERS TO DEFAULT VALUES RETURN TO MAIN MENU DRAW SHADE MAP

3 0

.2500 1.000 .1000 1 0 NO > >

> > > >

Many of the options control the size and placement of the column key to the colors that appear on the map. Other options affect the size and characteristics of the map itself and are identical in operation to options with similar names under the PO - POST... and CO - CONTOUR. .. menus.

424

RISKMAP

Manual

SL - CONTOUR LINE SMOOTHING. See under CO command, above. LB - LOCATION OF LEGEND BAR. If 0, no legend bar will be produced. Values greater than zero will place a legend bar beside the map inside the neat line, if any: 1 = lower left, 2 = center left, 3 = lower right, 4 = center right. HL - HEIGHT OF LEGEND BAR. Specifies the height, in inches, of the color bars in the legend. WL - WIDTH OF LEGEND BAR. Specifies the width, in inches, of the legend color bar. HN - HEIGHT OF NUMBERS ON LEGEND. This functions identically to option HL under the P O command, above, but effects only the size of labels on the legend. See the following options under the P O command, above. PL - PEN NUMBER FOR DRAWING LEGEND DP - NUMBER OF CHARACTERS TO RIGHT OF DECIMAL OV - OVERLAY THIS MAP ON PREVIOUS MAP SI - SET SHADE INTERVAL. This option brings up the following submenu, which controls the assignment of colors to specific intervals of Zin the grid matrix. Many of the commands are identical to those in the CI submenu of the CONTOUR option. BL - BASE CONTOUR LEVEL .0000 CI - CONTOUR INTERVAL 10.00 SR - MAXIMUM NUMBER OF CONTOUR INTERVALS 0 IF 0, CONTOURS WILL BE GENERATED TO COVER RANGE SI - SHADE COLOR 1 1 S9 SD RE GO -

SHADE COLOR 9 SET ALL PARAMETERS TO DEFAULT VALUES RETURN TO PREVIOUS MENU SET SHADE CONTOUR INTERVAL PARAMETERS

9

Options BL, CI, and SR are described under the submenu CI - SET CONTOUR INTERVALS, above. Option SR is functionally equivalent to Option MI of that submenu. The shade color options SI through S9 are color codes for the shaded intervals, from lowest to highest. Options SD, RE, and GO have the same functions as in previously described menus. The following options of the SH - DRAW SHADED CONTOUR menu are explained under the P O command, above. MS - SET MAP SIZE BL - LABEL INDEX MARKS BX - SET BOX EXTREMES NE - DRAW NEAT LINE AROUND MAP BO - DRAW INDEX MARKS AROUND BOX The three final options on the DRAW SHADED CONTOUR menu have the same functions as in previously described menus. After entering G O J , the screen will clear and a contour map of the specified grid in the currently active grid file will be drawn, representing the contours as colored bands having the characteristics specified in the options and suboptions of the DRAW SHADED CONTOUR menu. A columnar key to the colors on the map will be drawn at the specified location adjacent to the map. TR — TTIANSECT PLOT OF GRID MATRIX: A transect plot is a perspective block diagram in which the shape of a surface is represented by a series of profiles. Producing a transect plot requires specification of parameters that control the orientation and viewing angle of the block, as well as the dimensions of the map. Most of these options are self-explanatory. Enter T R « J to bring up the TRANSECT P L O T menu:

425

Computing Risk for Oil Prospects — Appendix C VS FL VI LI SD RE GO

-

VERTICAL SCALE - 7. OF X-Y RANGE DIRECTION OF LINES ON SURFACE SET VIEW POINT SET NUMBER OF FORM LINES TO DRAW SET ALL PARAMETERS TO DEFAULT VALUES RETURN TO MAIN MENU DRAW TRANSECT PLOT

50.000 0 > >

The apparent vertical exaggeration of the surface is controlled by option VS. Values between 40% and 60% usually produce acceptable results. If option FL is 0, form lines will be drawn parallel to both the X and Y axes. If 1, lines are drawn parallel to the X axis and if 2, parallel to the Y axis. Option VI brings up the following SET VIEWING LOCATION submenu: AZ EL DI SD RE GO

-

VIEWING ANGLE LOOKING FROM SOUTH ELEVATION ABOVE HORIZON DISTANCE TO VIEW POINT SET ALL PARAMETERS TO DEFAULT VALUES RETURN TO PREVIOUS MENU SET VIEWING PARAMETERS

45.000 30.000 10000.00

The orientation of the block is specified as though seen from a satellite in space. The bottom row of the grid matrix represents the "south" side of the block, and the viewing angle is measured in degrees counterclockwise from the south. The height of the viewing position is given in degrees above the horizon; a 90° angle is looking directly downward and yields a map view. The distance of the viewing position is given in numbers of grid cells; the smaller the distance, the greater the perspective effect. The three final options on the SET VIEWING LOCATION submenu have the same functions as in previously described menus. LI - controls the number of profile lines drawn on the block to represent the form of the surface. Parameters for LI are set in a submenu whose options are selfexplanatory. To help choose the appropriate number of lines, the number of rows and colimins in the currently selected grid matrix are displayed. SD - has same functions as in previously described options. RE - has same functions as in previously described options. GO - has same functions as in previously described options. PS — CONVERT PLOT TO POSTSCRIPT FORMAT: This option converts the map displayed on the screen into a PostScript file. This file can then be downloaded to a PostScript-compatible printer using an appropriate download utility. Options in the CONVERT PLOT menu are self-explanatory.

426

APPENDIX D - RISKTAB MANUAL R I S K T A B : Linked c o m p u t e r programs that perform financial calculations in t h e RISK software

Running R I S K T A B Once the RISK software is loaded according to the installation procedure, RISKTAB can be accessed from the RISK main menu. A RISKTAB menu will appear listing five options: RISKTAB 1 2 3 4 0

CUM CASHFLOW

RAT DECISION EXIT

Ciimulative Production Function Discounted Net Cash Flow Risk Analysis Table Decision Table

Each option can be activated by entering the associated number. For example, if you wish to calculate the discounted net cash flow for a well, enter 2 and press J [the E N T E R or R E T U R N key]. Physical and financial units must be defined in a file called UNIT.DEF. This file can be edited to adjust units to the needs of an individual analysis. These units will then be used in all RISKTAB programs. An example of a UNIT.DEF file appropriate for use in the United States is: BBL O i l volume u n i t 1000 Oil volume unit multiplier (for output only) MCF Gas volume unit 1000 Gas volume unit multiplier (for output only) ACRE Area unit 1 Area unit multiplier (for output only) $ Monetary unit 1000 Monetary unit multiplier (for output only) The imits must start at the leftmost character position and the maximum length of a unit description is four (4) characters. In this manual, each option of RISKTAB is explained by guiding the user through the operation. Input to programs is explained using examples from the diskettes. Option 0 - E X I T RISKTAB will return to the main RISK menu if you enter 0 J . Option 1 — C U M

Cumulative P r o d u c t i o n Function

CUM creates a production stream from a parametric decline curve and writes the record to a file. Note: CUM input files should have names without an extension. CUM data files are text files and can be edited directly by an experienced user. However, it is simpler to change parameters in the files interactively using option CUM.

Computing Risk for Oil Prospects — Appendix D Enter data f i l e name :

CUMINP

If the data file cannot be found, CUM will ask if a new analysis is to be performed, t o which you respond y (yes) or n (no). Cannot find f i l e CUMINP I s t h i s a new a n a l y s i s (y/n) ? y A n input table appears. In general, input to RISKTAB is organized in tables. Table entries can always be changed interactively by entering the row number of the entry and the new value. Decline functions can be entered as cumulative production or as the parameters of two different production rates. To use the cumulative production option, enter Enter your choice followed by input : 1 1 WELL PRODUCTION FORECAST

1 2 3 4 5 6 7 Five -

Decline based on cumulative production (1) or based on a second production rate (2) : Exponential (1) or hyperbolic (2) decline Production of f i r s t operation year Average o i l production (BBL/day) Average gas production (MCF/day) Ultimate cumulative production Ultimate cums for o i l (BBL) Ultimate cums for gas (BBL) Life span of well (years)

1 1 55. 0. 200000. 0. 30.

parameters must be entered: initial oil production in oil volume units per day initial gas production in gas volume units per day ultimate cumulative production for oil in oil volume units ultimate cumulative production for gas in gas volume units projected life span of the well in years.

