Dielectric Constant and 

Electrical Resistivity of 

Natural-State Cores

GEOLOGICAL SURVEY BULLETIN 1052-H

This report concerns work done on behalf 

of the U.S. Atomic Energy Commission 

and is published with the permission of 

the Commission

Dielectric Constant and 

Electrical Resistivity of 

Natural-State Cores

By G. V. KELLER and P. H. LICASTAO

EXPERIMENTAL AND THEORETICAL GEOPHYSICS

GEOLOGICAL SURVEY BULLETIN 1052-H

This report concerns work done on behalf 

of the U.S. Atomic Energy Commission 

and is published with the permission of 

the Commission

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON i 1959
UNITED STATES DEPARTMENT OF THE INTERIOR 

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington 25. D.C. - Price 20 cents (paper cover)
CONTENTS

	Page 

Abstract--..-..-_______________________.__________ 257

Introduction... ____________________________________ _ ________ 257

Acknowledgments. _______________________ _ 258

Experimental work_ _______ _____________________ _ 258

Preparation of core material.,__________________________________ 258

Presentation of data for natural-state cores ___________ _ 25S

Measurements on dry cores_-______-___________________ _ 263»

Evaluation of the data in terms of dispersion theories._____ __ 265»

Summary and conclusions_____________________-__-______ 283

Literature cited._______________________________________________ 284

ILLUSTRATIONS

Page 

FIGURE 68. Dielectric constants of natural-state Morrison cores_____ 259

69. Resistivity of natural-state Morrison cores_____________ 260

70. Loss angles measured on natural-state Morrison cores.____ 262

71. Phase angle versus water content at a frequency of 10 kc__ 263

72. Dielectric constants of dried Morrison cores measured in

vacuum. ____________________ _ 264

73. Equivalent circuit for electrode polarization______________ 266

74. Dispersion curves showing effect of electrode-polarization

capacity decaying as an inverse power of the frequency.___ 267

75. Static dielectric constant versus water content.____________ 267

76. Distribution times as a function of the distribution parameter b- 27O

77. Wagner's equation for dispersion of the real dielectric constant 

in a substance with a normal distribution of relaxation 

times. ____________________________________._, 271

78. Equation for dispersion of the imaginary dielectric constant 

in a substance with a normal distribution of relaxation 

times. _________ ___________ ____________ 271

79. Dielectric ratio as a function of distribution parameter and

frequency________________________.._,.......... 272:

80. Example of the graphical determination of Wagner's parameters 6 and z0 -------------___________________ 273

81. Theoretical dielectric constant as a function of distribution

parameter and frequency_____________________________ 274

82. Graphical determination of the maximum dielectric constant

and the parameter k of Wagner's equation______________ 275-

83. Dielectric constants in core Mo-18_---------__-___.__ 275-

nz
IV CONTENTS

Page

FIGURE 84. Equivalent circuit for a dielectric material having more than

one relaxation time_________________________________ 277

85. Dielectric constant for a laminated mixture of quartz and

water. ______________________________________ 278

86. Idealized pore structure_______________________ 278

87. Tracing made from a photomicrograph of a stained thin section, showing the pore structure of a typical sandstone in 

the Salt Wash member_1___________________ 279

88. Number of immobile cations in a typical sandstone of the

Salt Wash member..________________________ 281

TABLES

Pag« 

TABLE 1. Characteristics of the dielectric dispersion of natural-state

cores______________________ ______ ---.___ 276

2. Evaluation of dielectric constant under certain hypothetical

conditions.-..______--__-------_-------_-------_------- 282
EXPERIMENTAL AND THEORETICAL GEOPHYSICS

DIELECTRIC CONSTAN'TAND ELECTRICAL RESISTIVITY 

OF NATURAL-STATE CORPS

By G. V. KELLEB and P. H. LICASTRO

ABSTRACT

As part of a program to obtain basic data on the physical properties of the rock 

in and near uranium-mineralized zones, measurements of electrical resistivity and 

dielectric constant were made on parts of 27 cores from the Morrison formation 

in the Colorado Plateau uranium province. For frequencies between 50 cycles 

and 30 megacycles per second, resistivity ranged from 102 to 1012 ohm-centimeters, 

and the dielectric constant ranged from 4.0 to more than 10fl. The water content 

of the cores seemed to be the controlling factor: the high resistivities were associated with low water content and the high dielectric constants with high water 

content.

INTRODUCTION

The electrical properties of the Morrison formation in the Uravan 

mineral belt of the Colorado Plateau have been measured in place by 

electric logging and in the laboratory on cores with restored water 

saturation (Keller, 1959). The resistivity of different members of 

the Morrison formation in drill holes ranged from 8 to 2,000 ohmmeters. The resistivity of the sandstone members of the Morrison 

was found to be highest in areas that were considered most favorable 

for the occurrence of ore, probably because of the lower water saturation or lower salinity of the water in the favorable areas.

