[p.t] modeling and inversion methods for the interpretation of resistivity logging tool response

8/8/2019 [p.T] Modeling and Inversion Methods for the Interpretation of Resistivity Logging Tool Response

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Modeling and Inversion Methods

for the Interpretation of

Resistivity Logging Tool Response

Barbara Ina Anderson


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Modeling and Inversion Methods

for the Interpretation of

Resistivity Logging Tool Response

PROEFSCHRIFT

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.F. Wakker,voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 15 oktober 2001 om 13.30 uur

door

Barbara Ina ANDERSON

B. Sc., Western Connecticut State Universitygeboren te Danbury, Connecticut, USA


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Dit proefschrift is goedgekeurd door de promotoren:

Prof. dr. ir. H. BlokProf. dr. ir. J.T. Fokkema

Samenstelling promotiecommissie:

Rector Magnificus, voorzitterProf. dr. ir. H. Blok, Technische Universiteit Delft, promotorProf. dr. ir. J.T. Fokkema, Technische Universiteit Delft, promotorProf. dr. ir. P.M. van den Berg, Technische Universiteit DelftProf. ir. C.P.J.W. van Kruijsdijk Technische Universiteit DelftProf. dr. S. Luthi, Technische Universiteit DelftProf. dr. ir. C.P.A. Wapenaar, Technische Universiteit DelftDr. T.M. Habashy, Schlumberger-Doll Research, USA, guest

Published and distributed by: DUP Science

DUP Science is an imprint of:Delft University PressP.O. Box 982600 MG DelftThe NetherlandsTelephone: +31 15 27 85 678Telefax: +31 15 27 85 706E-mail: [email protected]

ISBN 90-407-2231-5

Copyright c 2001 by Schlumberger Technology Corporation

All rights reserved. No part of the material protected by this copyright notice may be repro-

duced or utilized in any form or by any means, electronic or mechanical, including photocopying,

recording or by any information storage and retrieval system, without written permission from the

publisher: Delft University Press

Printed in The Netherlands


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Contents

1 Introduction 11.1 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Introduction to well logging . . . . . . . . . . . . . . . . . . . 3

1.3 Computer modeling in log interpretation . . . . . . . . . . . . 7

1.4 Anisotropy in log interpretation . . . . . . . . . . . . . . . . . 15

1.5 Inversion in layered anisotropic media . . . . . . . . . . . . . 19

2 Electromagnetic relations for logging 27

2.1 Overview of logging environments . . . . . . . . . . . . . . . . 27

2.1.1 Borehole effect . . . . . . . . . . . . . . . . . . . . . . 29

2.1.2 Coaxial layers; invasion . . . . . . . . . . . . . . . . . 30

2.1.3 Thin beds (bed boundary discontinuities) . . . . . . . 32

2.1.4 Invaded thin beds . . . . . . . . . . . . . . . . . . . . 33

2.1.5 Dipping beds . . . . . . . . . . . . . . . . . . . . . . . 34

2.1.6 3D geometries; horizontal wells . . . . . . . . . . . . . 35

2.1.7 Anisotropy in layered media; laminated formations . . 36

2.2 Description of logging tool configurations . . . . . . . . . . . 38

2.3 Electromagnetic field equations and notation . . . . . . . . . 41

2.4 Time domain equations . . . . . . . . . . . . . . . . . . . . . 43

2.5 Frequency domain equations . . . . . . . . . . . . . . . . . . . 44

2.6 Anisotropic media; the conductivity tensor . . . . . . . . . . . 45


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vi CONTENTS

2.7 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 47

2.8 Transform for axisymmetric configurations . . . . . . . . . . . 49

3 Electrical well-logging measurements 51

3.1 What do resistivity tools measure . . . . . . . . . . . . . . 51

3.2 Induction tools . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 Two-coil sonde response . . . . . . . . . . . . . . . . . 56

3.2.2 Early induction tools; focused sondes . . . . . . . . 64

3.2.3 6FF40 and the Dual Induction tool; a standard is set . 70

3.2.4 Phasor processing and deconvolution . . . . . . . . . . 803.2.5 Array Induction Tool (AIT) . . . . . . . . . . . . . . . 87

3.2.6 Russian induction tools . . . . . . . . . . . . . . . . . 95

3.3 Propagation tools . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.3.1 2-MHz tools for logging while drilling . . . . . . . . . 100

3.3.2 Deep Propagation Tool (DPT) . . . . . . . . . . . . . 108

3.3.3 Electromagnetic Propagation Tool (EPT) . . . . . . . 112

3.4 Electrode (laterolog) tools . . . . . . . . . . . . . . . . . . . . 116

3.4.1 The Normal . . . . . . . . . . . . . . . . . . . . . . . . 117

3.4.2 The Lateral . . . . . . . . . . . . . . . . . . . . . . . . 124

3.4.3 Russian BKZ tools . . . . . . . . . . . . . . . . . . . . 127

3.4.4 Laterolog 7 (LL7) . . . . . . . . . . . . . . . . . . . . 130

3.4.5 Laterolog 3 (LL3) . . . . . . . . . . . . . . . . . . . . 136

3.4.6 Laterolog 8 (LL8) . . . . . . . . . . . . . . . . . . . . 139

3.4.7 The Dual Laterolog tool (DLT) . . . . . . . . . . . . . 143

3.4.8 The Spherically Focused Log (SFL) . . . . . . . . . . 153

3.4.9 High Resolution Laterolog Array (HRLA) . . . . . . . 160

3.5 Microresistivity tools . . . . . . . . . . . . . . . . . . . . . . . 166

3.5.1 The Microlog . . . . . . . . . . . . . . . . . . . . . . . 168

3.5.2 The MicroLaterolog . . . . . . . . . . . . . . . . . . . 170

3.5.3 The Proximity log . . . . . . . . . . . . . . . . . . . . 171


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CONTENTS vii

3.5.4 The MicroSpherically Focused Log (MSFL) . . . . . . 173

3.6 Imaging tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

3.6.1 Formation MicroScanner (FMS) . . . . . . . . . . . . 175

3.6.2 Azimuthal Resistivity Imager (ARI) . . . . . . . . . . 178

3.6.3 Resistivity-At-the-Bit tool (RAB) . . . . . . . . . . . 179

3.6.4 Oil-Base MicroImager tool (OBMI) . . . . . . . . . . . 180

3.7 Resistivity through casing . . . . . . . . . . . . . . . . . . . . 182

4 Modeling of tool response 185

4.1 Analytical methods . . . . . . . . . . . . . . . . . . . . . . . . 185

4.1.1 Dolls induction geometrical factor theory . . . . . . . 186

4.1.2 Induction skin effect in homogeneous media . . . . . . 190

4.1.3 Induction real axis, spectral integration . . . . . . . . 197

4.1.4 The induction Born response function . . . . . . . . . 205

4.1.5 Laterolog response . . . . . . . . . . . . . . . . . . . . 212

4.2 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . 214

4.2.1 The finite element method . . . . . . . . . . . . . . . . 216

4.2.2 The finite difference method . . . . . . . . . . . . . . . 225

4.3 Hybrid methods . . . . . . . . . . . . . . . . . . . . . . . . . 2334.3.1 Fast semi-analytic (mode matching) . . . . . . . . . . 233

4.3.2 With/without skin effect hybrid . . . . . . . . . . . . 244

4.4 Glossary of computer codes . . . . . . . . . . . . . . . . . . . 247

4.4.1 Induction codes . . . . . . . . . . . . . . . . . . . . . . 247

4.4.2 Laterolog codes . . . . . . . . . . . . . . . . . . . . . . 249

5 Using modeling in log interpretation 253

5.1 Relating resistivity logs to rock physics . . . . . . . . . . . . . 253

5.2 Early 1D plus 1D inversion efforts . . . . . . . . . . . . . . 257

5.2.1 Deconvolution and boosting . . . . . . . . . . . . . . . 258

5.2.2 Correction chartbooks and departure curves . . . . . . 263

5.3 A 2D iterative forward modeling case study . . . . . . . . . . 274


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viii CONTENTS

5.4 A least squares inversion example in thin beds . . . . . . . . 279

6 Parametric inversion 285

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

6.2 Forward modeling . . . . . . . . . . . . . . . . . . . . . . . . 289

6.3 2-MHz inversion in layered media . . . . . . . . . . . . . . . . 293

6.3.1 2-MHz tool response in anisotropic media . . . . . . . 293

6.3.2 The inversion algorithm . . . . . . . . . . . . . . . . . 295

6.3.3 2-MHz inversion results . . . . . . . . . . . . . . . . . 305

6.4 Triaxial inversion in layered media . . . . . . . . . . . . . . . 307

6.4.1 Triaxial tool response in some limiting cases . . . . . . 307

6.4.2 Triaxial inversion results . . . . . . . . . . . . . . . . . 318

6.5 Summary and future plans . . . . . . . . . . . . . . . . . . . . 326

Bibliography 333

Summary 361

Samenvatting 365

About the author 369

Acknowledgments 373

Index 375


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Chapter 1

Introduction

Summary: This chapter introduces the reader to the world of borehole well logging

from a historical perspective. In addition to describing the evolution of resistivity

measurements, nuclear and acoustic measurements are briefly summarized as well.

