Table Of ContentGEOST ATISTICS AND
PETROLEUM GEOLOGY
COMPUTER METHODS IN THE
GEOSCIENCES
Daniel F. Merriam, Series Editor
Computer Applications in Petroleum Geology
Joseph E. Robinson
Graphic Display of Two- and Three-Dimensional Markov
Computer Models in Geology
Cunshan Lin and John W. Harbaugh
Image Processing of Geological Data
Andrea G. Fabbri
Contouring Geologic Surfaces with the Computer
Thomas A. Jones, David E. Hamilton,
and Carlton R. Johnson
Exploration-Geochemical Data Analysis with the IBM PC
George S. Koch, Jr. (with programs on diskettes)
Geostatistics and Petroleum Geology
Michael Edward Hohn
Related Titles
Statistical Analysis in Geology
John M. Cubitt and Stephen Henley (eds.)
Cluster Analysis for Researchers
H. Charles Romesberg
Analysis of Messy Data, Volume 1: Designed Experiments
George A. Milliken and Dallas E. Johnson
GEOSTA TISTICS AND
PETROLEUM GE01,OGY
MICHAEL EDWARD HOHN, West Virginia Geological
and Economic Survey
tnm5I VAN NOSTRAND REINHOLD
~ New York
Copyright © 1988 by Van Nostrand Reinhold
Sotlcover reprint ofthe hardcover 1s t edition 1988
Library of Congress Catalog Card Number 87-37155
ISBN-13: 978-1-4615-7108-7 e-ISBN-13: 978-1-4615-7106-3
DOl: 10.1007/978-1-4615-7106-3
All rights reserved. No part of this work covered by the
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Library of Congress Cataloging-in-Publication Data
Hohn, Michael E.
Geostatistics and petroleum geology/Michael Edward Hohn.
p. cm.-(Computer methods in the geosciences)
Includes index.
1. Petroleum-Geology-Statistical methods. 2. Petroleum
Geology-Data processing. l. Title. II. Series.
TN87!.H536 1988
553.2'8'015195-dcl9
87-37155
CIP
Contents
Series Editor's Foreword ix
Preface xi
1
Overview of Geostatistics 1
A Few Definitions 2
Geostatistics on the Computer 3
A Simple Geostatistical Case Study 5
2
The Semivariogram
Basic Calculation and Principles 13
Anisotropy 19
Practical Effect of Distance Tolerance 21
Stationarity 22
Modeling an Observed Semivariogram 25
Theoretical Models 25
Nugget Effect 29
Nested Models 31
Admissibility 33
A Simple Example 34
A More Complex Model 38
v
vi Contents
Yet More Complex Models 43
Hole Effects 43
Geometric Anisotropy 46
Zonal Anisotropy 50
Autocorrelation 51
Autoregressive Processes 52
Outliers, Normality, and Robustness 59
The Role of the Computer 73
Using a Spreadsheet 73
Nested Anisotropic Model 77
The h-Scattergram 91
Summary 98
3
Linear Estimation 101
Kriging Equations 102
Terms in the Kriging System 105
Properties of Kriged Estimates 107
Examples Along Transects 112
Examples 117
Thickness of Clastic Section 117
Initial Potential of Upper Devonian Gas 120
Huron Shale Thickness 132
Summary 136
4
Multivariate Geostatistics 139
Coregionalization 140
Cokriging Equations 141
Details of Computations 143
Example: Synthetic Data 144
A Stratigraphic Application 150
Initial Potentials and Cumulative Production 159
Principal Components Analysis 165
Summary 167
5
Nonlinear Estimation 169
Methodology 171
Normalization 171
The Experimental Semivariogram 172
Coefficients of Hermite Expansion 172
Estimation 174
Normalization: Examples 175
Initial Potentials 175
Huron Shale Thickness 177
Contents Vll
Semivariograms of Normalized Data 180
Estimation 185
Summary and Parting Comments 192
6
From Indicator Variables to Nonparametric
Estimation 195
Nonparametric Estimation 200
Functions of the Estimates 203
Examples 204
Shale Thickness 204
Initial Potential in Barbour County 210
Initial Potential in Southwestern West Virginia 214
Probability Kriging 230
Summary 231
7
More Detail, Less Certainty: Conditional
Simulation 233
Nonconditional Simulation 234
Conditional Simulation 237
Case Studies 237
Initial Potentials in Lincoln County 237
Cumulative Gas Production 242
Related Methods 253
Multivariate Simulation 253
Conditional Indicator Simulation 255
Summary 256
Index 259
Series Editor's
Foreword
This is the sixth contribution to the Computer Methods in the Geosciences series
and it continues the tradition of being practical, germaine, and easy to read.
