A MBGRATORY ENVESTIGAUON 0F
CONDUCYWETY AND DIELECTRIC
CONSTANT TENSORS 0F ROCKS

Thesis for the Degree of Ph. D.
MICHiGAN STATE UNWERSITY
DONALD GARDNER HILL
1969

vHESlE

This is to certify that the
thesis entitled

A Laboratory Investigation of

Conductivity and Dielectric Constant Tensors

in Rocks

presented by

Donald Gardner Hill

has been accepted towards fulfillment
of the requirements for

Pll.D. degree in 680105”!

  

/ " I.
Date 4 ..

0-169

9 ”MARY
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ABSTRACT

    
    
    
   
   
   
 
  
 
  
 

{A LABORATORY INVESTIGATION OF CONDUCTIVITY
‘-;, AND DIELECTRIC CONSTANT TENSORS OF ROCKS

BY

Donald Gardner Hill

3il} . The dielectric constant and conductivity of a material
_§?€e.symmetric second-rank tensors. These particular tensors
grrepresented mathematically by a symmetric 3 x 3 matrix
>igeometrica11y by an ellipsoidal surface. Six inde-

ikqt coefficients must be determined to completely define
% Ligxtensors. Thus, measurements of these properties must
I'lnfide in at least six different directions, to completely
”‘e the tensors. Electrical anisotropy is studied by

the orientations and magnitudes of the conductivity

dielectric constant tensors. An ordered arrangement,

”*‘g3 and dielectric constant tensors.

.,Laboratory A.C. dielectric constant and electrical

Iggagn~o£ Michigan and Ontario. Sample preparation
L]

,gmant followed A.S.T.M. standards D 150-65 (1965)

 

 

 

 

gastric materials .
:iielectric consta:
7.2:; range from 30
The results of
arenas, particuld
fixing, are Chara
gszzri-cal properties
510%: frequencies ,
231.9. prospecting.
electric data gene
Titexhibit strong or
Elite: and minimum
~shou1d be exerci

’rnzW
_--...rlon methods for

 

Donald Gardner Hill

   
  
   
  
   
 
 
 
  

'Iaterials. The anisotropy of the conductivity

Lgéric constant tensors was studied over the fre-
- I)"-

firings from 30 to 100,000 cps.

pawn U"
“=The results of this laboratory study indicate that

_,;j¢ffiocks, particularly those with pronounced lineation,
57§Inding, are characterized by strongly anisotropic

M Dvical properties. This anisotropy tends to increase
{L alowur frequencies, in the range of those used for E.M.

I.P. prospecting. Theoretical methods of interpreting

ate, and minimum principal values) anisotropy. Thus,

 

 

memo“ 1‘? “

in DIELECTRIC .

C)
Q
3

 
  

in . Micr
Partial fr;

 

 

BY

Donald Gardner Hill

A THESIS

. Submitted to
.1 Michigan State University

in partial fulfillment of the requirements
for the degree of

  

DOCTOR OF PHILOSOPHY

Department of Geology

1969

 

 

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I.‘
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The anti. 1‘ wishe:

3r. w, J. Hinze,
a;a:::e:t of Geology,
':.agement, and cm

:r. H. M. Mooney

_-__:a:::e:t o

r—fi
C)
(D
0
FJ
O

to

u"

sec-r With the ins i :33:

~: Silly ,

Jr. H. B. Stone?
a State Univers

..;:::.lOI‘.S and critic
7‘": Of this Study.

Mr T
' ° Do l‘iaggO}

“o
I‘VE."

“h for his aid

5. \
.9.

mp _
rue N '
llChlgan St
‘312‘éerin
9' for the

‘i‘ECtr.

The l

L

.‘i
~«nanc'
1a1
support

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cnf.‘("
1‘ 53.!‘1

-

  
  
  
  
  
  
  
 
 
  
 
   

r. f 5- . i ‘ ACKNOWLEDGMENTS

é‘ex

. P:
.:.:ys The author wishes to express his sincere appreciation
slilvr Dr. W. J. Hinze, of the Michigan State University
§w.-n:nt of Geology, for his patient guidance, advice,
inn agement, and criticism throughout this study.
't ”Dr. H. M. Mooney, of the University of Minnesota
f: tment of Geology and Geophysics, for providing the

riayjor with the insight and inspiration needed to undertake

Dr. H. B. Stonehouse and Dr. J. W. Trow, of the
fgan State University Department of Geology, for their

:Btions and criticisms concerning the geological as-

‘ ' in
Ir. addition, the

.n-r- with Mr O J . D .

infir. 2. L. Hill-5'

LA; u
.

agar-err of f-iat‘rxersdt
‘:.::aio School of Mi.”
Zla:allister, of The
2:12, of the ..ic..ig;

22?.atatics to be ver

   
  
  
  
  

.AnflCowen, of the Michigan State University Depart-

14l’l'tysics: Mr. M. Halverson, of the Anaconda Com-

ftment of Mathematics; Dr. G. V. Keller, of the

~5€prdo School of Mines Department of Geophysics; Mr.

vfifi} of the Michigan State University Department of

:fdmatics to be very helpful.

iii

 

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a 3'-
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1. 131.41JUUL; IUA‘ .

N

nationals 5
Previous Ho
Purpose of

II. THE EFFECT OF
EHSOTROPIQ

Introductir
The Isotrc;
The Isotro;
The liOmOgc:
Apparent It
Review of
Horizon
Value P

Image 8
Lays
Transfo
Co-c

The Anisot
and BQL
The HOmOgs
Space ‘
The Anisoi
Value ]

TABLE OF CONTENTS

Page

   
  
  
  
 
 
 
  
  
   
  
   
 
  
 
  
  

:' NTS . . . I O . O C U I O I ii
OF TABLES . . . . . . . . . . . . vii

fT OF FIGURES . . . . . . . . . . . . ix

INTRODUCTION. . . . . . . . . . . l

Rationale for the Present Study . . . 1
Previous Work . . . . . . . . . 3
' Purpose of the Investigation . . . . 5
THE EFFECT OF ORTHORHOMBIC ANISOTROPY--THE
ANISOTROPIC LAYERED EARTH PROBLEM. . .
Introduction. . . . . . . . 7
The Isotropic Differential Equation . . 8
The Isotropic Boundary Conditions. . . 9
The Homogeneous, Isotropic Half—Space . 10
Apparent Resistivity . . . . . . . 11
Review of the Homogeneous, Isotropic,
Horizontally Layered Earth Boundary
Value Problem . . . . . . . . 14
Image Solution to the IsotrOpic
Layered Earth Problem. . . 16
Transformation to Cylindrical Polar
CO‘ordinateS o a o o o o o a 17
The Anisotropic Differential Equation
and Boundary Conditions . . . . 20 a
The Homogeneous, Anisotropic Half— ,fi
,-. Space . . . 23 I
~3 The Anisotropic Layered Earth Boundary
‘ ' Value Problem . . . . . . . . 24

iv

 

ConVCrSiC
Co- :3
The Methc

Qualitative
Results

L r“ ‘. '
hnroduct-o:
m' '\' l

the ul€-eCtI

The Conjuc:;
.t HEIDSSY DIELE

Introductirx
lbcroscoyi;
Polarizalil:
Inssy Dielec
Qualitative

for Rocks
Predictions

Electric

: Y‘xv
., LIPEETVE‘""‘ p
HAH~ .“hL :
a .

Instrumentat
Laboratorv 3

Procedur.
Rational

meats
EXperimo

The La

hora:

 

 

 

:gq.
SQ’w'f'

Page
J'y'

'.:hva 1Conversion to Cylindrical Polar

Co-ordinates . . . . . . . . 25
The Method of Images . . . . . . 27

  
 
  
   
   
  
  
  
  
 
  
 
 
  
  
  
  

Qualitative Extension of Buchheim's
Results . . . . . . . . . . 29

'THE CONDUCTIVITY AND DIELECTRIC CONSTANT
TEtJS ORS O O O I O O O l O I O U 3 4

Introduction . . . . . . . . . . 34
The Dielectric Constant Tensor . . . . 37
The Conductivity Tensor . . . . . . 40

THE LOSSY DIELECTRIC MODEL FOR ROCKS . . . 45

Introduction . . . . . . 45
Macroscopic Electrical Model. . . . . 46
Polarizability . . . . . . . 49
Lossy Dielectric Frequency Dependence. . 50

Qualitative Microscopic Electric Model

for Rocks . . . . . . 53
Predictions Based on the Qualitative

Electric Model for Rocks . . . . . 56

EXPERIMENTAL PROCEDURES . . . . . . . 60

5Instrumentation . . . . . . . . 60
Laboratory Measurements . . . . . . 60

Procedure . . . . . 60
Rationale for Two-Electrode Measure-
ments 0 O O I I 0 O O O I 66

Experimental Limitations . . . . . 67
.33CNDYT TThe Laboratory Measurement Samples. . . 68
‘ — s

Sample Preparation . . . . . . . 68
Rationale for Measurements on Dry

Samples. . 70
Limitations of the Sample Preparation 0
“ Procedure . . . . . . . . . 74 ' ”

 

“ SPCA A‘JALYSIS .

Introductio:
Least-Square
Second-Fr
Principal C<
for a Ge:
Rank Ten:

Presentatio:
Conduction:
Magnitue!
Discussion l
AniSOtrOQV
Tense: SIZE";
ROCk Fabric
prlnCipal l)
parallehsr
ReSODEHZCe F
Effects Of
Consequence

.3. CONCLUSIONS.
l. RECOP-l‘lETIDATIOIJ
:rn~~ ~ .

. or HEFLRERCES .
M...
;"‘“.::Y N v

.....,, ILFEREI‘IIES .
sonar

n, ‘ ‘
‘ A
'- hlfl‘

.fs.““x

 

   

computel' Pro

ESCripti

 

   
  
  
 
  
 
 
 
 
  
 
   
 
   
  
  
   
  
 
  
  
 
  

«U,'
pn,NQATA ANALYSIS . . . . . . . . . .

Introduction . . . . . . . .
Least-Square Determination of. Symmetric
Second-Rank Tensor Coefficients . .
Principal Coefficients and Directions
for a Generalized Symmetric Second-
Rank Tensor. . . . . . . . .

SAMPLE LOCATIONS AND DESCRIPTIONS . . .

DISCUSSION AND INTERPRETATION OF THE
LABORATORY MEASUREMENTS. . . . . .

Presentation of the Data . . . .
Conductivity and Dielectric Constant
Magnitudes . . . . . . . .
Discussion of Error . . . . . .
Anisotropy Ratios. . . . . .
Tensor Symmetry . . .
Rock Fabric and Structural Control .
Principal Direction Dispersion . .
Parallelism of the O and K Tensors .
Resonance Frequencies . . . . .
Effects of Moisture . . .
Consequences of the Laboratory Results

CONCLUSIONS. . . . . . . . . . .
RECOMMENDATIONS FOR FURTHER STUDY . . .
. ’ a? or REFERENCES. o o o o o o o o o o

RAE REFERENCES. . . . . . . . . . .

4:1. :55 ,
Lugggl Review of Polarization Mechanisms . . .

Computer Program and Subroutine
Deacriptions . . . . . . . . .

vi

Page
75
75

76

81

86

95
95

120
124
127
133
137
139
140
141
142
144

146
149
151
160

164

172

 

L15

Electrical C135;

Instrumentat in.
Pr0perties St

Mineralogy of t~'.

:irection Anglg
Axes , .

M'1 (Meta-Argi'
Data . .

“'2 (Greenston
Data .

M (mphiboll
Data

8-4(Syenite (
Data

”'5 (Granite
Data

M‘6 (Sub G
- ra“
Data . J
M‘7V(GrayWac}
;‘IOt a Leaf
DEtErminat

trepy Data

M-9 (Hemlock
trOpy Data

MO (Amphi‘:

LIST OF TABLES

  
   
  
  
   
   
   
  
   
    
  
  
   
   
   

Page

Electrical Classification of Materials. . 45
Instrumentation for the Electrical

Properties Study . . . . . . . . 62
VMineralogy of M—7. . . . . . . . . 90
Direction Angles for Tensor Principal

Axes . O O O O O I I O O I I 9 5
M91 (Meta-Argillite) Electrical Anisotropy

Data 0‘ O I 0 O I O O O I I O 96
M—Z (Greenstone) Electrical Anisotropy

Data 0 O O O O O O I I O O O 98

Mr3 (Amphibolite) Electrical Anisotropy
Data C O I O D O I O I O I O 100
8-4 (Syenite Gneiss) Electrical Anisotropy

Data . O C D O I O I C I O I 102

M-S (Granite Gneiss) Electrical Anisotropy
Data 0 O I O I I O I I I O O 1 o 4

'M-6 (Sub-Graywacke) Electrical Anisotropy
Data 0 O C I O I l O O I 0 I 1 o 6

fwg%"M-7 (Graywacke) Electrical Anisotropy Data
~ (Not a Least Square Tensor Coefficient
_Determination) . . . . . . . . . 108

T°VM58 (Staurolite Schist) Electrical Aniso—
.tr0py Data . . . . . . . . . . 110

.;IM-9 (Hemlock Formation) Electrical Aniso-

;j._ tropy Data . . . . . . . . . . 112
,1: “7?. ,

"Irma-'10 (Amphibole Schist) Electrical Aniso-

‘rfl 4‘; tropy Data 0 o o o o o o o o o 114

vii

 

 

a:

34-11 (Siltstor
Data 0 0

HA (Greenstc
Moisture) 53

Electrical Ami

 

 

Table
8.12.

 

_..

M—ll (Siltstone) Electrical Anisotropy

Data . . . . . . . .

M—2A (Greenstone with Atmospheric

Moisture) Electrical Anisotropy Data.

Electrical Anisotropy Summary.

viii

Page

116

118

128

:12

‘l
J...

t4
’4
(I

Esmc'qeneous, lsr
Model . . .

'vs‘exner Electra;

‘ ‘ ‘l.‘
Scnl‘amerger
Two-Laver Bart:

Reamer Configu
Curves for
Layered Cart
and Cook, 1.

Schl‘mberger <
Master Curr
tropic, TWC
Orellara a:-

Wenner Elect;
Arbitrary (
Principal .

LCSSY Dielec
Circuit

LOSSY DiElEC
prOpertieE
964; Kel

Block Diagr

Analog Cir:

Analo

9 Cir:
H01

d‘er v;

"1

LIST OF FIGURES

  
   
  
  
 
 
   
  
   
  
   
  
  
   
 
  
  
  
 
  

Homogeneous, Isotropic Half—Space Earth
”we 1 O O C O O O C l I I O O

Wanner Electrode Configuration . . . .
Schlumberger Electrode Configuration . .
Two-Layer Earth Model . . . . . . .

Wenner Configuration Resistivity Master
Curves for Homogeneous, Isotropic, Two-
Layered Earth Models (After Van Nostrand
and Cook, 1966). . . . . . . . .

Schlumberger Configuration Resistivity
Master Curves for Homogeneous, Iso-
’tropic, Two-Layered Earth Models (After
Orellana and Mooney, 1965) . . . . .

Wanner Electrode Configuration at Some
Arbitrary Orientation to the Horizontal
Principal Tensor Directions. . . . .

Lossy Dielectric Response and Analog R-C
Circuit 0 O O O I O I O I O D

LosSy Dielectric Dispersion of Electrical
v Properties (After Wert and Thompson,
1964; Keller and Frischknecht, 1966). .

Block Diagram of Electrical Properties
” Measurement Circuit . . . . . . .

Analog Circuit for Micrometer Sample
”'Holder Containing Sample. . . . . .

“7y§nalog Circuit for Micrometer Sample
1.W”Kolder'Without Sample. . . . , , ,
. L. .k 1

ix

 

Page

10
12
13
15

18

21

30

47

52

61

63

65

 

Cross-Section 1"!
Surface in tzu
u and I . -

.
:00

Eorthern Michig
Microscopic Pet
Euhedral Staurc

E.'.. 24-1 (Meta-Argil
tropy Data .

M-Z (Greenston-
Data . . .
33. M'3 (:‘flphlbOll
Data . . .
5-" 5‘4 (Syenite C
tropy Data .
”‘3 (Granite C
trOpy Data

m. M'6(SUb-Gravn
trOPY Data‘

W7 (Gray‘dacI-z

 

 

Data .
..c. M—S (Stauroli
Anisotropy
5.9.
Amigotroby
.. M‘lolAmphib
”1i ‘1
”‘11 (ants.
Data . ‘
€42. . '
a a h
”:33. '
(driatio g

 

'ffl;
*4:
ISPQBs-Section Through the T Reference

' Surface in the Plane Containing Both
L n and v I I I I I I I I I I I

  
  
  
  
  
 
  
  
   
  
  
  
  
   
  
  
  
   
   
   
  
  

Northern Michigan Sample Location Map. .
Microscopic Petrofabric Analysis of M-7 .
Euhedral Staurolite Crystals in M-8 . .

M—l (Meta-Argillite) Electrical Aniso-
tropy Data I I I I I I I I I I

M-Z (Greenstone) Electrical Anisotropy
Data I I I I I I I I I I I I

M-3 (Amphibolite) Electrical Anisotropy
Data I I I I I I I I I I I I

3-4 (Syenite Gneiss) Electrical Aniso-
tropy Data . . . . . . . . . .

M-S (Granite Gneiss) Electrical Aniso-
tropy Data . . . . . . . . . .

M-6 (Sub-Graywacke) Electrical Aniso-
tropy Data . . . . . . . . . .

M—7 (Graywacke) Electrical Anisotropy
Data I I I I I I I I I I I I

M-8 (Staurolite Schist) Electrical
Anisotropy Data . . . . . . . .

M-9 (Hemlock Formation) Electrical
Anisotropy Data . . . . . . . .

M-lO (Amphibole Schist) Electrical
Anisotropy Data . . . . . . . .

'-'Mrll (Siltstone) Electrical Anisotropy
Data I I I I I I I I I I I I

W ‘EPZA (Greenstone) Electrical Anisotropy
~ Data I I I I I I I I I I I I

F
Yariation of the Less Determined Data
- ;. About the Results Obtained Using A11
’f\.Directiona1 Measurements . . . . .

x

Page

83
87
91

93

97

99

101

103

105

107

109

111

113

115

117

119

125

. i
LI

,0

 

Exisotropy Rat

w Order for

his—v“

 
   

V - >$“ Page
'1m2MiIotropy Ratio Relationships . . . . 131

. ”nook Order for Program ELECT. . . . 175

“4%” Bloc!
W 00.1mm“
ml: at '
w Thomgso-A
-II‘IIII \ca
mu, 0, 2:7

. i am

 

 

* .

     

xi

 

 

Rationale

 

(‘

ElectricaNYr I“
j:: semiconductor S .
11:: of the actual i3:
:if'mnpson (1964) 511
isssification given
;:;5 tost practical
17:3, 3, of rocks a
3. 32:11 the dielect:
Eterial are symmetr:
:"ienpletely defim
5‘52 presented as

-"::i:‘.1ar v '
‘ Szm’netrlc

. .
~_'

‘A
“4.:

‘14

graphically L‘

-~:Yfi31‘;fl€l that thfi

3118-31 pro '
pertles c

"3318, the SYYu‘e‘

I
It“.

i" 4 '
A‘Jfar.

was they I‘Ep‘

11% Ch.»
‘1 Stallogras‘qi.

.451
‘4ig’. m

CHAPTER I

INTRODUCTION

Rationale for the Present Study

 

    
 
  
  
 
   
  
   
   
 
  

hjfirfisemiconductors. This is true, both from the stand-
u-t of the actual physical definitions, as given by Wert
;£Vfgtflhompson (1964) and Beam (1965), or the rule of thumb
-U£§§§hsification given by Keller (1966). For this reason,
\ ifLS most practical to consider the electrical conduc-
!'*ty, 0, of rocks along with their dielectric constant,
'Yfifloth.the dielectric constant and conductivity of a
axial are symmetric, second-rank tensors. As such, they
“completely defined by six independent coefficients
‘€.presented as a symmetric 3 x 3 matrix). These two
3» lax symmetric second-rank tensors also can be repre-
f;éigraphically by ellipsoidal surfaces, in much the
.;nanner that the optical indicatrix represents the
'lnprOPerties of transparent materials. Within

’1; the symmetry of these surfaces and that of the

 

H yighns; we can orient crystals by the symmetry of
‘}L91V v '
12"

 

 

‘

.w;:ysica1. E
"22:325.“.1p 11‘?

‘ abr 1 c .

_-fiwiv'b -

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”O‘Or‘ina

08$ nOt
3L3: _ '
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their physical properties. One would expect a similar
relationship in rocks, particularly those with a pro-
nounced fabric.

Theoretical methods of interpreting geoelectric
data generally assume that earth materials do not exhibit
strong orthorhombic anisotropy (distinct maximum, inter-
mediate, and minimum principal values). For an example
of this, we need look no further than the many papers on
the potential distribution about a point electrode oVer
a horizontally layered earth (cf., Stefansco g£_gl., 1930;
Peters and Bardeen, 1932; Grant and West, 1965; Keller and
Frischknecht,1966). In the isotropic layered earth pro-
blem, the assumption is made that the earth consists of
parallel, homogeneous, and isotropic layers. The differ—
ential equation (D.E.) to be satisfied is Laplace's
equation and the model has cylindrical symmetry. Thus,
the solution is in terms of exponentials in z, the verti—
cal distance, and zero order Bessel functions of the first
kind in r, the radial distance from the source.

If, however, the requirement that the layers be
isotropic is eliminated from the B.V.P., it apparently
has no analytical solution. Conversion to cylindrical
polar co-ordinates, which is so helpful in the isotropic
case, does not yield a D.E. which has an analytical solu-
tion. A co—ordinate transformation so that the layers

appear to be isotropic in the transformed system

 

:-::iinate syst

If the el»:
zayce possil
33:55 based c.
:3. The put
"_EEZ-SSlblE O
:::j.e:ties c-i

.. . Q _ q. . - ’
‘--: D L- ‘d‘i ‘A T;

2:3“: :HV
‘0?! (cf
‘12.:9'.’

\ ,Ea+
°~ gation
31435»
ti USES f
"3318

   
 
  
  
  
   
    
 
  
   
  
  
 
 
 
  

'fi

ay~ hate systems at the boundaries.

';%F‘ If the electrical anisotropy of rocks is not great,
’ in? be possible to use theoretical interpretation

9 Dds based on the assumption of isotropy with little

‘gggior. The purpose of this study will be to investigate

}
ޤ$ possible orthorhombic anisotropy in the electrical

't‘é
rties of vacuum dried rocks. Rocks were selected for

4 ’7 t
‘( ”file'study which are most suspect of exhibiting anisotropy
i“"‘i.e., those rock types which exhibit strong fabrics).

3;!53 absence of anisotropy in the rock types studied would

e1iminate the possibility of strong electrical aniso-

‘Qy in rocks. It would, however, cast doubt on its

§£22122§_fl2££

' Pyd_There has been very little previous work on electrical
ifyjiifotropy of rocks. Those papers which did consider
1 “hibotropy (cf., Schlumberger et al., 1934, Rao, 1948;

-3‘“
Vfioy, 1961) did not attempt to completely define the

L7,“.

£0r_ properties. There has, however, been more work on

”1931 properties in general. An early method of

 

_..-1.
\Ir
II. '

mint ele

e (2

_=:.:-_ strap potent
.‘--OV6AQ “E“.il‘

1.:CUIO‘JUV .

22513.“. as to w‘r.

.....

"’“Ctlvity
n: ;i
3
- 5
:"‘Jr“‘ .
:1 10:1 ‘
f"".s~h‘
a ‘blSL
LS
1".
SE‘A‘ ~
bu“
l‘ran}
‘\
’5. +
5‘ .
03 net
‘ E
‘Sgs‘,
‘sr‘x
U -..
by 1

  
  
  
  
   
    
   
  
 
  
  
  
  
  
  
  

'*}f Sfigap potential electrodes and plate current
;;zodes. While he succeeded at this, there is some

;;tion as to whether or not he completely eliminated

‘zeffects of polarization at the sample-electrode con-

. Ward (1953) and Orr (1964) used inductive measure-

: JQBntB. Keller and Licastro (1959), Arbogast et a1. (1960),

 

gavell and Licastro (1961), Stacey (1961), Simandoux (1963),
2 and Liessman (1964) used capacitance measurements. Keller
- 3H(1966) and Parkhomenko (1967) gave the most recent com-

JV flyilations of electrical properties determined for rocks

1 wk? 7

‘fand minerals. Heiland (1940) and Parkhomenko (1967) have
gatalogued measurement methods in some detail.

A very thorough literature search, as well as written
:joral discussions with industrial research laboratory
51 ”sonnel, revealed that apparently no one has attempted,
lia‘now attempting, to completely define the electrical

u.:'vnctivity or dielectric constant tensors of rocks.

:ioists normally do not consider a and K as symmetric

,--rank tensors and thus fail to define them completely.

‘1’

 

“1.333.“ it does no

:::;';i°rank tensor .

Purccsa
The purpose of
SELEClOYI underl‘dng
221:: rpretation,
:atleast have cyl;

:..es') in their axis
52:;2ion will only

Before come no

1:32 tnat a theoreti

’Mp.
.-

f.....rty to be SUN:j
2::e present st‘dd‘
L'ww‘n
“Mieal model f0
15$. ihlS model i
Name for meaSL
--q.».:e qualitative
wage of usinc
labo
r
atory pro
nation theorv
111 yi ‘
eld estima‘
iiseé mod
91 and t‘

“my would be t

  
   
    
   
  
  
     
   
  
  
  
    
   
 
  
   
 
  
  

:fgyuit does not completely define the symmetric
.1;fs:isnk tensor.
.‘.' '
v1.5
Purpose of the Investigation

 

9 The purpose of this study is to test the fundamental

‘J 75!} l

.1 aisumption underlying theoretical methods of geoelectric
.33 ,

, -§h&a interpretation, that rocks are essentially isotropic,
,.

- F6! at least have cylindrical symmetry (two equal principal

-y

1."..Jx'v

.*E guinea) in their anisotropy. In the present study, this
‘ “.3.ch-

v -}absumption will only be tested for vacuum dried rocks.

.‘c.

. 1’.
‘1 .. Before commencing a petrophysics study, it is impera—

. 6

~ ~ ' tive that a theoretical model be developed for the rock

-"7 5‘5

fipropérty to be studied. Thus, much of the early portion
7‘;‘.; 3; the present study was devoted to developing a qualitative
. ‘gieétrical model for rocks based on sound physical princi—
-; gé1;;.- This model is then used to design a laboratory

«. ma}. E

C

‘ procedure for measuring electrical anisotropy of rocks and
1:9 give qualitative predictions of expected results. The
'%%£Vantage of using a theoretically based model to design

II"- in-“ ,lfin

V$\« i laboratory procedure is that it allows the use of
.-,
t .’

tion theory in the interpretation of the data. This

model and their accuracy. Opposed to estimation

would be tests of significance. This approach

 

.=2:e used when littl
:catheareticall)’ “3
3:315 more app/[Opt
The theoreticall
alliethat of the 1,3
gzratcry and data It
1:12 will completely
:i??ter.sors. Care
sixes which will a]
areas-snable amount .
The efiects of
3-73. used in them
3150 considered
ir‘iso'tmpy to a
FEESible' ‘10 Solve

2711a '
0 ~¥ a
9 mad by p:

«
.9
n.
‘s I

fut,-
‘ -Ca1 Solutio
Tiled .
b1 Simplify
52:3:0 .
n 15 Possik

kmapllcated 1

  
  
  
  
  
  
    
   
  
  
  
  
  
  

gigretically valid model is available, estimation
‘ajis more appropriate.

’.:fwhe theoretically based model developed for rocks

I; be that of the lossy dielectric. Thus, suitable

1 atory and data reduction procedures must be developed
will completely define the symmetric second-rank c
?K tensors. Care must be exercised to develop pro-
‘r.rg@§hres which will allow measurements to be repeated with
'nsonable amount of precision.