The important role that mathematical modeling has played in the design and in-

terpretation of resistivity measurements is illustrated with computed log examples.

The unique log interpretation problems created by resistivity anisotropy are traced

back to experiments performed by Conrad Schlumberger in the 1920s. Paramet-

ric inversion is proposed as a method for quantifying resistivity anisotropy fromborehole measurements.

1.1 Overview of the thesis

The purpose of this thesis is twofold:

1. To provide an overview of the use of mathematical modeling in resis-tivity log interpretation, and

2. To describe a new inversion method for determining formation electri-

cal properties in anisotropic layered media.

When computationally efficient 2D modeling codes were first developedin the 1980s, the author of this thesis began to experiment with incorporat-ing forward modeling directly in the log interpretation process [14, 22]. One


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2 CHAPTER 1. INTRODUCTION

of the first practical uses of modeling was to improve estimates of reserves in

dipping, thinly-bedded reservoirs [117, 19]. When these improved estimateswere validated by production history, the use of modeling exploded.

Today, forward modeling and inversion methods based on forward model-ing are routinely used to improve the accuracy of log interpretation and evento steer the drilling of directional wells in complex reservoirs [169]. Modelingaccurately accounts for the multidimensional and often nonlinear aspects oftool physicsaspects that were previously corrected on a point-by-pointbasis by using 1D charts or algorithms.

The foremost reason for opening this thesis with a modeling overview is torespond to requests to take the mystery out of modeling, that is, to bridge

the gap between log interpretation and computational physics. There aremany excellent books that explain the geophysical and petrophysical aspectsof log interpretation [253, 86, 147, 208, 111]. There are also many excellenttexts describing methods for solving problems in electromagnetic theory [161,241, 68, 153, 243]. However, log analysts and petroleum engineers oftencomplain that there are no texts that explain how to go from Maxwellsequations to computing resistivity tool response in understandable terms.The objective of the overview is to fulfill this need in sufficient detail foran interested reader to construct elementary codes for computing syntheticresistivity tool response.

Another important reason for including and overview is to examine in

detail the various environmental effects that can complicate inversion. Theseinclude effects of the borehole, invaded zone, shoulder beds and formationdip. Often some nearly linear effects, such as those of the borehole, can beaddressed separately prior to the inversion process. This prior treatmentcan simplify the inversion for formation anisotropy.

The overview begins with a description of typical logging environmentsand a review of the fundamental electromagnetic field equations and rela-tions for logging tools. There will be an examination of the electrical loggingtools under consideration (e.g., laterologs, induction, 2-MHz for logging whiledrilling, propagation tools), including a description of the measurement char-acteristics and volumes of investigation of each tool. Next, there will be areview of forward modeling methods and computer codes commonly usedto simulate logging tool response. The evolution of inversion in log inter-pretation will be illustrated with several case studies demonstrating simpleinversion methods, such as the use of departure curve charts and iterative


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1.2. INTRODUCTION TO WELL LOGGING 3

forward modeling of multiple tools with user intervention.

These studies eventually lead to the parametric inversion of resistivitytool response in layered anisotropic media. The problem will encompass boththe forward model development and the inversion. In the forward model, thegeneral problem of anisotropy in planar layered media will be considered inthe frequency domain. The axis of anisotropy is assumed to be oriented ina general direction which varies from the direction of the tool axis. Theanisotropy is assumed to occur in conductivity, permittivity and magneticpermeability of the layered-earth formation. In the inversion, the formationwill be described by layers whose parameters are to be retrieved through anonlinear optimization scheme.

The potential applications of the inversion will cover tool response indipping beds or deviated wells. Environments where invasion contributessignificantly will not be addressed.

1.2 Introduction to well logging

Electrical well logging was the first logging method used below ground inboreholes by the petroleum industry. Of all the rock parameters measuredby logging tools, the electrical resistivity is of particular importance. Re-sistivity measurements are essential for determining the relative amount of

hydrocarbons in a formation. In simplest terms, high resistivity indicates thepossible presence of oil or gas in rock pores, since hydrocarbons are insula-tors. On the other hand, low resistivity indicates water, the other fluid thatmay be present. The specific formulas that are used to determine the exactamounts of hydrocarbons and water present from resistivity measurementsare given in Section 5.1.

Borehole logging was an outgrowth of prior techniques for exploring theunderground from the surface by means of electrical measurements. Thefirst electrical surface prospecting experiments were carried out in 1912 byConrad Schlumberger. These experiments consisted of sending an electricalcurrent between two metallic rods driven into the earth and drawing a map

of lines of constant potential observed at the surface. The shape of these linesindicated the nature and geometrical configuration of subsurface geologicalbodies permeated by the electric field (equipotential lines elongated whenan adjacent resistive body was approached [5]). From 1912 until World WarI measurement techniques were progressively improved, and in 1920 Conrad


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Figure 1.1: The first log: points plotted on graph paper by Henri Doll andannotated with a description of the formation layers.

Schlumberger and his brother Marcel founded a surface prospecting companybearing the family name.

The first electrical log in a borehole was recorded on September 5, 1927,by the Schlumberger brothers and Henri Doll (Conrads son-in-law and thecompanys chief theoretician) in the Pechelbronn field in Alsace-Lorraine. Aportion of this log is reproduced in Figure 1.1 [220]. The electrical resistivityof the rock formation cut by the borehole was recorded at approximately onemeter depth intervals and plotted by hand.

The concept of apparent resistivity allowed the data to be scaled inabsolute units that are independent of the electrode configuration and theintensity of the current. This generality was the major reason for the com-mercial success of the Schlumberger companys logging methods. The mea-surement configuration is shown in Figure 1.2 [238]. In early literature this


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1.2. INTRODUCTION TO WELL LOGGING 5

Figure 1.2: Electrode configuration of the first electrical well logging tool.

is referred to as a lateral or inverse sonde.

Three electrodes, A, M and N were lowered into the borehole, each at theend of an insulated conductor. The current (I) emitted by A flows throughthe mud inside the borehole and spreads across the formation as it returns

to B near the surface. The difference in potential between M and N (V) istransmitted to the surface and measured. The apparent resistivity (Ra) isevaluated using the formula

Ra = K V/I, (1.1)where K is a tool constant determined by the geometry of the electrodesystem AMNB. Ra characterizes resistivity of the formation layer at theM N level.

Carrotage electrique was the name that the Schlumberger brothersgave to borehole electrical prospecting. This translates from the French aselectrical coring, meaning that sensors lowered into a well on a cable were

a replacement for the time-consuming and expensive cutting of cylindersof rock (cores) as wells were drilled. Until that time coring was the mainmethod of learning about the formations that the drill bit had penetrated. Inthe 1930s the term electrical coring was replaced by electrical logging.The word log referred to the strip of paper on which the curves were


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plotted, borrowing from nautical science where it denotes the recording of

the position of a ship in terms of time.

The petroleum industry quickly recognized the usefulness of resistivitylogging for the identification of potential hydrocarbon-bearing zones and forcorrelation purposes. In 1929 electrical resistivity logging was introducedon a commercial basis in the United States, Venezuela and Russia. In 1931the spontaneous potential (SP) measurement was included along with theresistivity curve on the electrical log, after it was discovered (by chance) thatthe M N circuit could measure natural potentials of electrochemical originwhich indicated permeable layers.

In 1931, the Schlumberger brothers also perfected a method of continuous

recording and the first pen recorder replaced the point-by-point system. Theautomatic photographic film recorder (single galvanometer) was introducedin 1936, eliminating tedious hand-copying of logs. By that time, the log-ging suite consisted of the SP curve, the lateral and long and short normals(normal tools have the N electrode at or near the surface). This combina-tion dominated logging until the 1950s, when focused electrode tools andinduction tools (described in Chapter 3) came into use.

