01 resistivity theory

CHAPTER

1

RESISTIVITY THEORY Introduction to Resistivity, Physics of the Measurement and Resistivity Tools

RESISTIVITY THEORY

TRAINING MANUAL i

TABLE OF CONTENTS

TABLE OF CONTENTS............................................................................................i,ii

PREFACE....................................................................................................................iii

INTRODUCTION TO RESISTIVITYLOGGING.......................................................1

History Of ResistivityLogging...............................................................................1

Evaluation OfHydrocarbons...................................................................................5

InvasionProfile........................................................................................................7

PHYSICS OF THEMEASUREMENT.........................................................................10

AbsolutePotential.................................................................................................12

PotentialDifference...............................................................................................13

EquipotentialSurfaces...........................................................................................15

CurrentDensity.....................................................................................................16

RESISTIVITIES FOR DIFFERENT CURRENT FLOW GEOMETRIES.............17

One Dimensional, Planar Flow - Mud Cup Analysis...........................................17

Spherical Current Flow -UnfocusedDevices........................................................19

Cylindrical Current Flow - GuardedDevices........................................................22

RESISTIVITY THEORY

ii TRAINING MANUAL

ELECTRODE LOGGINGDEVICES..........................................................................24

Defining the Tool - Resistivity vs Conductivity...................................................24

RESISTIVITYTOOLS..................................................................................................25

Unfocused Devices (ESTools)..............................................................................25

LateralTool...........................................................................................................25

NormalTool..........................................................................................................27

Problems Associated with the ESTools...............................................................28

Focused Devices (LaterologTools)......................................................................29

DUAL LATEROLOGMEASUREMENTS.................................................................35

Dual Laterolog ToolPrinciples.............................................................................35

Depth OfInvestigation..........................................................................................36

Radial ResistivityProfile......................................................................................37

SPHERICALFOCUSING.............................................................................................42

Spherical Focused logging Tool (SFL).................................................................42

Micro-Spherical Focusing (MSFLTheory)..........................................................45

Microguard Tool(MG).........................................................................................48

Invasion Effects - The Butterfly Chart.................................................................49

Dual Laterolog “Fingerprints”..............................................................................51

The MicrologTool................................................................................................52

REFERENCES...............................................................................................................54

RESISTIVITY THEORY

TRAINING MANUAL iii

PREFACE

In 1942 G. E. Archie of Shell developed the following equation that is known as ARCHIEEQUATION.

WATER SATURATION EQUATION

S = c R / R /w w t φ

Where c = 1.0 for carbonates and 0.90 for sands.

This is the basic equation of log interpretation. The whole well-logging industry is builtupon this equation.

The equation shows that hydrocarbons in place can be evaluated if there are sufficient logsto give interstitial water resistivity ( R w ), formation resistivity ( R t ), and Porosity (φ). In

practice Rw is obtained either from applying the equation in a nearby water sand (Sw = 1)

or from the SP log or from catalogs or water sample measurements; and φ is obtainedfrom porosity logs (Density, Neutron, or Sonic). R t is obtained from deep resistivity

readings (Induction or Laterolog).

RESISTIVITY THEORY

TRAINING MANUAL 1

INTRODUCTION TO RESISTIVITY LOGGING

History of Resistivity Logging

The history of resistivity logging is the history of electrical wireline logging. Wirelinelogging started with a resistivity device connected to a rubber insulated copper wire andsuspended in a borehole drilled to 1,500 feet. The well, the Diefenback 2905 was ownedby the Pechelbronn Company in Pechelbronn France.

The date was September 5, 1927. Using the principle's first conceived and later put intopractical use by Conrad Schlumberger for surface electrical prospecting. Henri GeorgesDoll designed the "sonde,” and with the aid of two assistants produced the first electrical"log." At the time Doll, with a B.S. degree from the University of Lyon, France, was aresearch Engineer for the Schlumberger Company. The first sonde consisted of fourelectrodes wired into an insulating mandrel made of Bakelite. By weighting the sondedown with lead pellets, and descending to the bottom of the well and slowly pullingupward, a resistivity profile of the subsurface formations was obtained. This firstoperation was not a thing of beauty. In his own words, Doll explains some of theproblems.

“We had no collector, instead, we had a plug, much like a common wallplug, at the side of the winch flange. When the winch had to be turned,the cable connection to the potentiometer was unplugged so the turncould be made. Then the cable was plugged back in so that we couldmake the reading. We began making our measurement. Someone hadto unplug the connector, someone else turned the winch, someone hadto run on the rig floor to look at the counter on the sheave...there was alot of running back and forth. I wrote down the measurements on apad, together with the depth reading. Then it was unplug, roll up onemeter to the next station, and plug back in. Make the next reading, andso on, one meter at a time”

The primary aim of this first log was merely to define the geological (lithology) crosssections, not locate the hydrocarbon-bearing zones. The idea was to use this wireline toolas an "electrical coring device" to compensate for the shortcomings of mechanical (drilltest) coring

HISTORY OF RESISTIVITY LOGGING INTRODUCTION

2 TRAINING MANUAL

FIG: 1 The First Logging Operations (Pechelbronn 1927)

RESISTIVITY THEORY HISTORY OF RESISTIVITYLOGGING

TRAINING MANUAL 3

After the initial operation, which wasconsidered successful, Doll plotted hismeasurements on a strip of graph paperand drew the first of the typical diagramsthat were to become familiar to thepetroleum industry. This figure shows thefirst “log."

After logging a number of wells inthe Pechelbronn field in the weeksto follow, positive conclusionswere drawn. Hard formation layersappeared on the diagrams as peakscontrasting clearly with the soft andconductive marls (sands). Whenthe log results were confirmed byactual physical core samples,electrical coring was firmlyestablished as a valuable tool forgeological surveys

FIG 2 The First Resistivity Log (Pechelbronn1927)

HISTORY OF RESISTIVITY LOGGING RESISTIVITY THEORY

4 TRAINING MANUAL

During the logging operations at Pechelbronn, Doll and his associates observed that evenwith no current emitted in the borehole by their tool, a potential difference was measuredacross a pair of monitor electrodes on the sonde. After integrating this self potential[called the Spontaneous Potential (SP)] over depth, one of the logs runs at Pechelbronn in1931 was the first to demonstrate that the SP could clearly delineate shales frompermeable formations. With the additional permeability and lithology informationprovided by the SP log, the basic electrical coring log in the mid 1930s provided enoughresistivity, porosity, lithology and permeability information that (in most cases)hydrocarbon-bearing formations could be distinguished from hard, low permeability zones.Thus the initial electrical coring diagrams were subsequently replaced by the lessrestrictive "electric survey logs" that included the SP curve. Well logging, as we know ittoday, had begun.

