OF THE
COLORADO SCHOOL OF MINES
Electrical Prospecting for Oil
George V. Keller
VOLUME 63, NUMBER 2
APRIL 1968
4
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Price X0.00
Professor of Geophysics
Colorado School of Mines
GEORGE V. KELLER
by
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CopyriahtO' 1968 b}~ the Colorado School
of ~~Iines. Printed in the United States
of America. Sherman W.Spear, Director of
Publications.
Number 2
Orlo E. Childs
President
aF
o~'~~ "S~
~
~N
eo~oRAOo
ELECT}~ICAL P&OSPECTII_~G FOR OIL
April 196$
Published quarterly at Golden; Colorado v0-101
Second-class posta~>~e paid at Golden
Colorado
Volume 63
QUARTERLY OF THE
COLOII~.ADO SCHOOL OF MINES
ELI
Three areas are covered: the matter of electrical properties of rocks in
and around oil fields, the basic electromagnetic theory for current flow in the
ground, and practical design considerations for a variety of electrical methods
that mibht be used in prospecting. In no case, is the discussion complete.
Rather, for more complete coverage, we refer the reader to one of a number
of texts which have been published on electrical prospectinb durinb the past
few years. For a beneral discussion of electrical prospectinb methods, we
refer you to "Electrical Methods in Geophysical Prospecting," by G. V.
Keller and I'. C. I'rischknecht, published by Perbamon Press, Oxford. For
more detailed coverabe of the direct current methods, I recommend three
books published durinb 1966: "Interpretation of Resistivity Data" by
Robert G. Van Nostrand and Kenneth L. Cook, published by the U. S. Geolobical Survey as Professional Paper X99; "Dipole Methods for Measurin
Earth Conductivity" by L. M. Al'pin and others, published by Consultants
Bureau, New York; and "Principles of Direct Current Resistivity Prospecting" by Geza Kunetz, published by Gebruder Borntrae~er, Berlin. Coverabe of electromagnetic prospecting.; methods is more sparse, but I recommend Part III of "Interpretation 'Theory in Applied Geophysics" by F. S.
Grant and G. F. West, published by McGraw-Hill in 1965, "Electroma;netic
Depth Soundings" by L. L. Vanyan and others, published by Consultants
Bureau in 1967, and volume 2 of "Mininn Geophysics" published by the
avenues for research.
This monobraph is a collection of notes on the topics covered in a
graduate course, "Special Problems in Electrical Prospecting Methods," which
has been offered at the Colorado School of Mines each spring semester for the
past six years (1961-1966). Generally, the problems chosen for consideration
have been related to the use of electrical prospectinb methods in a layered
environment, such as one visualizes in a sedimentary basin. Our interest has
been directed toward the possibility of applyinb electrical prospecting methods
in the search for oil, inasmuch as discovezy of new fields, particularly in the
United States, is becomin; more difTicult and more costly using the methods
which have been in use for several decades. Inasmuch as the consideration
of each phase was terminated abruptly at the end of a semester, the material
covered tends to be inconclusive and disjointed. However, this monograph is
being offered to stimulate interest in the possibility of usin electrical prospecting methods in the search for oil, and to point out a number of possible
1OREWORD
lv
Society of Exploration Geophysicists in 1)67. Additional references to the
journal literature are included in the body of this monobraph.
The topics covered here represent an interchange of ideas with the
members of the class, and especially from ideas nurtured by my colleagues
J. I. Pritchard, Hans A. Meinardus, R. L. Gray, J. J• Jacobson, and N.
Harthill. I am also grateful to the Colorado School of Mines foundation,
which provided financial support for some of the digitized electric log studies
and a field program during the summer of 1966 when some of the design
ideas described here were tested, and to 1VIrs. Ona Steinke, who typed the
manuscript.
-- -- -
...
-------- ------ iu
79
-------- --14~
1~0
162
Evaluation by series expansion _____.__ ._________
v
---------161
__._____-_--
---- ----- ---14.6
General properties of the I-Iankel transform _______
The IIankel transform
The K and Q functions for thin layers __
The Q and K functions
Asymptotic behavior for small o, ____.____ --- ------------ — ------ —145
Asymptotic behavior far• small m ____
Behavior of the R and R"' functions ______._____
A uniform earCh and the problem of defining apparent resistivity ____._122
Solution of lZaxwell's equations for a vertical axis dipole source___117
Solution of Maxwell's equations for a short ;rounded wire source_._ 95
Solution of the boundary value problem __.____ __________________....____---_- 93
Part II: "Theory of Electrical Sounding __-----.._--__--_---_--.._-----.----------- v9
Properties oI oil .fields ---- ___------. -----______--------------------
Electrical properties of the weathered layer ..___ _-____--_-_-_-_-____-- 73
Electrical properties of other rock sequences _.____. _______________ 7
Electrical j~roperties of the Paleozoic to Cenozoic section of the
llenver Basin and High Plains area __.. ______.._______..______..______ 52
Electrical properties of the Paleozoic to Cenozoic section of the
Colorado Plateaus ------_ _._----------- _-------------------.._---------- 46
Electrical properties of the Mesozoio-Cenozoic sedimentary se36
quence of the east Gulf Coast area ____._____________----__
Electrical properties of sedimentary rocks ______..________.._______.__---__--.-- 11
Part I: The Properties of Oil fields and Their Environment _________--__ 11
Introduction..— --- - ------ ---- -------- ---- ---- ----------------------------- 1
Foreword ---
TA}~LL OF CONTENTS
169
________189
____211
_____...2-17
Summary and Conclusions—Strategy - ---- - - --- --------------______263
Geologic noise and precision of measurements _ ___________..____.__________252
Current requirements in AC methods __._________._
Current requirements for zero-frequency methods ________________._________230
Noise above 1 cPs --------------------.- -- --- ------- - --- ----- --- 220
Noise below 1 cPs -------------.--------------------223
Noise considerations and power requirements ____________________________218
Depth of probinb with electromagnetic methods _______________. _
Comparison of arrays used in direct-current resistivity surveys ______.__189
Part III: Design Criteria for Electrical Surveyinb _______ _.
An alternate approach to the zero frequency case ______....______._...___________._173
Evaluation using numerical quadrature ___________ _____________.....______.___170
Evaluation usin ortho;onal polynomial approximation
1
when the Schlumberger Company was engaged to work for the Roxana Petroleum
As far as the use of resistivity methods in oil work is coneer'ned, the Schlumberger
Company has proved tl~e practicability of the method in structural studies since 1921.
Work was then begun in the Pechelbron oil region and continued until 1926. In
Roumania, conunerci<~1 work was carried out, among others, for the Steaua Romana in
1923-26. Salt domes were located in the Alsace region in 1926-1927.
Resistivity prospecting for oil structin~es was begun in the United States in 1925,
And later in the paper
—the interest in the possibilities of electrical pr~ispecting leas been aroused again of
late, due to the perfection of the resistivity end PDR method—. Tl~e object of the
writer is to give a summary of the whole field of the resistivity methods, with particular
reference to tlic recent developments--.
Reiland (1932) believed that structure might be mapped effectively,
but also considered that it might be possible to detect the resistivity anomaly
associated directly with oil saturation. In his abstract, he states:
ness in prospecting for oil.
At the present time electrical methods of prospecting for oil seem to he in disrepute.
This is partly due to cost of electrical surveys as compared with other geophysical
methods and partly due to the failure of the extravagant claims made for the process
to materialise. However, the electrical method of prospectinn for oil cannot he
forgotten because it is one of two prominent geophysical methods in which it is
possible to control the field being ernployed. Improvements in methods of interE~retation and in field techniques should dive electrical iuethods a definite field of useful-
And in their final parabraph:
The depth which can be obtained reliably in electrical prospecting is a much debated
question. Up to the present time (1932), the depth at which structures have been.
reliably mapped by electrical surveys has not exceeded 1,500 feet.
also say:
The concept of using electrical prospecting methods in the search for
oil is by no means novel. Consideration has been liven to the use of electrical methods from the late 20's to the present, with the intensity of interest
varying markedly from time to time. To place the present work in proper
perspective, it is perhaps wise to quote some of the early contributions to the
topic of electrical oil finding. Peters and Bardeen (1932) reviewed early
attempts to use electrical prospectinb methods, concluding that they mi;ht
best be used in mappinb structural relief of electrical marker horizons. They
INTRODUCTION
QU9PT~P,LY OP' THE COLORADO SCHOOL OF MINES
It should be remembered that these comments concernin; less-thanintelliaent applications of electrical methods and the distrust of advocates of
such methods were made on the basis of direct-current resistivity surveys
used during the 1920's, prior to the period when electrical transient and
radio wave methods were widely ballyhooed.
Durinb Che 1930'x, considerable interest was developed in the "Eltran
method," an electromagnetic approach to studying a layered earth, based on
a patent by L. W. Biau (U. S. Patent 1,911,137, issued in 1933). The
Eltran method consisted in the generation of an electromagnetic field with a
current dipole excited with a current pulse, and detected with an electric
dipole situated in line with the source dipole. It was hoped that energy
reflected from boundaries between layers with different conductivities would
be detected on the recorded transient at the receiver in much the same ~vay
that acoustic reflections were detected in the seismic reflection techniques.
The method aroused considerable interest among oil companies for about 10
years. with a series of papers appearing which described results of field
trials (Karcher and McDermott, 1935 _ Statham. 19 6: West. 1930; Hawley, 1938; White, 1939; Klipsch, 1939; Rust, 191.0; and Evjen, 194$).
With all this experimentation, there was remarkably little theoretical consideration of the method reported in the literature. A careful theoretical
evaluation of the Eltran method was not reported until the work of the
Socony Mobii laboratory was published 1 Yost. 1952; Yost and others, 1952:
Orsin~er and Van Nostrand, 1954). It was then apparent that for the conductive rocks normally found in sedimentary basins, the transient response
to impulse excitation contained such low frequencies that it would be difficult to obtain the resolution needed to identify individual reflected
events.
One in whom statistical sense is lackin~~ is likely to extol or condemn a method after
too brief a trial.
The quack and the shy.ter seem to have a strong prediliction for' electrical vestmenis.
An attitude of disfavor toward geoclectric methods has perhaps sprung in con~iderable part fi~om subjective rather than objective sources.
Although some ~eoelectric methods may under favorable conditions detect oil
directly because of its l~igki insulati~ pro~~ei•ty, geoelectric as well as other methods
are efficaceous iii the location of oil only to the extent that they disclose subsurface
structural features which in turn may indicate oil-bearinn structure.
Gish also surve}red the use of electrical methods for oil exploration in
1932, and made the following comments.
Corporation and the Shell C:umpany of California. Tl~c ~~~oi-k for the latter co~itinued
to about 1929.
2
3
Despite the fact that an impressive success ratio has been claimed for the
use of radio-wave methods, the extenC of use has declined markedly in the
past 10 years. The present-day explanation of why the technique may work
on occasions is that in some cases, oil-fields may coincide with near-surface
anomalies in electrical properties, coincidental or not, which give rise to tl~e
characterisCic pattern observed on the intensity decay curve. Accordinb to
Barret (1949), 00.6 percent of the observed patterns interpreted as indicating
oil were proved to he correct by drilling, while all condemned prospects
which were later drilled proved to be nonproductive. Professor Harold ~'I.
iYIooney (195-1~), in a study carried out for the U. 5. Atomic Energy Coznmission, concluded that the success ratio for the method is about one to ten,
or about the same as would he expected from a completely random clrillinb
pattern carried out in a petxoliferous pro~~ince. On the other hand, the high
degree of success reported for condemning prospects is a natural consequence
of the fact that seven out of eight randomly located wells will be diy.
During the late 19 ~0's and early 1950'x, interest in the Eleran method
vas displaced by an intense interest in radio-wave methods. For several years,
many of Che major oil companies experimented with the use of radio-waves
in the 1 to 2megacycle-per-second frequency range. The most common
approach was to make use of a horizontal antenna laid on the ~,~round, with
the field-strength decay curve being measured along a inline radial. Presumably, an oil field at depth ~~-ould lead to a characteristic pattern beinb
superimposed on the deca~~ curve (Barret, 1.91-9~). F~Io~vever, simple computations of radio waves in a normal oil-field environment shoved that skin
depths at radio frequencies were Cens of feet 1t most, and that radio sinnals
reflected from depth would be attenuated so se~rez~el~r that there ti~-ould be no
possibility of the formation of an interference pattern on the decay curve
(Haycock, ~VIadsen and Hurst, 194)). Advocates of the use of radio-~a~ave
methods put forth the argument that at radio frequencies there might be
"windows" where, because of some peculiar behavior of the electrical properties of rocks. attenuation might be much less.
Inasmuch as the electrical properties of rocks were relatively unknown
over the radio frequency range at that time (and are still known only rather
poorly),this argument could not be answered simply. However, several carefully controlled experiments were carz-ied out using transmission paths between surface antennas and antennas in mines or caves (Pritchett, 1952;
McGehee, 1954,j, which showed thaC high attenuation rates do in fact apply
for radio transmission through wet rocks. 1'he intensity of feeling over the
matter of radio transmission through rocks is demonstrated by the len~Chy
and severe discussion of Pritchett's paper (following; Pritchett, 1952).
InTronucTio`~
QUARTERLY OF TIIr COLORADO SCHOOL OF MINES
Since the mid-fifties, there has been some indication of interest in the
use of the magnetotelluric method in oil prospecting. The basis for the
magnetotelluric meChod, which consists of determining •resistivity from
simultaneous observations of variations in the natural electric and magnetic
fields of the earth, was .first described by Ca;niard (193). The frequencies
used are as low as a thousandth of a cycle per second, assurinb penetration
even in the most conductive sedimentar} basin. The advantage seen for the
maanetotelluric method is the supposed sim~~licity of measurement, which
would allow determinations of basin character and confiauraCion, even in
remote areas not easily accessible with seismic equipment. A disadvantage
appears to be that the resistivity profile with depth cannot be determined with
a hibh debree of accuracy.
The experience with electrical procpectin~~, methods in other countries,
where less exotic applications have been developed, has not been as disappointinb as the experience in the United States. The telluric current method
has been used quite effecCively since it was first employed by the Sehlumberaers in the 1930's, first in the french cozninunity and later in the Soviet
Union. Accordinb to d'Erceville and Kunetz (1962j, the French firm
Compa~nie Generale de Geophysique, carried out some 565 crew months of
activity telluric ~rospectina over the period from 1941 to 1955. According
to Berdichevskiy (1965) the number of crews enbabed in telluric current
surveys in the Soviet Union grew from a few experimental crews in 1955 to
79 crews in 1960. At present, a variety of electrical prospectinb ir~ethods are
being used in the Soviet Union in the search for oil, including the telluric
method, the magnetotelluric method, the mannetotelluric profiling method
and the electroma~,netic sounding; method (Alexeyev and others, 1961). In
1965, use had grown to the point where 1'39 crews were enbabed in these
three forms of electrical surveying, in addition to crews enbabed in the more
conventional direct-current curve}'ing techniques. According to Smith
(1962), some 200 crews were enbaaed in electrical prospecting in the Soviet
Union in 1961, with about two-thirds using the direct current methods.
Apparently, the amount of direct-current exploration has been sharply reduced with the introduction of the new electromagnetic techniques, which
provide much more accurate information. According; to Berdichevskiy and
Fomina (1966),the use of various combinations of the new electrical exploration techniques to study basin structure and locate areas most likely to be
profitable prospects for subsequent detailinb with seismic methods has reduced
the cost of exploration by a factor of ten.
Exploration costs are a significant factor in determininb the ultimate
cost of produced petroleum. The money spent on geophysical surveys has
~~
5
Percent of effort
33
20
17
30
$~l53
~'1~57
"~`'33o
X685
Cost
millions
millions
millions
millions
These costs inay be divided into two parts—those involved in exploration
to the point of finding a drilling; location, which are the costs For ~~eolo~y ai~cl
beophysics; and those involved in testinn these locations, which are costs of
land acquisition and wildcaC drilling. It is interesting to note that approximately equal amounts are spent on the two parts. This reflects the fact that after a
certain point, it is cheaper to drill uncertain prospects than to continue exploration to a more definitive conclusion.
'Ihe high cost of exploration and wildcat drillin~~ reflects the inherent
difficulty (one might say impossibility I in locating; oil. All exploration techniques in use today are indirect: they are not designed to detect oil directly,
but only to locate areas which are most favorable for oil occurrence. These
areas are tested by drilling, and if enough areas are tested, enough oil fields
are found to make money. The factors to be considered in assessing; favorability seem to he we11 known, afCer a century of experience. It is kizown, for
example, that marine sedimentary crocks are most favorable, particularly those
laid down in a platform area I a shallow-water basin) rich in marine life.
Under these favor~b(e~ circumstances, the basin would he tectonically quiet, so
that great thickness of organic-rich rock may accumulate without interruption
oeer geolo~~ic time, without loss of organic material by sub-aerial erosion.
Better yet. the basin should rise and fall gently by a few feet, so that sediment
varies in texture from place to place, sometimes being sand, sometimes beinb
mud. This then wi11 allow oil to he separated from water by capillary pressure
differences between fine-trained and coarse-grained rocks. Deposition of sediment n~uat go on for a prolonged period of time, to provide at least 6,000 feet
of sediment thickness, which seems to be the minimum overburden to provide
the pressure needed to starC converting, or~~anic matter to oil. Oil is found at
shallower depths, but as a resulC of later uplift and erosion of overlying materials. Later tectonic activity, such as foldinb and faulCing, may assist in moving oil into reservoirs, but severe tectonic activity may destroy oil.
T}ape of action
Geology
Geophysics
Land acquisition
Wildcat drilling
increased at a yearly rate of about t percent since the late 1940's, and now
stands at 0.6 to 0.7 billion dollars per year in the Free World (the level of
exploration acCiviry in the Soviet bloc is about the same). According to
Newfarmer (1962), in 1959 geophysics accounted for about 20 percent of
the exploration budget, with beoloay, land acquistion and "wildcat" drillin
accounting for the rest of the exploration costs:
IvTRODUCTION
QUAP.T~RLY OF TII~ COLORADO SCHOOL OF iVIIN~S
Thus, a sedimentary basin is considered a candidate for exploration only
if it has adequate thickness of sedimentary rock, and only if these rocks have
not been too severely folded. The bulk of the effort in exploration is the search
for reservoirs in which oil may accumulate, within these basically favorable
basins. Many types of "traps" may be recognized, but they may be grouped in
two general classes—structural and capillary. In a structural trap, oil is
localized as it migrates upward under the influence of bravity through heavier
water until it collects a~aiiist an impermeable barrier. The classic structural
trap is one in which a permeable sandstone has been folded into a gentle arch,
or anticline, so that oil can accumulate at the top of the structure. Other examples ai•e fault traps, where a dipping, permeable bed has been broken by a
fault which brings the truncated permeable bed up against an impermeable
bed. Oil may then migrate up the bed until it is trapped against the fault.
The same sort of truncation occm•s in strati~;raphic traps where a bed may be
truncated b} erosion and then sealed by later deposition of impermeable sediments, followed by emplacement of oil. Almost all exploration is based on
the search for such structures in hopes that when they are found they will be
oil bearing. The principal exploration method in use today is the seismic
method, which uses travel tithes for reflected acoustic waves to map relief of
marker horizons—those which are characterized by an acoustic impedance
contrast so t1~at the}' reflect a detectable portion of the incident acoustic energy.
Usually, these marker horizons are not the beds iu which oil accumulates, but
are nearb}~ beds which ire probably concordant with reservoir beds. Gravity
and magnetic methods are also used to some extent in mappinb relief of the
surface of the denser and more ma~nelic rocks which form the basement
beneath the sediments. These structures may then be continued into the sediments, where oil is found. In the Free World, more than 90 percent of the
beophysical effort is devoted to seismic surveys; less than 10 percent is devoted to gravity and n7agnetic sur~~eys.
In many cases, capillary pi•essui•e drive has been observed to be more important than gravity forces it separating oil from water. Rocks usually are
preferentially water-wet, so oil will move to coarser rocks, being displaced
from finer openings where water is held more strongly. Oil will then gocumulate in zones of porous, permeable rock, independent of structure. Many
oil fields of this type have been discovered accidentally in drilling; for supposed structures. Many combined reservoirs are known—those in which both
structural elevation and capillary pressure segrebation cause oil accumulation.
The abundance of dual-force reservoirs may represent only the fact that exploration is done for structures, and that there is currently no exploration
method available for reservoirs with onl}~ capillary drive.
Thus, at present, exploration is highly indirect; only structures in potential-
E)
7
on which electrical prospecting; is ba~ec3, with particular reference to practical
As a result, overall statistics on the success of wilcleat drillinn are not
particularly meanin~~ful. For a province at its peak, the success ratio may he
as high as 1 in 2, as reported by V. V. I'edinskiy 1 personal communication,
196r) for the Siberian lo~vlai~ds. Similar hinh success raCios were enjoyed
durin~~ early oil booms in Pennsylvania, Oklahoma, and Tesas. On the other
hand, success in wildcat drillin.;~ in mature nil provinces is much lower, perhaps as low as 1 in 150, as speculated by J. C. GriFfiths 1 personal communication, 19671. Accordinn to a survey by the American AssociaCion of Petroleum
Geolo~~ists, duri~ig 1966. one wildcat well in ten was successful.
Even these success ratios tend to be optimistic. Griffiths (1.962) has made
a sC~tistical analysis of the sizes of oil fields and found ghat half of the oil
fields in several major oil provinces in the I~SA have values less than a mi1lion dollars each, while 1 percent have values in excess of a billion dollars each.
Obvious~~~, the cost of developing, the less valuable fields is bor7ie by the few
lar~~e fields. The success ratio in finding lal•~;e fields ~s veiry much less than the
overall success ratio of l iu 10. Accordin~~ to i1L T. Halbouty, president of
the flmerican Association of Petroleum Geoloni~ts in 1966,"—the riuznber of
iieti+~-field tivildcats drilled for each discovery of a ne~v field of sufficient size to
return a profit on the operator's investment has i-an~~~ed between ~6 and ~9
wildcats, Prior to 1950. the ratio was between 2~ and33." Thus, the exploration proI~le~Y~ is severe and becoming more so each year.
In view of the inereasin~ c(ii~iculty with which ne~v reserves of oil ai'e being
found, it seems appropriate aC this tinge to reconsider the possible role of
electrical exploration in a petroleum exploration program. However, to
avoid the pitfalls associated ~~ith the fascination wiCh exotic applications which
promise much but produce Little, it tiaould be wise to review the entire theory
in,;.
2. After ~ period ran~~in~ from a fe~v years to 10 }'ears, tl~e wildcat drilling
success ratio will begin eo decline, as the more promising prospects are
ex~~ausCed and less ~romisin.;~ prospects must he tested.
1. In the early stages of development, the success i•aCio for wildcat drilling
will increase, ~s additional information provided by di-illin~ is used to
improve the interpretation of ~ eoph5%sical surve}~s to reduce risk on <irill-
ha~es as follows:
15' }>etroliferous basins are sought, and then these are cli-ill~d to find the ones
~~hicli leave oil. ~Chis requires careful ~~~ei~;hin~; of risk anainst possible return,
a subject ~~-hich hay been treated by- Kaufman (1963). The hest prospects are
tested firsC; then as more confidence is built up, the less likely prospects are
drilled. "lhis leads to an exploration history for an oil province which be-
I\TPODliC.TIOti
QUAPT~RLY OF THE COLOR9D0 SCHOOL OF 1VIINES
Alekeyev, A n7., Berdichevskiy, ?~4. N., Bezruck, I. A.,
I'omina, V. I., Nickitenko,
K L. Polshkov. 11 K., and V~inyan, L. I., 1967
Application of clech~oma netic
m <•thods in oi] find ~~is eaploratiun in the IiSSFi:
tiVorld Petroleurn Con,. ltli,
Trans., Mexico City 1967, T15evier.
~1'pin, 1.. l'l., and others, 1966, Dipole ~::;~thods for
~neasurin~ cau[I~ conductivity:
1~c~a lurk. Plen~~m Press, 302 p.
Barret, ~~. '1I. 19'1~9A. Explorin;,~~ tl~e cartl~ with radio
wives: World Oil, 9pri1.
--- 19~9h, Advertising brochure entitled "Tl~e Kaduil
lZethod," 33 p.
I~~rrr~~~~c~s
exploration for oil. Considering the present structure of exploration ~eophysics, electrical methods might conceivably be applied in three ways:
1. In the search for structures. Such an application would be in
direct competition with presently used methods, and to be successful, structures
would need to be found more economically than with present methods, or• the
methods would have to be applicable in areas where present methods are ineffective. This is the type of application which is bein; done successfully in
the Soviet Union.
2. In the search for fine litholo~,ic chanties associated
with
litholoaic oil traps. Oil which is localized by chances in the texture of
rock independently of structure may occur in a thick sequence of
rocks whose
properties differ to some extent from those laterally away from the oil field.
Such an environment can be postulated for oil trapped in strand-line
or• bar
sands which represent the boundaz•y between shallow water and deep
water in
the depositional basin. If the shoreline remains relatively fixed
throubh lonb
periods of deposition, a Chick column of rock may have properties
diagnostic
of the shoreline environment. This thick sequence may then
present a reason
able tarbet even i'or electrical methods which have poor resolution.
3. In the recognition of resistivity anomalies
associated darectly
with the presence of oil. 'Che presence of oil in the pore
structure of a
rock, displacing water, may increase the resistivity of that
rock markedly.
Detection of resistive zones caused by oil saturation would provide a
powerful technique in-.oil exploration if it is possible. A direct oilfindin; method
could he considerably more expensi~~e than present seismic
methods and still
provide more economical overall finding by reducinb waste drilling.
In the follo~vin~ three sections, various aspects of electrical
prospecting are
considered. first the electrical properties of oii-bearinn rocks and
the sedimentary sequence in which they are found are treated, primarily
to provide a
basis for estiinatin~ the feasibilit}~ of reco~;nizina structures,
litholo~ic trends
or the resistivity anomaly associated with oil saturation,
usin~~ the theory
developed in the second section. Finally, the requirements for
instrumentation
are discussed in the last section.
a
9
Comm., Contract AT-(4~9-1)-900.
~Vewfarmer, L. R., 7.962, Geopkitisic's share of the expinratiuu ~lullar in the LSA and
Canada: Geophy~ic~, v. ~7, nu. 1, p 113-120.
Orlin cr, A, and t' in \ost~ find Pi.. 1914, A field eealuation of [fir electruma~netic
~~flection method: (;eupli~_ic~, ~. 19, no. 3, p. ~~1~78-489.
PetcrS, L. J., and L'~udccn, Jolui, 1932, Sumc i~liccts of ~1r,cUic~i1 prosp~ctin~~ applied
in locating oil structures: Suc. Pe:trolc~irn Gcop1~~- ic~5 l~riui~, v. 2, 11~ii~L, p. 1-Z0.
Pritchett, a/. C.. 192, ~Attenuatioii of rxtlio fre~~ucncy wa~~es tl~rouh the earili:
Geopliy~ics, v. ]7, nu. 2, li. 193-217.
247-257.
Haycock, 0.(,!bladncn, ~~. C., and Horst, S. K., 1949, Prop~ination of electromagnetic
waves in t firth: (F ophysic~, ~~. 14, no. 2, p. 162-171.
Reiland, C. A., 1932, Advances in technique end application of resistivity and potential-drop-ratio methods in oil prospecting: Am. Assoc. Petrolewn Geologists Bull.,
v. 16, no. 12, p. 1260-1336.
Karcher, J. L., ancL ~Icllermott, E., 7.93 , llecp electrical prospectinn: Geoph~ ;acs,
v. 19, no. 1, p. 64-77.
Kaufman, G b1., 1963, Statistical decision xnd related tecf~niques in oil and has
exploration: ~n~lewood Cliffs, A.J., Prentice-H~Il, 307 p.
Keller, G V., and Friscl~knecht, F. C., 1966, Electrical inethuds in geophysical
prospecting: Oxford, Per~amon Press, 526 p.
(;copl~y~ics, v.
Klipsch, P. W., 1939, Recent developments in I:ltran prospectin
4 no. 4, p. 283-291.
Kunet~, Geza, 1966, Principles of direct-current resistivity prosp~ctin~~: Perlin, Cebruder 13orntragei', 103 p.
McGehee:, F. b~l., Jr., 1954, Propagati~,~n of radio frequency entr y through the earth:
Geopf~ysics, v. 19, no. 3, p. 459-417.
Mooney, H. h1., 19~~~, The status of (non-direct-current) elecu'ical e~ploratiun ~k'itli
special reference ro uranium prospecting: final report to U.S. Atmuic Energy
1952, 1~ ute on the radio-trausmission demonsh'ation at Grand Saline, Texxs:
Geophysics, v. 17, no. 3, p. X44-549.
Berdich~vskiy ~I. N., 1965, ElECtrical prospecting with the telluric current method:
Colorado School Mines Quart., v. 60, no. 1, 216 p.
Berdichevskiy, 1~1. N., and ~Fomina, V. ?., 1966, Routine application of new methods
for structural electrical proepectin,: Exploration Geopl~~~ics, v. 9~7 (English
trains., I'rikltidnaya Geofizika, New York, Plenum Press)
Ca:;niard, L., 1953, Basic theor}~ of tl~e ma~nerotelluric method of geophysical prospeeting: Gcophysic~, v. 18, no. 3, p. 605-635.
d`Lrceville, I., ~u~d Kunetz, G., 1962, Tl~e effect of a Exult on the earth's clectroinagnetic field: Geophysics, v. 27, nu. ~, p. 627-650.
Evjen, H. 1L. 1948. `Theory and practice of low -frequency electromagnetic prospecting:
Geophysics, v. 13, no. 4, p. 584-594.
Gish, 0. I~I., 1932, Use of ~~eoelectric methods in search for oil: Am. Assoc. Petroleum
Geologists Bull., v. 16, no. 12, p. 1331-1348.
Grant, F. S., and West, G. F., 1965, Interpretation theory in applied ~;eopl~ysics:
New York, McGraw-Hill, 583 p.
Griffiths, 1. C., 1962, frequency clistnbutions of some natural resource matcnals:
Tech. Conf. on Petroleam Produ~t5, 23rd, Sept. 2h-28, Yennsylvanii State L:niv.
Min. Tnd. ~xp. Sta. Circ. 63, p. 174-198.
Hawley, Y. F., 1938, I~ransients in electrical prospecting: Geopl~}sic;~, v. 3, no. 3, p.
IiVTRODUCTION
QuART~rr.Y or Tx~ Cor,o~Ano ScxooL of VIiNFs
SmitL, A'. J.. 1962, Geoph}~sical activity in 1961: Geophysics, v. 27, no. 6, p. 859-886.
St<ithani, L., 1936, Electric earth transients in geol>hysi~al prospecting: Geophysics,
v. 1, no. 2, p. 271-2~7.
Van \'usUand, li. G., x~d Cook, K. L.. 196f>, Interpretation of resistivity data: U.
S.
Ceul. Surre}~ Prut. Papci' 499, ~/ashii~e~tun, D.C.. L'. S. Gov. Pi'intin~ Office, 310 p.
~'an~'an, L. L., and otl~er~, 1967, ~lech~omas~nrtic depth soundin
1~'ew York, Plenum
Press. 312 p.
Ne t, 5. S., 1938, Electrical prospectirz ~vitli non-sinu~uidal alCernating currents:
Geophy ~c~, a. 3, no. L, p. 157-1(4~.
White, G. L.. 193), A note on the rcl,itions o[ uddenly applied DC earCh transients
to pulse response transients: Ceop6~~ics, v. <}, no. 4, p. 279-282.
dust, ~. J., 19 2, "I~he interpretation uE electroma~netic reflection data in
~eopl~ysicxl
esploratiuir—Part I, Genera( tl~eury: Ceoph}°sic, v. 17, tio. 1, ~~. 89-108.
Yost, ~V. .J. Caldwell, li. L.. Beard, C. L., 1~IcClure, C. D., and Skontal, E. N.,
19 2. "Chc interpretation of electromaan~ tic reHectiun dtita in geophysical exploration Part II. Metallic iuodel experiments: Geophysics, v. 17, no. 4, p.
806-826.
Rost, W. :l1., Jr., 1940, 1'}~pical electrical pruspectiu~ methods: Geopl~ysic~, v. 5,
no. 3, p. 243-249.
10
—m
rock- a~ws ~
—n
(1)
~ -frock/I'~w - a.s n~ -m
~2~
~l
structures. In the presence of water, these minerals electrolyze, adding their ad-
in table 1.
All rocks have appreciable ion exchange capacity—Chat is, a supply of
ions adsorbed more or less loosely on mineral surfaces or within mineral
The variation of formation factor with porosity, for complete water saturation,
is shown graphically in figure 1.
Typical expressions of Archie's law for various litholobies are summarized
a measure of porosity
inhere p,.~,,.~; is the bulk resistivity of a rock, usually measured in 1VIK5 units,
ohmmeters; p,,. is the resistivity of the water fi11in~; in the pore spaces of
that rock; ~ is the volume fraction of pore space in a rock; S is the fraction
of that pore space occupied by water; and a, n, and m are empiricaIly determined parameters which seem to depend on the texture of the rock.
Archie's equation indicates that the bulk resistivity of a rock is proportional
to the resistivity of the water contained in it; this has led to the use of the
ratio of rock resistivity to water resistivity, termed the formation factor, as
relationship:
The relationship between rock resistivity, water content, and texture has
been studied in great detail over the past quarter century by well-lo; analysts,
with the result that the }~i•obable resistivity of a rock in a marine sedimentary
sequence can be estimated with a high de~~ree of reliability. It is Generally
accepted that Archie's law 1 Archie, 19-1.2) inay be used to describe this
ELECTRICAL PROPEPTIF.5 OF SEDIMENTARY ROCILS
In discu~sin~; the feasibility of locatii~n oil fields using electrical exploration
methods, we need to describe t~~e properties not onl~~ of oil fields, which are Che
tar~~et, but also of their ens-ironment, which constitutes the background anainst
which this tar~eC must be seen.
THL PROI'I~:R"CIEs OF OIL rIF.LllS Atill 'THEIR E~VIRO~`'IE~T
PART I
3
100
5
RATIO OF ROCK RESISTIVITY TO WATER
RESISTIVITY
~2
4
1000
QU9RTERLY OF THE COLORADO SCHOOL OF MINES
sorbed ions to solution, though with reduced mobility. These ions do not
move with the water when a sample is extracted for anaylsis, and so are not
seen as part of the salinit5> of an extracted sample. They may add significantly
to the conductivity of the pore water if the free salinity of that water is low,
but if the normal salinity is high, the added salinity may have only a negligible
effect on the water conductivity.
The usual connate water in a petroliferous sequence of sedimentary rocks is
quite saline. However, in many places, the section over an oil field may have
continental sedimentary rocks which are saeurated by much less saline waters,
and in such rocks, salinity added by desorption from minerals may be sianificant. The added salinity from adsorbed salts will limit the maximum
resistivity a rock may assume, as the free salinity is decreased. The change
in resistivity of pore water with channe in free salinity is indicated by the
curves in figure 2, for various assumed cation exchange capacities in a rock.