Experimental dipmeter tools were used in the 1930s to help identifymajor geologic structures. They were greatly improved during the 1940s andbecame the principal logging tool for describing internal lithologic features.Dipmeter tools consist of four evenly spaced pads that are pressed against

the borehole wall. Each pad contains a short micro-resistivity device. Thefour micro-resistivity curves are correlated to find the difference in depthbetween bedding markers around the borehole, which yields the magnitudeand azimuth of formation dip [225].

The major uses of electrical logging tools were to infer geologic structureand to determine the nature of fluids in sedimentary rocks from the measuredresistivity. Over the years, other types of tools were introduced to determineadditional physical properties, such as rock density, porosity, radioactivityand sound transmission.

Nuclear measurements were developed after World War II. The gamma

ray and neutron tools were the first borehole measurements of radioactiveproperties (they were also the first tools to use downhole electronics). Unlikeresistivity tools, nuclear tools are able to log formations through steel casing.

The basic gamma ray (GR) log was introduced in the 1950s as a per-meability indicator. It measures the natural formation radioactivity which


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1.3. COMPUTER MODELING IN LOG INTERPRETATION 7

reflects shale content.

In the 1960s the compensated neutron tool (CNL) gained acceptance asa porosity measurement, inferring porosity from the energy loss of emittedneutrons (compensation refers to the use of two sets of antennas whose re-sponses are averaged together to cancel errors from sonde tilt and hole sizechanges). The compensated density tool (CDL) was also introduced in the1960s. It infers bulk density, a property primarily dependent on porosity,from the attenuation of emitted gamma rays.

Since the 1930s, geophones had been lowered into oil wells on loggingcables to measure long-interval acoustic travel times from sound sourcesat the surface. In the 1950s the borehole sonic log gained acceptance as

a reliable porosity log because its travel time measurement is essentiallyindependent of fluid saturation.

It may seem redundant to have three porosity tools in common use whenonly one porosity value is needed. However, the three tools respond not onlyto porosity but also to the type of the rock matrix and the nature of thefluid filling the pore space. When the rock and fluid types are unknown, allthree measurements are needed to sort out parameters [86].

From the 1950s through the 1980s, a typical logging suite consisted ofa focused resistivity tool (laterolog or induction), SP, a neutron/density logand a sonic log. The advent of digital signal processing and transmission inthe 1980s lead to the modernization of existing tools and the eventual intro-duction of array induction, laterolog and sonic tools in the 1990s (modernresistivity tools are described in detail in Chapter 3). Other currently evolv-ing measurements include nuclear magnetic resonance, nuclear spectrometryand electrical and acoustic imaging.

The 1990s also saw the growth of logging while drilling (LWD), that isthe placement of electrical, nuclear and acoustic tools on the drill string torecord measurements just behind the drill bit as it cuts through the forma-tion. This early time data is used along with information from explorationwells to steer drilling in the direction of hydrocarbon-bearing zones.

1.3 History of computer modeling in log interpretation

Technological progress in tool development was accompanied by the evolu-tion of the new discipline of log interpretation. Very few of the petrophys-


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Figure 1.3: Sample chart for interpreting an early lateral tool. Lateral ap-parent resistivity is plotted in ordinate and distance relative to bed thicknessis plotted in abscissa. Four ratios of bed thickness (e) to tool spacing (L)are shown. The resistivity of the central bed is 25 ohm-m and the resistivityof the surrounding beds is 5 ohm-m. Note that the shapes of the logs areconsiderably different for the four bed thicknesses. Also note the large dif-

ference between the log resistivity and the actual resistivity in each of thecentral beds. In addition, note the large overshoot above 25 ohm-m on thetop left log.


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ical properties needed to evaluate the amount of hydrocarbons in place in

a reservoir can be measured directly. The most important petrophysicalproperties used in formation evaluation are porosity (pore volume per unitvolume of formation), water and hydrocarbon saturation (fraction of porevolume occupied by fluid) and permeability (ease with which fluids flow).Log interpretation applies known physical relationships to the parametersmeasured by logging tools (resistivity, bulk density, travel time, radioactiv-ity, etc.) in order to obtain a quantitative evaluation of the above mentionedpetrophysical properties.

Mathematical modeling has been intimately associated with electrical loginterpretation since the time that the first log was run for two basic reasons:(1) electrical tools survey large volumes of formation making it necessary toquantify parasitic effects caused by regions adjacent to beds of interest, and(2) electrical tool response is highly nonlinear.

The Schlumberger brothers used small electrodes in saltwater baths toperform early experimental modeling. Soon afterwards, mathematiciansfrom the Ecole des Mines in Paris were enlisted to solve the problem of apoint electrode tool logging perpendicular to vertical layers using Maxwellsimage theory [5]. This solution served as a basis for calculating numeroussets of theoretical departure curves for normal and lateral tools which werepublished in booklets throughout the 1930s (departure, in this case, refersto the difference in resistivity between tool response in a thin bed and the

unperturbed response in an infinitely thick bed). Interpretation consistedof superimposing a transparent chart over a log and finding the theoreticalcurve that gave the best coincidence. However, this method was only de-pendable for at most three layers. An example of an early chart is shown inFigure 1.3 [183].

Chartbooks of theoretical departure curves were routinely used to inter-pret resistivity logs from the 1930s through the 1970s (some of the mostcommonly used charts are described in Chapter 5). After induction toolswere introduced in the 1950s, charts were produced to correct these toolsseparately for both the effect of shoulder beds and invasion of the boreholemud into the formation [215]. These charts were generated using computer

programs that modeled 1D analytical solutions of Maxwells equations [194].2D interpretation was achieved by applying 1D corrections in sequence; alayered media correction for shoulder bed effect was performed first, followedby a cylindrical media correction for invasion effect.


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Figure 1.4: The resistor network.

In 1950 a resistor network [131] was introduced for simulating electrodetool response in more realistic 2D logging environments consisting of a bore-hole and multiple thin beds with invaded zones. The network, shown inFigure 1.4, consisted of tens of thousands of resistors and was in effect ananalog computer. Charts generated by the resistor network soon replacedthe earlier layered media charts, which suffered from inaccuracy caused byignoring borehole effect.

Starting in the late 1960s, work was begun on 2D axisymmetric finiteelement and finite difference codes for modeling both electrode tool and in-duction response [175, 182, 176]. Although these numerical methods hadbeen successful for small-scale problems in the power industry, they proved

impractical for simulating resistivity tool response at that time because ex-isting computer memory and speed were insufficient for modeling electriccurrents that penetrated tens of meters from the borehole.

Large improvements in computing capabilities in the late 1970s reducedthe time required to compute finite element and finite difference simulated


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logs from weeks to hours. In 1980, the resistor network was retired to

the Schlumberger museum in France and replaced by a 2D finite differencecode [119]. Shortly after this, a 2D finite element code for modeling inductiontool response came into common use [62].

These codes were at first used to aid in tool design and to generate inter-pretation charts. However, two changes occurred in the petroleum industryin the 1980s which led to computer modeling assuming a more active rolein log interpretation. The first was the growth in economic importance ofthinly bedded reservoirs. Resistivity tools of that time were designed tobe relatively free of effects of adjacent layers in beds thicker than six feet.After the era of easy oil was over, one-to-two foot beds needed to be in-terpreted. The application of 1D plus 1D chartbook corrections proved tobe highly inaccurate in these thin beds.

The second change was the advent of horizontal drilling. All publishedcharts had been generated for vertical wells, with tools logging perpendicularto bed boundaries. These charts no longer applied when tools logged parallelto boundaries in horizontal wells. As more and more charts became obsolete,it became clear that another approach to interpretation was needed.

Fortunately the 1980s also saw a continued evolution in computer power.Personal computers were introduced that could run 2D modeling codes whichpreviously required large mainframe parallel machines. Continued advancesin numerical techniques [70] made it possible to compute simulated logs

in minutes instead of hours. This set the stage for the integration of toolmodeling with log interpretation.

In the 1980s several papers were written by the author of this the-sis [14, 15, 16] which demonstrated the power of iterative forward modelingin log interpretation for the first time. These papers showed how forwardmodeling could be applied to accurately determine formation resistivity incomplex formations that were beyond the scope of chartbook interpreta-tion. Soon after this, a user-friendly electromagnetic modeling package calledELMOD [19] was made available to Schlumberger log analysts for use onpersonal computers at regional log interpretation centers.