During 1952 a program of natural-state coring was begun by the 

U.S. Geological Survey in the Long Park area (Manger, Gates, and 

Cadigan, 1954) to obtain basic data on physical properties of rocks 

in and near uranium-mineralized zones. Drilling was done using air 

and oil-base drilling mud so that the water content of the cores could 

be preserved as nearly as possible in its original condition. These 

cores offered an opportunity to obtain truly representative values oi 

the resistivity.

Parts of 27 cores of sandstones and mudstones of the Morrison formation, taken from 2 drill holes in the Long Park area, were used in

257
258 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

this study. Both dielectric constant and electrical resistivity were 

measured from frequencies of 50 cycles to 30 megacycles per second.

ACKNOWLEDGMENTS

The work herein reported was done in part under the program 

carried on by the Geological Survey on behalf of the Division of 

Raw Materials, U.S. Atomic Energy Commission. Some of the 

laboratory work was carried out at the Pennsylvania State University, 

where equipment and laboratory facilities were made available by 

the University.

EXPERIMENTAL WORK

PREPARATION OF CORE MATERIAL

For measurements of dielectric constant and electrical resistivity, 

a thin wafer was cut from each of the cores. The plane faces of these 

wafers were coated with conductive silver paint to form a pair of 

electrodes. The capacity and resistance of each piece of core were 

determined by using'the core as one arm of a bridge circuit. A second 

arm contained a capacitor and resistor which were varied to match 

the capacity and resistance of the core. The bridge was operated at 

discrete frequencies from 50 cycles per second to 30 megacycles per 

second.

The core holder consisted of a brass plate placed in contact with 

the bottom of a core and a point electrode on a micrometer adjustment in contact with the top. Use of a conical-point electrode, 

which has practically no capacitance of its own, and wafer-shaped 

cores eliminated the necessity, for Kelvin shielding of the cores and 

reduced considerably the complexity of the bridge adjustment.

PRESENTATION OF DATA FOR NATURAL-STATE CORES

The dielectric constants measured on the 27 natural-state cores are 

presented graphically in figure 68. The data are presented as dielectric ratios: that is, the capacity of a core divided by the capacity 

that an air condenser of the same dimensions would have. In order 

to obtain dielectric constants in cgs units from figure 68, it is necessary to multiply the ratios by the factor 8.85 X10~4 farads per centimeter, the dielectric constant of air.

The values of dielectric constant are strongly dispersed, ranging 

from 4 to more than 106 at the low frequency end of the spectrum. 

At frequencies above 10 megacycles per second, the dielectric constants 

are all less than 80, as might be expected. In cores with very low 

water saturations, such as Nps. 10 and 12, there is almost no variation of dielectric constant with frequency.
DIELECTRIC CONSTANT OP NATURAL-STATE CORES 259

CM »

000 

1NV1SNOO OW10313IQ 3AI1V13U
10

10

10*

I05 I04 10s 

FREQUENCY,IN CYCLES PER SE

FIGURE 69. Resistivity of natural-state Morri
DIELECTRIC CONSTANT OP NATURAL-STATE CORES 251

The results of the resistivity measurements on these same cores 

are presented in figure 69 in cgs units, ohm-centimeters. In the dry 

cores (see, for example, Nos. 10 and 12) there is a linear inverse 

relation between resistivity and frequency over practically the entire 

range of frequencies studied. On the other hand, in the wet cores 

such as Nos. 21 and 23 there is no variation of resistivity with 

frequency.

The dispersion of resistivity and the anomalous behavior of dielectric constant raise a question as to which is the more important 

in controlling current flow. When a voltage is impressed on a material, the current that flows can be considered as consisting of two 

parts: one in phase with the applied voltage, resulting in ohmic 

conductivity; and another in quadrature with the applied voltage, 

termed the displacement current. The ratio of these two currents 

is the tangent of the phase angle between the applied voltage and the 

resultant current and is a measure of the relative importance of resistivity and dielectric constant in determining the response of a 

material in an applied electric field. The ratio of ohmic current to 

displacement current was determined for each of the 27 cores by

tan 8=

where 8=the phase angle

e=the relative dielectric constant of a core 

eo=the dielectric constant of air 

w=the frequency in radians per second 

p=the electrical resistivity

The phase angles are shown graphically in figure 70.

Conduction in the cores studied ranges from predominantly ohmic 

(tan S>1) to predominantly displacement conduction (tan 5>1). 

The cores with the highest dielectric constant are those in which conduction is almost entirely ohmic; and in those with very low dielectric 

constant, conduction is almost entirely by displacement currents.

The transition from displacement conduction at low water content 

to ohmic conduction at high water content is relatively sharp. This 

is shown in figure 71 by the phase-angle data at 10 kilocycles per 

second (kc). At this frequency the transition occurs at a water <x>ntent of 3 percent.. At lower frequencies the transition takes place at 

lower water contents; but even at frequencies as low as 50 cycles per 

second, the quadrature component of conduction was important in 

some of the sandstone cores studied.