For production rate parameters, you enter Enter your choice followed by input : 1 2 1 2 3 4 5 6 7 8 9

WELL PRODUCTION FORECAST Decline based on cumulative production (1) or based on a second production rate (2) : Exponential (1) or hyperbolic (2) decline Production of f i r s t operation year Average o i l production (BBL/day) Average gas production (MCF/day) Production of other operation year Average o i l production (BBL/day) Average gas production (MCF/day) : Other operation year (years) Oil production at econ. l i m i t (BBL/day) : Gas production at econ. l i m i t (MCF/day)

2 1 55. 0. 20. 0. 5 1. 1.

Seven parameters must be entered: - initial oil production in oil volume units per day - initial gas production in gas volume units per day - mid-year oil production for some other year in oil volume units per day - mid-year gas production for some other year in gas volume units per day

428

RISKTAB

Manual

- year for which the two production rates are entered - oil production at economic Hmit in oil volume units per day - gas production at economic limit in gas volume units per day. The decline function is defined either by an exponential or constant rate decline, or by a hyperbolic decline. For hyperbolic decline you enter Enter your choice followed by input : 2 2 Two additional parameters need to be entered, the decline exponents for hyperbolic decline of oil and gas production. 10 Decline exponent b for oil and gas : 0.5 0.5 You can now save all the changes to a new data file : Do you want to save your CUM changes ? y Enter f i l e name : CUMINP Data f i l e is CUMINP Production curves have been written to CUMINP.CUM CUM creates an output file that has the same name as the data file but with a different extension: .CUM contains the decline curves and cumulative production curves for oil and gas. This data file is written in RISKSTAT format and can be viewed graphically using RISKSTAT. Option 2 - C A S H F L O W

Discounted Net Cash Flow

CASHFLOW reads various financial parameters and a production forecast to calculate the Discounted Net Cash Flow (DNCF) and Internal Rate of Return (IRR) for a single producing well. First, enter a CASHFLOW input file name: Enter data f i l e name : EX9-1 Four -

example applications are provided as data files with the RISKTAB software: EX9-1 describes cash flow of an operator with a 100% working interest EX9-2 describes cash flow of a mineral rights owner EX9-3 describes cash flow of a person with some working and revenue interest EX9-4 describes cash flow of a person who purchases mineral rights. N o t e : CASHFLOW input files should have names without an extension.

CASHFLOW data files are text files and can be edited directly. However, it is recommended that parameters be changed interactively within CASHFLOW. If the data file cannot be found, CASHFLOW assumes that a new analysis is being performed: Cannot find f i l e EX9-1 Is t h i s a new analysis (y/n) ? y The interactive input to CASHFLOW consists of several tables with different options. You can scroll forward by simply entering J or you can scroll backward by entering O j . If no parameters are changed, CASHFLOW will immediately perform the calculations. If you wish to change parameters, enter: Do you want to change any CASHFLOW parameters (y/n) ? y For a new analysis or to change parameters, CASHFLOW will go to the first input option and ask for the title of the cash flow analysis:

429

Computing Risk for Oil Prospects — Appendix D TITLE OF CASHFLOW Ex Appl 9 - 1 , lOOy, working i n t e r e s t , 87.5*/, revenue interest If OK h i t RETURN. Otherwise type in new t i t l e : If you wish to change the title, type in the new title for the cash flow analysis followed by ^ . Otherwise, just press J . The first table lists revenue and working interests. REVENUE AND WORKING INTERESTS IN PERCENT 1 2 3 4

5

Royalties paid to mineral rights owners Overriding royalties paid Royalties we receive Revenue paid to other working interests Revenue we receive as a working interest owner Sum Our working interest Working interest of others Sum

before 12.50 0.00 0.00 0.00 87.50

100.00 100.00 0.00 100.00

after payout 12.50 0.00 0.00 0.00 87.50 100.00 100.00 0.00 100.00

You can change any of these parameters interactively. For example, to change the royalty rate before payout to 15%, enter : Enter choice followed by input: 1 15 12.5 Revenues we receive as a working interest owner are automatically adjusted. If we receive royalties, our revenue and working interests automatically go to 0. Note that payout is defined as the year after which the operating income of the aggregate revenue interests exceed all costs incurred by the aggregate of working interests. The next table lists tax and discount rates. TAX AND DISCOUNT RATES 1 Income tax rate (7.) : 28.00 2 Oil and gas severance tax rate (7.) : 5.00 5.00 3 Discount rates (none up to 4) (7.) : 6.00 9.00 12.00 Up to four discount rates can be entered in CASHFLOW. For example, if you want two discount rates of 10% and 15%, enter: Enter choice followed by input: 3 10 15 If you do not want any discount, enter: Enter choice followed by input: 3 0 The next table lists tangible and intangible capital costs for the well. TANGIBLE AND INTANGIBLE COSTS 1 Tangible costs at year 0 ($) : 70000. 2 Intangible costs at year 0 ($) : 82000. 3 Number of years for which a schedule of tangible and intangible capital costs is to be entered 0 There are tangible and intangible costs that occur before the well starts producing (that is, in Year 0). They are entered as single numbers in the fiscal units specified in the file UNIT.DEF. Costs occurring in subsequent years are specified by creating a table of years and corresponding costs. For example, to specify costs for the first 5 years after starting operations, enter: Enter choice followed by input: 3 5

430

RISKTAB

Manual

Capital costs for subsequent years ($) Year >^ Tangible >/ Intangible

1 2 3 4 5

0. 0.

0. 0.

5000.

7000.

0.

0.

18000.

16500.

A table will appear with five rows corresponding to the next 5 years and three columns corresponding to year, tangible costs, and intangible costs. If more than 5 years are specified, the screen will hold until you depress J and so on for all consecutive years. To enter tangible costs of 10000 monetary units and 0 intangible costs for Year 1, enter: Enter choice followed by input: 4 1 10000 0 The table of years will be updated accordingly. The next table lists depletion parameters. DEPLETION 1 2 3

Percentage depletion 15.00 Limit for depletion C/,) -.65.00 Dominant phase for unit-of-production depreciation (O«oil, G=gas) : 0

The dominant phase for unit-of-production depreciation is only activated if both oil and gas are produced simultaneously. Otherwise it is taken automatically as the phase that is produced. The next table lists depreciation parameters for tangible cost of the well. DEPRECIATION OF TANGIBLE COST 1 Depreciation function

2

3

Straight line (1) Unit of production (2) Empirical depreciation (3) : Depreciation of tangibles before subtracting salvage value (0) or after subtracting salvage value (1) : Number of years to depreciate :

1

0 0

Options include straight line depreciation, unit-of-production depreciation, and empirical depreciation. To select empirical depreciation, enter: Enter choice followed by input: 1 3 Tangibles are depreciated either including their salvage value or after subtracting the salvage value. You must provide the number of years for which straight line or empirical depreciation will be calculated. For example, to depreciate within 5 years, enter: Enter choice followed by input: 3 5 A table will appear with five rows corresponding to the 5 years of depreciation and two columns corresponding to year and percent depreciation. Empirical Depreciation 4 Year / Percent depreciation (sum«100y,) 1 30.0 2 20.0 3 15.0

431

Computing Risk for Oil Prospects — Appendix D 4 5 Sum «

10.0 5.0 80.0

<

NOT correct!