The systematic application of forward modeling in log interpretation isillustrated in the flowchart in Figure 1.5. Estimates of formation resistivitiesand bed boundary dimensions are obtained from either visual inspection orcomputer algorithms (i.e., bed boundaries from log inflection points and re-sistivities from maximum/minimum values). These parameters are used to


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Figure 1.5: Flowchart illustrating the use of forward modeling in log inter-pretation.

set up an initial formation model for a given section of log. The modelingcode is then run to simulate tool response in this formation in an attempt togenerate a computed log that overlays the field log. If the two logs disagree,then the formation model is refined, either by systematically varying param-eters or by incorporating additional information from other logs or cores.The process is repeated until reasonable agreement is achieved. The finalformation model provides the resistivity values in each layer. Even thoughsolutions obtained in this way are not necessarily unique, modeling can nev-ertheless serve to eliminate impossible scenarios and validate the most likelyinterpretation.

The first successful use of ELMOD was in improving the determinationof hydrocarbon reserves from induction logs in deviated wells in the NorthSea. A series of ELMOD runs was used to find a squared resistivity profilethat would reproduce an induction field log. The steps involved in findingthe solution are illustrated in Figure 1.6 [19], using a section of an actual

field log.Dipmeter logs in this well indicated that the combined hole deviation and

formation dip gave a total dip of 38. In order to determine the characteristicresponse of induction tools at that dip angle, the log analysts involved firstconsulted published examples of dip effect [32, 143]. Inflection points on


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Figure 1.6: Three iterative forward modeling runs are used to find the for-mation resistivity in a North Sea case study.


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the induction curve were used as initial bed boundary locations. Using

this information, a trial formation was set up and induction response wasmodeled.

In Figure 1.6, the log on the left (simulation 1) shows the first assumedformation resistivity profile (square Rt), along with the field log and thecomputed log (IDPH is the deep Phasor induction tool). Although the twologs agree fairly well in the center of most beds, the shape of the computedlog isnt correct near the bed boundaries. The second model in the middle(simulation 2) adjusts bed boundary locations and fine-tunes some resistivityvalues, making the computed log agree more nearly with the field log. Thefinal model on the right (simulation 3) adjusts for overcompensation andfurther refines the shape of some beds. The square formation now gives acomputed log that agrees very closely with the field log.

Note the difference in resistivity level between the field log and the finalsquare formation resistivity in the two resistive pay zones (40 ohm-m versus200 ohm-m at 1040 feet, and 60 ohm-m versus 150 ohm-m at 1100 feet).This difference is a result of dip effect. If the resistivity read by the toolwas used in reserve calculations, the amount of hydrocarbons in place wouldbe severely underestimated. The log analysts involved in this study citedan additional benefit of modeling: it gave them a better insight into toolphysics which they could apply to future interpretations.

The iterative forward modeling process could of course be replaced by in-

version. Indeed, in the 1980s several authors proposed inverse solutions [170,106, 115, 138, 71, 267] for resistivity logging. However, computers at thattime were still too slow to make inverse solutions practical. In addition,inverse solutions for the tools of the 1980s were plagued by nonuniquenessto an even greater extent than iterative forward modeling solutions.

The problem of nonuniqueness is illustrated by the two logs in Figure 1.7.The log in the 2 ohm-m bed on the left is identical to the log in the alternat-ing 1100 ohm-m laminated zone on the right. Nonuniqueness caused by atools poor vertical resolution, such as in this case, is not a major problem initerative forward modeling. During the iterative process, formation modelscan be severely constrained by local knowledge from cores or higher resolu-tion logs (such as nuclear or imaging logs). Commercial inversion softwarefor resistivity logging is not implemented to access non-resistivity informa-tion, although this problem is currently receiving considerable attention.

The introduction of high resolution array tools with multiple depths of


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1.4. ANISOTROPY IN LOG INTERPRETATION 15

Figure 1.7: Identical 6FF40 logs generated by two different formation modelsillustrating the problem of nonuniqueness in resistivity log inversion.

investigation in the 1990s has made reliable inverse solutions possible. Re-cently, maximum entropy log inversion (MERLIN) [49] was developed for the

Schlumberger AIT Array Induction tool to provide more accurate Rt and in-vasion interpretation in highly deviated wells. AIT response to invasion invertical wells has also been inverted to generate fractional flow logs whichdisplay saturations [207]. For the Schlumberger HRLA Array Laterolog, 2Dimaging inversion [237] is used to obtain Rt and the invasion profile. Baker-Atlas has also developed and documented inversion algorithms for both theirarray induction [248] and array laterolog [142] tools.

1.4 Anisotropy in log interpretation

Anisotropy (the variation of properties with direction) is not uncommon insedimentary strata. Many solid particles have flat or elongated shapes thatare usually oriented parallel to the plane of deposition as shown in Fig-ure 1.8 [21]. This results in a pore structure that allows electric current toflow more easily parallel to the bedding plane than perpendicular to it [112]


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Figure 1.8: Scanning electron photomicrograph showing aligned grains in alimestone sample.

(the conducting medium is the water saturating the rock pores). Sedimen-tation of this type produces transversely isotropic (TI) anisotropy, that is,the horizontal resistivity (Rh) is the same in every direction in the horizon-tal bedding plane, while the vertical resistivity (Rv) normal to the beddingplane is different. Particle shape anisotropy is most commonly found inshales, and may also occur in sands and carbonates.

Although we are concerned with electrical anisotropy, it is important tonote that the same sedimentary processes that cause electrical anisotropycan result in anisotropy in other physical parameters. Permeability anisot-ropy is particularly important in determining hydrocarbon flow in reservoirs.Currently work is being carried out to find relationships between electricalanisotropy and permeability anisotropy [254, 159].

Anisotropy depends very much on scale. In addition to microscopic an-isotropy occurring at the particle scale, formations consisting of a series ofisotropic beds of different lithology (such as sequences of sand and shales)

also behave anisotropically if a logging tool is significantly longer than thebed thickness. This is referred to as macroscopic anisotropy. The two logsin Figure 1.7 are identical because the eight foot induction tool averages theone foot resistive and conductive layers (on the right), reading an effectiveresistivity which is equivalent to the resistivity in the thick bed (on the left).


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1.4. ANISOTROPY IN LOG INTERPRETATION 17

When logging perpendicular to bed boundaries in cases such as this, resistiv-

ity tools read the effective horizontal resistivity, Rh, which can be calculatedfrom the volume average of the layer conductivities (inverse resistivities),

1

Rh= Vsand 1

Rsand+ Vshale 1

Rshale, (1.2)

where resistivities are expressed in ohm-m and Vsand and Vshale are the bulkvolume fractions (percentages) distributed throughout the layered region(layers are all assumed to be approximately uniform in thickness). Theeffective vertical resistivity, Rv, can be calculated in a similar manner fromthe volume average of the layer resistivities,

Rv = Vsand Rsand + Vshale Rshale. (1.3)

As early as 1920, Conrad Schlumberger recognized that anisotropy af-fected surface prospecting measurements [223]. In 1932, Maillet and Doll [181]presented a method for interpreting surface potential measurements in aniso-tropic formations. They showed that a TI anisotropic medium could berescaled to an isotropic medium using the anisotropy coefficient , definedas

=

Rv/Rh. (1.4)

The isotropic medium was assigned an effective resistivity (geometric mean)

denoted as R, withR =

Rv Rh. (1.5)

These results were used to design an experimental electromagnetic surfaceprospecting device that determined the direction of formation dip from mea-surements of the horizontal and vertical components of the magnetic field [5].

For both electrode and induction tools, the apparent resistivity (Ra) ina TI anisotropic medium can be calculated using the approximation [193]

Ra = R/

sin2 + 2 cos2 , (1.6)

where is the angle between the tool axis and vertical. For = /2(surface prospecting or horizontal wells), Ra = R. For = 0 (verticalwells), Ra = Rh. Thus the vertical resistivity cannot be detected at all byconventional resistivity logging tools in vertical wells. This is sometimesreferred to as the paradox of anisotropy.


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Although both electrode and induction tool response is theoretically the

same in homogeneous anisotropic media, in the 1950s it was noticed that16 inch Normal logs sometimes read higher resistivity values than inductionlogs in shales. This prompted Kunz and Moran [165] to investigate boreholeeffect in anisotropic formations. Adding a borehole filled with conductivemud to the vertical well model, they showed that Rv can affect electrodetool response since current has a considerable vertical component as it travelsbetween the source and the return.

Twenty years later, this work was extended by Moran and Gianzero [193]to model both induction and electrode tool response to dipping beddingplanes (with no borehole). In the same paper, they proposed a technique formeasuring anisotropy using a combination of horizontal and vertical coils,since vertically oriented coils are sensitive to Rv. However, they concludedthat borehole and bed boundary effects would make the method impractical.In a later paper [112], the same authors proposed a sidewall pad device toovercome borehole effect.