493911 69 2
262 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

co in _

o 

o 8

C)

o 

o 

o

C)
DIELECTKIC CONSTANT OF NATURAL-STATE COEES 263

Ul </)

a: 75°

UJ 

Q_

tn

UJ -J 60°

45

< 30°-

15° -

_ j,]

o

_c

o

- S at

I '

7" Predominately

|

u

 o

i 

1t 

 ^ Coarse- 0 

 grained sandstone jtAver^ae sarv

1 1 

«0 /

S° /

1 x

 ' """"""l lo 

0.0001 O.OOI 0.01

 J) U.I «i 1 

A 

A

y°°°

t

o

___ I _____ I

0.1 1.

FRACTIONAL WATER CONTENT

FIGURE 71. Phase angle versus water content at a frequency of 10 kc.

MEASUREMENTS ON DRY CORES

The behavior of resistivity and dielectric constant described in the 

preceding sections seems to be intimately associated with the presence 

of water in the cores. As a further check on this, measurements of 

dielectric constant and resistivity were made on the same group of 

cores after their fluid contents ^had been removed. The oil in the 

cores taken with oil-base mud was removed in a Soxhlet extractor 

with pentane. Most of the water was removed by baking the cores 

at 160°C for 72 hours. The cores were evacuated while still hot, 

resaturated with dry nitrogen, and then stored in a dry nitrogen 

atmosphere. Measurements were later made with the cores in a 

vacuum. This procedure ensured the removal of all water except 

that closely associated with minerals in the cores, and it prevented 

re-entrance of moisture from the atmosphere while measurements 

were being made.

Some dispersion is evident in the dielectric constants of the dry 

cores (fig. 72). The dielectric constants at low frequencies are as 

much as 20 percent more than those at the high frequencies, but the 

tremendous variations observed in the cores in their original state 

are absent. It is apparent that the observed dispersion was entirely 

dependent on the presence of water.

Determinations of resistivity and phase angle were made on these 

dried cores; but as might be expected, the ohmic conductivity is 

practically negligible. Because the in-phase current is such a small
264 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

1NV1SNOO OI«iD313ia 3AliV13y
DIELECTRIC CONSTANT OF NATURAL-STATE CORES 265

part of the total current in the dried cores, only a very low. accuracy 

was achieved in these determinations. It was found, however, that 

all dried cores behaved as did cores number 10 and 12 in their naitural 

state; that is, the resistivity of the dried cores was inversely proportional to the frequency. :

EVALUATION OF THE DATA IN TERMS OF DISPERSION

THEORIES

Examples of dispersion of electrical resistivity and dielectric 

constant are abundant in the literature, though few measurements of 

dielectric constants as high as those observed on some of the cores 

in the present work have been reported. The most pertinent of these 

earlier measurements are those of H. M. Evjen (1948), who estimated 

a dielectric constant of 108 to explain transients observed in a sedimentary section. Dielectric constants as large as 103 or 105 have been 

reported by Fricke (1925, 1926) for certain biological-coil suspensions,, 

by Fricke and Curtis (1935) for kaolin-gelatin suspensions, by Koopa 

(1951) for artifically prepared ferrites, by Von Hippel (1954) for 

bottom-round steak, by Dunlap and Makower (1945) for dehydrated 

carrots, and by Smith-Rose (1934) and Tarkhov (1948) for rocks and 

other earth materials.

Several explanations for such anomalous behavior of dielectric 

materials have been advanced by the various authors. Smith-Rose 

(1934) and Tarkhov (1948) have suggested that the high dielectric 

constants observed on moist rocks are caused by electrode polarization. It is a well-known fact that a pair of metallic electrodes 

immersed in an ionic solution will act as a condenser of large capacity, 

if current is caused to flow between the electrodes. This capacity 

is confined to a narrow zone near the electrode surface and is caused 

in part by a change in concentration near the electrodes during 

current flow, resulting in a change of the normal electrode potential 

in opposition to the direction of current flow. In addition, the inincreased density of cations near the cathode and of anions near the 

anode results in a concentration potential, also opposed to the current. 

flow. For steady-state direct currents, the capacity of a smooth 

electrode surface is of the order of 6 to 12 microfarads per square, 

centimeter. The capacity of metal electrodes with a large specific 

surface area may be much larger, and polarization capacities as large 

as 0.02 farads per square centimeter have been reported (Jaffe and 

Rider, 1952). With an alternating current, these reactions are 

effective only if the ions have time to concentrate at the electrodes 

during each half cycle. It can be shown theoretically that if the 

diffusion potential is the only factor contributing to the polarization 

capacity, then that capacity should decrease as the square root of
266 EXPERIMENTAL AND THEOBETICAL GEOPHYSICS

C 2 R 2 Z( -L + -L P P w

RP

p * '

O i

 -

C.B. JoMJffe,J923

FIGURE 73. Equivalent circuit for electrode polarization. s

frequency. In a study of polarization capacity at intermediate and 

nigh frequencies, Jolliffe (1923) found that the inverse square-root 

relation held good for platinum electrodes. Other investigators have 

found that the capacity dies off less rapidly than the square root of 

frequency, and still others have found an inverse proportionality.