FIG 3 Electric Coring Operations (California 1932)


TRAINING MANUAL 5

Evaluation of Hydrocarbons

The manner in which the presence of hydrocarbons in pore space is sensed is through theelectrical resistance of the formation. The formation consists of rock matrix and porespace occupied by fluid. The resistivity of this combination of matrix and fluid is termed“true resistivity," R t . For normal logging situations, the rock matrix is considered a

perfect insulator; it conducts no electricity, therefore, the formation’s conductivity is onlya function of the fluid in the pores.

At depths below 2,000 feet, the water found in formation pores is generally saline, whichmakes it quite conductive. The conductive (resistive) property of water is a function ofthe salinity (number of ions' present) and temperature. The higher these two variables, themore conductive the water and the lower the resistivity of the formation that contains thewater. The illustration on the next page shows the relationship between the fluid resistivityand salinity and temperature.

Note that at constant temperature, the greater the salinity, the lower the resistivity. Also,at constant salinity, the greater the temperature the lower the resistivity. Remember thatresistivity is the reciprocal of conductivity. The resistivity of the formation water istermed R w .


6 TRAINING MANUAL

FIG 4 Resistivity Salinity Temperature Of Aqueous NaCl Solutions


TRAINING MANUAL 7

What happens when some of the saline pore water is replaced by hydrocarbons? We havealready mentioned that the rock matrix is an insulator. We should note, as can be seenfrom the table below, Hydrocarbons (oil or gas) are also insulators. Our resistivitymeasurement can not distinguish one insulator from another. Since these hydrocarbonfluids do not conduct electricity, their presence means there is less pore fluid available forconduction. We can say that the resisitivity of a sedimentary formation with hydrocarbonsdepends primarily on the resistivity of the water in the pores and the quantity of waterpresent. This resistivity; to a lesser extent, will also depend on the formation texture(grain size, distribution of pores, etc.), clay content, and lithology.

Table 1

TYPICAL RESISTIVITY VALUES

MATERIAL RESISTIVITY (ΩΩ·m)

Marble 57 9

x 10 10→

Quartz 10 31 2 1 4

→ x 10

Petroleum 2 x 1014

Distilled Water 5 x 1 0 3

TYPICAL FORMATIONS

Clay/Shale 2 → 10

Salt-Water Sand 0.5 → 10

Oil Sand 5 → 1 0 3

“Tight” Limestone 1 0 3


8 TRAINING MANUAL

Invasion Profile

The formation resisitivity, R t , that we have been discussing is for the virginal zone. This

is to say, R t is assumed to be the resistivity of the undisturbed reservoir beyond any

invasion.

FIG: 5 Idealized Invasion Profile


TRAINING MANUAL 9

In the borehole you have the drilling mud of resistivity ( R m ). The effect of this drillingmud on permeable formations causes mudcake of resistivity ( R mc ) to build on theborehole wall and invasion of these formations by mud filtrate. There are twocomponents of the invaded zone, one fully “flushed” by mud filtrate and the other a“transition” between fully flushed and uninvaded. The transition zone is neglected, andthe diameter of invasion ( d i ) is measured to the edge of the flushed zone. The resistivityof the zone, whose pore space contains only mud filtrate of resistivity ( R mf ) and residualhydrocarbons if present in the formation, is denoted as ( R xo ). The associated watersaturation isSxo . Beyond that zone is the undisturbed formation with resistivity( R t ),interstitial water resistivity ( R w ), and water saturation (Sw ).

The existence of invasion has forced the development of resistivity logging tools that makedeep measurements in an effort to read R t uninfluenced by mud filtrate. However notool has been developed that can read deeply enough under all circumstances and stillmaintain good vertical resolution. Consequently, the standard is to run three resistivitycurves simultaneously with three difference depths of investigation:

• Deep resistivity curve.

• Medium resistivity curve.

• Shallow resistivity curve.

The reading of the deep investigation curve can be corrected for invasion effects toprovide the R t value. The flushed zone resistivity and the diameter of invasion can alsobe estimated, using the three measurements in a set of simultaneous equations with threeunknowns ( R t , R xo , and d i ).

RESISTIVITY THEORY PHYSICS OF THE MEASUREMENT

TRAINING MANUAL 11

PHYSICS OF THE MEASUREMENT

As a starting point for resistivity theory considers the cylinder below. Suppose that apotential difference is established between the two ends of the conductor of length L andcross-sectional area A. For a given material, experiment shows that the resistance for thissimple geometrical configuration is directly proportional to the length and inverselyproportional to the cross sectional area.

FIG: 6 Resistivity Of A Cylindrical Conductor

Here V = Voltage

I = Current

L = Length

A = Cross Sectional Area

In an equation form, we can write the resistivity as:

1-1. ρ ρ= R L

A , = resistance

PHYSICS OF THE MEASUREMENT RESISTIVITY THEORY

12 TRAINING MANUAL

ere R is the resistivity. It is a property of the material of which the conductor is made,but does not depend on the conductor's physical shape. Resistivity has the units of ohm-m2/m (or ohm-m) and is affected by temperature; exact resistivity values are always statedalong with the corresponding value of temperature.

The resisitivity values for two metal conductors are:

copper = 1.7 x 10-8 ohm-m at 20° C

nickel = 6.8 x 10-8 ohm-m at 20° C

If we define:

1-2. A/L = K

then

1-3. R = Kρ

K can therefore be considered the constant of proportionality that converts resistance toresistivity. K has the units of length. From Ohm's law we know:

1-4. V = Iρ..

By combining equations 1-3 and 1-4 we obtain the general resistivity equation:

1-5. R = K V

I

Resistivity can be computed if the voltage drop and current are known. The K constant isonly a function of the geometry of current flow (which is controlled by the specific shapeand arrangement of electrodes for a current emitting resistivity device in the wellbore).The geometry of current flow defines the shape of the equipotential surfaces associatedwith the flow. Since an understanding of the idea of equipotential surfaces is necessary forderiving the resistivity equations for more complicated flow patterns, we need to reviewsome basic electromagnetic ideas.