43nd “The effects of strong orthorhombic anisotropy on a
QQWRPQ used in theoretical geoelectric data interpretation
, %aiso considered. As previously indicated, the addition
5“: endsotrOpy to a B.V.P. makes it difficult, if not im—
v-le, to solve analytically. However, some insight
4befgained by proceeding as far as possible toward an
lktical solution. Qualitative predictions may also be

Vlffby.simplifying the B.V.P. so that an analytical

fer

_.’, m C‘ s~
1.25 EFFECL o. ‘-

AXISOTRCE’I L

The eleccrical c!
segmetric second-r

arrest to measure

2:5;ier the effect t3
Laztrapy has on the
:ter‘retation methOj
rim.) problem is
inseffect. Other <
3;: this particular

Eistically the prC
Iterhombic anisotr
illiary value probl

~~~ .oRdUCtiVi ty 1

2:: .. .
“vat?
1L

Jlty (Conduc

(?)0 Or t
rssist

CHAPTER II

   
  
  
  
  
  
  
  
 
 
  
  
  
  
  
 
 
  

. THE EFFECT OF ORTHORHOMBIC ANISOTROPY--THE

‘ 1’ 2 ANISOTROPIC LAYJERED EARTH PROBLEM

.y.-_-
j'.1 Introduction

-1$rr The electrical conductivity and dielectric constant

3 aéée symmetric second-rank tensors. Before designing an

lgy$§g6riment to measure the electrical properties of rocks

;Qi3ustudy their electrical anisotropy, it is helpful to

.;;_;E§sider the effect.that orthorhombic (low symmetry)

- Entropy has on theoretically based geoelectric data
g~:erpretation methods. The layered earth resistivity
I. P. ) problem is chosen as a vehicle to demonstrate
., ; effect. Other geoelectric problems could be used,
jfthis particular problem is useful because it shows
urtically the problems involved when the condition of
irhcmbic anisotropy is added to the layered earth
gary value problem (B. V. P. ). In this chapter, only
gaflductivity, a, will be considered. However, the
fiivity (conductivity) results of this chapter can

1.4,":
' verted to I .P. response by use of the complex conduc-

fig} 3 jwe + a = ooejg, where do = / wig: + o2 and

 

ziroiel will first i
aprablen will be co:
irisctropy. The dis.
:25: about the bounda
::;~::ential distribut

:erei earth. A bO‘JT-u

 

sired bv a differenti

 

;::.iary conditions (1:

1 0". ~

”4.11101". which 5 at i s

2-" M

of interest ar.‘

‘1

The lsotr:
N
Electric curre:
.and Lorrain,

or Sinks, t‘:

spot, .1, vanisheS

  
  
  
  
   
  
  
  
  
  
 
  
 
  

f 6111 first be briefly reviewed. Then, the
1‘1em will be considered with the added condition
is ;0py. The discussions of this chapter will
' 'tfihout the boundary value problems concerned with
s7i'ipotentia1 distribution about a point source in a
“prered earth. A boundary value problem (B.V.P.) is
.Jij‘ined by a differential equation (D.E.) and a set of
'13:: Eary conditions (B. C. ). The solution to a B.V.P. is
” §Unction which satisfies the D. E. at every point in the

u.)
w sion of interest and the B.C. at all boundaries of the

 

Vfg = 0. 2.1
'1. ‘ 19(4)}. c'.

scurrent density vector, i, is related to the electric
éiinfiensity vector, E, by the vector form of Ohm's
3'8 GE, and E is related to the electric potential,

.31! 1W- For isotropic materials, a is a scalar,

hV-aEa-OV vv--ovv=o,

 

 

Li“. is Laplace‘ s equ;

:s;s::vity problem po:

glace's equation (2. .

The Isotrc
--—-—--—|

The basic B.C . f

 

5.2::ric current in ma
:rnent of g and tin
2113113215 across all
:fiLarrain, 1962) .

7;:zent of Ti, it i
futon, V, should
mes

I .
\Cr ant and \x‘ e:

353:3,"
‘ ““9 material

   
   
 
 
  
 
  
  
  
 
    
  
  

The Isotropic Boundary Conditions
The basic B.C. for a B.V.P. involving the flow of
;:;;3ctric current in materials are that the tangential
ViHJ-nent of E and the normal component of i must be
‘o§_'nuous across all boundaries in the model (Corson

. ”kééhorrain, 1962). From the continuity of the tangential
Ir (Qwent of E, it follows directly that the potential
£1 on, V, should also be continuous across all bound-
_Txicrant and West, 1965). For example, at a boundary

‘ sting material 1 from material 2, the B.C. would be:

"VI a v2. 2.3

* 73WTrdk an isotropic material, a is a scalar. Thus,
fin:isotropic material, the normal component of the

': density vector may be written as:
‘ .".-“k r.) ‘ V .'
1" I

 

The above B.C. (Of;
355183. for all i
a, there are usually
:sztial function at ti.
:22 from the source .

"axial function .

l

The homogenec
Consider a currer.
:azeé on a plane hour:
:22 of conductivity,
43: in Figure 2.1. '1
5:;2-3 conductivity is

‘V-
; .i‘
s

""7" It is thus

 

3'“, go 2
Ni mOdEI .l.“HOr

   
 
   

10

.éove B.C.

     
   
  

(equations 2.3 and 2.4) are the most

 
 

,ifixyirs.c. for all isotropic earth models. In addi-

.y~¥h¢re are usually B.C. involving the values of the
t .

  
   
   
  

f}{.;' The Homogeneous, Isotropic Half-space

A» rConsider a current (point) source electrode, S,

‘i«; of conductivity, 0, from a half-Space of o = 0, as

It} in.Figure 2.1. Because only the half-space with

     
 
        

l‘?’te conductivity is of interest, the model has spherical

 
 

thry. It is thus convenient to convert Laplace's

  

ftion (2.2) into spherical polar co—ordinates. Because

‘t-isotropic, the potential function will be a_function

g:i I x2 + y2 + 22 only and Laplace's equation becomes:

1.

' o

l" ‘-
p'.‘, i

" Ida-o».

5’, W‘

. LEM

dV
3-133 0. 2.5

 

.,;:.i;:ion of the 8""

iii the earth model

‘A

51. 1765‘? killer and

M‘ l
at“) : m I

62:21 is the current

.
mpsnn
. n

..... e resistivity a:

‘
r - m

.: measured at t

‘5..." .. ‘ 2 A
......lr.g r - r X + Y

:.:;an about a point e

 

‘«’(r0 = I
') T7;

In the deve loo:

1”
P!

N

3“: ‘i
..s.med to be in
4'." f‘eld

. operation
- stint electrode

~- on potential 9

ZLstanceg
) o For mu

Sir“) the
Potential
‘2 2:1: .
‘t Point Cine

Ely (beCaUSc

SE of +7
“’0 Potenti-
u:: <
u.~er8nh
ue
I [N ‘
0 W.'“

A

:5 Wu .
Stiv‘
lty 0f

The solution of the B.V.P. defined by the D.E. of equation
2.5 and the earth model of Figure 2.1 is (cf., Grant and

West, 1965; Keller and Frischknecht, 1966):

WR) =fi%§ '

where I is the current introduced at the source, S. For
surface resistivity and I.P. prospecting, the potentials
are only measured at the earth's surface, or at z = 0.

Defining r = V x2 + y , then the surface potential distri-

 

l bution about a point electrode is:

I V(r,0) . 2.6

I
nor

 

i Apparent Resistivity

In the development of equation 2.6, the current sink
was assumed to be infinitely distant from the source.
Many field operations, however utilize variations of a
four-point electrode configuration (two current electrodes
and two potential electrodes, all separated by finite
distances). For multiple current electrodes (source and
sink) the potential at a point is the sum of the potentials
at that point due to each of the current electrodes acting
separately (because Laplace's equation is linear). The
use of two potential electrodes yields the potential
difference, AV, which allows the direct calculation of

_the resistivity of the half-space. For a generalized

 

 

;:-;:int electrode cor.

 

"Instant r1 and r2 f:
(respectively) and P

:sgecti’mly), the resi ,

T:s:‘:.l<:.cht, 1966; Vaul

L=s =......“ (.1
l r‘
1

 

:gtactical iield W01"
L—L'iie'fich simplify 91
taintisn, or both. "-
ignetions are the l4
i of these conf
iis:raight line a:

u.
.5
~

. neutlguration .

in“.

‘_ H

i§‘:
.

12

   
    
   
 
 
  
    

upractical field work, certain geometrical patterns are

ea which simplify either the field procedures, data

  

l‘ u
. 5‘.-

‘7' 4- . .
uo-ction, or both. Two of the most common electrode con-

Elf ations are the Wenner and Schlumberger configurations.
, §1both of these configurations, the electrodes are arranged
__Efla.straight line and are symmetrical about the center of

(iconfiguration. However, there are significant differ—

l." _h

. ,‘ ,_ .' . -' ’..:
,g'?’ yea-«r rte—flu - ..

 

 

 

 

nah,

 

.-

K

.1g3‘2.2.-—Wenner electrode configuration.

' e equally spaced and their common spacing,

 

:rent 8 lectr
""*‘¢*berqer <

13

fl and %¥ measured. For this electrode con-

  
       

;fi,1equation 2.7 becomes (Grant and West, 1965;

 

       

 

 

 

 

AV
P2 0 Pl SI
Me A” L/2__,..h

$339. 2.3.--Schlumberger electrode configuration.

fgpnfiential electrode spacing, a, is much less than the
gaaitTGIectrode spacing, L. Usually, L/a>>5, for the
3yerrger configuration, as opposed to L/a = 3, for

fifienner configuration. For the Schlumberger configur-

9“ IAV L2(1‘%2) . 2.9

«than pas-f (ii) 291! and in the limit, a + 0, this

 

t.)
<3

0"
HI T“
Jr

C

(A
'1

The resistivities
..’.C are true resistiv
253351" approximate
”.5333: arenent site .
fries yield apparer

agited averages of l

tzpcrerts near the in
722.5, where an anal

:.~.?. exists, this is
“._‘;gn 2.10 can be U
Iis;s*'v'

.itity master C‘

41'3th of the at)?

.Zslstiv' '

samensron of ti.
" d a manner ma‘
”tease

 

 

 

 

l4

  
   
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

peg—(hzég. 2.10
"' The resistivities given by equations 2.7 through

2:10 are true resistivities only if the earth can be
reasonably approximated by the model of Figure 2.1 at

the measurement site. Otherwise, the above resistivity
formulas yield apparent resistivity values, pa, which are
weighted averages of the true resistivities of the earth's
components near the measurement site. For simple earth
models, where an analytical solution to the appropriate
B.V.P. exists, this B.V.P. solution and equations 2.7
through 2.10 can be used to obtain theoretical apparent
resistivity master curves. These are usually plotted as
the ratio of the apparent resistivity to one of the true
resistivities versus the ratio of the electrode spacing to
some dimension of the model on logarithmic paper. Plotting
in such a manner makes the master curves very flexible and

increases their usefulness.

Revieykof the Homogeneous, Isotropic,
Horizontally Layered Earth
oun ary Va ue Pro em
The B.V.P. for the potential distribution about a
point electrode on the surface of an earth model consisting
of homogeneous, isotropic, horizontal layers was solved in

the 1930's and 1940's (cf., Stefanesco et al., 1930; Roman,

{.P. are given bY (
tiszhknecht (1966) ,
7:12 results have *2
iasnany as four <
race, the importan
21121 two-layered
elf-Space) model,
Inties, ol and c
aerial above lay
trait c = 0' St

Put the boun
1.32;“ be satis
3:“: l'Jvder bounds:
LSQ' the Potent:
1‘22 behave like

I
'U

The D.E. whi
2.2:.

L

Pi

15

  
  
  
  
  
  
  
  
  
   
  
  
   
 

' “' While results have been obtained for models consisting

of as many as four or more layers overlying a half-
space, the important concepts are covered by the so-
called two-layered (single horizontal layer overlying a
half-space) model, shown in Figure 2.4. The conduc-
tivities, 01 and 02, are finite and isotropic, while the
material ab0ve layer I (i.e., for z < 0) is taken to be
air with c = 0, so that no current will pass through it.

At the boundaries above and below layer I, equation

2.3 must be satisfied. Equation 2.4 must be satisfied at

3V 3V
the lower boundary and at the upper boundary, 1fi% = 15% = 0.

Also, the potential function, V, must vanish at infinity
and behave like equation 2.6, with o = 01, for points near
8. The D.E. which V must satisfy is Laplace's equation

(2.2).

 

 

 

 

S a
I] 0" // f X
s”/ '
/’ Ir
y’flllz 0'2 1
Zr 5

Fig. 2.4.--Two-layer earth model.

 

 

The methOd 0f

grain B.V.P. ”it?
:2 correct Green'1
:ecorrect Green'
iglace's equation
:: iz‘age located
23$. of the m
:3;-:-s has been u
Erical Optics ,

:iels with simpl
Bind is straiq‘.

Tit. multiple b.

wiled
the met}

at, n
Emblem a

D
a/0 ‘ '

s the .
. “enher

‘4

functic

,1..
.a‘
‘.

 
 
 

., ‘1 '
- ‘lq

   
   
  
 
 

16

_L

Nae Solution to the Isotro-ic

;ere- 'ar 'rOo em

The method of images is a direct method for solving
‘W"certain B.V.P. with simple geometry by attempting to guess
the correct Green's function for the B.V.P. By definition,
the correct Green's function for a given B.V.P. satisfies
Laplace's equation at all points except at its pole and
the image located outside the region of interest, and all
the B.C. of the model (Kellogg, 1929). The method of
images has been used with considerable success in geo-
metrical optics, seismology, and potential theory for
models with simple geometry. For single boundaries, the
method is straight forward and gives a single image. How-
ever, multiple boundaries result in multiple images and
the method can not always be used.

Hummel (1932) and Keller and Frischknecht (1966)
applied the method of images to the isotropic two-layered
earth problem and obtained:

1
”f

pa/pl = {1 + 4 kn [(1 + (2nt/a)2)

IIM8

n 1‘

1
— (4 + (2nt/a)2) 2.1} 2.11

   
 
 

as the Wenner configuration theoretical apparent resis-

‘tivity function. The term, k, is a reflection coefficient

a

’4. 3 gainer: by:
'! ‘.\‘ f '

is tbs-layer Wenne
obtained f r om

Keller and K1

 

ates-layer model

The theoret
itiations 2.11 an
3;:st term is the

“at?! C] l p wh i l!

n "L
~- see two-layer

fsformation t
"‘ar 0‘01: inat
The earth

"‘2?“ tr‘." ab on t

5:“.- e
b.“~fi‘

t0 conv

I":
.‘s‘r

”Pic
9 t

he p04

QLEJL

“:9

17

 

  
  
  
   
  
   
  
  
    
   
  
  

fThe two-layer Wenner master curves shown in Figure 2.5
were obtained from a form of equation 2.12.

Keller and Krischknecht (1966) obtained:

pa/pl = {1 + 2 02° k“[1 + (nt/L)2]’3/2}, 2.13
n=1
as the Schlumberger theoretical resistivity function for
a two-layer model, using the method of images.

The theoretical apparent resistivity functions of
equations 2.11 and 2.13 both consist of two terms. The
first term is the resistivity of a half-space of resis—
tivity, 01, while the second term (series) modifies this

to the two-layer model.

Transformation to Cylindrical
0 ar o-or inates
The earth model shown in Figure 2.4 has cylindrical
symmetry about the 2 (vertical) axis. Thus, it is con-
venient to convert Laplace's equation into cylindrical
polar co-ordinates. Because the conductivities are iso-
tropic, the potential function will be independent of

azimuth, B, and the D.E. becomes:

 

 

 

 

 

 

18

 

f =l

 

(m
///n

 

\\\\\

\u

4,, Mi l\\

//

.0
tn

OJ

3——-5

(100

b
m

wasps? 8

o '0: mo moo
0 «084cm

 

   
 

L, (gig. 2.5.--Wenner configuration resistivity master
~es for homogeneous, isotropic, two-layered earth
i:§after Van Nostrand and Cook, 1966).

 

 

ease the variables
zgirable (See Kreyst
.3122 method of seal?
glysis (see Byerly
iii-e Hankel variet
:ssform was used ‘1
:iSardeen (1932) .
aresults obtaine
=-.'-i':alent to those
1'13 analysis has

415 different frc
“in": it requires
31:, HOWEVer' it
:‘iariables appro
255501-115 1f the

using the Ila

ital
\‘- (1930) ' Pet

m (1955) obtaii

D
6‘[31 = (2c

    
   
  

19

Z’ause the variables, r and z, in equation 2.14 are
separable (see Kreyszig, 1967), the B.V.P. may be solved
‘by the method of separation of variables and harmonic
analysis (see Byerly, 1893) or by an integral transform
of the Hankel variety (see Tranter, 1966). The Hankel
transform was used by Stefanesco g£_al. (1930) and Peters
and Bardeen (1932). Stefanesco gt_gl. (1930) showed that
the results obtained by using Hankel transforms were
equivalent to those obtained using image theory. Har—
monic analysis has not been applied to this problem, for
it is different from the Hankel transform approach only
in that it requires the develOpment of the Hankel trans—
form. However, it is helpful to remember the separation
of variables approach, for it is difficult to use integral
transforms if the variables cannot be separated.
Using the Hankel transform approach, Stefanesco
gt_al. (1930), Peters and Bardeen (1932), and Grant and

West (1965) obtained:

 

where:
°° J°(>\r) '
G(r,k) = 1 + 2kr 1TE— d},
e - k
c

 
 
 
 

k is defined by equation 2.2, and J0(Ar) is a zero order

‘Bessel function of the first kind (see Gray and Mathews,

I
,
1.;
K
r
I «
.1 .

 

:2: Watson! 19'
ration theor‘

Mooney et

51:5 Schlumber

”"5. Q P],

.-.--.. for a t‘

is a firs

;:i. The two-l

camparinq 1
25? consist Of

""3? of a half.

‘4
u-‘ in ‘

“my; 4- .
“em (lflte

--_.
‘1.:i

<1

[C4
II

20

    
 
  
   
   
  
  
  
   
  
    

'1theoretica1 apparent resistivity function.

“a_~éney et a1. (1966) obtained:

., L2 ” J1(AL/2)
pa/pl = 1 + T m dk' 2.16

e
O

VE-Ahe Schlumberger theoretical apparent resistivity

‘ f~ ion for a two-layer earth model. The function,
lit‘gJ) is a first order Bessel function of the first

. The two-layer master curves of Figure 2.6 were

sbned from a form of equation 2.16.

‘ Comparing equations 2.15 and 2.16, it is seen that

“fhonsist of two terms. The first term is the resis-

Lt of a half-space of conductivity, 01, while the

term (integral) modifies this to the two-layer

“The Anisotro ic Differential E nation
and Eounaagy Eonditions
: 7qfis was true for the isotropic case, the starting

“Eor’the D. B. will be equation 2.1. However, in

«ghee, a is not isotropic. Thus the anisotropic D.E.

2
_ _ 3 V -
‘3. 5.53 'I V 0-3- -V O'VV — O'ij-s-m— - 0. 2.17
’ \ i 3

 

 

 

30..

 

 

0_|..

0.05..
0.0}.

 

 

 

 

21

 

t=|
pl'l

P
1-:*L/2

/m

3
m

-— —4v

)

 

21)

p—————Iqo— I. 25

 

U'l

 

l

99997.“!“9‘
muamouuumuomo

 

 

 

     

L2.6.-—Schlumberger configuration resistivity
for homogeneous, isotropic, two-layered
.fiafinsr- Orallana and Mooney, 1965).

r-v,
.;

 

 

32322.17 is elliP‘
mare still involv
satiation of a B.V-
ssi:;le as for thOS-‘i
tazzaroWest (1965)

xfelzpaent only as f
LI}. At that point

:zéizions to simplii
L325: apply synmetrg
Sizever obtain an e
33563 shortly, the
E3331M for equatio
311nm: certain s
isotropic, half-5;;
taster, inSight Shc

ISOteriC
gentlal E
‘iids t .
the COntix

 

  

 

22

fijéuation 2.17 is elliptic (Courant and Hilbert, 1962),

, so we are still involved with potential theory. However,
the solution of a B.V.P. involving equation 2.17 is not

as simple as for those involving Laplace's equation (2.2).
Grant and West (1965) carry the generalized anisotropic
development only as far as their equivalent to equation
2.17. At that point they impose additional symmetry
conditions to simplify the D.E. Keller and Frischknecht
(1966) apply symmetry restrictions at an earlier point

and never obtain an equivalent to equation 2.17. As will
be seen shortly, the reason for this is that an analytical
solution for equation 2.17 apparently cannot be obtained
except for certain simple models, such as the homogeneous,
anisotropic, half-space considered by Buchheim (1947).
However, insight should be gained by pressing the analytic
anisotropic development as far as possible.

The basic B.C. for anisotropic problems are the same
as for isotropic problems. However, the anisotropic pro-
blem B.C. cannot always be simplified as much as is possi-
ble for isotropic problems. The first B.C., the continuity
of the tangential § component across all boundaries again
leads to the continuity of the potential function across
all boundaries (equation 2.3). The second B.C., the
continuity of the normal component of g across all bound-

aries cannot be further simplified, except under the very

i -£ortuitous circumstance where the boundary is normal to

 
 

.QQQQ of the co-ordinate axes. Thus, the second B.C. is:

(L;

 

The Homooe
The earth mode
"if-space is that I
razriction that C
.3 tensor. The '2
-519 B.C. to be
ii: 1540' as X,j
;:‘.t.:ipal axes are
li-zriirzate axes.

Mama the D.E. :

e]

23

I. I» I. g 0

15.“ '33. .- -»
i : The Homogeneous, Anisotropic Half- -Space

31‘

  
  
  
 
 
  
  
  
  
  
  

g‘The earth model for the homogeneous, anisotropic

igpace is that of Figure 2.1 with the additional

a iction that c be a generalized symmetric second—
”vtensor. The D.E. to be satisfied is equation 2.17

'Ftthe B.C. to be satisfied are equations 2.3 and 2.18,

V-—*0, as x,y,z,—4 w. For convenience, the tensor

i grginate axes. With this situation, Buchheim (1947)

'ea the.D.E.:

5; 18 a normalization constant which will cancel
Y o + c + a
Tv’aht in the end, e,g,, “o = 11 22 33

—:yi(i947) then defined the transformation:

0
“"3 Y=/'—E: y, z = 00 2.20
3.: J"-'— 2 J 33
siL.‘ ‘.

 

.

WR' = /x2 + Y'
:;:s::ial function,
ir-of the anisotr

.gl‘fi .
.311), 1.6.:

- v,“ —
~1I - a
My “31
’§

‘I 2 VIC

Pig'si:ally, equati
resistivity is mea
fixation paralle
resistivity) dire
133-eat resistiv
Etric mean of th

“-“~lons 2.21 in

333*; -
lvlty measu

24

‘ i / "' o O 0
,‘e,R' g x2 + Y2 and am = _ll__§£__gi. With this
0

. gotential function, Buchheim (1947) obtained the general

 
  
      
   
 

F .-" I.
." form of the anisotropy paradox of Schlumberger et a1.

("1934) , i.e.=

pa||.x = “922 933' Ua||x = '022 C33'
pa||y = ””33 911' °a||y = '°33 011' 2'22
pa||z = '911 922' 0a||z = /011 022'

Physically, equations 2.22 indicate that if the apparent
resistivity is measured with a four point electrode con—
figuration parallel to one of the principal conductivity
(resistivity) directions, in an anisotropic medium, the
apparent resistivity in that direction will be the geo-
metric mean of the other two principal values. Thus,
equations 2.21 indicate a possible serious error for
resistivity measurements taken over anisotropic areas.

i ‘ The Anisotropic Layered Earth
Boun ary Value Problem

 

 

The earth model for this B.V.P. is that of Figure

2.4 with the added restriction that 01 and 02 are sym-
metric second-rank tensors. The D.E. to be satisfied
'is equation 2.17. Equation 2.3 must be satisfied at
both boundaries of the model. Equation 2.18 must be

.Q'satisfied at the lower boundary and at the upper boundary:
-h' 3

Zereaaining B.C. are
:35 near the source
according to e:
:eE.‘.='.P. as much as

igeczicrs in both ma

:3 noel co-ordinate

grersion to Cylindr

M

Despite the fac
:.3e re symmetric S
is have rotational
"u‘ill be a function
Izzzferting equation
fixthe case where 1

3.523% -
‘ «Dclpal axes I

 

 

25

Ir. = o. 2.23

   
   
     
   

L ,

;IG?ining B.C. are that V -+0, as x,y,z a-w, and for
7' ..

4::neer the source, S, the potential function must

ffe according to equation 2.21. In order to simplify

":Ision to C lindrical
’o-or-1nates

"'J ‘ Despite the fact that the conductivities in this

'U'u‘)
r-ug are symmetric second-rank tensors, the earth model

y...

ting equation 2.17 to cylindrical polar co—ordinates

ljthe case where the conductivity tensor is relative to

O 2
= 011(cosze + 333 sin26)2Jg
11 3r

0
sinze + GEE cosze
‘ 11 8V

+0 5—;

 

11 r

 

c
2cose sin6[1- 532] 2
_ a 11 a V
7: » 11 r 5156

 

 

 

care the only requil
liczzi 333' Th‘
riot be separated.
eased. Also, this
firssible to use if
ms done for the
3.341s in a form w?
Icanore symmetric
33% problem with
are cylindrical s;

.h)~ (i) _l
'1 ‘ 022 ‘ ‘

 

53159011 2.24 beco:
. 32v ‘
ii ET -.
1 xi
he” layer w1~
e: cient of anis
flat and West

 

 

 

 

26
- o
2cose sin6[1 - GEE]
+ o 11 av
11 2 56
r
- o
sinze + —33 c0526
0 2
+ o 11 8 V
11 2 2
L r so
2
3 V _
+ 033 3:7 — 0, 2.24

  
    
 
    
   
  
     
 

where the only requirement on the principal values of o is
all f 022 # 033. The r and 6 variables in equation 2.24
cannot be separated. Thus, separation of variables cannot
be used. Also, this will make it very difficult, if not
'impossible to use integral transforms to solve this B.V.P.
as was done for the isotropic case. However, equation

‘2.24 is in a form which allows consideration of the approach
‘to a more symmetrical form of the tensor. Consider the

‘above problem with the additional condition that 01 and 02

”have cylindrical symmetry about the z axis, i.e.,

(i) (i)

”11 = 0'22 3(1) = av”) (i = 1, 2). Then

_ (i)
'“t 7t°3

2 2 2
V 3 V 1 8V 1 8 V
a “5—5—‘=("'2'+ 5—)+—7——§=0, 2.25
:_ i x x 3r r r A 32

.:_in each layer, where A = J pv/pt is the classical co—
; Lea . .

: fiégfgicient of anisotropy (cf., Schlumberger et al., 1934;

we and West, 1965; Keller and Krischknecht, 1966) .

~

Liking a mean resist
.=I :3 , the Wenne
: Vt

171:: is given by:

= I1)

'5 3m (2il(a,

l (1)
k‘: 13(33)

h .L
.I o ’
l (33)
-.I.:.;on 2.26 is ver'
anion 2.26 shows
mung master cur
Ue more appro:..

32221:. .
‘ at reSlstivit.

tit“
unv'ros

.y Coeffici.
ec(n

resultS Ca»

 

 

27

  
 
 
  
 
  
   
  
  
   
   
    
  
   

Ejré J pvpt, the Wenner configuration apparent resis-

'eivity is given by:

- ' ' pa = pm(1)(2H(a,k') — H(2a,k')), 2.26

 

where:
J (Ar)
H(r,k') = 1 +2k'r “ITBTE'""' d),
e l _kl
O
(1) (2)
k. = A10(33) ' A20(33)
(1) (2
A1°(33) + A20(33)

Equation 2.26 is very similar to equation 2.15. However,
equation 2.26 shows that resistivity interpretations
utilizing master curves based on equation 2.15, when 2.26
would be more appropriate, will yield thicknesses and
apparent resistivities which will be the product of the
'anisotropy coefficient, Ai, with the true thicknesses

and pt(i), respectively (Keller and Krischknecht, 1966).
Similar results can be obtained with any other four point

electrode configuration.

7

“ghe Method of Images

The method of images cannot be applied directly to

., l-

”-Irthe anisotropic layered earth B.V.P., for the conduc-

 

.:*:s-sible to set up
azilayer. However, I
lZCC'OIdlI'ldte syste:
tits-ordinate system
75.1 complicate the 'r
£7.51: that the spec;
.fficult, if not in;

F3! the condit;
“"i'ltY (resist

)—' '5?

.3.. configuratior.

L:1:E“tl'!0

-(4+(_

er “d Kriscr

iKT
e Situation, t"
mag isotropic ma

anism
IOPY Coe

or; Equa:

L
H
\

equation 2 .11

' ‘ieslmilar

5:1“

 

   
 

 

28

'ssible to set up the generalized image series for
.y;édch layer. However, because each layer will have its
bun co-ordinate system, there will be discontinuities in
the co-ordinate systems at the two boundaries. This
will complicate the B.C. at these boundaries to such an
extent that the specific solution to the B.V.P. may be
difficult, if not impossible to obtain.

For the condition of cylindrical symmetry in the
conductivity (resistivity) tensor about the z axis, the

Wenner configuration apparent resistivity function is

apparently:
(I) w n 2 _ 1
pa = pm {1 + 4 E k' [(l + (2nAlt/a) )
n—l
-_1.
- (4 + (2nAlt/a)2) 21}. 2.27

Keller and Krischknecht (1966) indicate that for the
above situation, the resistivities and depths obtained
using isotropic master curves will be the product of
. the anisotropy coefficients with the true thicknesses
‘and the pt. Equation 2.27 does have that relationship
to equation 2.11 and it does satisfy equation 2.25.

hgain,'simi1ar results may be obtained with other four

a point electrode configurations.
"E.

 

 

Qualitative
"H..—

The results of ’1
111:; the condition 0
earth B.V.P.
iff;:ult to obtain.
rezeatpears to be i

n rolution to t3.

 

1:35:11 (1947) may 1.
Ease qualitative EX’.
fife expected resuj

The apparent r.
$2110.15 2.22 are t
filation where the
1: :ne of the princ;
:EPraCtical situ:
litensor principal

.1 not be Paralle

.ections. Consic
: “E r
I'dis
1. 3a
T3089, . 343
S‘
2 a

*1 a. a
“' 0 Q. .
22 sq ‘Slr
:i‘l 2
use

 

29

  
  
 

Qualitative Extension of Buchheim's
“Results

 

The results of the above section indicate that
adding the condition of orthorhombic anisotropy to the
layered earth B.V.P. makes its analytic solution very
difficult to obtain. Short of using numerical methods,
there appears to be little else that can be done to ob-
tain a solution to this B.V.P. However, the results of
Buchheim (1947) may be extended in a qualitative manner.
These qualitative extensions may then indicate the form
of the expected results of the layered earth B.V.P.

The apparent resistivity relationships given by

equations 2.22 are useful only for the very fortuitous

: situation where the electrode configuration is parallel
} to one of the principal tensor directions. A somewhat
more practical situation will now be considered. Again,
the tensor principal directions will be parallel to the
co—ordinate axes. However, the electrode configuration
{ -need not be parallel to one of the principal tensor
t directions. Consider the situation in Figure 2.7, where
the Wenner electrode configuration line is at an angle,

} 0, with the x-axis. For current electrodes,

 

3a 3a . 3 3a .
$1(- 1:0056, - Traine, 0) and 82(7gcose, ETSIne, 0), of

~strength, I, and potential electrodes, P1(- gcose, — gsine, 0)

 
   
 

39d Pzigcose, gaine, 0), the Wenner configuration apparent

gestativity is g

 

Fig. 2.7.--We"’
itztrar'y orientatior

"VA.
-

.-.:tions .

 

.r‘cgg/

Equation 2 . 2 8
mg structural tr

151(8 .
g., glaCial

--:.u:tiv

ity (resist

--~~lgurations or i e r

K

'-"\

center point
iElite
nt reSistivit'

N}. ‘91"!
nor semi-axes

Czar“ '
310118.

 

U
| 32 //
I P a 2"
________ ;J9<:g:l}2i____-___
o,/' |
/o/ P. |
/’ sI :

   
  
  
  
  
  
  
  
  
  
  
   
 
  

Fig. 2.7.--Wenner electrode configuration at some
arbitrary orientation to the horizontal principal tensor
directions.

cos 26 sin 26
2.28
pa(6) = p33 / p22 p11

Equation 2.28 suggests a field procedure for deter—
mining structural trends in areas where they are not well
known (e.g., glacial drift and alluvium covered areas).
Conductivity (resistivity) measurements with electrode
configurations oriented in more than one azimuth and a
common center point are taken. Then, a ground surface
apparent resistivity ellipse may be defined, whose major
and minor semi-axes should correspond to major structural
directions. For example, these structural directions may
gcbrrespond to the principal axes of the stress or strain
,ellipsoids used in structural geology (Billings, 1954).

_- 1519 approach may be used even if the earth does not fit

the Simple anisotropic half- -space model at the measurement

 

 

stiefine the tensor
92:21:: ellipse, beca

lgeneralized s
:agzeserted nathemat‘.
ninhas six indeper.
7:;Letely define the
seeequation 3.5) If“

Liferent directions

 

:six different 5-
2;be used to deter
I:a_:‘.er‘JI). If it
E‘.S‘.i'£ity (conducti‘.
25:35, there will t
:zese trends when 5

‘~
3
‘-~

- general form c

. ...E orlgln ' must

387:6"
1' The genera

ET-Iered
at the or

I A"

    

.5.

    
  
 

 

lunpt define the tensor representation surface cross

_ground surface apparent resistivity ellipse will not be

section ellipse, because of the anisotropy paradox.

A generalized symmetric second—rank tensor is
represented mathematically by a symmetric 3 x 3 matrix,
which has six independent coefficients. In order to
completely define the tensor, the resultant property
(see equation 3.5) must be measured in at least six