Jolliffe (1923) has suggested that electrode polarization can be 

represented by the equivalent circuit shown in figure 73. The effects 

of electrode polarization are represented by a resistor and by a variable 

capacitor in series and shunted with a second resistor. The capacitance is assumed to decrease exponentially as frequency increases, 

the factor s representing the rate of decrease. Such an equivalent 

circuit would result in an apparent dispersion of the dielectric constants 

as shown by the curves presented in figure 74. For comparison, 

One of the observed dielectric-dispersion curves is shown. It is 

apparent that the observed dielectric constant decreases less rapidly 

with frequency than any of the cases where the dispersion might be the 

result of electrode polarization.

Further evidence that the high dielectric constants are not caused 

by electrode polarization is given by extrapolating dielectric constants 

to zero frequency. The apparent dielectric constant that would 

result from an electrode polarization capacity of 10 microfarads 

per square centimeter is shown as a function of water content by the 

solid curve in figure 75. The observed values of static dielectric 

constant are 2 to 5 orders of magnitude smaller than would be expected 

if the dispersion were caused by electrode polarization. This indicates 

that in the frequency range used in the present work, the effects of 

electrode polarization are relatively unimportant.

Actually, all solid and liquid dielectrics possess abnormal or anomalous properties at low frequencies beyond those that can be explained 

% electrode polarization (Hartshorn, 1926). In addition to the normal charging current and conduction current, such dielectrics exhibit 

aii anomalous charging current when subjected to a steady-state
STATIC DIELECTRIC CONSTANT

P

o o

6

I

m

oo 

ooz ~

H 

m

z

-H

Z 

TJ

& m

<j Z'

1 o

Or~

s 

rn

o 

b
268 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

voltage. When a static voltage is applied to a material with anomalous 

behavior, the current flowing may be expressed as

/<=io+*'.(0-H2 (0 (2) 

where

to = the ohmic current flowing in the medium 

«i(0 the normal capacitative charging current 

«a(0 the anomalous charging current

This anomalous charging current is comparatively large at the instant 

the voltage is applied but decreases rapidly with tune. However in 

some materials an appreciable part of this current may persist for 

months. As the anomalous current is proportional to the applied 

voltage, its characteristics may be expressed by the equation

t2 (0 = C0W(0 (3) 

where

Co= the geometric capacity of a system

S=the applied voltage

0 a proportionality parameter specifying the magnitude of the anomalous

current in a particular material

jF(Z)=the characteristic time function, describing the manner in which the anomalous current decreases with time

The form of F(£) is a matter of some dispute. Many early investigators found that within the limits of their measurements the characteristic time function was a simple power function

F(0=j8t-f«<l (4)

It is obvious, though, that this expression cannot be physically correct, because if one considers zero tune, the current is infinite, and if 

one considers the accumulated charge during an infinite charging 

period, this is also infinite.

In an alternating field the anomalous current will increase both 

the apparent capacity and the loss; the amount of this increase depends 

on the length of a cycle. According to Hartshorn (1926) Schweidler 

has shown that the fractional increase in capacity caused by the 

anomalous current is

coswtdt (5) 

o

where

w=the angular frequency of the alternating field 

An exponential form for F(t)

F (()=&!- (6)

has been suggested on theoretical grounds by Maxwell and by Pellat 

(Hartshorn, 1926). Maxwell arrived at this expression by considering
DIELECTRIC CONSTANT OF NATURAL-STATE CORES 269

the discharge current from a material composed of alternating layers 

of different electrical properties, and Pellat arrived at the same result 

by considering a viscous displacement of charge following the initial 

elastic deformation in an applied field.

Unfortunately, observed data do not ordinarily follow simple 

exponential tune functions. Schweidler has extended Pellat's analysis 

by considering a characteristic tune function consisting of a sum of 

exponential terms

F(0 = Z0»<r ' (7)

where there is an infinite variety of arbitrary coefficients |3n, « .

This equation may be applied in a practical case by specifying a 

law of distribution for these arbitrary factors. Several such laws 

have been suggested (Fuoss and Kirkwood, 1941; Macdonald, 1952; 

and Davidson and Cole, 1950), but the law which is most applicable 

to our data is a log-normal distribution law suggested by Wagner 

(1913). Wagner's equation for the characteristic time function is

TdT (8) 

where

y(T)dT=the sum of the amplitudes of all terms in the range from T to dT, 

with the distribution function replacing the individual amplitude 

parameters /3B of equation 7

T= relaxation time, replacing the reciprocals of the parameters an in equation 7

For a log-normal distribution, the distribution function is

-v^-fr2 *2 7' 

y(T)dT=?0e dz; z=en ± (9)

VT y «

where

TO = the principle value of T, the time constants

6= a parameter expressing how closely the values of the time constants are 

grouped about their principle value

Curves showing the distribution of time constants for various values 

of b are presented in figure 76.