Note that in this example, depreciation does not sum to 100% and you cannot proceed without correcting the table. To correct the table, one or more of the percentages must be changed. For example, you can enter: Enter choice followed by Input: 4 1 50 Empirical Depreciation 3 Year / Percent depreciation (sum*«1007.) 50.0 1 20.0 2 3 15.0 4 10.0

5

Sum *

5.0

100.0

The sum is now correct and you can proceed to the next table which lists additional costs. ADDITIONAL COSTS 1 Leasehold cost ($) : 32000. 2 Mineral rights cost ($) : 0. 3 Abandonment cost as proportion of well's aggregate capital cost (7,) : 5.00 4 Salvage value as proportion of well's tangible cost (7.) : 6.00 Leasehold costs and mineral rights costs usually are mutually exclusive. Abandonment cost is expressed as a percentage of the tangible and intangible costs that occurred in Year 0. Salvage value is actually an income and is expressed as a percentage of the sum of all tangible costs over the life span of the well. Abandonment cost and salvage value are taken into account only at the economic limit. The next table lists depreciation function for leasehold cost, or overriding royalty cost (ORR), or mineral rights cost. DEPRECIATION OF LEASEHOLD, OR ORR, OR MINERAL RIGHTS COST 1 Depreciation function Straight line (1) Unit of production (2) Empirical depreciation (3) : 2 Options include straight line depreciation, unit-of-production depreciation, or empirical depreciation. Depreciation is entered in the same way as depreciation of tangible costs. Note that unit-of-production depreciation is given as cost depletion in the output tables. The next table lists print options for CASHFLOW results. PRINT OPTIONS 1 2 3

Print cashflow tables (y/n) ? y Print results up to economic limit (1) or up to specified year or economic limit whichever comes first (2) : 2 Last year of printout 30

You may wish to write the DNCF tables to a file which then can be printed. Results can be printed up to a specified year or to the economic limit, whichever comes first, or results can be printed up to the economic limit. Note that the

432

RISKTAB

Manual

options are mutually exclusive. CASHFLOW can calculate a DNCF for up to a maximum of 200 years. The next table lists operating costs. OPERATING COSTS 1 Year in which well s t a r t s operating 1 2 Number of years for which specific operating costs are entered : 3 Operating costs for these years 3 Year / Operating costs ($) 1 4000. 2 4200. 3 4400. 4 Ratio of change in operating costs (costs of current year/previous year) : 1.04 Operating costs are entered in the form of a table similar to the tables for tangible and intangible costs. The number of years for which specific operating costs are entered yields the number of rows in the table. The first column gives the year and the second column gives the operating costs incurred in that year. For example, to specify operating costs of 5000 monetary units in Year 3, enter: Enter choice followed by input: 3 3 5000 Operating costs also can be specified as a constant ratio of change in operating costs during the current year divided by the operating costs of the previous year. For example, if operating costs of 20000 monetary units constantly increase by 10% (that is, 20000 in Year 1, 22000 in Year 2, 24200 in Year 3, .. .), enter: 2 1 Enter choice followed by input 3 1 20000 Enter choice followed by input Enter choice followed by input 4 1.1 T h e next table lists the production forecast. WELL PRODUCTION FORECAST 1 Production stream from file (f) or by parametric decline function (p) : 2 Production stream file name : The welPs production stream can be entered either as a decline function by specifying the necessary function parameters (f), or by an external data file containing a production stream for oil and gas (p). An arbitrary production stream from a data file is chosen by entering f and a file name: Enter your choice followed by input : 1 f Enter your choice followed by input : 2 CUMINRCUM WELL PRODUCTION FORECAST 1 Production stream from file (f) or by parametric decline function (p) : f 2 Production stream file name CUMINRCUM The production stream must be in a file written in RISKSTAT format. This file can be generated with CUM or manually within RISKSTAT. It must contain at least three columns: - Column 1: Year starting with Year 1 of production - Column 2: Oil production in oil volume for that year - Column 3: Gas production in gas volume for that year. The headers for columns 2 and 3 must contain the same units as specified in file UNIT.DEF. Units are case sensitive and therefore have to be in upper- and

433

Computing Risk for Oil Prospects — Appendix D lowercase exactly as defined in file UNIT.DEF. An example of a production stream file is: 1 1 20 3 Year Oil Prod BBL Gas Prod MCF 1 2 3 4

30593.080000 21233.830000 14737.830000 10229.130000

518445.300000 249663.200000 120228.200000 57897.250000

To specify a parametric decline, enter: Enter your choice followed by input : 1 p Parameters are defined and entered as described above for module CUM. The next table lists price forecasts. HYDROCARBON PRICE FORECAST

1 Number of years for which specific price forecasts are entered (>«1) : 2 Price forecast for these years 2 Year Oil price ($/BBL) Gas price ($/MCF) 1 19.00 1.50 2 19.50 1.55 Change (price 3 Change 4 Change

of price in subsequent years of current year/previous year) in oil price 1.03 in gas price 1.04

The price forecast is entered in a manner similar to operating costs. The oil and gas prices are entered separately in the fiscal units specified in file UNIT.DEF. Oil and gas prices either can be entered for each individual year or as a constant ratio of change. For example, if oil and gas prices are assumed to be constant for the lifetime of the well at 15 and 1.5 monetary units respectively, enter the following sequence: 1 1 Enter choice followed by input 2 1 15 1.5 Enter choice followed by input Enter choice followed by input 3 1 Enter choice followed by input 4 1 This concludes the input tables of CASHFLOW. All of the changes that have been made can now be saved to a new data file: Do you want to save your CASHFLOW changes ? y Enter f i l e name : EX9-1 CASHFLOW then calculates the discounted net cash flow for the well and presents a summary result on the screen:

434

RISKTAB

Manual

DISCOUNTED NET CASH FLOW FOR A SINGLE PRODUCING WELL Ex Appl 9-1, lOOy, working interest, 87.57, revenue interest ECONOMIC LIMIT REACHED AT YEAR 30 PAYOUT DURING YEAR 1 INTERNAL RATE OF RETURN « 141.387, NET PRESENT VALUES (in $ X 1000) : 2726. Undiscounted 07, 1573. Discounted at 10.07. 1269. Discounted at 12.07i 1052. Discounted at 14.07. Input tables have been written to EX9-1.INP DNCF tables have been written to EX9-1.TAB Production curves have been written to EX9-1.CUM DNCF curves have been written to EX9-1.NCF CASHFLOW creates four output files that have the same name as the data file but with different extensions: - .INP contains the input data in table form - .TAB contains the DNCF tables - .CUM contains the cumulative oil and gas production curves - .NCF contains the cumulative net cash flow curves at specified discount rates. The cash flow and production rate data files are written in RISKSTAT format and can be viewed graphicaUy using RISKSTAT.

3 - RAT

Risk Analysis Table

RAT calculates a Risk Analysis Table (RAT) from a field size distribution, performs a discounted net cash flow analysis for a single well in each field size probabifity class, and calculates an expected monetary and utility value for the entire field or leasehold. Five example applications are provided as data files with the RISKTAB software: - EXll-1 is a RAT pertaining to the cash flow of an operator with a 100% working interest - EX 11-2 is a RAT pertaining to the cash flow of an operator with a 100% working interest but whose leasehold is constrained in area - EXll-3 is a RAT pertaining to the cash flow of a mineral rights owner - EX 11-4 is a RAT pertaining to the cash flow of a person with some working and revenue interests - EX 11-5 is a RAT pertaining to the cash flow of a person who purchases mineral rights. N o t e : RAT input files should have names without an extension. RAT data files are text files and can be edited directly. However, it is recommended that parameters be changed interactively within RAT. Enter a RAT input file name. If the data file cannot be found, RAT assumes that a new analysis is to be made.