different directions. If measurements are taken in more
than six different directions, a least square procedure
may be used to determine the tensor coefficients (see
Chapter VI). If it is desired to use the apparent re—
sistivity (conductivity) ellipse to determine structural
trends, there will generally be no prior knowledge of
these trends when setting up the field survey. Thus, the
most general form of the equation for an ellipse, centered
at the origin, must be considered when setting up a field
survey. The general form for the equation of an ellipse

centered at the origin is:

2 _ 2
+ 822k2 + 2 l. 2.29

Sijxixj = S11x1 S12x1x2 =

To completely define the surface apparent resistivity
ellipse, apparent resistivity measurements must be taken
in at least three different azimuths. For a least square
determination, four or more must be taken. Because of

the anisotropy paradox (equations 2.22 and 2.28), the

 

 

:sresistivity tenso:
arm ellipse. T‘ne
rained from surface

713'. symmetry COW:-i

L.I3:hrougn 2.27) Ci:-
xenon the true It
:3: in advance.

If the apparent
stsbe completely L
zectional apparent
Zita-ten in at least

4: acrizontal. The

 

udf'fiilt attitUde S
'l

..
J‘V'a

.

"Hut";

+ .
wation surfa

\v

Can be deterrr
Liza:
:11?)

tensor Prim

ifall
. el t
0 th
are; e St
.‘ne
an
. app
‘2': l'r. are

32

the resistivity tensor representation surface cross-
section ellipse. The true resistivities cannot be
obtained from surface measurements alone, unless addi-
tional symmetry conditions (such as those of equations
2.25 through 2.27) can be applied, or additional infor-
mation on the true resistivity principal values are
known in advance.

If the apparent resistivity representation surface
is to be completely defined by field measurements alone,
directional apparent resistivity measurements must also
be taken in at least three additional directions out of
the horizontal. These could be made in drill holes at
different attitudes. After the apparent resistivity
representation surface has been defined, its principal
values can be determined (as in Chapter VI) and used with
equations 2.22 to determine the true resistivity (conduc-
tivity) tensor principal values.

The above field procedure might be helpful in an
area where the structural trends are unknown but suspected.
The field geOphysicist might equally well be faced with
the converse problem, i.e., setting up a field program in
an area where the structural trends, or anisotropy, are
fairly well known. In this case, measurements normal and
parallel to the structural trends would be sufficient to
determine an apparent resistivity ellipse. However, if

the true resistivities are desired and no additional

fixation available

agitation surface mus

 

 

33

information available, the apparent resistivity repre-

sentation surface must be defined as above.

 

THE CON;

The theoretica
tr:;erties of rocks
211's: of an anisot
allrely heavily up
5313.3, and conduc
iiiscrs. Solkolnikc

itste discus 3 ions

iirscrs. Nye (196 4)

mine second-rat

:" 2 y '

~~ . symmetric se
..‘_

‘32:)

. DeGroot and

7itcha15}
.y and Curr

 

CHAPTER III

THE CONDUCTIVITY AND DIELECTRIC

CONSTANT TENSORS

Introduction

 

The theoretically based model for the electrical
properties of rocks to be developed in Chapter IV will
be that of an anisotropic lossy dielectric. This model
will rely heavily upon the fact that the dielectric con-
stant, K, and conductivity, 0, are symmetric second-rank
tensors. Solkolnikoff (1951) and Nye (1964) have very
complete discussions on the properties of second-rank
tensors. Nye (1964) also establishes that K and o are
symmetric second-rank tensors. More thorough proofs that
o is a symmetric second-rank tensor are given by Casimir
(1945), DeGroot and Mazur (1954a, 1954b), Callen (1961),
Katchalsky and Curran (1965), and Smith etflal. (1967).
Because the remainder of this paper is dependent upon a
knowledge of some properties of the symmetric second-rank
K and o tensors, a short review is now included.

One of the criteria for defining second-rank tensors
is that they obey the rotational transformation laws

(Nye, 1964):

34

lj h:l m=l

3T"-
o

I
0—4
._3
H

 

k3

H
H
>-3
H
‘

.511; matrix notat i 0

¢

::<:ensor, (11].) i

1:2 angles between t

u a

.....

u
'''''

 

35

3 3
T'.. = . . = . .
13 kil millikamljm l1kamljm'
3.1
3 3
= v = I
Tij hi1 mEllkiTkmlmj 1kiTkmlmj'
using the Einstein summation convention (Nye, 1964), or:
T'=lTlT,
3.2

using matrix notation, where T is a generalized second-
rank tensor, (lij) is a 3 x 3 direction cosine matrix for
the angles between the co-ordinate axes, Xi and Xj' and
the T superscript indicates a matrix transpose. The
second-rank tensor, T, is represented mathematically by

a 3 x 3 matrix. If T.. = T

1] ji' the tensor is said to be

symmetric and the representation matrix is symmetric about
its main diagonal. This reduces the number of independent
coefficients from nine to six. If the co-ordinate axes
are rotated so that the off-diagonal terms of the repre-
sentation matrix vanish, the tensor is said to be referred
to its principal axes. The remaining main diagonal terms
are called the principal values of the tensor.

Symmetric second-rank tensors also may be represented
geometrically by a second order surface. The coefficients,

8.

ij' for the general equation of such a surface, centered

at the origin:

S...XX-=1' S
131]

isacbey t'ne rotati
L.'.and3.2. The ti;
':.c:.the principal \'
w... 1

, ..... gal values art

.sareal ellipsoid.

“xvii- .
_.:...»e, the repr’|

‘
b

 

:59: two sheets I 1
.21... '
its are negative
rmary ellipsoid
Tue magnitude
..:.sitrary direct

35 Tie. 1964)

T . . “
111 liT‘l:

1"
x"

A

 

36
Sijxixj = l, Sij = Sji 3.3

also obey the rotational transformation laws of equations

3.1 and 3.2. The type of representation surface depends

upon the principal values of the tensor. If all three

principal values are positive, the representation surface

is a real ellipsoid. If one or two principal values are

negative, the representation surface is a hyperbaloid of

one or two sheets, respectively. If all three principal

values are negative, the representation surface is an

imaginary ellipsoid.

The magnitude of a symmetric second-rank tensor in

an arbitrary direction indicated by the unit vector, 1,

is (Nye, 1964):

A

T11 l = liTijlj' 3.4a

using the Einstein summation convention, or:

using matrix notation. If equations 3.4 are expanded,

the result is:

>

H)
H
LA)
0
U1

1 1 + 2T12 1 2°

$210135 is of ba
515527 as a model tc
in laboratory measr

:5 grincipal values

;:e»::icns have been

The Diei

m

The relations‘n
:;.;'~.rec-tor, g, and ‘
assent, vector, [‘3

-: arson and Lorra

J3 30 that D
" 15 anith
L: 571‘

mation 3.6 hr.

 

. v
. .
— —

1351K a
. I'e reprea‘
who I

 

37

Emiation 3.5 is of basic importance to this study. It
is used as a model to obtain least square coefficients
frcnn laboratory measurements. It is also used to obtain
the principal values of the tensor, once the principal

directions have been determined.

The Dielectric Constant Tensor
The relationship between the electric field inten-
sfity vector, E, and the electric flux density, or dis-
;flacement, vector, Q, in a dielectric material is given

kw (Corson and Lorrain, 1962):
D = CE = E KE, 3.6
_. _ 0....

where so = 8.85 . 10.2 farad/m is the electric permit-

tivity of free space, 6 is the permittivity of the

material, and K = 5L is the dielectric constant of the

0
material. If the material is isotrOpic, e and K are
scalars so that g is parallel to E. If, however, the

material is anisotropic, 2 is no longer parallel to E

and equation 3.6 becomes:

which is of the correct form for a second-rank tensor
relating a dependent vector quantity to an independent
vector quantity (Nye, 1964). For the anisotrOpic case,

a and K are represented mathematically by 3 x 3 matrices.

The dielectric constant, K, and permittivity, e, do obey

 

:5 rotational tra
3.3. Thus, they a
In addition
sacrite a materia
:zlarized in the ;
a:::2;lished with

.iefined by (Co;

'15 the 3 x 3 is

”I! -t is Clear

K

H
H
+

‘u
.\
“F,

:::$.‘

atlon laws
.8 ‘1

ago a Secc
‘lit'n

Elsi ‘
5‘0
Th
e .
=1 disig

38

the rotational transformation laws of equations 3.1 and
3.2. Thus, they are second-rank tensors.

In addition to K and s, it is often desired to
describe a material's willingness to become electrically
polarized in the presence of an E field. This is
accomplished with the electric (volume) susceptibility,

x, defined by (Corson and Lorrain, 1962):

Q = cg = e E + E = 60(I + X)§: 3.8

where E is the electric (volume) polarization vector and
I is the 3 x 3 identity matrix. From equations 3.6 and

3.8, it is clear that:
K = I + x. 3.9

In addition to relating a dependent vector, 3, to an
independent vector, E, x also obeys the rotational trans—
formation laws, given by equations 3.1 and 3.2. Thus,
x is also a second-rank tensor. The electric suscepti—
bility, x, is a more fundamental property of the material
than K or 6. However, by tradition, laboratory measure-
ment results are usually given as K rather than x. The
results of the laboratory portion of this study will
also be given as K, though x will be used during the
development of an electrical model for rocks.

The dielectric constant, K, permittivity, s, and

susceptibility, x, have all been shown to be second-rank

:zszrs. However, i
2: are symmetric,
m‘Straignt forwa:
$213.5 If a the:
:gerature, 9, ele:
tithe electric di:

:_':a:.ic work at C0“

combined fir s t

zuzu 3y (Mac I nne s
I

at? =

dds + E.
J

J is the tote

39

tensors. However, it has not yet been established that

they are symmetric, e.g., . This is done in a

K.. K..
13 31
very straight forward manner, using classical thermo-
dynamics. If a thermodynamic system is defined by
temperature, 6, electric field intensity vector, E,
and the electric diSplacement vector, 2, the thermo-

dynamic work at constant E is given by:

dW = -E.dD..
1 1

The combined first and second laws of thermodynamics are

given by (MacInnes, 1961; Moore, 1964):

d0 = eds + EidDi,

where U is the total internal energy of the system and S
is the entropy of the system. The Gibbs free energy of

the system is defined as (MacInnes, 1961; Moore, 1964):

G = U + w - Q,

where Q 68 is the heat transferred. Thus:

dG = dU - E.dD. - D.dE. - eds - sae
1 1 1 1

"' DodEo '" Sdeo
l 1

At constant temperature, this becomes:

dG = - DidEi = - (DldEl + [)sz2 + D3dE3).

3:3 is a function
:fierential (Maclnn

3:3:9 and DiPrima ,

{ED} (3:)_‘1

1.9.] -‘ l
3E- . ' ‘53.
{I}? K 1.

s..
N‘ N

i

'1
Ti‘ ‘
3:“ r 1
‘ ~0r .
raln' 19621
E =
s DJ,

 

40

But G is a function of state. Thus, dG is an exact
differential (MacInnes, 1961; Moore, 1964), so that

(Boyce and DiPrima, 1965):

BDi BDi
BEj 6 8Ei 8

or:
E (6) em. = E (a) 3B.
1 BE. 1 8E ’
3 J 3 3
so that:
(e) _ (e)
Kij — Kij , q.e.d., 3.10

where the 6 superscript indicates the condition of con-
stant temperature. While the above proof of symmetry was
for K, similar proofs could also be given for e and x.
Experimental measurements of K (Nye, 1964) have
always given positive principal values. This was also
the case for the laboratory portion of the present study.

Thus, the representation surface for K is a real ellipsoid.

The Conductivity Tensor
The vector form of Ohm's law is given by (Corson

and Lorrain, 1962):

E = pi, 3.11a

 

(.4
ll
(’1
| [11
‘

IBIS _ is the curre
reaszivity of the T
2:3:ctivity of the
L.‘.‘;aor 3.11b is d»:
'2 independent vec:
sample shape.
For isotropic
£15 parallel to E.

2)-
I.

”“9 longer para:

 

 

41

g = 0E, 3.11b

where g'is the current density vector, p is the electrical
resistivity of the material, and o = p-1 is the electrical
conductivity of the material. The choice of equations
3.11a or 3.11b is dependent upon the choice of E or g as
the independent vector. This, in turn, is governed by
the sample shape.

For isotropic materials, 0 (or p) is a scalar and
1 is parallel to E. For anisotropic materials, g and E

are no longer parallel and equations 3.11 become:

which is of the same form as equation 3.6. In addition
to relating a dependent vector to an independent vector,
0 and p also obey the rotational transformation laws of
equations 3.1 and 3.2. Thus, 0 and p are second-rank
tensors.

The establishment of the symmetry of o and p is not
as straight forward as it was for K. This is because
electrical conduction is a transport phenomenon. Since
transport phenomena deal with irreversible processes,
classical thermodynamics cannot be used. A more sophis-
ticated approach which can be used for irreversible

processes is Onsager's principle of microscopic

 

reversibility. Thf
;s;iven in the ori
.331; Casimir, 19‘
inmost advanCe
r1.,Callen, 1961
i_‘._al_., 19671 .

To be able t
3.15:1: system and
::::'itions. These

agrees is consider

I :

2.":1’gg‘

mg small f

eldilibrium e n

S :
p101!

“ and the Q_

1

system, BeCa
521111

Uflum' ther
~..r:p

 

42

reversibility. The justification of Onsager's principle
is given in the original papers (cf., Onsager, 1931a,
1931b; Casimir, 1945, DeGroot and Mazur, 1954a, 1954b)
and in most advanced texts covering statistical physics
(cf., Callen, 1961, Katchalsky and Curran, 1965; Smith
g£_gl., 1967).

To be able to use Onsager's principle, the thermo-
dynamic system and its variables must satisfy certain
conditions. These conditions will now be reviewed. The
system is considered to be in statistical equilibrium
undergoing small fluctuations about this equilibrium.

The equilibrium entrOpy must be given by:
S = P.Q., 3.13

where the Pi are generalized intensive parameters of the
system and the Qi are generalized extensive parameters of
the system. Because of the local fluctuations from
equilibrium, there will be an entropy current, JS.
Entropy (disorder) is a maximum at equilibrium. Thus,
JS, is not conservative and a local entropy creation per
unit volume, per unit time, ¢, can be defined as (Callen,
1961; Katchalsky and Curran, 1965):

<1> =%%=FiJi, 3.14

where g is the local (per volume) entrOpy, the Fi are

generalized affinities (forces) due to the non-equilibrium

::-:ilticns
flies (cur
;\, flowing

j::;er choi

'le inter)
111;,
’I Is.

43

conditions within the system, and the Ji are generalized
fluxes (current densities) of the extensive variables,

01' flowing in response to the Pi. The criteria for the
proper choices of fluxes and affinities are (Smith et al.,

1967):

. a.
J _ J
aiJi ‘ TE'

where the qj are local (per volume) extensive variables.
The fluxes, Ji' and affinities, Fi' are related by the

so-called phenomenological equations (Onsager, 1931a) as:

J. = L..F., 3.16

where the Lij are called coupling coefficients and vary
for different thermodynamic systems. Onsager's principle
states that if the parameters of the thermodynamic system
satisfy equations 3.13 through 3.16, the cross coupling

coefficients are equal, i.e.:

L..=L... 3.17

For the case of electrical conductivity, the exten-
sive variable of the system is the electric charge, Q,
and the intensive variable is the electric potential, V.
With this choice of variables, equation 3.13 is satisfied,

for by dimensional analysis (Halliday and Resnick, 1961):

1011"] = Q :

its: equations 3 .

d,

.1?

U1

“M
H

m.v‘

mg is the 961

“June equation 3
“50109 of syst.
...lltions'

Onsage:

.: ametry 0f the

 

Fij = Uij
‘ Experimental
aways given p0
case for the 1
“‘3’ the 0 repre=
alipsoid.

44

2 2
ML ML
[Q][V] = Q —I— = '17 = [S], q.e.d.
t O t

From equations 3.15, the flux and affinity are given by:
F = E = - VV,

v-._I_

II
I
a}:

where q is the per unit volume charge density. This

choice of flux and affinity satisfies equation 3.14, for:

[any ===%L—-%--3‘-3- 3%) q-e-d-

t Q L t Lt

Because equation 3.12 is of the same form as 3.16, and
the choice of system parameters do satisfy the Onsager
conditions, Onsager's principle can be used to establish
the symmetry of the conductivity (resistivity) tensor,

i.e.:

0.. = 0.. q.e.d. 3.18
13 13
Experimental measurements of o (Nye, 1964) have
always given positive principal values. This was also
the case for the laboratory portion of the present study.

Thus, the a representation surface will also be a real

ellipsoid.

 

 

THE LC

Keller' 5 (1
:szicn. of materia

:2 be physical 1y

A— 7'

Ellison (l 9 6 4) a

.n.

in an applied

2 4.1, the re;
:Itgerties of dry

cozes, 1942; Jak<

filer, 1966; Par}

32125 r '
a.e high res

229:1

“ atOrS
.
=::3

CHAPTER IV

THE LOSSY DIELECTRIC MODEL FOR ROCKS

Introduction

 

Keller's (1966) rule-of-thumb electrical classifi-
cation of materials is given in Table 4.1. While it may
not be physically as sound as those used by Wert and
Thompson (1964) and Beam (1965), it is perhaps more use-
ful in an applied sense. Using the classification of
Table 4.1, the reported measurements of the electrical
prOperties of dry rocks (cf., Heiland, 1940; Slichter and
Telkes, 1942; Jakosky, 1950; Keller and Licastro, 1959;
Keller, 1966; Parkhomenko, 1967) indicate that most dry
rocks are high resistivity semi-conductors, or lossy di-
electrics. Thus, it is desirable to consider the prOperties

of lossy dielectrics.

TABLE 4.1.--E1ectrica1 classification of materials.

 

 

Material Type Conductivity Range (mho/m)
-8
Insulators o < 10
Semi-conductors 10“8 < o < 105
Conductors o > 105

_ v v v f v Viv f r‘ 'T—fi

45

 

Macr

 

The laborator
:erzed with the bul
2:;Les of dry rOC‘ri
22ndnddual dipc
:3is)composing
;:‘:es:igation will
Patrcphysics sense
titerial science 8'

If a paral 1e

z... separation , d r

‘1 across a si

5-
N

“Sltase will 1
a 3.?

-.
v.1

~idea
1: there v

:' -
(‘h

,
I l
r:
o
H

:J‘
(D

[C4
II

 

|__

46

Macroscopic Electrical Model

 

The laboratory portion of this study will be con-
cerned with the bulk electrical properties of small
samples of dry rock, but not directly concerned with
the individual dipole centers (atoms, molecules,
crystals) composing the rock. Thus, the laboratory
investigation will be microscopic in nature, in the
petrophysics sense, but macrOSCOpic in nature, in the
material science sense.

If a parallel plate capacitor, with plate area, A,
and separation, d, containing an ideal dielectric is con-
nected across a sinusoidal voltage source, V = Voejwt,
the voltage will lead the current through the capacitor
by a phase angle of 90°. If the dielectric material is

not ideal, there will be a loss current, I in phase with

1!
the voltage, as well as a charging current, IC, lagging
the voltage by 90°. The total current, I, will then be

the complex sum of the charging and loss currents (von

Hippel, 1954a, 1954c):

I = 11 + 310 = (c + ij)V, 4.1
where j = V—I, C = K80 3-13 the capacitance of the system,
A

and G = o a-is the loss factor of the system. The vector

form of equation 4.1 is:

g = £1 + 32c = (0 + Jw€)§; 4.2

: “3’

!!~- ~ at"

at: is the vector
aletzrics. The for:
Ezraparallel R-C
7:21.15 reason, a g

22:10:; for a los-

 

 

 

47

which is the vector form of Ohm's law for lossy di-
electrics. The form of equation 4.1 is the same as that
for a parallel R-C circuit (von Hippel, 1954a, 1954b).
For this reason, a parallel R-C circuit is often used as

an analog for a lossy dielectric.

'an

 

V=Voej“”
C R= l/G

 

 

 

 

l8. (6+ jwcw

Fig. 4.1.--Lossy dielectric response and analog
RrC circuit.
Thus, the total current, I, for such a lossy capacitor
(capacitor with a lossy dielectric), or its R—C analog
circuit, will be inclined at a power angle, 8 < 90°,
against the voltage, or a loss angle, 6, against the
positive imaginary axis of the phasor diagram (see
Figure 4.1). A common method of describing the loss of
a.lossy capacitor or dielectric is to use the loss tan-

gent, or dissipation factor, D, given by:

The frequencl
;=.:erally not agree
:;::'.“.ts. This is
is: exclusively to
2:11;; energy cons
fiant-of-phase a

a;

‘6-.. frequency . I

._‘.ttance var i e S

 

Because Of
251353 and char
Incorrittally b‘;

:aescribe a 105

(
N

m

l

j s

. ch .

. ECtlve ' (
“hit

165 lnStea<

tiriCa l
. meth
i the 1
a boratg
'Rstint
and
.5. SE
3.3 ace
. 0rd
2 1n

48

The frequency response of a lossy capacitor will
generally not agree with that of any of its R—C analog
circuits. This is because the loss current may not be
due exclusively to the migration of change carriers, but
to any energy consuming process. Thus, both the in-phase
and out-of-phase admittances for the lossy capacitor vary
with frequency. By contrast, only the out-of-phase
admittance varies with frequency for the R-C analog.

Because of the ambiguity of the R-C analog circuit,
the loss and charging currents are often lumped together
noncommittally by the use of a complex permittivity, 6*,

to describe a lossy dielectric material, i.e.:

6* E - 3.6" 4.4

where e' = % is called the specific loss factor. This
is the approach used by von Hippel (1954a, 1954c) and
Sharbaugh and Roberts (1959).

Another approach to the ambiguity of the R-C analog
is to realize that the conductivity will be frequency de-
pendent for a lossy dielectric and call it the dielectric,
or effective, conductivity. The use of effective conduc-
tivities instead of complex permittivities seems more
realistic for geophysical problems because most geo-
electrical methods involve A.C. measurements of some kind.
In the laboratory portion of this study, the dielectric
constant and effective conductivity of rocks will be deter—
mined, according to the lossy dielectric model of equation

4.2.

 

 

From equat 101

:5 expressed as

2:2 volume polar i 2

1:;ige between the
;::;erties of a ma

:‘._:-:ile species of
1':"., ‘1
....n:1y through

" 52.... .
1...: polarizati

49

Bolarizability

 

From equation 3.8, the polarization vector, 3, can

be expressed as

g = 60X§° 4.5

The volume polarization vector, 3, provides a convenient
bridge between the macrosc0pic and microscopic electric
properties of a material. If a material contains a single
dipole species of microscopic dipoles, p, distributed
uniformly throughout the material with a volume density,

N, the polarization vector may also be written as:
E= Ne- 4-6

These microsc0pic induced dipoles are related to the in-

ducing field, E', by the polarizability, a, as:

p = aE', 4.7

where a may be due to a number of different mechanisms.

The inducing field, E', is a local field which takes into
account the effects of the other dipoles within the material.
The local field is given (von Hippel, 1954c; Corson and

Lorrain, 1962) as:

meg; l is a co:
3 3 _ 3 II
:::1e dipole distr'
Lazritutions, g =
gzeral form of the

11239. 1954c; Cor s

aerials is obta 1r.

 

 

50

where g 2 l is a coefficient which describes the symmetry
of the dipole distribution. For highly symmetric dipole
distributions, g = 1 (Corson and Lorrain, 1962). The
general form of the Clausius—Mossotti equation (Von
Hipple, 1954c; Corson and Lorrain, 1962) for isotrOpic

materials is obtained:

Na K - l

Beo-gK-i-TI-g) '

 

which, for highly symmetric dipole distributions, becomes:

Na __K - l
35 — K 2 ' 4'9

 

0

Equation 4.9 is included at this point for later comparison
with a similar result which can be obtained from the quali-

tative electric model prOposed for rocks.

Lossy Dielectric FreqpencyfiDependence

r—VP—v—V—v—vr—

 

Electrical polarization is seldom due to a single
mechanism. Three mechanisms common to homogeneous, lossy
dielectrics are called electronic, ionic, and dipole
relaxation polarization. These mechanisms are due to
the shift of electron clouds, relative shift in ions,
and physical orientation of permanent dipoles in the
presence of an electric field, respectively. Inhomo-
geneous lossy dielectrics also exhibit interfacial polari-

zation, due to charge build up at phase (inhomogenity)

2:2:‘aries. Some C
scianisms are desc
2:2.arisms, and the
lzadiition, they c
:‘1: frequencies ne
aerial will have
Twill decrease ra
F3: frequencies at

C<<g
‘ s

c' Also, 5'

:2. it is for f

2:: is an inertia
:ires a definite
fieestablished in
‘C field at
"iii the inducinc

.fc,the dipo]

.

.. F0): f >> f
[3425 ‘
i: dp with El
123:, ‘

“Rt' when t
Eiin.

he resona

2

51

boundaries. Some common theoretical models for these
mechanisms are described in Appendix A. Each of these
mechanisms, and their models, are frequency dependent.

In addition, they exhibit resonance characteristics.

For frequencies near the resonance frequency, fc, the
material will have a high specific loss factor, 6', and

K will decrease rapidly with increasing frequency, f.

For frequencies above fc' K is much less than it is for

f << fc. Also, 6' is much smaller for f >> fC and f << fc
than it is for f ~ fc. Physically, the resonance phenome—
non is an inertial one. Each polarization mechanism re-
quires a definite relaxation time, T, for its dipole to

be established in the presence of the inducing field. For
an A.C. field at f << fc, the dipole can polarize in phase
with the inducing field, and 6' assumes small values. As
f——>fc, the dipole begins to lag E}, creating an increased
6'. For f >> fc' the dipole can no longer even begin to
keep up with E' and the polarization mechanism becomes
dormant. When this happens, 5' becomes insignificant
again.

The resonance frequency of a polarization mechanism
can be determined empirically from plots of K and 8' versus
f. The specific loss factor, 5', will show a symmetric
peak about the resonance frequency, fc, and K will decrease
sharply at fc. From equation 4.4, o = e' , so that a 0
versus f plot will be a steadily increasing function of

f, with local maximas near the fc. Because of the

 

tensions and mas svr
itrtie four polari'

tiely separated fr

   

issuance frequenci,
ate'ials, however,
electronic polar 12a

aim of the elect

 

zation, EC general]
:21: relaxation £0:
3321011, but in sol
iiefc for interfé
1112.9 range of 1

.= Shown in F igur

0! K’ (I /
Fi

3m: .9. .

‘Ftles 4 .3

52

dimensions and masses involved, the resonance frequencies
for the four polarization mechanisms described above, are
widely separated from one another (see Figure 4.2).
Resonance frequencies for a given mechanism in different
materials, however, are generally close together. The
electronic polarization fC values occur in the ultraviolet
region of the electromagnetic Spectrum. For ionic polari-
zation, fc generally occurs in the infra-red range. Di—
pole relaxation for fluids has its fC in the microwave
region, but in solids it is much lower, if present at all.
The fC for interfacial polarization appear to be very low,
in the range of 100 to 1000 cps. This frequency dependence

is shown in Figure 4.2.

l i l i '

I
I
I l
l l
i : K ' I
| I
——-—+-———J|————+————
I |
e’ '
9‘” (”2) (av)
f

Fig. 4.2.--Lossy dielectric dispersion of electrical
properties (after Wert and Thompson, 1964; Keller and
Frischknecht, 1966).

 

Qualit

 

A quantitati‘:
2:15 be very diff;
glaxity needed for
icrcsc0pic model ,
fixing laboratory
:terpreting them

2:31:01: to this ch

 

 

 

 

h.
'11
V

appropriate
cfriisotropy and
2:1 lossy dielectl
3:: the purposes

2’1"! M
a .u 14a)?

be consi

azan'
"“193 0f polar'

._ale
of the Sn
«555,:
t~C
t to Com
9%,. F
wrvmes.

53

Qualitative MicrosEOpic Electric
Model'fOr Rocks

 

 

A quantitative microscopic electric model for rocks
would be very difficult to develop because of the com-
plexity needed for a realistic model. A qualitative
microscopic model, however, can be very helpful for pre-
dicting laboratory results before experimentation and
interpreting them afterward. As indicated in the intro—
duction to this chapter, the lossy dielectric model is
very appropriate for dry rocks. However, the conditions
of anisotrOpy and inhomogenity must be added to the classi-
cal lossy dielectric model to make it applicable to rocks.
For the purposes of the qualitative theoretical model, a
rock may be considered to consist of its constituent
grains, each species of which would constitute a separate
species of polarization mechanism. Such a model is defi—
nitely heterogeneous, however we will require that the
rock model be statistically homogeneous. Thus, we will
only consider rock samples of sufficient size that the
statistical composition will not vary significantly from
the sub-sample to sub-sample taken from the parent sample.
This implies that each mineral species be distributed
uniformly throughout the parent sample (at least on the
scale of the sub-sample). Borrowing terms, we can say
that the rock model is statistically stationary with
respect to composition. For such a model, equation 4.6

becomes:

11
°‘ ”191'

- i-l
izranodel consist“
. -I
:zzriined) of dipo-
22::‘ne model with

The second 91

 

i.electric model iii
§::s«::ropy may be '
iis:ri"ution of iS
istri'o‘ution of. a:
iistribution of a
22:15 may he p‘n‘y
ieuse of the m:
Shires the cal

2.3:: Silrrmetry i s

il-‘A'. ‘-
‘5,
.V..

in model is

~='.‘se 0f the 1

.221

i and 0. Of
31 3.
qr‘zation t
12945

\nges

2 a S‘

54

n

E = iiiNiEi' 4.10
for a model consisting of n > 1 species and mechanisms
(combined) of dipoles, pi, distributed uniformly through-
out the model with densities, Ni'

The second generalization from the classical lossy
dielectric model involves the condition of anisotropy.
AnisotrOpy may be achieved by an anisotrOpic (low symmetry)
distribution of isotropic dipole sources, an isotropic
distribution of anisotropic sources, or an anisotrOpic
distribution of anisotrOpic sources. The first and last
models may be physically more realistic, but they require
the use of the most general form of equation 4.8. This
requires the calculation of the coefficient, g, using what—
ever symmetry is present in the dipole distribution. The
second model is mathematically much simpler, for it allows
the use of the high symmetry form of equation 4.8:

g
E. = E + 33— . 4.11
o
This is the approach used to develop the qualitative
electric model for rocks.

For isotropic polarization models, 2 is parallel
to E' and a of equation 4.7 is a scalar. For anisotropic
polarization models, 2 and E' are no longer parallel and
a becomes a symmetric second-rank tensor. Then, equation

4.7 becomes:

3‘..E-o
“Si 3111

is polarizabil ity
:azsfcrmation law
- tensor. The

ii: for K. Inger

n \ n —
VOILE: Z l
1:1L

rn

= ‘ Z
Li=l

55

The polarizability, a does satisfy the rotational

ij’
transformation laws of 3.1 and 3.2 and thus is a second-
rank tensor. The proof of its symmetry is similar to