Ftc;oaE 1. —The relationship between rock resistivity, water resistivity and
water
content_ given by Archie's law for various types of rocks. Curves 1, 2 and
3 represent clastic detrital rocks of progressively older aces, as indicated
in table 1; curve 4 represents porous volcanic rocks, and curve 5
represents low-porosity rocks in which fractures play an important role
in the conduction of electricity.
•~
r-
FRACTIONAL
POROSITY
12
l~
Our primary concern here, however, is not with the detailed variation of
rock resistivity as a function of water salinity and content, inasmuch as such
information is not easily available in sufficient quantity to permit a valid description of the whole sedimentary section. We wish to know reliable average
values for the resistivity of thick portions of the sedimentary column, so that
we may evaluate the effectiveness of various surface-based electrical measuring
techniques. A wealth of information about electrical properties of oil-bearing
sequences of rock is available from electrical well logs, with hundreds of
thousands of logs beinb available in well log libraries and oil company files.
Utilization of data contained on electrical well logs has been hindered by
the volume of data which exists, even on a sinnle lob. The usual method used
in interpretinb well lobs has been a detailed analysis of zones of potential interest, with the rest of the lob; being; ignored, except for noting the depths to
formation toes. With the availability of high-speed computers, such deter-
—i.~a
Pr/P.~~ = 1.4. ~
rocks, where joint porosity is important:
5. Rocks with less than ~~ percent porosity, such as ibneous and metamorphic
—~.4:~
PtIP~~ = 3.5 ~
'~~. Highly porous volcanic rocks, with a porosity in the range from 20 to 30
percent:
prlP~~ = 0.62 ~—i.s:~
3. Well cemented sedimentary rocks with a porosity range from S Co 25 percent, usually Paleozoic in Abe:
pt/P~~~ = 0.62 ~—~.7~
2. l~Toderately well cemented sedimentary rocks, includinb sandstones and
limestones, with a porosity range from to to 35 percent, usually Mesozoic in
age:
—i.3~
Pc/P ~,~ = O.bB ~
1. Weakly cemented detrital rocks, such as sand, sandstone. and some lime
sands, with a porosity ranbe from 25 to 45 percent, usually Tertiary in aae:
TABLE 1. — Archie's law expressions for various rock types
These curves stress the idea that fresh-water saturated rocks need not be nearly
so resistant as world be predicCed by Archie's law, if values for water resistivity
are determined from extracted water samples.
PROPERTIES OF OIL FIELDS AND TI~IF,IR ENVIRONMENT
0.9
10 102 103 104
BULK RESISTIVITY OF
WATER, ohm -meters
I
Si/tstone
Sandstone
floe-ing vertically through the length of the column, the transverse resistance,
These pai•aineters are defined by considering the resistance to current
flo~vin~ either vertically or horizontally throubh the column. For current
3.
minations have been extended to larger portions of tl~e well section, but only
rarely have interpreters been interested in the resistivity recorded on such
lobs per se.
"Clie large number of individual Dyers apparent on an electric log cannot
be resolered Fvith any of the surface-based electrical prospecting methods, and
so, it is necessary to devise a realistic way of defining the average resistivity
of an assembla~~e of relatively thin beds. It has been shown (Schlumbezber
and others, 1934) that the average electrical properties for a finely I~yered
sequence inay be described with a set of 5 parameters, these parameters beinb
defined in terms of a column of rock one meter square cut from the sequence
of layers. This column consists of m horizontal beds. each wit~i its own
characteristic resistivity, p;, and thickness, t;, as shown schematically in figure
I~icu~te 2.— Relationship between tl~e resistivity of water in place in the
pore sti-uctui•e
of a rock, and the resistivity of the same water measured in bulk.
Intersction between the pore water end the solids of the rock cause the
two values to differ, with the departure bein~~ greatest at low water salinities, and for the finest rocks.
0.1
10
102
103
104
No interoction
QuarT~~LY~ o~ ~rfrr Cor.oitn~o ScirooL of Ml~~rs
PORE-WATER
RESISTIVITY
1=~
average
I~zcuaE 3.— Column from a ]a}~ered sequence of rocks, used in defining
longituc{inal resistivity, avcra7e [reinsverse resistivity and anisotropy arisin~; from layering.
~ P,t;
~ t.
~~~
(s)
~6~
Thus. an assernbla~e of thin layers, each of which is isotropic, will appear
to he anisotropic when considered as a macroscopically uniform medium.
This type of anisotropy might be termed macro-anisotropy. It is also conceivable that the individual layers might be anisotropic on a microscopic
scale as a result of soiree preferential orientation of brains. If the direction of
maximum conduction is parallel to the bedding planes, as would normally be
the case, it can be seen from the defining equations for the average electric
x
Linless the resistivities of the individual la}>ers are all exactly the same, the
lonbitudiral resistivity is smaller than the transverse resistivity. This dependence of resistivity on the dix•eceion of current flow constitutes anisotropy.
The coefficient of anisotropy for a layered sequence of rocks is defined as:
/~i
1Jt = s = ~t,'
Again by assuming the longitudinal conductance, S, applies to a column
which is macroscopically uniform, we can calculate an average lon~a~turlinal
resistivity, pi:
~ =t
S = ~ p,
(H being the total thickness of the assemblabe of fine layers).
F'or current Ilowina laterally throw h the column, the longita~di~aal resistance
is that of each of the layers considered to be connected in parallel. With parallel circuits, it is more convenient to talk about the conductance, which is
the reciprocal of the resistance:
~tY
T
H
If the bed resistivities and thickness are biven in ohm-meters and meters,
respectively, the transverse resistance is expressed in ohms.
By assuming that the transverse resistance applies to a column which is
macroscopically uniform, we can calculate the averabe transverse resistivity,
pt,., which is seen by current flowin; vertically through the column:
T- ~I~~~~
(3)
QUARTERLY OI' TIIE COLOFADO SCHOOL OF 1~~IN~S
T, is the sum of the resistances met in each of the individual layers:
16
17
Gaussian distribution curve shown in figure r. defined by the expression:
A well known e~.ample of a probability densiCy curve is the bell-shaped
ject to statistical uncertainty.
Experience has sho~~=n that it is prefera}>le to compile such histograms with
lo,~arithmicall}' incremented class intervals, so that the tapering off of probabiiity densities as higher resistivity classes are considered may he reduced.
Also, the data may be used to compile a cumulative frequency of occurrence
curve for resistivity values, as shown in fibure ~ (for the same well from
Adams County', Mississippi). The points on a cumulative frequency of occurrence curve may then be fitted with a smooch curve, and Chis curve nzay be
differentiated to provide a continuous probability density curve such as the
one shoti~n in figure 6. The resulting curve is less sensitive in appearance to
Che samplin~~~ than is the histonram presentation.
A probability density curve is a generalized form of a histogram; it is a
plot of the z•atio of the number of samples per class to class width, as the class
width is reduced to zero in the limit. Avery large number of samples may
be required to establish the shape of the curve in detail. Less than infinite
7iumbers cif samples provide an estimate of the probability density curve sub-
Evaluation of the average defined in equations 3-7 by scaling the thickness and resistivity of each layer apparent on an electric lob; is a more tedious
procedure than is actually necessary. It should be noted that the summations
are not ordered—all beds with the same resistivity may be grouped tobether
in the summations and treated as a single layer with the cumulative thickness
of the indiaidual la}'ers. "Thus, one approach to synthesizing the highly detailed information contained on an electric log is the use of a probability
density curve for values of resistivity sampled randomly from the lob. Such
probability densit~~ curs-ee are histograms for the frequency of occurrence for
resistivity values within a series of ranges, as shown in the examples of values
of resistivity read from ari induction lob; from a section of Pleistocene sedimenCs penetrated by a well in Adams Countq, Mississippi (fig. 4,).
properties that the total or general anisotrop}~ will be the product of the
macro-anisotrop}~ and the aeez-a~e micro-anisotropy.
The primary source of data for evaluatinb these average electrical properties
is an electric log. Electric logs have a limited resolution, and may fail to
distinguish between layers which have appreciable thickness. Therefore, the
distinction made between macro-anisotropy and micro-anisotropy is usually a
practical one—macro-anisotrop}> arises from laverin~ coarse enou:h to be
distingushed on electric logs, while micro-anisotropy may be used to describe
both the microscopic anisotropy inherent in a rock and layering anisotropy
from layers too fine to be disCinbuished oii au electric low.
P~orERTZ~s or Oii. FlrLns aiv~ Tx~;ix E~vvzao~M~nT
2
~P
~~)
When Yl.~ resistivity samples from an elECtz-ic lu,~ are
eom~~iled into ~~zobability densit}~ curves, designated as functions G I lo~~ ~p~),
the a~-erage electrical
F ~ ~P~ -f F ~~p) dC~P)
~
where ~r is the standard deviation of Che values fc~r log p.
A cumulative frequenc~T of occurrence curve fora Gaussian distribution,
also s~~own in figure
r, is obtained b5~ integrating the probal~~ility
densit}~ curve from t ie left:
2/~
I~ictitiF: 4.— Linear I~i~to~ram fur values of conductii~it~~
sampled From a FiFF=1~0 induction lob from a Gcell in 11i ~i~si~~pi.
0
100
200
300
400
500
600
700
CONDUCTIVITY, mhos/meter
QUaRT~RLY OI' THE COLORADO SCHOOL OF MINES
FEET OF SECTION
PER CONDUCTIVITY INTERVAL
800
la
0.1
RESISTIVITY, ohm -mefers
I
10
19
~
~ _
for longitudinal conductivity:
0
l
f >~ ~~p p~
dp
~ = ffa G(~p)~p
(11.j
(1oi
~~roperties for the sequence from which the resistivities were sampled ina}~ he
e~~aluated usinn the expressions:
for transverse resistivity:
I~icoitE :i. — Cwutilative fre~~uency of occurrence curve for tlic resistivity values
tttbultUed in tl~e liisto_~ra~» of fi~urc 9~.
10
20
30
40
50
60
70
80
CUMULATIVE Ff~EQUENCY
OF OCCURENCE, /a of total
Pi~on~rTZrs or OiL Fz~l~s ~i~~~ 'lxci~ E~viror;tiz~~~T
RESISTIVITY, ohm- meters
(~
The resistivity density function ~va~ first used as a computational convenienc~e (Keller, 196-1), but they are of intez-est in their o~~-n ri~~ht The resisti~rity density curves have properties of t~~ o types: shape and position on
the
resistivit}r axis. The shape of a resistivity probabilit}' density curve is usually
I ic[;rr; G.--- Generalized hi~t<~~ram or r~:isti~ity probability cicii~ity
curee compiled
For the seuit lo_~ ns tl~e delta in (i~~uc 4. Tlie dat~i z~-ere sorted into
class
intE.i~~als with tividtha iner<~as~n~ clE~one~~tially, rather than linearly, tts in
figure 4~.
1
,~
40
80
120
20C
24C
2so r
~
QL~rTExLi of Tx~ Co~or~no Scxoor, of l~Tl~~s
RES3STIVITY PROBABILITY
DENSITY, percent per decade
20
21
~
~
~v
1o~.~er bound
O~ c~455
upper boU~d off' c~o~55
(1'21
x
Cumulative
probobility
F(a)=/ f(x)dx
-~
0
Ftcoite !.- li,~~u~iple~~ ~~f Ifu~ ~~ruhal~ilit}~ d~~n>ily <urc~~ ,ind tlic cinuulali~~~~ ~nr~lialiility
cun~~~ I~,r tlic ~~~ell kn~,~~°n (:au~=tan dis[rihutiun.
Probobility
density
f(x)=(2rr62)~2 exp(-x2/~Z)
C~'AUSSIAN DISTRIBUTION
tchere ~' i~ the fraction ~~f the total number of sample resi~t~t dies t+~hich fa11
~a~ithin the class hounds under consideration.
"I,he inultirnodal ~~a[ure of the pro}~abilit~ densitq ctn-~es results from the
presence of ~e~er~l ~~referred litholozic types in the section. such as shale and
Milestone. for e~~ample. 't he ~~~tiltii~~odal form is charactEristic noC only of
seditnentar~~ enek~. but als~~ of c~th<'r layered sec~uenees o[ rock, including
~
GI\ ~vq JL~~
decade of ~~la~~ width:
not that of ~ normal or lo,~-normal distril~u~ic~n. ,l~lore commonly, the curve
appears to represent a multimodal type of log-normal distribution. Several
values for resistivity apf~ear to occur more frequently in a particular sequence
of rocks than other values, and the sampled re~istivities tend to ~~roup about
these several cenh~~l values.
1 ~ainples of clearly ~nultimodal distributions are shown in figure o, a
probability density cure for 16e Pierre shale from eastern Colorado, and
figure: >, a distribution curve for a sequence of Paleocene anci Eocene limestone aucl <lolo~nite~ beds from the Sirte Bain in Lib}~a• In these density curves,
the probabilit~~ densities are et~n-essecl as the fi~acCion of the total footage per
Pro~~~rTZrs or OiL F'i~:r.i~s -~~i~ Tlirn~ 1.n~~~Iro~~?~rLU•r
2.22
8,g$
Resistivity, ohm - meters
4.44
f~ t~,tntt~. ).
0,5
I
2
10
20
~~~7~~ i~~~~ d Y% O~1~T!-PY1e~@rS
~
~ Hesi~(icity p~~obabilit} den;ii~ curve Ior ~~alue~ of ri ~i~tivit} sampled
fruni an elerciric los in ~i 1900 f~~ut ectiun of i;uceiie Limestone from the
Jh~te f3a~in, Libya,
~V~
r~•
RESISTIVITY PROBABILITY
DENSITY, percent per decade
Ficua~ 8.— Resistivity probability density curve for values of apparent resistivity
measured with a short normal electric lo, in a 4300 foot section of
Pierre Sliule in eastern Colorado.
O ~.I
2
4
G
8
10
percent per 0.05 range in log ~
r~esi5i ivi ry aensiry runction,
23
1~
~
~
o- 0
b0
y~ X11
a_o~
o
r,
~,
9
I /
i
~
j
~
~
'~ '
~~
i ~~
9~ ~
Oft
~~
~i
I
\~
4~
~
`~
~
,~
i
~—
10,000
RESISTIVITY, ohm-meters
1000
,~
a
a--~ ,~
,~
~
~\
\
l ~o ~~ 6GY~ -4/00
/~
q
~~
',~69GY~-10,660
~.Section from 4/00 to 6
sarupled
FtcU~E 10.— frequency of occun-ence histocram for values of iesistivit}frmn an rle.ch'ic .lon run in ei wall in ha~alt nn the (:olumhi~i River
Plate~iu 1 fr~~m .Iacks~~n. ]9681.
5
15
PERCENTAG E OF SAMPLES
IN EACH RESISTIVITY CLASS
recorded re~istivitY (apparent icsistiviti~ 1 <~e~>end~ not only on the resisti~~ity
volcanic rocks. An example of a probability densilti~ curee for a sequence of
volcanic rocks has been given by ,Jackson f l ~)6r l , as shown in figure 10. "This
curve was compiled from an electric lob; run in Rattlesnake I~'o. 1, drilled to
about 10.000 feet in basalC and tufT in the Columbia Ricer Plateau area, near
Yakiml, Washington. The petrographic ~i~nificance of the cur~-e shape is
not clear. but indicates t6aC there were }~referrec3 textures involved in the
fm~mation of these basalt layers.
Resi~ti~~ity densit}~ curves min*ht be of some use in determining; the ratios of
IiCholo~ic components which ~~~i~~e ripe to the various modal peaks. Ho«~ever..
before considering the meaning of cur~~e shapes. tie inu~t consider the degree
of reliability of values for resistivit}- taken from well logs. The accuracy of
a resistivity probability density curve depends nn the sizes of errors of two
t}apes: Chose involved in the ot~i~~inal measurement <~f resistivity during; lo,~~~in~, and those associated with the sampli~~~,~ procedure. In log in., the
Proi~LrlTir-.s of Orr. Fi~L~s ;~~~~v 1'icria E~vz~on~~r~,n~•r
Quni;zl~~zY or Txr Cor_o~ia~o ~c~zoor. or MI~~Ls
of the rock around tl~e ~~~ell, lout aleo on the resistivit}- of Che muc3 in
the well
bore, the well diameter; bed thickness, the degree of flush~n~~ of the rock by
drilling; fluid and the type of lo~;g~in~; device used. Corrections ina~
be made
for each of these effects ~~~hich ire more vi le,s satisfactor}_ 1~ut the ~~ork
iiivolved in correction procec~ui•es is prohibitive if a vei•~~ lar~~~e volume of
data
is to be handled. It appears to he more ~~eeisonable to make use only of
those
types of logs tihich have minimum corrections fur- a given set of conditions
than to aYteinpt to defei~mine the enact i~esisti~~ity of each zone penetrated
b3~ a
well_ The resiativity densit}~ curve obtained ~~-it6out making corrections
will
he lar from exact, but if the errors are consistent, comparison bette~eeu
sets of
resistivity density- curves niay be made with some confidence.
It is therefore necessary insofar as possible to choose the electric lo~.t~inn
curves which come closest to ree~rding the actual reeistivit~ o~ the rock
around
the well. In the examples which follow, three t}pes oI electric lob; ~~~ere
used;
16-inch normal spacinn lo~~s, 61I'-]0 induction lo~~s and laterolo,~s.
The latter•
tsvo types are considered to require virtually no corrections for the
shunting;
effect of the mud column, and to pi•o~~ide cssentiallt tl~e cor~recl rc~istivity
for
beds more than a few we11 diameters thick. Ho~~-ever. reliable induction
and
laterulo ~ s}~stem~ have been available only for the gist decade.. and
iu ma~~y
areas, these t~~pes of Ines may not be available except f~~urn
recentl} drilled
wells. In such areas, spaciiin lobs must be used, and these are su}~ject
to
Larne errors resulting from current shuntiu~ through tl~e mud column, and
the
effect of bed thickness is appreciable even for beds tens of feet thick.
In sections where invasion is ne~;lig~il~le. the s}~orf normal spacing
is preferred to the lone noi•rnal oi• lateral sl~acin~;s, because the cf~ect of
bed thickne~c on the lob is simpler. An exaiuple of the correction
procedure which
may be used with the spacing lobs is ~,>~i~~en in table 2. ~Che left-hand
columns
are the distributions of resistivit~~ values taken at 10-foot intervals
i❑ the
Pierre shale in a well from NIor~a❑ County-, Colorado, at de}~ths between
120
and ~~,4~70 feet. These values were r;rouped in class inter~~als with
the up~~er
bound of each interval being 1ar~;er than tl~e lover bound b} the ratio
L08.
The class mark is the aeome[ric a~-erase of the upper and lower bounds.
On
the right-hand side of the table, the class marks ha~~e been corrected
for the
effect of the mud column (for a mud resistivity of 1.2 ohm- meters),
using the
departure curve shown in figure 1l. "The lon~~itudinal conductivity a~~d
transverse resisti~>ity are then computed for the corrected distribution
curve, and
are found to differ only slightly from the uncorrected values.
See pane 25.
It is apparent that if the mud resistivity is not ~;reatiyr different than
the
crock resistivit}~, the correction for mud resistivity is nol particularly
important.
The effect is greater on the value determined for transverse resistivity
than on
24.
25
1.43
.025
.022
.OS2
.063
.071
.170
.175
.286
.365
240
.~63
.017
.14.0
.140
.017
.008
.008
.001
.319
.171
.171
.021
.006
.227
.243
4.32
4.63
4.63
5.32
5.83
6.30
6.75
.015
.36t
.007
.19~
.007
.197
.001
.025
,pp7
.005
262
.005
2d2
.35r
3288
(~~)
(pt~~)
A =1.08
1.02
1.9%
4~.5fo
0.374. mhos/m
0.357 mhos/m
Coefficient of
anisotropy
1.06
9.5 ~o
Percentabe
difference
1.402 ohm-m
Corrected
.019
.039
.034
.0-18
.053
.025
.051
.042
.011
.010
.069
.163
.167
274
.347
.187
.4.27
.015
.122
.126
1.93
2.04
220
2.38
2.55
2.72
2.89
3.12
3.31
3.52
.O1S
.037
.033
.04.6
.051
.024.
A~,7
.002
.010
.049
Uncorrected
.022
.062
1.67
.006
.005
.374
3.002
(Pty') (~~')
~ =1.06
.025
.051
M~
pM'N N/per'
3288 ohm-m
.036
0
.037
0
.036
.080
A76
.115
.136
.069
.14~
.005
.037
.036
~
4
.074
.037
.037
.004
.ppl
.036
.036
1.000
Corrected
class mark,
N/pM
pM\
resistivity
Lonbitudinal
conductivity
Transverse
Sum
1.45
1.57
1.70
1.83
1,9g
2.13
2.30
2.49
2.69
2.9p
3.13
3.38
3.65
3.9~
426
4.59
4.96
5.35
5,7g
624
6 74
7,27
7.g5
Uncorrected
N
class mark, (.fraction of
samples)
pNI, ohm-m
TASLF 2.-Example of the correction of a resistivity density curve for the
ef~jects of mucl resistivity (depth interval from 120 to 4,470 feet
t:~a the Pierre shale, Fort ll~organ County, Colorado)
PROPERTIES OF OIL FIELDS ANll THEIR ENVIRONMENT
Apparent resistivit
Mud resistivity
~~
the values determined for longitudinal resistivity of the coefficient of anisotropy.
It is not so simple to correct for the effect of limited bed thickness when a
spacing lob is used. In ~n aCtempt to estimate the errors caused
b}' departure
of resistivity values in thin layers, the curves shoti=n in figure 12 were
prepared. A hypothetical secCion consistin, of layers to~ith two resistivities, 1 and
100 ohm-meters, was considered, with the znud resistivity being; 1
ohmmeter also. Only the 16-inch normal spacing curve and an 2-inch well
diameter were considered. High resistivity beds, with a thickness of 4. well
diameters (32 inches) in one case, and 16 well diameters (12S inches) in
another case, were assumed to be distributed uniformly throubh the low
re-
I'rcUaE 11.-- Iiesistivi[y departure curve for correcting resistivity
measured with
the short normal array for the effect of the mud column in the
example
given in the text.
/~~ Line of no
correction
~
'QUART~fiLY OP THT COLORADO SCF3007. OI' ~VIIN~S
Effc-mot of mud
0.~
True resistivit
ud resistivity
10
26
0.1
`Z7
sistivity is dominated by the presence of high resistivity beds in the section.
sistivity rock. The fraction of hi,,h resistivity beds in the section was varied
by changinb the interval between them.
F'or this section, the correct values of longitudinal and transverse resistivity
would lie between 1 and 100 ohm-meters, depending on the ratio of the t~vo
components. The value for longitudinal resistivity is close to 1 ohm-meter until the fraction of hibh resistivity beds is quite large—the value for longitudinal
resisti~~ity is dominated by the more conductive beds in the sequence. On the
other hand, the transverse resistivity increases rapidly with the addition of
even a small fraction of high resistivity beds—the value for transverse re-
Ficaa~ 12. —Tr~n~verse and longitudinal resistivities which would be obtained using.
apparent resistivity values recorded for a sequence of layers with altei~natin~ resistivities of 1 and 100 ohm-meters.
FRACTION OF HIGH RESISTIVITY BEDS IN SECTION
0.01
~•
PROP~RTI~S OF OIL FIIsLDS AND THEIR FNVIRONMfi'_VT
QUARTERLY Or TFIr COLOR9D0 SCHOOL OF MINES
10
Longitudinal Resistivity
~•
AVERAGE BED THICKNESS, EXPRESSED IN HOLE
DIAMETERS
i
~~ _
~~
Transverse
Resistivity
~~ ~
~~--
or 10:1.
F'iceite 13. — tivera~e transverse and lono~i~udinal
resistivities which would be camputed from apparent resistivities me~isured in a
section contxinin~ equal
numbers of beds with two resistivities, the
contrast being either 100:1
w
Q
w
Q
w
r
~•
The departure of apparent resistivity as recorded on the lo;
from true resistivity is most significant in the high resistivity beds. As a
result, the value
for lonbitudinal resistivity is little affected by considering
this departure. On
the other hand, the departure in resistant beds contributes a
large error to the
determination of transverse resistivity from the lob;, with the error beinn
nreater with thinner beds. The error also is lamer when the hiz;h
and low resistivity
beds are present in equal portions than when the section is
made up mostly of
one or the other.
It is reasonable to expect that the error caused by
limited bed thickness
will be more serious if the contrast in resistivities is lar;e.
In order to determine the nature of such a dependence, the curves in figure
13 were prepared.
Here, a fixed ratio of high resistivity and low resistivity
beds was considered,
with each being equally abundant, a condition assumed to
insure a maximum
28
`Z9
mediate in sensitivity to errors, inasmuch as it is the square root of the ratio
2. Induction and laterolobs provide better• compiled resistivities, though the
induction lob is limited to use in sections where the maximum resistivity is
less than 50 or 100 ohrn-meters, and the laterolo~; is limited to use in areas
where there is no sinnificant invasion. These two limitations tend to be complementary, inasmuch as tight rocks not subject to invasion tend to have
higher resistivities.
3. The value for lonnitudinal resistivity is less subject to error than is the
value for transverse resistivity. The coefficient of anisotrop}' may he inter-
ciently tight that invasion is not a problem.
effect from the departure of measured rock resistivity from true resistivity. As
in the ~~revious case, the resistivity of the mud column and of the conductive
beds was taken as 1ohm-meter, but the resistivity of the resistant beds was
varied, as well as the thickness of the layers. Again, the error in longitudinal
resistivity is generally neali~ible, but the transverse resistivity departs seriously from the correct value for• bed thickness less than 30 to 50 well diameters.
The error is considerably az•eater for a contrast of 1O0 to 1 between the resistivities of the two types of beds than for a contrast of 10:1, while the error
in lonbitudinal resistivity does not vary much with resistivity contrast.
The error caused by limited bed thickness is more serious with the spacing
lops of greater dimensions than with the short-normal spacing lon. However,
if there is significant invasion of the rock around the well by mud filtrate, the
short-normal log may give an erroneous value for both the longitudinal resistivity and the transverse resistivity. In some areas, the overall permeability
of the section is low so that invasion is not a serious problem; in other areas,
as for example the Gulf Coast, the entire section may be so permeable that
invasion will seriously affect the resistivity density curves taken from a short
normal log. Typical of the errors which may develop are the effects shown
by the data in figure 14~, which consists of two cumulative frequency of occurrence curves for resistivities sampled from a lob run in a Cretaceous section in Mississippi. The two lo;s used here were the 16-inch normal and the
6FI+4~.0 induction log. The resistivities taken from the induction lob are lower
than those from the short normal lob generally by a factor of 3. This difference may be attributed to the effect of invasion, inasmuch as the drillinbmud filtrate resistivity in this well was approximately 1..30 ohm-meters, considerably higher than the connate water resistivity.
Considerinb these various errors in compiled resistivities, some generalizations may he made:
1. Of the spacin; lobs, the short-normal spacing is preferable, but may be
used with confidence only when none of the recorded resistivities differ from
the mud resistivity by a factor• of more than 10, and when the section is suffi-
PROP~RTI~S OF OIL FIELDS AND ~I'H~IR ENVIRONn4ENT
20
A~3~~ t~ ?S
2D
30 40
50 60 70
80
JO
CUMULATIVE FREQUENCY OF OCGURENCE
X09
o~ ~°g~
of the transverse and longitudinal resistiviCies.
Having considered l ow the compiled resistivity parameters depend on the
errors involved in logging, we may now turd to the relationship between Che
compiled parameters and the ~~eological nature of Che section for which they
are compiled. If a sequence of rocks consists of layers of only two kinds, each
characterized by a sinble resistivity value, the coefficient of rr~acroanisotropy
is a simple function of the ratio of one component to the other, and of the
resistivity contrast between the two components:
Ftctiit~ 14.— Cwnulative frequency of occuucnce curves for valurs of i~_istivity
sampled fruin a (FF~9~0 induction Iu~ and i spurt noruuil electric lob for a
common interval in a well in 1Ii_sissiplii. The dif~erenccr bcriti~een the
two sets of values may be atU~ibute~d to u~-erafl invasion of the section by
drilling mucl filtrate.
•
Mud resistivity, 0.78 ohm-m, BHT
~t;on
►nd~
Snot
N~~m
3to {0 7a~0
QU9RTERLY OF TF3E COI.OR9D0 SCHOOL OF ~TINP:S
Hole diameter, 8 5/e in.
RESISTIVITY
10
3O
~ ~ (~a2+~C~ —a)z+a (t—cx) t (32a(~— oc)~~
/a
(13)
3Z
(l~)
•~
0.1
~~3
10~~
20 ~~~
50 ~~50
(151
~
OC, RATIO OF COMPONENTS
1 %z
2 L p {' z+ ~
Ft~u~te 15.— Variation of the coefficient of anisoh~upy, arising from layerin~> as a
function of the ratio of two components with different resistivities, as
indicated on the curves.
:~]
2
3
4
COEFFICIENT OF
ANISOTROPY
max
as the fraction of siltstone beds with a resistivity p~ = 4.5 ohm-meters in a
shale with resistivity p, = 3.0 ohm-meters is varied. As one would anticipate,
the anisotropy is a minimum tivhen one or the other component is Che only one
present, and the anisotropy is maximum when both components are present in
equal abundances. This maximum coefficienC of anisotropy is
~1 - ~a +1~Ct-a)~
w~iere a is the fraction of the section made up of beds with a resistivity pi,
and (3 is the ratio of resistivities between the two types of rocks, pi~p~>. F'i~;ure
15 shows the variation of the coefficient of anisotropy and of the lonbitudinal
resistivity
~_
PROPERTIES OP' OIL FIELDS A1VD TE3EIR ENVIPONMENT
'~~~
a
A'/P, = a•
FrcuaE 16A. — Variation of tl~e coefficient of anirotropy with the ratio of components,
a, in a ttiro-component system of layers having resistivity contrast,
• t ~~
•••
' 1
2, i2
3, pis
5~ ~i5
10, ~iio
20, ~i~
~,~~50
/d = 100, ~iioo
'
Fic[~et: 16T3.
10
100
0.1
a
~/
Ps/Pt = /3•
Variation of loii~itudinal resistivity with the ratio of comp~inents,
~y, in a two component system of layers lien°inn a resistivity contrast,
0.01
i/In
~3ai/5
I/3
I/2
2
3
5
10
20
Q =50
P~i
3j
REDUCED
ANISOTROPY
10 ~
PROPERTIES OF OIL FII;LDS AND THEIR EnVIRONM~i~T
resistivity• Usinb such a plot, we might he able Co postulate the control exerted
by chanties in lithology or other factors on the electrical properties of a sequence of rocks. One typical type of marine sequence mibht consist of essentially mud or shale, in which there are included varying amounts of
thin sandstone, limestone or evaporite beds. The properties of such a sequence could chan;e laterally either because the relative portions of shale and
other rock types changed, or• because the nature of the second rock type, and
therefore the resistivity contrast, changed. Three curves relating; the reduced coeEFicient of anisotropy and the longitudinal resistivity ire shown in
figure 1l. The parameter for each curve is the resistivity contrast, /3, which
QliAPT~P,Ll OF TIII: COLOPADO SCHOOL OP' MINES
or, approximately 1.02 for the assumed case.
It is difficult to see the behavior of the coefficient of anisotropy from the
curve in fi ure 15 because the values ire so close to unity. It is more convenient to subtract 1. frozn the coel~icient of anisotropy before plottiti~; it
so that Che variations are more Basil}' observed. A family of curves for the
coefficient of anisotropy plotted in this manner is shown in figure 16a, for
various contrasts in resistivity between tl~e hvo members of a simple sequence.
The corresponding, curves for the variation iii lon~itudii~al resistivity are
shown in fi~~ure 16b.
A rock sequence rniaht be characterized by a pattern plot of the value for
the reduced coe~cient of anisotropy, or A-1, and The value for longiCudinal
~`Z
100
1000
Paths followed by re=i_tivit}~-anisotropq field plots as the ratio, ~y, of a
]~ia~l~-ri i tnit~~ cum ponent to a ]ow re t tnity componetlt is increased.
"I~he luw resisti~ity component leas a rr5istivity of 1 ohmaneter; the
hi<rli ~E i~tivity component has x resistivity of l~~j.
LONGITUDINAL RESISTIVITY
10
Qua~Tr:E~LY or Txl~ CoLOl,.a~o Scxooz of 'VIin~Es
vas a~~umed to be U.1 I correspondin~ to a shale-sand sequence), 0.01
f corresponding to a shale limestone sequence) or 0.001 (corresponding to
a shale-e~~aporite sequence). 1'he zatio of non-shale to shale (assumed to
have a resistivity of 1ohm-meter) increases in travelling along these curves
from left to right. The maximum point on each of these curves corresponds to
a ratio of 1:1 between shale and the other litholobic component It is apparent
Ftc~re: lt.
0.010.1
REDUCED
ANISOTROPY
100I-
~~
3J
that the coefT~icient of anisotroPS~ rises with lout little change in longitudinal
resistivity-. so 7on~ as the shale remains the dominant litholo~ical constituent
of the eection. Ho~~~ever, when the other component becomes tl~e more abundant, lon~ituclinal resisCivity v~ ries quite sirnificantly, while Clue coefficient of
anisotrop}' drops slowly.
9nother set of ~;eolo~ical conditions which minht affect the electrical characteristics of a sequence in a consistent manner is a change in the salinity of
the connate water•, perhaps with no change in the litholo~;y or litl~ologic ratios.
Considerin~~ Archie's lava, it is conceivable that a uniform change in Clue salinity of the eater in both the shale and iron-shale fades in a sequence ~voiild
change the lon~~itudinal resistivitti~ ~~~ithout changing the coefficient of anisotrop~ —that is, the resisti~>it~ of each bed in a sequence would increaee in the
same proportion as the water salinity decreased. Such a situation would lead
to the translation of a field plot of anisotropy vs. resistivity parallel eo the resistivity axis, as sho~~~m h~ line A in fi~~ure 18. It is more likely, though, that
~~-ater s~li~~ity will not chai~~>e b}' the same amount in both the shale and the
uon-hale beds. If the change in salinity across a basin is the result of depositional environment, the salinity in shale beds may- chanu~e less rapidly than
the salinit}~ ~n non-shale beds because of the moder~tin~; influence of aclsorbec~
salinity in clay minerals This salinity is essenCiall}~ a function of the cation exchan~e capacity of a rock, and not particularly sensitive to the salinity of the
~~~ater in the pore spaces. "I~herefore, if the salinity of the water at time of
deposition varies across a basin, the vlriation in salinity for shale members
~~~ill be leis than wi11 be the variation in salinity for non-s~~ale members. As a
consequence. the contrast in resistivity het~veen shale and non-shale beds will
increase as i~he con~~ate water salinitj- decreases, and the coefiic~ient of anisori~opy will vary with the re~istivity as indicated by line B in figure lo. At
high salinities. the change in the coefficient of anisotropy will be small, because
the effect of the added salinity from cation exchange capacity will be relatively
small. At lo~~~ salinities. the chance in resistivity and the coefficient of anisotropy will be ;i•eater.