435

Computing Risk for Oil Prospects — Appendix D Enter data f i l e name : EXll-1 Cannot find f i l e EXll-1 Is t h i s a new analysis (y/n) ? y The interactive input part of RAT consists of several tables of options. You can scroll forward by simply pressing U or you can scroll backward by entering O J . If no parameters must be changed, RAT will immediately perform the calculations. If parameters must be changed, enter: Do you want to change any RAT parameters (y/n) ? y If a new analysis is begun or parameters are to be changed, the first option of RAT will ask for the title of the new RAT: TITLE OF RAT Ex Appl 11-1, lOOy, working interest, 87.57, revenue interest If OK hit RETURN. Otherwise type in new t i t l e : Type in the desired title followed by J . The first table lists some basic RAT input. RAT DATA 1 Field size distribution from empirical distribution (1) or from field size data (2) : 1 2 Name of field size file : DJSIZE.DAT 3 Name of CASHFLOW data file : EX9-1 4 Discount rate ('/.) : 10.00 RAT accepts two different types of field size distribution data. The distribution can be read from a file containing the probability distribution associated with field volumes and areas, or RAT can calculate the probability distribution from a file containing field volumes and areas in RISKSTAT format. For example, if you wish to use field size data for the Denver-Julesburg basin to create a field size distribution, enter: Enter choice followed by input: 1 2 Enter choice followed by input: 2 DJSIZE.DAT The first few lines of DJSIZE.DAT are shown below: 1 98

Date area ACRE volume BBL 1955.000000 1958.000000 1973.000000

1 3

1240 000000 460 000000 960 000000

3140000 000000 2249200 000000 345000 000000

Be sure that the descriptions of the volume and area columns contain the words "volume" and "area," as well as the units that are defined in UNIT.DEF. Units are case sensitive and therefore have to be in upper- and lowercase exactly as defined in file UNIT.DEF. If you wish to use a previously defined field size distribution, enter: Enter choice followed by input : 1 1 Enter choice followed by input : 2 FSIZE.DAT RAT will look for file FSIZE.DAT and read the field size distribution. The file may contain a manually generated field size distribution, but must be in RISKSTAT format. The distribution may contain from two to seven probability classes and must consist of at least the following three columns:

436

RISKTAB

Manual

- Column 1: Probability expressed as a decimal fraction - Column 2: Field volimie - Column 3: Field size. The probabilities must sum to 1.0 and the headers for columns 2 and 3 must contain the units defined in UNIT.DEF. The number of rows in the file corresponds to the number of probability classes. An example of a field size distribution file is: 1 5 probability volume BBL area ACRE 0.06 0.21 0.46 0.21 0.06

103590 459263 1177970 2559437 4049464

1 3

352 513 650 791 888

Cash flow parameters are provided by CASHFLOW data files. For example, to run a RAT with cash flow parameters from Example Application 9-1, enter: Enter choice followed by input : 3 EX9-1 In RAT, only one discount rate is specified, whereas in CASHFLOW up to four discount rates may be specified. If you want to calculate RATs with different discount rates, RAT must be run repeatedly for each discount rate desired. The next table lists RAT's production forecast: RAT PRODUCTION FORECAST 1 Type of hydrocarbons ( o i l * l , gas«2) : 1 Cumulative/Initial oil production function 2 Linear (1) or polynomial (2) f i t : 3 Number of data pairs 4

Num 1 2 3

(BBL) 17000. 200000. 600000.

1 3

(BBL/day) 20. 80. 175.

5 Life span of field (years) : 6 Exponential (1) or hyperbolic (2) decline :

30 1

In RAT, only a parametric cumulative production function is allowed because the ultimate cumulative production of a well is derived from the field size and the number of producing wells. A RAT can be generated only for one type of hydrocarbon at a time. For example, to create a RAT for gas fields, enter: Enter choice followed by input : 1 2 The production stream can be entered as an exponential or hyperbolic decline curve in the same way as in CASHFLOW. The production function is entered as a table consisting of two columns, the cumulative production per well in hydrocarbon volume, and the initial production per well in hydrocarbon volume per day. The number of rows is given by the number of data pairs in the table. For example, to add the production for an additional field size of 1 million bbls cumulative production and 250 bbls/day initial production, enter the following sequence:

437

Computing Risk for Oil Prospects — Appendix D Enter choice followed by input : 3 4 Enter choice followed by input : 4 4 1000000 250 The cumulative well production data should cover the spectrum of cumulative productions expected from field volimies and number of producing wells. The actual production stream for an individual well is calculated from interpolated values of this fimction. Interpolation is done either by piecewise linear interpolation of the function parameters or by a fitted second-degree polynomial function. Note: For linear interpolation at least two data pairs are required and for a second-degree polynomial interpolation at least three data pairs are required. The next table lists field parameters. RAT FIELD PARAMETERS 1 2

Well spacing (ACRE) Area limit for potentially productive leasehold (0 - no limit) (ACRE) 3 Dry hole probability (7.) 4 Dry hole cost ($) 5 Cost of dry development hole ($)

:

40.

: 0. : 80. : 80000. : 65000.

The total number of wells is determined by dividing the field area by the well spacing. Field area may be limited to all or part of the leasehold area. In such circumstances, the number of wells in areas greater than the leasehold is limited by the size of the leasehold. 6 7 8

Dry development holes function Linear (1) or polynomial (2) fit : 1 Number of function data pairs 3 Num Wells Dry 1 6 1 2 16 2 3 30 3

The function relating total number of wells to number of dry development holes in a prospect is entered as a table similar to the production table. The table consists of two columns containing the total number of wells and the number of dry development holes. The number of rows is given by the number of data pairs in the table. The total number of wells should cover the range of given field areas divided by the well spacing. The actual number of dry development holes is calculated from interpolated nearest integer values of this function. Interpolation is done either by a piecewise linear interpolation of the function parameters or by fitting a second-degree polynomial. The next table lists physical access cost. PHYSICAL ACCESS COST 1 Tangible physical access cost ($) : 10000. 2 Intangible physical access cost ($) : 5000. 3 Abandonment cost as proportion of aggregate physical access cost (7.) : 5.00 4 Salvage value as proportion of physical access tangible cost (*/,) : 6.00 Physical access cost is prorated on a per-we 11 basis for cash flow purposes and is added to tangible and intangible well costs in Year 0. Note that abandonment cost and salvage value for physical access are different from abandonment cost and salvage value for tangibles.

438

RISKTAB

Manual

The next table lists depreciation of physical access cost. DEPRECIATION OF PHYSICAL ACCESS COST 1 Depreciation function

2

Straight line (1) Unit of production (2) Empirical depreciation (3) Number of years to depreciate :

1 10

Depreciation of physical access cost is entered in the same way that the depreciation of tangible well costs are entered in CASHFLOW. However, depreciation of physical access cost is treated separately from depreciation of tangible well costs. In addition to the Expected Monetary Value (EMV), RAT may also optionally calculate an Expected Utility Value (EUV). The next table lists the necessary utility data: 1 2 3

UTILITY FUNCTION EUV analysis (y/n) ? n Linear (1) or polynomial (2) fit : 1 Name of utility function file : UTILE.DAT

A Utility function must be provided in RISKSTAT format. This file must contain at least two columns: - Column 1: Utility value, given in utiles - Column 2: Monetary value in fiscal units. The header for the monetary column must contain the fiscal units that are defined in UNIT.DEF. The number of rows is given by the number of data pairs. An example of a utility function file is: 1 1

6 2 utiles money

$

-500.0 -100.0 .0 100.0 300.0 450.0

-2500000.0 -1000000.0 .0 1000000.0 5000000.0 10000000.0

The actual utile value is calculated from interpolated values of this function. Interpolation is done either by piecewise linear interpolation of the function parameters or by fitting a second-degree polynomial. This table concludes the RAT input. RAT will then offer the option of saving the input as a new file. To save the input, enter: Do you want to save your changes ? y Enter f i l e name : EXll-1 RAT will then either read a predefined field size distribution from a data file and proceed, or it will read a data file containing field sizes and determine the field size distribution. In the latter case, the number of field size classes must be specified. Between two and seven classes may be chosen. RAT will use logarithmic transformations of the volumes and create size classes by dividing the span of log volumes in the data file into the specified number of classes. Each field size class is represented by its midpoint. The midpoints are then back-transformed into the original units of volume. For example, if seven classes are specified, enter: Enter number of field size classes (from 2 to 7) : 7

439

Computing Risk for Oil Prospects — Appendix D RAT then displays the field size distribution: Field Size Distribution •obability

(•/.)