that for K. Inserting equations 4.12 and 4.11 into 4.10

results in:

 

 

 

“ (1) n <') P
B=ZN.o: B'=£N.als+" ,
._ 1 — _ 1 - 36
1-1 1—1 o__
or:
n 7 (1) x
e xE - X [N.a I + — E]
0 i=1 1 3 —
___ gNau) K+2IE
i=1i 7'
so that:
n .
3%” = Z Nia(l) = (K - I)(K +2I)—1, 4.13
0 i=1

which is the Clausius-Mossotti equation for anisotropic
material with n > 1 species and mechanisms (combined) of
dipoles. In equation 4.13 and the steps leading to it,
a(i), x, and K are 3 x 3 representation matrices for the
appropriate symmetric second-rank tensors, I is the 3 x 3

identity matrix, and the -1 superscript indicates matrix

inversion.

    
    
   

Predicti

A qualitative
electrical characte
in alternating 1_-
2::oscopic study,
Life: is not neede:
asthat developed a
glaring macrosc0p

For a unifor
SPacies of isotrO‘
Ersect to obtain

3131 a model wou?

Lian
“SC.

3i the pola

u..‘

..:-».3uencies woul

56

Predictions Based on the Qualitative
VT ElectriE'Moael for Rocks

 

 

A qualitative working microsc0pic model for the
electrical characteristics of dry rocks in the presence
of an alternating E field has been deve10ped. For a
macroscopic study, such as the present, a quantitative
model is not needed. However, a qualitative model, such
as that developed above does aid in predicting and ex—
plaining macrosc0pic observations.

For a uniform, isotropic distribution of a single
species of isotropic polarization centers, one would
expect to obtain dispersion curves similar to Figure 4.2.
Such a model would give unique resonance frequencies for
each of the polarization mechanisms present. If the
polarization centers were anisotrOpic, unique resonance
frequencies would still be expected but K and 0 would
now be symmetric second-rank tensors. The result is
that for the anisotropic model, there are three disper—
sion curves for K and 0, indicating the maximum, inter-
mediate, and minimum principal values of the tensor
properties. One would not, however, for a single
Species of dipoles, expect the principal directions of
K and o to vary with the frequency of E.

For single crystals, the principal directions of
tensor properties are controlled by the crystal lattice,

as stated by Neumann's principle (Nye, 1964): "The

pastry elements c

 

:3: include the 5;
II
:f:i*.e crystal.
For heteroge

rid expect to be

 

;::.:ciple, which n
1:15 of any physf
symmetry elem.
:i the rock.
Turner and

iiiect rock fabr

t-
".3?

u.."‘

no d(We 10

2C they attempt

.little mOre

57

symmetry elements of any physical property of a crystal
must include the symmetry elements of the point group
of the crystal."

For heterogeneous materials, like rocks, one
would expect to be able to use an extension of Neumann's
principle, which might be stated as: The symmetry ele—
ments of any physical property of a rock must include
the symmetry elements of the statistical symmetry (fabric)
of the rock.

Turner and Weiss (1963) mention that they would
expect rock fabric to control physical properties, but
offer no deve10pment in support of this statement, nor
do they attempt to follow it up. Let us now consider, in
a little more detail, the relationships one might expect
between the rock fabric and the principal directions of
the conductivity and dielectric constant tensors.

Single lineations and S-planes in rock fabric are
parallel and normal, respectively, to rotary symmetry
axes for the fabric (and rock). Using the extension of
Neumann's principle, the principal K and 0 values would
be expected to parallel any lineations and be normal to
S-planes of the rock fabric. Since these relationships
are statistically based, this correspondence might not
be very good for small numbers of macroscopic fabric
measurements on the field sample. Also, multiple S—plane
and lineations present in some rocks may further confuse

these relationships.

Let us now c
:zclrziodel prOposc
szazistically uni:
:Lriple species
Farthis model, 11
:ftre four types
553‘.) would not )
retire that th
different pc

.er fortuitous

‘ n
',_.-u. A
0o- ‘u-
.

-ncy Spect:
““13, it is re
:J“On

.1118 to the

:22»: from diffe
;;er.:ies. This
resonance free
::e rocks. Th
iii the m

axima

is 2..
tuey are it

58

Let us now consider the expected results for the
rock model prOposed in the previous section, i.e., a
statistically uniform and homogeneous distribution of
multiple species of anisotropic polarization centers.
For this model, unique resonance frequencies for each
of the four types of polarization mechanisms (if pre-
sent) would not be expected. For them to be unique would
require that the corresponding resonance frequencies of
the different polarization species would be identical; a
very fortuitous occurrence. Thus, even for the short
frequency spectrum used in the laboratory portion of this
study, it is reasonable to expect that the dominant contri-
butions to the dielectric constant and conductivity may
come from different dipole species at different fre-
quencies. This should result in spreading out the
resonance frequencies of the bulk characteristics of
the rocks. That is, the change in slope of the K curve
and the maxima on the 0 curve will not be as pronounced
as they are in Figure 4.2. Also, there is no reason why
the principal tensor directions for all the dipole species
should be coincident. In rocks, different minerals, or
even different sized grains of the same mineral, may
exhibit different fabrics.

Combining the variations in anistropy of the indi-
vidual dipole species and the variations in their fre-
quency responses, one would logically expect variation

in the principal directions of the bulk rock tensors

 

 

 

L...
AI 'l'
g. -.

 

 

ox.

qv.

ea.

59

with frequency. Also, the relative o and K contributions
of the individual dipole species may vary with frequency.
This may, in turn, lead to the condition where the princi-
pal directions for the o and K tensors do not coincide
for some frequencies of investigation. However, one
would, in general, expect that lineations be sub-parallel
and foliations (and other S-planes) be normal to the
tensor principal axes, unless multiple S—planes and
lineations lower the symmetry of the rock fabric.

The model deve10ped above gives us considerable
latitude in anticipated results. However, we must remember
that the model is an analog for a material which, in it-

self, is quite variable.

 

A block C1139
1:5 laboratory POI
3.1. The elements
Tie shunt capacit
2.2. Keller (19‘;
:4r;:se is twofo'
:ition factor to
5:32 the holder
:73 Circuit for

wring data I er

CHAPTER V

EXPERIMENTAL PROCEDURES

Instrumentation

 

A block diagram of the measurement circuit used for
the laboratory portion of this study is shown in Figure
5.1. The elements of the circuit are given in Table 5.1.
The shunt capacitor was included at the suggestion of Dr.
G. V. Keller (1967), of the Colorado School of Mines. Its
purpose is twofold. It decreases the holder system dissi-
pation factor for high loss samples and provides sufficient
system capacitance for balance when the sample was removed
from the holder. Since the shunt capacitor remained in
the circuit for all measurements, its value canceled out

during data reduction.

Laboratory Measurements

 

Procedure

 

The laboratory measurement procedure used for the
present study was governed by the inhomogeneous, aniso—
tropic lossy dielectric model developed for rocks in
Chapter IV. The electrical properties of lossy di-
electrics are determined by simultaneous measurement of

the dielectric constant, K, and (effective) conductivity,

60

.uflsouflo ucmEmusmmmE mofluuwmoum HMUfluuoon mo Emuwmflc xoonII.H.m .mwm

 

61

 

 

 

oaoomzzomo

   
   

oouzm

HHHHHHHH 522354 ....... W oczocom

 

 

 
  

 

 

 

 

 

   

 

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
o

lg
fl 3.2.33 .36

 

 

 

 

EEC CC C

 

 

 

 

QUCGUUQEH
UDQUJC

I >133!“

iiiii‘i

®UCNUQQEH CORUQRHUWOQ UCTEQHW

USQCH

mas.“ .«-H.mvnu—AU\.HAM HQUHMUUUHQ

32d

COflUQUCGEUNUQCHIlifloW Eqn<h

62

i i Ill Iii

nouflommmo

 

 

on oseuomm In: in- unamumo oooamhnm hasnm
nopaom mamfimm

ofluuomamflo «lemma nooHom mHosmm

1:: III In: oflomm Hmumcmo umuoEouon
omoomfiaaflomo
momma Hmso eom

III III 8:02 a xflcouuooh mmoomflaaflomo
anMHHmE<
macs H omNflHmhmum oma

umoo OH Ego omH ago: a oumxommuuoasmm umaoaaoea
mmox oom omcflum

imam om 12:0 com is moamuaommmo oumHn mmoaum

.ha ooaauooa 30H ago ooo chasm Hmuocoo mcaumnom
HoumHHHomo

moo: OH chose «some nonmaafiomo

umao 0H Ego com in- ounxomanumasmm .3 .u

mocmomQEH oocmcomEH
mmcmm usmuso usmcH coaumwuomoo ucmeHm
.mosum mmfluumdoum HMUfluuomHo can now coaumucoenuumcHll.a.m wands

 

 

vie: Hipeelv 19:-
32: low frequency
glished by measur
.aier of material
at an A.C. b1
1331.2; Sharbaugh
2e 21.8.1314. sta
iiige measuremt
fillowed, the e:
2:: and Ward ,
Prerts, 1959;

'y‘mg twO meas

63

G (von Hippel, 1954a, 1954c; Sharbaugh and Roberts, 1959).
For low frequency measurements, this is usually accom—
plished by measuring the electrical properties of a thin
wafer of material in a micrometer sample holder as one

arm of an A.C. bridge (Hartshorn and Ward, 1936; Field,
1954b; Sharbaugh and Roberts, 1959; A.S.T.M., 1965). If
the A.S.T.M. standards (D-150-6ST, 1965) for low frequency
bridge measurements using a micrometer sample holder are
followed, the edge and lead effects are eliminated (Harts-
horn and Ward, 1936; Field, 1954a, 1954b; Sharbaugh and
Roberts, 1959; A.S.T.M., 1965). This is accomplished by
taking two measurements at every signal frequency of
interest. The sample wafer is inserted into the sample
holder and the movable electrode brought into contact with

the sample face for the first measurement. For this

 

(P

measurement, the analog circuit shown in Figure 5.2 is

11121
1.1 T T

CH 6H c): 6:: CEd

Fig. 5.2.--Analog circuit for micrometer sample
holder containing sample.

assumed for the sample-holder system. From the initial
balance, the system capacitance, Cl, and dissipation

factor, D are obtained as:

1'

 

C-‘-C +Cx

 

O

H
3 = —— +
1 13C]. ‘“

care C.“ is the Cl

Leads, (5 is the
x

actress, d, anc

$3.2, CH is the

£113; C is thf
x

2:. air capacit

1+1 SeparationI

‘h
C.
._‘-

5‘1916 is

“Ge 18 st
'2: 15d 6.183
=:e

64

A
= = ———)—(-.
Cl CH + CX + CEd + CS CH + CX + Cod(l AE) + Cs'
G1 = CH + GX + Gs, 5.1
G G G
H x s
D=-——-+-——-+ ,
l wCl wCl wCl

where GH is the combined loss factor for the holder and

leads, Gx is the loss factor for the sample wafer of
thickness, d, and area, AX, GS is the loss factor for the

shunt, C is the combined capacitance for the holder and

H
leads, Cx is the capacitance of the sample wafer, COd is
the air capacitance of the capacitor with plate area, AB,
and separation, d. After the initial balance is obtained,
the sample is removed from the holder and a second balance
measurement taken with the same electrode separation, d,

and an air dielectric. The analog circuit for the second
balance is shown in Figure 5.3. The system capacitance,

C and dissipation factor, D2' for the second balance

2'

are then:

C2 = CH + C5 + Cod,
GZ = GH + G3, 5.2
D = EE— + EE-

2 wC wC °

,_
Il-M

 

Eig o S O 3 0 _-‘
1.5.333: without sa

fze sample dielei

ugh“,
“'"yl '3

,aret

X
‘ t e diamet
‘li’e‘ter of 1

Jt 5513 S‘
k 18 VQI
17‘...
“33?)
' E1
:23
l‘ Scots
"C’Perti
R3-

 

ii: i
0.6.cu 6.. c...

Fig. 5.3.--Analog circuit for micrometer sample
holder without sample.

The sample dielectric constant, K, and (effective) conduc-

tivity, o, are then obtained from equations 5.2 and 5.2 by:

K_de_(C1’C2)d+l
‘eA‘ eA '
OX OX

 

 

O _ Gxd _ wd(DlC1 - D2C2)
"A " I‘ A ' °
x x

 

If the diameter of the sample wafer is smaller than the
diameter of the holder electrodes, the edge effect is
eliminated by the use of this procedure (Field, 1954a,
1954b; Sharbaugh and Roberts, 1959).

The measurement procedure described above was used
for this study. It follows the A.S.T.M. standards (1965)
and is very similar to that used by Keller and Licastro
(1959), Howell and Licastro (1961), Simandoux (1963),
and Scott g£_21. (1967) in their studies of the electrical
properties of rocks. Measurements were made on vacuum

dried sample wafers at room temperature and pressure.

 

.. .A:ithflI these

:six, or more, d
Iris resulted in c'
1:23;". allowed the
2:53: coefficien
Measurement

user. 30 and 300C

assurements we r1

. ..
,-0v h

. ~H
a.-J-'¢

ale for TW

.3:'
.-...-.ements

'3‘
X

For aniSO1
sand-rank ten
tie to specify
:, for the pre
i: done most e.

-~...= dlSk‘ Sblab

&

~5£ faces ,

66

In addition, these measurements were taken on wafers cut
in six, or more, different directions from each sample.
This resulted in directional measurements of K and o,
which allowed the least square determination of the
tensor coefficients according to the model of equation
3.5. Measurements were made at several frequencies be-
tween 30 and 30000 cps for all samples. In addition,
measurements were made at 100000 cps on two samples.

Rationale for Two—Electrode
Measurements’w' W*' I

 

 

For anisotropy measurements involving symmetric
second-rank tensors, it is of utmost importance to be
able to specify the direction of the independent vector
(E, for the present study) within the sample. This can
be done most easily with two-electrode measurements on
thin disk-shaped samples, for then, E is normal to the
disk faces. A.S.T.M. measurement standards for a given
physical prOperty represent the combined efforts of many
individuals concerned, primarily, with measurement
accuracy. The measurement procedures outlined above do
follow A.S.T.M. standards (1965). The material prOperties,
K and o, are not measured directly. Instead, the direct
laboratory measurements are of C and G for the sample-
holder system. The choice of the thin wafer sample shape
increases the system C and G values, which in turn, will

increase the measurement sensitivity for K and 0. Another

 

2::siieration invc
Bailey (1948) , Cor
L934), and Knox
1155 on eguipoten‘
talues. Because
Eases in the vici

1
. 3”?

......odes should

 

 

 

 

 

 

mposite faces
222314111 be at
21:5 method is 11

2::genities . F

I, -

ants were used

.
,.
T
.

~Zerir.enta 1 L 1'

Tne great
"“‘e outlineqf

Hi“
‘i
.0

Host Sat

:1
E amounts

”3' _
.41“!

the 7 16_C

1161

g T1111 1
“,2 251:1 .
11> MOtrle‘

67

consideration involves the effects of inhomogenities.
Bewley (1948), Cook and Van Nostrand (1954), Deppermann
(1954), and Knox (1964) discuss the effects of inhomogeni-
ties on equipotential surfaces and apparent resistivity
values. Because of the distortion of equipotential sur-
faces in the vicinity of inhomogenities, the use of point
electrodes should be avoided. The two large electrodes
on opposite faces of the thin wafer sample, as described
above, will be at constant potentials at all times. Thus
this method is least susceptible to the effects of in-
homogenities. For these reasons, two-electrode measure-

ments were used for the laboratory study.

Experimental Limitations

The greatest problems with the experimental pro-
cedure outlined above lay not with the procedure, itself,
but with the instrumentation with which it was carried
out. Most saturated samples and those containing appreci-
able amounts of metallic minerals could not be measured
with the 716-C bridge because of their high loss. This
problem appears to plague all capacitance bridges. Im-
pedance bridges will allow the measurement of higher loss
samples, but balance is then obtained by means of a
sliding null (Stout, 1960) which is very difficult to
obtain.

Another limitation to the above procedure is that

it operates very close to the limits of sensitivity of

    
  

its instrumentati
Stunt, 1960) she

: values may be

(}\

:erxeen 15 per Cr
:12 same point w
:31. Greater an
aim measureme:
E33,:‘agated error

:T::~

.-»..est When K

 

... .
~b\:'

"maiStiC in r
:9? Should $1";va

{LS and the re‘

:clsk of green
we measured
The A;

L

q n

:11
ca 185 s t

:Cpagation

(:1

‘ s
“vital .
1n
She
a°ide

68

the instrumentation. A statistical prOpagation of error
(Stout, 1960) showed that the error in the directional

0 values may be as high as 49 per cent, though most were
between 15 per cent and 25 per cent. The K errors at
the same point were fairly constant at about 10—15 per
cent. Greater accuracy in the capacitance and dissi-
pation measurements would help this situation. The
prOpagated error for the directional o and K values is
greatest when K is very small (K ~ 5). For larger K
values, the errors become much less. It should also be
remembered, that a statistical propagation of error is
pessimistic in nature. If the errors are indeed random,
they should show up in repeated measurements. To check
this and the repeatability of the measurement procedure,
a disk of greenstone schist and one of syenite gneiss
were measured repeatedly over the period of a month.

The discrepancies between the different measurements were
all much less than that predicted by the above error

prOpagation (AK ~ 3%, A0 ~ 7%).
The Laboratory Measurement Samples

Sample Ereparation

The samples collected in the field were roughly
cubical in shape, approximately ten to twelve inches on
a side. Care was taken to collect only from fresh, non-
weathered, surfaces. Thus, most of the samples were

collected from road cuts.

The Sample
szzz‘yclosely fol
1951) and the A'

:1: make some mo

3
:_i

was more a}:-
From the fi
Lie different d:
:crresponded to

13;:thetical cub
ins reference 5

.a. ‘
a?
«:Bg‘393 f

abriC

fie-w”; a

tely succe

cut in this

we tne measure

cetermlna
actions dist
tents uniformly
rezerence syste
Disks 1

it, and lapped

 

69

The sample preparation procedures used for this
study closely followed those of Howell and Licastro
(1961) and the A.S.T.M. standards (1965). The author
did make some modifications so that the sample prepar—
ation was more applicable to the present study.

From the field samples, slices were cut normal to
nine different directions in the rock. These directions
correSponded to the face normals and face diagonals of a
hypothetical cube (co-ordinate system). Whenever possible,
this reference system was oriented with reSpect to an
observed fabric in the rock. However, this was never
completely successful because of the mechanical problems
involved in slabbing such large rock samples. The rocks
were cut in this manner because the slab normal then be—
came the measurement direction for the directional tensor
value determinations. The use of these particular nine
directions distributed the directional K and 0 measure-
ments uniformly over the quadrants of the laboratory
reference system.

Disks, 1.6 in. in diameter were cored from the slices,
cut, and lapped down to constant thicknesses, which ranged
from 35 to 235 mills. During this process, the opposite
sides of the disks were taken to a 600 grit polish. The
samples were then vacuum dried at 60° C for 24 hours,
coated with conducting silver paint, and stored in a
desiccated container until electrical measurements could

be made.

?2:ionale for Meaa
'T 5 Samples

The samples

   
  
  

:15 study were \i'.
.iair natural sta
alactrolyte solut
551:: tion of the

:: the saturating

::;sci".l<necht , l9

“filo This Enfl

if electrical p,

“=12“ Conditj

33:9 up
\.Lat eVe n

*«ifie

1‘1 that

Z‘.
-
‘\

3 W111 dupl

‘S’Jratory s arr

lOn durj

‘

. Q

\ .

r. i."'v

i‘l' SuffiC'

asureme]

lit L‘a
dx. reerea
$3"in
and
“Ja
and; .
«thnS
$51.“ '
Uglons

(f)

70

Rationale for Measurements
on Dry Samples

 

 

The samples measured in the laboratory portion of
this study were vacuum dried. However, most rocks in
their natural state contain moisture in the form of
electrolyte solutions. The effective porosity, per cent
saturation of the available pore space, and the salinity
of the saturating solution can drastically affect the
bulk electrical properties of the rock (cf., Jakosky,
1950; Wyllie, 1963; Grant and West, 1965; Keller and
Frischknecht, 1966; Parkhomenko, 1967; Ward and Fraser,
1967). This presents a strong case for the measurement
of electrical properties in saturated rocks to duplicate
in-situ conditions. However, Simmons and Nur (1968) indi-
cate that even with the most careful precautions, it is
unlikely that laboratory measurements of physical proper-
ties will duplicate in-situ results. This is because the
laboratory sample is generally altered from its in-situ
condition during sample preparation. For the present
study, sufficient time elapsed between sample collection
and measurement that most of the in-situ water evaporated.
Also, during sample preparation, the rock was thoroughly
flushed with cutting fluid so that in-situ solutions could
not be recreated by saturating with distilled water.
Keevil and Ward (1962) did not attempt to recreate in-situ
conditions, but studied the effects of various electrolyte

solutions. Such a study is certainly worth doing, but it

 

  
  
 

;ssuch a large l3

u.
. o
. :
DJ

o<:j:

The presenc
iiitional consid
airlines, l9 6 1'.
Line electrica‘
:ata‘ring place.
"arsible, or can:
12:12 the current
i’ili become pole
ethin the rock
Viich we are mo
Iitain valid me
:agligible, an}

q
“v‘

‘*‘901arizino
5152‘:
‘ 05498 are

to Predict

2133
E boundarx

r5
Effort ha

”i” SCOtt
E
“3°10? non
\Y‘
Sc'sca
lled 1'10»-

71

is such a large problem that it constitutes a separate
study.

The presence of aqueous solutions requires the
additional consideration of electrochemical problems
(MacInnes, 1961; Moore, 1964). At every phase boundary
in the electrical circuit, electrochemical reactions will
be taking place. If these reactions are not truly re-
versible, or cannot take place fast enough to keep up
with the current flow, the junctions at which they occur
will become polarized. When these polarizations occur
within the rock, they are a prOperty of the rock (I.P.)
which we are most interested in measuring. However, to
obtain valid measurements, we must eliminate, or make
negligible, any polarizations occurring at the sample-
electrode contact. This is accomplished by the use of
non-polarizing electrodes (Moore, 1964). Non-polarizing
electrodes are not universal and there is apparently no
way to predict, theoretically, whether or not a particular
phase boundary will be polarizing in advance. Much time
and effort has been spent by various workers (cf., Mandel

et al., 1957; Collett, 1959; McEven et al., 1959; Mayper,

 

1959; Keevil and Ward, 1962; Simandoux, 1963; Quraishi,
1967; Scott 25.2lxt 1967; Scott and West, 1969) trying to
develop non-polarizing electrodes. The great variety of
so-called non-polarizing electrodes now in use stands as

mute testimony to the fact that most of these have not

 

   
 
  
 

:aanconpletely s:
:tavery carefu.
selecting a non-p.
:fzhair study in
argla system mus
:a electrodes ar
Se‘nering b:
12:91 C, and 61
11:5: generally

Tza bridge used

(u.
.-:

353311)! 290 . S

:15 Very easl

.

'1‘“.
:~ 4591'

ties Of l
1:.szrumentatio

The rock
Eta-sediment
3:37:15. have ve

u
‘

u». reSiStivj

72

been completely successful. Scott et a1. (1967) carried
out a very careful and detailed investigation aimed at
selecting a non-polarizing electrode system. The results
of their study indicate that each electrode-electrolyte-
sample system must be tested before it can be assumed that
the electrodes are indeed non—polarizing.

Schering bridges are designed to measure the capaci-
tance, C, and dissipation factor, D, of real capacitors
which generally have a finite, but low, loss factor, G.
The bridge used for this study could measure a maximum
dissipation factor which corresponded to a loss angle, 6,
of only 29°. Saturating samples with electrolyte solu-
tions has the effect of increasing the loss angle. Thus,
it is very easy to reach a condition where the electrical
properties of the rocks can no longer be measured with the
instrumentation available.

The rock types selected for this laboratory study
(meta-sediments, meta-volcanics, and meta-igneous rocks)
should have very low porosities. As a check on this,

D.C. resistivity measurements were made on a core of
greenstone schist (M-2). This core was measured in both
the vacuum dried condition and saturated with tap water.
In both cases, it was impossible to make measurements with
the low current densities used for the bridge measurements
(i.e., 1 ~ 20 ma/mz). However, with much larger current
densities (Q ~ 3-4 amp/m2), a dry resistivity of 2.28 - 102

ohm-m and a saturated resistivity of 1.66 . 102 ohm-m were

 

 

   
   
  
  

:‘tained. Both 0
23362 obtained US
:asistivities PIC
2:59 of the hiq‘?
1'3; needed to {:1
Szilar results
fixation collec
aisles used for
fat these sampI
:ciation of the
Lake Superior r
iffaumemogi (j
753M931 appa
35this area r
HOWell

ilfficulty re
is” extended

”3:!!! Constrlh

I

.19. authOr ‘ S
1“; Sample d

1’ i .
311 time S

73

obtained. Both of these resistivities are much lower than
those obtained using the A.C. bridge. However, these D.C.
resistivities probably represent dielectric breakdown be-
cause of the high current densities and the high voltage
dr0p needed to produce a measurable current (V = 500v).
Similar results were also obtained for a core of iron
formation collected from Northern Michigan along with the
samples used for this study. These results would indicate
that these samples do indeed have low porosity. Extra—
polation of these low porosities to the Precambrian of the
Lake Superior region appears to be confirmed by the results
of Tammemogi (1969a, 1969b) and Dowling (1969) who observed
very high apparent resistivities in the Precambrian rocks
of this area using magneto-telluric measurements.

Howell and Licastro (1961) encountered considerable
difficulty repeating rock electrical property measurements
over extended periods of time. To overcome this problem,
they constructed a sample holder which could be evacuated.
The author's solution to this problem was to vacuum dry
all sample disks and store them in a desiccated container
at all times except when taking electrical measurements.
Using the vacuum dried samples, measurements were repeated
within the accuracy described above (AK ~ 3%, A0 ~ 7%).

Until now, electrical measurements on vacuum dried
samples has been considered as the best alternative to a

bad situation. As such their results should be applied

:2 field resm
tase result
11:5 the Apc
Lararautics

£315 on d:

Ira: exglc
he '

i§§liiable

sate: aterl

.
“‘3" .
a ‘ lc
- K
u..‘
1

74

to field results in only a qualitative manner. However,
these results may be used directly for dry environments.
With the Apollo and post-Apollo programs of the National
Aeronautics and Space Administration, laboratory measure-
ments on dry samples should be directly applicable to
lunar exploration (see Strangway, 1969).

The lossy dielectric model for rocks is particularly
applicable for dry rocks, however, it may also be used for
saturated rocks (cf., Bacon, 1965). With the very low
porosities found in the rocks chosen for this study, the
use of vacuum dried samples is not an unreasonable simpli-
fication.

Limitations of the Sample
Efeparation ProcedUre

 

 

The greatest limitation to the sample procedure given
above is that it is very time consuming. The author found
this to be the most frustrating aSpect of the study. A
minimum of forty-two man-hours was required, in sample
preparation, to produce measurement disks for nine different
directions. For most samples, the preparation time was
even longer because of damage during one, or more, of the
preparation steps. While it is possible that a slightly
altered preparation procedure might reduce the preparation
time somewhat, it is unlikely that electrical anisotropy
measurements will ever be made on a production basis be-

cause of the preparation time involved.

 

 

The data 0‘L
:agacitance and

renal frequenci
1st orientatior
12a K and a ten
re the desired

.azcratory date
31'1‘: -

u values

.::rcation of

“EP‘endent t
5 see
i Squai

‘ ‘3‘: Coef‘
‘i‘rz‘n

especi
It {13
2‘3 -

CHAPTER VI

DATA ANALYSIS

Introduction

 

The data observed in the laboratory consist of
capacitance and dissipation factor values at different
signal frequencies, sample disk dimensions, and sample
disk orientations. This data is greatly removed from
the K and a tensor principal values and directions, which
are the desired information of the study. The original
laboratory data must first be converted into directional
K and o values. The directional K and o values give an
indication of anisotropy, but do not define the appropri-
ate symmetric second-rank tensors. However, if six, or
more, different directional values are available, the six
independent tensor coefficients can be obtained by use of
a least square reduction process based upon equation 3.5.
The least square tensor coefficient process will be
developed in this chapter. The output of the least square
tensor coefficient reduction process defines the tensors
with respect to the laboratory co-ordinate system, however,
it does not give the tensor principal values and directions.
This is accomplished by a successive approximation pro-

cedure to be described later in this chapter.

75

Thus , the

    
 

excessive reduc
ation is obtain

at least, would

available at Mic
his grogram wa:
Lie successive

:aiiate outputs
reduction. Thl

«are t

nen pres
Leasi
3

While t

mnrical ar

,
ll
1'1

.
.J.
‘

G

Stacey

2:419 te 1y (3

fr."
‘\

i ‘1’):

“1"! goal

1n‘-’EStiqé

76

Thus, the original data must pass through three
successive reduction processes before the desired infor-
mation is obtained. The last two reduction processes,
at least, would be very tedious and error prone if done
by hand. For this reason, all of the reduction pro-
cesses were programmed for the CDC 3600 digital computer
available at Michigan State University (see Appendix B).
This program was written so that the results of each of
the successive reduction processes were output. The inter-
mediate outputs allowed the data to be monitored during
reduction. The tensor principal directions and values
were then presented in graphical and tabulated form.

Least-Square Determination of Symmetric
'"'Second}Rank Tensor Coefficients

While there have been previous investigations of
electrical anisotropy of rocks (cf., Rao, 1948; Parasnis,
1956; Stacey, 1961), none of these were concerned with
completely defining the tensor prOperties. One of the