A similar phenoi~lenon may he observed if the salinity is changed by
circulation of fresh water into a marine sequence sometime after deposition.
In this case. the ~~~ater resistivity will he most increased in the layers which
are most pei•rneable. ~Che effect will be that of increasing the contrast in resistivity between the shale and non-shale beds. This will result in pronounced
changes in the coefficient of anisotropy with lesser c~a~~ges i~1 the resiskivity (if
the shale is the dominant rock type), with a transition of Che anisotropy-resistivity field as shown by line C on fisure 13.
Let us now consider some examples of the behavior of the ~~ross electrical
properties of sedimentary sequences.
pPOPI:RTIFS OF DIL 1'II:I.DS ~\D ~~III•:IP LnVIP,ON~II:NT
to a minimum resistivity at a depth of several thousands of feet, increasing
Induction-electric loac were selected from wells in 1VIississippi, southern
Alabama and Che Florida panhandle, at locations indicated on the map in
figure 19. The contours on this map are the depths to the base of the
~VIesozoic, taken from the Basement ~Iap of North America (IJ. S. Geol. Survey, 1967j. The sediments in this column are chiefly sand, clay, marl,
limestone, and chalk, all rather poorly consolidated. Calcareous sediments tend
to be more abundant in Che lower part of the column than in the upper part,
and increase in importance toward the Gulf Coast (Earcllev, 1962; Sloss,
Dapples, and Krumbein, 1960). The section eai•ies rapidly in character over
short distances, so that con~elations in detail are difficult. The sequence in
southeastern Mississippi—the cenCer of the area of the Ion study—has been
reviewed in a recent paper by F.arble (1963).
Despite the difficulty in correlating; beds in detail, ali lobs from the area,
except those in the far north which penetrate pre-Mesozoic rocks, exhibit a
similar overall character
:surficial resistivities are high, decreasinb gradually
Electrical properties of the MesozoiaCenoaoic sedimentary sequence o~ the
east Gulf Coast area
I~icuaF; 18.— Hypothetical shifts- in tl~e resistivity-anisotropy pattern for ~ change in
cunnate water salinitq as mi~~lit be caused by a change in depositional
environment (A) or by flooding from ~tirface waters IF3).
LONGITUDINAL RESISTIVITY
A. Migration of pattern for
o uniform change in waster
so/in~ty in each bed
B. Migration of pottern if
salinity chonges /ess in
sho% than in other beds
QUARTERLY OI' THI: COLORADO SCHOOI. OF MINES
COEFFICIENT OF ANISOTROPY
36
J7
5~
~2 \ ~~
j
~—
37 ~ ~ ~
~ 29 2O
-2B
\
~ 32
~ -36
-40
\
~~~ ~
~4 ~~~~~
~1
o
25
/~
l
1
\
-8
\-/2 ~
-/6 I
^~ ) l
—~
~A.
pre.-n~Iesozoic surface, in tl~ous<inds of feet (fi~om ~3asement VIap of the
United States).
F`icuttE 19.— ~VIap of the eastern Culf Coast area, s}iowin~ locations of wells for
which electric lobs were compiled. Contours are the elevation of the
-40
A~A~
~ ~°34 \
20 0~
-36 ~— — _
32
-2a
-24
32 21
16
~~
~ ~
MISS. —~i
\~~
~
X20
-4
-20 ~/
in table =1~.
main slowly at greater depths. In picking; depth intervals for' averabinb the
electrical properties, zones which appeared to be consistent in electrical properties were selected without attempting to assign formation boundaries. In this
process, four ~;eoelectric units could usually be picked consistently. The
units are designated in table 3 with the letters IJ, 1VI or L, indicating; that they
are in the upper part of the section, the miciclle part where resistivity is a minimuin, or in the lower part, where the resistivity increases with depth. If one of
these sections is further subdivided, the letter designator is subscripCed to
indicate the sequence from top down, within that section. Thus, if the section
above the interval with inin~imum resistivity has been broken into two parts, Che
upper part is desi~;natecl U~, and the lower part, U_. Litholo~;ically, Che upper
sequence, U, consists of formations of Miocene and latter age, while the mid
part of the section consists generally of formations of Eocene and Oligocene
aae, including; Che Wilcox, Claiborne; Jackson and Vicksburg formations. The
lower part of the sequence, L, includes beds of Cretaceous ale, though not the
whole Cretaceous section in any of the wells included in this study. A summary of foi•mati~m names and litholo~~y, taken from Ear~1e (1960 is given
PROPEFTIGS OF OIL TI~LDS A~rD TIIFIR ~NVIRONIIENT
QunxT~~LY o~ TtiE CozoxA~o ScxooL or Mr~vEs
1.76
0.59
1.0~
0.66
U
Mr
VI•~
M;3
1,550- 2.500
2,510- 5,770
5,7t0- 8, 90
1,720- 3,650
3,650- 4,140
4-,140- 3,650
5,650-12,010
9,870-13,050
830- 2,3i0
2,30- 5,540
5,540- 7,950
7,960- 8,570
5. Stone County,
Mississippi
Sec. 1, 3S, 12W
6. Wayne County,
Mississip~~i
Sec. 16, 7N, 9W
7. Green County
Mississippi
Sec. 10, 4.N, SW
8. George County,
Mississippi
Sec. 15, 3S, 6W
22~
0.61
1.26
2.06
3.53
I.•>
X1.56
0.76
1.66
101
3.4~~
O.US
0.90
0.66
093
1.62
U
Ml
L.t
L•~
U
M
L
Ul
U~~
112j
M~
1001,7502,=1`0'06,360-
9•. Marion CountS~,
Mississippi
Sec. 3, 4N, 19W
1,7'~-0
2,470
6.350
7,X00
MY
M~
L
3. Forrest County,
2.500- 5,330
Mississippi
5,330- 5,830
Sec. 1r, 1S, 13W S,o30- 0,380
2.44
0.39
2.20
7.4ti1.
U
7VI
Li
L~
3.19
0.71
1.25
2.20
0.59
0.57
0.97
1.42
3.22
2.Oo
0.60
0.)3
1.36
2.56
0.69
lA2
3.59
2.16
0.71
4.93
0.64
0.30
L51
2.00
O.o6
1.73
6.35
1.70
0.66
1.21
1.50
1.06
1.02
1.0~
111
1.0~
1.0'1.
1.01
1.17
1.23
1.26
1.05
1.08
1..65
1.27
l.OS
1.02
1.01
1.08
1.04
110
1.02
1.15
1.02
1.37
1.04
1.02
1.23
tropy
aniso-
Lon~itu- Coeffidinal
dent of
resistivity resistivity
Transverse
2,500- 5,350
5,350- 6,020
6,020-11;0°0
11,090-14,000
I)esi;nator
2. Smith County,
Mississippi
Sec. 6, lON, 16W
.Depth
interval
U
M
Ll
L~~
Location
1. Walthall County, 2,130- 3,150
Mississippi
3,160- 6,300
Sec. 13, 2N, 12~ 6,010- 7,650
7,660-1 L000
Index No.
of well
TnBr.E 3.- Electrical resistivilies summarized from inclzcction log's of wells
penetrating Mesoaoic-Cenozoic rocks of the east Giclf Coast
38
1.31
1.02
1.01
1.20
121
1.01
12-1
1.~1
1.13
2.2 ~•
0.63
0.81
1.91
~.-15
0.8 ~,
l.Ob
1.36
223
3.>4~
0.66
O.oi
2.r5
5.02
0.86
1.67
3.11.
2.86
U
M,
_~VI~
L1
ti
M
L,
L~
L;;
U,
U.,
NT,
iVI_
L,
L~,
15. Pearl River County, 1,250- Z.O7ti
2;070- '~,t>5t)
iVIississippi
8,77.0-1~,20U
2,00- 5.~1~ci0
5,480- fi,i=~U
U
NI
L,
L~>
2,500- 5.3<~0
14-. Smith County,
5,»0- fi.(lZO
~'Iissiscippi
Sec. 6, lOI~T, 16W 6,020-ll.tl,°,0
11,094-1 ~~~_01)0
Sec. ll_, 2S, 15~
Ul
U~
M
L~
6602,2604,0304,660-
9,7'0-11,000
7~j~~~- ~~7`~~~)
1,150- 3,830
3,830- 4,600
~~~,600- 7,530
2260
'1,03O
4, 60
7,500
13. Scott County,
Mississippi
Sec. 15, ~1~~, 6E
12. Madison County
I_l~lississippi
Sec. 1, BN~, 2E
6,650-1.0,820
~~.I~?
2.61
O.66
LU3
L.63
1.02
7.58
3.20
O.o2
1.05
1.~~
1.78
STA7'~ ~b~.LEG~ ~a~F~A
~4€~~VILLE, ~ti~5~l7~i
i~5t~t9~d
l.11
1.12
111
1.01
1.06
1.32
1.0°
0.~~
0.<y~'t~
2.~2
225
0.57
1.55
5.03
~~0~'f9-~inJ~S7'
1.44
1.02
1.28
1.3~~
2.75
2.31
0.61
1.19
4.25
3.34
0.63
lA•5
1.2-1~
1.20
1.02
1.10
118
1.25
0.96
1.76
1.33
2.67
Ll
L•>
5,0-1.0- x,790
8,800-12,500
10. Rankin County,
Mississippi
Sec. 14, 5N,5E
1,760- 2,40
ll.. Lamar County,
2,490- 5,900
Mississippi
Sec. 19, 2N, 16W 5,91.0- 6,650
1.01
1.02
1.17
121
0.'76
0.65
1.04
1.63
trop}r
aniso-
Longitu- Coeffident of
dinal
resistivity resistivity
Transverse
0.rr
0.67
lA~l
2A~3
Desibnator
M,
M.>
L~
L~>
Depth
interval
1,920- 2,670
2,600- 5,760
5,770- 8,650
x,660-12.3 0
Location
3~
9. PerrS~ County,
Mississippi
Sec. lo, N, 9W
Index No.
of well
rrABI.I: 3 (cont.)
PROPERTIES OF OII. FIELDS A~'D THLIR ENVIRONMENT
Location
U
M
Li
L•a
U
M
Li
L•>
19. Covington County, 3,050- 5,750
Mississippi
5,760- 6,7~~1~0
Sec. 4, 7N, 17W 6,750-10,650
10,660-13,4.60
1,700- 3,400
3,410- 7,700
7,710- 8, 70
3,080-11,20
2,110- 3,130
3,140- 7,110
7,110- 7,970
7,980-10,900
10,910-12.5<10
2,520- 2,930
2,980- 3,570
3,5°0- 8,50
8,560-1.4-,330
2,410- 6,O~i0
6,Oo0- 6,780
20. Lincoln County,
Mississippi
Sec. 32, 5N, 6E
21. Copiah County,
Mississippi
Sec. 9, 9N, 7E
22• Jasper County,
Mississippi
Sec. 12, 3N, 11E
23. Sirnpson Couury~
Mississippi
U
1VI
U
M
Li
L~
U
Mi
M.>
Li
L~
U
M
Li
L~
18. Jones County,
2,390- 5,020
iVIississippi
5,034- 5,660
Sec. 16, 7N, lOW 5,670-10,020
10,030-14,060
M
Li
L~
U
Desinator
U
M
Ll
L•,
5,900- 6,820
6,830- 9,720
9,730-11,100
3,210- 5,900
Depth
interval
TA~T,E 3 (cont.)
L20
723
4.63
2.81
0.79
1.0~~
4.51
x.04.
0.76
0.~0
2.03
3.68
3.13
O.o2
1.33
3.05
1.04
0.82
220
3.59
1.4.9
0.68
1.36
3.04
427
0.80
I.31
3.15
0.90
1.89
4.03
6.39
0.62
2.50
0.77
0.79
L39
3.74.
0.73
0.r9
L49
3.55
1.54
0.73
L30
2.70
0.76
0.80
1.6c~
3.01
0.85
0.66
1.04
2".84,
221
0.70
L23
2.87
0.96
1.33
2.83
0.88
1.06
1.01
1..06
1.03
115
1.80
L0~
1.02
1.01
L16
1.02
L42
1.06
1.01
1.0~
1.17
1.01
1.14.
1.09
1.32
1.01
114.
116
1.39
1.07
1.03
1.05
1.01
L19
L19
L17
troPY
TransLonbitu- Coeffiverse
dinal
dent of
resistivity resistivity aniso-
QUARTERLY OF TIII: COLORADO SCHOOL OF MINIS
1,500- 3,160
3,160- 7,350
7,360- 8;500
8,500-ll,100
17. Pike County,
Mississippi
Sec. 1, 'IN, 7E
Mississippi
Sec. 30, 6N, 4W
16. Hinds County,
Index No.
of well
~~~
€
-
Location
Depth
interval
M
Ll
L~
1..990- x,070
x;070- 8,270
82r0- 9,870
2-10- 1.,200
1,210- 4-,340
4,350- 9,620
9,630-10,560
1,4-00- 5,200
5,210- 6,720
24~. Green County,
~'iississippi
Sec. 10, 4~N, SW
25. Franklin County,
I'lorida
26. Baldwin County
Alabama
Sec. 16, 2N, 4,E
1VI
Li
L•,
U
~VI~
L,
L~
U
VI
1,300- ~=1,100
4,160- 6,990
6.990-1.3,000
1,140- 4-,300
4-,310- 7,820
7,30- 9,-110
9,110-1.0,810
X60- 3,000
3,000- :3,93O
30. Adams County,
Mississippi
Sea 4~, 6N, 2W
31. Clarke County,
Alabama
Sec. 11, l_ON~, 2W
Ui
U~
M
520- 2,380
2 394- 4~ 1`20
~4~,130- 6,120
23. Clarke County,
Mississippi
Sec. 6, 2N, 15F
29. Adams County,
1l~Iississippi
Sec. ~6, ~N, 1W
U
M
Li
L~
1.,X60- 2,4.30
2,~~'lO- 6,000
6,010- 9,510
9,520.10,620
27• Amite County,
Mississippi
Sec. 28, lN, ~F.
U
M
U
Mi
NI.,
Li
L~
Li
Desinator
9,990-12,980
Sec. 1.U, lON, 19~ 6,790- 9.990
Index No.
of well
7ABLF. i (cont.)
Lonbitu- Coeffidinal
dent of
41
125
1.06
1 ~~
~
6.17
1.39
U ~~
9.7~
L57
118
1.1
1.21
0.7~1~
0.70
118
3.1r
-1.0r
(1.62
1.35
0.39
1.4,9
4. ~~6
S.t2
O.92
1.16
1.1~
113
1.19
0.38
0.~>6
1.51
0.54.
1.65
2.30
110
I.1~0
123
119
l.2)
1.12
1.03
1.11
1.30
0.33
0.42
1.02
0.71
6.03
0.5~
L11.
2.49
1.34
1.18
]..=1.-1~
1.19
2.13
0.7~~1~
0.53
2.95
3.d2
1.0~
112
4.32
8.52
1.01
1.40
2.6=1~
1.05
1.10
1.23
0.81
1.10
0.39
0.89
1.33
1.36
l.lv
11 ~,
1.13
2.94.
3.00
1.60
tropy
resistivity resistivity aniso-
Transverse
Pxor~~Tir•.s of Oir. Fz~L~s a?v~ TxEir EvvrROhM~~T
1,840- 5, 00
5,310- 8,720
8,730-13,030
2,050- 8,800
35. Mobile County,
Alabama
Sec. 16, 3S, 2W
36. Clay County,
L~
10,=X30-12, 00
7.35
0.9-1,
1..57
3.39
00.0
1.05
2.09
10.E
33.6
190.
2.65
123
0.89
0.93
315
x..59
0.80
1.=~•7
2.52
~~3.3
0.90
1.54
5.~5
14~.4~
155.
1.45
0.76
0.52
0.96
2.06
All resi~tivitie~ compiled iii table 3 were taken from 6FF~1-U inductio~i logs.
(See Prison, 1963, for a discussion of induction log charactu~istics.l The
short-normal electric lobs were also sampled, but ~~enerally~ gave resisticities
several times lar~,er than those on the induction long. This departure is probabl~ caused l~v invasion oc~er most of the section, and s~>. the induction lob;,
having a greater depth of investigation than the short normal lob, probably
indicates snore closely the true formation resistivity.
9nisotroE~y-resistivity field plots are shown in figures 20-23 for Che U, 1VI,
Li and L~> porti~~iis of the section. The lower three parts of the section e?chibit field plots that are elongated vertically, indicative of a sequence that is
126
1.03
1.0~
1.2-1~
1.35
L05
116
L42
1.52
110
1.35
1.27
L31
1.01
1.23
TransLonbitu- Coeffiverse
dinal
dent of
resistivity resistivity anisotropy
Note: All resistivities are compiled from Che 6FF'-1-0 induction lo~~~.
V1
Li
L~>
L~3
M
Li
L~
3t. Escambia County, 2.350- 1,~~0
I'lorida
1,580- 7.590
Sec. 31, 2S, 31W 1,600-10,-120
Sec. 2-1~, 165, 5E
-~'T(~S1SC1~~1
80- 3,950
3,960-10,020
34. Sumpter County,
Alabama
Sec. 9, 23N, 3W
L;3
L~
L
1,200- 6,150
llesianator
33. Decatur County,
Georbia
Depth
interval
U
M
Li
L~
Location
TnBL~ 3 (cont.)
QU9RTERLY OF THE COIORADO SCHOOL OF MINES
32. Claiborne County, 2,220- 3;760
Mississippi
3,770- 6,910
Sec. 27, 13N, 2E 6,934- 8,050
8,060-1.0,430
Index No.
of well
~2
~
t
`
~
~
~
~
t
~'
~~~j
_
Selma, Eutaw,
Cretaceous
Tuscaloosa, Dantzler, Andrew,
Paluxy, 1Vlooringsport, I'ei-ry Lake,
Sligo, Holston
L
PaleoceneEocene
~VIiocene
Abe
Vicksburg, Jackson, Claiborne.
Wi1coY, VIidwa}'
Citronelle, Pascagoula, F~Iattiesburg, Catahoula
_
l?'urmation
naives
M
U
— -electrical
designator
5,000
3,2006,000
1,7.003;100
Thickness
_.
~ha(e, limestone,
anhydrite
cla}', shale,
sand, marl,
limestone
sand; gravel,
clay, marl
_
Lilholo~y
T~~LL -l~. — 1~'on~ac~tio~z nairaes from Ifae easC Gt~~l~ Coast
To summarize, the ~edi~nentarti~ column along; the east Gulf Coasl is characterized b~~ the presence of thick sections of hirhh coi~ducti~~e rock, ~~-itli
thousands of feet of section ha~~inn a resistivity of less than one ohm -meter,
~~~d ~+ith the first 2 to 3 miles of section having an average iESistivitg of less
thou ~ ohm-rncters. "Chic lo~v resistivity is accompanied by values for the
coefficient of anisotropy ran~in,~ from ].Ul to 2.40.
quite uniform, exce~~t for the addition of a small fraction of resistant beds.
The ~>attern for the uppermost rocks is not elonnate. but this probabl}~ reflects
the non-uniform nature of these rocks. It is interesting; to note that there is a
tendency for the anic~trop~~ Co be ~;re~zter fox• the more conductive sections withiii a siu~rle plot. Such behavior is co~~trary to that postulated earlier iii this
section for increasing amounts of noi~shale members in a shale sequence. It
nay be etplainecl by a tenden~~~~ for shale to become more conductive under
the same circumstances that ~le1d to the introduction of resistive beds, such as
limestone oi• eva~~orites, into the sectio~~. In a near-shore environment, ii is
snore likcl~~ that tl~e section will consist maii7ly of cl~stics_ so that there is less
range in Iesisti~-sty for the members of a sequence, and the shale wi11 tend to
be less co~~ductive. 1'urther from shore. the section ivay include more conductive shales and carl~onites ur f~-aporitc5, resulting in a section which is more
conducti~-e in < cneral, I~ut wiCli a lii,h~r anisotropy Chan near-shore deposits.
Pg01'L'R'CI1~:S OF OII. I'Il~:[.DS ;1\U ~I'7II;Ili F~_A'VIRO~IIENT
I
d
~
~
b
~
I
100
—
I
1000
~
j
I
I I
0
I
I I I I IIII
io
Exen~
action,
Eosf Gu/f Coosl
_-_ ~
0 0
omo
0 o
o
I I I I I III I
ioo
I
I I (I
i000
LONGITUDINAL RESISTIVITY
I~ict.~ar•, 21.— Scatter ~~lot for values of lonnitudival resi~ti~~ity and coefficient of
anisotropy fur tl~e Paleocene-Loce.ne section p~,netrate~d b}' wells in the
east Cult (;oast area. "I~h~t values for the coefficient of anisotropy have
been reduced by subtracting 1 from them.
0.01 ~
o.~
~. ~
0
LO(VGITUDIIVAL RESIS7IVI7Y
I'icutte: 20.-- Scatter plot of values for longitudinal to
istivity and c~n~~fftcient of
anisotropy for tl~e '_1~Iiocene section penetiatcd b} .:ells in tli~
easC
Gulf C<~a5t areti Ilse ealues for the coetTi~ient of anisotrop}~
leave been
reduced by subtraetin~, 1 fi~om them.
REDUCED
ANISOTROPY
10
0.!
000i
O,i
10 ~
~rvsso~r~oPv
i
}
0.1
I~
~-- —
o
0
0
I
I__.
a
0
~ 8O o
00
~o 0
0
Ooo0 Qo 0 0
Eosl Gu/r Coosl
0
10
100
LONGITUDINAL RESISTIVITY
1000
ai
--
i
0
0
0
o
0
0
0
oao
0
0
ao
0
io
GNoceous Srclioq
EosJ Gu/f Coasl
o°
0
0
o
ioo
_..
LONGITUDINAL RESISTIVITY
_ _. '_
i000
Fictia~ 23. - - Scatter plot of values for Ion~iCUdinal resistivity and coefficient of
anisotropy for the Cretaceous section penetrated by wells in the east Gulf
Coast area. Values fur the coefficient of anisotropy have been reduced by
subtracting 1 from them.
..
~.~
10
REDUCED
ANISOTROPY
FtruttF: 22.- Scatter plot of ~ilues Ear lon~~itudinal ~c istivity and the coefficient of
~inisutropy for . E.ctions of Paleuct~tne rocl.s penetrated by wells in the
east Gulf (:oast tn~ezi.
~.
~.~
10
ANISOTF SPY
QII~PTTI:LY OP 'I'H1•: (,OLORADO SCftOOL OF 111NFS
Tlie 1-elatiael~ hi~;li resistivity of 1•ocks c~t~ the C.oloracio P1aCeaus makes the
determination of a~era.e electrical properties less certain than in the case of
t1~e east Gulf Coast area. Wherever possible, laterolo~ or induction electric
lo_>~s +ere used its the compilation.
It is i~~teresting to note that the normal sedimentary column on the Colorado Plateaus exhibits the same overall character as that for the east Gulf
Coat—the re~i~tivity~ passes through a minimum with increasing, depth from
~~'est.
Jackson 1.1.96 , 1.96"ll I~as described a similar ctud~- of electric logs from
the Colorad~~ Plateaus area, in the stales of Color~clo. Utah. New Mexico, and
Arizona. ~~~ith ~~e11s aE the Ic~cations indicated on figure 2-1-. The contours on
this map aie the ele~~ati~~n of the Precambrian ha~ement, again taken from
the F3a~ernent lrlap of 1oi~th Ainei~ca. As may° be seen From these contours,
the Colorado Plateaus area is much snore complex structurally than the east
Gulf Coast az•ea.
Accordin~~~ to Jackson (196:ij; in compilin~~ the average electrical properties for rocky of tl~~ sedimentary section on the Colorado Plateaus, it was
found flat the section could lo~icall~- be divided into four dross geoelectrical
units, though not all four aie present over all of the area because of the complex structure. These units are:
1. Sec3iinentar~ socks. pi-edoininantl}' shale and sandstone, of Upper
Cretaceuus Ind lotier Cen<~LO~c a~,e, Fvhich exhibit low rc~ietivity. The most
conducti~ e portions of the sc ction ai•e~ rocks such as tl~e Mancos and Levis
shalt ~. Gencrall~. thr sandstone beds in these formations have lower resisti~~it~« [bars indstonE beds in the lo~~~ei• part of the sequence.
2 SedimEntar~~ socks_ piim~rily sandstones, of Permian to Lower Cretaceous i~~~e, ~~~huh are fne enl nearly e~er~~~~here across the Plateaus. These rocks
ha~~f ~ome~~h~t ~~ig}i~~ i~~~~t~~it~~ than the u~-eil~-ing 7ock~ Neal• t~~e southern
niai~ ~u of the Paradox ba~~n, ~vliere the e~a~~ozite fades of the Cudei• Forn~ation is ~~ell developed, i~~ese ~•ocka may better be ~~rouped with the hibh
iESi~ti~~it~~ bcd~ just I>Eneath them in the section.
:3. In ~~lace ~. the Paradox I~o~~ination, of Pennsylvanian ale, is sufficiently
thick and has such a high resistivit}~ in cotnj~arison to beds above and below
that it must }x. concidered a5 a separate electrical unit. Nearly all Che evaporite
beds in this section I~~ve i~e~istivities of a thousand ohrn-meters or more.
1, Sedim~ntar}~ socks, mainly limestones and dolomites, of Cambrian to
Permian a~;E ~vl~ich lave high to eery High i~esistivities. This part of the
~edimentar~ sequence thickens from the east tide of the Plaeeaus towards the
Electrical properties o~ the Paleo.:oic to Ceno:;oic section of the Colorado
Plateaus
-~6
,~
Ficca~•: 24.
=~7
~Iap uE the I~uurvCnru~ rs area in tlic states of Culortido, [Jtah, Arizona
end \ew )l~~~ico, with locations of wells fur which electric; lri~.~ were
i~om~~ile~L Cun~uurs reEn~r~ent the el~°vtition of the Prccambrixu ba~cmen[
stu'fxcc, in tlwu=quids of Curt irum sea-level ltaken from I SCS Basenic~t !4iap of thy• llniird Stat~•.1. Shaded eirett one areas of basement
outi'rup.
ARIZ.
PROPEPTII~S OI' ~IL 1'~II:LDS ;~1D ~THI:IIi ~.NVIROIVPIIsNT
Qu;~rTrri_1 0~ T~~ir Colorn~o Scriooz, of l2zn~Fs
Cambrian
to
Permian
Per2ni~n
Permian
to
Cretaceous
to
2,000'
to
2,000'
to
-1,.000'
to
6;000'
"Thickness
limestones
limestone,
evaporates
sandstone,
shale, limestone
shale. sandstone
Lithology
Compiled resicti~~ities for cells from the Colorac~~ T'lateaus province are
listed in table 6. Resi~tivit~-anisotrop~~ field plots for these same data are
~~iven iti figures 2~-2r (the L~ and L;; data are plotted together on figure 277.
IC i~ quite obvious that the sedimentary rocks on the Colorado Plateaus have
much higher re~istivities and anisotropies than rocks from the east Gulf Coast.
This reflects the greater heterogeneity in porosity from bed to bed in the sequence. Phis hetero~eneit}~ inay be a consequence of the greater age of the
l~~wer pant of the section treated for the Colorado Plateaus, with var}~ina degi•ees of post-depositional cementation resulting in a ride difference in resistivit~ het~veen the various litholo,~ic elements comprising the section.
Hermosa. Cutler,
Ignacio, Elbert,
Ouray
Paradox, Cutler, PennsylvaKaibab, Coconino
nian to
L~;
L;3
Cutler, Dolores,
La Plata. Ntci?ln~o, Dakota, _l~toenkopi, Shinarump. C}iinle,
Glen Can~~on, San
Rafael, Morric_on
L,
Cretaceous
and
tioun,~er
1Ia~uos. ~'Iesa
Verde, I..e~~ is
Shale, I'ruidand,
Kirtland, Aniina~
l'I
A~;e
F'oi~mation
names
Electrical
designator
T.at3LF: 5.-I'ornaation names ~rona the Colorado Plateaus
the surface, and then tends to increase continuall~~ ~~ ith de~~th, if bed to
bed
variations are ignored. The surface la}-er of high i•esistivitt~ is
relativel}' thin
on the Colorado Plateaus, and so, the section might be indicated as
the sequence of design~tor~_VI, L,, I.~ and L;i, defi~~ed anal<~~;ously to those used
for
the east Gulf Coast area. A summar~~ of formation names and
litholo~~y,
taken from Eardle~ 1 1.962). is g~i~-en in table 5.
1~8
-19
~~I
_
L,
~'I
l,,
VI
L,
. ~~icKinle~ G>u~~~~_ I.Z00- Z.l-(~~
2.150- 2.2:~U
1.1i.
Z_~:'>G- Ei2O6
1 I_ 1 1~A'. ~,W
L~00- x.10O
5.1~>0-11.Z~0
~)~~O- 1,~~UU
(i.~)2O-]0.025
~I. ~~t~,ntru~F~ C~,unt~~.
Cul~~.
'? l. I i\. 1')~~?
5. I,a Plata C„unt~.
Colo.
1 ~.:31~. 1 l~V
6. 1 ~)~V, 231:
~>,OZZ- 3.;> IU
.~9.6
J.l
uT
2~~. ~~
1.~~
2:>.~
Z~2
5')0
~~)2- 1,~)iO
l.lU9. tiavajo County,
L,
1'I
L,
lI
11S
1.7~~
].0:3
1.2~>
1.07
L~>6
3f>.0
1rii.
1 r9.
623
15G.
62~.
l:1.00- -4,~U5
(.~Ui- G.Uc>U
6.Ot~Z- <~,36U
~,. Garfield (:~iunt~.
Utah
18..365. lOF~~
LU7
1.31
l.:>1
1.SU
i,.1
ZZ.O
J. 1~l~l.<,
1~ZU- 7-]3
~ (5- 1,330
1.33Z- 3,7cU
:3.776- i.UU-I
r. San Juan County°.
N.II.
3U, 2E~1~~. 1~)~%
L,
9.7
3~).8
].f2
13~.
1.Z0
2. Z
I~.Z
~~:>.Z
I.U~>
l .Z 1
LZO
1.Z0
1.11
Zl.~>
1Ifi.
19.E
11-Z.
1.1(>
1.;3'?
1.'?5
l.(i>
1.1 r
').10
SOU.
21.2
11,6
~~1.;>
l.Z~)
1.16
1.-1')
1;,.•~
:3~.0
6')1.
'~2.0
~1 ~.l
1.5 O.
11~.c
1.5 0.
52.E
110
l.Z l
1.35
l 1.~
Z I.S
~~,Z.
1 i.~~
;> ~.~)
1.OSO.
trop
l;>.2
:~5.2
l ~.9
1i.9
I~).~)
Z»- :~:;5
5 2- t~O~
~~0~- :~,I;>7
.~12i- S.~i6
5.52<;- i_~,~)0
f~. San Juan C~~unt~.
Utah
7, ~(~OS. 26I~;
1~1•i7,.
r~~si~ti~it~~
r~~~i~ti~il~
ani~~~-
I_un~~~itu- Cue(Ticieut e~f
cli~ial
~l~ran~~ ei~~e
7~2
51.
Z~.(i
25.r
800.
~I
lT
L,
i._
:>10- 1.9~~0
L932- :i.3~0
i2- .l~0O
Z. Grand C~~unt~.
Lftal~
Z0, 21 S. Z F~;
lle~~ig~nat~~r
1I
L,
I.~
UrE~th
intr~e~ al
31.0- 2.3~)<>
2.10O- 1.I Z:~
b.lZS- I_i-I~
Lo<ati~~n
1. Grand G>u❑t~ _
Utah
2:i. 2OS. Z11~:
Inde.t \~~~.
of n~e°ll
'I'_~t;[,~: (>. - /l~vera{;e resisilci[ies com~~iletl ~roin. elechic logs r~ui i~~ zcells opt
the Colorado Plateau,c
Proi>r~;Tn;S or OIr~ I'i~:i.us ;~~~» 'I'EU~:ir E~~ n~~~~~t~:~T
L<,cati,,n
l.(-~J- Z.I~J
r-10- 1,r-Ij
(-~U
X60.
128.
>~i.0
1;920.
li
11
L.~
9~.
-1,1~-1,.
jd.~
J /.7
1.0.0
32.0
10.8
~.~
6.:~
231
11.6
;3<'>.3
X3.1
28.6
X10.
2,<;~U.
9~.
1-1.5.
1280.
<,9.
~iJ.
11.3
2L~>
10.6
1 ~.9
1.1.0U.
6
J~).
83.
13.2
97.
76.
6~.
:382.
IJ.~1~
~~~~.~J
f3.~0
26.-1~
9.9
iii.
~.r
18.6
9.6
.~-1,3
6.7
2~.8
X10.
2,12.
6~.
120.
200.
6~3.
:1i.
(
~0.-1~
1-1~.2
8.0
21.2
610.
63.
Z~(.
--.. _
1.2~
1..:1.1
-1.50
2.71
1.17
LO-1
1..~~~
l.lZ
DLO ~
1.10
1.7.2
I.JJ
L0~1~
1.23
1.:10
1..06
1.10
1.06
1.0-]~
1.16
1.1 o
1.10
2.52
l.l.-1~
2.~~J
1.16
1.2-1
115
l .-1-~.
1.x,2
L03
.l.U~
tro~~)
BIII?O-
Lon~itu- Coef~i~3ina1
c;<~nt ~~f
1'(-'S 13I ll~lt~- YPSI~(IA~1~V~
"francverse
L,
1'I
L~
Z.~)~~J- (.i~~ ~.
~10-
l~I
L,
lI
L,
L..
L;
1.00- 9~0
9i0- L202
I.ZU?- 2.02
2,195- x.023
16. Sa~~ .luau Count~~,
7U- 835
Utah
3c ~- 5.050
3, 315, 2z~
~,0~0- 7,800
>. .ir~_ ~LJ~,
U[al~
~;). ~iltl ~Utitl ~OUIIt~~.
11. llelta Count.
Colo.
1 Ci. l ~S. 9~~
1i5- X00
i00- 1,290
1.290- 2,30
2.30- i.i10
~1~,710- ~,1~0
7,10- ~,~~7U
.5,97(}- f.-13~;
6,-138- G.<,2~;
6,82t~- 8.0-10
1?>. Garfie~lci Count~~.
Utah
12..65. ll~;
.~)-~~)- ii. i C) /
U- 1,0'?0
1.022- ~>,6 ~~
3.6~ r- 1.1~~0
I,.~
ii.~~~Z- ~~.~Z~1
1 Z. 1~>>ner~ Cr,unt~ _
Utah
6. »S. 12i:
~~I
L~,
I-;
j~~
I)e~i~~nr~t~,r
L,
2_>72- 3,090
3.09L 3.5'?2
G~)~- Z.,~(()
Depth
inre~r~•~al
"I'_~tt~.t: fi I coat. i
QL;~~t~r~:ar.r or ~r~~l~: Cor.or;_~~o Sciioo~ or ~Ir~~rs
11. ~Icltinle~ Count~~. ~1-~U- 1.160
I.