When laterolog tools were introduced, it was assumed that they measuredRh with negligible influence of Rv since these tools use bucking currents toforce the survey current laterally into the formation (see a yet unnumberedfigure in Chapter 3 showing current lines). However, discrepancies betweeninduction and laterolog measurements were still noted in shales and also inlaminated sandshale sequences. Chemali, et al. [64], showed that laterologs

still responded appreciably to Rv, although to a lesser degree than unfocusedelectrode tools. They generated charts for evaluating from differences be-tween laterolog and induction logs in dipping and horizontal beds. However,the method is seldom used because in most cases the difference is so smallthat it is less than the precision of the measurements.

Anisotropy can also enter into the interpretation of ULSEL logs. TheULSEL (Ultra Long Spaced Electrical Logging) tool was developed in the1960s [214]. It is used to locate distant resistive anomalies such as salt domeswhich act as traps for hydrocarbons. The ULSEL tool consists of four to sixlong normal arrays with spacings ranging from 75 to 2400 feet. The depthof investigation of ULSEL is approximately 2000 feet from the wellbore.

Accurate location of a salt dome (normally to the side of a well) with suchlong arrays depends on knowing both Rh and Rv. Since no measurement ofRv is available, an induction or laterolog log in the same well is used to set upa layered model of the formation. The theoretical ULSEL response is thencomputed in this formation. The presence of a lateral salt dome is indicated


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1.5. INVERSION IN LAYERED ANISOTROPIC MEDIA 19

when the ratio of the actual log (with salt dome) to the computed log (no

salt dome) is significantly greater than one. A comparison of differencesbetween the ratios of the various normals indicates the distance to the saltdome.

In the early 1990s ULSEL started to be used in large-scale reservoirdescription [197], and ULSEL interpretation was updated. Borehole seismicmeasurements, dipmeter logs and modeling codes including anisotropy nowhelp ULSEL predict distance and direction to any resistive or conductiveanomaly more accurately.

1.5 Parametric inversion in layered anisotropic media

From the 1920s through the 1980s, anisotropy was regarded as a secondaryeffect on resistivity logs. Even though papers were written describing themathematical modeling of anisotropy and occurrences of anisotropy wereflagged on logs, anisotropy effect was rarely included in routine log inter-pretation. Because most wells drilled up to the mid-1980s were vertical oronly slightly deviated, resistivity tool sensitivity to Rv was negligible andthe effect of anisotropy was masked. Therefore modeling and inversion toevaluate parasitic effects on beds of interest from adjacent zones (borehole,neighboring beds, invasion) received primary attention.

However, the increased use of horizontal drilling in the late 1980s andthe subsequent introduction of 2 MHz LWD resistivity tools revealed thatanisotropy could not be ignored in horizontal well interpretation. In fact,anisotropy effect was often surprisingly larger than shoulder bed or invasioneffects in horizontal wells.

The interpretation of horizontal well data is a multi-step process. Priorto drilling a horizontal well, potential hydrocarbon-bearing zones are firstlocated using vertical exploration wells. Then a horizontal well is drilledtoward a target bed, with marker beds used to maintain the wellbore tra-

jectory. Resistivity logs recorded behind the bit are compared to logs fromthe exploration wells to identify the marker beds. Computer modeling of

predicted resistivity tool response at different well deviation angles (calledgeosteering [9]) is used to modify the well path as needed. After a horizontalwell penetrates a hydrocarbon-bearing bed, drillers attempt to keep it insidethe bed for as long as possible. This procedure allows the well to drain alarge area, making a horizontal well more cost effective than several vertical


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Figure 1.9: Wireline induction (left) and 2-MHz CDR (right) response toanisotropy for Rv/Rh = 10 with Rh = 10 ohm-m.

wells.

When comparing resistivity logs in a horizontal well to logs from a verti-cal exploration well, it was noticed that the resistivity values often differed inshales and in laminated zones. This made identification of beds ambiguous,posing a problem in steering a horizontal well toward a target bed. After acloser examination of all available logs, cores and modeling, these differenceswere attributed to anisotropy for the first time in 1991 [169].

Figure 1.9 illustrates typical differences between resistivity tool readingsin vertical wells (0) and horizontal wells (90) caused by anisotropy. At0 dip, both the dual induction and Compensated Dual Resistivity (CDR)tools accurately read Rh. As the dip (or deviation) angle increases, the deepand medium induction curves both increase in the direction of Rv with littleseparation between them. The CDR curves also increase in the directionof Rv, with the phase shift resistivity reading higher than the attenuation

resistivity (this curve order is also characteristic of CDR response for valuesof Rv/Rh other than 10).

The induction and CDR tools both generate azimuthally polarized elec-tric fields which induce current loops that are tilted with respect to thetransverse anisotropy. These tilted current loops sense a weighted average


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of Rv and Rh which depends on dip angle. The response for induction tools

can be approximated from Equation (1.6). The low frequency (20-kHz) in-duction response is fairly linear and not strongly sensitive to anisotropy. Incontrast, extensive modeling and analysis of the higher frequency (2-MHz)CDR response by Luling, et al. [180], using the approach of Moran andGianzero [193], has demonstrated that radiation effects control the phaseshift measurement more strongly than the attenuation measurement. Thusseparation between 2-MHz phase shift and attenuation logs provide a goodindication of anisotropy (in the absence of invasion and shoulder bed effect),with sufficient resolution for inversion.

Resistivity tool sensitivity to Rv revealed by horizontal well interpreta-tion prompted a reassessment of the phenomenon known as low resistivitypay [59], which in turn led to proposals for tools that could measure Rvdirectly. In some reservoirs, particularly in the Gulf of Mexico, hydrocar-bons are produced from vertical wells in zones with resistivities between 0.5to 5 ohm-m, values usually associated with fresh water production. Withsuch low resistivities, these zones were often bypassed. However, high reso-lution resistivity imaging tools introduced after the late 1980s (such as theFormation MicroScanner and the LWD Resistivity-At-the-Bit tool) revealedthat many of these low resistivity zones consisted of laminated conductiveshales and resistive oil-bearing sands. The conductive shales were loweringthe average resistivity read by the induction tools. Occasionally these reser-voirs were penetrated by horizontal wells, and resistivity tools read higherthan in the vertical wells, confirming anisotropy. The effective resistivity inhorizontal wells was influenced more by Rv, which was higher and nearer tovalues normally expected in hydrocarbon-bearing zones.

Naturally, this generated interest in designing a tool that could measureRv in vertical exploration wells so that these productive zones would notbe bypassed. Calculations of vertical coil response in homogeneous aniso-tropic media [193] have demonstrated that a transverse magnetic dipole tool(TMD) is moderately sensitive to Rv in vertical wells. Unfortunately, morerecent calculations [201] have shown that TMD antennas are extremely sen-sitive to borehole effect. Methods are currently being investigated to cancel

TMD borehole effect, either by means of hardware or software. With bore-hole effect removed, layered media inversion algorithms are more accurateand easier to implement. Triaxial antennas, which provide more informationfor inversion, are also being investigated using a 3D anisotropic media finitedifference code [82].


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Figure 1.10: CDR response at 0 dip (left) and 80 dip (right) as the toollogs an isotropic bed above an anisotropic bed.

Historically, the first method used to solve for Rh and Rv in horizontalwells was iterative forward modeling using a laminated formation model [20],

both for modeling laminations and to approximate bulk anisotropy usingEquation (1.2) and Equation (1.3). Subsequently, a code was written tomodel induction and CDR tool response in anisotropic layered media [137],eliminating the tedious task of setting up a lamination model. Examples oftypical CDR synthetic logs are shown in Figure 1.10. The log on the rightin Figure 1.10 illustrates separations between phase shift and attenuationcurves that are typically seen in anisotropic media in highly deviated wells.The log on the left shows the insensitivity of the CDR tool to anisotropy ina vertical well in the same formation.

In the early 1980s, software to invert CDR response for Rh and Rv basedon the homogeneous anisotropic media solution of Moran and Gianzero [193]

was implemented for commercial use by Rosthal [211]. Results obtained byapplying this inversion to the 80 log in Figure 1.10 are shown in Figure 1.11on the left. Since the log input to the inversion was generated by a layeredmedium code [137], it is free of noise. The known information used in thehomogeneous medium inversion is the relative dip angle and the apparent


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Figure 1.11: Inversion for Rh and Rv for the 80 log of Figure 1.10. Results

based on the homogeneous medium solution are on the left and parametricinversion results are on the right.

phase shift and attenuation resistivities. The closed form analytical solution

for tool response in homogeneous anisotropic media is solved iteratively bya Newton-Raphson algorithm. An initial guess for Rh and Rv is obtainedfrom the log apparent resistivities and used to compute the correspondingphase shift and attenuation resistivities at the given dip angle. The iterationscheme uses the computed resistivities and their gradient with respect tochanges in Rh and Rv to obtain the next estimate. Iteration continuesuntil a solution is found to a specified accuracy or a maximum number ofiterations is exceeded. Typically about five iterations are required to reachconvergence.