3. 4. 9. 22. 45. 16. 1.

/

Field Size / (bbls * 1000) 28. 79. 216. 591. 1617. 4422. 12094.

Field Area (acres) 255. 329. 424. 547. 705. 908. 1170.

This distribution can be saved to a field size distribution data file: Do you want to save the distribution (y/n) ? y Enter f i l e name: FSIZE.DAT If a file with this name already exists, RAT will ask if it should overwrite the old file: File exists already. Do you want to overwrite i t (y/n) ? y The field size distribution is written in RISKSTAT format with three columns containing the probability, field volume midpoint, and field area midpoint. After generating the field size distribution, RAT will read the CASHFLOW data file. CASHFLOW data can be interactively changed in the same way as described under the CASHFLOW option. If none of the CASHFLOW data are to be changed, enter: Do you want to change any CASHFLOW parameters (y/n) ? n RAT now performs the cash flow distribution. DNCF results for 1. field size class DNCF results for 2. field size class DNCF results for 3. field size class DNCF results for 4. field size class DNCF results for 5. field size class DNCF results for 6. field size class DNCF results for 7. field size class

analyses for all classes of the probability calculated calculated calculated calculated calculated calculated calculated

RAT writes the complete Risk Analysis Table to the screen. You can scroll through the RAT simply by pressing J . RISK ANALYSIS TABLE Ex Appl 11"-1, 1007. working interest, 87.57. revenue interest 6 5 1 2 3 4 7 a) Field size probability

.031

.041

.092

.214

.449

.163

.010

b) Field size BBL X 1000

28

79

216

591

1617

4422

12094

c) Field size ACRE

255

329

424

547

705

908

1170

5

7

10

12

16

21

26

d) Producing wells

440

1.00

RISKTAB Manual A t a b l e of your input has been saved in Tlie RAT has been written t o Production curves have been written t o DNCF curves have been written t o Functions used in RAT have been written t o Cash flow t a b l e s have been written t o

EXll-l.INP EXll-l.RAT EXll-l.CUM EXll-l.NCF EXll-l.FNC EXll-l.TAB

RAT creates six output files that have the same name as the d a t a file w i t h different extensions: - < F I L E N A M E > . I N P repeats the input data in table form - < F I L E N A M E > . R A T contains the RAT - < F I L E N A M E > . T A B contains the DNCF tables for all probability classes - < F I L E N A M E > . C U M contains the cumulative oil and gas production curves for all probability classes - < F I L E N A M E > . N C F contains the cumulative net cash flow curves for all probability classes - < F I L E N A M E > . F N C contains the three user-specified functions used in RAT. T h e cash flow and production curves, and the three functions, are written in RISKSTAT format and can be viewed graphically in RISKSTAT. T h e three user-specified functions are: - Cumulative versus initial production - Total number of wells versus number of dry development holes - Utiles versus monetary value. A t the end of a Risk Analysis Table, an Expected Monetary Value (EMV) and optionally an Expected Utility Value (EUV) are calculated. For different RAT runs, different EMVs and EUVs are obtained that can be compared in an EMV or EUV decision table. T h e EMV and EUV can be saved in a d a t a file: Do you want t o save EMV/EUV i n data f i l e (y/n) ? y Enter f i l e name: DECISION.DAT T o create a decision table, different RAT outcomes must be assembled into a single DECISION input file. To add RAT outcomes t o an already existing DECISION input data file name, make the following response: There i s already a f i l e with that name. Do you want t o append EMV/EUV t o f i l e (a) or overwrite the f i l e (o) or write a new data f i l e (n) ? a RAT will then append the current EMV and EUV to the data file with the message EMV/EUV appended t o DECISION.DAT

4 - DECISION

Decision Table

DECISION creates a decision table of Expected Monetary Values (EMVs) or E x pected Utihty Values (EUVs) for a series of RAT outcomes. A DECISION input data file that has been created with RAT must be provided: Enter name of EMV/EUV data f i l e : DECISION.DAT DECISION then will ask you whether you want an EMV table or an EUV table. To specify a decision table based on EMVs, enter:

441

Computing Risk for Oil Prospects — Appendix D 1 Create an EMV d e c i s i o n t a b l e 2 Create an EUV d e c i s i o n t a b l e 0 Exit Enter your choice : 1 DECISION will then display the EMV decision table on the screen. FSize Class

1

Field size BBL X 1000

28

Field size probability

.031

EMV DECISION TABLE 2 3 4

5

6

7

79

216

591

1617

4422

12094

.041

.092

.214

.449

.163

.010

Dry

Sum

0 0.80

1.00

EMV $ X 1000 Action 1

Ex Appl 11-1,

looy, working

interest

1721

N e x t , DECISION will ask if the EMV table is to be saved in a decision table file: Do you want t o save t h i s d e c i s i o n t a b l e ? y Enter f i l e name: DECISION.EMV Decision t a b l e has been w r i t t e n t o DECISION.EMV This procedure can be repeated, with appropriate changes, to create an EUV table or to quit DECISION.

442

INDEX

bid 342 Big Horn Basin (Wyoming) 26, 33, 41, 46 binomial distribution 57 a priori expected field size 109 graphs of 62 a priori probability 100 binomial equation 102 abandonment costs 238 binomial expansions 58 absolute deviations 38 binomial probabilities, table of 60 absolute frequencies 39 bivariate scatter plots 72, 75, 77, 79, 85, ad hoc procedures for introducing depen177, 205, 292, 295, 337, 341-43, dence 134 BOE (barrels of oil equivalent) 29, 81, 287 ad valorem taxes 228 bonus 324, 327, 342 additive schemes 8 bounded distributions 130 adequacy of seal 132 box-and-whisker plots 151 adjustment of probabilities 101 buildups, carbonate 27, 142 aerial photographs 12 burial, depth of 9 allowable depletion 237, 253 alternative actions 306, 312 American Petroleum Institute 132 capacity of prospect 113 analogue region, mature 19, 181, 327 capital costs 230, 232 anticlines 13 carbonate buildups 27, 142 area of closure 81, 326 carried interest 285 areal constraint 281, 298 carrier beds 8, 113 arithmetic mean 37 cash flow 215 assessments 6 analysis 218, 303 autocorrelation 139, 149 projection 227 automated semivariogram calculation 157 CASHFLOW see "module C A S H F L O W average cumulative production per well 288 Central Limits Theorem 185 average production in initial year 288 centrality, measures of 37 average ultimate cumulative production centroid 176 291 chance fork (decision tree) 312 aversion to loss 265 characteristics of oil and gas fields 132 aversion to risk 25 charge 113 class intervals 36 of probability distributions 287 Bakant Basin (Magyarstan) 27 barrels of oil equivalent (BOE) 29, 81, 287 classes, field size 34 classes of trend surface residuals 98 Bayes, Reverend Thomas 90 closure, structural 80, 324, 335 Bayes' theorem 89, 93, 104 coal seams 17 Bayesian estimates 141 Bayesian revision of expected field size 108 coin tossing 266, 272 experiments 59 Bayesian revision of field size probability combining geological variables 112, 174, distributions 107 189, 337 Bayesian revision of regional success ratios combining individual geologic maps 182 99 computer contouring 7, 137 Bayesian statistics 89 algorithms 139 bedding index 27, 142, 176, 192, 197 map of 197 computer programs 25 Belaskova region (Magyarstan) 27, 323 concession 2, 324 bell curve 35 cost of 237 best estimate 84 conditional analysis 97 bias 30 conditional dry hole probability 331