primary goals of the present study is to develop a method
of investigation which will allow complete definition of
the dielectric constant and conductivity tensors in rocks.

The K and 0 values obtained from a single disk of
rock are directional values of the tensor (after equation
3.5) in the direction normal to the parallel disk faces.

Directional measurements in several directions give an

indication of anisotropy but do not, by themselves, define

   
   
 
    

”re tensor. The
:aused to obtai

Consider f

:33 unknown
1 Fl

:cafficients, it
rexgeriment i
experimental er
Talus of a.
Now, if 0
351511;) to measu
1: this will re

{‘F‘fi‘.“
““4“ "'8 .

It is

:M

aetermine t.

= lea
St Square
“39“ Such tl

n' a -
«inemled'

(*1
*U

H M

77

the tensor. These directional measurements can, however,

be used to obtain a set of least square tensor coefficients.
Consider first, the general problem where there are

{zi}2=l unknown quantities which, when multiplied by the

q
coefficients, {bi}? 1, give the quantity, a = Z b.zi. If

l= . 1
an experiment is set up to measure a, there willlbe an
experimental error, 6 = a - m, where m is the measured
value of a.

Now, if other experiments, p in number (p > g), are
set up to measure a, each with a different set of bi and
m, this will result in a system of p equations and q un-
knowns. It is desired to use this system of equations
to determine the best set of values for the zi. If p > q,
a least square procedure can be used, i.e., the 21 are

chosen such that the sum of the squares of the errors, E,

is minimized, where:

a. = z z b..z - m. = (BZ — M)T(BZ - M),

2 p q 2
1 i=1 j=1 13 3 1

to
II
"row

1

where B = (bij) is a p X q matrix, Z = (zj) is a g X 1
column vector, M = (mi) is a p x 1 column vector and the
superscript, T, indicates matrix tranSpose.

Equation 6.1 is positive definite, and will now be
minimized. The classical approach to the least square

problem would be to differentiate equation 6.1 with

 

:assect to each

 

rations equal

. ' a
grlnowns which

as just such a

anal to zero d:

seen give a mi

I“.

93'

shun, or infl

“I-
3:.CI‘I

Lee have a mi n i

.-

 

d derivatis

s‘rstitute in t

1;:ferentiation

..
\n‘fih
"'U'ao

d derivati

astray-mm is a r

.m‘
.. 373
n ‘¥b

"‘ure is ,

'3: it does no

“-1.8 Thus .

t

.. Emmi 2e EC

\

§

The mat:

a "
I‘D

“ p idei

n;‘

U

Sides of

M=(I

nu“

. a Shortha

78

respect to each of the 2j and set each of the resulting
equations equal to zero. This would give q equations and
q unknowns which are then solved for the zj. Nye (1964)
uses just such a procedure. Setting the first derivatives
equal to zero does not insure that the zj so determined do
indeed give a minimum, but only an extremum (maximum,
minimum, or inflection point). To insure that we do in-
deed have a minimum using this approach, we must take the
second derivative of E with respect to each of the 2j and
substitute in the values of the zj determined by the first
differentiation to test for the type of extremum. If the
second derivative is positive at an extremum point, the
extremum is a minimum (Sherwood and Taylor, 1957). This
procedure is, at best, tedious and, at worst, tenuous,
for it does not protect against local minima and saddle
points. Thus, matrix Operations will be used exclusively
to minimize equation 6.1.

The matrix, B(BTB)-lBT, is a p X p matrix. Thus,
the p X p identity matrix, I, can be written as:
I = (I - B(BTB)—1BT) + B(BTB)-1BT. Post multiplying

both sides of this equation by M yields:

1 l T

M = (I - B(BTB)- BT)M + B(BTB)— B M.

For a shorthand notation, define:

l T

G = (I - B(BTB)- B )M,

H = (BTB)-lBTM.

.......

n=<3+ Bi

  

.:.i equation 6 . ~,

('11
II

(M -

(BZ -

BTG = ET
13,9quat10n

E: (82

inch wi 11 be

.v“ '
.-.: r1

ght ham
.aIiIiite and

‘*~~H.s. of
azed Only if

79
Then:
M = G + BH

and equation 6.1 can be written as:

E = (BZ - BH - G)T (Bz - BH - G)

T

(82 - BB)T (BZ - BH) + G
But:

1 T

Thus, equation 6.3 becomes:

B = (BZ - BH)T(BZ - BH) + GTG,

B G = BT(I - B(BTB)' B )M = (BT - BT

T

o + 2(z - H)TB G. 6.3

)M = 0.

which will be minimized by varying Z. The first term on

the right hand side (R.H.S.) of equation 4.6 is positive

definite and therefore non-negative. The second term on

the R.H.S. of 6.4 is independent of Z. Thus, E is mini-

mized only if Z is chosen such that:

z = H = (BTB)’1BTM.

Then E becomes the least square error and

T l

E = G G = ((I - B(BTB)'

BT)M)T((I - B(BTB)-

is given by:

lBTHVI).

6.6

 

 

I
Now that tl'

 

 
 
 
  
  
 
 

:ltiple indepen
s;a:ific case of
:a be considere
iazarnination of
LS equation 3.5 .
else is assume
:ratnan six '1
11a above least
ifineasured va
b. = "2
il ‘1

b12=1

‘th Summat
iii:

V Squar
““PEctiVe
75%) is o

s2 =
1:8 .
N Lne I

80

Now that the generalized least square problem for
multiple independent variables has been solved, the
specific case of the symmetric second-rank tensor model
can be considered. The model used for the least square
determination of symmetric second-rank tensor coefficients
is equation 3.5. For this model, the directional tensor
value is assumed to be measured parallel to 1. Thus, if
more than six (FILE were measured for different 1, then

the above least square procedure would be used, with mi,

the measured value to Tllii and:
bil = Ail' bi4 = 2iizii3' 21 = T11' 24 = T23'
biz = 132' bis = 2ii3iil' 22 = T22' z5 ='T31' 6'7
bi3 = ii3' bi6 = ZiiliiZ' 23 = T33' 26 = TlZ'

where the repeated subscripts in equations 6.7 do not indi-
cate summation. The least square tensor coefficients and
least square error are then given by equations 6.5 and 6.6,
respectively. The mean square error (Jenkins and Watts,

1968) is obtained from equation 6.6 by:

 

 

 

   
    

The rms er
error. Physical
inactional val‘
sing the least
rs error can be
’aIii-tipal tenso

is least squar

The deri
5358 above is
its Of this
we (1964)

£22339 deve 1 DE

53:0qu tha
The le

“mation 1

T- J
“‘2‘“

9058i]

'7-.
V

- A

w .
““ltative

5‘, 1‘
«Se

‘.

S the

4: 33a 1Y8 1

-.
‘5‘
‘5’.

imation

Tob

‘-

.‘ reeks
O

81

The rms error was chosen as the tensor determination
error. Physically, it represents the probable error in a
directional value of T given by equations 3.4 and 3.5,
using the least square tensor coefficients. Thus, the
rms error can be used as an uncertainty factor for the
principal tensor values, for they will be determined from
the least square tensor coefficients by equations 3.4 and
3.5.

The derivation of the least square tensor coefficients
given above is somewhat academic. This is because the re-
sults of this development are the same as those obtained
by Nye (1964) using the classical approach. However, the
above development serves to confirm the results of the
classical development with none of the uncertainties which
surround that approach.

The least square procedure developed above uses
estimation theory. The model of equation 3.5 is the only
model possible for a symmetric second-rank tensor. The
qualitative electric model developed for rocks in Chapter
IV uses the symmetric second-rank K and o tensors. Thus,
the analysis of the laboratory data should consist of the
estimation of the parameters of equation 3.5.

Principal Coefficients and Directions

for aiéfierwalized'éigmnetric

Second-Rank Tensor

 

 

To be able to compare the electrical anisotropies

of rocks, it is necessary to determine the orientations

 

 

:5 values of ti“
second-rank K a.
a: the eigenva'
sentation matr l‘.
:23. tensor , th
iii-3‘) is a cubi
salve on a rou
133-‘9 procedure
égietion, Th:
:ESiS are out

COUS ide

iii-er
e ‘3. = v!

1': r n .
‘e‘Lathe

k
g;fi‘
‘i .

._cr

Ere r .
l

3);; .

if.

"I:

82

and values of the principal coefficients of the symmetric
second-rank K and o tensors. This is equivalent to find-
ing the eigenvalues and eigenvectors of the 3 x 3 repre-
sentation matrix. For a generalized symmetric second-
rank tensor, the characteristic equation (Marcus and Minc,
1965) is a cubic equation, which is rather difficult to
solve on a routine basis. Nye (1964) suggests an itera-
tive procedure as an alternative to solving the cubic
equation. This iterative procedure and its theoretical
basis are outlined below.

Consider the tensor relationship:

where !'= v(11, 12, 13) is the independent vector and T
is relative to its principal axes. Then, the dependent
vector is 2.: v(Tllll, T2212, T3313) and the representation

surface for T is:

11x1 22

Figure 6.1 shows a cross-section of the representation
surface for T containing both E and 2. If the vector,

p, in Figure 6.1 is parallel to y, then: p = r(ll, 12, 13),
where r is representation surface radius at the point, p.
Equation 6.11 is of the form for an isotimic surface,

f(x1, x2, x3) = C, where C is a constant (Davis, 1961).

 

Fig. 601.
surface in the

 

 

 

83

 

Fig. 6.l.--Cross-section through the T reference
surface in the plane containing both _u_ and y_.

The (unit) normal to such a surface at the point,

P(rll, r12, r13) is given by:

1 12, T3313)

2

Vf(P) _ (T11 1' T22

P

D)
l

 

2
/(T1111) + (T + (T

2212) 3313)

which is parallel to 3. As can be seen in Figure 6.1,
E is more nearly parallel to the minor semi-axis of the
cross Sectional ellipse than 2.

Now, consider the case of a generalized symmetric

Second~rank tensor, T (Tij # or for all i! j). The

13'
information from Figure 6.1 and equations 6.10 and 6.12
Can now be used in an iterative sense to determine the

aXimum prinCipal direction of the tensor (minimum semi-

”is of the ellipsoid). The iterative, or successive

a - -
ppro’flmation procedure 13 as follows:

i. Take a
and pi
matri:
whose
minor
maxim

if 11

(a): wi
-l

 

Paral

F%

s.“

each SuCce

:ze orientation

 

tem‘axis of t},

84

1. Take an arbitrary unit column vector, 11,

and pre-multiply it by the representation

matrix for T to obtain a new vector, wl'

whose orientation is closer to that of the
minor semi-axis of the ellipsoid (i.e.,
maximum principal axis of the tensor), e.g.,

if 11 is parallel to X in Figure 6.1, then

21 will be parallel to 3.

ii. Normalize ml to obtain a new unit vector, 12,

parallel to 31 and repeat step i.
With each successive application of the above procedure,
the orientation of 3i is closer to that of the minor
semi-axis of the ellipsoid (maximum principal tensor

direction) than was the case for w

. . Also, with each
—i-1

successive 11, the vector, dli = 1. - l approaches

1 i-l’

zero. Because of this, an appropriate cutoff point for
the iteration procedure can be chosen on the basis of
the decreasing magnitude of 911° The cutoff used for

3

2 Ti

i=1
109

the maximum principal tensor direction (minor semi-axis

2
i
this study was

 

. The last li is then taken as

of the ellipSOid), lmx' Once lmx has been determined,
the maximum principal tensor value is determined by

inserting lmx and T into equation 3.4, i.e.:

AT A
= . . 3
me 1mx T 1mx 6 l

To find the
:9, the above it
aazrix inverse of
item principal
1:5? (13% T-l) i

The interme
is obtained by t;
and normalizing
Tis then found

3.4 {6.13)

85

To find the minimum principal tensor direction,

imn’ the above iterative procedure is applied to the

matrix inverse of the representation matrix for T. The

minimum principal value of T is found by inserting lmn

and T (not T-l) into equation 3.4 (6.13).

A

The intermediate principal tensor direction, lint'

is obtained by taking the vector cross product, imx X lmn'
and normalizing it. The intermediate principal value of
T is then found by inserting iint and T into equation

3.4 (6.13).

 

sAflpLE

Most Of the
.-':::ion of this 51
its Precambrian O
.:ited for a deta
ragional geology
1337; Van Hise ar
’59; Boyum, 1963

accessible to r o

\
a?“
‘5

a, there is

re: , '
aniorphic zone

‘,-_p
.3“

.‘
LJ'

). Also, pr

~~=‘- Carried ou1

CHAPTER VII

SAMPLE LOCATIONS AND DESCRIPTIONS

Most of the samples considered in the laboratory
portion of this study were collected by the author from
the Precambrian of Northern Michigan. This area is well
suited for a detailed petrOphysics investigation. The
regional geology is quite well known (Van Hise and Bayley,
1897; Van Hise and Lieth, 1911; Martin, 1937; Bayley,
1959; Boyum, 1963). Outcrops are abundant and easily
accessible to roads. Within a rather small geographic
area, there is a wide variety of rock types in several
metamorphic zones (James, 1955; Henrickson, 1956; Willar,
1965). Also, previous petrophysics investigations have
been carried out on rocks from this area (Merritt, 1963;
Whitaker, 1966). The results of these previous petro-
physics studies were used as a guide for rock type
selection in the present study.

Figure 7.1 is a map of a portion of Northern Michigan
showing the sample collection sites for this study. The
broken line in Figure 7.1 indicates the Pre-Animikie-
Animike contact (after Martin, 1937; Boyum, 1963). The

Marquette synclinorium is the major structural feature

86

 

o-IEO ._ \

 

 

87

.mmE coaumooH mamsmm cmmflsowz cumnuHOZII.H.> .on

 

032.55

 

n .

252003}

 

 

 

. do 435..

.lll..l.l-l|l bug 200 w. £2.24le

 

 

  

.3 22..

 

 

2

 
  

 

 

:i the area, extend

estaxis from a PC
1‘. ler'quette. The

n
I‘.‘

tiaikie meta-igne

$1;

underlain by a
,teisses, meta-vo

.enawan sedime

Paleozoic se<

u
":32
“in.

Sample “‘1
s;ie of ms. 14
Earaqa County 1
:e formatiOI

o

in

. "Q

Lvlr“
'5

3 0E James

Q“! I
"“51

Sts of a

'7‘graywacke

88

of the area, extending roughly forty miles along an east-
west axis from a point on the Lake Superior shore south

of Marquette. The rocks filling the basin consist of
Animikie Series meta-sediments, meta-volcanics, and post
Animikie meta-igneous rocks of Keweenawan age. The basin

is underlain by a Pre-Animikie complex of granites,
gneisses, meta-volcanics, schists and minor quartzites.
Keweenawan sediments and volcanics outcrOp to the west

and Paleozoic sediments to the east of the sample collection
area.

Sample M-l was collected from a road cut on the east
side of U.S. 141, about one mile south of Covington in
Baraga County, Michigan. This sample is from the Michi-
gamme formation and the site is located in the biotite
zones of James (1955) and Henrickson (1956). The sample
consists of a light gray, fine grained, meta-argillite to
sub-graywacke. Slaty cleavage (C in Figure 8.1) is sub-
normal to the [001] (z) laboratory direction.

Sample M—Z was collected from a road cut on the
north side of U.S. 41-M. 28, just west of the Carp River
and about one mile east of Negaunee in Marquette County,
Michigan. This sample is from a Pre-Animikie greenstone
schist (possibly Mona) containing pillow structures. The
collection site is in the chlorite zones of James (1955)
and Henrickson (1956). The sample is a fine grained,

chlorite schist with calcite in veins. Epidote and some

 

EE’Tite are also PI
Egan-normal t0 I
sit. cleavage (C
:tersect forming
i313}.

Sample M-3
sis of county r
section 10, T475
sizple is from
iclite. The sa
ifJames (19551
Staurolite—qua
Easies. The 5

am careen am]

89

pyrite are also present. The foliation (f in Figure 8.2)
is sub-normal to the [100] (x) laboratory direction. The
rock cleavage (C in 8.2) is sub-normal to [001]. These
intersect forming a lineation (L in 8.2) sub-parallel to
[010].

Sample M-3 was collected from an outcrop on the east
side of county road 607 near the west section line of
section 10, T47N, R30W in Marquette County, Michigan. This
sample is from a medium grained, dark greenish-black amphi-
bolite. The sample site is located in the sillimanite zones
of James (1955) and Henrickson (1956) and Villar's (1965)
staurolite-quartz sub-facies of the almandine amphibolite
facies. The sample consists primarily of medium grained
dark green amphiboles (probably hornblende) with some
chlorite or epidote and pyrite. There is a fracture set
labeled F in Figure 8.3.

Sample M-5 was collected from the rubble of a road
cut on county road 607 near the north section line of
section 10, T41N, R30W in Dickinson County, Michigan. The
sample is a granite-tourmaline gneiss, containing quartz,
two feldspars, tourmaline, and mica. The collection site
is located in the chlorite zone of James (1955). The
gneissic banding is labeled B in Figure 8.5.

Sample M-6 was collected from a road cut on the
north side of U.S. 41-M.28, just west of Tioga Creek in

Baraga County, Michigan. This sample is from the

Echigaxmne
:"te zones
5.1.18 is
j: in Fig‘
Erection

Sax

[:1

i

If)
n

(I,

9O

Michigamme Formation and the site is located in the bio-
tite zones of James (1955) and Henrickson (1956). The
sample is a light gray sub-graywacke with slaty cleavage
(C in Figure 8.6) sub-normal to the [001] laboratory
direction.

Sample M-7 was collected in the pit of the Republic
Mine, Marquette County, Michigan. This sample is from
the Negaunee formation and the site is located in the
silliminate zones of James (1955) and Henrickson (1956)
and Villar's (1965) staurolite-quartz sub-facies of the
almandine amphibolite facies. The mineralogy of sample

M-7 is given in Table 7.1. A microscopic petrofabric

TABLE 7.l.--Mineralogy of M-7.

 

 

 

Mineral % Volume
Quartz 63.6
Hornblende 12.5
Epidote 8.6
Augite 6.3
Grunerite (?) 4.3
Hemitite and other 1.9
Total 100.0

analysis of quartz c-axes based on 400 grains in two thin
sections is presented in Figure 7.2. The > 5 per cent

maximum indicates a strong lineation (L in Figure 8.7) in

91

 

g.

.....
'-

__lLNII
a":-

"Erniivf"

 

- 25-434
[:10 '% Quartz C-axes
[I] I-2% Plotted on 00: [HEM-5%
Ez-sx I > 5%

 

 

 

Fig. 7.2.--Microscopic petrofabric analysis of M-7.

 

I |

:is direction -
gtdle (G in Fig
:iicate a SYmme
tare are faint
Laeiea a in Fi-
Sample M‘
as: side of a
1Earaga Coun
gate formatiO
1:5(1955) 5
firm muscovi1
staurolite or

are shown in

.a sub-normal

92

this direction. There also appears to be a petrofabric
girdle (G in Figure 8.7) sub-normal to [110] which may
indicate a symmetry plane in the rock. In addition,
there are faint alternating quartz and epidote rich bands
labeled B in Figure 8.7.

Sample M-8 was collected from an outcrop on the
east side of a secondary road in section 36, T48N, R3lw
in Baraga County, Michigan. The sample is from the Michi-
gamme formation and the site is in the staurolite zones of
James (1955) and Henrickson (1956). The sample is a medium
brown muscovite-staurolite schist with large euhedral
staurolite crystals. These euhedral staurolite crystals
are shown in Figure 7.3. The foliation (f in Figure 8.8)
is sub-normal to the [010] laboratory direction.

Sample M-9 was collected from an outcrop on the west
side of a secondary road near the east section line of
section 27, T43N, R31W, Iron County, Michigan. This sample
is from the Hemlock formation and the site is in the chlor-
ite zone of James (1955) and the green schist facies of
Bayley (1959). The sample consists of a fine grained
grayish-green matrix containing many small feldSpar
crystals and some possible epidote and chlorite. The
rocks at this collection site all had an over-all leached,
or washed out, appearance. There are three fracture sets,

and F

labeled F in Figure 8.9. Their intersections

1' F2' 3
form lineations, labeled L1' L2, and L3 in Figure 8.9.

93

 

Fig. 7.3.--Euhedral staurolite crystals in M-B.

Sample M—10 was collected from the road bed of a
logging trail near the north section line of section 5,
T49N, R27W, Marquette County, Michigan. The sample is
a dark green amphibole schist containing quartz, horn-
blende or actinolite, two feldspars, chlorite, and
epidote, with calcite in veins. The collection site is
in the chlorite zone of James (1955). The foliation (f
in Figure 8.10) is sub-normal to [001] and the lineation
of the amphibole crystals (L in Figure 8.10) is sub-
parallel to [010].

Sample M-ll was collected from a road cut on the
west side of U.S. 141-M. 28, about 0.3 miles south of

their intersection with U.S. 41 in Baraga County, Michigan.

94

This sample is from the Michigamme formation and the site
is in the biotite zones of James (1955) and Henrickson
(1956). The sample is a massive, light-gray, sub-graywacke
to siltstone. There is a fracture set marked F in Figure
8.11.

Sample B-4 was the only sample investigated in this
study which was not collected from Northern Michigan. It
was collected by the author from the Cardiff complex,
Cardiff Township, Ontario (Hewit, 1959; Stonehouse, 1967).
The collection site was at stOp number eight on the 1967
Institute on Lake Superior Geology, Cardiff Complex field
trip (Stonehouse, 1967). The sample is a syenite gneiss,
containing two feldspars, some quartz, agerine-augite,
micas, and opaques. The gneissic banding (B in Figure
8.4) is sub-normal to the [010] laboratory direction and
there is a slight lineation (L in 8.4) due to the apparent
alignment of the agerine-augite crystals sub-parallel to

[100].

 

aa—
"«

‘5."
his

 

«.44
‘N i

Ihs

 

 

CHAPTER VIII

DISCUSSION AND INTERPRETATION OF THE

LABORATORY MEASUREMENTS

K

Presentation of the Data

The electrical measurement data for each sample
investigated in the laboratory portion of this study are
presented in Tables 8.2 through 8.13 and Figures 8.1
through 8.12. The tabulated data consists of the K and
(effective) conductivity, a, principal values, rms errors
and direction angles for the tensor principal axes. This
data is given for each measured signal frequency for each
of the rock samples considered in this study. The signal
frequency, principal value, and rms error presentation is
self explanatory. The presentation of the direction angles

for the tensor principal axes is demonstrated by Table 8.1.

TABLE 8.1.--Direction angles for tensor principal axes.

 

“11 0‘21 0‘31
“12 “22 0‘32
“13 “23 “33

95

96

TABLE 8.2.--M-1 (meta-argillite) electrical anisotropy data.

 

 

 

1"“: _ae- . ‘—

 

. K . l). (J r)

K 1 y. _ Princ1pa1 rms ._ y

. . ‘ Prlnulpnl nxcs . Princxnal AXQs
f Pr1nc1pa1 rms Direction \n 1‘s Values eror 0‘ ‘t ‘ , 1‘s

(cps) Values Error (0)] g L x109 x109 lrec 1on)nng L

(mho/m) (mho/m)

6.480 137.4 58.8 64.2 1430.8 115.5 154.2 86.3
100,000 6.376 0.227 99.0 60.2 148.6 1008.9 174.0 154.2 64.1 87,9
6.281 48.8 45.9 73.4 653.1 86.5 87.5 4.2
6.741 146.5 86.7 123.2 588.2 11 .4 155.0 84.9
30,000 6.416 0.196 123.4 91.3 33.4 461.0 62.0 15'.6 )5.6 90.6
6.332 88.0 3.3 87.1 299.1 8 .4 85.1 5.1
6.967 117.8 77.4 129.4 239.5 114.0 154.7 82.6
10,000 6.645 0.240 110.8 90.9 40.8 193.4 29.6 115.9 66.1 92.6
6.552 81.1 12.6 81.1 127.4 89.3 82.7 7.8
7.342 135.4 45.5 87.8 94.42 74.0 162.3 82.6
3,000 6.868 0.285 134.1 135.4 84.8 73.13 9.16 16.0 74.3 92.7
6.833 84.9 87.8 5.6 50.00 89.4 82.2 7.8
7.754 131.2 41.3 92.1 37.24 79.6 168.9 86.2
1,000 7.222 0.315 138.1 130.1 80.0 28.61 3.50 10.5 79.7 92.0
7.017 83.8 81.9 10.2 19.28 88.7 86.0 4.2
8.261 125.1 144.6 85.6 14.860 84.6 8.0 96.0
300 7.657 0.372 144.8 54.8 90.3 11.404 1.595 6.7 95.7 93.4
7.345 87.8 86.2 4.3 7.000 86.0 84.4 6.9
8.629 121.3 148.4 86.2 8.020 89.6 172.4 82.4
150 7.974 0.410 148.6 58.6 90.5 6.082 0.889 2.3 90.0 92.3
7.533 88.5 86.5 1.8 3.312 87.7 82.4 8.0
8.862 118.6 151.1 85.9 5.675 88.0 2.5 91.5
100 8.173 0.430 151.4 61.4 91.4 4.485 0.561 4.0 92.1 93.3
7.654 80,2 85 7 4.3 2.493 86.6 88.6 3.7
9,168 117.1 152.8 87.7 1. 73 92.2 3.8 93.0
60 8.422 0.459 15.3.9 62.9 90.2 2.954 0.406 4.6 88.0 94.1
7.798 89.1 87.8 2.1 1.627 86.0 86.8 5.1
9.465 66.1 155.9 86.8 2.2012 94.4 175.4 88.7
30 8.844 0.518 23.9 66.2 91.7 1.7899 0.2736 6.0 94.5 94.0
8.021 89.7 86.4 3.6 0.9569 85.9 89.0 4.2

 

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TABLE 8.3 --—-M-2 (greenstone) electrical anisotropy data.
K K K p . °. 1 f!" 0

f E’r’incipal rms Principal Axes régitgg Hirer Principal Axes

(ops) \lalues Error Direction Angles x109 x188 Direction Angles
( ) (mho/m) (mho/m)

1.0.776 109.4 119.2 36.1 1064.8 99.8 76.7 163.4

1(30,ooo 8.866 0.196 160.5 78.5 105.5 490.6 29.0 166.5 83.2 78.4
8.126 88.2 31.8 58.3 378.0 80.8 15.0 78.3
112.316 107.0 133.6 48.5 389.6 94.0 90.0 4.0
30.000 9.542 0.276 162.4 74.5 98.2 158.8 17.2 174.4 86.1 94.0
9.040 85.4 47.7 42.7 148.2 86.1 3.9 89.7
14.015 102.1 138.8 51.4 148.61 94.7 69.6 159.0
10,000 10.301 0.313 167.3 78.2 94.7 57.86 4.30 172.2 85.8 83.4
9.754 86.2 51.2 39.0 52.72 83.8 20.9 70.1
16.15 99.8 130.0 41.7 58.88 97.7 78.9 166.4
3.000 11.10 0.32 169.4 80.6 94.8 23.86 0.73 169.7 84.8 81.1
10.54 86.0 41.6 48.7 18.11 83.2 12.3 79.8
1 18.71 99.2 119.4 31.1 27.299 99.4 80.2 166.4
'000 12.14 0.37 169.7 81.4 95.6 10.442 0.782 167.2 83.0 79.3
11.42 85.6 30.9 59.5 8.266 81.4 12.0 81.6

22.48 98.9 96.0 10.8 13.482 103.1 75.7 160.4

300 13.49 0.40 169.3 83.1 98.2 5.703 0.478 163.7 83.8 75.0
12.79 84.1 9.1 83.1 3.734 80.5 15.6 77.7
1 25.68 99.5 85.0 169.2 9.224 103.6 72.1 157.2
50 14.75 0.42 168.5 04.4 30.0 3.780 0.453 161.9 32.9 73.5
13.84 83.6 7.5 86.0 2.121 78.4 19.3 74.8
1 28.18 99.5 89.6 9.5 7.664 105.0 74.1 157.9
00 15.77 0.49 168.2 83.1 99.5 3.078 0.450 160.6 82.3 72.4
14.60 83.2 6.9 89.2 1.574 78.0 17.7 77.1
6 32.07 99.0 92.5 9.3 6.106 104.1 72.0 156.8
0 .17.21 0.56 168.3 82.1 98.6 2.382 0.370 160.7 81.8 72.6
16.00 82.6 8.3 86.3 1.297 77.1 19.8 75.2
30 38.29 99.6 79.1 165.4 4.6528 104.7 68.3 153.4
20.16 0.59 167.3 83.7 79.0 1.8300 0.2757 159.4 82.0 71.1
18.23 81.9 12.6 80.4 0.8812 75.9 23.2 71.9

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TABLE 8.4 .--M93 (amphibolite) electrical anisotropy data.
X K K c o o
f I>z'incipal rms Principal Axes Principal rms Principal Axes
(eras) \Jalues Error Direction Angles Values Error Direction Angles
(°) x109 x109 (°)
(mho/m) (mho/m)

1().29O 20.8 108.5 80.9 2193.9 24.8 114.2 95.2
30.000 9.276 0.802 99.4 89.3 9.4 1110.8 623.3 94.6 112.5 23.0
8.416 71.6 18.5 87.7 395.0 65.7 34.1 67.6
11.220 23.2 112.3 84.2 769.5 26.4 115.8 95.0
10.000 9.689 1.035 99.2 97.2 11.7 381.2 210.3 95.5 112.6 23.4
8.857 68.9 23.6 79.9 130.8 64.3 35.5 67.2
12.310 24.9 114.7 87.1 257.84 27.2 116.8 94.4
3:000 10.235 1.312 98.2 101.0 13.8 126.39 68.36 96.2 111.6 22.6
9.066 66.6 27.4 76.5 47.44 63.6 35.6 67.9
13.462 25.3 115.3 88.7 94.64 28.6 118.4 93.4
14000 10.761 1.629 97.8 103.7 15.8 45.09 24.44 97.4 110.7 22.1
9.290 66.1 29.2 74.2 19.05 62.5 36.3 68.2
14.775 26.2 116.2 89.6 34.952 29.7 119.5 92.9
300 11.386 1.972 97.6 104.8 16.7 16.982 8.333 97.9 109.7 21.4
9.556 65.1 30.6 73.3 9.147 61.6 36.7 68.8
15.725 26.6 116.6 90.2 18.618 30.0 119.9 91.4
150 11.813 2.211 97.4 105.3 17.0 9.201 4.404 98.4 107.3 19.3
9.761 64.5 31.3 73.0 5.303 61.4 35.5 70.7
16.237 26.9 116.9 90.0 13.344 30.4 120.4 90.9
100 12.088 2.337 97.8 105.5 17.5 6.465 3.095 98.8 106.7 19.0
9.912 64.4 31.7 72.5 4.006 61.1 35.6 71.0
17.03 27.0 117.0 89.9 8.929 31.7 121.7 90.0
60 12.44 2.51 97.7 105.1 17.0 4.207 2.040 98.6 104.2 16.7
1.0.10 64.3 31.5 73.0 2.931 59.8 35.4 73.2
1.8.18 27.5 117.5 90.7 5.232 32.7 122.6 87.6
30 1.3.04 2.78 98.0 107.0 18.8 2.645 1.216 100.0 101.3 15.2
10.35 63.9 33.1 71.2 1.923 59.2 35.0 75.0

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TABLE 8.5».--B-4 (syenite gneiss) electrical anisotrOpy data.

 

 

K K K 0 a o
. . Principal Axes Principal rms Principal Axes
f Pr 1nc1pa1 rms . . . .
k:ps) \lalues Error Direction)Angles Viiggs Eiigg Direction)Angles
(mho/m) (mho/m)

7.516 148.6 62.9 104.8 591.8 143.4 77.9 56.1
30.000 7.417 0.159 73.0 89.4 163.0 350.9 123.3 65.3 118.1 39.0
6.761 64.4 27.1 81.8 235.4 64.8 31.0 73.1
7.842 150.2 65.1 105.4 287.35 148.8 80.6 60.6
10,000 7.524 0.199 72.9 89.8 162.9 162.42 40.12 74.4 132.1 46.3
6.836 66.3 24.9 82.8 51.78 63.8 43.7 58.0
8.228 151.6 68.6 107.7 116.88 153.1 84.8 63.7
3:000 7.692 0.256 72.2 93.3 161.9 53.65 20.84 72.7 128.5 43.6
6.974 68.7 21.6 86.5 16.59 70.1 39.0 58.1
8.802 151.2 70.4 110.3 55.177 149.6 82.4 60.7
1:000 7.887 0.349 71.0 96.9 159.7 18.668 9.142 72.2 129.4 44.8
7.075 69.1 20.9 89.8 7.951 66.2 40.4 59.5
9.755 146.4 66.1 67.8 25.063 148.8 83.8 59.5
300 8.189 0.512 73.4 105.7 23.2 9.222 4.376 71.7 132.2 47.8
7.184 61.6 29.2 83.6 3.006 65.6 42.9 57.3
10.322 148.1 70.3 66.0 15.114 148.9 82.2 60.1
150 8.271 0.590 71.6 107.1 25.6 5.292 2.646 73.3 132.6 47.3
7.372 64.9 26.6 81.8 1.450 64.6 43.6 57.3
10.732 149.6 73.4 65.3 11.454 149.0 83.0 60.0
100 8.428 0.684 72.0 110.9 28.2 3.817 2.028 73.4 134.1 48.8
7.387 66.4 27.2 77.3 0.927 64.6 45.0 55.8
11.357 149.3 74.4 64.2 8.0440 149.0 84.1 59.7
60 8.612 0.778 71.8 113.5 30.4 2.6606 1.476 72.6 134.2 49.4
7.450 66.1 28.7 75.0 0.5644 65.2 44.8 55.5
12.427 149.2 75.7 63.3 5.4058 149.1 83.8 59.8
30 8.905 0.966 72.4 117.6 33.6 1.6591 0.9937 73.5 135.8 50.4
7.566 65.6 31.7 71.1 0.2925 64.6 46.5 54.2

 

 

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TABLE 8.6 . --M-5 (granite gneiss) electrical anisotropy data.

 

G d

 

 

K K K Principal rms O
f Pr incipal rms I)P::2::p:l;\2x1§5 Values Error “PriniipalAAx135
(cps) Values Error 1 f0) 9 L x109 x109 ”QC “(’2’ ng es
(mlm/m) (mhn/m)
6.763 98.9 28.9 117.2 1135.9 90.0 168.2 78.2
30.000 6.197 0.053 105.1 65.7 29.1 732.4 135.3 116.1 79.4 28.4
5.873 17.6 75.3 80.5 030.7 26.1 84.8 64.5
7.300 97.9 26.9 115.6 488.4 89.9 154.8 64.8
10.000 6.515 0.145 107.4 67.8 28.8 327.9 66.1 126.1 69.9 42.9
6.185 19.2 75.5 77.6 255.1 36.1 75.4 57.8
8.126 95.9 21.7 110.8 211.91 90.2 11.1 101.1
3.000 7.004 0.257 110.2 72.5 27.3 124.38 33.32 116.9 80.2 28.9
6.596 21.2 77.6 73.1 99.53 26.9 84.8 63.7
1 9.179 94.3 18.9 108.4 88.82 39.0 17.6 107.6
.000 7.612 0.428 112.0 74.5 27.4 51.63 16.00 115.6 73.7 31.0
7.073 22.5 79.3 70.4 39.08 25.6 83.4 65.4
10.556 92.9 19.9 109.7 ‘2.88 88.4 7.0 96.8
300 8.382 0.678 113.0 73.1 29.1 19.58 6.51 116.1 83.2 27.1
7.643 23.2 79.8 69.4 15.46 26.2 88.4 63.9
11.458 92.1 17.8 107.7 17.197 83.5 168.8 99.1
150 8.931 0.850 113.2 74.6 28.3 10.364 3.650 114.3 101.0 26.9
8.096 23.3 81.3 68.6 8.079 25.2 87.9 64.9
11.943 92.8 15.7 105.4 11.779 85.0 173.8 93.6
100 9.239 0.917 114.5 77.1 28.1 7.132 2.677 114.8 95.4 25.4
8.363 24.7 81.2 67.1 5.724 25.3 87.0 64.9
12.708 92.5 19.7 109.6 7.434 81.4 171.3 91.2
60 9.613 1.136 113.8 73.2 29.8 4.594 1.732 114.0 94.6 24.5
8.691 24.0 80.0 68.4 3.640 25.7 82.6 65.5
13.695 90.3 14.9 104.9 3.986 75.0 163.1 82.3
30 10.308 1.384 113.3 76.4 27.4 2.608 1.010 113.6 83.8 23.7
9.186 23.3 83.8 67.6 2.011 28.5 73.1 67.8

 

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TABLE 8 .

7.--M-6 (sub—graywacke)

1(36

electrical anisotropy data.

 

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K K K Principal rms 0
f Principal rms p::t:::::l\£fffi Values Error HT:E?:;pAIAAX§:q