1,162- 2_ ll5
1-I. 1~I:1'. ;~~/
1.11 r- r."r r6
r.~ic- <.600
Ariz.
3;~. 2~;\. t~1'
CUllllC1-_
_ _-..
1 ~). (.,000III[l0
--
Inclel \o.
~~f sell
50
Lc~cati<~n
Depth
1.09
1..0
L11
22t
12.E
29.2
1.5.6
3L~~
15.2
1~~.7
19.2
163.
L,
~~00- 1,720
1.722- 2,920l'I
x,000- 6,-100
L,
6,-102-13.050
~~~U
6(j
~~3~- 3,930
,93~- 5,156
~,'~E)O- (, I~56
f)`ciU-
(U-
21. San Juan Count}', i~0- 3;10
3,110- 6.1.16
[Jtah
7,5Ei0
6,15020E
1~1S,
36,
19 29~, 16W
1i.1'1.
~J. J811 JLIaTI GOUIIty;
~>.S
30.6
X0.3
1-12
56.2
106.
L,
L•,
It
L~
L;;
l.~
1.20
1.~6
123
l.l~
1.0~
2.-~~
Z~).Z
~U.1
6.-I,
ll.7
11.6
Z.~.J
J.2
~.6
8.-1~
1~.0
2JO.
~'1
~)JU.
I.,
Z.I,~i
1.0:3
1.5r
-1.1.6
8L~
1-12
200.
~I
L~
-1~0- 1,;10
1,313- 2,50
2,~.~i l.b'iO
22. Navajo Coun[r',
~
Ariz.
1.2 Il \'. 18}s
Z1~).
222
93.
-1~0.
9c~i.
L.,
L-lZ
1.~~
1~.6
6-~.
31.-1iVI
Lt
x.310- 7,21.0
~.~~~- J~~~J
28, ~7\, 11.I
ATIZ.
00. 2,93
20. La Plata CounCV~•
Colo.
5, >:~y> 7W
1.06
1.1.1.
l.0r
1.5-t
69.6
19J.
X9.5
1=1~3.
NI
r~.8
24~.
57.1
3-1~0.
10- 670
670- 1,200
1,200- 1,rr0
1,r r0- ~{,~>55
M
L,
L_
L~
2,>i~
1,1.60
1;000
~ix)00
19. Garfield County>
Utah
3, 325, 15E
1.1r
2.56
2.~2
2.0L
15.r
3~.2
85.5
~1,~~.r
21.6
232.
~ ~5.
].77.
Li
L.
5002,871,160~,O0O-
l.Z 1
2.50
3.r
I29
18. Sari ,Juan Count~~
Utah
~~, 27S, 191:
1.1~
~J.
(
(J•c~i
II'Op)'
3J.Z
25.0
39.5
I(1.
L5821-,061-5,0316.~Of>-
21. Coconino Count`.
C~]Ilftl
flRl~O-
CoeffiCIP.17t 0~
Lon~;itu-
J-I.O
17~>.
5$0.
,2J~.
~I
VP.I'~8
I11CC)7'
PE:S1Ft7V7t}~ Tf',57 SC1V1tV
'brans-
1)esi~;-
r
~l
I,O61
5;03~1~
6,506
Y,,(UO
O-~' 1-~C~2
ltl(-CCV81
COIO.
19, ~9N, 1-IW
17. ~OIICeZtIlYIa C011Ilt}`-
pf ~y~~~
Index 1'0.
T~~Lr: 6 (cont.~l
Pror~r.~;~rn~;s or Ori, Tlrt.ns .a~n "I'it~:zr; I~~~n;~~~~~IL~T
L~>cation
701,6552,3053,303~~,X03-
1,650
2,300
3,300
-1,700
5,~r ~0
L~
L~~
M
L,1
L•,
~~-~9J~.~- O~~JO
_.
__--..-
26.2
6.4
22.E
53.7
~r0.
~9J.
i~u.J
8~.°
31.3
40.0
1~.6
2r0.
720.
_. _.
_._.
_... ___
22.2
~.S
21.0
1'3.1
92.E
~-~.S
JI.J
7~.3
229
34..7
8.3
9=~~.
~11~~.
1.09
1.20
1.0~~~
1.29
2.00
2.-~7
I.JI
LO-1~
l.iS
1.07
L37
L69
1.32
__..__._
t2'OIJ
'FransLon~itu- Coef~iverse
dinal
dent of
resistivity re~istiv~ty aniso-
-- -.___.
basis of atera~~e electrical ~~i•operties:
Both Jackson l 1962) and Hai•thill (1'>(iG, 196r) s-eco~;riize a series of four
anit~ in the section which one might expect to differentiate
readil}' on the
section.
7~he most deeailed study to date of aeera~;ed electrical resistivities from
electric logs is that which has been done for the Denver basin area of eastern
Colorado and the surrounding High Plains area of ~~~ebraska and W5 oming.
This ~tud~ consisted of two parts; zn initial study in which logs fr~in deep
~cclls, those penetrating; the Paleozoic section, were selected from the thi•eestate area, follo~ced b~• a detailed stud}- of 1o~~s fz-om wells in eastern Coloradu. The locations for the ti~ells used in the ii~itiai stud}' are indicated nn
the ina~~ iri fi~ui•e 2i3. 'I~he contours are elevations of the Precambrian surface,
take❑ from the Basement_l'Iap of North America IU.S. Geol. Survey, 196~j.
"I~he ~ecor~d part of the study ~~as a detailed eealuation of a~~ei•a~;e resistivities
fr~,~m approxin~atel~~ 25J ~~ells located iu "I~ownships 1 and Z ilort6, in eastern
Cr,lorado r fib. Z9). These ~~~ells ~;enerall~ did not penetrate to the Paleozoic
Electrical properties of t{ie Paleo_oic to Ce~iozoic section of the Denver basi~a
crn~l Hi~~h Plains area
2r. Aplche Count,
Ariz.
12, 10i~'~, 23I?
Z(, ,~(~, 1 / W
Z,GJ~- -~~,~JO
M
2, 135, ~>~TV
litah
26. 1'IontezumaCouuty, L160- L~iO
Colo.
1,-17~- 2,60
......
M
L,
L;;
____
llesi~;nator
~ (COI1t.)
_---
~~A13LI
D~~~>tli
interval
...
~U9hTi,RLP OF' 1'HL COLORADO SCF300L OF tiTINF,S
1,020- 2,39
2,400- ~-,8=b0
1,8-1•~- x;500
x,770-10,060
2~. Kane Count}.
Index 1c~.
of ~s~ell
~~
i
~"
t
1
_._
—
1_._. _'
-.
E-` r
,i~
~
f_I._.. _._
.- _..
.1_.. _~.._...~_.. _
~__
~ __.
__
10
-
~ o
-~—
o d3000°
°
o
-
--- --
100
Creloceous and Cenozoic,
Colorado P/oleaus
p
~
00
~~
i
O.I
I'ictsit~: 2<.
C~1I
10
REDUCED
ANISOTROPY
10
I00
~OCG
LONGITUDINAL RESISTIVITY
Scatter plot fur values of longitudinal resistivity and coefficient of
anisotropy for the Paleozoic section (exe]tiding the Paradox Formation)
penetrated by wells on tfie Colorado Plateaus. The values for the coefficient of anisotropy have been reduced by subtracting 1 from them.
I
LONGITUDINAL RESISTIVITY
Ficce~ 25. — Scatter plot for values of longitudinal resistivity and coefficient of
anisotropy for the Cretaceous and Cenozoic sections penetrated by
wells nn the Colurado Plateaus. The values for the coefficient of anisotropy have had 1 subtracted from them.
0.01
p.~
~~~
~
~~
H~'!i'~v I rtVr i
100
LONGITUDINAL RESISTIVITY
0
o
0
~G~;~
~s,
_,
Scalier plot fur ~ailucs of ]un~itudinnl re~istivity and coefficient of
ttnisutrupy fur the Paradox I`ormation where it ~+~as peneu'atr:~d by wells
un tl~e C~,lurado Ylatcau~, "I7~e c<iltie fur the cuc~fficient of tuiisotropy has
Lee❑ reduced h~~ suhtructin~ 1.
10
~
0
o
u
~
c, ,00~
QU9PITFRLP Ol~ TIfI; COLORADO .SCI~IOOL OF ~TIV`I:S
Iayer U (sirface layer): The upper rocks are discontinuous anc~ vary
si~;i~ificantly in thickness across the area. In the ~~e~t, alone the front Range
in Colorado, this unit consists of the I'ox I-Iills, Laramie. Denver, and 9rapahoe formations of Cretaceous a~~e. 1~he I~ox Hills anci Laramie formations are
made up of sandstone beds separated by shalee, ~~-hile the Deliver and Arapahoe formations are a hetero~~eneous mixture of sandstones, clatstones, acid silesCones. "[~he upper la~~er is the O~all~la Formation ui the eastern part of the
re.;ion. bein~~ an asseml>la~e of continental conglomerates, sandstones, siltstones, ciao stories, and ceinentstoiies. The upper layer varies in thickness front
as much as 1.000 feet in the tivesC to as 1itt1~ as S00 feet in the east.
Iaver 12, the Pierre Slagle: "Cliffs unlit is composed mainly of shale end silt~tone, the silC l~ein~~ found in Che "Cran~itiou Zone at the Cop of the formation.
T(~e Pierre Formation is remarkably uniform in character over lame distances.
It thi~~~ from about -1,000 feet in tl~e central part of the Denver basin to about
X00 feet over Che eastern part ~f the area under consideration.
Ftcuitt: 2i.
5~~
05405
~ ~
oI~~12
~ °3
~~
\~
\ ~
~~ ~
~~ 52 I.
oI56~~~
f~ ~~
\\
K5.
`"
r
~
~L-I
J'
~ o
~~
°1021
/,~--~
l,~-~
titatr:~,
AI,iE~ n~ th~~ Eli_h I'I,iin~ r~~;~inn sh~,~vin~~ the Iuc.Ui~~n~ of ~n~ll~ fur ~~hicli
~~I~~clri~~ I~~_: ~c~v~~~ ~v~m~iil~~~l. (:untuun .n~ ~ I~~~~atir~n~ of thy• I'rccaminian
~urlai~~, in ~h~~u=.in~i~ ref fret ilrr~m tli~~ ~tru~tin~ul 11u~~ ..I tl~r~ ['Wiled
COLD.
~/
~~
2
~
~~18
~
ag
_2
o~
N
\
~
— — -~----~
—~
~
- ~'S3 ~~
37 34~ 3~ 4° 04~~
~
~4
~3
~39
~_
J
1
0 36 0 °~,~2
'~~48
aa~ , ~ ~
'7
~2 05 \
\6
NEOR~
~~
4
J~
l~lie L~ layer: "Che ~o~cet~nu~~t layer in the ~eclimentar~' column includes all
be~7s from the 1>asc~nen~ surface lu the top of the ,Jura ~~ic 1~Iorrison Formation.
"Che beds are maii~l~~ of L'eiiiis~Ivanian. Permian. and )uia~~ic ales. Relatively,
fei~~ ti~~ells have heern drilled tlir~~u,~~h the: Paleozoic section. but logs froi~l these
cvelle, a~ cell as r~utcrr,~~ ir~f~~rmation from the edges of the I)enrer basin in
The 1,, lati~er: he~ieath the PiE:rre Fc~nnation, tk~~ resi~ti~~it~~ tends to increase~ ~~~ith incic ~~in~~ depth. tllou~h this }~1rt of the section ~i~a~~ ~:er~erally be
di~~ided into an uE~l~er and lu~~~er part on [h<, basis of resistieit}~. "I~he la~~er
irninediatel~~ under the Pierre. the L~ layer, con_i~Cs of the Dakota. Penton, ai d
iViobrara (~~i~inatic>i~s, of Jur~is~ic to Cretaceous ale. 7~he third layer is raised
sandstone. shale ar~d limestone. which despite the l~etero~~eneit~ of rock t}peg.
varies uniTorml~~ over the area. TEle thickness is con~iste~itly about 1.100 feet
in eastern C~~l~~radu l>ut ihin~ to the north and east.
I~'rct~it~~: 28.
~/~
Z~al ~ 8
~~~
I
I'~ /4I\
~'
-~
\~~
_5
I~IiOI'I:IiTIhS OI' ~IL 1'IP:I.D~ :1~'D ~~f[IiIR l~A~'II;O\CIE:A't
~
,~
~
o
V
I
~6
x
—
,
n
I o. -~m~z z~
N
~~
----
a
~
~ W
~ o
~~
_.
I j:`o
~
o❑ I--I—
— -~
~
a
i
~
__ __._..~f , ~
a
a
I
I
o
Q
~~
a
I
a
W
o
=
~
I
cn
~
a
3
8881
-oa~ -~
~ a
~--~
=
o~° °a
c~
Z
o~
0
..~~
o
o
r
a
oo'
l
z
—~~
~
7
~
°a~ _
a
i
O
O i
9
o~i
~_
.
= '__
~
W
~
~
a.
J
N
z
v
3
Qu_Lr.~rLrLl~ or ~r~ir; Co~_or~~vo ScxooL or- l-li~rs
~
k
5t
sk~ale, sandstone,
limestone,
1,000'
Carboniferous,
fountain, Lyons;
Lykins; Morrison
L~
"I'he averae~e electrical parameters for- 5-1 wells which penetrated at least
part of the Paleozoic section are listed in table ~~ (from Jackson, 1962). Although the wells are sparsel}' distributed over the area, and the reliability of
individual values for the avera :ed pro~~erties are subject to errors comparable
to the change from well to we11, it is possible to contour the areal variation
in properties for several of the units. The most z~eliable data are those for the
1~2 and L~ units, primarily because the lobs in Che li anc~ L•. snits ire usually
i~~complete. Contour maps showinb variation in longitudinal resistiviCy and
the coefficient of anisotropy over the Higb Plains area are biven in figures 30
and 31. It is interesting to note that Che resistivity tends to increase both to the
east and to the west from the center of the area. ".Che minimum resistivities are
not found in the center of the Denver basin, however, but considerably to the
anhydrite
shale, sandstone;
limestone
1,100'
Cretaceous
Dakota. Benton,
Niobrara
Li
Jurassic
s}iale,. siltstone
-1~,(~0'
Cretaceous
sandstone. shale.
siltstone
I,ithology
Pierre
1.000'
"Chickiie~s
M
Cretaceous
A~;e
Fo~c Hills, Laramie, Denver,
Araf~ahoe, O~;allala
T'ormati<>n
names
I~'ornaatiora ~irimes ~ro~rz the 1)e~i~uer basi~a
U
electrical
desi,~nalor
T_nt~Li•; r.
sun~rnariz~cl in table r.
the ~~resC. and outcrops of similar rocks in Kansas and Oklahoma, indicate
that the section is i~~ainly composed of claystone~ and siltstones interk~edded
with limestones, doloin~l<~, auc3 anh~~drites. Wells withi❑ the area under consideration indicate that tl~e strata. except for ehe evaporites, are continuous
across the basin. Althoug~l~ the re~i~tivity~ varies drasticalh~ from bed to bed.
the layer comprises a uniform electrical unit about 1.000 feet thick.
As in other cages. the section n ay be divided into a set of ~;eoelectric units
~u ~tihich the resistivitt fiat de<«ases with de~Ch from the surface, and then
i~~cic.lses ~a~ith depth once a zun~ of mini~7~um resi~tiviLv Ithe _l'1-lay~erj is
gassed. 7~he formation names ~ssinned to [he electrical L. ~~I and L la}~ers are
PROZ>rrTtr:s or Oit_ I'rj:r.~~s ;a~~» '1'fii:lr I~~~vtao~~~L~1•
QU9RTERLY OF TAF, (,OLORADO SCHOOL OF
III\TS
Location
510- 2,6b01 LL)
2,685- 3,7-15 t LL)
3,750- 5.550(LL)
1,220- 2,3-~~6(SiV)
2,350- v,r10(SN)
1,112- 5,650~LL)
4, Chase County,
Nebr.
5. Garden County,
Nebr.
12, 22N, 46W
7. Cherry County,
Nebr.
23, 25N, 35W
6. Grant County,
Nebr.
1, 23N, 3<W
640- 1,4861 SN)
1,490- 3,070(SN)
3,075- 3,i81 I LL)
3,705- 4,681(LL)
1,160- 1,936(SN)
1,94- 3,356(SN j
3,360- 4-,0~0(SN)
450- 2.010(S1V)
2,015- 3,~-1,1 I SN j
3. Perkins County,
Nebr.
23, lON, 39W
io, ~~r, ~-iw
730- 2,880(SN)
2,885- 4-,400(SN)
4~,4~05- 4->590(SN)
4,405- 4~,590(LN)
4,405- 9~,5901_LL)
4,,590- 6,310(SN)
x:,590- 6,310 LL)
5,090- 5,720(LL)
5,725- 7,45r(LL)
67~- 095(S~~
900- ~1,Oo0(Sl~T)
llepth
inter~xl
2. iVIorrill County,
Nebr.
I, 21N, 4,9W
1. Banner Count}',
Nebr.
15, 19N, 53W
Index iVo.
of well
iVI
Lr
L~
L_,
VI
Lt
L_
iVI
Li
L~
11~'I
L,
L~
1I
L,
U
NI
L,
L~
L•~
715
12.1
51.2
2.05
3.=11
291.
2.74•
113
562
3.30
5.r2
2.05
4.95
4,~8
24,5
37.5
3.37
313
4,.2Z
4..76
221
2.56
~~~.82
8.54
162
1.97
2.09
12.3
1.83
2.r5
5.0~~
2.52
3.-1.7
1.39
2.50
2.4.7
719
220
2.7-12.65
5.<13
2.62
3.81
4,27
16.3
L14.
1.20
3.60
7.71
122
1.19
L78
1.02
1.20
4~.9
1.22
1.22
3.34
L15
1.2~
1.22
1.38
1.36
1.84.
2.14=
111
L02
L02
1.03
1.31
1.27
2.4.0
dine] cicnt
resistivit} of
anisot ro p}•
verse
resistivity
5.63
2.70
6.5~
6.92
9-1.1
I.Oil r,~ll l]- ~.Ot'~1-
T P$[ls-
5.50
7.86
2~.7
152.
De~i~nator
Tat3r,E 8. — Reszslivr~ties co»apiled ~rona logs ri~n in
wells on the High Plains
east of the ceiitei•. The change in resistivit~r a~~~ay
from the center of the area
possibly reflects decreases iu water salinity near
outcrop areas.
~~
~o
Lu~~atiun
16. Lincoln County,
1Vebr.
2, 9N, 32W
x-50- 700(Sid j
70~- 1;87:i(SN)
1,~~;0- ~1~;3961,S~1~)
160- 1,530(SN)
15. Hitchcock County>
1,335- 2,5301,SN j
1Vebr.
2,55- -1,120(SNj
23, -1\, 33W
U
1~I
Li
M
Lr
I,.
M
Li
L~
L~
Z,i~(J- J,U~~U(Si~T ~
8.89
~.c,l
16.E
2.22
~~,56
102
3.4-U
4-.8=1
1=1.0
IJ.i
2.b5
6.82
12.5
r.-10
>.38
511
1.~5
3.29
3.80
L10
1.06
1.79
1.10
11~>
1.61
1.03
1.211.55
1.~~
7.U1
X18
x.13
5.~,-1
1.1l
1.37
2.30
3.6=~
1.42
6.18
L~
3,102- x,165(S1~~ j
M
L~
1.02
1..0 ~~
529
2.33
U
M
560- 900(SN)
905- 2,300 t SN j
150- 1,6-10(SN)
l,fi95- 2,70(SN j
1.79
x.14•
1.6.5
L~
3,570- ~1~,7101 S~)
~.4~5
3.09
1-•~2
1.0-1~
l.~.1~0
1.mil_
L3~
1.30
L96
116
11-1~
2.5h
=~.9~
2.22
1•.80
6.92
3.30
615
1.~1
5.1.~-1,
0.01
9.15
t ~„i~y,
Lun~~itu- Cocffident
dine]
rc~i~tiei[y ~~f
aniso-
5.1.0
2.30
9.6-1•
1L9
6.01
10.1
57.7
6.9~~
10.=1
61.0
"Crxnsverse
rc~istivity
U
VI
M
Li
1NI
L~
L•~
iVI
L,
L~
lle~i~nator
5~
730- 7.,1.60(SN)
1,165- 1,700(SVj
160- 1,141(SN)
1,612- 2,]10(SN j
7.,270- 1,930(S1V}
1,93- 3,365(SN)
;>,X70- 3,9601 LL)
1,70- 2,370(SiV)
2,',r5- 3,621 I S1V)
3,625- -~,2-15 I SN j
Uc~,th
interval
14, Hitchcock County, 190- 1,3=1~0(SN j
1,3-15- 2,550 ~ SN)
Nebr.
2,554- -1,61o(SNj
26;1N, W
iJ, Z1~, J / W
13. Dundy County,
Nebr.
12. Chase County,
Nebr.
1L Lincoln County.
Nebi•.
10. Brown Count}',
Nebr.
13, 26N, 22W
9. Hooker County,
Nebr.
5, 23\, 31 W
2. Sheridan County,
lVebr.
17. 26N, 46W
Indcr h~o.
ofwell
~~:~I3LF: `ci (COrit.~
pFOPI:RTIF,S OF ~IL 1'IISLDS AtiD ~T1II~.IR ~1~VItiO'~AII~:1'T
Location
25. Laramie County,
Wyo.
15, 19N,67W
2~1. Goshen County,
~30.
lU. 20N, 6~1~W
23. Goshen County,
W~~•
32, 30~~, 60W
~,150~SN)
4,300(SN)
5;200(SN)
7,020(SN)
500- 1,6=~-01 SN)
l,G9S- 8,2r5(SN'1
200- 2,7.50(SN)
2,155- 7,95(S\'j
7,9-1a7- 9,0201 S~~')
1,0003155-1,305,205-
U
M
U
AZ
L~
M
L,
L.,
L.~
M
Lr
22. l2cPherson County, 950- 1,450(S_1V)
~~ebr.
1, 65- 'x,3501 SN)
25, 19N, 33W
U
M
U
~~I
L,
U
~T
L,
L,
U
M
L,
L~
510- 994.(51~~ ~
1,000- 3,c374-(SN)
146- ~0=~-('SN)
t,10- 1,26 I S\)
1,20- 3,510(SN)
650- 750(SN)
7~0- L1301S\')
l,l»- 1,57~(S1'1
1,80- ~,/:~0 ~ S~)
M
L,
319
9.72
39.3
4.56
17.8
58.5
132
10.0
29.5
-1.~r3
2~.7
16.9
3.9~
9.7-19.90
x.59
13.0
x.72
5.06
J.=1~4
5.04
10.3
2.29
4-.41
25.6
3.8~~
112
17.9
13.0
~.6o
11-1
3.82
10.~1~
15.~,
3.-18
5.61
J.~~
3.97
6.25
5.21
=1-.01
6.10
4,10
6. ~~8
L29
1.12
L24~
110
L26
1.05
L13
1..26
1.60
1.1~
1.4 3
1.2-1,
1.08
L26
1.81
1.01
L4-6
]..-1~5
1.U7
1.66
1.05
1.06
1.32
1.29
1.03
LEI<:1,
troPY
Lon~itu- Cocftidin<tl
dent
rcistivity of
ani~o-
2.11
.1~.r3
13.0
50.8
transverse
r<~si,tivit~~
2.32
6.10
2U.r
129.
nator
interval
675- 1,020(LN)
1,02 - 3,960(S1V')
~~`~Q
7~r~~t~~
~~ABI,E ;(Cont.~
QUARTERLY OF THE: COLORADO
SCHOOL OF ~TINL'S
90- 74~0(SN)
715- 1,24-0(S~~)
1,260- 2,720(SN)
2,722- 3,3601 SN)
21. Blaine County,
Nebr.
22, 2-1-V, 21W
20. I'i•ontier County,
Nebr.
21, ~_~, 29W
19. Blaine County>
1`ebr.
23, z~N; 27W
~1~. Lo6an County,
Nebr.
~6, 181. 2c3A~
1~~. Lincoln County,
i~ebr.
2~; 1`21, 30W
Indeh No.
of w<~11
GO
F
Locati~~~i
U
9c>0l.,r25~,21U<>.9l 2.
1c.9
:>~~.~1
1~•~
J.~~~~
7.07
C>Z~~.
L<,1
I.61
G.c7
1.7.0
13.2
i~.,
:iU
6.~1
1~~.~~
I.~G
3.1~
2.39
11.-1
33.<>
5.;')
6.6Z
1..11
1.16
1.30
x.39
;>.92
10.0
I.JU
116
I..Uj
1.01
j.~(
1.13
1.1~
1.68
1.35
1.12
1.08
1.02
1..10
I.3
L39
6.0~
l..~c~
1..02
LU5
1.0-1
L0~>
L09
1.33
1.0<>
1.1r
11-1~
~~20
~~-.j~~
5.99
L32
3.c2
1~~.3
3.IG
6.0r
6.:~2
2.9~
~.~~
6.6&
`?.~0
5.~-1
r.2-1,
26.7
Ei<3.~
61
Longi~u- Cucffidinal
dent
re~i=ti~~itc of
6.)2
2.c~2
1U.~
1 J,2
10.E
x.26
16.~~
>O1.
f.1~
6.2
2.03
]~.02
16.E
3.i7
10.r
7.6U
-1,06
7.65
"Craos.
cer,e
re~i~ti~~ity
L,
1'1
L,
~~~~~OJ- iii=~Jl I LL~
1,72O I S~ j
72~»(S ti)
~~.921 ~ S'
02101 ~ti
1I
Lr
L,
<>6U '~,I Bbl S1
l.l~>0- X22O I ~\
522;- 55.9U1(LL)
32. l~Ior~aii Count}',
bolo.
32, :>.\. »~~;'
33. dS'eld Count},
bolo.
8, 2N. fi7~~'
L,
L_
7,07- ~>.~>-11 1 ~V 1
t>,~> l~- 9.xO;3 i
li
'?O. ~\. 66~~~
950- .11~i01 ~\~ i
I
~1
L,
L.
~l-0U- L32O I S~ i
:1.325- ii.101 I ~ti i
:30, l'Ior~;an ~:ount7'.
Colo.
31. ~✓eld C:o~nCy.
ti
1I
L,
~1~- 2,1 1 I ~~TI
2.195 ~>.r3] I ~!1
3.735-11.06 1 g1 ~
2~. Feld County,
X010.
~, 1 I ~. (>fit~
I~~
I,~,
~
~~T
1.1 G(? I S~ 1
1,15(}~
Ci.lfill5\1
3.111 t LI,i
6Z0l.l~i~l,.l~i~Ci.165-
1,>n- ~.~5O1~~~
i.~SZ- 8.~5:~ ~
l3. 1 ~\. 6~;~4'
1~I
l%T
lI
I_.,
U
lI
L,
llc~i~n~itor
20. L,~~r ati Counh`,
~olc~.
2b. 11 \. ,~t`v'
1,900 3,995(5~')
~,9~2 1.:,7O15 1
~2~- 2.~1~IS11
2.,20- c>.66U(S\ ~
~~.fiCiS-10.30t)I ~\ i
llcpth
inter~~al
Z7. Laramie Counh-_
~Vti o.
2E. Laramie C:ount}.
~I}o.
2, 1:>1, bf`~%
Codex h~u.
vt n~ell
'r_st;r.~~. ~> (Coin. i
Pror~zitrli~:s or Oz1, Fl~~:t.~>s ~~~~ TEi~~:zf~ E~~~1};oti~~t1;~~[•
Location
JJ. J~ W
~I.l. Yuma CountS~,
Colo.
21, IN, ~ 1~W
-1-0. Logan County;
Colo.
23, 10~~, 53W
2~~~.
39. Adams County,
Colo.
1,1 ~~02,9753,7505,0 ~~0-
2,9r0(S_~)
~~,i-1~ 151)
5,0361 LLj
6,6~c> I LL)
~I
LI
L_
L,~
li
ibI
L,
L_
L_
400- ~4~0(SV)
8-~-5- x,955 ~ ~~'1
x,970. 6,110f S1')
6,110- 7,.636 I LL j
1.~,
(.Jii~~~ J~')
~1
I,,
U
11~I
L,
U
VI
L,
1~I
L,
L.,
U
'V1
Lt
L_
L:.
Lt
~I
U
ll.6
GC~.ii
2.65
715
5.59
3.30
13.~-1.
G71
2.61.
8.90
3.~9
192
~~'>._-1
-1.r2
2.15
.<;8
2~.9
201
~.!~1
I.-ICJ
4,53
1.53
~.3~
=1~0.~
50.3
2.38
1~.~13
-1.21
2.9c3
1L2
=1..r5
2.33
~.~5
3.39
10.P>
21.-1-
;1.34
L96
3.0-1
2.28
1.0.3
~-I~.JJ
l.`LO
~1~22
L34~
2.21
3.01
9.69
5.5°
1.52
2.92
3.1.9
9.46
1.07
122
3.6b
2.20
1.03
1.17
1..40
6.85
1.11
l.J-~~
1.06
1.26
115
1.05
1.27
1.19
1.06
12i
1..07
L33
1.3-1-
1.0=1•
1..04•
1.60
3.60
1.3g
1.ZJ
I.IO
1.01
anisotropy
Lon~ihi- (:oeftic1'ina I
cent
resi~ti~~ity ~~{
IJ.J
resistirity
Trans~~~>>~;~~
5.91
2.11
5.82
561..
Dcsi;~nator
7,=120- 7.512 i SN)
(,1-~~2
x-30- 5,2i0(Sl\~)
~,2l5- i,l>SIS~')
2~0- 1,720(SI~~)
1 725 r,::1-95 I S1~T)
~,~00 9,2~0(SiV)
500- 960(SN)
960 6,231(SN)
6,23~c3,l-1lIS'~')
3r. Adams County,
Colo.
1-1-, 2S, 62W
~~~,. Adazns Count}',
°°
Colo.
ir, 2S, 65W
450- 6,510(SN)
6,:i1~- 8,221(" SN)
3,222- ~3,6001SN)
~6. field County-,
Colo.
Z1. 1~. 63VJ
1,101(SN j
3,So5(S~V 1
5,286(S~')
5,000 i SN)
7,270- 7,866(SN)
J.3JJ- (.Z~~ (, SN J
I.~OS- J.JJI(S~' I
420- 1,600(SiV)
Df~pth
interval
TABLE $ ~CoTlt.~
QuarTE2zr,Y or Tiir CoLOF~a~o Scxooz. or
~Iz~vF;s
3~. ~rashin~;ton Count}', 505Colo.
11057, 2S. ~2W
3,:1905.290-
1-~-. IN, J~~~
COIO.
3-1~. Vlor~~an County,
Index Na
uf ~~-e11
~Z
I.i~c~rtiun
LO')
1.03
8.~r3
~~.05
102
5.r>>
LOl 5- 1.7i~ I S1~ I
1,7~>O- .Ci!(115\ 1
L'
Vi
50. Lariii~er Cc,untST,
Cnl~,.
LO'?
1.06
1:12
613
6.92
I.~,S
l~. l
>~3.~,
7.22
U
5.~{
~'I
L,
~0.~>
L., 1.-180.
1.01:>- 1,6911 LL I
1,695- 7,.1O11LL)
1.105- 8,90 ~ LL 1
>,910-10..6161 LLl
9~9. Weld Count~~.
Colo.
2"r, ~~, 66W
7.02
1.O~t
1.26
I .G7
5.1~
I.;~>
(i.l ~
12.~>
7.39
2.IU
').~;(
;>::>. ~
C;'
~I
Li
L_
L3f301 5\' 1
~.lZl ~ S\ ~
(i.001 15` ~
u.~>O"r 15:1 i
1<~f>1.3<3~5.12tiCi,00:i=1<~. P~eld Count°.
Colo.
lc,. 1O\'. ~EiV~~
l . l (1
1.1
].~U
3.6~;
1.12
2.G1
6.Z1
19.x;
5.3Z
:~.(i
1.3.9
26<>.
~9U- L~>~,6 i S\ I
L~90- 6,5101 S\ I
G_515- ~.2-13 ~ LL l
>21.-J-- ~1022<~ t LL I
'1~7. Weld Count}'.
Colo.
19. ~>N. 61 W
li
~'I
L,
I..
~'1
L,
L,
L.,
2.10~f S1')
2,<~001 S.V')
~ 650(LLl
:~.~~51 ~ LLl
9702.1102.900
3,6~~>
-1-6. Yuma Count.
Colo.
~l, I.N. -1 ~W
1.02
1.1
7.19
2.11
LOi
1.>1L89
:>.fif>
21r
Ci.60
4T
Li
92U- 2,~i61 S\' 1
Z.5~>0- 6.;~~~OISV~1
-15. Yuma C~~unt~~,
Co(o.
31, 1\. 1fiW
'?.1~1
I .~6
Z. 16
:~.~l
l.'?(,
1.1.51
1.~0
.>2
l 5.>
2.10
]0.0
;~~.L
l°T
L,
I,~
50~- 2.~~7,~ ~ S~~ ~
2,x320- -1.300 r S1~ ~
l.:>O:i- 6,U0c I S1 I
4~-1~. Yui~~a Count~~.
Colo.
1`0', 7 S. 1,~~T1~
2.~:~~
5.73
:x.19
`33.:>
1.06
1.ZO
1.')1
ii.:~Z
1.06
Z.20
1.2~
1.<<~
3.:>(i
1-~.J
2.03
~.1~
L~~n_itu- Curtii~lival
cunt
resi~tici~~ ..1
aniruir~~~~c
1:10
2.51
1Z.>
~ iii).
2.»
2~.7
'Cr.in~ccr<c
re~si<ticit~'
~'I
L,
I~,
L~
'~~I
L~
I)~;i~,ntito,'
250 1.50(5~ 1
1.59 - ?x.15] ~ S~'1
:>.l~_~ 5,55] 1 S\ J
~~~.ZJ~)- ~);JJU (I,1 ~
900- 2.5101 S~~ 1
G_5~1~- 12-1~f5,11
I)~yrth
intr~rcal
~)
=1.3. ~'uma Countti'.
Colo.
1 t. :>S. 12~~
4~2. Yuma Count~~_
(.ol~.
Index V'i~.
ofwell
AI;LE ii 1 Cont.
~POPI:RTI1?S OF ~II. 1'IP:LDS :1 ~'U TfIISIR 1'.tiVIRO\III;AT
_.
-.
I.~~catinn
j-~~
_....
L,~~
__...
__
3.6361 S'~'1
-1~.7~0 i 31'1
,x.£,96 t LT 1
~. 1 161 LL 1
l~l~
L,
L!
I„
___
L3~1-1.96
I -l-2
6~.,3
)Z]_
.~.<<~
r.-1-3
29.-1-
2.Z~
Zj.l
_._
x.37
2.`~Z
~n t>
1~~.2
~I'ran:~~ci~.:c
re;i~ti~it~
o
_...
1.15
2.~~9
~1~.r l
~~.1 i
~Z ~
~~.~s
3.r9
13.)