C

B

Computing Risk for Oil Prospects conditional probability 10, 86, 88, 90, 141, 350 distribution 190 of success 180 conditional relationships 24, 84, 89, 348 conditional success ratio 96 confidence bands 75, 85, 78, 79, 337 confidence intervals 78, 337 Conservation Division, U.S. Geological Survey 80 constructing contour maps 137 constructing decision tables 312 contingency tables 94 continuous function 180 continuous probability distribution 23 continuous surfaces 137 contour mapping 350 contour maps 137, 139, 351 of bedding index 146, 197 of discriminant scores 184, 198, 203, 209, 212 of probabilities 188, 201, 204, 210, 213, 214 of seismic reflection time 325 of shale ratio 145, 196, 329 of standard error 163, 172, 185 of structure 123, 143, 171, 193 of thickness 144, 162, 195, 330 of trend surface 147 of trend surface residuals 148, 174 reliability of 138, 184 contour-drawing procedures 139, 159 contouring as a forecasting tool 139 contouring by computer 7, 137 contouring grid 140 contouring programs 139, 149 corrections for interdependence 133, 350 correlation 73, 133, 335 coefficient 73, 335, 339 correspondence between dollars and utiles 316 cost depletion 229, 236, 253 cost of a concession 237 cost of dry development wells 289, 293 covariance 72, 175 equation for 72 cumulative curves 51 cumulative discounted net cash flows 223, 225, 250, 257, 259, 261, 263, 303, 305, 309, 344

444

cumulative exploration effort 20 cumulative histogram 42, 45 cumulative hydrocarbon volume 20 cumulative percentage scale 44 cumulative probability 42, 334 distributions 87, 125, 334 cumulative production 22, 229, 241 cumulative undiscounted net cash flow 254 currency unit 226 curve, cumulative probability 125 curve, J-shaped 20, 29 curve, lognormal 36 curve, normal 35, 42

D decision analysis 216, 305 decision fork (of decision tree) 312 decision table 220, 266, 279, 306, 311, 313 decision tree 221, 306, 312, 315 decisions 1, 215, 313, 342 decline curve 239, 257, 296 decline exponent 229, 240, 242 decline functions 229 decline rate 239 degree of belief 5, 99 degree of exploration maturity 19, 192 Delphi method 7 Denver-Julesburg Basin 26, 41, 71, 286 depletion 225, 236, 285 depreciation 225, 229, 235, 253, 285 depth of burial 9 deterministic nonstationary trend 168 development wells 220, 288, 293, 301 deviation, standard 27 discount factor 225 discount rate 222, 230, 235, 254, 284 discounted net cash flow (DNCF) analysis 218 discounted net cash flow schedules 254 discounting 219, 221 discovery efficiency 30 discovery probability 50, 53, 93, 99, 116, 186, 199, 204, 210, 294, 328, 331 discovery sequence 51 discrete probability distribution 23, 220 discriminant function analysis 175, 190, 196, 202, 331, 332, 351 discriminant index 176 discriminant score 176, 180, 199, 331 map 184, 198, 203, 208, 211

Index discriminating between discoveries and dry holes 173 diskettes 25, 39, 81 dispersion, measures of 37 distance-weighted averages 149 distribution, bell-shaped 35 distribution, binomial 57 distribution, error 164, 182, 187 distribution, exponential 126 distribution, field size 29, 82 distribution, input 134, 287, 301 distribution, log-gamma 128 distribution, lognormal 44, 115, 133 distribution, normal 35, 115, 129, 161, 162, 172, 185, 333 distribution, output 134 distribution, Pareto 128 distribution, probability 1 distribution, risked 116, 125 distribution, triangular 11, 129 distribution, uniform 130 distribution, unrisked 116, 125 distribution parameters 131 distributions, bounded 130 distributions, frequency 19 distributions, selecting 128 distributions, spatial 26 distributions, statistical 26 distributions, tails of 131, 134 DNCF (discounted net cash flow) 218 dollar-to-utile conversions 315 downstream capital costs 231, 238 drift 169 drift coefficients 170 drill-stem test 192 drilling at random 21 drive mechanisms 18 dry hole probability 22, 55, 87, 100, 116, 308, 328, 329

E early discovery of large fields 49 economic limit 225, 229, 241, 244, 252, 260 efficiency of discovery 30 empirical conditional probability function 180, 185, 187, 332 empirical depreciation 229, 235 empirical production stream 238

EMV (expected monetary value) 219, 285, 290, 294, 342 criterion 267, 315 decision tree 313 endowment, hydrocarbon 15 enhanced recovery 18 entrapment of hydrocarbons 8 entries in risk analysis tables 288 envelope (semivariogram models) 154, 164 environmental damage 4 error bands 85, 182, 337, 338 error distribution 164, 182, 187 error functions 86, 164, 187, 331 error maps 163, 172, 185, 189 error reduction 13 error variance 161, 169 error-weighted conditional probability 186 essential geologic factors 8 estimating discovery size 71 estimating endowments 18 estimating "Q" 29 EUV (expected utility value) 220, 285, 290, 296 decision table 221, 314, 318 decision tree 317 expected field size, Bayesian revision of 108 expected monetary value (EMV) 219, 285, 290, 294, 342 expected success ratio 100 expected utility value (EUV) 220, 285, 290, 296 decision table 221, 314, 318 expected value of field size 109 experience, group 7 experience, individual 7 experimental semivariogram 151, 158, 167, 168, 170, 183, 207, 210 exploration effort 20 cumulative 20 exploration maturity 19, 20, 192, 209 exploratory drilling 20 exploratory dry hole cost 289 exploratory wells 3 exponential decline 229, 239, 286, 340 exponential distribution 126 exponential model (semivariogram) 156 expropriation 3 extrapolation 51 extreme outcomes 131 extreme tails of distributions 131

445

Computing Risk for Oil Prospects

F field 17 area 71, 286, 291, 326 size classes 34, 46, 288, 289 size data 26 size distributions 21, 29, 82 Bayesian revision of 107 statistics 53 volume 71, 286 field-wide averages 121 final stage (of target area exploration) 192, 209, 211 financial analysis 2, 25, 215, 340, 352 financial consequences table 311, 312 "financial masochist" 266 financial overview 215 financial risk 1 first-order drift 170 fitted regression lines 74, 185, 292, 296, 337, 341, 343 floating-point arithmetic 226 forecasting future prices 228 FORTRAN 25 frequencies 1, 5, 22, 89 absolute 39 marginal 92 discriminant score 205 frequency class 42 frequency distribution of volumes 286 frequency distributions 4, 11, 19, 29, 33, 44, 57, 82, 115, 128-31, 134, 153, 161, 172 frequency, relative 4, 39 function relating number of dry development wells 293 future prices, forecasting 228

G "gambler's ruin" 62 gas-oil ratio 115 Gaussian model (semivariogram) 156, 207 equation 156 generalized form of risk analysis table 280 generation of hydrocarbons 8, 113 geologic factors, essential 8 geologic maps, combining individual 182 geological properties, mapping 137 mapping combined 183 geological uncertainty 12 geological variables, combining 174, 337 geometric mean 38

446

geostatistics 138, 149 gradient, temperature 9 graphing a utility function 272, 275, 278, 295, 343 graphs of binomial distribution 62 grid, contouring 140 gross pore volume 10 gross rock volume 11 group centroids 178 group experience 7 guesses 6

H height of closure 97 height of reservoir 10 held-by-production (HBP) clause 3, 237 histograms 24, 33, 40, 63, 177, 178, 187, 206, 283 hull (semivariogram models) 154 hydrocarbon endowment 15 hydrocarbon entrapment 8 hydrocarbon generation 8 hydrocarbon migration 8 hydrocarbon volume, cumulative 20 hydrocarbon-occurrence uncertainty 15 hydrocarbon-volume probabilities 24 hyperbolic decline 229, 239, 242, 286

I IBM-compatible personal computers 25 immature stage (of exploration) 192, 196 income tax 225, 234, 253 incorporating risk (in simulation) 124 independence 9, 59, 102, 133 index, bedding 27 index score 176 individual experience 7 inflows, cash 225 information service companies 30 initial capital costs 230 initial producing (IP) rate 229, 239, 291 initial tangible costs 238 input probability distributions 116, 120, 134, 281, 287, 301, 339, 346 intangible costs 286 intangible physical access cost 286 intercept 74 interconnections between financial modules 217 interdependencies between geological variables 9, 133, 350