Often a solution does not exist or it is physically unrealistic. Many otherenvironmental effects exist (invasion, borehole effect, response to dielectric

rock properties, shoulder bed effect) that cause separations between phaseshift and attenuation resistivity curves similar to those caused by anisot-ropy. Note that in Figure 1.11 (left) the solution for Rh and Rv in theanisotropic bed is only correct at distances greater than eight feet below thebed boundary. In this case, the homogeneous medium inversion cannot ac-


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curately account for shoulder bed effect and the polarization horn [20] that

occurs at bed boundaries at high dip angles. The height of a polarizationhorn depends on resistivity contrast and horns are a common occurrencenear resistive hydrocarbon-bearing zones. In fact, horns are often used inGeosteering as an indication that the well path has crossed into a target bed,so they must be accurately taken into account in the model.

Parametric inversion based on a layered-earth model provides a means ofaccounting for shoulder bed effect and polarization horns more accurately.Results obtained using parametric inversion in the same 80 formation areshown on the right in Figure 1.11, and will be described in greater detail inChapter 6. In this case, the known information used in the inversion is therelative dip angle, the bed boundary location obtained from a boundary de-tection algorithm and the apparent phase shift and attenuation resistivities.It is assumed that the bed boundary location is known within an accuracyof 2 inches. Errors greater than 2 inches will degrade the inversion.

Triaxial measurements are proposed as a means of overcoming this dif-ficulty. Triaxial measurements have sufficient sensitivity to anisotropy todirectly solve for the bed boundary locations and dip angle, in addition toRh and Rv.

The general geometry considered in this thesis consists of multiple, dip-ping anisotropic thin beds. Borehole effect is not taken into considerationbecause it is fairly linear and can be decoupled from the problem (com-

mercial software exists for pre-processing resistivity tool response to correctfor borehole effect). Invasion is also not considered here because it is nor-mally shallow at early times during logging while drilling, the area whereanisotropy interpretation is of most interest.

The objective is to invert for the horizontal and vertical resistivitieswithin each bed from the apparent resistivity log. Two cases are consid-ered. For the CDR inversion it is assumed that a fixed deviation angle canbe obtained from a dipmeter or imaging log. Fixed bed boundary locationsare obtained from inflection points on logs for small dip angles, or from peakvalues of polarization horns for large dip angles. For the triaxial inversion,the bed boundary locations and dip angle are not fixed, but are includedin inversion solution. In both cases, the initial guesses for Rh and Rv areobtained from measured center-bed resistivity readings.

The inversion algorithm is an iterative approach based on the Gauss-Newton method that employs a quadratic model of the cost function. The


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cost function is defined as the square of the sum of the relative residual er-

rors given by the difference between the log data and the estimated responsenormalized to the log data. The step length is adjusted by line search to suf-ficiently decrease mismatch between measured and predicted responses aftereach iteration. The method is based on constrained minimization where up-per and lower bounds are imposed on the inverted parameters. The forwardmodel is generated from the code ANISBEDS [137] which is an AC modelfor arbitrarily oriented point dipoles. The same general parametric inver-sion method has been applied to laterolog tools in isotropic invaded beds invertical wells by Habashy, et al. [136].


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Chapter 2

Basic electromagnetic field relations for

logging tools

Summary: This chapter relates Maxwells equations to resistivity tool antenna

configurations and the borehole logging environment. Basic electromagnetic con-

cepts such as notation, boundary conditions and the conductivity tensor representa-

tion are defined as they apply to resistivity measurements. The logging environment

is characterized in terms of both geometry and geology, with emphasis on the de-

positional processes that give rise to anisotropy. The need for accurate modeling

and inversion is demonstrated by showing how readily resistivity measurements in

beds of interest can be corrupted by adjacent media because of the large volumes

of investigation of resistivity tools.

2.1 Overview of logging environments

The parameter of greatest interest in evaluating a reservoir for its hydrocar-bon content is Rt, the resistivity of a bed under consideration which has notbeen contaminated by borehole fluids. Logging tools measure the over-all

apparent resistivity, Ra, and in order to accurately determine Rt, perturba-tions caused by adjacent regions must be taken into account. These regionsare shown in Figure 2.1 [222], and include:

- The borehole of diameter dh (6 to 16 inches), filled with drilling mud


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28 CHAPTER 2. ELECTROMAGNETIC RELATIONS FOR LOGGING

Mud

Adjacentbed

AdjacentbedInvasiondi

ameters

Borehole

diameter

Uninvaded

zone

Transition

zone

orAnnulus

Invadedzone

Bed

thickness

Mu

dca

ke

di

h

dh

dj

di

hmc

Rm

Rs

Rxo

Rs

Rt

Figure 2.1: The logging environment.

of resistivity Rm,

- Zones encircling the borehole flushed by the borehole mud called in-vaded zones, with resistivity Rxo and diameter di (ranging from dh to200 inches, and occasionally larger),

- Adjacent layers of differing resistivity called shoulder beds, with resis-tivity Rs and thickness h (ranging from several inches to 100 feet).

The effects of the borehole and adjacent beds can be decreased by de-signing tools to minimize their effect or by computer processing. Invasion

can be resolved by using tools with several depths of investigation.The first half of this chapter addresses the geometry of the logging envi-

ronment and the formation electrical characteristics which affect resistivitytool modeling and inversion. The second half defines the subset of Maxwellsequations used for modeling resistivity tool response.


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2.1. OVERVIEW OF LOGGING ENVIRONMENTS 29

2.1.1 Borehole effect

Because well logging is carried out with the tool immersed in the boreholemud, mud properties and borehole size can affect the accuracy of the mea-surement of Rt. For example, highly conductive mud can short-circuit lat-erolog currents and prevent them from penetrating deeply into a formation.Therefore it is important to accurately account for borehole effect.

Most wells are drilled with a rotary bit located at the end of a longstring of drill-pipe. A liquid mud is pumped down inside the drill-pipe andout through holes in the bit, and returns to the surface in the annular spacebetween the drill-pipe and the borehole wall. The mud lubricates the bit

and carries cuttings to the surface. In addition, the mud prevents blowoutsby providing a weighted column of liquid whose hydrostatic pressure can beadjusted to exceed that of the pore fluids in the formation [253].

The majority of drilling muds are water-based. These muds containweighting materials (usually clays) for adjusting the density, chemicals formaintaing a desired pH and gels to adjust flow properties. The resistivity ofwater-based mud is dependent mainly on its salinity. Muds made from seawater can be very conductive, ranging from 0.005 to 0.1 ohm-m at downholetemperatures. Muds made from fresh water are less conductive, rangingfrom 0.01 to 5 ohm-m, depending on the blend of the additives [177].

Oil-basedmuds are also commonly used. These muds consist of a complexmixture of oil, water, salt and surfactants necessary to keep the oil-watermixture in emulsion. Although oil is the continuous phase, some oil-basedmuds may contain as much as 40% water. The resistivity of oil-based mud istypically about 1000 ohm-m or greater. Oil-based muds usually do not invadethe formation very deeply. However, high down-hole temperatures and theeffects of the surfactants can sometimes combine to produce moderately deepinvasion of either the water-phase or the oil-phase [168].

Borehole sizes commonly range between 6 and 10 inches in diameter, butmay be as large as 20 inches. The larger the hole, the greater the volumeof mud around the tool, and therefore the stronger its effect on the tool

response. Corrections for borehole size and mud resistivity are performedeither on-line on the logging truck by means of computer algorithms, or afterthe log is recorded by using correction charts (see Section 5.2.2).

In soft or poorly cemented formations, the borehole may be eroded toa diameter much larger then the bit size by the action of the mud flow.


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This enlargement is called a cave. Caves may increase borehole effect either

smoothly or irregularly with depth.

Because a pressure drop is maintained across the borehole wall, a mudsliquid phase (mud filtrate) displaces the movable connate liquid in permeableformations. Particles in the mud are filtered out and adhere to the boreholewall to form a mudcake. Filtrate flow diminishes rapidly at first and thenmore slowly until it reaches equilibrium [86]. The mudcake formed usuallyranges from 0.1 to 1 inch in thickness. The thin mudcake has little effecton the response of mandrel tools, such as induction or laterologs. Mudcakecorrections are only needed for pad-type tools, which are applied against theborehole wall and have shallow depths of investigation.