(cps) Values Error 'L (o), 3 95 x10) x10” LL :3? .ng “
(mho/m) (mhn/m)
7.240 165.2 102.1 81.7 1198.6 155.2 114.3 85.5
30.<30() 6.640 0.078 104.1 21.5 105.9 410.0 32.5 114.6 24. 92.8
5.952 85.6 72.5 18.0 172.0 87. 89.4 5.2
7.818 161.6 106.5 82.0 528.77 157.9 111.5 85.3
10.<)oc3 6.839 0.102 108.0 22.2 102.5 189.28 12 56 111.8 22.1 93.3
6.040 86.2 75.6 14.9 66.57 86.8 85.2 5.8
8.726 159.8 109.1 83.8 206.53 159.6 109.8 85.1
3.00c) 7.116 0.127 109.7 20.5 95.6 78.12 6.48 110.0 20.1 92.0
6.156 86.0 82.6 8.4 25.66 86.1 86.5 5.3
9.731 160.4 108.5 83.9 80.294 162.1 106.9 84.2
1.0013 7.495 0.154 109.1 20.0 95.6 31.951 2.505 107.3 17.6 93.0
6.261 86.0 82.7 8.3 8.711 85.4 85.4 6.5
10.931 169.6 108.3 83.9 28.958 163.9 104.2 82.6
30C) 7.936 0.188 108.8 19.5 94.7 12.644 0.751 104.6 14.9 92.7
6.380 85.8 83.6 7.7 3.480 83.5 85.5 7.9
11.698 161.4 107.6 84.0 14.855 166.9 101.2 33.4
15C) 8.327 0.214 108.0 18.4 93.8 7.011 0.696 101.7 12.2 93.5
6.480 85.5 84.5 7.1 1.838 84.3 85.3 7.4
12.175 161.7 107.2 84.0 10.265 168.4 99.3 83.2
1°C) 8.535 0.228 107.6 18.1 93.9 4.804 0.511 99.8 10.3 93.3
6.512 85.4 84.5 7.2 1.292 83.8 85.6 7.6
12.775 161.9 106.9 83.8 6.6091 170.2 96.7 82.9
60 8.362 0.264 107.4 17.8 93.6 3.1410 0.3811 7.3 8.6 94.5
6.583 85.1 84.7 7.2 0.8413 83.5 84.7 8.4
13.576 163.6 105.1 83.8 3.8716 169.4 95.8 81.2
30 9.287 0.316 105.6 16.0 93.4 1.7982 0.3157 06.6 8.1 94.7
6.685 84.9 85.1 7.1 0.5235 81.7 84.4 10.0
m

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108

TABLE 8.8.--M-7 (grayWacke) electrical anisotropy data (not a least square tensor
coefficient determination).

 

O

 

K . . 0
. K, Principal Axes Principal Principal Axes
f PrinCipal Direction Angles le“85 Direction An les
(cps) Values 0 x10 ’ (o) g
(mho/m)

7.108 76.4 110.8 154.8 562.3 71.9 65.3 148.6
30,000 6.155 99.0 25.2 113.4 301.8 101.4 24.9 68.2
5.570 16.4 76.4 81.0 160.0 21.6 86.9 68.6
7.297 76.8 106.1 159.0 211.61 80.0 56.0 144.1
10,000 6.274 99.7 21.2 108.6 112.32 109.9 36.3 61.0
5.652 16.5 76.7 80.5 58.81 22.4 79.0 70.7
7.572 76.9 101.7 162.3 86.95 78.8 55.6 143.3
3,000 6.394 101.6 18.7 104.5 52.84 113.7 37.6 62.6
5.789 17.6 75.6 80.0 23.32 26.5 76.5 67.6
7.979 76.8 94.3 166.1 41.747 77.1 2.2 139.4
1,000 6.578 103.6 15.6 97.4 22.406 108.6 38.6 57.6
5.946 19.1 75.1 78.4 9.162 22.9 82.9 68.3
8.598 76.3 85.5 155.6 21.173 72.6 49.6 134.4
300 6.823 103.5 13.6 88.8 10.127 105.0 40.4 53 5
6.147 19.4 77.2 75.6 4.565 23.3 89.8 6.6
9.084 76.4 79.6 162.8 11.988 75.5 51.7 138.1
150 7.030 104.6 16.2 83.1 5.978 106.6 38.6 56.3
6.302 20.1 77.7 74.3 2.828 22.4 85.8 68.1
9.446 76.2 77.3 161.1 9.091 76.6 51.7 138.6
100 7.171 104.4 17.2 80.8 4.521 108.5 39.0 57.1
6.397 20.2 78.6 73.6 2.089 23.2 83.6 67.8
9.978 75.8 73.8 158.2 6.580 75.8 51.5 138.0
60 7.398 104.7 19.5 77.5 3.225 108.4 39.1 56.9
6.520 20.7 79.4 72.4 1.494 23.6 84.2 67.2
10.841 76.1 68.1 153.6 4.3749 76.5 52.1 138.9
30 7.817 106.3 24.5 72.3 2.1417 109.3 38.8 57.8

6.706 21.7 79.7 71.1 0.9717 23.9 82.8 67.3

109

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110

’NxBLE 8.9.--M-8 (staurolite schist) electrical anisotropy data.

 

 

K K K Pringipal r25 O
f Principal rms DiiegzigglAfixizs Values Errog D§:in:ipalAAxes
k:ps) Values Error (0) 9 x109 x10‘ ec 1??) nq es
(mho/m) (th/m)
10.427 17.7 95.6 73.2 4551.4 12.9 97.2 79.4
30,000 10.071 0.013 106.8 90.4 16.9 3085.2 68.7 100.8 90.2 10.8
5.746 84.8 5.6 88.0 213.8 83.0 7.2 88.4
12.613 16.2 96.2 75.1 1843.8 12.4 96.1 79.3
10.000 11.537 0.017 105.0 90.1 15.0 1229.4 21.6 100.8 90.4 10.8
5.843 84.0 6.2 88.2 62.1 84.0 6.2 88.4
15.495 15.5 97.4 76.5 676.38 11.7 94.2 79.1
3,000 13.459 0.020 103.6 89.9 13.6 449.34 3.63 101.0 91.5 11.1
5.927 82.8 7.4 88.3 17.74 86.2 4.4 87.8
18.663 13.9 95.2 77.2 259.55 11.1 91.3 78.9
1,000 15.543 0.056 103.0 90.7 13.0 174.29 1.01 101.1 92.4 11.4
6.026 85.1 5.3 88.1 6.00 89.2 2.7 87.4
22.450 13.0 93.8 77.6 94.622 168.7 88.0 78.9
300 18.120 0.057 102.5 91.3 12.6 64.798 0.022 79.0 94.0 11.7
6.102 86.6 4.0 87.9 2.638 87.3 4.5 86.4
25.043 12.5 92.7 77.8 50.406 167.8 85.8 78.6
150 19.894 0.065 102.3 91.7 12.4 34.582 0.110 79.0 94.8 12.4
6.175 87.8 3.2 87.8 1.392 84.9 6.4 86.1
26.725 12.1 91.5 78.0 35.655 167.3 84.2 78.7
100 21.042 0.034 102.1 92.2 12.3 24.247 0.023 79.3 95.5 12.1
6.188 89.0 2.6 87.6 0.895 83.2 8.1 85.7
28.868 12.0 90.6 78.0 23.541 166.3 82.2 78.7
60 22.531 0.028 102.0 92.5 12.2 15.632 0.158 79.5 95.7 12.0
6.230 90.0 2.5 87.4 0.542 81.2 9.7 85.8
32.100 168.2 89.0 78.2 13.268 164.5 79.6 78.6
30 24.662 0.030 78.3 93.1 12.1 9.076 0.080 80.0 98.2 12.9
6.319 88.4 3.3 87.1 0.353 78.2 13.3 83.9

 

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112

TABL£28.10.--M-9 (Hemlock formation) electrical anisotrOpy data.

 

 

K K K Pringipal r35 0
f Principal rms “E:i2::::lA:x::s Values Error ”Prtni3gglrgxis
(C‘ps) Values Error L a) 9 x109 x109 irtc 1(0)‘ q es
(mho/m) (mho/m)
11.474 133.6 79.5 134.5 1350.1 48.2 86.1 138.0
30,000 10.566 2.458 45.4 93.8 135.1 1012.6 300.3 138.2 85.8 131.5
5.954 79.8 11.2 85.3 637.2 89.5 5.7 84.3
12.048 134.1 81.0 134.5 458.5 48.4 88.8 138.4
10,000 10.964 2.583 45.3 92.8 135.2 319.8 101.1 138.1 83.8 131.2
6.225 81.7 9.4 85.6 209.0 86.2 6.3 85.0
12.688 135.0 84.2 134.4 153.82 50.4 84.4 139.8
3,000 11.394 2.743 45.2 89.5 135.2 113.61 32.21 140.3 82.6 128.7
6.483 86.2 5.8 85.6 66.61 87.9 9.3 80.9
13.354 134.6 84.4 134.8 60.72 128.1 81.0 140.4
1,000 11.853 2.888 44.8 89.0 134.8 45.34 12.26 38.1 83.2 127.3
6.777 86.7 5.7 85.3 24.74 89.7 11.3 78.7
14.201 134.2 84.2 135.2 26.28 52.0 83.3 141.2
300 12.459 3.050 44.4 88.6 134.3 18.99 5.14 141.9 81.8 126.8
7.116 87.0 5.9 84.9 10.43 87.7 10.6 79.6
14.844 133.9 84.5 135.5 15.377 52.7 80.8 141.2
150 12.964 3.181 44.0 88.1 134.0 11.261 2.912 142.7 81.2 125.9
7.391 87.5 5.8 84.7 5.790 88.6 12.8 77.3
15.291 133.8 85.0 135.8 11.628 53.0 82.3 141.9
100 13.259 3.265 43.8 87.3 133.7 8.525 2.122 142.9 81.6 125.8
7.560 88.5 5.6 84.6 4.383 87.9 11.4 78.8
15.902 133.4 85.0 136.1 8.201 54.4 82.5 143.3
60 13.739 3.382 43.5 87.3 133.3 6.168 1.497 144.2 81.0 124.3
7.820 88.5 5.7 84.5 3.023 87.0 11.7 78.7
16.946 133.2 85.3 136.4 5.490 53.0 83.3 142.2
30 14.517 3.585 43.2 86.2 133.0 3.815 0.960 142.6 79.0 125.2
8.222 89.6 6.1 83.9 1.953 85.2 12.9 78.0

 

113

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'NKBLE 8.11.--M-10 (amphibole schist) electrical anisotropy data.

 

 

K K K PrinZipal r85 0
r vans.
ucps) Values Error (0) 9 x10 x109 (0) g
(mho/m) (th/m)

8.976 65.3 139.7 60.4 777.1 84.7 143.1 53.6
30,000 8.748 0.395 38.0 88.1 128.0 423.0 224.9 18.1 96.2 107.0
7.983 63.0 49.7 52.1 316.5 72.7 53.8 41.4
9.292 67.7 140.3 59.0 298.7 84.9 152.7 63.2
10,000 8.914 0.506 33.8 87.8 123.7 147.0 88.9 15.0 91.9 104.9
8.129 66.0 50.3 49.2 107.9 75.9 62.8 31.2
9.708 71.8 143.8 59.9 110.32 96.6 165.8 77.4
3,000 9.103 0.633 29.9 88.4 119.8 60.42 26.71 15.9 99.4 102.7
8.283 67.2 53.8 44.9 46.67 75.6 79.4 18.0
10.002 74.2 145.3 60.0 56.30 90.3 156.2 66.2
1,500 9.269 0.715 28.0 89.6 118.0 33.76 14.86 14.0 95.9 102.7
8.393 67.5 55.3 43.3 24.64 76.0 67.0 27.3
10.166 75.6 147.4 61.5 40.55 91.2 156.6 66.6
1,000 9.396 0.752 26.9 89.5 116.9 25.92 9.75 17.7 98.0 105.7
8.511 67.9 57.5 41.0 18.85 72.3 68.2 28.7
10.341 77.3 147.5 60.6 32.48 90.4 157.5 67.5
600 9.456 0.800 26.0 91.2 116.0 22.52 5.92 20.0 97.9 108.3
8.583 67.7 57.5 41.2 17.07 70.0 69.0 29.7
10.681 78.8 147.9 60.4 16.036 98.2 159.1 70.9
300 9.743 0.901 24.3 91.7 114.2 11.973 3.069 23.1 104.3 107.7
8.797 68.8 58.0 40.0 9.401 68.6 75.1 26.5
11.113 80.5 150.6 62.5 9.046 105.2 156.7 72.8
150 10.019 0.970 24.4 92.6 114.2 6.911 1.447 30.6 111.2 110.9
9.021 67.8 60.8 38.1 5.633 64.2 80.8 27.6
11.370 82.3 151.2 62.5 6.757 109.1 154.8 74.2
100 10.237 1.010 24.2 94.2 113.8 5.297 0.962 36.9 114.8 115.5
9.174 67.2 61.6 37.8 4.092 59.8 85.6 30.6
11.692 83.6 152.8 63.6 4.712 115.3 28.0 78.8
60 10.525 1.088 23.7 94.7 113.2 3.892 0.582 42.7 62.3 119.6
9.385 67.3 63.2 36.4 2.906 58.3 86.0 32.0
12.210 86.0 153.9 64.3 2.996 118.3 28.3 88.9
30 11.015 1.168 23.8 96.6 112.7 2.572 0.342 49.3 68.5 131.6
9.766 66.6 64.9 35.5 1.929 53.7 72.4 41.6

 

115

 

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TABLE 0. 1 2 .--M-ll

¥

(Siltstone)

116

electrical anisotropy data.

 

 

O O
K . . L}
K K 7 . Pr1nc1pal rms . ‘. .
. pm...
(Cps) Values Error 0 ‘ x10 x109 0 ‘
(th/m) (mho/m)
7.865 20.0 117.9 05.0 1429.1 170.0 92.0 00.5
30,000 7.452 0.519 115.0 149.9 75.3 1047.4 150.0 00.9 103.9 74.2
6.087 70.1 79.7 15 0 209.5 09.3 74.2 15.0
8-500 21.0 110.5 9.3.9 525.1 174.9 94.3 07.3
10,000 7-930 0.557 100.9 155.0 75.2 350.5 30.1 05.1 101.3 72.0
6-220 01.2 77.5 15.3 122.3 00.0 71.0 10.2
9 -240 14.0 104.0 92.4 100.43 172.5 90.4 80.0
3.000 8 -441 0.504 103.4 159.0 74.2 113.50 14.71 82.0 150.9 07.2
6 -400 03.0 75.3 10.0 40.71 00.0 00.0 23.2
10 - 057 12.5 102.5 91.1 00.39 12.8 102.0 09.0
1.000 8 - 942 0.570 101.0 159.4 73.3 40.25 0.45 101.9 155.2 00.0
6-053 05.4 73.9 10.0 1.30 05.0 09.1 21.4
10- 997 11.9 101.9 90.2 25.011 15.0 104.5 00.2
300 9 - 480 0.010 101.2 150.0 72.0 14.973 2.001 104.0 149.4 03.7
6 - 979 00.2 72.4 10.0 0.507 00.9 03.0 20.0
11- 650 12.0 102.0 09.0 13.911 10.7 100.3 00.4
150 9- 840 0.000 102.0 157.7 71.5 7.709 1.347 108.5 140.0 04.5
7- 200 00.2 71.9 10.5 4.750 05.3 04.7 25.0
100 12. 101 13.0 103.0 90.2 10.104 20.5 110.5 00.4
0- 052 0.009 102.8 100.5 70.7 5.397 1.040 109.2 140.0 05.4
7- 430 80.4 71.3 19.3 3.307 03.0 00.5 24.0
00 12-646 13.7 103.7 09.5 0.074 22.2 112.2 90.0
0- 405 0.750 103.0 150.3 70.0 3.530 0.032 110.4 150.0 09.0
- 583 05.9 71.1 19.4 2.120 01.5 71.0 21.0
30 13- 520 15.1 105.1 00.0 4.202 25.9 115.0 93.7
- 800 0.050 104.3 155.7 70.0 2.000 0.014 113.3 150.4 72.0
- 867 85.4 71.4 19.2 1.218 79.4 70.2 17.0
\—_—\

 

117

 

 

 

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118

77‘5LE 8.1;3.--M-2A (greenstone with atmospheric moisture) electrical anisotropy data.

 

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K . . a
K K . . Pr1nc1pa1 rms .. .
f - . Princ1pal Axes . Pr1nc1pa1 Axes
(c z>z-J_nc1pal ‘rms Direction Angles Valugs Error Direction Angles
PS) \lalues Error 0 x10 x108 0
(mho/m) (mho/m)
11.107 92.4 114.6 24.7 1291.0 95.2 69.8 159.1
100.000 9.042 0.180 16.7 100.0 94.6 589.2 13.7 13.2 99.6 99.0
8.170 73.5 29.9 65.8 430.1 77.8 22.6 71.3
12.886 90.6 112.7 22.7 503.3 80.1 81.0 100.6
30,000 10-074 0.200 9.0 98.5 92.8 242.4 11.4 170.1 88.1 99.7
8-819 81.1 24.4 67.5 162.5 89.6 9.2 80.8
15-077 92.7 103.4 13.6 222.12 77.8 87.0 100.4 : .'
10,000 11-040 0.223 7.5 97.4 89.0 106.19 9.04 106.7 87.8 102.1 1's
9-619 83 0 15.4 76.4 64.92 88.4 3.7 86.6 B
18-16 95.6 98.1 9.9 105.57 74.9 89.7 104.9
3,000 12-58 0.24 7.0 94.9 85.0 48.09 5.52 164.8 88.1 105.1
10-62 85.9 9.5 81.4 27.63 88.3 1.9 89.2
23-14 98.5 94.5 9.7 56.27 72.4 80.0 102.1
1.000 14.73 0.53 8.8 93.1 81.7 24.03 3.41 162.4 88.0 107.5
12-01 87 6 5 5 85.0 13.06 89.2 4.0 86.1
“-42 99.2 93.2 9.7 48.16 72.2 86.0 101.7
700 15.28 0.63 9.4 92.0 81.0 21.17 2.87 162.2 88.6 107.7
12.30 87.9 4.2 80.4 11.35 89.9 4.3 85.7
00 27-50 100.3 91.4 10.4 40.318 107.0 82.4 161.2
5 :5-50 0.70 10.9 93.9 79.8 17.583 2.119 17.6 92.0 107.5
2-79 86.4 4.2 88.0 8.545 85.8 7.9 83.3
300 $8.69 102.6 91.3 12.0 28.740 108.5 82.5 100.0
4°28 1.24 12.0 90.4 77.4 12.820 1.664 18.8 91.2 108.8
~05 89.9 1.3 88.7 6.248 86.5 7.6 83.3
200 38-22 70.0 89.6 166.0 22.320 109.3 83.8 159.7
“-40 1.50 106.0 89.2 103.9 10.036 1.536 19.7 91.9 109.6
~95 89.3 0.9 89.4 4.748 86.1 6.5 84.8
100 gig-48 74.3 87.3 164.0 14.345 109.3 84.3 159.8
”~96 2.35 104.2 88.2 105.6 6.540 1.245 20.2 94.1 109.7
'07 89.0 3.2 86.9 3.237 84.3 7.0 86.0
70 33-38 72.8 80.4 102.4 11.394 108.6 84.2 160.4
”~04 3.00 102.8 87.0 107.1 5.218 1.147 20.4 96.4 109.3
-07 88.8 4.3 85.9 2.669 82.0 8.7 80.6
50 $952 72.5 86.7 162.2 8.923 107.1 85.2 102.2
19.31 3.43 102.4 87.2 107.3 4.291 1.035 21.4 101.6 107.8
.40 88.3 4 3 86.0 2.262 77.5 12.6 88.9
30 1:956 108.4 83.4 100.3 6.193 105.9 85.8 10.5
21-11 4.33 18.4 88.1 108.3 3.063 0.878 27.5 111.5 73.0
A 89.7 6.9 83.1 1.024 68.2 21.9 87.9

119

 

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120

The direction angle, a in Table 8.1 is the angle be-

ij'
tween the ith principal direction (i = 1, 2, 3, for the
maximum, intermediate, and minimum principal axes, re-
spectively) and the jth laboratory coordinate axis
(j = l, 2, 3, for the x, y, and z axes, respectively).

The graphical presentations of Figures 8.1 through
8.12 consist of dispersion (frequency variation) curves
of the principal K and (effective) 0 values and principal
axis poles on equal area, or Schmidt net, projections.
The Schmidt net projections are relative to the laboratory
coordinate system and utilize the lower reference hemisphere.
To indicate the systematic principal axis dispersion ob-
served for some samples, successive principal axis poles
on the Schmidt net projections are connected with arrows
proceeding from low to high frequency. Also, each princi-
pal axis pole set is identified at the highest frequency
pole. The maximum, intermediate, and minimum principal
values and axis poles are indicated in Figures 8.1 through
8.12 with the subscripts l, 2, and 3, respectively. All
structural controls (e.g., lineations, foliation, cleavage,
etc.) observed in the field sample are plotted on the con-
ductivity Schmidt net projection for each sample.

Conductivityfiand Dielectric
" Constant magnitudes

 

 

The dielectric constant values obtained for the

vacuum dried rocks considered in this study ranged from

121

a low of 5.570, for the M-7 graywacke (see Figure 8.7

and Table 8.8) at 30,000 cps, to a high of 38.29, for

the M-2 greenstone schist (see Figure 8.2 and Table 8.3)
at 30 cps. The dielectric constant values at 30 cps
ranged from 6.319 to 38.29, while at 30,000 cps, the
values were spread from 5.570 to 12.316. The K values
obtained in this study are consistent with those reported
by Keller and Licastro (1959), Howell and Licastro (1961),
Keller (1966), and Parkhomenko (1967) for rocks measured
under similar conditions to those used for this study.

The lowest K values were observed for the sub-
graywacke of M-l (see Figure 8.1 and Table 8.2) and the
graywacke of M-7 (see Figure 8.7 and Table 8.8). The
highest K values were observed for the greenstone schist
of M-2 (see Figure 8.2 and Table 8.3) and the staurolite-
muscovite schist of M-8 (see Figure 8.8 and Table 8.9).
The coarse grained samples (M—3, B-4, M-5, M-9, and M-lO)
had intermediate K values, as did the M—6 graywacke and
the M-ll graywacke.

Most I.P. models for rocks yield a maximum polari-
zation condition for an Optimum grain size, due to the
combined effects of surface resistivity and surface
polarization (Keller and Frischknecht, 1966). The Optimum
grain size will depend upon the electrical properties of
the model medium and the polarizing inhomogenities (Sillars,
1937; Wait, 1959). The combined effect of grain size and

electrical prOperties on the bulk electrical polarization

122

explains why there seems to be no consistent grouping of
the K values with respect to rock type or grain size,
based upon magnitudes alone.

The effective conductivity values obtained for
these same rocks ranged from a low of 2.925010—10mho/m,
for the B-4 syenite gneiss (see Figure 8.4 and Table 8.5)
at 30 cps to a high of 1.0648-10-5mho/m, for the M—2
greenstone schist (see Figure 8.2 and Table 8.3) at
100,000 Cps. The effective conductivity values at 30

10 8

mho/m to 4.653'10- mho/m,

7

cps ranged from 2.925010-

while the range at 30,000 cps is from 1.600-10' mho/m

to 3.896.10‘6

mho/m. The O values obtained in this study
are consistent with those reported by Keller and Licastro
(1959), Grant and West (1965) and Parkhomenko (1967) for
rocks measured under similar conditions.

The lowest 0 values were generally observed for
the B-4 syenite gneiss (see Figure 8.4 and Table 8.5)
and the M-6 sub-graywacke (see Figure 8.6 and Table 8.7).
The highest 0 values were observed for the M-2 greenstone
schist (see Figure 8.2 and Table 8.3). The 0 data for
sample M—8 are rather unusual. Because of the very large
0 anisotropy for this sample, its minimum 0 values are
among the lowest of those studied, while its intermediate
and maximum principal values are among the highest. The
0 values for the remaining samples (exclusive of M-2A) were

intermediate in nature, although those for M-1 and M-7

tended to be lower than the others.

123

The electrical measurements of this study were made

on vacuum dried rocks containing little conducting minerals.
Thus, the (effective) conductivity values obtained repre-
sent displacement current power loss, rather than the
tranSport of charge carriers. For this situation, the
lossy dielectric model of Chapter IV is very apprOpriate.
Thus, the (effective) conductivity may be discussed with
the aid of the imaginary components of lossy dielectric
polarization models (cf., Sillars, 1937; von Hippel,
1954a, 1954c; Wait, 1959; Wert and Thompson, 1964; Beam,
1965; and Appendix A). These models yield effective
conductivities which depend upon the polarizing inhomogen-
ity size and spatial density, as well as the electrical
properties of the medium and polarizing inhomogenities.
As was the case for K, the bulk 0 represents the combined
effects of the grain size and electrical properties. This
does not allow the rocks studied in this investigation to
be grouped on the basis of 0.

The lossy dielectric model for rocks (see Chapter
IV) predicted K and o diSpersion curves similar to 4.2,
With only the lowest resonance frequency within the range
0f the present study. The dispersion curves of Figures
8.1 through 8.12 do confirm this qualitative model. The
0 Values increased with frequency while the K values
decreased with frequency, as predicted in Chapter IV.

With some exceptions (e.g., B-4, M-6), those samples with

Tfigh K values also had high 0 values and those with low K

124

values also had low 0 values. This may be explained with
the aid of the lossy dielectric model of Chapter IV.
Because the effective conductivity is due to displace-
ment current power loss, those materials with large

polarizations would be expected to exhibit large loss.

Discussion of Error

 

It is desirable to have an estimate on the vari-
ability of the tensor principal value and directions
determined by the methods of Chapter VI. The principal
value uncertainty is the rms error, given by equation 6.9.
The reduction procedures of Chapter VI offer no estimate
of the uncertainty in the principal directions. However,
some indication of the principal direction uncertainty may
be gained from the following approach. The principal
values and directions are calculated for all possible
combinations of the directional measurement data using
one less measurement direction than actually measured.

For the purposes of this study, these results will be
referred to as the less determined cases. The results
for the less determined cases can then be compared to the
results using all the directional measurement data for
the sample. This can also be used to check the useful-
ness of the rms error. The combined results for such a
study using the directional data for M-2, M-3, M-6, and
M-ll at 1000 cps is presented in Figure 8.13. The fre-

quency of 1000 CpS was chosen because it is in the center

125

 

 

 

 

 

 

( i

60‘ 60
r1
40‘ 4O r—:ir
r—
7. 7.
20‘ 20 4“
O A T—L-JL-l—A
0' IO. 20' 30° '28 O 2!

a. Directional variation b Principal voluo voriciion

Fig. 8.13.--Variation of the less determined data
about the results obtained using all directional measure-
ments.
of the frequency spectrum used for this study. These
samples were chosen because they exhibited low symmetry
at 1000 cps and used either eight or nine measurement
directions.

The data of Figure 8.13 indicate that about 80
per cent of the principal values for the less determined
cases lie within one rms error of the principal value
estimates obtained using all measurement directions. The
data of Figure 8.13 also indicate that about 75 per cent
of the principal direction estimates for the less deter-
mined cases lie within 20° of the principal direction
estimates using all measurement directions. The direc-
tional scatter of M-3 (which has large rms errors) contri-
butes heavily to this apparent poor correspondence. If
the M-3 data are removed from consideration, about 87

per cent of the less determined principal directions are

126

within 20° of the principal direction estimates obtained
using all directional measurements.

If the above variations of the principal value and
direction estimates are a good indication of random error
in these estimates, then the random errors should show up
on the dispersion curves of Figures 8.1 through 8.12 as
scatter. There is some apparent scatter on these plots
(e.g., the K2 and K3 values and directions of Figure 8.1,
the 02 values and 03 directions of Figure 8.5, and all
the 0 directions of Figure 8.10). However, the apparent
scatter in these cases is rather small. In particular,
the directional scatter is much less than the 20° dis-
cussed above. In general, the dispersion curves of
Figures 8.1 through 8.12 vary smoothly with frequency.
This would indicate that the errors discussed above are
not random, or that they are much less than the above
discussion would lead us to believe. This is similar to
the conclusions reached in the discussion of experimental
limitations in Chapter V. The actual variation observed
in repeated measurements was much less than that pre-
dicted on the basis of a statistical propagation of error.
If such large errors are indeed present, they are apparently
not random.

The ratios of the rms errors to the principal values
vary from sample to sample and, to a certain extent, with
frequency within any particular sample. The coarse

grained samples, such as the M-3 amphibolite, the B—4

127

syenite gneiss, the M-5 granite gneiss, the M—9 Hemlock
formation, and the M-lO amphibole schist consistently

exhibit large error ratios. By contrast, the finer

grained samples, such as the M-1 sub-graywacke, the M-2
greenstone schist, the M-6 sub-graywacke, the M-8 staurolite-
muscovite schist, and the M-ll graywacke consistently ex-
hibit small error ratios.

The 0 error ratios are generally larger than the K
error ratios. This can be traced to their magnitudes at
input to the least square tensor coefficient determination
procedure (see Chapter V).

Only six measurement disks were prepared for the M-7
graywacke. Thus, its principal value and direction esti-

mates do not involve least square determinations.

Anisotropy Ratios

The electrical anisotropy information for this study
is summarized in Table 8.14 for measurements at 30, 1,000,
and 30,000 cps, which effectively Span the three decades
of signal frequency used in this study. The results for
these three signal frequencies can be used to summarize
the results for the study because the dispersion curves
of Figures 8.1 through 8.12 are very smooth.

It is desirable to have criteria with which to com-
pare the electrical anisotropies of rocks. No precedents
for these criteria were found in the literature, because

the present study appears to be the first in which the c

128

TABLE 8.ll.--£lectrical anisotropy summary.

 

 

 

(Approx.) 30'000 Cp'
Sample Lithology Structural Controls c K 3K 1K
(cp.) o '0 :0 Haxinu- Symmetry
l 2 3

H-l Michigamme fm. Slaty cleavage & x3,o3 350 l.0£7nl.000:0.984 Isotropic
aub-graywacke-- (except for hig l.27641.000:0.609 Orthorhombic
meta-argillite f K3)

M-2 Mona fm. Poliation 1 K3, 0 200 l.2l9:l.000:0.947 Cylindrical, ii K1
greenstone slaty cleavage 1 i2, 02 2.454:l.000:0.934 Cylindrical, (I 01
schist lineation ii K1, 01

H—3 Amphibolite Fracture set 350 l.lO9:l.000:0.907 Cylindrical, ii K1 or x;

(no good correlation) l.975:1.000:0.356 Cylindrical, ll 01 or 03

8-4 Syenite gneiss Gneiasic banding 1 K3,03 400 l.013:l.000:0.912 Cylindrical, il K1 or K3

l.686|l.000:0.67l Cylindrical, il 01 or 03
M-S Granite gneiss Gneissic banding l K3,03 400 l.09l:l.00010.940 Orthorhombic
l.551|l.00030.861 Cylindrical, il 01

H-6 Michigamme tn. Slaty cleavage L K3,o3 300 l.090:l.000:0.896 Orthorho-bic
sub-graywacke 2.923:l.00030.4l9 Orthorhonbic

H—7 Negaunee fl. Quartz C axis 300 l.l$5:l.000:0.905 -----
quartzite- lineation ll K1, 01 l.86311.000:0.530 -----
graywacke

M-B Michigamme tn. Poliation 1 K3, 03 200 l.035:l.000:0.57l Orthorhombic
staurolite- 1.475¢l.00040.693 Orthorhonbic
muscovite
schist

H-9 Hemlock fm. One fracture set 400 l.086:l.000:0.564 Cylindrical, (I ll or K,

(no good correlat ons l.333:l.000:0.628 Cylindrical, ii 01 or 03
with the other fracture
sets)

M-lo Anphibole- Amphibole lineation 600 l.026:l.000:0.912 Cylindrical, I1, I or R
chlorite ii K1, 01 l.837:l.00030.740 Cylindrical, ||. 01 or 03
schist

- Hichi same In. Fracture set K 300 l.055:l.000:0.017 Cylindrical, (I K

H 11 sub-ggaywacke- l 3' 3 l.36hl.000|0.257 Orthorhombic 3

siltstone
- a. Same as for M-z 500 l.279:l.00040.875 Orthorhonbic
M 2A Hons f 2.077:l.00040.670 Orthorhonbic

greenstone schist

 

129

 

 

 

 

1,000 Cpa 30 cps
:12K2:K3 Maximum Symmetry 51:K2:K3 Maximum Symmetry Comments
: :0 6 :3 :3
l 2 3 l 2 3

1.074:l.000:0.972 Cylindrical, ii Kl or K3 1.09l:1.000:0.907 Orthorhombic

l.302:l.000:0.674 Orthorhombic l.230:l.000:0.610 Orthorhombic

l.S42:l.000:0.94l Cylindrical, Ii K1 l.900:l.000:0.904 Orthorhombic

2.61‘:l.000:0.792 Orthorhombic 2.542:l.000:0.484 Orthorhombic

l.251:l.000:0.863 Cylindrical, ii Kl or K3 1.394:l.000:0.793 Cylindrical, ii K1 High 3, K errors

2.099:l.000:0.422 Cylindrical, ll 01 2.012:l.000:0.727 Cylindrical, Ii :1

l.ll6:l.000:0.897 Cylindrical, ii K1 l.396:l.000:0.850 Cylindrical, ll K1 High J errors

2.956:l.000:0.462 Cylindrical, il 01 3.238:l.000:0.l76 Cylindrical, II 31

l.206:l.000:0.206 Cylindrical, ll K1 l.329:l.000:0.89l Cylindrical, Ii K1 High 3 errors

1.720:1.000:0.7S7 Cylindrical, ii 31 1.529:l.00010.77l Isotropic

1.298:l.000:0.835 Orthorhombic l.462:l.000:0.720 Orthorhomhic

2.763xl.000:0.273 Orthorhombic 2.153:l.000:0.29l Orthorhombic

l.213:l.000:0.904 ----- 1.387:l.000:0.858 ----- Not a least squarw

l.863:l.000:0.409 ----- 2.043:l.000:0.454 ----- tensor coeffic1ent
determination

l.201:l.000:0.388 Orthorhomhic l.302:l.000:0.256 Urthorhomhic

l.689:1.000:0.344 Orthorhombic l.462:l.000:0.389 Orthorhomhic

1.12711.000:0.572 Cylindrical, il K1 or K3 1.167:1.000:0.566 Cylindrical, ll Kl or K3 High 3, K errors

l.339:l.000:0.546 C lindrical c or c l.439:l.000:0.512 Cylindrical, | 3 or C

y .Ii1 3 '1 3

l.082:l.000:0.906 Cylindrical, il KO or K3 1.108:1.000:0.886 Cylindrical, ll K1 or K3 High 3, K errors

l.9l2:l.000:0.727 Cylindrical, I) C1 or 03 1.165:l.000:0.750 Cylindrical, || 01 or 03

1.125:1.000:0.744 Cylindrical, ii K3 1.244:1.000:0.724 Orthorhombic

l.650:l.000:0.456 Orthorhombic Ii 2.122:l.000:0.606 Cylindrical, il 01

1.57l:l.000:0.815 Orthorhombic l.982:l.000:0.606 Orthorhombic Sample Mf2 at

2.285:l.000:0.530 Orthorhombic 2.022:l.000:0.530 Cylindrical, ll 01 equilibrium with

atmospheric
m01sture

 

130

and K tensors were completely defined. As a result, two
criteria were established during the present study. One
of these, here called tensor symmetry, will be discussed
in the next section. The other criterion is the use of
anisotropy ratios. The anisotropy ratios used in this
study and presented in Table 8.14 are the ratios of the
maximum and minimum principal tensor values to the inter-
mediate principal value (R and R

12 32’
greater the departure of these ratios from unity, the

respectively). The

greater the anisotropy.

The R32(K) range from 0.256, for M-8 at 30 cps to
0.984, for M-l at 30,000 cps, and the R12(K) from 1.013,
for B-4 at 30,000, to 1.900, for M-Z at 30 cps. The
R32(o) range from 0.257, for M-ll at 30,000 cps to 0.934,
for M-2 at 30,000 cps, and the R12(o) range from 1.165,
for M-lO at 30 cps, to 3.258, for B-4 at 30 cps.

The anisotropy ratios of Table 8.14 indicate that 0
generally shows greater anisotropy than K at all frequencies.
The conductivity of materials may vary over several orders
of magnitude, while the dielectric constant varies over
Only two or three (Wert and Thompson, 1964; Beam, 1965;
Kaller, 1966). Thus, the physical property which has a
Wider range in possible values might be expected to show
greater anisotrOpy (as judged by anisotrOpy ratios).

With some exceptions (e.g., B-4 and M-ll), the aniso—

trOpy of both K and c is generally greater at lower

131

frequencies than at the higher frequencies. The occur-
rence of resonance frequencies between 200 and 600 cps
for all samples indicates that one of the polarization
mechanisms present at low frequencies is dormant at high
frequencies. These polarization mechanisms will add their
anisotropic polarizabilities to the bulk polarization of
the rock, which will cause greater K and o anisotropy at
low frequencies over that at high frequencies.

Figure 8.14 shows plots of R(o) versus R(K) taken
from Table 8.14. There appear to be linear relationships

between these ratios for each of the three plots. These

 

 

 

 

 

 

 

 

i i i
40- 40L 40L
30 *- o 30 c 3.0 ‘-

o,3
R(a) 2.0 - R(a') 2.0~ R(o) 2.0 r
I0~ a lo- L0-
V a

0.0 ‘ Pr 0.0 0.0

00 L0 20 00 L0 20 00
R (K) R(K)
(8 30.000 cps f= i,OOO cps

Fig. 8.14.-—Anisotropy ratio relationships.

linear relationships pass very close to the point (1.0,
1.0). The scatter about the least square lines for each

Of these plots is also slight. The 510pes of these linear

132

relationships vary with frequency. At 30,000 cps, the
slope of the least square line is 6.182, while at 1,000
cps it is 3.714, and at 30 cps it is 2.019. This change
in slope is a reflection that both the K and c anisotropy
are greater at lower frequencies and that the a variation
is greater than that for K.

Plots such as those in Figure 8.14 are helpful for
identifying poor data. Points 1, 5, and 11, in Figure
8.14, are from M-9 (Hemlock formation), which has very
large 0 and K errors. Points 7, 9, and 12 are from B-4
(syenite gneiss) and point 4 is from M-S (granite gneiss),
both of which have large 0 errors.

Points 2, 6, and 10 in Figure 8.14 are from M-8
(staurolite-muscovite schist). Both the o and K errors
for the M-8 data are quite low. However, the minimum
principal 0 values (sub-normal to the foliation) are more
than an order of magnitude smaller than the maximum and
intermediate principal values at all frequencies. This
is a much greater spread than was observed for any of the
other samples. The K principal value separation, by con-