Index interest carried t o casing point 285 log-log plot 72, 342 intermediate stage (of target area explora- log-probability plot 44, 48, 52, 84, 87, 334 logarithms 34 tion) 192, 202 internal rate of return (IRR) 235, 252, 260 lognormal curve 36 lognormal distribution 44, 115, 133 inverse of the square of the distance 140 lognormal equation 36 investment scenarios 296 Louisiana and Texas offshore 26 investment schedule 253 lowest closing reflection-time contour 81 IP (initial producing rate) 229, 239, 291 IRR (internal rate of return) 235, 252, 260, luck 57 303 isopach maps 144, 162, 195, 330 Magyarstan 26, 31, 50, 69, 97, 118, 121, iterations 116, 126 131, 159, 176, 177, 323 Magyarstan target area 191, 193 Magyarstan training area 141, 162, 183 J-shaped curve 20, 29 map, bedding index 197 joint probability 10, 90, 115 map, kriged 184, 186 joint venture 342 map, prospect 123, 325 map, trend surface residual 148, 194 map, standard error 161, 163 Kansas 13, 150 map, structure contour 12, 97, 143, 148, 193 Western Shelf area 26 map, thickness 144, 162, 165, 330 Kansas Geological Survey 26 map, trend surface residual 97, 148, 194 kriged estimates 164 map, universal kriging 171 kriged map 184, 186 map error 192, 197 kriging 151, 159, 179, 202, 350 map of scores 184, 198, 203, 208, 212 algorithm 160 map of seismic reflection time 80,196, 325 equations 160, 169, 182 map of shale ratio 145, 196, 329 standard error 169, 185 mapping, computer contour 137, 350 universal 169 mapping combined geological properties weights 170 183 mapping geological properties 137 lags 150 maps, contour 351 land position 2, 281 of bedding index 146, 197 of discriminant scores 184, 198, lease cost 285 203, 208, 211 lease delay rentals 237 of error 165, 166, 189 leasehold 2, 229 of probability 188, 201, 204, 209, costs 232, 237, 294 212, 213 mineral rights or O R R cost 289 of seismic reflection time 325 least squares 74 of shale ratio 145, 196, 329 criterion 142, 174 of standard error 163, 172, 185, regression line 75 198, 203, 206, 209, 212 life span, producing 229, 238, 239, 242 of structure 123, 143, 171, 193 limestone reservoir rocks 27 of thickness 144, 162, 195, 330 line fitting 74, 187, 292, 295, 337, 341, 343 maps, geologic (combining individual) 182 linear combination 174 maps, reliability of contour 138 linear discriminant function 175 maps of optimum usefulness 213 linear regression 84 marginal frequencies 92 linear semivariogram 154 marginal probability 90 linear trend 144 market for mineral rights 238 log-gamma distribution 128

M

J

K

L

447

Computing Risk for Oil Prospects mature analogue region 19, 119, 132, 191 mature areas 119, 132 mature stage (of target area exploration) 19, 20, 192, 200, 209 maximum acceptable losses 276 maximum search radius 140 mean, arithmetic 36, 37, 78 mean, geometric 38 measures of centrality 37 measures of dispersion 37 median 37 mid-year discounting 235 migration 8, 113 efficiency 132 mineral rights 227, 229, 237 minimum acceptable number of control points 140 misc lass ificat ion 176 table 179 mode 37 model semivariogram 151, 170 modeling prospects 111 module CASHFLOW 221, 226, 279 applications 255 input 245 output 248, 252 module RAT 216, 219, 220, 279, 285, 340, 342 and CASHFLOW, relationship between 280 cages 319 input 299, 306 output 301, 307 monetary units 230 Monte Carlo simulation 11, 112, 116, 126, 182, 350 mud logging 15 multiple linear regression 337 multiplicative procedures 9, 349 multivariate statistics 175

N NFDP (net for depletion purposes) 237 negative utility 271, 290, 295 net flows 225 net for depletion purposes (NFDP) 237 net operating income 254 net physical access cost 286, 290 net present value (NPV) 218, 279, 291 net revenue interest (NRI) 230 net taxable income 254

448

net utility 271 Newton's method 241 nonflows 225 non-geological aspects of risk 3 nonstationarity 168 nonstationary geological properties 171 normal curve 35, 42 normal distribution 35, 78, 85, 115, 129, 161, 162, 172, 185, 333 standardized 78 truncated 123 normal error distribution 164, 185, 187 NPV (net present value) 218, 279, 291 NRI (net revenue interest) 230 nugget effect (semivariogram) 156, 207 null point 271 number of discrete class intervals 287 number of dry development wells 288, 292 number of field size classes 36, 286 number of fields 287 number of iterations 126 number of producing wells 288, 291 numerical probabilities 1 numerical weights 8, 175, 332, 337

O obtaining a utility function 268 occurrence probabilities 50 octant search procedure (semivariogram calculation) 158 offshore Louisiana and Texas 26, 81 oil and gas originally in place 19 oil and gas price forecasts 244 oil field populations 33 oil-in-place 126 omnidirectional semivariogram 158 operating costs 238 operating income schedule 252 optimal combination of geologic variables 179 optimum action (decision tree) 313 organic carbon 17 ORR (overriding royalty) 227, 229, 231-33, 262 outcome probabilities 23, 87, 270, 282, 287, 301, 328, 337 outflows 225 output distribution 120, 134 overhead 238

Index limits 162 mapping 180 maps 188, 201, 204, 209, 212, 213 marginal 90 paleotemperature indicators 15 occurrence 50 Pareto distribution 128 of correct assignment 99 Pascal's triangle 57 of discovery, maps of 188, 201, 203, payoff 272 210, 213, 214 payout 227, 231, 252 of discovery, risked 125 percentage depletion 236 253 of success 22, 93, 186 percentiles 39, 51, 125, 132 plots 48, 52, 63, 64, 84, 180, 336 Permian Basin 20, 29 scale 4 petroleum potential, remaining 29 subjective 6, 99 PHV (producible hydrocarbon volume) 10, theory 1 119, 122, 124 transformation 180, 192, 209, 332 physical access costs 285 unconditional 329 physical disasters 4 probability distributions see "distribution" piecewise linear interpolation 292 continuous 23 play 2 cumulative 87 "Pleistocene Trend" 80 discrete 23 point estimates 111, 349 input 120 polynomial 144, 169 of discounted net cash flows 297, pool 17, 50 305, 309 populations, oil field 33 of field areas 283 pore volume, gross 10 of field volumes 87, 114, 135, 287, porosity 10, 119, 121 334, 337, 339 positive utility 271 of probabilities 132 PostScript 26 risked 117 potentially producible hydrocarbons, two-part 22 endowment of 18 unrisked 84, 117, 122, 125, 287 Powder River Basin (Wyoming) 21, 26, 39 probability function 180, 192, 196, 207, 332 prediction error 76, 84, 185, 337 empirical 332 predrill/postdrill comparison 82, 212 selection of 114, 120, 124, 199, 331, presence of producible hydrocarbons 8 328 present value 218, 221, 288 producer probability 23, 93, 186, 331 of producing wells 293 producible hydrocarbon volume (PHV) 10 of field size classes 290 producible hydrocarbons, endowment of 16 price declines 3 principal streams in cash flow analysis 224 producing life span 229, 238 production rates 229 probability 4 production taxes 228 adjustment of 101 production, ultimate cumulative 19, 239, a priori 100 243, 326 conditional 10, 86, 88, 90, 141, 186, production-decline curves 240 331, 335 projected subpopulations 51 cumulative 42 proportion of dry holes 55, 93, 292 discovery 22, 50, 55, 89, 93, 100, prospect 1 173, 186, 275, 310, 312, 328 map 123, 325 error-weighted conditional 186 ranking schemes 5 highs 202 "proven oil finders" 7 hydrocarbon-volume 24 psychological experiments 131 joint 10, 90, 115 overriding royalty (ORR) 227, 229, 231-33, 262