TRAINING MANUAL 13

Absolute Potential

The absolute potential at a point in space is the work done against electrical forces incarrying a unit positive charge from infinity to that point. Hence for the geometry definedbelow, the absolute potential at "r" is the work done in bringing a unit positive chargefrom infinity (∞) to r. The potential at ∞ is considered to be zero.

FIG: 7 The Position Vector


14 TRAINING MANUAL

tential Difference

The potential difference from point 2, to point 1, is the work done against electrical forcesin carrying a unit positive charge from 2 to 1. See below.

FIG: 8 The Potential Difference Between Two Points


TRAINING MANUAL 15

The potential difference (V1 - V2) is therefore defined as a work/charge. This ratio has thespecific units of Joule/coulomb and is called the volt.

Potential Difference is defined by the line integral equation:

1-6. V - V = - = - Cos 1 2 r r

E dr E dr•∫ ∫2

1

r2

1 r Θ

In this case E is the electric field vector, and dr is an element of length along the currentflow, and Θ is the angle between E and dr. The minus sign arises because the potential isdefined as the work done "against" the field E. For all practical cases, E and dr areparallel and Equation 6 reduces to the scalar form.

1-7. V - V = - E dr 1 2 r

r

2

1

∫By definition you are able to choose V = 0 for an infinitely distant point (say, point 2),Equation 1-7 becomes:

1-8. V = - E dr

r

∞∫Where V is now the absolute potential at the point r.


16 TRAINING MANUAL

Equipotential Surfaces

Consider a simple case of a spherical current source electrode of radius ( ro ) in an infinite,

homogenous, and isotropic medium centered at the origin (see below).

FIG: 9 Spherical Current Source

If the potential on the surface of the electrode is maintained constant with the currentemanating from the electrode to infinity, then the electric field is radial and the potential atany point where r > ro will depends on the distance [V = V(r)]. Therefore, on a givenradial distance of r, all points of the sphere of radius r will have the same potential. Thistherefore defines an equipotential surface. Indeed, we can imagine the electrode to besurrounded by an infinite number of concentric spherical surfaces, each point on a givensphere having the same potential, and each spherical equipotential surface differing fromits neighboring surfaces in potential by a constant amount. The electric force lines,radially outward from the positive current source, are perpendicular to the equipotentialsurfaces. It should be understood that the electric force lines must be perpendicular to theequipotential surfaces. If this was otherwise, a component of the electric field (E) wouldbe tangent (parallel) to the surface and current would be able to flow on the surface. Ifthe current is flowing on the surface, the entire surface cannot be at the same potential(current flows from a higher to a lower potential). Therefore, if the surface isequipotential, only radial current flow is possible and the electric force lines must beperpendicular to the surface.


TRAINING MANUAL 17

Current Density

The magnitude of the current divided by the unit cross-sectional area of current flow(defined by the flow geometry) is called the current density. To take into account thedirection of the flow, this current density must be a vector quantity.

1-9. J = I

A n

In this case n is a unit vector along the direction of current flow, “I” is the current and A isthe cross sectional area. The magnitude of J is simply:

1-10. IJ =

A

*

It has been found that throughout a wide range of conditions, in solids and liquids, therelationship between J and the electric field E is a linear one.

1-11. E = RJ

Here R is the resistivity. Equation 1-11 is the microscopical equation of Ohm's law, andholds true for any point within a conductor of any shape. It should be obvious fromEquation 1-10 that the exact expression for J is a function of the current flow pattern,which is dictated by the equipotential surfaces defined by the shape of the current source(and the spacing of the return electrode).

* NOTE: The current density “J” is not to be confused with the pseudo geometrical factor “J” in the ofRadial Resistivity Profile Section.


TRAINING MANUAL 18

RESISTIVITIES FOR DIFFERENT CURRENT FLOWGEOMETRIES

One Dimensional, Planar Flow - Mud Cup Analysis

Referring to figure 6 , we can apply Equation 1-6 and the microscopical form of Ohm'slaw (Equation 1-11) to obtain the expression for the resistivity of the cylinder. We mayassume that the configuration obeys the coordinates as shown below:

FIG: 10 One Dimensional Planar Flow

Since the equipotential surfaces are circular planes, the area and current density are givenby:

1-12. A = πlo2 = constant, J = Io /πlo

2

The vectors J and E are parallel. And an element of current flow length dr is parallel to Eand J.

RESISTIVITY THEORY RESISTIVITIES FOR DIFFERENT CURRENT FLOWGEOMETRY’S

TRAINING MANUAL 19

From Equations 1-6 and 1-11:

1-13. V L00 0

L

L

- V = - E dr = - ∫ ∫• RJ • dr

Since J and dr are parallel, the dot product J • dr becomes just the scalar product Jdr andEquation 13 reduces to:

V Jdr I

L00 0 0

02

L

L

- V = - R = - R dr∫ ∫

π

V L00

02

0

02

0

L- V =

I R dr =

I R L

π πl l∫

defining Vo - VL as V:

V = I R L0

02πl

or in terms of the resistivity

R = K V

I K =

L =

A

L0, 0

2π l

these are just equations 1-5 and 1-2 respectively.

RESISTIVITIES FOR DIFFERENT CURRENT FLOW GEOMETRY’S RESISTIVITY THEORY

20 TRAINING MANUAL

A practical use of the previous Equations is made by the measurement of resistivity in the"mud tester" on the logging truck. This tester is a cylindrical shaped hollow container inwhich fluid is drawn. A d.c. potential is maintained across the ends of the tester andcurrent flows through the fluid within. Resistivity is computed from Equations 5 and 2.As an example, suppose a mud tester with a cylinder of 0.5 inches and a length of 4 inchesis filled with mud filtrate. The voltage across the sample is 10 volts with the current equalto 0.5 Amps. The resistivity of the fluid is easily computed as follows:

A πlo2 (3.14) (0.00635)2 m2

K = = = = 0.00127 m L L 0.10 m

V 10VR = K = (0.00127 m) = 0.0253 ohm-m

I 0.5A


TRAINING MANUAL 21

NOTE: The dimensions in inches had to be converted into meters in the above equations to keep K andR in the correct units.

Spherical Current Flow - Unfocused Devices

The downhole formation resistivities are not so easily computed as the directmeasurements made by a mud tester. Still the approach to obtaining “apparent”resistivities of the formation again uses equation 1-5. The earliest popular resistivitydevices (by their unfocused nature) approximated the case of a spherical electrodeemitting constant current I o radially in an infinite, isotropic and homogenous medium.