Z.O6
1-~~.J
3.96
l.(~
~
11.6
___-_
1.08
1.31
1.r-1~
2.63
l.oF
,~ ~ ~
L-I-O
L=1~6
I.~J
1.33
1.~G
l.11
~.-~~
1.~9
tro;~y
c,linal
cicnt
re~i<ticit~~ of
aniso-
L.un_~i(tt- ~.ue{~i-
U IIl
~1-e11 ]p~S {I'OII7 d~)jJ 1-OXIIYldYe1V
Itt CUI]t~)jlE'C] CICCtl'IC81 ~)L'O~1C 1'11ES
c1i'e ~i~171fiC8T1 ~, OI- If t~7ey
~Etutcn iu ~i~;ure 3~.
lun,~ituc~it~ll tESi~ti~ih anc~ anisotrop~~ for the
Dakota to ~rio})ra2'a seCtioll is
~»f~~'1~~ rfpie ~ent ~~~attei i, the rc..,ult of erc•ors in compilation. A
profile ~~~ith
t}~c ~~alue~ f~~i• resi~tieit~~ ~n<I aui~otrop~~ of tl~e
Pierre Shale is shown iti figure
:~5. al~~n~~~
~. ~~ith a~~~c ~ier~alized crops section prepared by Harthill, using
lugs
fr~~in "I'uicuship 2 1oi-tli 11~arth~ll. 196~~1. A similar
set of profiles for the
tt> 11~E'~1 C~I~~c CCt1Ce
2~0 ~cell~ iii "I~o4~n hips 1 and 2 \oit}~ across t}~e Deny-r,i
basin fi-~m the Trout
R in~~e to the 1~ebi i~ka border (fig. 2~)j pi imaril}
to determine whether well-
J)t C 31~E(~ ~U~~- CUtllj)1~cit1U11: 11'P,le IIl BC~f
1~~ i~ti~it~~ i~u ~,t~op~ fi~lcl plots for the X11. L, and L,
unit_ listed in table
f~ i~~ ~ho~~n to fi utc~ 3~ `;.I. 7'}le rf i tivihcs
end inisotropiPS are interiucd~ate t~~ thu~c ob~r•~ red ~❑ the past Gulf (oast and on tl~e
Colorado I~lateaus,
rc(lECting the fact ifiat tl~e. de~,iee of ~o:t-depositional
d~astrophi,in in the
I)u~~cr hi-in 1~a~ h~~e❑ mole intense than that in th~~ Gulf
Coast, but considerahl~ I~.~~ than th~~t in the (oloi~ado Plateaus
--
5 I. L~, an Count~~.
Colo.
1. ~ A. ,~Z~
~-10'3.6-101.7. 0~_9UU-
J_.,_
I~•6 i>- 6.~~Jc`i f L,I.)
~-~~~~~•
>~~• ~5. ~i~W
_
L,
:.loo- ~.~~>~ ~ Lr_ ~
:~:~. ~-~~„~~~ c~,u~,r,~.
L,
L..
1 6~~- 3.72 (S1 I
3.X30- 6.OG1 I S1~)
~~1
L,
3.5ll i S\)
1~I
-
1~~
i
L,..
llr~i_
nator
(
?.)(
(S~~ I
!~)U- l.~)IUIS.~)
i Jlj
1,36
1. ~~(S~~l
t,jU:il ~~ 1
j2U
7.J~~1 (5~~~)
i.~-~J
7.J~J
l)c ~>th
inter~~al
Qli:1RTF.RLT OF THF, COLOP,ADO SC~IOOI. OF
1`II11':S
C~~l~~.
(i, iS. 16W
.~'~. ~IC ~,Hl'tiUtl ~.OUtlt1".
i:i. ~)S.
O~U.
J~. f11~ ~,Fjl'~utl ~.OUtlt}
~~. ii~. 6ii~~
...-..-
in~l~ x \~,.
.,f ~~cll
~ ~'
/
~
056
1.04
°.i5
/.52
0
°/.82
~ ~ ~
0
Z
~.o~ ~ ~~
°~,
~
~
0%~25
o
~
~
2
~
~ 230
3
°
3.~s
4
°"~
3.3/
0
/
0
NEBR.
{~5.
3.97
~
/
3.48
o
~L~S~
os3
2.52
0
~
2.22 ~
2.29
X3.82
s.ae
~~ ~
10 7 5
C0~0.
~°~ ~ ~ ~ 4.43
~
3.70
3.~s
~
3.32
I
~
3.36
5
oss
7
~
6.25
595
34~
~°
~a ~~
4.z3~
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o5.a4
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6.92
7
KS.
NEBR.
~--io2~10
lO 4
//~
I~Q
s.i5 ~~
/ o
2.75
a~ 0.57\ o
246
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~2.3~o.O/ 3.68
304 5.is
~33~
o
o
a
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~
/O.~
5.74// 2.89 ,-,
0 / ~
0
s.n ~ ~3
~
oz.sz
~
~
~
3
~
\
4.76
F~cuae 31.-- Contour map of values for tl~e longitudinal resistivity of the prePierre Cretaceous-Jurassic section in tl~e High Plains, compiled from
electric Ions. Contoiu~s are in ohm -meters, and the individual values are
indicated next to the well locations.
~°~
/1.4
goo
15.4 ~~.oi
~~~ o
\
I
.84
S~Qa~
s.ai ~
~
/02.58
/~.09
~
~
8.54
~~
7
~~_
°I
a.o4
/
j
~
10
/
648
°
~ ~
WYO.
'o~
Fzcox~: 30.-Conto~r map of values for lonnitudinal resistivity of the Pierre Shale
(or ec uivalent) over the High Plains, as com piled from electric logs.
The contours are in ohm-meters, and the individual values are indicated
next to the well locations.
COLD.
~.3B
~
~
z.a n
L83
~°
x/.50
L89
o
/.20 [96
2.03
298 oJ.~_ _~~-O\
~ 4
0
5.30
~/~3 3.09
~~
~
, J \ 2o/
-.~
/.78
~0
°~
~
2.60
/l.97
\ ~~~ -~
0/.39
o.s2
4.22
°
5
5.is
~~-5
~~~
/ ~-~
~ ~ 2.64
4.dg
0
o
3 Gb~.95
03.92
5.05
5
°
4.10
4.0~
°
~
WYO.
2.
i
io
LONGITUDINAL RESISTIVITY
ioo
i000
QUARTERLY OF' THE COLORADO SCHOOL OP' 'VIIN~S
The detailed studies show clearly the trend from hinher resistivities along
the Front Rance to lower resistiviYies east of the center of the Denver basin,
and a reversal in the trend still further east. It is interestinb to note that the
data for the lower unit, consisting of shale, sandstone and limestone beds, show
considerably more scatter than do the da±a for i~he overlyin~~ Pierre Shale. The
avera~~~e scatCer in resistivity values for the Pierre Shale is approximately -!- 7
percenC about a trend line, while the average scatter in the underlyino~ unit is
± 25 percent, nearly =1 times greater. In past this ma}' be cau~ecl b }~ a diffei-ence in the nui~iber of values sampled from tl~e lo~~s for the two electrical units.
inasmuch as samples were read from the logs at intervals of 1.0 feet, providing
-100 to 500 values iu the Pierre Shale and 90 to 120 values iv the Dakota to
\Tiobi•ara inCerval. The statistical reliability of an average computed fror~i a
lai•~~e number of samples varies as the square root of the number of samples.
Therefore, if the scatter in t)ie two cases r•e$ected only variations caused by
sainplin ,the averages for the Dakota-to-Niobraz~a sections would be expected
to show about t~~ice Che scaCCer of the a.vera~~es for fhe Pierre section. Because
the scatter is considerably more Chan twice as ~~reat, the higher degree of
I'~cuse 32. — Scatter plot of values fur ]on~itudinti] resi=tivity and the coefficient of
anisotropy for sections of Pierre Shale pencU'atcd by wells in the High
Plains at~ea.
o.~
io
REDUCED
ANfSOTROPY
66
67
0
0
10
0
o
0
100
1000
LONGITUDINAL (RESISTIVITY
0
o
° o moo° °
0 0 00
00 0
°
ao
~
o o
a
0
0 a
o
a
Fronts 33.— Scatter plot of values for lona~itudinal resistivity, and coefficient of
anisotropy fur the Jurassic-Cretaceous section from the Dakota to the
Benton formations for wells in the High Plains area.
i
REDUCED
ANISOTROPY
10
scatter for longitudinal resistivity in Che llakota-to-Niobrara section most
probably reflects a greater real variability in resistivity from bed to bed. This
in turn should contribute a higher coefficient of anisotropy in the Dakota to
Niobrara section than in the Pierre Shale, and this appears to be the case.
Values for the coefficient of anisotropy in the Pierre Shale also show coiasiderably less scatter than do the values for the Dakota to Niobrara section.
An interestin feature of these data is the well-defined maximum in values for
the Pierre Shale in the area of Rannes 52-53 West. A similar maximum is
noted also in the data from Tier 1 North, and possibly may be present in
PROPERTIES OP' OIL FIF,LllS AND THEIR ENVIRON'_VI~NT
I
0
a
o
10
o
oo
o
p0
~,.
—1
Pa/soroic Section,
High Ploins
—~
LONGITUDINAL RESISTIVITY
°~
°
o a
o
Qo
a
°o
o
0°
~
0
o
0
0
0
0
o
00
T~i
1000
QliAFiTERLY OI' TII~ COLOPADO SCHOOL
OF MINES
1`he behavior of the electrical properties foi•
the Pierre Shale snay he betCer
understood by considering; typical resisti
vity pi•obabiliCy density curves for
typical wells along the ati•ip under consideration
(fi~,~. 3r). As one ~~oes east
towards lower Range numbers),the probability
density curves have proni•essively hi~~hei• peaks at lower resistivities. The
higher the peak of a distribution
curve, the lower will be the coefficient of ~nisoti
-opy. Thus, it appears that the
Pierre becomes more uniform and more conduc
tive to Che east.
The anisotropy-resistivity field plots for
data frozr~ Tier 2 North show
considerably less scatter than do similar plots
for the entire High Plains area
(see fins. 38 end ~9j. This sug~~es~~s that the lar~~~
scatter shown b}~ the other
such pattern plots represents a real variation
of electric properties over large
areas and not erz•ors in compilation.
the data for the Dakota to Niobrara sections also.
'Chis ivaxin~iuin apparently
reflects a trend line 11on~; which the Jurassic
and Cretaceous beds contain a
higher than normal pzoportion of sandstone beds.
FtcuaE; 3~.— Scalier plot u1 vtilures for longitu
dinal resistivit}~ end ~~o<~f}icierit of
anisotropy for sections of Paleozoic rocl:
penetrated b}~ webs in the
Hilt Plains area.
0.1
io ~—
REDl10ED
ANISOTROPY
~~
o
~n
wOa
U
oQ
~~
wz
ti p
N —~~
o
~- r
0
0
°
O
0
0
0
o
o
o
~
0
0
~
o
o~
o~
ro
~~
00 o
~e~
ooQ
°~ o
rl 1 1
~
I
O
¢
0
3
c
N
~
3
m
in
¢
~ J
Q
~ O
F—
Q
3z
O
F-
~ ~
Q
N
3 ~
~ O
Z
~
~
~
\
~ ~ ~
~ ~ ~ ~
O
~
O
h
69
~ 3
G
C t°
.~ :6
~
~ 3
r
J
^"~
W
y-f.
w U .~
J~
~~;
~~.~:
J
~ ~,
;,;~ —
a
~~
~ ~ y
'm
J r
Y C. ~
O~~
s.
v ~.. ~
~~ .~
~
c~
V
~'
y~
v
~ v
~.
w y
F.
"~ c~
~ v
~
:tl .d
~ J
O •^
~~
II'IIIIIII ~jO
O
Illlllll~lo
°o
v
° ~r'
I
PFOP~RTIES OF OIL FIELDS ADD THEIR ENVIRONMENT
LONGITUDINAL
RESISTIVITY
35
-~1
O
20
io —__
5
_~
_
—_
i
_ 7
I
—y—
o
r ~ __~-- ~ ..
0
0°
o 0o ~
o0
1
0
~ 0~ g~
o
°~o
~--_ —~ ~
oo °o
0 0 0 0
i -~~o
0
z
COEFFICIENT OF
AN I SOT ROPY
i.ao
0 0
000
o
R70~N
R68W
a
0 ° o °o ~
0 ~° aoa~
R64W
R62w
~
°° a °
o$ oa° 0 o~ o ~o
$ ~ ~a~
o 0 a ~aa 0o
-~1~ 1_
R66W
R60w
R59w
R56W
R54W
~ o
R52W
o ~`-~
ao~ °—
—
RSOW
R48W
LOCATION IN TOWNSHIP 2 NORTH
s000—
—
—5000
Seo /eve/—
—
i
—Sro /eve/
—
-5000—
—
—
— -5000
Ficua~ 36.— Profile of values for
longitudinal
resistivity
and
the
coefficient
anisotropy
of
determined
for wells along
T2N in eastern Colorado, for tl~e portion of the section from the Renton to Niobrara
formations. The interval o[ secY~ion for which these values were compiled is shown as shaded in
the cross section (taken from
Harthill, 1967).
-x.E:-.4
c
w
.'v
N
c
,~. ~
r ~,~.,
n
✓
n
~ ..
G
~
,~_
m
• f
H
C:
v
~
G
~
~
c. ° ~ ~
c
~ ~,~~
N
~
G~
~
~
M
o v; ~ 3
~
~
N
~
F ~
C:
r
~
.=- `G
H
i
QUARTEPLY OI' THI: COLORADO SCI300L OF NIIVIS
6,500- 3,190
1,400- 6,5r0
6,600- 3,760
2,020- 6,620
910- 4.,620
4,630- 5,&50
4,820- 6,510
6,5r0- 7.360
7,3r0- 8,200
1,X00- 3,570
3;600- 9,060
1,300- 3,090
3,100- x,000
County
3. Indiana
County
4-. Indiana County
5. Cameron
County
6. Mercer
County
7. Payette
County
~. Fayette
County
1,380- 6,460
2. Indiana
442
79.1
28.1
73.7
170.
1,800.
606.
65.3
204.
'11.2
39.1
4.4..0
72.0
39.5
190.
~A1
205.
resistivity
3,00- 3,050
0,100- 3,4.00
~,4,r0-12,310
Transverse
Depth
interval
L Sullivan
County
Location
37.6
65.7
22.6
63.6
139.
1,490.
303.
50.2
154.
31.4
31.1
38.5
56.0
32.7
151.
2.33
72.3
resistivit}'
Coefficient of
L09
1.09
l.11
1.0?
L10
1.11
1.42
114
L15
1.14
L12
L11
1.12
1.12
L12
125
1.68
anisotropy
Longitudinal
TaBLL 9.-Resistivities compiled froan lobs run in deep aoells
in
Pennsylvania
In addition tc~ the detailed studies made of electric log resistivities
in these
three areas-the east Gulf Coast, the Colorado Plateaus, and the High
Plainsa few scattered compilations of re~istivities have been made, covering
other
types of rock sequences. Anderson (1965) has compiled averabe resistivities
from lobs run in eibht deep wells in Pennsylvania, at the locations
indicated
on the mad in figure 4~0. The sections lowed were mainly of
Paleozoic abe,
consisting cif Cambrian to Pennsylvanian shale and limestone beds.
The
compiled resistivities and anisotropies for these eight wells are listed in
table 9.
It may be noted that, while the resistivities are very high, the
coefficients of
anisotropy are low to moderate.
Electrical properties of otJaer rock sequences
~2
r~
l3
resistivity
11,300
resistivity
12,700
1.07
anisotropy
Coefficient of
The Tederal Coi7~munications Commission requires that radio broadcasting
stations submit measurements of the ii~itensity of their broadcast field as a
In the application of surface-based electrical prosj~ecting methods, the
electrical properties of. the weathered layer at the earth's su1-face are of
parCicular importance, inasmucl~i as ehis is the material closest to the equipment, and it exerts a particularly large effect oii the results. No detailed studies
of the electrical properties of the weathered layer lave been reported, but in
19 4, the National Bureau of Standards published a catalob of 7,237 values
of earth conductivity determined from radio-wave inCensity patterns abouC 621
broadcast stations in the ~=1-0 to 1,600 kilohertz frequency ranbe. Radiowaves
in this fret{uency range penetrate the earth to some tens or hundreds of feet,
so that the conductivity seen by the radiowaves is largely that of the surficial
weathered zone. The skin depth for radiowaves in the broadcast spectrum is
shown as a functio❑ of earth conductivity for an assumed uniform earth in
figure 41.
I'.LliCTRICA(. PIiOPI~;FiTIG:S OF THIi WL:ATFITI{ED LAYEh
in petroleum e?~ploration.
A few data have been reported for non-sedimentary rocks but the data are
sparse. It appears reasonable to assume, though, that igneous and metamorphic
rocks will htive lzi~;h resistivities, but relatively moderate anisotropies, inasmuch as the conduction in such rocks is controlled lamely by joint patterns
which tend to be rather uniformly distributed in preferred directions. Volcanic rocks, which tend to be porous and have a layered structure, might be
expected to have a moderately iii~;h anisotropy.
All of the foregoing plots of anisotropy and longitudinal resistivity might he
combined nn a con~~mon ~~raph to provide a generalized approach to summarizin~~ the electrical properties of layered rocks. Such a plot is shown in fibure
=~1. This nenei•alized plot should provide us with the information we need in
discussin~~ the applicaCion o~f surface-based electrical prospecting methods
Longitudinal
Transverse
Keller (1960j has reported resistiviCy values from 11 dri11 holes which
penetrate Paleozoic carbonate rocks in the, zinc-mining district in eastern
Tennessee. The rocks penetrated by the holes were of the Knox broup, of
Cambrian oz• Ordovician a~;e, being; almost entirely limestone and dolomite
beds. The loaned section was abouC 1,000 feet thick, and the 17. holes provided
Che following average ~~alues for electrical properties:
PROPERTIES OF OIL FIELllS ~n'll TH1'sIR ~,NVIRO\n~tE~iT
L
~
~~o
P'erre Sha/e, T2N
l29 De/erminolions,
0
10
1000
LOPJGITUDINAL RESISTIVITY
100
0.1
I
~~~
~N~N~,
~~~~/111
10
10
goo
~~~
1000
LONGITUDINAL RESISTIV;TY
I00
i
~~
~~
~~y0~~
RESISTIVITY, c~hm-meters
~~~
of earth iesisticity.
Ficoar. 41.--Skin depths at bro~idcast frequcn~ies as a function
'~.~
01
~
SKIN DEPTH, meters
■
■~IIII~
■
1
■■~Nq'
~1~~~~~~
1~~~
~~~~~~~
~~~11
~~~n~~~■~~n~
~
-8/(/ ~ //~~//
-l2 /G ~
-24 ~ -32
comFtcottE 40.-- Locations of wells in Pennsylvania for which electric logs were
piled. Contours xre elevations of the Precambrian surface in thousands
of feet.
-4
0
~
~/~
`~
/
~—
/~_'_`
~
—~~-
~~/ _ _ I i
~
~ ~~li~~e
J~
/
/
~~i~~~■
■~~II
nn~~~~■nn
FicuttL 39. —~- Scatter plot of values for longitudinal resistivity and coefficient of
anisotropy for the Jw~assic-Cretaceous section from the Dakota to the
Benton formations for wells in the strip in Township 2 North, in
eastern Colorado.
0.01
'~~~
~~~~~~
■•~~"
~~~■~~~1~
~~~~■~~n~
~~~~~~~~~~~~~~~~~~~~~~~H~~~~~~~~~_
~~~~~~~\t
~N~~~~~~~N~~
~~IIIIIII~~1111111~~111~11~~1~1111
~~~~~~~~~~~~~~1~~~~~~~11~~~~~~~~1~~~
~~~■~~~~~~~~■~~~~~~~~~~~~1~~~~~~0~
~~~~~~~~~~~~~~~~~~~~~~~N~~~~~~~N~~
~~~~~~~~~~~~~~~~~~~~~~~Y~~~~~~~M~/
REDUCED
ANISOTROPY
10
FicuaE 38. — Scatter plot of values for ]onitudinal resistivity and coefficient of
anisotropy for the Pierre Sf~ale, compiled from electric lo~~s from wells
in "Township 2 North in eastern Colorado.
~• ~
ANISOTROPY
10
Qu.~r.T~xLY or Trlr CoLO~aDO ScriooL of MiNrs
which could be assigned to ~ single oeoloaical environment of this sort.
Conductivity values determined along radials about a h•ansmitter represent
the averabe characteristics of the subsurface rocks over a horizontal distance
of from 5 to 50 miles. Therefore, the geologic units with which
conductivity
values are correlated must be chosen large enou~~h in size that any particular
radial will lie entirely within a siu<~le ~;eolo~ic unit. The smallest subdivision
which can be used appears to be a ~~eologic period, though in cases where
rock type varies radically within a period, further subdivision is
necessliy.
About half the data reported by Kirby and oChers I195~1.) were for radials
1932).
Between 1)~~-r and 1953, the National Bureau of Standards compiled a
catalog of 1,237 around conductivity determinations made in such a manner
about 621 broadcast stations for the purpose of constructing a ground conductivity map of the United States. Such a map was published by Fine
(1951.). However, t~~e correlation between conductivity and soil type was
not impressive, probably for two reasons. First, the Department of Abiiculture
soil classification system used in this correlation lists 256 soil classes, and is
based on a number of factors which might not be expected to relate to soil
conductivity. Secondlq, the radiowaves penetrate to depths of lOs to 104s of
feet, while the soil snapped by the Department of Agriculture may be only a
few feet thick. Althoubh surface soil types reflect the, properties of rocks at
depth in the case of residual soils, this reflection of subsurface properties by
surface soil characteristics is riot alwa}s to be expected. However, radiowave
conductivities do appear to correlate reasonably well wieh bedrock geology as
liven on the Geolobical Map of the Ignited StaCes (Stow and Ljun~stedt,
part of sCandard operating practice. Such field sk~en~;th measurements are
usually made alonb a number of radial traverses about a radio tr•ansmitYer.
At least 13 or 20 point measui•en~ents must be made along each radial to
establish a decay curve, and generally, measurements are made alonb eight
or more radials about each station. In evaluatinb these measurements, a
transparent overlay consisting of a group of theoretical decay curves for various ;round conductivities is placed over an observed decay curve. The
theoretical curee which most closely corresponds to the experimental curve is
chosen, and so, a value for ground conductivity is assigned. Variations in
transmitter power, inhomobeneities in earth conductivity and various experimental errors cause some scatter to the data. Also, the theoretical curves do
not vary greatly for different values of earth conductivity. As a result, the
value of earth conductivity assigned by the visual comparison of theoretical
curves and experimental data points ~~robably has an average error of -!- 30
percent.
~6
77
The verti~ lo~~~ rest ~ti~•ities observed for surficial rocks of late Paleozoic to
1~Iesoz~i~~ a,~e rF~flE~et the f~ct~ that these r~x~ks are 1ar;.;~ely marine in origin, in
\~'[iocene a~~d Pliocene sediments have a ;rester resistivih thtin most
Quaternary racks. ~~r~~l~abl~ representin,~ the effect of reduction of porosity
nn compacti<~n, as well a; the fact that 19i~~cene and Pliocene rocky are primarily fresh and hacki~h ~~~aCer deposits.
The ~~er}- lo~v resislivit~~ of soiree Quaternar~~ alluvial deposits i~ probablti~
a result of the lame amount of ~~~ater contained in such unconsolidated ~naterials. A~ a consequence. Quaternary deposits may have a lo~~ iesistivi<<
even "here saturated ~~itfi brackish or fresh eater. i~zceptions are Quaternar~~
windblo~+~n ~ancl tend ,ra~~el depr,siL ~~~hich are relati~°ely dry and hate high
resistivities.
normall}~ resistive rocks.
The variation in iesi~ti~ ih t~'ith a,~e must represent the results of 2 combination of bec'irock litholo~;y and modifications of the normal ~~orosit~ and
water content by weatheriu,~ processes. It is interestin~~ to note that the soil
resistivipes may vary conaiderabl~~ from the re~~sCivities in the un~~eatherec~
rock beneatl~i. The surficial zc~isCivitie~ vai-~~ from about ~0 ohm-~iietez•s o~~er
sections of l~i~hly conductive seditnentai~y~~ rock to about 2.000 ohin-meters
over highl}' zesistive inneous and metamorphic rocks. ~Che ~nediari surfici~l
resistiviC}' for all of the measurements rnzc~e at radio frequcncie~ is 1-1:3 ohmmeters. IC appears therefore that tl~e et~ecC of weathering; i? to iu~iease the
resistivity of nortnall}' c~~ncluctive rock_ and t~~ decrease the re ~~tivit~- of
The size of each g~eolo~ical unit used in [hip stuck is so ~re~at that the data
cannot lie used to defect dii}erences bet~ceen rock t}~pe~ I that is, betl~~een
sandstone and s}~a1e1, though it is p<~~sible to recognize dilFerence~ bet~ceen
the sedimentary sequences and the i~~neous anc~ metamorphic rocks. One may
also note a correlation bet~ve~n iE~isti~it~~ and the a~;e of the J~edrock, if only
the sedimentary sequences are considered. "I he iesistiviC~~ is not a monotonous
function o£ time_ but exhiUits maxii~~um ~ glues durin~ the Precambrian and
Cenozoic eras and mi~~ima during the ~~(esozoic era and Qualernarr time.
Similar correlations ~a~ith a~;e have; been noted b~ Card 11.9-101 u~in, conductivity determinations I~asecl on induction from power lines.
A coinpai~isun of the data ~~~ith >eolog~ti ~~~as carried out by tracin~~ the
radials from \ational Bureau of Standards Circular ~ ~6 (Kirby and others_
19~-] 1 onto the Geological 1Tap of the Unitec{ State. anc3 thin compilin~~ the
~~alue of conducti~it~ +~ithin each ~e~~lu;ic unit Nrobability densit~~ cur~~es
for 39 u iii. ~rou~~ed in ~e~~en ,~roup~ accordinn to aye or litholo,~ti~. are shown
i~~ figures 12-19, and the areas over ~~~hich the 39 units outcrop are sho~~~n on
the map of the United Slates in figure 50.
Prot>r•.r,Tn:~ or Olt. I~t~:~.i>s :~~u "I'iir:1~ I:~viron~i~~:~~~
~.vvi
v.vl
CONDUCTIVITY mhos/m
O.I
[heir conc~ucti~~il~~.
The inforn~atioit aUout the ~~cathered-la~~er iE~i~ticit~ ~~r~i~-icled b~~ these
radiu~ca~e e~~aluat;ons uf~c~s~iril~' i~ x~~Eia~e~d over tar e area. end sn does
❑ot }~ro~~ide an essential part o(' the infr,r~ntttioii ncede~l in desi~~nin;> surFaceba;ed electrical soundin, n~ethod~: the: small kale ~~ariabilitr in ie ~i~ti~~ity iii
the ~~eathered la~~er. Hr~~re~~er, these data du prop ide ~ feelin.;~ for tl~e surficial
in, i; not netirl~~ ~~~ e~f~ectivF,~ in incrca~i~~
co~itra~C to later rocks ~cl~ich are, l~li el~~ continental in origin. While these
rocks 6a~~~ leis j~or<~sit1' khan the later ~ediinei~t.~_ the salinity of Che connate
cater is much ~r~ater. c ~u~i~i the euck~ to he quite coliilucti~~e. eren after
~~ eatherin,~.
Iielati~~ely }n~~}~ resisti~it~e~ are noted o~~er area; ~iliere~ earlt~ P~ifec~zoic
rocky form the suboutcru~~. 'Chic ma~~ rf~flect the. fact that these rocks I~a~e
l:~een cell indurated prior• to esE~oaurc~ fur ~ceall~erin,. anal thereFure ti~eather-
Cure (i. Oa. Gulf C:u:t~t
f~ictrt::~.2 —_lie=i~li~~itit•: d~~~rrmin~~d from deca}~ ui radio wa~~c~ Geld ~h~en,th ovi~r
outcru~~s of Ou,ilernar}~ ~le~rosit.; in file l!nited States.
Cur~c L Q~, mcn~inc lime~ton~~s
Cin~ee. 7. Qa. Pacific coast
C:~uc~~ ;i. Q:i. K~~cl:~~ )I~~uniain cura
Cw~c~ 1 Oa. 1t(iu~ic (.u~i~t
••••
PROBABILITY
DENSITY
,~
~U_LR"T1~;Rl,Y O7~ Tkll; COLOR:IDO cJCII00L Ol 1'[I\ES
o.000i
o.00i
~.~~
~.
CONDUCTNITY, mhos/m
r9
of the environment of
We need not only to know the electrical properties
if we are to conthemselves,
.fields
oil fields but also the properties of the oil
of Che
consequence
a
as
directly
sider the possibility of locatinb oil fields
are
properties
physical
!Most
anomaly in resistivity associated with them.
rock,
porous
a
in
water
of
modified to some extent when oil is present in place
the basis for evaluation
and these chances in ~~hysical properties serve as
well lobs. Howtechniques used in interpreting a wide variety of geophysical
electrical resistivity. Aoever, the property which is affected the most is the
of an nil-bearing; rock
cepting a simple form of Archie's law, the resistivity
may be written as:
ProrcriTics of OiL FirL~s
exploration problem in
resistivity whic~rmay be encounte7•ed in a particular
the United State.
of radio wave fiield strengths over
FtcoaE 43.— Kesistivities determined from decay
in the United States.
rocks
sedimentary
~~Iiocene
of
outcrops
Curve 1. Mab, Gulf Cotist
Curve 2. b1y, Atlantic Coast
Cure 3. Mt, Gulf Coast
Geologic Map of the
Formation names and abbreviations taken firom the
United States.
PROBABILITY
DENSITY
~.NVIRONM~NT
PROPEFTIES OF OIL FIELDS AND THEIR
~.vv~
uvi
CONDUCTIVITY, mhos/m
o.~
QUARTERLY OF THE COLOR9ll0 SCHOOL OF MINES
~Ig~
~t;w = ~1 — So) ~
~17~
where p,~. is the resistivity of the water in the rock, ~ is the porosity
expressed
as a fraction of the pore space filled with water. S„ is the fraction
filled with
hydrocarbons, and m and n are empirical constants, commonly havinb values
close to 2 in oil reservoir rocks. If the surroundinb rock is similar to
the rock
in which the oil is trapped, differing only in that it is fully saturated with
water
with the same resistivity, p~~., the contrast in resistivity between
the oilsaturated rock and the water-saturated rock is:
—mow ~~m Sw~ —~"v ~ m ~] — So~ n
Ficot~e ~4. — Resistivities determined from decay of radio wave
field strength over
outcrops of Eocene sedimentary rocks in the United States.
Curve 1. Eocene formations of the Gulf Coast
Curve 2. Mgr, Eb, ~fu, Ews formations, Great Plains
Curve 3. Pacific Coast continental sediments
Curve 4~. Pacific Coast marine sediments
Formation names and abbreviations taken from Geologic Map of
the
United States.
PROBABILITI
DENSITY
so
00001
0.001
0.01
0.1
CONDUCTIVITY, mhos/m
b'1
would be T~ = Hpt ~ ,with I-3 being the depth to the oil field, and the
where p t is the true resistivity of the oil-saturated rock and~pt ` is the true
resistivity of the water-saturated rock.
In evaluating, the sensiCivit}' of surface-based electrical exploration techniques, we will find that the transverse resistance of a bed, defined as the
product of resistiviCy and thickness, is a descriptive ~~arameter needed in the
theoretical development. If we assume an extremely simplified model in which
the oil zone is contained in a rock which is completely uniform in resistivity,
the resistivity beinb that which the oil zone would have if it were completely
water saturated, the transverse resistance for the rocks above the oil field
I icor,F: 45. — Resi~tivities de!erroineci from decay of radio wave field strength over
outcrops of Cretaceous sedimentary rocks in the United States.
Curve 1. Krf, K~~. funnations, Texas
Curve 2. Kr, K~, K~, K~. formations, southeastern United States
Curve 3. K~, K~., K„~~. formations, Kocky iVlountain area
Curve 4. K~,~~., Ka, Kr.~, K~~ formations, Texas
Curve 5. K~~, K~~~ formations, Crcat Plains
Curve 6. K,,, Kip, Kai formations, Great Plains
Formation names' and ahbreviationc taken from .Geologic Map of the
United Srates.
PROC3ABILI TY
DENSITY
YROP~PTIES OF OIL I'IF.LDS 9ND THEIR ENVIPONMENT
0.01
0.1
CONDUCTIVITY, mhos/m
- Resistivitie~ determined from decay of radio
wave fleld stren~~th over
outcrops of late Paleozoic aedi~uentary rocks in tl~e
United States.
Curve 1. Ca, Lc, Cpv formations, Creat Lakes area
Curve 2. Cml, Cmm Eormatio~~s, midcontinent area
Curve 3. Pennsylvanian form~itions of the midcontinent
Curve 4, 1lississippian formations of the Great Plaiins
Curve 5. Permian formations of the mideontinent
Formation names and abbreviations ar-e taken from the
Geolo~~ic ~11ap of
the United States.
`
•~~
~~
4
QUARTEP,LY OI' TH1: COLORADO SCI300L OI' 1VIINES
(lEj
The anomaly in transverse resistance indicated in
this expression hay
been plotted in fibure 51 as a function of the
thickness of the pay zone,
expressed as a ratio to the thickness of the overlying rock,
and as a function
T
nt
-~-1 = ~i -5o~ H
transverse resistance fur tl~e oil zone would be T~ = tpr with t
being the
thickness of the pay zone.
The ratio of transverse resistances between the overlying
rock and the pay
zone is then
I'tcuite 46
•~~•
PROBABILITY
DENSITY
82
o.00i
o.oi
CONDUCTIVITY, mhos/m
o.~
83
of oil saturation. Usually an oil field must have an oil saturation in excess of
50 percent to be profitable. Also, even with favorable drillin; conditions,
the thickness of the pay zone should be at least 10 feet at a depth of 10;000
feet, in order to provide a marbinally profitable well. These considerations
provide a lower limit to the value for T_>/Ti which would be associated with
an oil zone worth producing, as indicated by the shaded area in figure
51. It appears that the lowest ratio T~/Ti which ini~ht represent a worthwhile
oil .field is 1.01 (a one percent contrastj. Of course, larger values for Tz/Ti
would be expected for larger oil fields.
A second factor to be considered in constructinb an electrical model of an
oil field is the areal extent. Some types of oil fields, such as shoestring sand
Ftc~rtr: 47. — Resistivities determined from decay of radio wave field strength over
outcrops of early Paleozoic sedimentary rocks in the United States.
Curve 1. Gg, Gl, Gu formations, Appalachian area
Curve 2. Silurian formations of the midcontinent area
Curve 3. Devonian formations of the Great Lakes area
Curve 4~. Din, Bml, 1~1 limestone formations of the Great Lakes area
Curve 5. SI, Sm, Su formations, New England
Curve 6. Ordivician and Cambrian formations, New England
Curve 7. Osp, 01, Cu formations, Great Lakes area
Formation names and abbreviations taken fi~om Geologic Map of the
United States.
o.000t
Pf~BABILITY
DENSITY
PROPEFiTILS OF OIL FIELDS AtiD TII~IR ENVIRONNI~NT
v.uvv~
o.00i
,~6
o.oi
o.i
QUARTERLY OP' TIIE COLORADO SCHOOL OF MINES
producers, ha~re very small areal extents, and will not be considered here.