Wagner has shown that with such a distribution of time constants, 

the capacity and phase angle of a material are given by the following 

equations :

AC 7be

493911 59

-fc^acos -z0 u r}

sv / 16 ;; &U C y,,. Jo cosh u

. ACV , 7&e~ &2 'v f °° _6j /3 cosh Vz0u , /inv + -TT Jtan « = - = e *- du (10) 

C / T J° co* U
270 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

where

Curves for this form of dielectric dispersion are presented in figure 77, 

and curves for the variation of phase angle with frequency are presented in figure 78.

An adaptation of a graphical method described by Yager (1936) 

was used to determine the parameters b, 7, and T0 for the dispersion 

data obtained with the natural-state Morrison cores. In this technique a set of curves relating a factor 5 (defined as the ratio of two 

values of dielectric constant, one at 10 times the frequency of the 

other) a factor z=lnT0/TT (T0 is the principal relaxation time, and 

Tr is the relaxation time for the frequency at which 5 is defined) and 

the distribution parameter b are used. These curves are presented 

in figure 79.

In analyzing a particular group of data, the ratio of two dielectric 

constants a decade apart hi frequency is formed; and by using the 

curves in figure 79, a curve relating z and b is constructed. This 

process is repeated with other pairs of dielectric constants taken at 

other frequencies. The final result is a series of curves of z versus b 

for a particular set of dielectric constants, as shown by the example

CO 

UJ

 z.

o

<

X

<

_l

UJ

oc 

u_ 

o

CO

UJ

o

UJ

UJ

oc

0.001 0.01 O.I I 10 100 1000 

FREQUENCY (RELATIVE TO THE PRINCIPAL FREQUENCY)

FrotTBE 76. Distribution of relaxation times as a function of the distribution parameter 6.
DIELECTRIC CONSTANT OP NATURAL-STATE GORES 271

,001 ,01 OJ I 10 100 

FREQUENCY RATIO, eu = UJTO

1000 10,000

FIOTTBE 77. Wagner's equation for dispersion of the real dielectric constant in a substance with a normal

distribution of relaxation times.

o.opi 0.01 0,1 I 10 

FREQUENCY RATIO,

100 1000 10,000

FIGURE 78. Equation for dispersion of the imaginary dielectric constant in a substance with a normal

distribution of relaxation times.
272 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

0.2 0.4 0.6 0.8 

DISTRIBUTION PARAMETER,b

\.Z

FIGUKE 79. Dielectric ratio as a function of distribution parameter and frequency.

in figure 80. The intersection of these trial curves determines the 

correct values of 6 and 20 for the data. The principal relaxation frequency may be obtained from the definition

where

wr =the frequency at which the first dielectric ratio was formed 

«o=the principal relaxation frequency
DIELECTRIC CONSTANT OF NATURAL-STATE CORES 273

0.8

of 

tu

(U

0.6

z 

o

O.4

m

IE H OT

o

0.2

CORE Mo-18

_Ratio taken at 50 and 50O cycles per 

"second, S = 0.272

Ratio taken at 100 and 1000 cycles per 

'second,8-0.252

Ratio taken at 500 and 5000 cycles per 

'second,8 = 0.166 o ^

IS

-4 -2 O 2 4 6

REDUCED FREQUENCY, *=//? £fc

FIGGRS 80. Example of the graphical determination of Wagner's parameters ft and ZQ.

The second stage in the graphical evaluation of the data is the determination of the low-frequency or static value of dielectric constant. 

For this purpose a group of curves showing the relation between b 

(the distribution parameter), 2=Z77to/&)0 (the frequency), and the dielectric constant (expressed as a fraction of the maximum value 

attained at low frequencies) is used. These curves are presented in 

figure 81.

As 6 and z are already known from the analysis described in the 

preceding paragraph, it is a simple matter to determine a theoretical 

value for the dielectric constant from figure 81 for each of the frequencies at which an experimental value was determined. When the 

experimental values are plotted as a function of these theoretical 

values, a straight line relation with a slope equal to the low-frequency 

dielectric constant is obtained provided the data satisfy Wagner's 

equations. An example is shown in figure 82.

This procedure was used to determine the dispersion parameters for 

each of the cores studied. The results are given in table 1.
274 EXPERIMENTAL AND THEORETICAL

0.8

GEOiPHYSICS

0.2 0.4 0.6 0.8 1.0

DISTRIBUTION PARAMETER, b

FIGXTBE 81. Theoretical dielectric constant as a function of distribution parameter and frequency.

1.2

An example of the agreement between the experimental data and 

Wagner's equations is given in figure 83. The observed values of resistivity are plotted in terms of an imaginary dielectric constant, 

defined as

c //=fop

where

tt=the frequency in radians per second

p=the electrical resistivity in ohm-centimeters

«o=the dielectric constant of a vacuum in farads per centimeter

(12)
DIELECTRIC CONSTANT OF NATURAL-STATE CORES 275

(2,500

10,000

in

z 

o 

o

o 7,500

o

U)

hJ.

o 

u

o:

W

V)

m

o 2,500

CORE Mo-18

0.05 0.10 0.15 0.20 O.25 O.3O

REDUCED THEORETICAL DIELECTRIC CONSTANT <t

. Graphical determination of the dielectric constant and the parameter k of

Wagner's equation.