P

449

Computing Risk for Oil Prospects "Republic of Magyarstan" 26, 31 rescaling of probabilities 56, 88, 289 reservoir 17 area 10 "Q" (total producible hydrocarbon endowheight 10 ment) 21 estimating 29 rocks 8 rocks, limestone 27 thickness 10 radial control 140 residual map 148, 194 radial segments (semivariogram calcularesiduals 97, 194 tion) 158 second-degree trend surface 148 random combination 112 trend surface 143 random drilling 21 revenue interest (RI) 227, 230 range 153 revising outcome probabilities 89 of discriminant scores 205 ©RISK 128 of scores 198, 210 risk 1 rare event 45, 131 aversion to 25, 219, 266 RAT see "module RAT" financial 1 ratio of change in annual operating costs incorporating in simulation 124 238 risk analysis tables 219, 266, 279, 346 ratio of dry and producing wells 292 risk averse utility function 269 "recoverable oil" 126 risk neutral utility function 269 recovery, enhanced 18 "risk seeking" 266 recovery factor 10, 18, 126 risk seeking utility function 269 reduced major axis (RMA) line 75 RISK software 25, 26, 352 reduction in error 13 "risked" (incorporation of dry hole risk) 281 reduction in uncertainty 353 distribution 23, 88, 116, 125, 219, redundancy 12 266, 279, 346 reflection-time contours 81, 325 dry hole cost 294, 345 regional component of structure 143 leasehold cost 294, 345, 348 regional dip 147 net physical access cost 294, 345 regional endowment 16 present values 294 regional seismic reflection-time maps 80, RISKMAP package (software) 25, 141, 157, 325 170, 180, 190 regional success ratio 58, 68, 87, 308 RISKSTAT package (software) 25, 30, 47, Bayesian revision of 99 73, 74, 78, 126, 175, 278, 291, 335, 337 regression 74, 85, 144, 291, 295, 337, 350 RISKTAB package (software) 25, 216, 221 regression line 74, 77, 79, 85, 335 RMA (reduced major axis) line 75 regression of log areas on log volumes 286 Roskoflf area (Magyarstan) 327 regression of log volumes on log areas 84 prospect 323, 340 regulatory agencies 30 target area 325, 329 relationship between area and volume 71 royalties 234 relationships, conditional 24, 84, 89, 141, royalty owners 227 350 relative frequency 4, 39 relative order of discovery 51 salt domes 80 reliability of contour maps 138, 149, 159, salvage value 237, 287 161 sample means 131 remaining petroleum potential 29 satellite imagery 12 "remaining reserve" 237 saturation, water 10, 119 psychological units 267

Q

R

S

450

Index scatter 175 score map 184, 196, 203, 209, 212 scores 5, 176, 210 seal 8, 113 quality of 132 second-degree polynomial 292 second-degree trend surface 145, 147 seismic closure 80, 325, 335 seismic maps 80, 325 reflection-time 82, 325 seismic surveys 12, 325 semivariance 150, 159 equation 150 semivariogram 150, 182, 190, 197, 202, 207, 209 experimental 151, 158, 167, 168, 170, 183, 207, 210 exponential 155 Gaussian 155 linear 155 model 151, 170 omnidirectional 158 spherical 155 sensitivity of EMVs to probabilities 308 sequences of decisions and outcomes 312 sequences of events 57 setting parameters 128 severance tax 228, 234 shale ratio 27, 93, 145, 176, 192, 194, 326, 329, 336 map 196, 329 sigmoidal curve 43, 180, 187, 332 sill (semivariogram) 153, 207 simulation 111 experiments 21 Monte Carlo 11 simultaneous equations 160 single-valued property 139 skewness 34, 40, 62, 119, 134 slope 74 source beds 8, 113 spatial autocorrelation 139, 149 spatial distributions 26 spatial rate of change 159 specifying realistic parameters 132 spectrum of possible discriminant scores 186 spherical semivariogram 155 spillpoint 115

standard deviation 27, 36, 39, 71, 78, 131, 132 standard error 86, 161, 164, 182, 184 kriging 169 map 161 of estimate 76 standardized covariance 73 standardized normal distribution 78 stationarity 166 stationary residuals 168 statistical correlation 71, 73, 133, 173, 335 statistical tests 177 "statistician's urn" 92 stepout well 3, 12 straight-line depreciation 229, 235 structural closure 80, 324 structural elevation 27, 192 structural trend surface residuals 148, 192 structure contour map 12, 97, 123, 143, 148, 171, 193 subjective estimates 93 subjective guesses 6 subjective probability 6, 99 success probability 22, 56, 87, 188, 201, 203, 210, 213, 328 success ratio 55, 88, 89, 93, 99, 103 conditional 96, 103 expected 100 regional 58, 68 "true" regional 69 unconditional 96 "successful" well 55 summaries of net cash flows 303 SURFACE III 26

T

table of binomial probabilities 60 tails of distributions 131, 134 tangible assets 237 tangible capital costs 235, 254, 286 tangible physical access cost 286 "target area" 27 target area (Magyarstan) 189, 191, 327 taxes 3 ad valorem 228 income 225, 234, 253, 254 production 228 severance 228 temperature gradient 9 thickness 27, 176, 192, 326, 336 maps 144, 165, 195, 330

451

Computing Risk for Oil Prospects of reservoir 10 three-point distribution 11, 129 time series analysis 139, 150 time value of money 219, 222 timing 8, 113 total hydrocarbon endowment 16 total organic carbon 114 total producible hydrocarbon endowment ("Q") 21 training area 27, 141, 162, 176, 183, 189, 191, 327 data set 161 transformation of dollars 290 trap 8, 113 "timing" 132 trend 169 trend surface 97, 147, 183, 193 analysis 142, 169, 174 residual map 97, 148, 194 residuals 97, 143, 148, 168, 176, 194 second-degree 145, 147 triangular distribution 11, 129 Troyska area (Magyarstan) 323, 327 "true" regional success ratio 69 truncated normal distributions 119, 123 two-part probability distributions 22

unit-of-production depreciation 229, 235 units for hydrocarbon volumes 230 universal kriging 169 equations 169 map 171 "unrisked" (no incorporation of dry hole risk) 88, 281 distribution 46, 116, 125 probabilities 88, 287, 288 probability distribution 117 updating assessments 189 utiles 219, 267, 271, 290 utility 265 equation 270, 276 family (of functions) 269 function 219, 265, 268, 286, 290, 295, 316, 343 graphing function 272, 278

V variance 38, 71, 78, 175 Venn diagram 105 virgin areais 132

W

wasting assets 238 water saturation 10, 119 weighted average 140, 159 U.S. Geological Survey Conservation Divi- weighting coefficients 160, 175, 352, 357 weighting function 159 sion 80 well, exploratory 3 U.S. Gulf Coast OCS 80 ultimate cumulative production 19, 30, 239, well, stepout 3, 12 well, "successful" 55 288, 289, 341 well data bases 138 ultimately recoverable oil and gas 29 well density 193 uncertainty 2, 4, 12, 113, 139, 160, 181, 184, Western Shelf area (Kansas) 26 228, 349 WI (working interest) 227, 230, 233 geological 12 owners 227 of hydrocarbon occurrence 15 Wind River Basin (Wyoming) 26, 41 reduction in 353 worth of money 265 unconditional probabilities 96, 329 Wyoming, Big Horn Basin 26, 33, 41, 46 unconditional success ratio 96 Wyoming, Powder River Basin 21, 26, 39 unconstrained guesses 349 Wyoming, Wind River Basin 26, 41 undiscounted cash flow 235, 254 uniform distribution 130 unit of currency 226 Zhardzhou Shelf (Magyarstan) 26, 141, 176

U

452