Accordinb to Griffiths and Drew (1965j, who have made statistical analyses
of the sizes and shapes of typical oil fields, a field is usually about equant in
dimensions, with the long dimension averaging only SQ percent ~,reater than the
narrow dimension. A summary of their data for the sizes of oil fields in
three U.~. provinces (west Texas. Denver basin and Indiana) is shown in
figure 52, plotted as cumulative frequency of occurrence curves for sizes
(az•eas j of oil fields. The west Texas fields are larger on the averabe (1.7
square miles) than either those in Indiana (average, 0.77 square mile) or
Chose in the Denver basin (0.62 square mile). The average depths at which
these oil fields are found range from somewhat less than a mile in Indiana, to
about a mile in the Denver basin, and somewhat more than a mile in west
CONDUCTIVITY, mhos/m
FicuaE ~8.— Resi~titiities deter~z~ined from decay of radio wave field s[ren~th over
outcrops of igneous and metxmorpliic rocks in the United States.
Curve 1. Carboniferous granites, Appalachian area
Curve 2. Paleozoic int~u~ives ~c~w England
(.:urve 3. Precambrian rocks of the Great Lakes area
Curve 4. Precambrian rocks of the Appalachian ~u~ea
Curve S. "Triassic rocks of iVew England
Curve 6. Aichean rocks of iVe~ti~ ~n~;land
PROBABILI
DENSIT
84
CONDUCTIVITY, mhos/m
gJ
1 percent.
Texas. It is apparent fioin the statistical summaries that the bulk of oil production must come from the fields biaber than average. Ililost of the smaller
than average nil fields are probably less than profitable. Therefore, we may
define the least of oil fields Chat would be of interest as having an area of about
3/4,-square mile at a depth of one mile.
Larger fields are required at greater depths in order to provide profitable
production, and so, it inay be reasonable to express the areal extent of a
minimum-size oil field in terms of the 5o1id ankle subtended when viewed from
the surface directly over the field. Usinb such a measure of area, the least of
oil fields would occupy an area of ~/4-steradian. In sum~iary, the least oil
field which mibht be of interest in exploraCion is characterized by a thickness
of at least one foot per 1,000 feet of burial, an areal extent of at least 3/ ~steradian and a transverse resistance contrast with overlying; rocks of at least
Frct;ttt~: 9~9. -- ResiStiviti~s determined from decay of radio wave. field stren~~tli over
outcrops of volcanic rock in the United States.
Qirve 1. }'D2v, Columbia River Plateau
C urvc 2. I v, Ev, R~,cky ~'Iountain area
]ormation names rind abin~e~'iations taken from the Geologic iVlap of the
[Inited State.
DENSITY
PROBABILITY
PROPERTIES OF OIL FIELDS .AND THEIR ENVIRONMENT
10
50
100
PERCENT OIL SATURATION
Frctia~ 51.. — Anomaly in transverse resistance as a fiuiction of oil saturation and
pay-cone thickn~ ss. "Clue pay-zone tliicknc .s is expressed as a ratio to
the thickness of the overhtn'de~~,
0.01
TZ -T,
TZ
CONTRAST IN
TRANSVERSE RESISTANCE
l~icuae 50. — ~'Iap showing the locations where resistivities of the surfieial rock are
hia~her than average, moderate, and lower than average, based on
determinations made at radio frequencies (from Keller, 1966; reprinted
by permission from the Society of Exploration Geophysicists).
7
AREAS OF OIL FIELDS, SQUARE MILES
•1•
87
Archie, G. E., 1942, Tl~e electrical resistivity lob as an aid in determining some
reservoir characteristics: Am. Inst. ~Tinin~, VIetall., Petroleum Engineers, Tech.
PaF~er 1422.
Anderson, L. A., 1965, 3?xperimeival deep resistivity probes in centrll and eastern
United States: U.S. Geol. Survey Tech. Let. Crustal Studies-31, 1_~Sarch 26.
Gard, R. H., 19~b0, Gon-elation of earth resistivity with geological structure xnd ane:
Am. In~t. ~`~~Iinin~, ~47eta11. Petrolewn Engineers Trans., v. 138, p. 380-398.
Eardley, A. J. 1962, Structural gcol~iay of l~rorth 9meriea, second edition: New York,
Harper' and Rciw, 743 p.
~ar~1e, U. H., 1963, Surface and subsurface strati,rapl~ic sequence iii souCheastern
141ississippi: Short Papers in G~oloay and Hydrolo~~;y: U.S. Geol. Survey Prof.
Paper 475-D, Washin~;tvn, U.S. Govt. Printinn Office, p. 43-48.
States:
fine, H., 1959, An effective ground conductivity mall for the continental United
Proc. IRE, v. 42, p. 1405-].408.
Griffithsr J. C., and Drew, L. J., 1965, Size, shape and arrangement ~f some oilfields
in the U.S.A.: Compti~ters and Computer Applications in Minim and Exploration
Symposimn, Tlniv. Arizr~na, Mardi 15-1.9, v. 3.
REFERENCES
States
Ftcuxe 52. ---- Statistical summary of the sizes of some oiI fields in the United
(from Griffiths and Drew},
• ~
••
25%
75%
CUMULATIVE FREQUENCY
OF OOCURENCE
I00°/d
PROPERTIES OF OIL FIELDS AVD ~I'HL:IR ENVIRONMENT
QL?AIiT~RI.Y OF TII~ COLOPADO SCHOOL OF MINP:S
~'o. 193-61.
1965, lleep resistivity probes in the soutlnvestern U~~ited States:
U.S.
Geo1. Surccy opus hle rcpt., Tech. Let C c astul Studies 29, 141arch 19, 97 p.
— — 196, Decp ic~isti~ity probes in the southwestern Linited
S[ates: Geopliy~ics, v. 32, nn. 1, p. 1123-114.
-- 1962, Division of the ~eolo~;ic column in the Rattlesnake
Hills, Wasliin~ton into three major ~;eoelectric sections: Lu_r Analyst, in print.
Keller, G V., 1960, Electrical pro2ierties of zinc-bearing rocks in
7efferson County,
Tennessee: Short Papers in tl~e Ceolo~ical Sciences: t?.S. Geol. Survey
Prof.
Paper 400P, p. 133(9-P9~00.
—~~— 1969. Co~upilation of electrical properties from electrical well lo;;s:
Colorado
Schaol _4lines Quart., v. $9, no. 9•, p. 91-11.0.
KirLp, P. S., H~ir~n~in, J. C., Capps, I'. A4., and ,Tones, K. N., 1954,
Effective radio
ground conducuvit}~ measurements in tlic United States: ~Vashingion, ll.
C., U.S.
Govt. Yrintin~~ Ol~ice. 1VraL F3ur. Standards Cira 516.
Pir~on, S. J., 1963, H<u~dbouk of well lob antivsis: Ennlewood Cliffs,
Prentice-Hall,
~I. J., 326 p.
5cl~lumber~er, C.. Schlumbe.~~er, 1~t., and Leonardon, E. C., 1939, Some
observations
concerning electrical in~•asm~eme~~ts in anisotropic media, and tl~cir
interpretauon: .1m Inct 1lmin~, ~Ieta~ll. Petroleum ~ninccrs 'I~rin~., v. ll0, p.
1~9-182.
Slu , L. L., llipples, ~,. C., and Ikrumbein, W. C., 1960, I.ithofacies maps:
iVew York,
,)ului Wiley and Sons, 108 p.
Stoic, G. W., and Ljun stedt, O. LI., 1932, Gcologic<il map of tl~e
[lnited States:
CI.S. GeoL Survey.
Li.S. Geological Survey, 19C>7, Basement ~~Iap of forth America.
Jackson, U. I3., 1962, Electrical properties of the sedimentary
section in the High
Plains area: Denver, Colo, II.S. Geol. Sin~vey 1'ec6. Let. Report.
ARIA Order
Harthill, N., 1967, An evaluation of tl~e audio-nza~netotelluric method: M.Sc.
"I~hesis. (:uloradu School11ine~s.
-- 1968, The (:S1~1 test area for electrical sur~e}~in~
methods: Geophysics,
~~>
~,9
2. Arrangements in which the g~radie~it of ~~otential (or electric field intensity) is said to be measured, usin a closely spaced pair of rneasurin
electrodes. tin example is the Schlumberger array, in which two closely spaced
measuring electrodes are placed midway between two current electrodes, and
in line with them, as shown in figure Sib. The measuring electrodes are
placed closely enough toether that the ratio of voltage observed between them
1. Arrangements in which the potential clif}erence between two widely
spaced measuring electrodes is said to be measured. An example is tl~e Wenner array, in which four electrodes are equally spaced alon; a straight line, as
sho~a~n in fi~~~ure 53a.
Direct-current sounding methods differ from one another primarily in the
way the electrode contacts are arran~~ed on the surface of the earth. Electrode
arrangemenes may be of three types:
By convention, electrical sounding methods have been classified as "DC"
methods or "AC" methods. The DC methods for measuring earth resistivity
have been used most widely, and there are a nuinbei~ of texts and inoi~o~raphs
devoCed to the subject of direct current methods (Kalenov, ].957; Lasfarbues,
197; Ta~~g, 1964; Van Nostrand and Cook, 1966; Kunetz, 1966, Al'pin and
others, 1966j. Generally, four-terminal arrays are used in order to minimize
the effect of maCeria( near the current electrodes. Current is driven through
one pair of electrodes; the potential estal~lisliecl in the earth by this current is
measured with the second pair of electrodes. Strictly direct current is not
used, but rather a lo~v-frequency alternatin~~ current is used so that the vo1[anes developed in the earth by this current can be easily reconized in the
presence o~f the other, miscellaneous voltages (ot~ self-potential) which arise at
electrode contacts. Howevez-, the frequency of the current is made sufficiently
lo~v that the assumption may he made that the flow of current in the earth can
b~ completely described by a ~oluCion to Laplace's equation.
The resistivity of sedimentary sequences appears to vary consistently over
large areas, so that the measurement of resisCivity from the surface of the
earth mibht conceivably provide information of interest in an exploration pro;ram. lYIoreover, the presence of oil in a rock normally increases the resistivity of that rock. If such resistivity chanties could be measured from
the earth's surface, it would be of considerable value in petroleum exploration. The purpose of this section, then, is to lay the theoretical foundation for
methods to measure resistivity from the earth's suz-face.
PART 2. — THEORY OF' ELrC'I~RICAL SOUNDIi~G
~
M
~ 11
o
rsi
~
ti i ~
~l
~ ~_bl-~
a
°~
N
GI'
DU
M
DU
I
B
I
Polar dipole orray
A
iii
L.-f--1
DU
N
°~°~° ~l
pia
Schlumberger array
c~
A
Wenner arroy
QuarT~T,Lr or ~cfrl CoLOrn~o ScxooL of Vllnrs
to their• sepai•atioii approximately equals the potential
gradient at the midpoint
of the current spread.
o. Arrangements in which a second spatial derivative of
the potential is
said to be measu~•ed, usin~~~ a closely spaced current
electrode pair as well as
a closely spaced measuz•in~ electrode pair•. ~n
example is the ~~olar dipole array, shown in finure 5~c. The voltage measured this
way is approximately
equal to the second derivative of the potential, after it has been
divided h}~ Che
disfiances AB aid VIN, provided these distances are small
compared to the
separation between dipole centers.
Any one of these arrays may be used to study horizontal
and vertical variations in resistivity. In studyinb the variation of resistivity
with depth, as in
the case ofa horizontally layered medium, the spacinbs
between the various
Ficuxe 53.— Electrode arrays commonly used in the direct-<:urrent
resistivity method.
A and F3 are current electrodes, Di and N arc measuring;
electrodes, and
a, b, and c are gray spacing factors (from Keller, 1966;
reprinted by
permission from the Society of Exploration Geophy;iristsl.
~~
91
electrodes are gradually increased. Lamer spacings accentuate the effect on
the measurements of material at deptk~i. Althounh it is difficult to specify how
deeply the resistivity is beinn sampled with a particular electrode array, it is
Generally accepted that the depth of sampling is less than half the total electrode
span, for most of the arrays. We will consider the sampling depth of all the
electrical methods in a later secCion.
In the AC methods for measurinn earth resistivity, atime-v~ryin~~ magnetic
field is generated by driving an alternating; current through an ungrounded
loop of wire or a grounded length of wire. If conductive material is present
within the magnetic field so generated, induced or eddy currents will flow in
closed loops along; paths norii~al to the direction of the n~a~;netic field. These
eddy cu~-reiits, in turn, generate their own magnetic fields so that at any point
in space, the total magnetic field may be tllou~;ht of as consisting; of two parts:
a primary, or nozmal field due to the source current and a seconclm~y or c3isturbin; field due to eddy currents induced in conductors. `Che electromagnetic
field may be detected either by measuring the voltage drop between a pair of
electrodes or by measuring the magnetic induction using a coil of wire as a
rnannetometer. The AC or electromannetic methods h2ve been used mostcomrnonly in mining exploration, where the objective is to detect an anomaly in
electromagnetic field strength caueed by a conductive ore body, rather than in
quantitative studies of earth-resistivity. As a consequence, there is less literature on the use aF the AC methods than of ehe DC methods for electrical
sounding (Vanyan, 1967).
In measurinn earth conductivity, one must first generate an electromagnetic
field and then measure or detect the distortion in this field caused by the presence of a conducCive earth. This may be done in many ways, and the variety of
way's of using an electromagnetic field in studyin~~ earth conductivitq has Gotually hindered the application of the methods.
The three common controlled sources for electromagnetic fields used in
Geophysical exploration are loops of wire, short mounded lengths of wire and
long brounded lengths of wire. A current flowing in a smell loop of wire
Generates a inannetic field which cannot be distin~~uished from that caused by a
dipole magnet, when the field is observed aC moderate distances (a moderate
disCance being greater than about five times the diameter of the loop). The
mabuetic field generated by such a current-carrying loop has a strength
equivalent to a dipole magnet with a moment equal to the product o~f the number of turns of wire in the loop, the area of the loop and the current flowing
in the wire. If the current is oscillatory, such a source is called a harmonic,
or oscillati~zg~ ~nag~netic dipole source. A steady current provides a magnetic
field which is constant in time. An abrupt termination or initiation of current
flow in such a loop leads to a transient m.n{;neti.c feld.
TxLO~Y or Er.~cTr~icaL Sovr;ninc
Quar.TLrLY or TFir CoLOraDO SciiooL of 1~Tzvcs
A loop ma5r be oriented arbitrarily with respect to t11e surface of the earth,
but normally, the plane of the loop is placed either• parallel to the surface
of
the earth, in which case it is called a vertical magnetic dipole, or perpendicular
to the surface of the earth, in which case it is called a horizontal
magnetic
dipole. The axis of the equivalent magnetic dipole coincides with the loop
axis.
For a vertical magnetic dipole source located at the earth's surface,
there
are only three electromabnetic field components which may be observed
at
the surface of a uniform earth: a vertical component of the magnetic
field,
Hz, a radial component of the magnetic field H,-, and a tangential
component
of the electric field, E~. With a horizontal dipole source, five of
the six ortho~onal field components may be observed over- a uniform earth.
The vertical component of the electric field is the only field component not
observed.
The vertical m~~netic field from a horizontal loop is the same as
the radial
magnetic field from a verCical magnetic dipole, as follows from
reciprocity.
A grounded wire ma}r serve ae the source of an electromagnetic field
as well
as acurrent-carrying Loop. In this case, if the lenbth of the grounded
wire is
short compared to the distance at which the field is observed, the
source inay
be termed a current cli~ole or an electric dipole. If observations
are made at
distances greater than about ten tunes the wire length, it i~ found
that the
product of wire lenbth and current is the only parameter describing
the source strength which is si~~nificant. This product, Ids, is called the
dipole
moment.
With a horizontal current dipole, all six components of the electromagnetic
field may be observed at the surface of a homobeneous earth.
A third idealized type of source for an electromabnetic field which
is used
in geophysical exploration is a long grounded wire. Field components
are
measured close enough to the wire so that it may be considered
to be infinitely lonb. Only two components of the electroma;netic field
from a lon;
wire may be observed at the surface of a uniform earth—the parallel
component of the electric field and the vertical component of the magnetic
field.
In recent years, a fourth source of energy has come into
use for making
electromabnetic depth soundings—the natural electromagnetic enerby contained in rapid variations of the earth's magnetic field. When such
energy is
considered to be a plane wave traveling downward into the earth,
the conductivity of the earth. if it is homogeneous, can be computed from
the ratio
of mabnetic field strenbth to electric field strength (Cabniard, 1953)
.
In measurinn the electric field in the earth, normally a short
grounded wire
is used. Magnetic field components may be measured with a magnetometer,
though this i~ rarely done except in the case of the mabnetotelluric
method.
1lilore commonly, the mabnetic field components are measured
with induction
coils, which detect the time-rate of change of the magnetic
induction:
92
at
dB
93
(19)
The .first throb we must accomplish is a solution of Maxwell's equations
for an earth made up of a sequence of horizontal layers. In fact, as it turns
SOLUTION OF THr BOUA'DARY VALUE PROBLEM
Thus, a voltage is measured, rather than a magnetic field component. In all
cases, this voltage is proportional to the strength of the source, or the current
in the source, and must be normalized for this stren;th.
It is apparent that there are a great variet}~ of techniques which might be
used in electromabnetic sounding. Four types of source have been listed, and
with various source-receiver• component combinations, 16 different combinations of source and receiver could be used in electromagnetic depth soundin;.
Depth soundinns may be made either by varying the spacing between the
source and the receiver, or by varying the frequency content of the source
current. The first is termed a ~~eometric sounding, and the second, a parametric
sounding. There are operating advantages to both approaches, and both are
used in practice. However, control of frequency is used more commonly than
variation of source-receiver• separation.
with a fixed separation, measurements may be made either in the frequency
domain (one frequency at a time. through a range of frequencies) or in the
time domain (use of transients containing a wide spectrum of frequencies).
Althoubh it is readily shown that time-domain measurements and frequencydomain measurements are uniquel}~ related through the Fourier transform, the
operatinb procedures and interpretation involved in the two approaches are
quite different.
As a result, there are 4.5 variants which might be used in the controlled
source methods, plus the magnetotelluric method, makinb a total of 46. Each
of the 46 methods requires somewhat different instrumentation and quite
different interpretation procedures and theoretical curves. Commonly, in
the literature, a single method is considered at a time so that comparison between methods is difficult. The variety of methods has led to confusion in
understandinb the basic principles of depth soundinb, and so, a unified approach to the theory of electromagnetic depth soundings is essential.
EMF = -~~nA~aH
If the source is harmonic—that is, if the current to the source is a sinusoid
at a specific frequency, ~~,—in the steady state, the derivative may he replaced by a multiplying term,iw:
EMF' _ —nA
TH~OPY OF ELECTRICAL SOUNDING
QUARTERLY OI' THF, COLOR9D0 SCHOOL OF iVIINES
FicuaE 54. — Definition of an anisotropic layered medium.
f'P~ ~Pti ~ ~i > h~
0%< H =A + ~f~c at
(201
out, we must do this twice iii order to have expressions which we can use
with each type of source—a short grounded wire oz• avertical-axis-coil—which
is of interest in oil prospecting. In so doing, I will follow closely the development used by Vanyan (1967j, and so lar as convenient, use his notation.
Let us first establish precisely the earth model ~~-e wish to use. The essential
features of the model are shown in figure 5-1-. The earth is assumed to consist
of a sequence of layers, each designated with an index p, which runs from
0 to N going downward through the sequence. The p=0 layer in reality will
be taken as a half-space representing; the air above the earth, inasmuch as we
are not planning to discuss layering in the ionosphere. We will allow each
layer to be anisotropic in two dimensions—that is, each layer is represented
by two values of resistivity, p~ in a horizontal direction and pt in the vertical
direction. Each value of resistivity also carries the index p to indicate the
layer to which it is assigned. Each layer is also characterized by a dielectric
constant ei„ and a magnetic permeability µ~„ thou h we shall ignore displacement currents and we will assume that all values of magnetic permeability are
equal to the value in free space, 4rrr X 10—% H/M.
In all cases, Che behavior• of an electromagnetic field is described by Maxwell's equations. The first of these:
9~
95
(21)
(23}
~~ _ ~
(25
two
tions in terms of a vector potential, which is allowed by the fact that the
soicrce
a. Solzetion o~ 1Ylaxzoell's equations for a s{aorl grounded wire
Maxwell's equaAs is usually the case, it is convenient to seek a solution to
V X E = i c.~ C3
P
multiplication by
This means that each time derivative can be represented as a
the coefficient —ion:
Maxwell's equations are:
e
— ~~,~t
value for a uniform medium close to the source.
steadyWe can siinplifq matters somewhat by seeking; a solution only for the
vary
to
assumed
is
components
field
the
of
each
which
state harmonic case, in
with time as
consider
Application of these diverbence conditions means that we are bomb to
inasmuch
fact,
In
space.
only solutions Co Maxwell's equations for a uniform
number of
as we wish to consider a layered space, we wi11 actually obtain a
of one of
properties
the
solutions, each of which is valid for a full space with
using
layers,
between
the layers, and then match the solutions at boundaries
requirof
consist
auxiliary boundary conditions. Tk~ese boundary conditions
fields be coning that the tangential components of the electric and tnabneCic
at large
tinuous at the boundaries, thaC the electromannetic field approach zero
the
distances, and that the expression for the electromagnetic field approaches
(22)
o(.inr 8 = O
and elecTwo additional equations which express the continuity of magnetic
charges,
electric
and
tric lines of force in the absence of free mannetic poles
equations,
VIaxwelPs
respectivel}'> are sometimes also considered to be part of
differently
though there are roan}' problems in which they must be expressed
O XE _— ~B
describes the fact Chat the observed magnetic field is generated either by current
equation
flowinb in the medium or displacemenC currents. Maxwell's second
induction:
relates the induced electric field to the fate of change of magnetic
T~~icoxY of EzFCTr~icaL Sou~~i~c
~
QUARTERLY OF THE COLORADO SCI300L OF MINES
~ 26j
(27)
~(2a)
(31)
(30)
=OX~f.~~clL'
(32)
~~L`~~x -a d?\ +kCa ~~ a ,~Q'~
dx `R )
x K
4n~I ~' ~ Yd~Z _ a ~ftx~ +s
R
2y s?~/ J
Expandinb the cross product indicated in this last equation, we
may recobnize
the expression for a curl:
B = 4n J I'~' ax ~RI +~ jy \K~ }k a ~R~~x:~2~
we can expand equation 29 as:
R3 = -U~R~
~s
r
exert a force on an element placed at the point for which B is being
evaluated.
Reco~;nizin~; that
where the integration is carried out over all the elements of current dl' which
~
4~fR, d '
~ x~
~
r
~
(29)
~
The dF in this equation differs from the one in equation 27 in that it represents
the total force on the element dl due to the presence of all other existin;
currents, while the dF in equation 27 represents only the force due to the
current
element dl'. Comparing equations 2r and 2v, we see that the field of
magnetic
induction is:
dF =Id~xg
which dives the force on the element cil due to the presence of dl. The two
elements are thin wires that carry currents I and I', respectively. The radius
vector R is directed from dl' to dl. The force on a single wire element in a
field of magnetic induction, B, is liven by the fundamental law;
dF = 4° R3 d2 x ~,~E~XR)
The nature of this vector potential can best be seen by considering the fundamental law of force between elements of current:
B - ~'~ A
diverbences are zero. On the basis that div B = U, we may always define a
vector potential, A, which satisfies the condition:
96
97
~X(~-~ituA~ = O
pX~=~~w~X A
(33)
_ ~~ -~cuaf~~~- aty~
(35)
ax
Using the designations
nexus earth)
f'i
as + aaZ _ ~ A
aZ
(37)
we obtain a reIntegrating the second of these equations with respect to y,
valid for this parlationship between the vector and scalar potentials which is
laterally homogea
over
source
element
ticular problem (an x-directed wire
~~
~(~ - as~ - a ~ aAZ~ _ ~~-'u'/,~oF~~~c.~A=- 3~
\aay~ - ~xl_~~
~y~_
Cartesian coordinates, we
Expanding equation 33 into three scalar equations in
have (with A, = 0,as explained above)
ax
r7Ax~- az ~~ - a~~ - ~ ~ - ~w£~ o ~ ~ icu Ax- au~
a
be represented as the
If the curl of a vector quantity is zero, that vector can
Thus:
potential.
~,radient of some ftmction U called the scalar
or
have
cases.
for a source which is a wire
As an example, consider the vector potential
other current flowinb in the
carrying current in the x direction. If there is no
Now, if we let
component.
xsystem, the vector potential will have only an
vector potenthe
earth,
current flow from the wire into a laterally symmetric
is symcurrent
the
Because
tial must also represent the effects of this current.
and
integral,
contribution to the
metrical in the y direction, there can he no net
is
flow
current
component. The
so, for this case, the vector potential has no y
not
is
distribution
the resistivity
not symmetrical in the z direction inasmuch as
The vector potential must be
directions.
z
minus
symmetrical in the plus and
allowed to have a z component.
equation (25), we
Substituting curl A in place of B in Maxwell's second
of being; the intebral of
Thus, the vector potential, A, has the significance
only those components that
effects of current elements, and so, A will have
in recobnizing from symassistance
of
is
current flow in the medium has. This
may be zero in certain
potential
vector
metry that certain components of the
THEORY OF ELECTKICAL SOUNDING
n
o
— u'z~~o = e z
~ ~~~
133)
aye
ax~Z
~ Pt~
x~
+ cuF~co)Ai — y
~ (~~— iwEt()(a2A;
ax + a~A")
anal ~~~~ )
~ _ ~~~*r
- per,
(17)
(=~~6
~
_ ~~tr ~PQ)
a2AX
azan
z
_.
(a2~xZz } a2~~~~ ~ ~.z~
~~wA x - ~z
XQ
a
axe J~
~2AX
A2 + a2AZ
aZ
~Z = -fez
;u,
'fhe third scalar re~prF~~eritation of ~1ia1ti~e11's first
equation fin 35~ can be
(p z- y~ z) Ax = o
Inh~uduun~, the Laplacian o}~e~alur notation. ~~r. h~~-e the basic
equation used
in fi~~din ~ the ~-cu~npon~~i~t of thf~ ~~ector Ex~lential.
A,:
ax
_a2a2 - a
x
(>> i
2
AZ
~ aAZ
As noted above, the second equation is inhomo~~eneous, but the degree of
inhomogeneity may be reduced with the foLlo~~in~; device. We wriCe the vertical component A,, as the derivative of a function W, ~~~hich has cv~indrical symmetry, with respect to x:
arz + r ar } ~~ aaZ,, - X~- AZ = ~~ - ~z aaZ
a aZ
ar
(50)
r
aAx ~ as - XQ2Ax = O
a2Ax !
+
arz
from the source.
In cylindrical coordinates, e~~uations ~1~=1 and <-19 for A, and A,take the form:
~1-~~~
(~~~~7
~chich ~in~~>lifies to:
~ azax
a~az
~ a~a~
axz + aza~
~~ + ~z
aZ~ - YtrzAZ = (I - ~z) ax
aZ
With these definitions. equation -t5 can l>e rewritten as:
(11'i0
~
At lo~v frec~ucnc~ie~. the coefficient of anisotropy assui~~es its usual form, which
i~ valid so Ic~n~ a~ disE~lacemenf currents can he ne~a~lectecl:
o.~
We can define another wave number, yr, in which pti replaces pi, anc~ a
coefficient ~~f' anisotropy ~~ follows:
`axz
~P.~ _ a2Az + a2P`z~ _ (~w
rearrannecl in a somewhat similar manner:
99
"Che ~econcl e~2~r<~ssio❑ is useful i❑ numerical evaluatiorn.
inasmuch as the ~ti~ave
nunibei i; separated into real and ima~.~~ii~ar~ parts.
~~''e n~~«' return to tl~e first ~~f the scalar rc~pre~entations
of 1~Ia~t~~el1's first
equati<~n in t 3~~ 1, a~~d rearrari~e teri~~s t<> obtain the follo~vin~;:
j~
(
z
~
~
f
4
Tr3rorY or IL~c•rrzcai. Sot,~~»i~~c
Unlike the first of the scalar equations. the' third does not reduce to a homo~eneous form, with n, and A, seE~arating. This will make the solution of
the differential equation for ~~, morc, difTicult than the solution of the dif~erenCial equation for A~. Il is interesting to notice that if, there is no anisotropy,
equation -1~~ induces to the same f~~rm of wave equation as liven in equation -1~~-.
So far ~~~e have noC done anythi~~~,~ except rean-an~e 1~IaYwell's equations into
a more convenient form for solution. We may now proceed Co this solution,
seekinb first an expression for A, i~l the form of. functions with cylindrical
symmetry—that is. depending only- on the vertical coordinate and the distance
a
~~~e ~~~i11 sec later that t ie fii~~t term iii equation 3o represents
the cuutribution
from cunducCion current. ~chilc~ the ~ec~~nd term rF~present~
the c~mtribution
from cii~placen~ent current. Aurinall~, displacement currents can
fie ❑~~~~lected
in c~>>lil~ari~u>> >cith c~,nductiun currents i ❑,coE~h~~~ical ~i~~plication~
of clectron~a~~uetic ~uundin,~ rncthc~d~. ant] if this is they ca~~~. the ~ca~e number is
tivrittf~n
L = 2,~/X
It mi~lit be ~x~inted ~>ut that the paramr~tee y, ~~l~ich has been
defined here
is tl~e ,:lon~itudin<31 radian ~ca~~e nuinbcr., at thc: frequent}~ ~,,.
T}ie actual
~~'a~'e length, L, of i}ie electroma~~netic ~~~a~~e is related to the radian
Heave number as:
1
2(. _ ~~
z c~v q
A
_ jw
Qti,~r,•ri:~;r.l- or ~rl~r CoLOr~_v>o ScflooL or _l-ti,~vl;s
ice can re~~~i~ite equatiuil 6 a~:
and
>o
Z - ax
A _ aw
f52)
QU9PT~P,LY Or TH1~ COLOIiADq SCE°IDOL OF
MINAS
2x ~Az~ az
(53)
2
ara
3
z
~
~r
~2
az2
azw
~ aqX
- X~2 W = ~'
~ ~Z~ ~z
(55)
(5~1)
(57)
w = Z Cz) • ~i2(r)
Di~ridin,~ both sides of the equation by X~~,
w<; find that the variables truly
separate. The teams to the left of the equality
sign are a function of r alone, and
not of z, ~~=hale the terms on the ri~~1~t of the
equality sign are a function of z
alone. end not of i-. Therefore, die eYpreseion
to d1e right of the equality sign
cannot Crary with the expression to the left of
the equality si~~n, and therefore,
~~~he~-e ~i and ~. are functions of r blrt not of z.
and ~tihere X 2nd Z are functions of z but not of r. Substitutin~~ the assumed
solution 56 into equation 50,
we find
(56)
aX = X Cz)'~Yi Cr)
~TOtice that the inliomobeneous tez•m has been
reduced $~om a second -order
derivative to a first-order derivative.
Equations 50 and 55 are the manipulated forms of
Maxwell's equations for
which we seek a solution. One of the commonest
methods used in solving equations o~ this type consists of "separation of
variables." ~e assume that there
are soluCions, a~ to equation 50 and w to equation
55, which may be written
in the fozms:
r
a2w } ~ aw } i
arz
2
t ~2 azan -qtr aw = ~ ~ -az~ as
Integrating this expl•ession ~a~itll respect to x
we get:
a +r
Substituting the appropriate loins of the function
W in the inhomo~~eneous
wave equation 61, we bet:
Q
u - ,~
~z
With this definition of W, equation ~9 which
relates U and Amay he rewritten as:
IOO
101
r ,4~i
~
X
= (~ —
Viz) 4~1 /~~
(~2)
+ r
~2
~
~tr2
~~
~z Z'
~
~- `'
~2~
(
~R
~1
Xi
rG
(6 )
(65j
(6~1~
It appears Chat the quanYiCies (m~ -I- y~~)'~ and (m~ -~- yc~) `~ will occur frequenfly, so it will he convenient to use the shorter notation, n and n for t}~ese,
respectively.
We now have three ordinary differential equations to solve:
Z'~ — ~z('m2 +~trz )Z = ~~z 1) X~
~~ + r ~z ~ + m2 ~i — ~
Inasmuch as the equation we have here for ~~ is precisely the same as the
equation for ~~ in 59, we may take ~~ and ~_ as being the same function of r,
and the separation ci~nstant in the two cases will he the same. The t~vo separated differential equations are:
72
Il
Again, dividing; by the product ~~Z to ohtain separation of the variables, we
have:
r2uZ +'r `z Z + ~i ~// — o/tt ~/zZ
The well known solution to equation 61 is an exponential term with either
positive or nebative power, but we will not bother to express this solution immediately. The solution to equation 60 is the Bessel functions of the first and
second kinds of order 0, ,~~„(mr) and H„(mr).
Substituting the assumed solution ~r into equation 55, we have:
`The constant m is called the "separation constant," and it has the same dimensions as y, or inverse distance. Writing the two separated equations, we have
the following; two complete differential equations:
"~/1
each expression must be a constant, which we will choose to call —m2, for
later convenience:
THEORY OF ELECTRICAL SOUNDING
X" -nZX =O
(67j
QUAPTERLY OF THE COLOPAllO SCHOOL OF VIIN~S
1 X,
m2
-m2
XI!/
+.
m
-n2 +z~h ~z
V/
~~
_ -m~- Yep +~m~ +az~t~ X,
rr~~
_
_ (~l~-I ~ X~
1
r~M \2"`~
(71~
Let us now recall that the assumed solutions to the two
original
b
partial
differential equations, (equations 56 and 57) are functions of the
separation
parameter m2. Any sinble value of m will generate a solution.
However, in
order to match boundary conditions, we want to express
the solutions in a
This form would be of little value if we did not have a homobeneous
equation in X that we can solve first. Now, we really have only two
hoinobeneous
solutions to bet, X and V, and then we have the inhomogeneous solution
by a
linear combination of V and X'.
Z = U _ ~~ x'
satisfy equation 6~. Therefore. we can write the general solution to
the inhomogeneous equation in the form:
which provides the correct tei•zn for the right-hand side of the
equality to
and so
~70~
(69)
However•, by differentiating; eyuatiorl h! once: with respect to z, we have
X ~v _ nz X,
dz
~ 2 x,
n z / _ ~ ~ X~
_ ~
~~~
~.~n 2 X
~ - ~ /~ ~ ~ m
~(Zz ~- m
~ —
m~ X + m ~
is the particular solution to the inhomogeneous equation by substitutinn this
solution in the left-hand side of equation 6v:
_
zero} and the particular solution to the inhomo~eneous equation. We can
readily see that the function
equation (equation 68 with the term to the ribht of the equality sign set to
The general solution to the iiiiiomo~~eneous equation for Z (6b) is made up
of the general solution, say V, for the corresponding hoino~eneous differential
lU2
~~
~
~.