2.1.2 Coaxial layers; invasion

In permeable formations, the mud filtrate flushes away most of the connatewater and much of any hydrocarbons that may be present in the region closeto the borehole. This flushed zone is referred to as the invaded zone (seeFigure 2.1). Further out from the borehole, the displacement of formationfluid may become less and less complete, resulting in a transition zone (formodeling simple invasion, the transition zone is normally ignored and stepcontact is assumed between Rxo and Rt). Saturations in the transition zonerange between those of the mud filtrate and the original formation fluid. Theextent of the invaded and transition zones depends on several parameters:drilling mud properties, formation porosity and permeability, the pressuredifferential and the time since the formation was first drilled [220].

Sometimes in oil and gas-bearing formations, where the mobility of thehydrocarbons is greater than that of water because of relative permeabilitydifferences, the hydrocarbons move away faster than the interstitial water.In this case, an annulus with high formation water saturation may be formedbetween the invaded zone and the uninvaded formation. Figure 2.2 showstypical saturation and resistivity profiles for an annulus region. Annuli prob-ably occur to some degree in most hydrocarbon-bearing formations. Their

influence on log measurements depends on the radial location of the annulusand the severity of the resistivity contrast. Annuli typically develop near theborehole shortly after drilling and gradually broaden and migrate outwarduntil they disappear in time through dispersion [7].

In fractured formations the invasion pattern is usually quite different.


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Figure 2.2: Saturation (a) and resistiv-ity (b) profiles for a representative ex-ample of annulus invasion.

Figure 2.3: 1D coaxial cylindri-cal geometry for modeling bore-

hole and invasion effects.

Unless fractures are very thin, they are generally invaded by bulk mud andno mudcake is formed [133]. Most shales have extremely low permeabilities,and it may be assumed that shales are not invaded (occasionally heavy oil-based mud can cause hydraulic fracturing of shales [18]).

Early 1D analytical codes for modeling borehole and invasion effect as-sumed coaxial layers with smooth cylindrical boundaries, as shown in Fig-ure 2.3 [27].

This simplification of the environment sometimes led to optimistic eval-uations of tool performance. Since the 1980s, 2D and 3D finite elementand finite difference codes have allowed features such as caves [14] and non-uniform invasion caused by gravity segregation [105] or permeability anisot-ropy [18] to be assessed more accurately.


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Figure 2.4: 1D layered formation geometry.

2.1.3 Thin beds (bed boundary discontinuities)

Most reservoir forming rocks were laid down in strata like a layer-cake. Theuniformity of layers is dependent on the conditions present at the time of de-position. For first-order interpretation purposes, the resistivity within a layeris assumed to be relatively uniform in all directions (i.e., anisotropy is not

taken into consideration). Boundaries between layers with different physicalcharacteristics are assumed to be planar and parallel to first approximation.This familiar layer-cake representation of sedimentary geological structureis shown in Figure 2.4 [27].

The main property that determines the resistivity of an individual layeris its porosity, since electrical current only flows through the water saturatingthe pore structure. The higher the porosity, the greater the amount of waterthat can be present, and therefore the lower the resistivity. The salinity of thewater also contributes to the over-all resistivity, with high salt concentrationsfurther reducing the resistivity.

Porosity of subsurface layers can vary widely. Carbonates (limestonesand dolomites) and evaporites (salt, anhydrites and gypsum) show prac-tically zero porosity [220]. Their resistivities are usually in excess of 100ohm-m.

Shales or clays may contain over 40% water-filled porosity. However,


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individual pores are so small that the rock is impervious to the flow of

fluids. Shale resistivities typically range from 0.5 to 5 ohm-m [86].

Well-consolidated sandstones have porosities between 10 to 15%; uncon-solidated sands may have 30% or more porosity. If sands are saturated withsalt water, as often occurs in offshore wells, the resistivity may be as lowas 0.2 ohm-m. Oil-bearing sands that are interspersed with shale lamina-tions (so-called low-resistivity pay) have resistivities averaging around 1 to2 ohm-m [59]. Normal pay sands have resistivities ranging from 2 to over1000 ohm-m.

Since tool response to a bed of interest can be strongly affected by adjacent layers, thin bed modeling has historically played an important role

in both tool design and log interpretation (for early tools of the 1950s, a 6foot bed was considered thin). The geometry shown in Figure 2.4 is assumedby 1D analytical codes that model induction response to thin beds with thetool logging perpendicular to bed boundaries (vertical wells).

1D codes have served well for the Dual Induction tool, which was de-signed to have minimal borehole effect and is often run in oil-based mudswhere invasion is shallow or nonexistent (borehole effect for laterologs is of-ten large and therefore cannot be neglected). 1D layered media codes wereused to evaluate the ability of early tools to resolve thin beds and to generateshoulder correction charts (described in Chapter 5.) In the 1980s, 1D thinbed modeling, supplemented by 2D modeling of beds with invasion, was

used to design Phasor processing which extended Dual Induction verticalresolution down to 2 feet [221].

2.1.4 Invaded thin beds

The 2D geometry for modeling thin beds with invasion, shown in Fig-ure 2.5 [27], is very much a combination of the 1D coaxial cylindrical ge-ometry (Figure 2.3) and the 1D layered formation geometry (Figure 2.4).Bed boundaries are assumed parallel to each other and perpendicular to theborehole axis (z). Radial boundaries (borehole and invasion, if it exists) are

perfectly cylindrical and centered around the borehole axis. This is the ge-ometry commonly assumed by 2D finite element, finite difference and hybridcodes for modeling induction and laterolog response in vertical wells.

Invasion that arises in thin beds normally occurs in the more porous andpermeable sandstones. Shales and tight carbonates usually do not invade


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Figure 2.5: 2D layered formation geometry with borehole and invasion.

and act as permeability barriers to prevent interaction between invasion indifferent beds.

2.1.5 Dipping beds

For the purpose of modeling tool response, dipping beds are considered to beany beds whose boundaries are not perpendicular to the tool axis. As such,dip has three causes: (1) geologic tilting of the formation, (2) deviation ofthe wellbore from vertical and (3) a combination of formation tilt and welldeviation.

The effect of dip on resistivity tool response was virtually ignored untilthe mid-1980s when horizontal drilling became common practice. Beforethat time, formation dips encountered were usually less than 30, and wereshown to have little effect on induction [32] or laterolog [65] response. How-ever, the 60 to 90 dips encountered in horizontal drilling often rendered

resistivity logs uninterpretable.Figure 2.6 [49] illustrates the reason for this complication. In vertical

wells, the volume of investigation of a tool is normally within the bed whereit resides. However, in horizontal wells, the volume of investigation mayextend over several beds.


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A

B

Figure 2.6: Induction response in vertical (A) and nearly horizontal (B)sections of a deviated well showing how dip causes the region probed by the

tool to cut across several beds.

Fast analytical codes for modeling induction response in dipping beds(without borehole effect) were developed in the 1980s [32, 143]. 3D finiteelement codes are required for modeling laterolog response in dipping beds,since borehole effect cannot be ignored.

2.1.6 3D geometries; horizontal wells

The drilling of horizontal wells has accelerated the development of 3D finite

element and finite differences codes. Indeed, if invasion is present in devi-ated wells, it is practically impossible to interpret induction and laterologresponse without 3D modeling. Two examples of the type of complex inva-sion geometries that can arise in deviated wells are illustrated in Figure 2.7and Figure 2.8.


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Figure 2.7: 3D deviated well with non-cylindrical invasion caused by gravitysegregation.

Figure 2.8: 3D horizontal well withthe wellbore passing below an im-permeable cap shale; there is annu-lus invasion in the pay sand below.

Figure 2.7 [27] shows a deviated well, where gravity segregation hascaused invasion to spread out above an impermeable bed. Figure 2.8 [18]shows annulus invasion which is truncated above a horizontal borehole bya cap shale. In addition to solving specific interpretation problems such asthese, 3D modeling is also prompting research in the areas of tool design, loginversion and invasion physics by identifying deficiencies in existing methods.