<ff 

O

° 

o 

a.

o

iUJ >to2

iUI *K>'

J0°

Observed real dielectric constant

 

Observed imaginary dielectric constant

Theoretical real dielectric constant

Theoretical imaginary dielectric constant

Theoretical curves are Wagner's equations 

with the following numerical parameters:

fc = 1.35 cycles per second 

b = 0.205 

e°=66,000 

e«= ' 6J6

r 0,000

10 I02 I03 I04 I05 I06 

FREQUENCY, IN CYCLES PER SECOND

FIGURE 83. Dielectric constants in core Mo-18.

to1 1C-8
276 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

The agreement between the real dielectric constant and Wagner's 

equation was excellent in all cases, but the agreement between imaginary dielectric constant (resistivity) and theoretical curves is not 

good.

TABLE 1. Characteristics of the dielectric dispersion of natural-state cores

Core No.

1A

2. __ .... _ .

3 ... ....... .....

4. . ____ ....

5... _ 

6. _ .... . 

7... .--_..- . 

8 _ .... 

9 _ __ . . _

10 - _ ........

11. _____ .. _ ........

12. 

13.... _

14. ___ .... ....... 

15 

16.. 

17.... _ . _ ....

18 _ ..... ____ .

19. 

20.... _ .... . ____ ....

21 

22 .................

23 . __

24...... _ ..... _ 

25... ____ 

26

27 

Water « 

content

0.088

.0089

.0012

.0114

.0031

.00179

.0004

.00103

.0609

.0189

.0106

.0240

.00796

.0340

.0316

.0613

.0180

.0519

.0730

.0545

.0556

.0833

.0415

.135

.0728

.0306

Relaxation 

frequency

61.3

1.75

.254 10 i

1.37

2.60

.87

.68

.338

1.29

1.01

.742

408

11.2

55.3

t <W

.331

1,510.

670.

2,470.

4,589.

9 97

14 9

.308

2.23

Distribution parameter 6

0.24

.26

.16

.24

.295

.20

.20

.275

.20

0

.135

0

.125

.175

.145

.35

.25

. ..205

.24

.60

.50

.76

.72

.25

29

.16

.27

Lowfrequency 

dielectric 

constant

7.0 X10*

8.0 X10>

3.95X10*

1.68X10*

7. 1 X103

76

8.65X10*

2.45X10*

6.0 X10«

32

28

72

1.41X103

2. 10X105

1.94X10'

6.3 X10*

1. 10X10*

1.63X10*

8.4 X10»

4.3 X105

2.88X10*

2,90X10*

1.04X105

3.75X10*

1.96X108

Highfrequency 

dielectric 

constant

10.0

6 0

3.8

3.56

9.0

3.67

4.65

6.1

7.9

4.2

3.6

4.3

3.75

4.0

4.6

9.0

9.0

7.1

5.6

6.7

7.5

5.3

fi 9

6.6

8.3

6.1

6.7

Dielectric 

increaseratio

7.0X18*

1 IRVIfll

104

- 4143X10"

7 a vifi*

21

188

4.0 Xlfl*

7.6 X10*

1.0

8.9

1.0

7.5

18.1

309

2.33X10*

2. 15X10*

8.9 XW

1.97X10»

2.86X10*

1. 12X108

8.1 X10*

4. 17X10*

4.39XW*

1.25X10*

6. 13X10«

2.88X10*

i Determined by U.S. Bureau of Mines, San Francisco, Calif.

Wagner's equations have provided a satisfactory description of 

dispersion of the dielectric constant and resistivity, but the problem 

of constructing a pragmatic model that can explain the physical 

processes involved remains.

Maxwell (1892) has shown that anomalous phenomena can result 

from a particular megascopic structure of the dielectric material. 

If a material is composed of alternating layers of two materials with 

different electrical properties, these layers may be represented by 

the equivalent circuit shown hi figure 84. The increase in low frequency dielectric constant over the high frequency constant for 

such a system is given by

K-- T)T 

where

T1T2

T =RiRt (Ci-\-Ct)l(Ri + Ra) and RI, Rt> d, C2 are the resistances and capacities 

of the alternate layers
 n-i _Len __6n-H i

n+i

00 cosh u

Wagner, 1913, Macdonald, 1952

FIGTOE 84. Equivalent circuit for a dielectric material having more than one relaxation time.

The ratio of the increase has been:plotted as a function of water 

content for a hypothetical rock consisting of layers of quartz and 

water (fig. 85). Although Maxwell's interfacial polarization can 

account for large dielectric constants, these constants are found only 

at very large water contents.

Von Hippel (1954) points out that there is a tacit assumption in 

Maxwell's derivation that the migration of charge carriers (ions) 

through the layers is unimpeded. McCardell (McCardell and 

Winsauer, 1953) has suggested that mudstones- or clay-bearing 

sandstones actually act as ion sieves. If a rock is considered to be a 

semipermeable membrane with fixed negative charges scattered 

through it, then it is-expected that positive ions may flow through 

the rock more readily than negative ions. The characteristics of 

such selectively permeable membraaes have been discussed by 

Sollner (1945).