~
f
i
~
~
~
~
s
f
~
s
i
103
(72)
(73)
(75)
and Z become vanishingly small as z is made lame.
(Note that the two sets of constants, d,, and C~„ are not necessarily the same
for the two equationsj.
In order to evaluate the constants in these solutions, we must apply reasonable boundary conditions, which are best defined in terms of the oribinal field
components rather than the modified vector potenCial functions X and V.
Continuity of the tangential components of the electric and magnetic fields
at the boundaries between la}'ers requires continuity of A„ A,; aA,/az and
U. Equations 167) and (62) must be solved with consideration of continuity of
X, Z, aX~az, and pr (X -I- aZ~az) at boundaries, as a consequence of
continuity in A" A„ aA,/az, and U, along with the conditions that both X
Vp = dPe nP~ + ~Pe
p
U. Solution to the ho7no~;ei~eous part of equation 68:
nPz
~'
a. Sol~~tion of equation 6!:
So, for each layer, we may write a solution for X and V which is a linear
combination of these two types of solutions, and then seek Co find values for
the arbitrary constants which suit boundary conditions.
No v we need only the forms of the functions. X, V and X' for a biven set of
layer parameters to complete our solution.
The two independent solutions to an equation of the type of 67 or 6o are
of the form
-~aPz
—al,z
and e
e
Az = ~ aw - T~ ax~ 2S°~'(Yl'r') ~m
ax
41'i
o
4t1'
(Note that the multiplying factor• Iµ„/-tor has been used only for later convenience.l The solution for the z-component of vector potential, A,,,, is:
AK - 4 ~ x-Jo (mr) dm
more general form. This can he done by takinb a linear combination 1 sum)
of individual solutions for individual values of m~ wiCh arbitrary ~+~ei~;htinn
factors. The parameter m~' can assume an}' positive value or zero. 7~he tnost
general sum of such solutions will then be an integral of the Stefanescu type
taken over a distribution of m From zero to infinity. Thus, the solution we
are seeking for the x-component of the vector potential, A„ is:
Tx~ohY or ~L~cTxlcnL Sou~flzvc
QU9PT~RLY OF TfIT' COLORADO SCHOOL Or NIIN~S
X
o
-
-
Xs - 2m
/
(76)
~77~
1
i
vn - X
2h'1
~ xg - Xi' = Xa ~G~) C mz
(78)
—
I
n~
dP~~r~PZ+Cpe~~PZ
dPe~nPZ_ CP e nPZ
dP e ~~Z + cP e ~PZ
np
R~
—
_ RP
(80)
X79)
~P) e
c~
nz
-~ ~dP~e P
RP — ~dP~ \y: -nPZ
J/i -r1Pz
Cp °
+ ~Gpd
i t)enPZ
`~~
~~l)
The ratio, R,,, may be expressed as a hyperbolic function
after some ai~ebraic
manipulation. first, we divide both the numerator and
denominator of the
ratio by the quantity (di,Ci,)'~:
v~ _' ~n~ d ~
~rtPZ_ C e~~PZ = --e
~nP
P
P
v
and the ratio V/V'is:
XP
x~ - _ 1
Tikhonov and Shakhsuvarov (1956), as well as Zhoaolev
and others
(1962) have subgested an approach to the solution of the
problem which involves operations with ratio X/X' which arises in the last
equation, rather than
by a solution for X.
The ratio Y/X'is then:
or
This means that a deri~rative of X may be taken merely by
multiplication by
the factor ire, and equation 7( rrii~;ht he w~•itten as
X~~i~ _ ~Co
mz
(o) e
at z = 0. Inasmuch as the solution for X in the upper
halfspace can contain
only the exponential term which tends to zero upward along
the z axis, we
have:
X~ alas a c~iseontinuity given by:
An addition condition is engendered for the function X
by the presence of
the dipole current source at the boundary z = 0. Tikhonov
(1950) has shown
that while X is continuous across the boundary z = 0, the
vertical derivative
lO=~
~'
~
'J
f
j
j
'
~
i
~
~
~
I
!
~
lO5
~
~ ~ ~(dp/CP)V1Yl~Z~-~ ~ ~'~d P~~PI'/Zt hpZ~
J .e~p ~en ~dP~CP~/z C1PZ~+?~7r~~ P,» CdP~Cp~'+11pZ~
(02~
(831
~P~2
r
~~ +
cQC~C
l
,1 J
~~ ~ ~~
p
alt
- ~C~CJJ
_ ~ L 1~ P ~ZZ
C
(~~'~
~s z~o = ~[n,h, + ea¢~
1
Rl Z_hl
~
~,
(051
Z_},~
R~ }
- nz I
-
Z~hZ
~06~
L,
~ ~~~"~ } ~
1
1 (`(~
`rig ~2,h~ /~
~d7~
And again, R•, at the top of the second layer can be written in terms of R= at
the bottom of the second layer:
R1,2=o
We maY notiv rePlace R, in equation 85 with its equivalent in terms of R
at z = ha from equation 86:
n1
~1 I
and second layers, we have the continuity condition
Inasmuch as X and Y' are conCinuous across the boundary between Che first
layer, at z,=h,
This relation is, of course, valid only if the two points, z~ and z_, are in the
same layer. The utility of equation c,4, is that we may compare values for the
ratio R,, at the top and bottom of t~~e pth layer. Normally, we will be concerned with field quantities at the surface of the earth, so we would like to
express the values for R, at z_ = 0 in terms of R~ at the bottom of the first
R P,~
Then, the ratio at depth z= can he rewritten with expression 33 used in place
of the constants:
/z
_fin ~dP;~~' _ ~-1 Rpi -nPZl
The constant Id,,/C~,)''~ may be eliminated by considering values for the ratio
at Cwo different depths, z~ and z•~, within a layer. At depth zi equation o2 is
solved for• In(d~,/C~,)'~:
~p
= e~°~~P~~P >~'~ ~ we have the form for the
Then > usin b the identity~ d,
i /C,)'~~
I
hyperbolic cotangent:
j'HLORY OF EL~CTRIC9L SOUNDING
`~
i
Rz >h i = ca-~( nzhz + c~- (R2,z= h ±fit= )~
188)
Qti~_~i,~rra;l_~~ or• ~rru: Col.orn~o Sclioor, or• 1IIV~:s
H =
c~[nN h N + c~-s ~~ Z=~~~
~
'
p-i
H = shF
hn-x
Ct~ = 1
N -t
N-I
N--i
R1 Zed = ~~~,h' +
P"i~y
w
,_/J i n,
COCK _ nz R2
H
/
i~~~
In effect, amulti-lad er sequence can be built up by adding layers to the
top
of the sequence, one al a time.
"This concept is important at this sia~;e; because of a diiliculty in the evaluation of the for~~iula as it stands. This di{~icult~- is apparent on considerins;~
the
behavior of the imei•se hyperbolic cotan~~eiit, ~chich is shown ~,~raphically
in
figure ~5. 1~he invez~se hyperbolic cotan~~~ent does riot exist ~lor ar:,~uinents between zEro and one. If such ar,~u~nei~t~ ~i•e n~ neratecl in evaluation of
the
chain in equation (901, it would indicate that not all conditions were
consic3ered ~~~hen the capression was writt~e~~ iu the form of a I~yperbolic cotangent
1 equations 81 and X21. The ar~~umer t of the inverse hyperbolic cotan;eat
can easily be less than unite—this ~voulc3 happen in the Last equation oi'
the
chain O0) if the wave i~uinber in the nth layer, n ~, were greater
than the
'f OT ftNO 18y P1'P:
Thin, ~~~~e have ~. sequential expics~ion for R
which can 1>e extended to
i,.—o
and nuit~ber of l ~~~ers, and ~~hich im-olvee ~r~lt the repetition of the ex~n~c:ssiou
RN
R~, at z = h, -{- }~_ t}~cn can be ~a~ritten in terms of Ii:; at the tofu of il~e third
(~i~er. 'l~liis pr~xc~s caii be continued to the top of the last laver, la}'er _A'.
`Chen, the ratio at [he top of the last 1a1~~~r, c~~~re~sed in terns of 1~ ~C the
h~>tt~~ni ~~F t6e layer. 4vhich is infinitely thick. is:
106
~
i
i
~
I~cha~~i~~r of ~h~~ h~~~~crl~ulic iunc~i~,nr.
Tonh -~
107
functions ++'ill b~~~
Ho~v~ver. it is «~e~l known tl~tlt if the reci~~rocal~ u[ the eLectriczil coutra~ts
reciprocals of
are considered, the ~~alues fowui for the K functions ~~~ill be tl~e
considerin~~~ two
those _'iven in equation 90. "Chip can be readil~~ gee❑ by
n~~n,=k
contrast,
electrical
an
has
~~-hich
of
one
sequences.
different tti~o-l~}'er
t~ao R
=Ilk.
The
n~
/i>>
contrast.
reciE~rucaL
the
has
~vhicli
alicl the other of
~ti~ave number in the o~erl~i~~,~ la}er_ n~ ~. 'I~hen tl~e ratio n~ ~;"n~ ~cuulcl be
less Chan unity.
Tlcciti~: 55.-
TxrorY or• Er.ic~rricni. Sov~~>i~c
_ k c~-~y; n,h, t 1
—
—
~~k~
k t c~ n h
i
R Ck)-- ~~
k
-
k ~~~1h~
~rt
~riN_~~N -~"
~~d)
n~ ~~
RPz = c~ C~n~ ~zzzl) +
,~
and to equation o4 as
CO-CX~i RP1~
~/J
- Vii, (~~1c.P)yz = cc~~-1 }=~P1 - ~ n QZl
and to equation 83 as
)
X93)
(92)
~
The same procedure can be followed with the V-function. If we desi;nate
the ratio for the V function as R~%, the analogy to equation 82 may be written
as:
-~
(Note that instead of writing the reciprocals, the equivalent step of using; the
tangent instead of the cotangent has been employed.)
ci'7 ~li~t
1
Qi-1 R31~3~
~p~nL/J
= c~A-Lft ~~'1N 11'~_+ LO'h jnN~ S~N~
RN =1
RN
~/
3
=O
°2 ~ 1
~n,~l, +-[e.x-fc.-' n 2 _i
T~2 = %v-~,E^ ~Ryh2 + ^~~~ -1 n? R3~ c2 ~c_`4~ [Cly~2 +
R~Z_o = :_cam l.h~hi t <'o~-i ~z kz~
n,h~
k + eo~c~ a,~,
!._
k ~o~'n,h,+ 1
~~ ~ k~ - ~~ Ln,h~ + Co~~-1 I ~ = k
k+
n h,t 1
(-Z <k) _ ~ ~T1~h~ +cc'~Q 1 k ~ = k `~
QUAPT~RLY OF THE COLORADO SCHOOL OF MIA'ES
This means that equation 90 might better he written as
so
lOg
~
,'
~
1.
I
~'
(9~)
lO9
^,.'"ti"' C n t'1 ~ +
2 ~"~
LQ~~
~
n
1}1~' ~
n,
n, s 1
n2~-
~ 1 ~ ~, RZ ~- 1
~ n,h, z
~
~
^U
(95)
,O ~t ~z < 1
z
~ n, RZ >
1
k
b
_
~Cc~ =
n1 RZ
1~ n`
1 - n;, R~
ny RZ
~'2
-1
n, ~z +
1
z
ZR
~ n~
1
~Z > 1
~ n`
ni 2
(96~
nh,-i~m~b
-2~kb~
- LR~h,
en,h,-~ P,,, k,e
Rah ~-z ~,.. ~~
This may also be written as:
Rs,z=o~ ~n,h,- 2 ZKkb }e Cn~h,-~~kb)
Rs,z=°
_ e,h,-z P,k~
t e Cn,h,-z ~.k~~
(97j
,~,/ n.
<1
(j n,, Rz
n`2 RZ>1
Next, the hyperbolic tangent or cotanDent can be expanded in its exponential
form:
1/k:
For convenience, we can designate the argument of the natural logarithm as
R 1Z=6-
p. 87)
The expression for the R'" and R functions in terms of hyperbolic functions
may sometimes be awkward to evaluate because of the necessity to interchange
hyperbolic tangents and cotangents as the arguments move to either side from
unity. An alternate expression for the R functions similar to that liven by
Sonde (1949) may be preferable in such cases. The development of the Sonde
algorithm may be seen by startin; with the first equation in 90. The inverse
hyperbolic cotanbent (or tanbent, as the case may be) in this expression may
be rewritten in terms of a natural loarithm (Abramowitz and Stegun, 1965,
~ n,hl + c~-I Rsz=h,~
R~ Z=~ coot ~
and finally, to equation 8~ as
THEORY OF ELECTRICAL SOUNDING
nh
nh
air,
i
~
?kb e ' - b e-
~- e ` ` - ~e
I
n,h,
Rh
n h~
D
nz
z
z
(9~)
QUARTERLY OI' THE COLORAllO SCHOOL OF MI~'ES
1 + kbezn~h~
1 - k0.e- zn, ~
-Zn~ h,
n~' ~
n,
~ n2 FZ2 > 1
(99~)
1,r-0
(1011
(100}
~zs~...N -
R~23...ri =
~
zaH N
e znxh2
e
1+ k 23H
— N z"~z
z
nz -n, RZ3.,..,,
Z +k,Z3..Ne
1- k~Z3..Ne
-z n,h,
zn h
The same process can be extended to the second equation in 90', as well as
the following; expressions. T~~is leads to the development of a recursive formula
for the R function similar to Sunde's resistivit}~ kernel expression:
1+ ke.
R1 Z_o = 1- k~_ ~~ n~ Rz
-zn,h~
we need only a single expression for R
2 ~ ~~Rz
However, k;,= —k~„ and so, these two expressions are the same. By~ convention,
n,
~ ~,,QQ n—~ Rz
if we define k as
k _ 2 - n2~2
R'~Z-°
1 +- k a- e2n`h'
We now divide boih the numerator and denominator by the first term in each:
R~Z-O
` `~2-°
110
~'Y7
~ni
~tr.1
''
i=h
~ iF I
_
z.h,
p~ I
~`~, "z
otr_ 2
(103)
(1.02)
111
z
~'z'~i.~z
n
N-1 ~ T~-1 ~N -i
+
~
~N
~10~)
e N-~
~nd nN /~QN
~N'\ ~N -i
~
RN
~~-1
~NnH~2N
n
i ,ZN1 n-~J~2,N-i
-
kiza...N =
~ZZ...,,
~
-z~,n,h,
-zz,n, h,
kiza-.N ~'.
~'z'~z~ez }~~n~PQ~
R23H' N
~2~2/-Q2 _ nl ~~LI ~23++~~N
~j,~,
1 + ~tiza Ne
1 —
This expression may' also be rewritten in a Sunde-type alborithm; as follows:
D-t
~/1
,CO✓X-K C ~
N1 + ~
,(^(
~ ~ ~ N-1nN -1
~anapQa
n hz+ ~~-i 7~zn2
R3
ce-~ t~ZRzhz + ~-1n.L
~n
~ax. :R3 ~ ~ ~~~aZz
a 3pea
~a'~',~tz
-i n pig R" ~ do
= co-~~~~R,h~ + co~ ~
t`~.xQ ~~,n,h,+ ~Q-1 ~'~n~~'~ RZ
We now see that the general N-layer expression for the R"" function equivale~lt
to 90 is written as:
1,z~s
'`N'l
-i
e nH-~ N-,
eznN_,hN_,
The boundary condition at z=h, for V is written as:
~ i~ H = 1
'~
I
1+ k
Rc,~-~~H
N
~'+S" N
N-,
nr,-nN-~
k w -~)N = nN
+nN
1-
k
2
^'
3
_ n~ - nz 8345
THEORY OF ELECTRICAL SOUNDIn~G
1,z°
R*
~ R#
~'
i
~I
-
~
~
~ 3N5~ ~ N
k
~k~uPeN + ~K-~~N-il~P,u-i
_ ~N-InN-I~P,N-i
- ~N'`~Nl-PN
~ CN-i)N
-2 7~N-~nN-~ hN-~
/`3~3~e3 + ~2R2/ Pz R3'iS~~N
n
~2n2p~z
~~]3 ~3 rp~ -
1 + kz3~ N
z~n~hi
ez~1n2h2
— 1 - k~N-,~N e
1 +- k cN -~~ ~ e-z ~ N_~nN-1 hN-i
*
•
~Z39~~-N _
z~...N
~ZQj~
QUARTERLY OI' THL COLORADO SCHOOL OF YIINES
(106)
Q AX
Z
aZ ~RAXO~~
— R R _
~,~ _ ~
t
~; A
t
X
(o~
(107)
where A ~°~ is the vector potential for a homobeneous medium, and
a°°
A.,<> ~ ~ > is the vector potential for the induction field.
In view of the symmetry of the problem, the electroma;netic field for an
electric dipole in a homobeneous medium ma~~ be written in terms of a vector
potential having only the single compor ant, A<<O~. Symmetry also indicates
that the differential equation for A, equation 54) should be solved in
spherical coordinates. With the spherical symmetry, the vector potential does
not depend on B or ~, but only on the radius vector, R, so that:
co)
~z)
Ax~o = /~xo + Ax,o
The reason foz this may be seen by assumm~ that the current dipole is
situated at a height h„ above the earth's surface, and by expressing the electromagnetic field in the upper halfspace as the sum of a primary field, which
increases without limit near the source, and a secondary field:
X~,'—~~' = 2m at z = 0
Equations 90 and 10~ are not sufficient Co determine the functions X and V'
completely, inasmuch as there are only N-1 boundaries to he used with each
of the two continuity conditions. Tikhonov (1950) has noted that while X is
continuous across every layer boundary, including the surface of the earth,
there ie a discontinuity in X' at the earth's surface which has the size:
112
~
,'
i
l
~s
~
~
r
~
~
I
'
~
t
c
-~( R
113
4'tt'rZ
_
L
L r/i
e
z
R "1J,Cmr)dm
°° mZ+~(s
rz
C .O~tM e
lll~~
~1~~~
Co)
—
r~j
~ m 'ho~z}~o~
3o(mr)dm
n~ ~
qs~
(111)
<s) _
x~°
IclC~µo ~c~e°~Scmr)clm
4n
o
X112)
x,o
0
the earth (z=0), so
Both A~ and aA,/a,, are continuous across the surface of
we can write the boundary conditions as:
4TY
~
,R Z
qZ
~ So~mr) cam
AX 1 = I~f ~C~e ~ -~-~10
(114)
'°
~z +h 1
+ Coen°Z 3oCmr)t~m (113)
= r~ f (n e,~°
41T
o
°
The vector potential, A„ in the first layer is:
A
for the upper
Combining the two contribuCions Co vector potential, we have
halfspace
A
(equation 74.)
In the upper half space, the coefficient d„ in the solution for X
Otherwise,
since the
downwards.
must be zero for an axis which is z-Ipositive
potential
vector
the
direction,
0-halfspace extends to infinity in the minus
potential
vector
the
for
expression
would increase without limit. Therefore, the
for the seconclaiy field in the upper halfspace is:
o
where h„ is the height of the source.
p,~ a
for the secondary fields: (equations v3 and 34~).
°
as our solution
to convert the solution in equation 10) to the same form
e
-1(~ trs+z~)`~"
_
ay ~cu -so
~A~,
We can make use of Sommerfeld's integral:
$x ~~
Zci-P~t(~.dVY~ B
considering the apThe constant of integration, C, may be evaluated by
placation of the Biot-Savart la~v at zero-frequency:
co>
The solution to this equation is:
ZII~ORY OF ELECTRICAL SOlitiDING
= ~~ +~~ = X1
X'~
ne - X
e
_n~ho
(115)
(112)
(117)
C 116)
Thus, at the earth's surface, there is discontinuity of X' of size
x' = n~ X - 2m e°°ham
X = Co .~ m enoho
no
X = no~o _ menoho
i
n~ho
2 me
no+ ~
2 rrze ~'°~'°
~ } n,
(X +Z')
Continuity of scalar potential requires continuity in:
mz
Z =~_X,
(119)
—2m e ~~0h0.
The other needed boundary condition at the earth's surface may be found
in V', the verCical derivaCive of the general solution to the inhomogeneous
equation in Z. This solution consisted of a particular and a beneral part:
or
and
X,
2 me n°h° _
~ _ x,~
X ~ _ - -~a' X = - n~
R
R
for x at z — o, we have:
and
X =
Thus the function X at the earth's surface scan be expressed in terms of the
ratio Xi/X,', which in turn is expressed in terms of the ratio function, R:
co =
rrt X;
ne X1
+m
-me"o o +nomo _ n`~~~ d~)=X1
ne n°h° + co
QUARTERLY OF THE COLORADO SCHOOL OF MINES
Solvinb for the constant C,,, we have:
114
115
~l`Q1 vl
z
(120)
Vo — Vl
=2
m e n°h°
°Z \ _ (Z ~ _'_'
= ~ Za + ~
and
vl -
C~L~~Ro
~o
\~Q~rro
~tti/
1/
~,R, J
~ (l""1
+
zeZ n°h°R*
2enho
r,-,
rr,(~z
R'~
vi = — ~ n
~~°no V~ ` -V~ = n2—,, e n°~~
e.
we have
Solving these equations for V~ and V,' ~t z=0,
2
Combining; equations 120, 121 and 122. we~ have
But
Vo' = no~o
(125)
l 1.2-11
(123)
(122)
(1211
finite for lame negative values of
Also, because the function V must remain
exponential n7ay be retained in the
z in the upper halfspace, only the positive
solution for the upper halfspace, and so:
U~ _~ ~/~' _ ~Zu~~
J'"l;ovO
boundaries, this indicates that the
Inasmuch as X is continuous across all
product p~V' is continuous. Thus
}~Ut t~71S 1~
Taro~Y or ELFCTrrcaL Sou~i~l~c
u
o
n
m
~ e
—Roho
n
-~-
°
a
no+~
dP
I
__~
2'n
_
R
_
~e~2
+ n,
no ~
(12~)
A,, -
~
aAx
~~z
_ ,zn ax
r~-
o
”
~
zn.
~
,~l~n~
elm
~o'mr)
`~
~~,~r10
~ tom*
2
~nz_ +
m~,n'
*
n }n, ~(mr) ckm
~
m
m
>.n~
---~+ ~~
~o
2
0
.._
I ~'~,
m (no }~-~-~
n.
— —
-- ~-
~:'tr
I ~~
~12l~
~Cmr)dm
i 126)
~
f
t
I
r
f
(129)
1
~ 1~~~
(lili
l Jo <rar) :gym
(130)
~A
-~
~
',
3
~
~
~
-oho
Ji Cm r) dm
e
S<mr) cam
1
1
1or a dipole at the earth's surface Ih„=0), these equations become:
1
---z ~~
'
r~~
z
~'
2tT'°
~(~_ + ~.*
e
n,
` Xe.2n0
7~,-
e nh~
m(~o1 } R'~
`YQZRo 1n
n~
no
e ~h°
m+
any
~*
m e—~oh~
n + ~ 3o(mr) dm
R
~~'? 2x~ ~ X, ~Z~~ J Cmr) cam
0
x
21Y ~ YR
n,
fix = — r~
~~z
~
a
I~
cox
21Y
~
~4
A _
z
Qu.~~TLf,LY or rFi~ Cor.o~_~DO Sc~rooL ar ~TIN~s
iVow that eve Dave the soluCion functions X~, X~', V~ and V~', we can write
e~preseions for the vertical and horizontal coinponent~ of ~~ector potential, the
vertical derivatives of the vector }:>otential and fir the sealai~ potential aC the
earth's_ stn-Face
116
Ce1 B
o
1
~Q2no
~
~~
—
~
~ei2
i
(133j
1 Cm r) cam
117
2
L
2~ ei2
x
ax r
o
o
Jo(mr
~~
—
R
~ n,
z
i
~
~,n,
T:m r; cam
1
l
(135 i
SoCmr~ dm
(13`~~)
32(mr)dm
~~~n
* _
1
]
R
np}~
YZ
,>
R
[ 161
~ ~ ~ n~ -~ ~ n
p'
~
~
~
n~
u
m n~
n°}R
(13!)
ic dipole
to distinguish fields due to a magnet
(note that the asterisk is used
dipole source 1.
source frotY~ fields clue to an electric
E~ = ILIJ~I~ {~~t
source
jor a z~erticc~l-axis min~~;netcc dipole
Solution of iVlazwell's equations
cal-axis
the same problem but with averti
Now let us pursue. the solution eo
sohiin
steps
the
ForCunately, many of
ina~netic dipole sereing as a source.
s
analysi
of
mass
s~~ we do not ~~ave such a
tion of the problem are the same,
a
(for
0
=
o~f t1~e condition that div ~'~
to ~o through. Takinn advantage
define a new
d we find it is covenient to
uniform medium ~vitl~ no source ,
vector• p~tPntial. A'°, such t~iat:
determinin~;~
the solution to our problem of
and so on. These equations are
an electric dipole as a source!
how to measure resistivity with
T
o
m
IdP;w f n ,~
~~
tl' ~
a?
~_L Sc(PµD ~w 2
Ex —
rd~
'D
8Z = — 1~° -~
integrals Ior electromagnetic field comWe are now in a position to write
the surface of a stratified earth:
ponents about an electric dipole at
T~ u"
SOUNllING
rlIIP:ORY OF ~LECTRIC9L
~
(
~i
~
~
~
f
V
~
~
~
~
~
'
~
~
~
~~
'- ~
i
i~
~/~_ MAGNETIC FIELD
FROM INDUCED
CUl~RENTS
~
/~
INDUCED CURRENTS
7
T ~ ~j'
~
~
1
i
— ~~
SECONDARY MCil~i
/
~
~
~~
~ ~
~~
~ ~~ —~
~--~ ~
~\
\,
~
~
~
~
/~ ~
\—~
~
`
I
1
I
~ l
~
~~
\
~
~~ _
~
i
PRIMARY
—~~MACiVETIC
~
FIELD
~~~\ Souceith ~~
magnetic moment, M
/ / ~_
i
1'~coit~ 56. — Primary magnetic flux from a
magnetic dipole source and secondary
magnetic flux from currents induced in the earth.
~
~
1
Quar,TLfiLY or rF~i~ CoLOra~o Scxoor. of Mzn~~s
A~ was the argument in the case of the electric
vector potential, A, Kee
would expect the inaanetic ~~ecYor potential, A''', to
have components parallel
to the flock of magnetic flux lines from the
source and in the medium. A
coil of wire energized with current behaves exactly
as a magnetic dipole directed along; the coil axis. Thus, the current flowing
in the loop contributes
on1~- a vertical component to the ~~ector potential.
Current flow induced in the
earth, provided the earth is laterally uniform, forms
loops in horizontal
planes, with axes coincident with the axis of the source
loop. Each of these induced current loo~~s also acts as a magnetic dipole, and
contributes only a vertidal component to the vector potential. "The magnetic
flux lines outside the dipole sources have other than vertical components, as
indicated in figure 56, but
inasmuch as we have assumed that all of space has
the same magnetic permaability, µ,,, these lines exhibit perfect radial symmetry,
and their contributions
to other than vertical components of the vector
potential cancel. Therefore,
fur a rertical-axis ma~~netic dipole source over
a laterally uniform medium
with no contrasts in magnetic permeability, the magnetic
vector potential, Ate,
has only a ~~ertical component, A,.
ll3
4~
}
~_
~
t
~:
~
~
~
~
~
~
~
%
~
`;,
~
~
jk
t
ll9
_ ~QzA~ + o-~~
oA~ ~ ou* =-gam
u~— —~'A*
i~ ~2i
Il ll l
it-~o)
11-131
b. The vertical mannetic field.
E~
a. The azimuthal electric field,
_ —~W ~~rz
(1~-1-j
Let us now consider the continuity conditions which may he applied to
further define the solution ~''. Inasmuch as the ~~ector potential has only one
component, we have only three field componc:nCS to be concerned with:
°
o~~,_X,Solmr) c~m
q
AZ_
~-
This is precisely the wine equlion as the one for A~ in the developi~~er~t
of the vector otential for- an electric dipole source. Therefore, the solution
is also the same:
~r
C~ A~ — 7 ~p • A~`~ = B~
Selecting; as our arbitrar}'~.;~~une condition [he relation
We ha~~P
~r
,*~ = g -~
o~ Co;~A
(1 >9)
1138)
1 The brad li'' term is [he most eneral constant of inter,ration term «hich can
~e introduced as a result of the inverse curl operation).
Replaci~~n ~"" b} i~„ ctu•1 A" in l~1ax«ell'ti second equation_ we have
~oin~ tl~e inverse curl operation, this equation becomes:
~
zy
B~ _ — XE; A* — D U
p/gam _ ~ ~~ ~- ~.c~2~~ctp~0 ~A~
P
= — ~F z OXA"
Substituting A °~ in ~Iax«eIL's first equation, ive have the following; result
for the field about a vertical tna~netic dipole:
THIAPl OF L'.LF.CTI:ICAL SOt \DI\G
araZ
(14,6}
(1~~5 j
Xp• =
rtP
RP
(147)
..'~~~
nN ~)
Z.~ O
ZD
(IZ~j
where l~I is the magnetic moment of the source, the product of the area, number of turns and current.
Using the Soinmerfeld inte~~ral, this can be expressed as a Hankel transform:
A?°~~ 4 ~eR
A ~">
where '
medium and A'' ~ i>
~~ is the vector potential for a homo<~,~eneous
~
~,
is the vector poCential for the induction field (analo~~ous to equation 106).
The erector potential for the ma,netic dipole source in a homogeneous
medium can he derived in exactly- the same manner as tiaras done for the electric
dipole (equations 101-10~) with the results:
Z O
Tor the upper halfspace (the atmosphere) we can se~~arate the vector
potential into primary and secondary parts:
Rp = ca~ {n`h' ~+ ~-1nz~cat~C(nzhz+ ~1 ....~-~
where
We require continuity of tangential components of electric field and
magnetic induction at the boundaries, so the sole cc~m~~onent of vector potential, A~', as well as its vertical deri~rative, aA'°/az is continuous.
"These are the same two boundary conditions we had for A~ in the case
of an electric dipole source, so we may follow the same reasoning; in eealuatin~;
the boundary conditions and write an expression for the ratio X~/X"' as in
~,
~~
the equation 1791, leading; ro the follo~vinn resulC for an N-layered anisotropic
earth:
~
,.,
r
g * _ 2zAZ
- -r a ~r 2AZ~
ar
~~ -- ~~~A~+ ~a~
z
Qu.arTLrLr or Txr~ CoLOrn~o ScxooL of NIz~~s
c. The radial magnetic field,
120
t1~0)
121
0
q~rr J~
~~~e
+nom.
~ 3acmr) dm
i 1521
X
*~
no — X
e
(1531
zn Jo z n,+
o
F2
'~` n, Jo Cm r) dm
no}
R
2n
ar o
'~`
J Crnr~ dm
no+ ~ °
BY =-~ arf R n~~ n, T ~mr)dm
CO
~ { r af~ no+
m ~- Ja(mr~ dm
t3Z = - ~~ ~
2n r ar ~ a~
~P
E ~ _ — ~° i w ~~~
and the electroma,netic fie1c1 coinpunents are therefure:
az
Az = M2.n
~~°
o
i1~~j
~1~~~
t 156 i
~i~~~
i le i.)
Thus, the fuiu•lions ~"" and ~i ~ at the ear ~l~ ~ surf ice arc t}~E saiuE 1s t rose
given in equatio~is ll6 and ]1~. 1~he ~rctor ~~otentill solution, ~~~e pie seel:in
for the ma~,~netic dipole source are therefore:
o
aA~';`az
at tf~e ~,rouncl
Applti~in~; Che conditions of con~inuit~~ i❑ A"
..
.. and
surface. a~ in equation 1 lam. S~~e can e~~~luat~
~,the a~n~tant C":
~~
AZ~°
so that the vector fx~tential in the upper }i~lf~pace is:
0
X ~ = C men°z
can contain unly a positive
Iii the u~~per halfs~ace. the ~~~lution for a"
<~
exponential tcrin if the ~ec~>n<lar~~ field i~ to disappear at <~~reat di_tances in
the —z direction:
/l~;~„ — Holz+l,,l
* ~°~
M
,~Cmr) dm
A~° = q J Hoe
TxLORY- o~ E1.LCraic:~L Sou~~rnc
QunrTrl~r.Y of TF~r~ CoLOfiano Scz~rooL of MiN~s
r ar ~ r arf n~n~ Jo cmr) dm ]
v
'1,~mr)dm
^~
zn ar,~ ~ nm+ n,
zcr
M
2n ar,~
no+ n, 30(mr)clm
D
(161)
(160)
(159)
2n
ar o
a
Yoa_~ 2
SoCmr) c1m
ocmr) c~rrt =
(r2+ z2)y2
e ~ ~ Y.s+ZZ~Yz
(162)
r3
-X r
(1 +Xr) -~- e~~~r~1
2~~a+~~z~ ~., [ra e
l
(16~~
(164)
+Xpir~,1
Carrying out Che differentiation indicated in equation 16~, we have the
following result:
"= -.w
E~
Usinb this identity to integrate equation 162, we have:
o
llifferentiatin~; both sides of this identity with respect to z and setting z=0,
we have:
-Xr
me
-ntZl
This is now the sum of two integrals, each of the Sommei•feid form:
~
The inte;ral in the first two equations can be converted to a known form by
multiplying, numerator and denominator by the quantity (n~,-n~)
gr*
Z
E,~~ _ -i w
It has become a convention in electrical geophysics to define "apparent
resistivity" as the resistivity one would compute from measured mutual impedance and receiver-transmitter geometry, assuming the earth is uniform.
Let us first consider equations 156-1~~ which Give the field components for a
vertical-axis magnetic dipole source. For a uniform earth, R becomes unity
and the expressions for these field components are:
A UNII'ORM EARTH AND THE PIiOBLED4 OI' 1)EI'INING APPAliENT
RESISTIVITY
122
I
1
2rr (~ ~ ~)
° -~`
~`~r
~Qir~~
X168)
~e know for a certainCy that y„rGGy~~i. Therefore, for small values of yiir
(say less than unify 1, only the exponential multiplier in 16H need be considered. Here y~, may he nenlectecl iri comparison with yiz and it is readily
3 ~-- 3~r +(Y,r)2 eCXo +Xei)r
3 +- 3eir ~ (xe~r)2
I'or sea water, the mabnitude of the inphase znd quadrature components of
wave number would be about 2.5 x 10-~;~,`= per meter.
In the first case, the wave number at any reasonable frequency for use in
electromagnetic sounding would be very small in the atrnosphere in comparison with that in the earth, and we should consider the possibility of
c3roppin; small terms from equation 165. Let us consider the ratio of the
surn of terms contatnin,~ y„ to the sum in terms containinb y~i:
term
Inasmuch as e~, = 8.85 x 10-1zF/m and µo = 4arX10-7H/m, the wave
number in Che atmosphere has a ina~xnitude of >.3 X 10-`'~, per meter.