For the spherical current flow shown below, we want to obtain an expression for theresistivity between two concentric equipotential surfaces whose radii are greater than thesource electrode radius ( ro ).

FIG: 11 Two Concentric Equipotential Spheres

The radius of the inner surface is AM, and that of the other surface is AN. Here A is theorigin. From our definition of the potential difference, the potential difference between thetwo equipotential surfaces is just the work done in moving a positive charge from thesurface N (the outer surface) to the surface M (the inner surface)


22 TRAINING MANUAL

1-14. V - V = - M N AN

AM

∫ •E dr

For the spherical geometry, the surface area and current density are given by:

1-15. A = 4πr2, J = Io / 4πr2

As before, J and dr are parallel and our dot product with E = RJ becomes a scalar product.

V R I dr

rM N AN

AM

AN

AM

AN

AM - V = - = - R dr

∫ ∫ ∫•J dr J = - R 024π

V VI R dr

r

I R

rM NAN

AM

ANAM - = - = - -

0

20

4 4

1

π π∫

1-16. V V VI R

r

I R

AM ANMN M N ANAM = - = + = - 0 0

4

1

4

1 1

π π

In terms of R,

1-17. K (VM - VN)R =

I0

Where K is given by

1-18 4π

K = 1 1 ( - ) AM AN

We see that if the potential at the two equipotential surfaces can be measured, and thedistances AM and AN is known along with the constant current output Io, “R” can becalculated from Equations 1-17 and 1-18. The "Lateral" and "Normal" unfocused resistivity


TRAINING MANUAL 23

devices with approximate spherical current patterns obey this equation. They will bediscussed in a later section.

Cylindrical Current Flow - Guarded Devices

If a resistivity device is designed to maintain lateral current flow around a cylindricalsource electrode emitting a constant current Io, cylindrical equipotential surfaces exist inthe ideal case of a homogenous and isotropic medium. See Figure 12 below.

FIG: 12 Cylindrical Current Flow

For this geometry the surface area and current density are:

A = 2πrh , J = Io /2πrh


24 TRAINING MANUAL

We want to find the resistivity between the source surface of radius ro and someequipotential surface a distant L from the center. The absolute potential at the surface ofthe source electrode is VO and the absolute potential at our reference surface is VL. Asbefore, E, J, and dr are parallel and we use Equation 6.

V0 - VL = -

o

L

r

∫ E •• dr

V0 - VL = -

L

ro

∫ E dr = - R

L

ro

∫ J dr = - R

L

ro

∫I dr

rh0

2π

V Vh

drr h

nrh

nrL L Lrr o

00 0 0

2 21

21 - = - = - =

π π π∫

1-19 ( )VI R

hn L rL0

002

1 - V = - /π

In terms of R,

1-20. Vo - VL

R = K Io

Where K is now given by

1-21.2πh

K = ln L/ro

If L is chosen at a sufficiently large distance (not ∞) the potential at L is negligible (VL ≈0). In this case:

1-22.K Vo

R = Io


TRAINING MANUAL 25

We see that for this type of device, “R” can be computed from the tool current flowpattern, the current, and the absolute potential at the source electrode (i.e., the potentialdifference between a surface ground and the source electrode).

ELECTRODE LOGGING DEVICES

Defining the Tool - Resistivity vs Conductivity

Equation 1-5 holds for all types of electrode tool systems in which current is emitted fromthe tool into the borehole. Obviously, electrode tools do not work in non-conductiveborehole fluids such as oil base muds, or air-filled boreholes. The definitions of thesetools as either resistivity or conductivity is determined by the measured parameter. If themeasured parameter is voltage, with the current held constant, Equation 1-5 can bewritten:

Vm

R = K Io

Here Vm is the measured voltage. It is directly proportional to R, (resistivity). This typeof device is designated as a resistivity tool.

If the measured parameter is the current with voltage being constant, then Equation 5 canbe written:

1 Im

1-23. = σ = K-1 R Vo

Here Im is the measured current. It is inversely proportional to R and therefore directly

proportional to σ, (conductivity). This type of device is designated as a conductivity tool.

If there is variation in current and voltage, then Equation 1-5 can be written:

Vm

1-24. R = K = K ratm

Im

This type of device is referred to as a resistivity tool; the more correct term might be“resistivity/conductivity” tool or simply “ratio” tool

RESISTIVITY THEORY RESISTIVITY TOOLS

TRAINING MANUAL 31

RESISTIVITY TOOLS

Unfocused Devices (ES Tools)

Lateral Tool

The early unfocused resistivity devices, called ES (Electrical Survey) tools, actuallyincorporated a four electrode system. The tools were not actually placed in an infinite,homogenous, and isotropic medium because (1) a borehole is required and (2) noformation is infinite, homogenous, or isotropic. Of the four electrodes, two were used ascurrent electrodes (one transmitted current and the other received the current), and theother two were used as monitor electrodes (they measured the potential at differentlocations). The illustration below is one ES type tool. It is called a Lateral Tool.

FIG: 13 Lateral Tool

RESISTIVITY TOOLS RESISTIVITY THEORY

32 TRAINING MANUAL

Current is emitted from the source electrode A to the return electrode B (on the surface).The potential difference between M and N is measured. The distance between M and N issmall compared to the spacing between A and the midpoint of M and N (the point O in theillustration). Originally, the distance MN was 32 inches and the spacing AO was 18 feetand 8 inches. The spacing AO defines the investigational region. With this large spacing,the lateral tool could record the resistivity of the uninvaded (virginal) zone. Because of itsunfocused nature, the current flow pattern exhibits approximately spherical symmetry andtherefore the equipotential surfaces are somewhat spherical in shape. We can useEquations 1-17 and 1-18.

K (VM - VN) 4πR = K =

I0 1 1 ( - ) AM AN


TRAINING MANUAL 33

Normal Tool

Another unfocused ES is a Normal Tool. Constant current is passed between the sourceelectrode A and returns electrode B (at the surface). The measured voltage (potentialdifference) appears between electrodes M and N.

FIG: 14 Normal Tool

From the illustration we see that the N electrode is sufficiently far from the sourceelectrode A (i.e. AN → ∞) that VN is negligible (i.e. VN ≈ 0). Equations 1-17 and 1-18are applicable for these types of devices (producing approximately Spherically symmetriccurrent patterns) and for a normal tool configured as in the above Figure, the equationsreduce to:

1-25.K VM

R = , K = 4πAMI0

The remaining distance AM is called the spacing, and determines the depth ofinvestigation. Originally for the Short Normal device a shallow investigation was achievedby having AM = 16". The Long Normal investigated deeper and had AM = 64".