A typical pore structure in a clastic detrital rock might fee considered as a relatively large, possibly equidimensional void, connected
278 EXPERIMENTAL AND THEORETICAL GEOPHYSICS 

I06| 

I05

o 

o

lO2

10'

10°,

High frequency

001 O.I I 10 

WATER CONTENT, IN PERCENT

FIGURE 86. Dielectric constant for a laminated mixture of quartz and water.

10O

< Connecting pore

Cloy particle Ion accumulation

FIGUBE 86. Idealized pore structure.

Storage pore
DIELECTRIC CONSTANT OF NATURAL-STATE CORES 279

to other such voids by thin fibrous channels. The essential features 

of such a pore structure are shown by the idealized "porelet" in 

figure 86. An example of the actual pore structure in a sandstone of 

the Morrison formation is shown in figure 87.

Clay particles positioned in constrictions in connecting channels 

will effectively block migration of negative ions because of the small 

size of the channels and the force of repulsions from the negative 

charges usually present on the clay particles. In such a situation 

the positive ions are free to move through the rock under the influence 

of an external field, but anions can migrate only until they come in 

contact with one of these selectively permeable pore constrictions. 

As a result there is an accumulation of negative charge as the anions

Black areas show pore 

structures impregnated 

by stained plastic

FIQCBE 87. Treeing made from a photomicrograph of a stained thin section, showing the pore structure 

of a typical sandstone in the Salt Wash member.
280 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

migrate from one side to the other of the free area in a pore. The 

average anion would migrate a distance equal to

i-=M-f" * J*=c D - (14)

where

M-=the mobility of anions

Vi(t) ihe voltage drop across a given pore 

D=the diameter of that pore 

<=the length of time that voltage is applied

The voltage drop across a pore would be a function of time because 

of the counter electromotive force set up by the concentration differences as anions collect at the barriers. The diffusion potential set 

up by the anion concentrations will cause an accumulation of positive 

ions and an associated polarization; but as long as the diffusion 

potential is small compared to the applied voltage drop, the polarization due to cation concentration will be small compared to the polarization of the anions.

If it is assumed that the cations are relatively unaffected by the 

anion concentrations, the charge stored by the anions in one porelet 

will be

q=^I-l (15)

where

I-£=the total quantity of electricity passing through the pore by means of anion 

motion

d-

 yr=the fraction of that motion which is not compensated by motion of cations

in the opposite direction.

The fraction of the total current carried by the anions is a function 

of the ionic mobilities and the relative concentrations of cations and 

anions. The clay particles act as immobile anions, so the number of 

mobile anions will be less than the number of cations. The part of 

the total current carried by anions will be

r M a **o r ftft\ f_== _ _ f (lo)

where

/*+ and j»_=the mobilities of the cations and anions, respectively

a=the concentration of cations

00=the equivalent concentration of the clay anions

Values of a0 measured on a group of typical sandstones of the Morrison 

formation, fully water saturated, are presented in figure 88. :
DIELECTRIC CONSTANT OF NATURAL-STATE CORES 281

30

25 HIU

o 

§20

t 15 

<o

8 

S,o \ % 

e \

\

I__ _______I

100 1000 10,000 100,000 

ION CONCENTRATION IMMOBILIZED IN DOUBLE LAYER, IN PARTS

PER MILLION

FIQDEE 88. Number of immobile cations in a typical sandstone of the Salt Wash member.

The volume of the central part of a pore, as presented in the idealized form in figure 86, would be irD3/8, and the volume of the small 

connecting channels can be called kirD^/S, so that the number n of 

pores in a unit volume of rock having a porosity, <£, is

If the connecting channels of only a fraction, N, of these pore 

structures are blocked by selectively permeable membranes, the 

charge stored in a unit volume will be

Q=-nNq (18)

Under the application of a unit voltage, the capacity of this unit 

volume will be equal to the charge, Q. Ordinarily, most of the 

resistance of a rock is contributed by the small pores because of their 

small cross section and great length. The larger, central pores will 

have a smaller voltage gradient, smaller by a factor, m, because of their 

smaller resistance per unit length.
282 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

Combining all of these considerations, it is possible to estimate the 

dielectric constant of a typical, water-saturated sandstone

Q ~C0~8.85X10-» 

where

C0=the capacity of a cubic centimeter of a vacuum, or 8.85X10"14 farads 

Using equations 14 through 18 to replace Q, we obtain 

9X1013

_ i ^v ) ' ~ : I

=0

(19)

(20)

It is necessary to know how the diffusion potential caused by the 

concentration of anions depends on charging time and pore diameter 

to evaluate the integral precisely. However, in a qualitative approach, the integral may be evaluated as

where

JEr=the externally applied voltage gradient, taken as 1 volt per centimeter for the

purposes of this discussion 

A = & constant depending on t and D

Also, Archie's law (Archie, 1947) may be used to express the current, I, in equation 20 in terms of porosity and water resistivity

T..... (22)

where

p«=the resistivity of the water in the pores 

/S=the fraction of the pores filled with water

Assuming that the pores are fully saturated, equation 20 becomes

- r*v/i/\m T* " *>! /A {(*~"~t*o/ /oo\

 = 9 X1013 ,, . ,.s ^ \ wsr \ (23)

An evaluation of this expression for several hypothetical sets of 

conditions is presented hi table 2.