trast, does not seem to be much different from that of
the other samples studied. Thus, the R32(o), R32(K)
points plot well off the least square lines for all three
frequencies. These anomalous results for one sample are
not sufficient for generalization. However, they do offer
the possibility that plots of R(o) versus R(K) may be

helpful in separation of rock types.

133

Tensor Symmetry

 

The tensor symmetry is the second criterion used to
compare electrical anisotropy of rocks. The maximum
symmetry of the symmetric second-rank 0 and K tensors is
dependent upon the number of distinguishable principal
values. This, in turn, is determined by the rms error.

For the purposes of the present study, the 0 and K
tensors will have orthorhombic symmetry at a particular
frequency, if their maximum, intermediate, and minimum
principal tensor values are separated by more than twice
the rms error at that frequency. The representation sur-
face (for both 0 and K) in this case will be a tri-axial
ellipsoid. This representation surface has the same
symmetry as the holohedral (2/m, 2/m, 2/m) orthorhombic
crystal class (see Berry and Mason, 1959). Orthorhombic
symmetry is the lowest symmetry which a symmetric second-
rank tensor can exhibit. Thus, a material which exhibits
orthorhombic symmetry in its 0 and K tensors need not
exhibit 2/m, 2/m, Z/m symmetry in its fabric or crystal
lattice. It may exhibit even lower symmetry in these
respects. However, it cannot exhibit symmetry higher
than 2/m, 2/m, 2/m in its fabric or crystal lattice (Nye,
1964). An alternative approach would be to follow the
convention used in optical crystallography and call this
symmetry biaxial (see Walstrom, 1962), for a tri—axial

ellipsoid has two circular (isotrOpic) cross-sections.

134

The author prefers not to use this term, for the analogy

tO Optical properties may not be a good one. The Optical
indicatrix is a geometrical representation surface for

the index Of refraction in Optical materials. The index

Of refraction in a given direction is proportional to the
radius vector Of the indicatrix in that direction. By
contrast, the value Of a symmetric second-rank tensor in

a given direction is proportional to the inverse square

Of the representation surface radius vector in that direc—
tion. The index Of refraction, which is the ratio Of the
velocity Of light in free space to that in the material is
not a symmetric second-rank tensor, but depends upon 0,

K, and the magnetic permeability Of the material (Corson,
and Lorrain, 1962). Electromagnetic radiation uses trans-
verse wave motion. Thus, the isotropic cross-sections
(isotropic directions) assume more importance for Optical
properties than they do for electrical prOperties. The
angle, 2V, between the two isotropic directions in optical
materials is Often measured directly and used as an aid

in identification. The author does not feel that the two
circular (isotrOpic) cross-sections Of the 0 and K repre-
sentation surfaces and their relative orientations will

be as important to electrical anisotropy as are the similar
features tO Optical anisotropy. The term, biaxial symmetry,
may be helpful for those with a strong background in crystal

Optics and is included for that reason.

135

If two Of the principal values (i.e., the inter-
mediate and either the maximum or minimum principal value)
are separated by less than twice the rms error, then they
are not distinct. If the remaining principal value is
distinct, in the above sense, then the tensor has cylindri—
cal symmetry about this unique axis. The representation
surface (for both 0 and K) in this case is an ellipsoid
Of revolution (about the unique axis). An alternative
name for this symmetry would be uniaxial, as would be used
for the parallel Optical discription. Cylindrical symmetry
is well established in the literature Of electrical proper-
ties Of rocks. Also, the above Objections tO the use Of
biaxial symmetry also hold for the use Of uniaxial symmetry.
For these reasons, the author prefers the term, cylindrical
symmetry. However, the term, uniaxial symmetry, may be
useful for those with a strong background in crystal Optics
and is included for that reason. If the intermediate
principal value is separated from both the maximum and
minimum principal values by less than twice the rms error,
but the maximum and minimum principal values are separated
by more than twice the rms error, then the tensor has
cylindrical symmetry about either the maximum or minimum
principal direction.

If the maximum and minimum principal values are
separated by less than twice the rms error, then none of

the principal values are unique and the tensor is

136

isotropic. The representation surface (for both 0 and
K) in this case will be a Sphere.

The maximum K and o tensor symmetry, based upon
the above definitions, are also given in Table 8.14 for
each sample at the signal frequencies Of 30,000 cps,
1,000 cps, and 30 Ops. The symmetry axes are denoted for
the cases Of cylindrical symmetry.

The tensor symmetry depends upon whether the indi—
vidual principal values can be distinguished above the rms
error. Thus, the tensor symmetry will be controlled by
the anisotrOpy ratios Of the principal value estimates and
the corresponding rms error estimates. The rms error
definitely increases the symmetry of those samples with
high error, even when the anisotrOpy ratios indicate
rather strong anisotropy (e.g., M-3, B—4, M-5, and M-9
in Table 8.14 and Tables 8.4, 8.5, 8.6, and 8.10, re-
spectively).

Both 0 and K tend to have higher symmetry at higher
frequencies. This tendency is much less pronounced for
0 than for K. This tendency is a reflection Of the com-
bined effects Of greater (ratio) anisotrOpy at low fre-
quencies and the relatively consistent error ratios. The
fine grained samples (e.g., M-1, M-2, M—6, and M-ll)
generally exhibit lower symmetry for both C and K than
the coarse grained rocks. This is a reflection Of the

lower rms errors found for these fine grained rocks.

137

Rock Fabric and Structural Control

The extension Of Neumann's principle developed in
Chapter IV predicted that the symmetric second-rank tensor
symmetry would be controlled by the symmetry Of the rock
fabric and structural control. Thus, the principal
directions should parallel any lineations and be normal
to any planar structural controls present in the rock.

If multiple structural controls are present in a rock
which are not mutually parallel or normal, then the above
statement breaks down. This is analogous to the orien-
tation Of symmetric second rank tensors for monoclinic
and triclinic crystals. The principal tensor directions,
which are, by definition, mutually perpendicular, cannot
simultaneously parallel all lineations and be normal tO
all planar structural controls.

In general, the principal tensor directions, as
shown in Figures 8.1 through 8.12, do relate to structural
controls very well. The maximum principal O and K
directions are sub-parallel tO the lineation Of the
amphibole crystals in M-lO (amphibole schist, see Figure
8.11). The correspondence between the maximum principal
directions and the quartz C-axis maximum for the M-7
graywacke (see Figure 8.8) is even better. This is very
encouraging in view Of the fact that only six directional
measurement sets were available for this sample. There

is also good correlation between the maximum principal

138

directions and the lineation due tO the intersection Of
the foliation and slaty cleavage in M-2 (greenstone
schist, see Figure 8.3).

For the sub-graywackes Of M-1 and M-6, the minimum
principal O and K directions are sub-normal to the slaty
cleavage, where these principal directions are distinct
(see Figures 8.1 and 8.6). This correspondence is very
gOOd except for the high frequency K3 directions for M-l,
where K2 and K3 are not distinct (see Figure 8.1). The
intermediate principal directions are sub-normal tO the
slaty cleavage in M-2 (see Figure 8.2).

The minimum principal K and 0 directions for the
greenstone schist Of M-2 and the staurolite-muscovite
schist Of M-8 are sub-normal tO the foliation with good
correspondence (see Figures 8.2 and 8.8). The corre-

spondence for K in M-Z is not as good as the others,

3
however.

In the syenite gneiss Of B-3 and the granite gneiss
Of M-S, the minimum principal directions are sub-normal
tO the gneissic banding (see Figures 8.4 and 8.5). How-
ever, the correspondence is not very gOOd. This poor
correspondence may be due tO the heterogeneity in mineral
content and grain size in both Of these samples.

There appears tO be very poor correspondence be-

tween the principal directions and structural controls

in the amphibolite Of M-3 (see Figure 8.3). The large

139

rms errors for the data from this sample may indicate
heterogenity. Such heterogenity, if present, may confuse
any relationship between the tensor principal directions
and the structural control. Also, the Observed fracture
set (F in Figure 8.3) is not a particularly strong one.
Thus, it may not be a true indication Of rock fabric.

The minimum principal o and K directions are sub-
normal tO fracture sets in the M—9 Hemlock formation sample
and the graywacke of M-ll (see Figures 8.9 and 8.10). How-
ever, the intermediate and maximum principal directions in
M-9 do not show gOOd correspondence tO the other structural

controls present in that sample.

Principal Direction Dispersion

 

For some Of the samples investigated, the principal
directions remained relatively constant (see Figures 8.3,
8.4, 8.8, and 8.9) with variations in signal frequency.
This, together with good correspondence between the o and
K principal directions (tO be discussed in the next sec-
tion), may indicate a strong preferred orientation Of the
constituent dipole sources in the rock. The lack of
directional dispersion also may indicate that the relative
contributions Of the various dipole sources to the bulk
o and K of the rock dO not vary appreciably with frequency.

Some samples exhibit considerable systematic vari-
ation in the orientation Of the principal directions with

frequency (see Figures 8.2, 8.5, 8.10, and 8.11). This

140

may be a manifestation Of the multiple anisotropic polari-
zation species model proposed in Chapter IV. The polari-
zation centers Of this model may have different preferred
orientations and resonance frequencies. Thus, their
various frequency dependent prOperties will contribute

tO the bulk o and K values, causing the orientation dis-
persion.

Some samples show uniformity for a single principal
direction with dispersion in the other principal axes (see
Figures 8.6, 8.7, and 8.11). This may indicate one common
principal polarization direction for the constituent polari-

zation centers Of the rock, with the others free tO vary.

Parallelism Of the 0 and_K_Tensors

 

The qualitative model for the electrical properties
Of rocks, developed in Chapter IV, did not require that the
0 and K principal directions be parallel. However, the
dependence Of the effective conductivity on the imaginary
component Of the lossy dielectric polarizability models
would lead us to expect that. Inspection Of Figures 8.1
through 8.12 shows that, in general, the 0 and K principal
axes are sub-parallel. The best examples Of this are
shown in Figures 8.4, 8.6, 8.8, 8.9, 8.11, and 8.12. Some
samples have good correlation for only one principal
direction. The best examples Of this type Of correlation
are shown in Figures 8.2, 8.3, and 8.7. In these cases,

the principal directions with gOOd correlation are usually

141

directly related to structural control. For example,
the maximum principal directions for M-2 and M-7 (those
with gOOd correlation) are parallel tO lineations.

GOOd correlation between the 0 and K principal
directions may indicate close alignment Of the principal
directions Of the constituent dipole centers of the rock.
Poor correlation, by contrast, may indicate misalignment
Of the principal directions Of the constituent dipole
centers. It would be difficult to gO beyond the above
comments because the bulk electrical properties Of a rock
represent a composite Of the prOperties Of the constituent

mineral grains.

Resonance Frequencies

 

The resonance frequencies, fc, for the bulk electrical
properties Of the rock are Obtained by Observation Of the
local maximas on the a dispersion curves and the inflection
points on the K dispersion curves. The approximate reso-
nance frequencies given in Table 8.14 were obtained from
Figures 8.1 through 8.12 by this method. These fc are
only approximate in nature because Of the lack Of detailed
control near most Of them. They are definitely real, as
can be seen from the data in Figures 8.10 and 8.12, which
have good control near the fc.

The fc in Table 8.14 vary from about 200 cps, for
the M-2 greenstone schist and the M-8 staurolite-muscovite

schist, to about 600 cps, for the M-lO amphibole schist.

142

These resonance frequencies are probably due to the mecha—
nism Of interfacial polarization because of their low
values. Interfacial polarization models, such as those

Of Sillars (1937) and Wait (1959), have resonance fre-
quencies which depend upon the electrical properties Of

the inhomogenities and the medium, the inhomogenity spatial
ciensity, and the grain size. The coarse grained samples
studied in this investigation (M-3, M—4, M-5, M-9, and
M-lO) have consistently higher approximate resonance fre-
quencies than the fine grained samples (M-1, M—2, M-6,

M-8, and M-ll). By contrast, Wait (1959) predicts that

for rocks Of similar mineral composition, those with

larger grain size will have longer relaxation times and
thus lower resonance frequencies. However, variations

in mineral content between rocks can cause the exact
Opposite to occur. This, then, may be the explanation

for the apparent fC-grain size relationships Of the present

study.

Effects Of Moisture

 

The great danger in using vacuum dried samples was
that they might show significant orthorhombic anisotropy,
while their saturated equivalents might not. In the early
portions Of the laboratory investigation, some Of the
samples were measured after being dried in air at room
temperature, so they were effectively in equilibrium with

the atmospheric moisture. This technique was later

143

abandoned in favor Of vacuum drying at 60° C to gain
repeatability in the measurements. While the data Ob-
tained from these early measurements were not repeatable,
it may be used as an indication Of the effects that
moisture might have on anisotropy. With this in mind,
the results for the M-2 greenstone schist (marked M-2A,
tO avoid confusion) in this slightly moistened condition
are included as Figure 8.12 and Table 8.13.

This slight increase in moisture content raised the
O values nearly one-half order Of magnitude and nearly
doubled the low frequency Kl values over their vacuum
dried equivalents. This served to increase the anisotropy
ratios for both C and K and increase the scatter Of the
R(O) versus R(K) plots Of Figure 8.14. Points 3, 8, and
13 in Figure 8.14 are from the M-ZA data. This anomalous
behavior Of the R(o) versus R(K) plot with the addition
Of water suggests the possibility that anisotrOpy ratio
plots may also be used tO evaluate the saturation Of rocks.
The resonance frequency was increased for the slightly
moistened sample over its vacuum dried equivalent and the
principal directions changed slightly.

The results for M-2 and M-2A are not sufficient to
generalize on the effects Of moisture on electrical aniso-
trOpy. However, they do Offer the interesting possibility
that metamorphic rocks may exhibit even greater electrical

anisotropy in the saturated state than in the vacuum dried

condition.

144

Consequences Of the Laboratory Results

 

Strong orthorhombic electrical anisotropy was Ob-
served for many Of the samples investigated in this study.
Thus, the assumptions Of electrical isotropy, or even
anisotropy with cylindrical symmetry, Often made in theo-
retical geoelectric data interpretation may not be valid
for some rocks. This implication is more important than
the specific values Obtained for twelve sets Of measure-
ments on eleven samples Of vacuum dried Precambrian rocks
Of Michigan and Ontario.

When strong orthorhombic anisotropy is present in
rocks, theoretical geoelectric interpretation methods should
be highly suspect. This is because the B.V.P. upon which
the theoretical master curves are based is nO longer analo-
gous to the physical problem found in nature. In such
cases, all geological control available should be applied
tO the curve matching results to Obtain a final interpre-
tation.

The laboratory portion Of this study was concerned
with micro-anisotropy, or that anisotropy which is Ob-
served in small samples. In a field situation, the macro-
anisotropy, due to the structural attitude and thickness
Of the various lithologic units, would also be Of interest.
The Observation Of strong micro-anisotrOpy in the labora-
tory does not mean that strong total (micro- plus macro-)

anisotropy will be Observed in the field. However, Dowling

145

(1967) and Tammemogi (1969a, 1969b) Observed striking
anisotrOpy in their reSpective magneto-telluric investi-
gations over Precambrian rocks in the Lake Superior
region. While Dowling was unable to make a direct corre—
lation Of his anisotropy with geological structure,
Tammemogi was able to make very gOOd correlations.
Schlumberger et al. (1934) indicate that the total aniso-
trOpy Observed in the field is usually much greater than
the micro-anisotrOpy Of the individual lithologic units.

A very significant consequence Of this study is
that it can no longer be safely assumed that three mutually
perpendicular O and K directional measurements serve to
completely define these tensor prOperties. In order to be
able tO use only three mutually perpendicular directional
values, the principal tensor directions must be precisely
known in advance. As indicated in Chapter V, the labora-
tory reference systems for the present study were oriented
with respect tO any Observed fabric in the field samples.
The principal direction plots Of Figures 8.1 through 8.12
indicate that the principal directions so determined did
not coincide with the tensor principal directions. Thus,
to completely define the symmetric second-rank c and K
tensors, all six Of the independent coefficients of the
representation matrix must be determined. This requires
that directional measurements Of these tensor properties

be made in six, or more directions.

CHAPTER IX

CONCLUSIONS

The addition Of strong orthorhombic anisotropy to
the boundary value problems used in theoretical geo—
electrical interpretation methods makes them very diffi-
cult, if not impossible, to solve analytically. This is
illustrated very well with the horizontally layered earth
B.V.P. commonly used tO develOp master apparent resis—
tivity curves used for interpretation of resistivity and
I.P. field data.

The lossy dielectric provides a very good quali-
tative model for a macroscopic (in the solid-state sense)
study Of electrical anisotropy Of rocks. The lossy di—
electric model requires that the dielectric constant, K,
and electrical conductivity, 0, tensors be completely
defined. TO completely define these symmetric second-
rank tensors, directional measurements must be made Of
the effective directional prOperties in at least six
different directions. Measurement Of the effective
directional properties in more than six different direc—
tions allows a least square determination Of the tensor

coefficients to be made.

146

147

Electrical anisotropy Of rocks is studied by deter-
mining the principal values and directions Of the O and
K tensors. The anisotropy is measured by the departure
Of the anisotropy ratios from unity. The degree Of
symmetry Of the c and K tensors is determined by the number
Of principal values which are distinguishable above the
rms errors.

There appear tO be linear relationships between the
corresponding 0 and K anisotropy ratios at each frequency.
Departures from these linear relationships may indicate
poor data, variations in lithology, or variations in mois-
ture content.

The coarse grained samples exhibited larger rms errors
in the principal values, higher symmetry, and higher reso-
nance frequencies than the fine grained samples.

The anisotropy Of both 0 and K tended to be greater
at lower frequencies. Also, the symmetry Of these tensors
tended tO be lower at lower frequencies. However, this
latter tendency was very weak for 0.

The symmetry Of the c and K tensors Of rocks is
related tO the symmetry Of the structural controls Of
the rocks. These structural controls can be statistical
microscopic rock fabrics or macrOSCOpically observable
features, such as cleavage, joints, and foliation. Thus,
electrical anisotropy may be used to infer rock fabric

symmetry.

148

Some Of the samples investigated exhibited signifi-
cant dispersion (frequency variation) Of the principal
axes. This may be related to the variation in the rela-
ative contributions of the various dipole centers to the
bulk electrical properties at various frequencies.

Generally speaking, the principal axes for K and 0
were sub-parallel. This may be attributed to the depend-
ence Of the effective conductivity on the imaginary com-
ponents Of the lossy dielectric polarizability models.

Comparison of the electrical anisotropy results for
a greenstone schist (M-Z) in the vacuum and air dried
states indicate that the presence of moisture in these

metamorphic rocks may increase the electrical anisotropy.

 

CHAPTER X

RECOMMENDATIONS FOR FURTHER STUDY

The present study did achieve its primary goals.

It demonstrated that strong orthorhombic electrical aniso-
trOpy may be present in some rocks. It also developed and
tested a method for the laboratory determination Of the c
and K tensors. In addition, it proposed a field method
for the determination Of these tensors. However, it did
not provide a comprehensive catalog Of electrical aniso-
trOpies for various rocks.

A study which would provide such a comprehensive
catalog Of rock electrical anisotropies of a given geo—
logical province is a logical follow-up study for the
present investigation. Such a follow-up study should
also include measurements on saturated and high metallic
content rocks. This would require slightly different
instrumentation than that used for the present study.
However, the problems involved are not insurmountable.

Studies Specifically aimed at investigating the
effects Of moisture content and electrolyte variations
on electrical anisotropy would also be very worthwhile.

Any investigation involving saturated rocks must deal

149

150

with the problem Of non-polarizing electrodes. A very
Significant advance for petrOphysics would be the develop-
ment Of a successful non-polarizing electrode system with
a high degree Of acceptability to those working in the
field.

Now that the possibility of strong orthorhombic
electrical anisotropy has been established, it is very
desirable to extend as many as possible Of the isotropic
boundary value problems, used for theoretical geoelectric
data interpretation, to include the condition Of aniso-
tropy. Numerical solutions may be necessary for these
extensions where analytical solutions cannot be obtained.

Finally, a very logical follow-up Of the present
study would be a field electrical anisotropy investigation.
It could also evaluate the field procedures suggested in

Chapter II.

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APPENDICES

APPENDIX A

REVIEW OF POLARIZATION MECHANISMS

REVIEW OF POLARIZATION MECHANISMS

Introduction

 

The present laboratory investigation is macroscopic
in nature (at least in the solid state physics sense). As
such, quantitative microscopic polarization models are not
required. However, qualitative microscopic polarization
models are very useful for predicting experimental results
before experimentation is commenced and explaining experi-
mental observations after eXperimentation is completed.
For these reasons, a catalog of the microscopic polari—

zation models Of others has been included as an appendix.

Electronic Polarization

The classical idea Of an atom consists of a dense,
positively charged nucleus surrounded by a diffuse cloud
Of negatively charged electrons. In the presence of an
E field, the centers of gravity for the nucleus and
electron cloud undergo a relative shift, which produces
a microscopic electric dipole. The magnitude of this
shift depends upon the field strength, E, atomic number
Of the atom, Z, and the radius Of the electron cloud,
von Hippel (1954b, 1954c), Wert and Thompson (1964),

and R. Beam (1965) assume a simple spherical atomic model

164

165

with the polarization displacement reaching equilibrium
between forces due to the external E field and the
coulombic attraction Of the shifted charge centers.

In the presence of an oscillating E field the polari-
zation Of this model is analogous to the motion of a
driven damped harmonic oscillator. With this somewhat
simplified model, they Obtain:

_ / ez/m

a — , A.l
e M? -707 4. j...

 

where: e and m are the electronic charge and mass,

respectively, j = V-I,

 

, _ 2 Ne2 .
w — mo - SfiE— , is the resonance (angular)
0

frequency Of the damped harmonic

oscillators, of density, N,

w = /R7ET is the resonance (angular) frequency of
the corresponding simple harmonic oscill-
ator model with elastic constant (bonding
strength), k,

20 = nmc is the electromagnetic attenuation

(damping) factor,

is the magnetic permeability Of free Space and

c is the velocity of light, respectively,

166

for the steady state electric polarizability. The relation-
ship Of equation A.1 yields a frequency dependent, or dis-

persive, polarizability.

For D.C. E fields, w 0 and equation A.1 becomes:

36 e 2
0 9..
k

Q
II
II:

 

408 R3, A.2
O

for low N. The polarizability models given by equations
A.1 and A.2 apparently agree with Observations quite well
(Wert and Thompson, 1964). The polarization mechanism
described by the models of equations A.1 and A.2 is called
induced, or electronic, polarization and its polarizability
is indicated by de. The term, induced polarization, is
undesirable, both from the standpoint that geophysicists
also use this term tO describe a different polarization
mechanism and because all polarization mechanisms due to
an external field are, in fact, induced. For this reason,
the polarization mechanism described by equations A.1 and
A.2 will be referred tO as electronic polarization in this

paper.

Ionic Polarization

 

When atoms combine into molecules, there is generally
an unequal distribution Of charge, giving rise to positive
and negative charge centers. In the presence of an E

field, these charge centers undergo a relative shift

167

from their equilibrium positions (in the absence of E).
This mechanism is really similar to that for electronic
polarization, however, the charge center geometries and
masses are different. The damped harmonic oscillator
model for electronic polarization was assumed to be iso-
trOpic and thus have three degrees of vibrational freedom.
A dipolar molecule requires an anisotropic damped oscill-
ator model, with only one degree of vibrational freedom.
Von Hippel (1954c) and Beam (1965) use an anisotropic

damped harmonic oscillator and Obtain:

2
a = 2 q /3m~ , A.3
I

w - w2 + ijG

as the steady state polarizability for multivalent di-
atomic molecules where wé, j, and 20 are as was defined
for the electronic polarization model, q is the dipole
charge, and 1/3 is the factor needed to convert from three
tO one degree Of vibrational freedom.

For static E fields, w = 0 and equation A.3 becomes:

2

cl = gf" A.4

for low N. This type Of polarization is called ionic,

or atomic, polarization and is indicated by al. The

term, ionic polarization will be used for this paper.

168

Dipole Relaxation Polarization

 

Some molecules, such as water and HCl, are highly
dipolar. In the gaseous, liquid, and, to a lesser extent,
solid state, such molecules attempt to physically align
themselves parallel to any E field. Opposing this align-
ment will be intermolecular and thermal forces. For
diffuse fluids with negligible intermolecular forces,

von Hippel (1954b) and Beam (1965) Obtained:

E2/3k0

OLd = + ij ' A.5

3
for the steady state polarizability, where T = £“E%"fl

is the relaxation time constant for a spherical molecular
model Of radius, a, k is Boltzman's constant, 0, is the
temperature in 0K, n is the viscosity of the fluid, and
Ep is the permanent dipole moment of the molecule. This
mechanism is called dipole relaxation, molecular, or
orientation, polarization and its polarizability is indi-
cated as dd. DipOle relaxation appears to be the most
commonly used name for this mechanism and will be used in
this paper. Dipole relaxation in solids will be similar
to that in fluids, however, the polarizability will be
more complicated than that given by equation A.5 and
include the effects Of intermolecular forces. I could

find no treatment Of this more complicated problem.

169

For static E fields, w = 0 and equation A.5 becomes:

for models assuming negligible intermolecular forces.
Models, which consider intermolecular forces, should have

more complicated static 0 than that given by equation A.6.

d

Interfacial Polarization

 

The above polarization mechanisms (electronic,
ionic, and dipole relaxation polarization) are sufficient
to describe electric polarization within a homogeneous
material. For such materials, the total polarizability
is simply the sum Of the above polarizabilities.

Most rocks, however, are far from homogeneous. For
such inhomogeneous materials, an additional polarization
mechanism must be added to account for the collection of
charge at the boundaries of the inhomogenities in the
presence Of an E field. Such a mechanism is called
space-charge, or interfacial, polarization and its polari—
zability is indicated as ai. This is one Of the primary
mechanisms involved in what geophysicists refer to as
induced polarization. Keller and Freschknecht (1966)
mention this phenomena, but do not pursue the matter
further. The term, interfacial polarization, will be

used in this paper, for its initials (I.P.) can then be

170