2.1.7 Anisotropy in layered media; laminated formations

Physical characteristics within a bed (i.e., resistivity, permeability) are usu-ally relatively uniform in all radial directions parallel to the plane of depo-sition and slightly different perpendicular to that plane. This gives rise tosome degree of TI anisotropy [220] (transversely isotropic anisotropy, whichdenotes having the same resistivity in every direction in the horizontal bed-ding plane, but a different resistivity normal to it). On the macroscopicscale (between grain-size and bed-size) there are two main types of deposi-tion that can cause anisotropy. They are: (1) alternating thin sandshale

laminae, and (2) alternating fine and coarse microlayering.Sandshale laminae are composed of fairly conductive shales and sands

that can be quite resistive if they are hydrocarbon saturated. The anisotropyresulting from this combination is one of the primary causes of low-resistivitypay, where hydrocarbons are recovered from zones that look like either shales


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Figure 2.9: Whole-core photograph from a well in the Gulf of Mexico showingthe relative distribution of shale (dark) and sand laminations. (Note that

the length of each of the three sections of core is slightly over one foot.)

or wet sands. The inherent conductivity of the shale contributes to the lowresistivity by reducing Rh read by resistivity tools in vertical wells (Equation(1.2)). Interpretation in deviated wells is further complicated because toolsrespond to both Rv and Rh as a function of deviation (Equation (1.6)). Thuslogs from a vertical well and a deviated well in the same reservoir will givedifferent values of Rt. Figure 2.9 [189] illustrates the relative size of sandand shale layers in a representative low resistivity pay reservoir. Individuallayer thicknesses typically range from a fraction of an inch to several inches.

Electrical and density image logs can be used to improve the interpre-tation of sandshale anisotropy in deviated wells. Image logs provide anestimation of sand and shale layer thicknesses and apparent dip. This infor-mation, along with resistivity from 2-MHz logging while drilling logs, can beused to derive Rh and Rv and to isolate the resistivity of the hydrocarbon-


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Figure 2.10: Photograph of a fluvial deposit of the Colorado river showingfine and course microlayering with crossbedding formed by ripples. (Notethe pencil near the top of the photograph indicating scale.)

bearing sand layers from the shale resistivity, giving a more accurate deter-mination of oil in place than traditional shaly sand methods [246].

Alternating fine and coarse microlayering can cause anisotropy in per-fectly clean sands with no shale content. If both hydrocarbons and water arepresent, the water saturation of the fine-grained layers will be higher thanthat of the coarse-grained layers, leading to alternating resistive and conduc-

tive layers with high anisotropy [159]. This type of anisotropy is often asso-ciated with crossbedding, that is, wind or water-deposited strata arrangedat different angles relative to the main bedding plane. In some cases theremay be thin cemented sandstone layers separating crossbeds [264], whichfurther complicates interpretation. Figure 2.10 [213] illustrates alternatingfine and course microlayering in a crossbedded dune.

It is also possible for fine and coarse sand microlayering to exist in com-bination with shaly layers or shaly sands. In general, pronounced electricalanisotropy in porous sediments is a good indicator of hydrocarbon pay.

2.2 Description of logging tool configurations; mandreltools vs. dipole approximations

Today, well logging is completely controlled by a computer located on alogging truck. Logging data are recorded and processed by the computer and


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2.2. DESCRIPTION OF LOGGING TOOL CONFIGURATIONS 39

output to either paper or magnetic media for additional processing offsite.

The logging tools themselves are composed of two main components: (1)a sonde containing the sensors used for making measurements (electrodesfor laterologs or coils for induction tools), and (2) a cartridge containingelectronics that power the sensors, process the measured signals and transmitthe data uphole [220]. Most logging tools are combinable, that is, the sondesand cartridges of several tools can be connected together in order to makemultiple measurements on a single trip into the borehole. The logging stringis typically 3.5 to 4 inches in diameter and 20 to 50 feet long [86].

In wireline logging, the tool is suspended from the end of a cable andlowered into the borehole by means of a powered winch-drum. The cableboth supplies power to the tool and digitally transmits recorded data upholeto the truck computer. In logging while drilling (LWD), tools are mountedon the drill string and powered by batteries. Data is either transmitted to thesurface in real time by pulsing the mud or stored in memory within the toolfor downloading when the bit is pulled to the surface [8]. LWD tools havethe advantage of acquiring early-time data that is relatively uncorruptedby invasion and can be used for steering the bit. However, the slow real-time data transmission rate of LWD tools (12 bits per second for mud-pulse compared to 500 kilobits per second for wireline) prohibits the use ofsophisticated array tools for LWD.

The transmitters and receivers of induction-type tools consist of coils

wound coaxially around a mandrel, as shown in Figure 2.11 (a). The mandrelof present-day tools is made of steel; early tools had a fiberglass mandrel.The entire tool is enclosed in an epoxy-composite housing. The inductiontransmitter coil is driven by a high-frequency alternating current (in the kHzto several MHz range) of constant intensity which creates a primary magneticfield around the tool. This magnetic field induces currents in the formationwhich flow in circular loops centered around the tool axis. These currentloops in turn set up a secondary magnetic field which induces a voltage ina receiver coil. This voltage is approximately proportional to the formationconductivity. Commercial tools consist of arrays of transmitters and receivercoils which focus induction currents in regions of interest (i.e., to make the

depth of investigation deeper or shallower.) Coil strengths are weighted byadjusting the number of turns and direction of winding (induction focusingis described in greater detail in Chapter 3).

Although induction coils are wound on a mandrel that is several inchesin diameter, calculations of magnetic fields generated by finite-size coils both


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(a) (b) (c)

Figure 2.11: Three source representations commonly used for modeling re-sistivity tool response shown in a borehole: (a) loop around a mandrel, (b)thin ring, (c) point dipole.

with and without a mandrel (Figure 2.11 (a) and (b)) show that coils maybe replaced by idealized point dipoles (Figure 2.11 (c)) for modeling mostcases of practical interest [33]. One notable exception is eccentricity effect inresistive formations with conductive boreholes [123, 178, 84]. In most othercases, tool effects are small in comparison to effects from the formation(such as anisotropy, shoulder-bed effect and dip). Therefore the extra timerequired for numerical analysis and modeling of finite-size coils and a mandrelis not justified.

Borehole effects for induction tools in general are also small. Conse-quently the borehole is often omitted from induction modeling in order tofurther speed up calculations. 3D modeling has shown [18] that the ar-ray induction borehole corrections algorithm [129] removes borehole effectso accurately that modeling tool response without a borehole is effectively

equivalent to the field performance of the borehole-corrected tool. This istrue even for the shortest spacings (under two feet).

Laterologs, however, cannot be accurately modeled as point sources [177],and the mandrel and borehole are always included in laterolog response cal-culations. Laterolog tools inject current into the formation from conductive


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2.3. ELECTROMAGNETIC FIELD EQUATIONS AND NOTATION 41

metallic electrodes which are directly in contact with the borehole mud. In

a homogeneous isotropic medium, the amount of voltage required to drivea unit current between two electrodes is approximately proportional to theresistance of the formation, as indicated by Equation (1.1). Currents radiateoutward from a source in straight lines, and surfaces of constant potential arespheres. However when a borehole is present, laterolog current lines bendas they cross the borehole wall, with the degree of bending being a func-tion resistivity contrast between the mud and the beds between the currentsource and return. Laterologs are often run in salty muds where the Rt/Rmcontrast is as high as 10,000. In cases such as these, borehole effect can belarge and highly nonlinear and cannot be neglected. In order to minimizethe effect of the borehole and shoulder beds, additional electrodes are in-

troduced to focus currents in regions of interest (various types of laterologfocusing are described in Chapter 3.)

2.3 Electromagnetic field equations and notation

The response of all electrical logging tools is calculated from numerical oranalytical solutions of Maxwells equations with the appropriate source andboundary conditions. Maxwells equations describe the behavior of electro-magnetic fields in space and time. Position in space is specified by (x,y ,z)coordinates in a right-handed Cartesian reference frame consisting of three

mutually perpendicular base vectors {iiix, iiiy, iiiz} that are of unit length each.(All vector quantities will be represented by bold-face symbols.) The posi-tion of a vector AAA is the linear combination of AAA = Axiiix + Ayiiiy + Aziiiz asshown in Figure 2.12. The Cartesian coordinate system is chosen so that itsx-axis and y-axis are parallel to planar beds in the logging environment (seeFigure 2.1). The borehole axis does not necessarily coincide with the z-axis;the borehole can be deviated as shown in Figure 2.7.

Electromagnetic quantities considered in this thesis and their units inthe International System of Units (SI) are, in the frequency domain:

EEE = electric field strength (V/m)HHH = magnetic field strength (A/m)

JJJe = volume density of external (source) electric current (A/m2)

JJJ = volume density of electric current (A/m2)


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Ax

Ay

A

z

Az

y

x

ix

[p.t] modeling and inversion methods for the interpretation of resistivity logging tool response

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