TABLE 2. Evaluation of dielectric constant under certain hypothetical conditions

t

 -- 

D....... ........

u+ (for Na*) 1

A...-. 

t ....... .........

Parameter

_. _- __ . __..______.ppm-

. . ... . .... ppmsec

Casel

0.10

100

.01

.1

50,000

40,000

67.6X10-8

45.6X10-8

12

20

.1

1

8.05X109

Case 2

0.16

2

,05

.01

20,000

1,000

67.6X10-«

45.6X10-8

34

50

.8

1

1. 19X108

CaseS

0.24

.2

.2

.001

10,000

100

67.6X10-*

45.6X10-*

65

100.

1.0

1

3. 2X10*

i Values expressed as

cm/sec 

volt/cm"
DIELECTRIC CONSTANT OF NATURAL-STATE COKES 283

The influence of any single factor on the dielectric constant cannot 

readily be evaluated because of the qualitative nature of this analysis. 

However, if values which are geologically probable are assumed for 

the various parameters, it is obvious that the existence of a membrane 

effect would lead to high values of static dielectric constant.

One may infer from such considerations that siltstones will probably 

have larger dielectric constants than any other rock types. In shales 

all the negative charges may be immobile, so that no polarization can 

take place; and in clean sandstones, there may be no anion traps to 

cause polarization. Siltstones will provide a large number of anion 

traps and at the same time have a sufficient number of mobile anions 

to make use of these traps.

Other mechanisms for the origin of large polarizations have been 

suggested (Dakhnov and others, 1952). In a metal-bearing rock, 

electrode polarization may take place at every interface between 

metal and electrolytic solution. The sandstones of the Morrison 

formation contain only small percentages of metallic minerals, so this 

mechanism is not of interest. A second phenomenon suggested by 

Dakhnov is the storage of charge by electroosmosis. When a voltage 

is applied across a rock, the water in the pores may be carried along 

with the electric current. Following the cessation of current the 

displaced water will return to its original position, giving rise to an 

electrokinetic potential. As the water in sandstones of the Morrison 

formation is highly saline, it is not likely that an appreciable electrokinetic potential could be generated or that an appreciable charge 

could be stored.

SUMMARY AND CONCLUSIONS

The experiments described hi this report suggest that the electrical 

characteristics and probably the nature of electrical conduction in 

earth materials is considerably more complicated than is commonly 

assumed in geophysical prospecting. Measurements on water-bearing 

sandstones and siltstones from the Morrison formation showed a strong 

dispersion of both resistivity and dielectric constant with frequency. 

The variation in resistivity is most pronounced in rocks with low water 

contents; and large dielectric constants are associated with rocks 

having the higher water contents, such as siltstones and mudstones. 

If a rock is considered to contain selectively permeable membranes, 

as has been suggested to explain natural potentials observed in electric 

logging, it is possible to explain extremely high dielectric constants.

These measurements were made on parts of only 27 cores from 2 

neighboring drill holes in a small area of the Colorado Plateau, but 

there is no reason to believe that these specimens are unique. The 

high dielectric constants observed with the water-bearing mudstones
284 EXPERIMENTAL AND THEORETICAL GEOFHTSICS

(as great as 106) might also be observed with any equally wet mudsfcoue 

or shale. The dispersion of resistivity observed on the sandstones 

may not be so common, as it appears only with water contents of less 

than 3 percent of the rock volume; but there are examples of sandstones 

in desert areas and in oil-wet oil reservoirs where such low saturations 

are the rule rather than the exception. For this reason the consideration of these phenomena hi electrical prospecting should be of general 

interest to geophysicists.

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Dunlap, W. C., and Makower, B., 1945, Radio frequency dielectric properties of 

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 1926, The electric capacity of suspensions with special reference to blood r 

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colloidal solutions: Phys. Rev., v. 47, no. 12, p. 974-975.

Fuoss, R. M., and Kirkwood, J. C., 1941, Electrical properties of solids, VIH^ 

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Jour., v. 63, no. 2, p. 385-394.

Hartshorn, L., 1926, A critical resume of recent work on dielectrics: Inst. Elec. 

- Engineers Jour., v. 64, p. 1152-1190.

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some semi-conductors at audiofrequencies: Phys. Rev., v. 83, No. 1, p. 121 

124.

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v. 198, p. 41-50.

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DIELECTRIC CONSTANT OF NATURAL-STATE CORES 285

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Smith-Rose, R. L., 1934, Electrical measurements on soil with alternating currents: Inst. Elec. Engineers Jour., v. 75, p. 221-237.

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o