used in geOphysical literature without causing further
confusion.

Wait (1959) describes an interfacial polarization
model consisting Of spherical particles of conductivity,
0p, and radius, a, coated with an insulating film Of
thickness, tm’ and dielectric constant, Km, suSpended
uniformly in a medium Of conductivity, 0. With this
model, he Obtained:

_ 3v 1 - 0
“1 "' “fir—T28" 1”

for the steady state polarizability, where v = 4 na3N/3,

t O
N is the inhomogenity density, and 0 = 2; + vJE—-.
O jwema

Sillars (1937) Obtained the results for an ideal
dielectric of dielectric constant, Kl, containing spheroidal
inhomogenities of conductivity, 02, and dielectric constant,
K , with semi-axes, a, parallel tO E, and b = c, normal to

2
E. With such a model, he Obtained:

I K N K NwT
_ 1 1 _ . 1

 

 

l + w T l + w T

Kl(n-l) + K2

 

for the steady state polarizability where 13 400
2

 

 

nle n(K2 - K1)
N = q Klihzii + K2 ' K = K1 1 I q Kl(n-l) + K2 '

:3
ll

4n/la,

171

q is the spheroid volume fraction of the dielectric,

 

2
1a = 4n{—%-- [Egl:§—l-sin-l%]}, for oblate spheroids
e e

40 when a << b, c),

III

(la

__ l _ l 1+e _
la - 4n[;§ l][2€ log 1:3’ 1], for prolate

b2 2a
40—2 [log T7 - 1] when a >> b, c),
a

ll!

spheroids (la

2 2
e = 1 - if = l - EE-is the eccentricity of the
c b

spheroids.

The coefficient, n, is a function Of the eccentricity Of
the spheroids and varies from a value Of unity, for a very
flat oblate spheroid, to infinity, for a very long prolate

spheroid. For a Sphere, n = 3.

APPENDIX B

COMPUTER PROGRAM AND SUBROUTINE

DESCRIPTIONS

COMPUTER PROGRAM AND SUBROUTINE

DESCRIPTIONS

Introduction

 

This appendix contains the program and subroutine
discriptions for only one rather involved program. The
input for this program (ELECT) consists of the Schering
bridge balance readings, the sample disk dimensions, the
disk orientations, and the measurement frequencies. The
final output is the principal values and directions, of
the K and O tensors, with respect to the laboratory axis
system.

The main program, ELECT, exists primarily to call
the semi-autonomous subroutines: KRD, TENSOR, and REDUCE,
which apply successive reduction steps to the laboratory
data. The program and subroutines of this appendix were
written in FORTRAN for the CDC 3600 computer available at
Michigan State University. They may need some revision if
used on some other system.

Some Of the subroutines described in this appendix
were Obtained from the book by Robinson (1967) and the
Michigan State University Computer Library, but most were

written by the author. Robinson (1967) uses the ingeneous

172

173

device of storing multiple dimensioned arrays as column
vectors in his calling program, or subroutines. The
author found this to be an invaluable aid in conserving
storage space, for then the respective arrays could be
dummy dimensioned in the called subroutines.

The three major working subroutines, mentioned
above, were written and debugged separately before inser-
tion into the main program. They were combined into a
single program, because the final principal values and
directions were what was most desired from the original
data. Combining these successive data reduction Operations
end-to-end in a Single program eliminated the need for inter-
mediate physical data handling. As an aid in interpretation
and quality control, the results of three stages in the data
reduction Operations are printed out.

Program ELECT and the three major subroutines were
written specifically for the problem described in Chapters
III, IV, and V of this paper. However, most of the smaller
subroutines are perfectly general in nature and may easily
be adapted for use in other programs. This is a great ad-
vantage in modular programming, for subroutines developed

for one program can then also be used for other programs.

174

'Program Description: ELECT

 

Purpose

To convert directional measurements Of O and K into
the respective tensor coefficients and to Obtain their

principal directions and values.

Usage

The input deck order for program elect is shown in
Figure 8.1. The input variables are:

N¢SPLs--number Of data sets which are to be pro-
cessed.

NAME--data set identification.

NODXN--number Of directions in which the electrical
properties were measured.

N¢FQ--number Of frequencies at which the electrical
properties were measured.

NM--measurement direction identification.

SDM--sample disk diameter, in inches.

T--samp1e disk thickness, in mills.

A(I,NTM)--direction angles of the NTM measurement
direction (NTM = l, N¢DXN; I = l, 3).

INDC--corrected-uncorrected capacitance value indi-
cator (if INDC # 0, the capacitance values are
pre-corrected).

F--the measurement signal frequency, in cps.

FO--bridge frequency range setting, in cps.

Cl--initial capacitance balance reading (see Chapter
V), in pf.

C2--second capacitance balance reading, in pf.

Dl--initial dissipation factor balance reading.

D2--second dissipation factor balance reading.

L(I)--direction cosines Of the arbitrary direction
used to initiate the iterative rotation procedure
Of Chapter VI (I = 1, 3).

Only cards 1 and 2 are read by the main program, ELECT. The

rest Of the data deck Of Figure B.l is read by KRD and REDUCE.

175

 

 

 

  
    

 

   

 

 

 

 

 

 

 

 

9 f//L(|L,|=L3. 6
0“ /~
.6 ‘ \Oo. (3Fll.5)
d“ ‘§’da~l/r
0° .9” 6“ F.Fo.CI.cz,0i,02. 5
o“ a . . . ,
(0° NM.SDM.T. 401.1: l,3,INDC.4 ’
(AB.F8.4J F8.243FIO.5 15) +-—4
NVDXN, NQFQ. 3
1215) i
NAME. 2 /
/ 110481 e—J
fuospts. , I
(i5)
__

 

 

 

 

Fig. B.l.--Deck order for program ELECT.

Method

Program ELECT is really a dummy program which com-
bines the Operations Of the three semi-autonomous major
subroutines: KRD, TENSOR, and REDUCE. It is these three
major subroutines which actually do the operations required

to accomplish the purpose Of program ELECT.

Restrictions
The carriage controls used in PERMAT statements 1

and 4 may only be valid for the Michigan State University

176

CDC 3600. This program requires the use of three scratch
units (here numbered 25, 26, and 27). Program ELECT
requires subroutines: KRD, TENSOR, and REDUCE. Other
restrictions will be listed with each Of the individual

subroutines.

177

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pouom z<mooma

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178

Subroutine Description: KRD

 

Purpose

TO convert the initial laboratory data Obtained from
the General Radio 716-C capacitance bridge, available in
the Michigan State University Department Of Geology, into

directional values for O and K.

Usage

The calling sequence is:
CALL KRD (N¢DXN, N¢FQ)
N¢DXN--number Of directions in which the electrical
prOperties were measured.
N¢FQ--number of frequencies at which the electrical
prOperties were measured.
Subroutine KRD reads NODXN, NOFQ, NM, SDM, T, A(I,NTM),
F(J), FO(J), C1(J), C2(J), D1(J), and D2(J) from the data

deck.

Method
The measurement direction angles are converted tO

direction cosines by:

l = cos-19.

If the Schering bridge balance C values are uncorrected,
they are corrected by subroutine CORRECT. The corrected C
values and D values are converted into CP and corrected
dissipation factors, D, by means of subroutine BRIDGE.

The CP and corrected D values for both balances (see

179

(flnapter V) along with the sample disk dimensions and
rmeasurement frequencies, are input into subroutine CAP

tn: Obtain the directional 0 and K values at each measure-
nmnit frequency. The measurement frequencies, direction
cussines, and directional values of c and K are then stored

cni scratch unit 25 until further use.

Restrictions

The variables, KB and SIG are specified to be double
Eirecision. Measurements can be made in no more than ten
ciirections, and at no more than twenty frequencies without
cfluanging the dimension statements. Subroutine KRD calls
smabroutines: CORRECT, BRIDGE, and CAP. Subroutine KRD
11585 a scratch unit (here, numbered 25). Care must be
«exercised to see that the data is read from the scratch
iinit.in.exactly the same manner in which it was written.
(the carriage controls used in the print formats may only

loe valid for the Michigan State University CDC 3600 com-

puter.

180

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182

 

Subroutine Description: CORRECT

Purpose

To correct the capacitance readings on the General
Radio 716-C capacitance bridge available in the Michigan
State University Department Of Geology, to calibrated

values.

Usage

The calling sequence is:

CALL CORRECT (RD)
RD--capacitance balance dial reading, in pf.

Method
If RD is between two calibration points, Di' and

D the correction, CR is given by

i+l’

SLP = (c - Ci)/(Di+l - 0.),

i+1 1

CR

C. + SLP(RD - D.),
1 1

where the Ci and Ci+1 are the corrections for the cali-

bration points, Di and D. respectively. The corrected

1+l'

value is then:
RD = RD + CR.

Restrictions

 

The Ci are valid only for the 716-C capacitance

bridge available in the Michigan State University

183

Department of Geology. They will vary for each specific
bridge and, perhaps with each calibration of the same

bridge. The allowable range in RD is: 100 pf g RD s 1150 pf.

184

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czm m 2mghwa

H »z~aa

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185

Subroutine Description: BRIDGE

Purpose

To convert the General Radio 7l6-C bridge balance
readings of C and D into appropriate dissipation factors

and parallel capacitances for the unknown element.

Usage

The calling sequence is:

CALL BRIDGE (F, F0, c, D, CP)

F--measurement frequency, in cps.

FO--bridge frequency range selector position, in cps.

C--corrected bridge capacitance dial reading, in f.

D--uncorrected bridge reading at input and unknown
dissipation factor value on output.

CP--unknown parallel capacitance.

Method

D is first corrected for the measuring frequency:

Then a correction for the dissipation of the standard

capacitor is applied:
D = D + 0.04/C.

The unknown dissipation factor, D, and the series capaci-

tance, CS, are obtained from:

.3.
F0

'71
O

186

The unknown parallel capacitance, CP, is then obtained
from the relationships between equivalent series and

parallel RC circuits (Stout, 1960):

CS

CP = —————7-.
l + D

Restrictions

 

This subroutine is intended only for use with a
General Radio 716-C Capacitance bridge with a type 722

internal capacitor.

Acknowledgments

 

The correction and reduction equations used in this
subroutine were obtained from the 716-C instrument manual

and verified by the author.

187

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tgbtt EQJCHIVQ UDX“GF DulA KtubblltJ attic

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CS
[I

9.01*:~(F/r0)

W * U.‘4/(8*(1.ONUUUFh+ld1)
= 3*\1.5402gonc30 - C.LZD*U~(*/FJ))
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CD = SS/(L,lHCZUObfinn + Drill

'

4+
E

N

‘fU?J

1‘
l

188

Subroutine Description: CAP

 

Purpose

To calculate 0 and K values for measurements using

a micrometer dielectric sample holder.

Usage

The calling sequence is:

CALL CAP (C1, C2, D1, D2, T, SDM, F, SIGMA, K)

Cl--initial balance unknown parallel capacitance,
in f.

C2--second balance unknown parallel capacitance,
in f.

Dl--initial balance unknown dissipation factor.

D2--second balance unknown dissipation factor.

T--sample disk thickness, in m.

SDM--sample disk diameter, in m.

SIGMA--sample disk effective conductivity, 0, in
mho/m.

K--sample disk dielectric constant.

F--measurement frequency, in cps.

Method

D2 values obtained from the bridge balances with and with-

out the sample disk in the holder,

Using the sample dimensions and the C1, C2, D1, and

and dielectric constant are obtained from:

K = (C1 - C2)T + 1,

O

 

_ wT(D1Cl - D2C2)
U - A ,

 

the sample conductivity

 

 

 

 

189

where w = 2nF is the measurement angular frequency and

A = 1r(SDM/2)2 is the sample disk area, and so = 8.85 -

 

10"12 f/m.
Restrictions
This subroutine should only be used to reduce data
obtained by using a micrometer sample holder in the manner r;
r-

described in Chapter V. SIGMA and K are specified as

double precision.

 

190

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191

Subroutine Description: TENSUR

 

Purpose

To calculate the K and 0 least square tensor co-
efficients and rms errors from directional measurements

of K and 0.

Usage

The calling sequence is:

CALL TENS¢R (N¢DXN, N¢FQ, NAME)

N¢DXN—-number of directions in which the electrical
properties were measured. -

N¢FQ-—number of frequencies at which the electrical £3
properties were measured. *

NAME--sample identification.

C A J. 'I A‘W‘nl' -

 

Method

Subroutine TENS¢R is really a dummy, or bookkeeping,
subroutine. The actual least square determinations and
rms error calculations are done by subroutine LSQ. Sub-
routine TENS¢R reads the direction cosines, frequencies,
and directional K and 0 values from scratch units 25 and
26 and inputs this information to LSQ for reduction. The
least square coefficients and rms error for each frequency
are then printed out. The measurement frequencies, the
least square K and a tensor coefficients, and the rms

errors are then stored on scratch unit 27 for later use.

Restrictions

 

There can be no more than twenty elements in arrays

K, KE, SIGMA, and C¢ND without changing the dimension

192

statements. Array DXN cannot have more than twenty-five
elements and NAME, ten, without changing the dimension
statements.

Variables KE, SIGMA, KR, SR, K, and C¢ND are
specified as double precision. Subroutine TENS¢R calls
subroutines LSQ and ¢UTPUT. Subroutine TENS¢R was in-
tended only to be used for the measurement technique {5
given in Chapters V and VI of the text. Its use for other
purposes may require revision.

This subroutine reads input from scratch units 25

*7.)
‘h .

 

1“.
1“ .

and 26. Thus, the data must be stored on these tapes in
exactly the same manner in which this subroutine reads it.
Subroutine TENS¢R also stores data on scratch units 26 and
27. Thus, this data must be read from these scratch units
in exactly the same manner that it is written by this sub-
routine.

Variables DXN, K, KE, SIGMA, and C¢ND are multiply
dimensioned arrays stored in column mode. This allows
the use of dummy dimensions for these variables in any
subroutine called from TENS¢R.

The carriage controls used in the F¢RTRAN statements
may only be valid for the Michigan State University CDC

3600 system.

193

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Subroutine Description: ¢UTPUT

 

P11172058

To print out a multiply dimensioned array as a

series of matrices.

Usage

The calling sequence is: *f

CALL ¢UTPUT (NRX, NCX, NX, X)

NRX--number of rows in each matrix of X.
NCX--number of columns in each matrix of X.
NX--number of matrices in X. y
X-—the array to be printed. 53

 

Method
A multiply dimensioned array, stored by columns and
in multiplexed mode (see Robinson, 1967) is printed out

as a series of matrices.

Restrictions

 

Subroutine ¢UTPUT assumes that the array, X, is
stored such that each matrix is stored by columns and the
different matrices are multiplexed (see Robinson, 1967).
This subroutine is general in nature and can be used to
print out any array stored in this manner. The only
change which may be necessary will be in the print format.

The total dimension for X, NRX°NCS°NX, cannot exceed
the dimension for this variable in the calling routine.

NCX cannot exceed seven without changing format statement 1.

196

The present format 1 will print out the elements of X
as floating point numbers. X is specified as double
precision.

The carriage control in format 2 may only be valid

for the Michigan State University CDC 3600 system.

H

10

197

suaRnUTINE OUTPUT (VRX.NCX:NX.X)
Tv=e UDUELF x

DIWENSIOV Y(NRY,NC¥,NX)

FuznaT (18x,7515.6)

FUQMAY (t t)
PRINT 2
DU 10 I = 1: NPX

PRINT 1. (Y(I.J.K), J = 1. NCX)
RETURN
END

198

Subroutine Description: LSQ

 

Purpose

To calculate least square tensor coefficients and
rms errors for a symmetric second-rank tensor, given

directional tensor values.

Usage

The calling sequence is:

CALL LSQ (D, B, N, A, G)

D--direction cosine array, stored by columns.

B--measured value column vector.

A--1east square tensor coefficient matrix, stored

by columns.

G--rms error (see Jenkins and Watts, 1968).

Method

The direction cosines of the measurement directions

are converted into a coefficient matrix, C, by:

2

“11 = dil’ C14 = 2dizdi3'
c - d2 - 2d d

12 ‘ 12' Ci5 ‘ 13 11'
c = d2 c = 2d d

13 13' i6 i1 12'

The least square coefficients are then given by:

A = (CTC)-1CTB'

where:

” .3

 

199

A11 = A1' A23 2 A32 = A4'
A22 = A2' A31 z A13 = A5'
A33 = A3' A12 = A21 = A6°

The least square error, E, is given by:

1 T )T

((I-C(CTC)-CB 1

(I — C(CTc)' CTB).

[31
II

and the rms error, G, by:

_ E
G " //; + 2 - 6 ’

where p is the number of directional values available.

 

 

Restrictions

 

Subroutine LSQ is written in completely general form.
Thus, it may be used to give least square coefficients and
rms errors for any symmetric second-rank tensor on which
directional measurements have been made. Subroutine LSQ
calls subroutines TRNSPS, MATMULT, INVERT, IDENT, and ADD.

The variables A, C, CT, Z, ZNV, C¢EF, G, and B are
specified to be double precision. All arrays are stored
by columns. The number of elements in D, B, and A cannot
exceed the dimensions of their respective arrays in the
calling routine. The number of elements in C, Ct, Z,
ZNV, and C¢EF cannot exceed 100 without changing the

dimension statements.

200

 

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202

Subroutine Description: TRNSPS

PUI’EOSG

To form a matrix transpose.

Usage

The calling sequence is:

CALL TRNSPS (N¢RWX, N¢COLX, x, XT) t
N¢RWX--number of rows in X. .
N¢C¢LX-—number of columns in X.
Xr-matrix to be transposed.
XT--matrix transpose of X.

 

Method B;
Given a N¢RWX by N¢COLX matrix, X, the NUCOLX by

NDRWX matrix, XT, is formed by:

XT

ll
X

where

Restrictions

 

Subroutine TRNSPS is written in a general form. Thus,
it may be used to form a matrix transpose under most cir—
cumstances. The variables, X and XT, are specified to be
double precision. This subroutine uses dummy dimensions
for X and XT. Thus they must be stored in column mode in

the calling routine.

p

203

SUQRDUTINE TRNSPS(NORNX, NUUULX, X, XT)
ct... MATRIX TRANSPOSt "***

TYPE UJUBLF X, XT

DI”EN$10\ X(NOPHX,NOCULX). X!(NOUOLX:NORWX)
DU 2 I = 1. NOCOLX

DU 1 J = 1, NORHX

XTClpJ) 3 X‘ng)

CUVTINJE

RtTUF‘N

END

 

204

Subroutine Description: MATMULT

 

Purpose

To perform matrix multiplication.

Usage

The calling sequence is:

CALL MATMULT (N¢RWX, N¢C¢LX, N¢RWY, N¢C¢LY, x, y, x, IM)
N¢RWX-—number of rows in matrix, X.

N¢COLX--number of columns in matrix, X.

N¢RWY--number of rows in matrix Y.

N¢C¢LY--number of columns in matrix Y.

X--input matrix.

Y--input matrix.

Z--output matrix.

IM--error flag. When N¢C¢LX # NURWY, IM = 0.

Method
Given the NURWX by N¢C¢LX matrix, X, and the NQRWY
by N¢C¢LY matrix, Y, the N¢RWX by N¢C¢LY matrix, Z, is

obtained by:
z = XY,

where:

N
zij = k:lxikykj, N = N¢C¢LX = N¢RWY

Restrictions

 

Subroutine MATMULT is written in general form. Thus,
it may be used to perform matrix multiplication under most

circumstances. The variables X, Y, and Z are specified to

 

 

205

be double precision. N¢C¢LX must equal N¢RWX. The
arrays X, Y, and Z are dummy dimensioned. Thus, they

must be stored in column mode in the calling routine.

 

206

 

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207

Subroutine Description: INVERT

 

PUI‘EOSE

To invert a real matrix by Gaussian elimination and

back substitution.

Usa e
___£_. r_

The calling sequence is:

' 31.31""

CALL INVERT (N, EP, B, x, KER)

N—-rank of the matrix.

EP--the zero test value for the singularity Check. .
B——the N by N input matrix. i
X--the N by N output (inverse of B) matrix. i
Ker--singularity flag. KER = 2 if B is singular. fir

 

Method

Given an N by N, non-singular matrix, A, it can be
transformed into an upper triangular matrix, B, by simple
row operations (Marcus and Minc, 1965). The matrix is
first searched to find the row with the largest (magnitude)
element in the first column. The magnitude of this element
is then tested against EP to insure that the matrix is not
singular. For non-singular matrices, the row containing
this element is interchanged with the first row of the
matrix. Next, the first column entries in all but the
first row are eliminated by subtracting multiples of the
first row such that their first column entries vanish.

The apprOpriate multiplication factor, mi, for the ith

row will be:

 

 

208

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Next, the above process is repeated on the (N-l) by E

(N-l) matrix remaining in rows 2 through N. The row of

this new matrix with the largest (magnitude) element in

 

the first column (column 2 of the original N by N matrix) E;
is tested against Ef, rotated into the top row (second row
of the original N by N matrix), and the first column ele-
ments eliminated in the remaining rows as above. This
process is repeated until an upper triangular matrix is
obtained, which will be called B. Simultaneously, the
same elementary row Operations applied to A to obtain B
are also applied to an N by N identity matrix, I, to obtain
an auxillary matrix, C.

When A has been transformed into B, back substitution
(Faddeeva, 1959; Weeg and Reed, 1966) is used to obtain
the elements of the inverse matrix, X. The recursion

formulas for this back substitution are (Bailey, 1961;

Weeg and Reed, 1966):

209

 

CN.
X = F1 ,
N3 NN
N
Cij ’ _¥ bikxkj
= k—1+l i < N . < N
l] bii I I 3 \ .

4b
Restrictions :1
,,

Subroutine INVERT is written in general form. Thus,

 

it may be used to invert any non-singular matrix, under

 

most circumstances. The rank of the matrix to be inverted Ej-
may not be greater than ten without changing the dimension I
for A. Variables X and B have dummy dimensions. Thus,
these variables must be stored in column mode in the calling
routine. The variables A, X, Z, RATI¢, S, and B are speci—
fied to be double precision.

The test for singular matrices will depend upon
the input value of EP. Care must be taken in determining
it.

Subroutine INVERT calls subroutine IDENT.

Acknowledgments

Subroutine INVERT was obtained from the Michigan State
University Computer Laboratory Library (Bailey, 1961), as

subroutine GAUSS. It was verified and modified by the

author.

210

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212

Subroutine Description:

IDENT

 

Purpose

To create an N by N identity matrix.

Usage
The calling sequence is:
CALL IDENT (N, X)

N--rank of the identity matrix, X.
X—-array in which the identity matrix will be stored.

Method

 

The N by N matrix, X, is first set equal to zero and

then the xii elements set equal to 1.000000.

Restrictions

 

Subroutine IDENT is general in form. Thus, it may
be used to create identity matrices in most cases. The
array, X, is dummy dimensioned. Thus, it must be stored
in column mode in the calling routine. The array, X, may
not have more elements than allowed by its dimension in
the calling routine. The variable, X, is specified to be

double precision. Subroutine IDENT calls subroutine ZERU.

L1

in

213

SuRROuTINE IDE“T (N.X)

itttt N X N IDENTITY MATRIX ‘fi'tfi
TYPE DOUPLF x

DIMENSION Y(‘J,’-J)

MN 8 NtN

CALL ZERO (0N.X)

DO 10 I = 1. N

X(I.I) = 1.000000

neTunv

END

 

214

Spppoutine Description: ZER¢

 

Purpose

To fill an array with zero elements.

Usage

The calling sequence is:

CALL ZER¢ (LX, X) P]
LX--number of elements in the array, X. a
X--the array to be set equal to zero.

Method

 

All elements in the array, X, are set equal to I;

0.000000.

Restrictions

 

Subroutine ZER¢ is general, so it may be used to
set arrays equal to zero in most cases. The array, X, is
dummy dimensioned. Thus all multiply dimensioned arrays
to be zeroed by this routine must be stored in column mode
in the calling routine. The variable, X, is specified to
be double precision. The number of elements in X cannot

exceed the dimension of X in the calling routine.

Acknowledgments

Subroutine ZER¢ was obtained from Robinson (1967).

215

SU'iRCsUTIH: ZpRO (LX.X)
TYRE DDU?LF X
Dl‘flé‘fiIO-‘J Y(LX)

IF (Lx .gt, 0) RETURN
X(I) = 0.00
RtTUL’N

EN’)

 

216

Subroutine Description: ADD

 

Purpose

To perform matrix addition.

Usage

The calling sequence is:

CALL ADD (NR, NC, x, Y, z, CNST)

RN--number of rows in the matrices to be added.
NC--number of columns in the matrices to be added.
X--matrix to be added.

Y--matrix to be added.

Z--matrix sum of X and Y.

CNST--a multiplicitive constant for Y

Method
Given the NR by NC matrices, X and Y, and a constant,

CNST, the NR by NC matrix, Z is obtained by:
Z = X + CNST’Y.

Restrictions

 

Subroutine ADD is general in form, so that it may
be used to perform real matrix addition under most cir-
cumstances. The arrays X, Y, and Z are dummy dimensioned.
Thus, they must be stored in column mode in the calling
routine. The arrays X, Y, and Z, may not contain more
elements than dimensioned in the calling routine. The
arrays X, Y, and Z must all be NR by NC matrices. The
variables X, Y, and Z are specified to be double pre-

cision.

1n

217

SUBROUTINE ADD (NR. NC. X. Y: Z. CAST)
..... MAYRXX ADDITION .....

0*... Z n x . CNST'Y tit't

TYPE DOUBLE X. Y: Z

DIMENSION xINR.Nc), YINR.NC). ZINR.~C)
DO 10 I 3 19 NR

00 10 J 3 10 NC

Z(I.J) 8 X(I.J) * CNST*Y(Io9)

RETURN

END

 

 

 

218

Subroutine Description: REDUCE

 

Purpose

To obtain the principal directions and values of the

O and K tensors by rotation to their principal axes.

Usage

The calling sequence is: . 4

CALL REDUCE (N¢FQ, NAME)

N¢FQ—-number of frequencies at which the electrical
properties were measured. .

NAME-—sample identification.

 

Method

The arbitrary initial direction, L, is read in and
stored on scratch unit 26. Then the measurement frequency,
0 and K tensors, and rms errors are read from scratch unit
27. The c and K data are processed separately.

The iteration cut-off value, CTF, for the tensor,

T, is obtained by:

 

The maximum principal direction for the symmetric second-
rank tensor, T, is obtained by inputting CTF and T into
subroutine MAXCF. The minimum principal direction is

obtained by taking the inverse of T, calculating a

219

new cut-off value, based on its main diagonal and in-
putting it to subroutine MAXCF.

The intermediate principal direction is obtained by
taking the cross product of the maximum and minimum
principal directions.

These principal directions are used as in equations

3.5 and 6.13.

This procedure is repeated for 0 and K at each frequency

and the results printed out, along with the rms errors.

Restrictions

 

Subroutine REDUCE was written specifically for the
problem described in Chapters II, IV, and VI of the text.
It will have to be rewritten if used for any other pur-
pose. The variables T, TIV, L, L1, and AT are Specified
as double precision. The variables T, TIV, L, AT, AA,
and L1 are multiply dimensioned arrays stored by column
mode.

Subroutine REDUCE calls subroutines ¢UTPUT, MAXCF,
INVERT, CR¢SS, and MATMULT. This subroutine reads input
from scratch units 26 and 27. Thus, care must be taken

to see that these tapes are written in exactly the same

manner as they are read.

 

220

The carriage controls in the print formats may
only be valid for the Michigan State University CDC 3600

system.

 

 

221

 

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Spbroutine Description: MAXCF

PUI’EOSG

To determine the maximum principal direction of a

symmetric second-rank tensor.

 

 

Usage

The calling sequence is:

CALL MAXCF (T, L, CLF, Ll)

T--the symmetric, second-rank tensor.

L--direction cosine column vector for an initial

arbitrary direction.

CLF--test cut-off value for the iteration procedure. ,

Ll--the maximum principal direction of T. fig
Method

The method used is the iterative procedure described
in Chapter VI of the text. Starting with an arbitrary
unit direction, L, the unit normal to the representation
surface at the point of intersection of L and the surface
is determined by:

fi‘TT‘¥'%‘TT'

This unit direction is now used to determine the unit
normal to the representation at its point of intersection.
This process is repeated, each time testing the difference
between successive representation surface normals

(Ilfii - fii+1||) against the iteration cut-off value, CLF.

 

225

When this difference falls within CLF, the last Si is
then taken to be the maximum principal direction of the

symmetric second-rank tensor.

Restrictions

 

Subroutine MAXCF is general in nature and can be
used to obtain the maximum principal direction of any
symmetric second-rank tensor.

Variables T, L1, and M are specified to be double
precision, while CLF is single precision. The arrays
T, L1, and L are dummy dimensioned and so must be stored
in column mode in the calling routine. Variable M is
a multiply dimensioned array stored in column mode as a
singly dimensioned array of 20 elements. Subroutine MAXCF

calls subroutine MATMULT.

 

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(.

 

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Subroutine Description: CR¢SS

Purpose

To perform the (vector) cross product.

Usa e
The calling sequence is:
CALL CR¢SS (X)

X--the 3 by 3 array containing all three column .
vectors. 3

 

Method

Given two three-dimension column vectors stored as

 

the first and third columns of a 3 by 3 matrix, X, the

cross product is obtained by:

[x1] x [X2] ' 811 x21 831 ' [le'

 

 

where the resulting column vector is stored as the second

column of X.

Restrictions
Subroutine CR¢88 can be used to perform the cross
product between any two vectors stored as above. The

variable, x, is specified to be double precision.

I

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34

 

 

 

10

11

228

SUBROUTINE CRJSS (X)

..... VEPIOQ (CRDSS) PRODUCT .....
TYPt DUUULP x ,

DIMENSION Y(3.?)

DU 111:1: 5

J = I + 1

[F (J OLE. 5) ”(1 T" 13

J = J - 3

K = I + 2 -

IF (K .LE. 3) 00 T0 11

K = K - 3

“1.2) = IID.1)v><('-’..:3)~ - X(’\oi)*x(vo3’
RETURN

END

 

 

 

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