Michael Hohn in his presentation, Geostatistics and Petroleum Geology, nicely
compliments the other books in the series and brings to the readers some new
techniques by which to analyze their data. New approaches always result in new
ideas or enhancement of old ones.
The French School of Geostatistiques (Fontainebleau, France) was founded
and developed by Georges Matheron in response to problems in mining explo
ration and exploitation. This approach has been used successfully in that
industry since the mid-1960s, but only recently applied to similar problems in
petroleum. Likewise, these applications have been successful in this applied field
as well and here Hohn gives examples.
Standard subjects of the field of geostatistics are explored and discussed-the
semivariogram, kriging, cokriging, nonlinear and parametric estimation, and
conditional simulation. These may be unrecognizable terms to the readers now,
but upon completion of reading the book, they will be fimiliar ones. Each subject
is discussed in detail with appropriate and pertinent case studies, taken from the
author's own research or from the literature.
The author notes the book is for working geologists in the petroleum industry.
These geostatistical techniques can be used in exploring for or in developing
mineral resources, whether metals or oil and gas. Practical applications such as
mapping reserves, locating areas to drill, computing success ratios, outlining
regional trends, or siting well locations are covered in the seven chapters of the
ix
x Series Editor's Foreword
book. Each of these applications contain abundant illustrations and a step-by
step outline just of "how to".
Readers of this book probably will be interested in several others in the series,
but in particular Robinson's Computer Applications in Petroleum Geology, and
Jones-Hamilton-Johnson's Contouring Geologic Surfaces with the Computer. Other
readings on the subject can be obtained from the reference list at the end of
each chapter.
Thus, this is yet another book written by a geologist for geologists in the field
of mathematical geology. It should serve well as an introduction to the subject
of the geostatistics for the novice, reference to the practitioner, or source of ideas
for the researcher; likewise it could serve as a regular textbook, notes for a short
course, or reference work. For whatever purpose the book is used, it will prove
to be informative and valuable.
D. F. Merriam
Preface
Resource estimation plays a primary role in the decision to explore or develop
a hydrocarbon prospect. The explorationist needs to know favorable sites for
locating a well, likelihood of success, expected level of production, and the
average or cumulative production across an area such as a lease block.
From knowledge of reservoir size, shape, porosity, permeability, and other
variables, the petroleum geologist can estimate reserves. When detailed infor
mation about a reservoir may be unavailable, or the mode of hydrocarbon
storage and the mechanism of release are poorly understood, resource evalua
tion must depend upon observations made from wells near the reservoir.
Maps are important in estimating resources. While constructing the familiar
contour map, the geologist assumes that the mapped entity varies continuously
across the study area except in the presence of faults or other discontinuities.
Two wells in proximity to each other should have about the same likelihood of
success and level of production.
But is this assumption true? How far can wells be separated and still yield
similar productivity? Over what distance can one interpolate for a meaningful
contour map? Is sufficient well control available? In the past, the geologist lacked
the statistical tools for answering these questions.
Geostatistics enables the geologist to measure spatial autocorrelation and thus
evaluate the nature and quality of raw data. The method of kriging provides
unbiased estimates that incorporate directional trends. After contouring these
estimates, the geologist can judge the quality of the resulting map through a
xi