For both lateral and normal tools, “R” as computed from equation 1-17 and 1-18 isreasonably accurate provided the formation is sufficiently thick and homogenous andborehole effects are negligible.


34 TRAINING MANUAL

Problems Associated With The ES Tools

Prior to 1950, resistivity logging consisted of running simultaneously a Short Normal, aLong Normal, and a Lateral tool. With this combination, three different depths ofinvestigation were possible, with the deepest being provided by the lateral tool. Althoughused very extensively for a number of years (approximately 20 years) the ES toolsproduced logs that were difficult, sometimes almost impossible to interpret. Extensivecharts were required to correct for borehole, bed thickness, and adjacent-bed resistivityeffects. It was found that the curves were relatively useless for bed thickness less than 1.5times the spacing, i.e. 28 ft. for the Lateral and 8 ft. for the Long Normal. The ShortNormal curve was the most usable, but it was severely affected by invasion. The basicproblem with the ES logs was that the direction of the survey current was not controlled.This current took the path of least resistance, favoring conductive mud and conductiveshoulder beds over resistive beds at the level of the tool.

FIG: 15 Unfocussed Tool - Possible Current Path

As a result of all the problems, the Normal and Lateral curves was replaced in the 1950sby focused logs in which the path of the survey current was controlled. The focusingminimized borehole and adjacent bed effects and provided good bed resolution. Twotypes of focused tools were introduced. One was the Induction tool that works byinducing (not injecting) a current flow of closed loops concentric with the tool axis in theformation. This tool works best in non-conductive or low conductivity boreholeenvironments (oil based muds, air filled boreholes or fresh mud systems). The other toolwas the laterolog device. This is an electrode type "Guard" tool that works well in veryconductive boreholes (i.e. salt mud systems). The Induction tool is fully discussed in theInduction Training Notes.


TRAINING MANUAL 35

Focused Devices (Laterolog Tools)

Laterolog systems utilize a multiple electrode array to force survey current to travellaterally across the mud and into the adjacent formation. The advantages of this techniqueare the ability to operate in very salty mud, while providing excellent bed definition,independent of neighboring bed resistivities.

There are two basic types of focused-electrode laterolog arrays. One is the 3-electrodesystem commonly called guard log or LL3 and the other is the 7 to 13 electrode system,with designation LL7, LL8, and dual laterolog (with 9 to 13 electrodes). Both systemsoperate on much the same principle, as illustrated below showing the LL7 and LL3.

FIG: 16 Laterolog 3 (LL3)

FIG: 17 Laterolog 7 (LL7)


36 TRAINING MANUAL

For an understanding of the focusing feature we will take an in-depth look at the laterolog7 devices. These types of logging tools are designed in such a way that the source currentis kept from flowing up and down in the drilling fluid. This is accomplished by placingfocusing electrodes on both sides of a centrally located source electrode as seen in theLL7 representation. All of the electrodes are maintained (essentially) at the same potentialand send currents out in the same sense (i.e. all currents have the same phase).

Since like charges repel and unlike charges attract, the current flowing out of the upperand lower guard electrodes (A1U and A1L) tends to repel the current flowing out of thecenter electrode (AO). The center current pattern (i.e. the survey current) is therefore keptfrom flowing upward toward A1U or down toward A1L. The survey current Io is forced toflow in a horizontal layer at right angles to the borehole before it begins to flow towardthe return electrode.

To keep the Ao, A1U and A1L electrodes at approximately the same potential, (which keepsIo focused) as the tool moves upward through the successive formation beds duringlogging, two sets of monitor electrodes are used. If a potential difference is sensed acrossa set of monitor electrodes (either across M1U and M2U or across M1L and M2L), indicatingthat the survey current lateral pattern is not being maintained, the potential of the guardelectrodes (A1U and A1L) is increased or decreased to maintain the focused cylindrical (discshape) path of survey current Io into the formation. Increasing the potential of the guardelectrodes tends to "push" the survey pattern away from the electrodes. Decreasing thepotential of the guard electrodes tends to "pull" the survey pattern toward the electrodes.

This push or pull feature is essential in maintaining the correct vertical resolution. As anexample, a very thin resistive bed across from Ao surrounded by conductive beds wouldrequire the system to "push" the survey current into the resistive bed since the currentwould separate and migrate above and below the bed. A very conductive bed whosethickness is less than the required disc thickness (usually 2 feet) surrounded by resistivebeds will require the electrodes to "pull" the survey current pattern toward the electrodes,thus increasing the disc height. This is illustrated below.

FIG: 18 Thin Bed Effect


TRAINING MANUAL 37

The resistivity of the formation is computed from the physics of the measurementassuming cylindrical equipotential surfaces. If we assume that the monitor electrodes aremaintaining lateral survey flow, the current distribution pattern (ideally) is a uniformcylindrical disc as shown below.

FIG: 19 Ideal Current Pattern For Laterolog Tools

The resistivity is computed from Equations 21 and 22 respectfully:

V0

R = K I0

2πh *K =

ln L/r0

Here h is the thickness of the cylindrical disc, L is the length and is dependent on the radialdistance this disc shape is maintained, and r0 is the radius of the cylindrical sourceelectrode. Equations 1-21 and 1-22 are applicable for all laterolog (guard) tools.


38 TRAINING MANUAL

* NOTE: The above equation only provides an approximated K value for laterolog tools. A morecorrect expression is obtained by computer, mathematical modeling with real boreholeformation effects considered

The original laterolog-7 (LL7) differed from the original laterolog-3 (LL3), in that it usedsmall guard electrodes, while the LL3 used long guard electrodes (5-6 feet) and the LL7used monitor electrodes to ensure lateral flow. At present, a shortened guard version ofthe LL3 (less than 18") is used for a shallow Rxo (invaded zone resistivity). The length ofa guard defines, to a great extent, the depth of investigation (we must also consider thereturn path).

The Dual Laterolog Tool system was introduced in the 1970’s. The system providedsimultaneous deep and medium curves produced from a dual focusing electrode array ofup to 13 electrodes. The focusing feature produces two cylindrical current discs ofdifferent radial lengths (L) but with the same thickness (h). The resistivity for both theshallow and medium system is again computed from Equations 1-21 and 1-22 with theappropriate input constants and measured parameters.

RESISTIVITY THEORY DUAL LATEROLOG MEASUREMENTS

TRAINING MANUAL 39

DUAL LATEROLOG MEASUREMENTS

Dual Laterolog Tool Principles

The dual laterolog system uses both small and large guard electrodes. The operationalaspect of this tool is the same as the LL7, except now we have both a deep and shallowsystem working simultaneously. This is the dual aspect of that tool. For both systems,horizontal cylindrical discs are maintained by the action of the guard and monitorelectrodes. The systems, using the same electrodes, operate at different frequencies andhave different depths of investigation. Both systems maintain the same vertical resolutionof 2 feet (disc thickness).

The deep tool needs long guards and remote current return, while for the shallow it isnecessary for the guards to be short and the return relatively near (on the tool). Figures 20and 21 shows the electrode arrangement and current patterns.

DUAL LATEROLOG MEASUREMENTS RESISTIVITY THEORY

40 TRAINING MANUAL

The shallow (LLS) current layer (disc) starts to spread after a short distance into theformation. The reason is the focusing currents are emitted from electrodes A1U and A1Land the returns are the nearby A2U and A2L electrodes. To keep the survey currentfocused, the shallow system monitors the values of the Electrodes M1 and M2 upper andlower. From these measurements, the system will control the upper and lower focusingroutine, with each one (upper or lower) being controlled independently.

The designations of the electrodes used here are for illustrative purposed and aredifferent from those in the DLLT Manual.

FIG: 20 Dual Laterolog Tool Shallow Current Pattern


TRAINING MANUAL 41

The deep laterolog measurement requires a long guard system.

The focusing above A0 is accomplished by combining the outputs from the A1U and A2Uelectrodes. Below A0, the outputs from the A1L and A2L electrodes are combined. Thereturn for the deep pattern is an electrode approximately 75 feet away from A0. Thecombination of long guards and the remote return, causes the layer of logging current tohold its horizontal thickness far into the formation. To keep this deep system focused,monitor electrodes M1, M3, and M4 (upper and lower are utilized).

FIG: 21


42 TRAINING MANUAL

Different frequencies are used by the different systems. For the shallow measurement afrequency of 1050 Hz is used; whereas for the deep (which requires a lower frequency),131 Hz is used. Because each system is sensitive to its own frequency, continuousrecording of data is attained. The tool measure's values of v o and I o and using Equations1-21 and 1-22, two resistivity values are determined. Two different K values are requiredfor the resistivity computations, due to the different radial investigations (this changes thevalue of L in equation 1-21).

Depth of Investigation And Vertical Resolution

For any specific arrangement of electrodes, the distance into the formation for which thecurrent maintains its lateral (disc shaped) pattern is the depth of investigation associatedwith that specific arrangement (see below). This is true because as the current lines beginto spread, the cross sectional area of the current path increases, which effectively makesthe resistance associated with that region of space negligible. The vertical resolution isapproximately the height of the disc. Figure 22 depicts this, and shows that for the DualLaterolog, the deep measures approximately 5-7 ft. into the formation; whereas, theshallow is 2-3 ft. The vertical resolution is approximately 2 ft. for both.

2 - 3f t

S h a l l o w

D e e p

2 f t

2 - 3 f t

S h a l l o w

5 - 7 f t

D e e p

I

I I

I

II I I

5 - 7 f t

FIG: 22 DLLT Depth Of Investigation And Vertical Resolution


TRAINING MANUAL 43

Radial Resistivity Profile

When using a dual laterolog, under normal ideal conditions, the radial profile ofresistivities is as shown below (i.e. Rm < Rxo < Rt). Between the invaded (flushed) zoneand the undisturbed formation, there is a transition zone that has a resistivity valuebetween the values of Rt and Rxo. Figure (24) is a diagram showing a plane view of ahorizontal slice made through the tool and the formation that surrounds it. This figureshows current flowing radially outward from the tool and passing through the mud, theinvaded zone, and the undisturbed formation before arriving at the return electrode.

FIG: 23 Radial Distribution Of Resistivities


44 TRAINING MANUAL

FIG: 24 Laterolog Current Paths

If held constant, the current will then develop a series of voltage drops across each zoneencountered. The relationship between these voltages can be simplistically written as:

1-26. Vtotal = Vmud + Vinvaded + Vundisturbed

Each voltage drop is proportional to the product of the current, the resistivity of the zone,and some relative weight function, Ji, that gives the percentage that zone contributes tothe total signal.


TRAINING MANUAL 45

1-27. Vtotal = Jm (IRm) + Jxo (IRxo) + Jt (IRt)

Here Vmud = Jm (IRm),

Vinvaded = Jxo (IRxo),

Vundisturbed = Jt (IRt)

Vtotal is related to the measured current I and the resistivity (RLL) measured by the toolby:

Vtotal = IRLL

Thus, the equation for resistivity measured by the laterolog (RLL) can be written as:

1-28. RLL = Jm Rm + Jxo Rxo + Jt Rt

Logically, Jm will depend on the hole size, while Jxo and Jt will depend on the invasiondiameter (di) and on the contrast between Rt and Rxo. The sum Σ Ji must be one bydefinition. An equivalent electrical circuit for a laterolog measurement is shown below.

FIG: 25 Equivalent Electrical Circuit For Laterolog Measurement

We see that in the laterolog tool the zones add in series. The tool therefore respondsprimarily to the most resistive zone. This is in contrast to an induction tool that seeszones surrounding the tool (including the shoulder bed) as adding in parallel and thusresponds to the most conductive zone (conductivity device).


46 TRAINING MANUAL

Referring to Equation 1-28, usually the mud column contributes a small signal and in mostcases can be ignored (Jm ≈ 0). The usual exception is in large boreholes, in which caseborehole correction charts are available to correct for this effect. Figure 26 shows thecorrection charts used for the laterolog deep reading. Therefore, after correcting for themud effect, the laterolog response in an invaded formation is described by the pseudogeometrical factors, Jxo and Jt. The term "pseudo" refers to the fact that, as previouslymentioned, Jxo and Jt are not only a function of the invasion diameter, but they also dependon the contrast between Rt and Rxo. From Equation 28 we now write:

1-29. RLL = Jxo Rxo + Jt Rt

Since Σi Ji = 1, Jxo + Jt = 1

RLL = Jxo Rxo + (1 - Jxo) Rt

Dropping the subscript on Jxo, J is called the pseudo radial geometrical factor, andEquation 29 takes the final form:

1-30. RLL = JRxo + (1 - J) Rt*

J is a variable with a value between 0 and 1; at di = 0, J = 0; at di = ∞, J = 1. Each of thetwo laterolog measurements (deep and shallow) has its own J-di relationship as shown infigure 27.


TRAINING MANUAL 47

*NOTE: In practice, along with the borehole correction, a bed thickness correction is also applied tothe log before invasion corrections are applied.

FIG: 26 Deep Laterolog Borehole Correction


48 TRAINING MANUAL

FIG: 27 Pseudo Geometrical Factor

SPHERICAL FOCUSING

Spherically Focused Logging Tool (SFL)

We have already discussed the problems associated with the early electric survey tools.The main problem being that the direction of the survey current was not controlled buttook the more conductive (less resistive) path (see Figure 15). The laterolog devicesimproved the situation by forcing the survey currents to flow in a lateral, disc shapedpattern by introducing guard electrodes above and below the center current emittingelectrodes (see Figure 17). Another focusing method was introduced by Schlumberger (asa replacement for the short normal) that forces the survey current to produce a sphericalflow pattern into the formation. The tool was called an SFL (Spherically Focused Log),and was designed to make a shallow measurement (primarily influenced by R xo ).

If a tool system is maintaining a truly spherical current flow pattern in the formation,equipotential surfaces are spheres. For this case, the geometry associated with the flowwill be that of Figure 11, and the associated equation for the potential difference (voltage)between two equipotential surfaces M and N is given by the lateral equation.

FIG: 28 Two Concentric Equipotential Spheres

RESISTIVITY THEORY SPHERICAL FOCUSING

TRAINING MANUAL 49

1-31. [ ]VM AM − V =

I R

4 N

0 -

1

ANπ1

same as Equation 16

Which as before reduces to:

1-32. R = K ( V )

IN

0

VM − same as Equation 17

or in terms of conductivity

1-33. σ =K I

( V )

-10

N VM −

SPHERICAL FOCUSING RESISTIVITY THEORY

50 TRAINING MANUAL

The SFL was designed to measure the conductivity of a region of formation just outsidethe borehole and extending vertically a short distance either side of the A o electrode.The actual electrode placement on the sonde along with the ideal current distributionpattern is shown below.

Figure 29 Current Pattern

To achieve the pattern above, two current systems are required. The bucking current I bleaves the A o electrode and returns to A1 and A1′ by traveling through the borehole.The survey current I o leaves A o and is returned to the remote electrodes (Cable Armorand lower tool body). The combined effect is that I o is forced into the formation sinceI b creates a barrier in the borehole (i.e. I o cannot cross I b ).


TRAINING MANUAL 51

Micro-Spherical Focusing (MSFL Theory)

Even though the results with the SFL tool were far better than the short normal, thespherical focusing deteriorates under very high R Rt m contrast because very largebucking currents are required to flow in the mud since I b is responsible for themaintenance of the equipotential shells inside the borehole.

The next logical step was to produce a pad-type miniature version of the SFL, to eliminateborehole effects and achieves superior shallow investigation. The result was the Micro-Spherically Focused Tool (MSFL). The MSFL is the state-of-the-art in R xo (flushedzone resistivity) measurements. It provides good results under a wide range of invasionand mudcake thickness. The MSFL electrode arrangement is shown in figure 30. Noticethe number and geometrical distribution of electrodes, as well as the resulting current flowpatterns, is the same as the SFL.

FIG: 30 MSFL Pad And Current Patterns

The actual physical description of the pad face is shown in the lower left corner. Theelectrodes are rectangular, metal strips concentrically molded into the rubber pad body.The electrodes are recessed in the rubber surface.


52 TRAINING MANUAL

In a manner "identical" to the SFL, the MSFL provides spherical focusing and a value forthe flushed zone conductivity. In describing the specifics of the tool operation, we canfirst start by simply explaining the functions of the two current loops produced from thetwo separate, but interacting, current feedback systems. I b (the bucking current) isresponsible for (1) preventing I o from flowing through the mudcake, and (2) establishingand maintaining (by interacting with I o ) the constant potential difference across thespherical shell. I o is responsible for (1) preventing I b from flowing across the monitorelectrode pair, M1 - M 2 (this keeps I b essentially traveling vertically through themudcake to A1 and A1′ ), and (2) providing the measurement proportional to flushedzone resistivity. Now we can provide a little more detail.

The current leaving the central electrode A o is I T . I T consist of the two components:

bucking current and the two components surveys (measure) current.

Here I b and I o represent both the upper and lower components, (it would probably bemore correct to refer to the bucking and survey currents as 2 I b and 2 I o , respectively).The bucking currents return to symmetrical electrodes A1 and A1′ . I b , which isrestrained to essentially flow in the mudcake due to the very small spacing of theelectrodes and its interaction with I o , establishes a current barrier for I o . I o is forced toflow deeper into the formation by the "bucking action" of the I b current lines (i.e. likecurrents repel). I o is returned to a remote electrode that is the mandrel body

The resulting equipotential surfaces due to the combined current paths are approximatelyconcentric spheres (or to be more exact, hemi-spheres) as shown by the dashed lines inFigure 30.

The I o current system is driven by the potential difference between the M1 - M 2electrode pair. The I o current is adjusted continuously and automatically to make thisdifference zero (i.e. v M1 - vM2 = 0).

The effect of this is to create an equipotential barrier between M1 - M 2 which furtherconfines I b in the vertical direction. The potential difference being zero insures thebucking current will not flow across the pair (current flows only across a potentialdifference).

The MSFL resistivity is computed across the spherical shell of thickness M o M1 (to beexact the shell thickness is between the M electrode and the M M1 2 center tap) usingEquation (17). Halliburton has two versions of the MSFL. The G-series tool maintainsthe potential across the shell at a constant reference value, v R . This makes this aconductivity tool. On the other hand, the W-series (DITS) allows both the current andvoltage to vary, measuring both. This is a ratio tool.


TRAINING MANUAL 53

The actual pad size and electrode spacing determine where in space the spherical shell islocated (i.e. the investigational region). The pad dimensions are such that this shell is deepenough to avoid the mudcake, but not too deep as to be significantly influenced by theuninvaded zone. The MSFL external pad system is shown in figure 31.

FIG: 31 MSFL Sonde


54 TRAINING MANUAL

Microguard Tool (MG)

The W-series Dual Laterolog Tool (depending on the version) can be run with the MSFLor another type microresistivity device called the Microguard or (MG). This tool is alsosometimes referred to as the FORXO (as in “for R xo ”). To make its measurement, the

Microguard uses a pad that contains a central “button” electrode that is encircled by aguard ring electrode. The current from the guard ring provides the focusing feature.Figure 32 below shows the features of the Microguards and current flow patterns. Noticethat a side view of the current flow pattern shows the Microguard to be like a pad typeLL3.

FIG: 32 Microguard Current Flow

Operationally, current from the guard ring focuses the survey current, the current from thecentral electrode, by forcing this current to only flow radially away from the tool. Thiscurrent flows into the first few inches of the formation (the flushed zone) and returns tothe Microguard Mandrel housing.


TRAINING MANUAL 55

Invasion Effects - The Butterfly Chart

Since the dual laterolog provides two resistivity values at different radial depths, Equation1-30 can be broken into the deep and shallow components (after borehole corrections).

1-34. RLLD = JD Rxo + (1 - JD ) Rt

1-35. RLLS = JS Rxo + (1 - JS) Rt

Where JD = JD (di), JS = JS (di)

The dual laterolog is usually combined with some kind of very shallow investigationdevice. This is usually one of the pad micro-resistivity system such as the MSFL (or MG)which allows us to determine a value for Rxo. When this resistivity is also boreholecorrected, we obtain two equations and two unknowns. With the borehole correctedMSFL resistivity assumed equal to Rxo, (as a first approximation) Equation 1-34 and 1-35

shows the unknowns is Rt and di. Note that JD and JS are a function of the same diameterof invasion. The Butterfly Chart (figure 31) is used for invasion effects. It graphicallysolves our two equations. The chart plots RLLD/Rxo vs RLLD/RLLS. From the chart,correct values for Rt and di are obtained, as well as a more correct value for Rxo.


56 TRAINING MANUAL

FIG: 33 Laterolog Invasion Correction Chart


TRAINING MANUAL 57

Dual Laterolog “Fingerprints”

The behavior of the dual laterolog-MSFL (or the Dual Laterolog-MG) in zones withmovable hydrocarbons permits simple quick-look interpretation. One good rule-of-thumbis the hydrocarbons are indicated where R LLD > R > RLLS MSFL . Conversely, thepattern R MSFL > R LLS > R LLD is a good indication that the zone is wet (100% watersaturation). The log example of figure (34) demonstrates these features.

FIG: 34 Finger Print Log


58 TRAINING MANUAL

The Microlog Tool

The Microlog tool is a simple three-electrode, non-focused electrical logging device. Itwas originally used to indicate porosity. When that use was supplanted by modernporosity devices such as the density, neutron, and sonic logs, use of the microlog fell offdramatically. It has seen increasing use, however, since the mid-1980’s. The tool isvaluable because it offers a superb means to identify mud cake and, therefore, permeablezones. The Microlog is normally run as a separate tool, but Halliburton has also combinedthis resistivity measurement with the density tool by placing a Microlog pad on the densitycaliper arm. This combination, of course, has the added advantage of indicating mud cake(and therefore confirming the Microlog measurement) when the caliper reads less thandrill bit size. The electrode configuration on the pad is shown below.

FIG: 35 Electrode Configuration


TRAINING MANUAL 59

Operationally, a constant survey current is emitted from the lower button. This currentflows through the mud cake, formation, and borehole before returning to the tool case. AMicronormal resistivity curve is recorded by measuring the potential between the upper ormiddle button and tool case. Similarly a Microlateral (also called a microinverse) isrecorded by measuring the potential between the middle and upper buttons. For this tool,the depth of investigation of the Micronormal is approximately twice the spacing betweenA o and the other button that is used. For the upper button, this gives a depth ofinvestigation of 4 inches. Using the measurement from the middle button we obtain a 2inch depth of investigation. For the Microlateral, the depth of investigation isapproximately equal to the spacing between A o and the measure point (the mid point ofthe upper and middle electrodes). This gives a value of 1.5 inch.


60 TRAINING MANUAL

ing the 2 inch Micronormal (with 4 inch depth of investigation) and the Microlateral,when no mud cake is present, as in impermeable zones, both curves should read the samevalue (i.e., they overlay). Thus the curves overlay in shales or in impermeable sands orcarbonates if resistivity is not too high. The presence of mud cake will cause the curves toseparate. The mud cake generally has lower resistivity than the flushed zone. Since theMicrolateral measurement has the shallowest depth of investigation, it should respondprimarily to the mud cake, and read a lower resistivity than the Micronormal. Thepresence of mud cake (permeability) is indicated by a positive separation between theMicronormal and Microlateral curves (Micronormal > Microlateral) and the Microlateralreads close to R mc . The Log example below shows this effect.

FIG: 36 PERMEABILITY INDICATOR LOG


TRAINING MANUAL 61

REFERENCES

1. Bateman, Richard M., Open-Hole Log Analysis and Formation Evaluation,IHROL, Boston, 1985

2. Dewan, John T., Essentials of Modern Open-Hole Log Interpretation, Penn WellPublishing Company, Tulsa, Oklahoma, 1983

3. Study Guide-Dual Laterolog, Gearhart Publications

4. Formation Evaluation Chart Book, Gearhart Publications

5. Pirson, Sylvain, Handbook of Well Log Analysis, Prentice-Hall, Inc, EnglewoodCliffs, N.J., 1963

6. Houston Chapter-SPWLA, The Art of Ancient Log Analysis, Houston, Texas,1979

7. Martin, Maurice H., and Louis A. Allaud, Schlumberger, The History of aTechnique, John Wiley and Sons, New York, 1977

8. Bueche, Fredrick J., College Physics, Schaum's Outline Series, McGraw-Hill BookCompany, New York, 1979

9. Sears, Francis W. and Mark W. Zemansky, University Physics, Addison-WesleyPublishing Company, Reading, Mass., 1977

10. Stratton, Julius Adams, Electromagnetic Theory, McGraw-Hill BookCompany, New York, 1941

11. Thomas, George B., Calculus and Analytic Geometry, Addison-Wesley PublishingCompany, Reading, Mass., 1969

12. Weidner, Richard T., and Robert Sells, Elementary Classical Physics, Allyn andBacon, Inc., Boston, 1965

13. Study Guide-Micro Resistivity, Gearhart Publications

01 resistivity theory

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