It is not necessary at this point Yo assume that the 0-th layer is the atinosphere. Rather it might be assumec{ that the 0-th layer is sea water, as would
be the case for measurements made on the yea bottom. In this case also, Che
wave number reduces ro a single value for both the longitudinal and transverse
directions. However, inasmuch as sea water is an excellent conductor, with
c7 — ~ to 5 mhos per meter, the displacement current term in the expression for
wave number could be ne~;leeted in comparison with the conduction current
0
z
[3 + 3~~r +~~i~r~,
(165}
-7e~r
r~ je ~°r [3+ 32Sr +(~r)z, - e
~
r ~ e~°r~~t~ Y'~
~
Here, it is assumed that a single value for wave number, y,,, describes the
properties of the 0-th layer. If the 0-th layer is the atmosphere, the conductivity may be taken as zero and the wave number reduces to a sinble value for
both the lon~;ituclinal and transverse directions:
_ - icu
+ ~e~e
123
e ~°r(i+~r)}~ e~°r- Xeie~~`r~ i + ~Q~r~
E ~` - „-`~
2t~' ~~a z- ~t~~) } r3[ -~
~
+ e~e~r C~ +
TFILORY OF ELliCTRICAL SOUNDING
(170)
I
1.2
0.9
l,p
0.6
0.7
0.8
Gr
Real component,
4~rr3U
multiplied b3~
-µOEM
.99975
.9912
.991~Ofi
.9681
.97 91
Gr
0.1
0.2
0.3
0.4
4.5
.004~73~
.017878
.037883
.063270
.092657
Imabinary component,
4-Trr3U
multiplied by
~.°wM
TABLE 10.-Tangential electric field about avertical-axis naa~g~netic dipole
These two expressions can he evaluated numerically without any great
difFiculty. It is interestinb to note that the only parameter to which values
need be assigned is the product Giir, which is a dimensionless spacing factor.
Numerical values for the real and imaginary parts of E~, computed from these
expressions (equations l!1 and 172) are given in table 10, while a single
cm•ve for each equation is shown iu figure 57.
(172)
(171)
~,a~{~~} = i [~3t 3 GA r~ ,o,c~x GQ~ r - ~3G~~r + 2 GP~2rz~ coy GQ~r, e ~'r
~~E~} = 3- ~.~`r ~ ~3+-3GQ~r) coy Gar - (3C-~~r .~ 2G~rz),o.ut
~ Geir ~
terms:
where
36909
10782
,:1330i~
.11601
~
!
~
~
-.00923
5.5
-.0063
6.0
-.003-1_l
6.5
-.00138
7.0
-.00022
7.5
wheze'VI is the moment of the source.
~
10~~1.6~
.0836
.069-1>
.0>><13
.05200
.135-12
'
4•.0
y.0
-.00962
.21204
.l0'9'1~
.16916
.IJI Zc~
.0003 ~~
-.~~.1~~-1.
4,A.
4~
~~.~,~,~tiT
___
.l-7Tt'3cr
.3209
-1.29103
.263 3
.2368~~~
-.UU`Ci~)ii
.Op~_~,p
_ ~}g;;2(~
~~~6~5
~~~~~ ~ 1
pl~~~;
.35003
11014
1Z~71
'1.03=~0
~<<9
.11.661.
.~ ~ l~ ~~~1
.:;!)r02
.32592
.26189
2060
.15723
,~ 2
~I~.O
~
J. ~,
~~
3a
J.`2
3.0
2.4
2.6
2.3
2.2
.<>> ~~
.226r0
.25960
•~"'1 ~
.1 ~279~
.12~~77
YI1 U~~7~.)~lf'(~ })~,
.71)U1.
.r>:3 r r]
.5~5U5
`~~ ~ ~~~~
12~
Imaginary component,
.<<~~»2
.o93Z5
.~;b3:~
.96109
•~>-1~22~
.91961
_ ,~ ~ ~12
---
~~
~:
~
~
f
`
-
Illll ~tl })~lf (~ }>): -
~171'r:t~r
Real coii~ponent,
7',~t~Lr.' 10 I Coin. l
THI:OP1 Ol~ I' 1' F,CTRI(:AL SOtitillI~'G
1.4
~
G ._ (~ 6/2)~~z
f
e.
1.6
~,
Thus, for the purpose of numerical evaluation, equation 169 can be rewritten
~
1'0
as two equations, one with only the real terms and one with only the imaginary
'~
Z•~
Xe1 = ~~ +- ~ G~~
~
~
~~
displacement current term can be neglected for most rocks, and we can
write the complex wave number as:
z
(
~
,~
2n'a- r4 (3 - e.
~3+ 31~Q~r + t~<<r) ~~ (169)
~
This equation is not really so simple as it looks, inasYnuch as the wave
number is a complex quantity. For frequencies below about 10 kilohertz, the
-Y~~r
~
s
apparent that we do not need to consider that y„ has any value other than zero
In the term (3 -I- 3y~r ~- y~,2r2), the second two terms can usually be neblected, inasmuch as y„rG<1 in most cases. Taking y~ = 0, equation 165
simplifies to:
E"= ML . ~ (
f
QUARTERLY OF THIS COLOFADO SCF300L OI' MINES
124
~
~j
3
a0
T
(~
~r
2~' o-
2 r'+
1
1
f
—~~r + ~ezrz~' ~ 3 +3X{~r +X~zrzl
11
—
3
r~
o C-ab~ ~
a
0
It is difficult to visualize the behavior of ~~' from the ec;nations (171
and 1r2),and so it is worthwhile to examine the asymptotic behavior both for
lame values of the dimensionless spacin;~, yiir, and for small values. I'or small
values of yi~r, which would be equivalent to a small source-receiver separation,
r, or to a low frequency or conductivity in the wave number expression, makin~> yii small, we have:
Ftcua~ 57. — C~i»p~iients of tangential electric field ~enerat~~ by a
vertical-axis
magnetic dipole source locat~~c1 on tl~c surface of a l~onwbeneou~ earth.
G is the real past of the wa~c number in the earth, and r is the separ<ition from source to receiver.
0.01
G.
iywM Em
Orr r
Qunr~Trr,LY of TxE Cor.or,~~o Scxooz or Mir;Es
NORMALIZED ELECTRIC FIELD
126
i
,;
~
j
1
(~
~
j
4
~~
Ili>1
127
ztt6-r4
`~
D'~~r -~co
— 3M~.c°
E~
1171)
11~~1
= icu~o~z ~ 1 + 3~{r ~
t 1.76 ~
vulta~;e foz~ small
Tf eve were to measure only the m -phase component of
r~>ot of con~qu~re
the
ui7
spacin„ we would have a yuantit~r which de~~ends
apparent
for
~i~n
exl~re"
ari
pion
i~1e
<~
ductieity, and which could lie ~olaecl t
conductivity a~ fc>Iln~rs:
Yeir ->o
E~"~
spacing.
<k~>re~sion for al~For small values of yr, it is possible to ohtiin an alteri~atE
terms ~n yr. those to the
parent ie~istivit}' by including the, Welt hi h<.i order
quation 17~:
third potiei•, iii the expansion crf the exponential in E+
of the actual
This is not entirel}~ ~atisfactor~, i~~asrnuch as some I:nowled~~e
is ~ati~fiecl.
ineyualit~~
conducCivity of the earth is required to assure that the
r, thaC
spacings,
(ar,e
Furthermore, quite commonl}' it is difficult t~> use such
of the
po~~~er
f<~urth
yr>6, inaamuch as the ~i~~nal strength decreas<~s a~ the
Mrr
6" _ ~
2Tt r'~ E~
a-
is entirely real.
In this case; the response does depend on conducti~ it}' and
and so define
conductivit~~;
We can solve this last ex~~ressio❑ explicitly for
In comlar~~e.
apparenC conduetiviCy for the case i❑ ~~hich yr is suf}icienth~
the
with
1~
1r
equation
p~irisi~ the curve for the a~>proxiinate e~pres~ioci i❑
in
voltage
curve for the true expression for the real part of the tan~;ei~tial
lame
sufficientl}~
ecjuaCion 171. as shown on fi,ure 5r. it is apparent that yr i~
in the earth.
~tihen it exceeds about fi, car about one rc:af wa~~e len~,th
The definition ~~f ap}>arent cc>nductivity f~~r yr>6 is:
with:
increases as
That is, the tangential elects is field for y~~ t = 0 is zero_ and
(y~ir 1 -' for sinall~values of the dimensu>nless sp~cin.
obseraed
Of primary concern is the fact that for ~m 111 ealues of yr, the
is of no
~o.
and
earth,
electric field is independent of the conducti~ity~ of Che
that
notE
also
~houlcl
value in aCtemptin~~ to measure that conc~ucti~-it~~. We
foi~ small yr, the respon~E is entirely out-of-phase.
c3iil~en~ionle~~
Next, let a consider what happens at laia~e values of the
w-e are left
spacing. Here. the expotienlial becomes ~~ani~hin~:l~ small end
uo ~,
_ ~M
9Tr,-z ~-o
TxF.oxY or ELrcTFZCa1. Sot;nr~IVC
QuarTrr,Lr or ~rFn: Colorauo Sc~~zooL or Mrr;~s
FCr) _ ~zr4 3 - E ~r ~3 + 3Xr +(fir
~zJ 1
M
(
~r
J
(1~>0
Ill9)
(l~lj
+5(~r)z+-~~r)3,~
(1o'2j
z
I Xr(« 1
(133
M
The apparent conductivity defined from this expression
is:
~
At the other- extreiue, for lame ealues of yr, the term
multiplied by~ the exporientialmay be neglected, with the result:
~" Xr-~o Ft~"'~ ~ ~~~ ~
.... 8tY zr9
"I,he real part of this second approsim ition contains a
term which is proportional to the square root ~f conducti~ its-, and ~o can
be used to define an
apparent conductivity:
uoM C1+
~2Xr~ - -i q~M3 ~1 + 2Gr +2i Gr)
~Z ~ -`9tTr3
It should be noted that the ma~rie.lic induction (the ~-ertical
com~~onentl is independent of the conductivity of the earth iu this first
approxim~tior~, and
that the received field is com}~letely ima<~inai-y. If first and
second order terms
are retained in the npproxitnation for sma11 yr, we have:
— ~ ~°M
-4tYr3
~
Z
~ ~'r)3
~Z = —~~.rs q—[1— Xr +~z~ — 6
+••]•~q +9?1r
For small values of yr, the expo~~ential may be replaced
with a short series.
Retaining only first order terms, equation 180~reduces to:
~
Cara ink out the dit~erentiatiou. we have:
where
Let us next consider the vertical compone~~t of magnetic induction from a
~rez-tical-axis ma,~~netic dipole source for a uniform earth, as Given in equation
160. "The iate~ral i~ the same as the one in ec{nation 1~9 so the same result
inav be taken. "Clue ma.~,netic induction is then:
lea
IXY~ >?1
2'Crcurs \ Bi~ i
(1051
131
M
— ~q + 9 Gr + 2 G3!r3~ ~.o, C,r~~"
(.187)
e Gr
[~9 Gr — 10 G~r2 — 2 G3r3~ cos Gr
(186)
+eG" [(9 +9Gr~-2G3r3~co~Gr
+(9Gr - IOG2rz- 2G3r3~~Gr,~
~ {q
p~s
2~rtc~a- rs ~
2f(cu6"
2
(188)
r.-~~ 0
integrates to
(The J„ function when weighted by an odd power of the dummy m
true when
is
same
zero over the infinite ranbe. If we are dealinn with Ji, the
only the
identity,
the weightinb factor is an even power of m). In view of this
first of the three integrals in 1t 8 has a value other than zero:
J
¢~
A very useful irtegral identity for Bessel functions is
-m~c~ ~3Cmr)dm
- ~i ar~ ~sn n~ -rrt3
~
-~M1~~
a
~ n,m Crt,- n,),Tcmr) dm
Br = z,~~,~Dz ~,cz~ arD
in~ the integ,~ral into three simpler rote;rals:
ma;netic
The numerical values for the real anc~ imaginary parts of the vertical
I.l,
table
in
listed
are
source
dipole
magnetic
induction from avertical-axis
and are ~.;~iven ~;~rapk~ically in figure 59.
field
The expression for the radial component of the magnetic induction
presents
161,
equation
in
given
about avertical-akis magnetic dipole source,
simple
a more difficult problem in that the integral cannot be reduced to a
radial
ma~the
for
expressions
closed fori~~. However, eve can ;et asymptotic
two
other
the
evaluating
netic field using an approach similar to that used in
dividthus
j,
integrals. We first multiply the intearand by (n~,—n~)/(n~>—ni
~?~'"°~
gz,nne.Q
~
In Chis case, the vertical component of ma~iietic inducCion is entirely real. In
numez~ical ~~aluation of the exact expression, liven in equation 100, it is
necessary to divide tl~e real and imaginary parts. "These are:
~°~ ~
~lr -q ~
T~i~orY or EL~:cTarcnL Sourrvr~Tc
1.0266
1.0391
1.054.0
1.0709
1.0°96
i.1306
1.1734
L214~
1.2502
1.27r9
1.2951
L3003
12925
1.2t17
12301
11927
1.136r
1.0718
0.9997
0.9223
O.o4:14.
0.7580
0.6762
0.5950
0.5166
0.3393
0.1962
0.0906
0.0203
-0.020
0.6
0.7
O.a
1.2
1.4
1.6
22
`~.2
4.4
5.5
~•~
~•5
7.0
7.5
`~•6
-}•~
5.0
3•~
3.8
4..0
2.4~
2.6
2.8
3.0
3.2
3.4.
1•g
2.0
0.~
1.0
~•`~
0.5
1.0002
1.0013
1.0041
1.0091
1.0166
/ao~
Real-component,
4~rr~
multiplied by
-
01
0.2
0.3
Gr
Q UARTERLY OF THE COLORADO SCHOOL OF MINES
-0.7737
-0.7003
-0.6100
-0.5160
--OA277
-0.7761
-0.7998
-0.8137
-0.8185
-0.14.7
-0.1328
-0.2062
-0.2839
-0.3632
-0.4412
-0.5157
-0.5846
-0.661
-0.6990
-0.7425
0.0792
0.0636
0.0334
-0.0106
-0.0667
1.0510
0.0614
0.0702
0.0769
0.4808
0.0022
0.4085
0.0175
0.0282
0.0396
multiplied by
~,oM
4ari~
Isnaainary component,
TnBLE ll.-T/ertical induction about a vertical-axis magnetic
dipole
132
•
gr =~ arf mzn,JoCmr) cQm
e,
Gr
(139j
Bessel functions. In order to convert
where I„ and K~ are the modified
one in equation 189 that we wish to
this interal to the same form as the
with respect to z, set z equal to
evaluate, it is necessary tc~ differentiate twice
Io~z'(rZz2 - z)
enZScr,~r)dm=
h~
o
}~Ko~z~~rt_ZZ'Z~~(19oj
the identity
This last integral may be evaluated using
00
~
•~
Z
133
dipole
induction about a vertical-axis Ana;netic
Fz~uee 59.-Vertical magnetic
uniforiu medium, and
a
in
number
wave
the
of
part
source. G is the real
r is the spacing from source to receiver.
M~la
4trr3 B~
NORMAI .'_~ ..`~~T,~
M AGNE'
i
THEORY OF ELECTRICAL SOUNDING
QII9PTF.PLY OF THl COLORADO SCHOOL
OF MIAT~S
2tYmr
11
—rRr`
— \2mrf e
('ft \yL
l6 iw~3M2
0.
+...~
...~.
(193)
(192)
zn~z
(195}
power, and so, are zero. The third ii.tebral is
of the type lisCed by Eredlyi
I. Higher 7'ranscenclental l~unctions, v. 2, p.
4~9, equation 19) and has the
~~alue:
~
~r f rrt3~T<rnr)dm +z2f m,TCmr)dm +$
f rs,.~JoCmr)drr~
T~~e first t~vo integrals ai•e of the type weighted by m
raised to an odd inteber
gr --
Substituting the first three terms into the expression fozthe integral, ti=e have
— mr ~
~+
1 + Xzrz
rn2r~, ~~/2 = rT'I1 + 2 Xr z
ni — r
~rnr~ + 8 ~ mrY~ + .'.7 (194)
In order to determine the behavior at small
values or- yr, we retuz-n to the
integral in equation 189 and expand the quantity n~
as follows:
~zr8Q~a
~ _
"I~his equation inay he solved fir apparent
conductivit}~:
~r = - ~—
~~,-rM-~°
so that the equation 191 may be written approxiniafely
as:
8mr
8mr+
i
1
2rt~mr ~ 1 + 8rnr + ...
Yi
K1 Cmr _ ~~~ ernr~1_
Ka(mr
So Cmr~ =
emr
f 191.
I+'or large values of the argument yi•, the modified
Bessel functions have the
asymptotic expressions:
Br =— g3 [ (Xr)2~I,K~-Io~~ +~Xr (I,Ko- IoK~~-f- 16 I,
K,1
zero and then differentiate twice t~-ith respect to
r•. 7'he differentiations are
somewhat tedious but lead to the result:
13~'
(
32tt' rz
(1977
—
T`—"LP
2tYa-
~
rq
~3 — e ~r ~3 + 3 2Sr +
1
(Xr)'']t
1198)
_
~r-~`b
31~
2R'6"r9
~r->~
6~~_
~ I~
2~crr4 F3Z
('202)
(201)
(2001
~ 199
evaluate
The other field components about an electric dipole are somewhat more
difficult to evaluate. From equations 13=1-136, takin; R=1 for a uniform
earth and y„=0 for practicality, we have the following interval expressions to
and
_ 32 rr2r~ / F3Z ~p\z
—
6~ 1Sr->o 9 w3jco ` rc?Q
The two forms for apparent conductivity are:
~z
and the expression for large yr from equation 176
13'r-9O
~ `~'~ ~~°
Likewise we have the expression for B, at small values of yr from equation
173:
z
~
Let us now turn to Che definition of apparent conductivity for field components measured about an electric dipole source.
The principal of reciprocity states that if the role of transmitter and receiver
are interchanged, there will be no difference in the measured field. This can he
used to evaluate the vertical mabnetic induction from an electric dipole source
in terms of the results we have already obtained for the tangential electric field
about avertical-axis magnetic dipole source. The com~~lete expression for I~,
is obtained from equation 1~9 by substitutinn R., for .C~,'~ and Tdl for Mµ,,:
r
B~ _ ~. ic„~oa-
jgZ
"
~
(136)
13j
Differentiating this expression, ~a~e have the simplified expression for radial
magnetic induction for small yr:
0
o
m ToCmr) dm = 2r
~'O — 2r
TH~OFiY OF ELLCTPIC9L SOliNDI1G
QUARTERLY OF THF, COLORADO SCHOOL OF MIA'ES
zn
aXay
n'
~ mCm +n,)
-~
m,3o(mr) c.~m
(205)
Z
~'
o
o
~
(206)
(208)
J m S~ ~m r) dvn = r,
0
(249)
We obtain the other intebral in 206 by considering the case with yi=0:
r2
~niJiCmr) c.~m - ~e~ r +ear
Differentiating; the left- and right-hand sides of Chis equation with respect to z
and settiii~; z=0, we have:
(207)
t -f 10Cmr)c~ ~e nZ)~'
r ~_e ~e~z + zf~ n en'Z3
<mr) dm3
Jo<c~tr) ~~
°
- XQ~ r~Zz
_ 2-~~z _ ze
r
r~-~
~Zz
_
-n,z
l ~e
rS
fe
n`Z cTl Cmr) c~m = - rf 2. n~Z c~ ~ SoCmr~
0
0
Consider the following; integral, which may be integrated by parts:
f~~n, - m+n,
~i~ ~,Tl (mr)dm =f(1n, +m- n,),3~1Cm~)dm
~
In the expression for ~„ the first integral is precisely the same as the
integral foi• E~p'~ ~shich was evaluated earlier. The second term in the second
interrand is multiplied by (m-n,1 /(m-n,) to make it snore tractable:
X
B =Imo° a2 ~x(
(204)
- misCmr)dm -rte?` ~f(~n,-.~~JcmY)dm
c. Parallel magnetic induction
EZ=~wI~ s2~~ n~
b. Perpendicular electric field:
(203)
dQ
x rn
r~
z
Ex= iu., I
~/~
Jo(mr) clm - ~ ~ rf (fin,- m?
}n~~,31~mr)dm
.l m +n,
a. Parallel electric field:
136
f
I
i
137
r
(210j
(:211)
(Polar dipole array)
(212)
L
z l
2
I Equatorial dipole array)
~
(213)
EX
2t~'r
IO~3 ~PQ
(214j
(Polar dipole array)
For very low frequencies, or for direct current, the parameter yr becomes
essentially zero, and these two equations reduce to:
~r
1 -(1+ ~-~-rl e der +? e~~r/~
In the second case, the ratio x/r is zero, and equation 211 simplifies to:
-~lQr ~ e-Xer/~(2,Z .4-~Qr~~
z~~3 ~1 - e
EX = r~
E _ - I
~~ r
X
eX~Y
/a~~_3~(r~z- Ye►'~r~2~~
Normally, the elecri-ic field about a cun~ent dipole source is measured along
the polar axis of the source dipole (as in the polar or inline dipole array) or
along the equatorial axis of the source dipole 1 as in the equatorial or broadside
dipole ai•ravl. In the first cage, the ratio x/r is unity. and equation 2ll simplifies to:
-
~x = I~~ ~
~3~r~z-z~ +e~r(1+X~r) +- eY~r~l~ 3~~~2- (~r)~r~z~
Carryin; out the indicated differ•entation, and collecting terms of the
same forYns, we have
rz~~r- ~~~r
+eft,►" } Cz~
~ J
1 3- e-Xe' [3 +- 3~~r + ~~'e~r~
- I~Q~- a r[ ~
EX~ iu~ r~. ~~
Replacing all the integrals in equation 205 by their closed-form expressions,
we have:
THEORY OF ELECTRICAL SOUNDING
E
I d~ ~
Qu.aaTrri.Y or TFir CoLOra~o ScxooL or VIrvrs
n.~-s Pp
IdP
;w
~ I ~~
~ i c.~
z
a
E ..,
L 2 -" 4 ~ F" ~
R.Y3
x
(Polar dipole array
1216j
~211~
(Equatorial dipole arra}')
2~ ~ 2~ + ~u r~ ~
Ex ~~ _ -i l~"~'_,~lo
(Equatorial dipole array)
(219)
lPolar dipole array)
(218
X ~Y~~~
EI
X2
2rrr3 3 Yz - 2~
~~
X220)
For the equatorial and polar dipole arrays, the correspondinn expressions
are:
ing approximation:
The variation of resistivity computed from equation 214, and 21~ as a function of yr is shotivn graphically in figure 60.
F'or high frequencies, where yr is large, equation 211 reduces to the follow-
and
EX L,«~.~ = i --~°
rdP~,
From these expressions, we may gay that the zero-frequency behavior is
valid so long as
~Zrz « 2
Thus, direct-current behavior should be expected with a ~~ood de~~ree of aocuracy for source-receiver separations up to half a radian wave length. The
imabinary component of electric field in the tt~o cases is:
and
Ex
or low frequencies, we call replace the esp~~nential in equations 212 and
213 with the first three terms of its series expansion, and so arrive at the
folio~viri~~ expressions valid at small yr:
x
I L:quatorial dipole array
1210
Thee, of course, are the standard equations derivable from Laplace's
equation for the direct-current resistivity problem, and can be solved for
resistivity to provide the usual defiiiiCion of apparent resistivity for Che polar
and equatorial dipole arrays. It should be noted that the resistivity which is
measured at DC is the quadratic average resistivit}' in a homogeneous anisotropic medium> pQ = f pr • p,)'= =App.
and
13o
+
v~.
or E~.t:c•rruc:~i. Sot;~nrvc
x
~ —
r~E
'~''r
E ~ I~p
x
2trr3
(Equatorial dipole array)
(222
I Polar dipole array)
(221)
frequrrnc~ for e~~uziU~rixl and polar
.Variati~m ~,t (i~~l~l ~tn•n_th with
the real part of the wave number
whi<~h
in
~•<n~tl~
ei~E~ulc: oe~~r a unif~~im
i~ G.
1Y
Gr
)RIAL
1J>
~. ,~
.;IF;
~~
frefrom the DG behavior to the high
It is int~iesCin~; to note that iii ~~~oin~~~
electric
the
and
for the polar arras- halves
c~ueuc} l~el~~~ ior, the ~ lectric field
after the inverse r~' behavior is taken
doubles,
field ~Ior the equatorial army
into c~>u~ideration.
Che
if~i~tivity can be defined with either
!is in the other cl c~, appaeent
lowthe
~~prosiinations. Here ho~~~ever,
high-fregtnc~~ or the lo~~~-frequency
decvidely than the l~i~~h-frequency
frequen~~~' definitions are used snore
current resistiviCy methode.
finiti~>n~, l~cin.;~ the b Isis for the directlow-frequency approximation is exthe
IC zs also iml~ot~tant to note, that
aiii~otropic
quadratic avera~~~e resistivity of an
pressed as 1 functi<in ~~f the
1i'equ~ncy approximation is expr/~ = p~~, ~~'hilE the hi~~hinediuin, dpi
i~~isdvit~ ~f the medium. It appears that
pre~sed in teiin~ of the lon,itudii~~ll
/electric-dipole
considered, only the electriodipole
o£ the methods ive hay e
information.
method p~~>vi~les thi; t+~ealth of
and
I~icirx 60_
AC FIELD STRENGTH
DC FIELD STRENGTH
'CII~:or
QliAPTI:PLY OI' TIDE COLORADO SCHOOL OI' MINES
There is a wide variety of electi•oma,~netic field techniques which might be
used in measuring the resistivity profile in a sequence of layers. From C1~ie
point of view of operational ease, two C~~pes of field source ire preferred to
others—the source may be a ;rounded ~vii•e or coil of wire with a vereical axis.
The first constituCes a current dipole source, if the len~~tl~ of the wine is small
compared with the offset distance at which field components are obseT•ved,
while the wire loop constitutes avertical-axis t~~a netic dipole, if the diameter
of the loop is small compared to the offset distance. with a cari•ent dipole
source, one might measure the vertical magnetic field, or the horizontal ma~netic field, or the horizontal electric field about the source. Depeiidina on the
B~zinvloa of TxL R nv~ K" FvvcTro~vs
Range B, 0.6< yr<6: At moderate distances oi- frequencies, the beha~~ior
of any of the field components about any of the sources becomes very complicated, inasmuch as there are contributions to the electromagnetic field both
by enei-~~y traveling; through t}~e earth and by enei•~~5~ traveling; through the
halfspace above the earth. While an apparent resistivit}' may be defined using
various graphical techniques ii7 this range, the definition is restricted in application, and is not normally used.
The relationship between earth resistivity separation and critical frequency
separatinb each of these ranges is shown graphically in figure 6]..
field.
Evaluation of other source-receiver combinations is straight forward, inasmuch as the forms of the I3essei integrals involved are all the same as the
types ahead}' considered. We will not at this point continue with the evaluation for the case of a uniform earth, but results are summarized in table 12.
Rarzge A, with yr>6: At large distances or hibh frequencies, the behavior
of any of the electromagnetic field components simplifies to an inverse r" behavior, with the value of the exponent n varying from 3 ro 5 depending on
the specific source and receiver being considered. The field components depend only on the longitudinal resistivity in a vertically anisotropic earth.
Range C, yr<0.6: At short distances or at low frequencies, the behavior
of any of the field components also simplifies. Except for the electric field
about an electric dipole, the field components become insensitive to the electrical properties of the eaz-th. With the electric field about an electric dipole
source, the method reduces to the DC resistivity sounding method. Generally,
the effect of the earth properties at this end of the spectrum is contained in
the term in quadrature to the principle term of the approximation, whether
this be the inphase or the nut-of-phase component. It is possible to define a
simple expression for apparent resistivity in this range, based on the expectation that the quadrature signal can be separated froze the i7iain part of the
1 -~0
c L
o
E ,~
N
~ i
n~o
N oO
U Y
°c' E
i7
THLOFY OF ~L~CTPICAL SOUNDING
0
c
T
U
a
0
•
O~
141
TaBr.~ 12. — Ezpressio~zs for al~pa~•ent, resistivity
Cou~~lin~; system
Vertical-axis coil
Vertical-axis
ina~neCic dipole
Receiver _
Source
`t"
5
GR < 0.6
3
z
Apparent resistivity
-----
_.
Gj{ ~ ~
2
p~ _
8rrzr'~ ~~2 ~Ur~
Radial-axis coil
a.
9 iu~2/~c~ Ar ~~s
2 ~'r r 5 Vr
16 I LU3~Jo3 Ar2 Ms
~,zr8v z
r
3 5
Tan~enti~l electric field ~~ ~
%~
Radial electric field
~`~
~_
d
~ ~`r3
I~7 Ex
`C
y
r
1
~1w/.~o MS
32trzr~`~ {~i'z.~E~~
~~ = 2 rr r y E~
3M5
E
__
Horizontal curz-ent
dipole
~~ - ~
2ri'r3
i TanbenCial electric field ~a - r~ Ex
4
2tYr3
rd~ co~6
r
r;
~~ - n"~2)
0
-tea. -
Vertical-axis coil
~"
32 ~,zr4 ~~~ ~v~~
o
3Ar ZdQ ,sa, e
~'
2t~' i r4 Vr
0
~ _
Vertical electric field
2t1'r3 ~z
r,
'~
x
o
TnBL~ 12 co~iGini~cerl--key to symbols
_.
-__ _.
'~
SY3r6 Vr2
Arwho(S~)''~2~
~~"
Radial-axis coil
~
;T;
__
r„ = frequene5~, rps
n
2,
6
-,~,
4tr 2 i r Ur
~0. - qr who (r~(Q)~'C~2(~
Tangential u~a~;ueCic field
__
__
_
_
_
cn
[
µ„ _ ~]~7rX10-'I3/tit
A,. ~- e~ffectine area of recei~~er induction coil
times effective area
NI, ~= moment of magnetic dipole source, current
r = separation beriveen source and receiver
V~. = voltage output of an induction coil receiver
l~, =real part of n~ieasured volta~~e
b5~ dipole
F = Can~~ential electric. field measured volt i~~e di~~icled
length
b}~ dipole length
Er = radial electric field; ~neasw~ed volla~~e divided
dipole axis
E~ = component of electric field parallel to cm~renC
and electrode
Ids - currenC dipole inoinent; product of current
fl
=~
separation
and tine radius
is tl~e angle bet~ti~eeu the current dipole axis
vector to the receiver location
..~
Qu:~rTL~~LY oE- ~rx1: Cot~otza~o Scxoor. or l~Ii~~rs
21
(22-1 ;1
~p,N-i
(22~)
.. ~-1 n
~ ~ ...\l}
i~ ~
I~hi~ limit provides a function which is relatively well-known in electrical
pio~pechn,~. Wait f 1962, p. 111 has designated this limiC foi~ tl~ie R function
as a Q function. used i❑ correctins the plRne-~tiave ~mpedauce at the sL~rface
of a ~ti-atified ezirth for the effects of laverin~~. Ca~niard 119531 has used the
Q function in definin, apparent iesi~ti~rit~ as measured with the ma~netotelluric method. A~ a consequence of the interest ~z~ the use oi' the maguetotelluiic
method over Che past decade, nur~eroLS computations of ~~~lues for the Q
funtion hate appeared in the literaCure (Wait, 1962; Jackson, Wait and
falters, 1962; Yun,u1, 1961.1.
~chich becomes in the limit
= c:o"L-R ~ R,R ~ + CoU( 1 R—~' C ca~i( (nzRz + Ce~1
"I~I~e lunitin~~ behavior of the R functions as the ~eparaCion parameter is
made yin ill is <~f particular cone~ern. T~~~r the R function eve havE;
=lsynz~~~otic I~ehavior ~or snacdl na
In spike of the ~°ariety of source-reccl~ er ,~E om~h•~es ~c}~ich mi;ht be u~ecL iu
the field, it is surpri~iu„ hor~T fe~v inde~~endene e~timate~ of the conducti~,it5 profile inay be obtained. As a conseque~lce of the use of the method of separation
of variables in the solution of bIax~~ ell's equations, each one of, the expressions
for a field component f as ~~iven by equatioi~~ 1:3-1-, 13~, 136. 156, l.5r and
lSv I consists of ~ Hankel transform of soi»e combi~~ation of R ~md R" funetion~. The R and R ~ functions depend on1~ on the electrical j~z~operties of
the medium land on the dummy. m. ~~~hic1~ ~~as the separation constant)_ a~~d
not on the ~ouice-reef ~r~ i ,~e~metr~~. The effect of ounce-i~ceiver geometry
enters only iu the detail~cl nature of the interr~l iran~form. 1'he 2~roblem of
d~_cussin~~~ ~~arious techniques is sim~~lified because ccc, can consider• soui~cereceiver aeometr~~ indel~endentl~~ of the c~lech~ical structure of the earth, within
the limits imposed b~ t}~e fact that ~~e ha~~e two functional repz-esentaCions of
the resi ti~~it~ structure of the earth. Let u~ now consider the propeeties of these
t~~~o functions, R a~~d P'`'
used to investi~~~ate the variation of s~e~»ti~~~ily ~~~itll c1eE~t1~.
~>~eometry of the source and recei~~er combination. the ~ ariety of methods
~k~hich mi~;hl be used is almost li~nitles~. ~WiC}i the ~na,netic ~otit-ce. the vertical
ma~rnetie field or the horizontal rna~~netic field or electric field can also he
1~1~=1~
!
E:
~C~(/~2 YtZhz
.!
(226j
(225}
+ cc~-i
145
_
?~i +l
~
— ~ ~~-'
—
'~z
p~>' ~
~/z
(2z<~j
(227}
Iimitina behavior of the
It is also of considerable interest to consider the
small values of the separation
R functions for low frequency, rather than for
variable, m. The Ii funcCion in this limit becomes
flsymptotic behavior for small o,
there is no differThus, if a sequence of layers is made up of isoCro~~ic rocks,
the other hand, in a
eiice bettiveen the Q function and the K,'' function. On
longitudinal
section of anisotropic rocks, the Q Junction depends only on the
resistivities.
transverse
resistivities. ~~~hi1e the R„” function depends on tie
in a
anisotropy
effects of
In princij~le, it shoul<~ be possible to detect tltie
the
both
evaluation of
layered iraecliuna with naeasua enzents u~hic7, permit tlae
Q fzcnctio~i. and tlae R„” f~c~action.
~i~l ~t,~,i ~.e,i ri
~~?~t,~ 1'Q>~
while the ratios in equation 226 are:
~,~''
~,e,;,~
similar, the
It is of interest to note that R„'' and the Q function are very
hyperbolic
inverse
each
of
argument
the
in
ratio
difference beiilg~ solely= in the
the form
of
exactly
are
226
in
hi
~~
yr~
cotai~~;ent. The terms oI' the form
22-1
ec{uation
in
terms
correspondinb
yip hi, which is idc;ntical with that of the
are:
22'~
equation
for the Q function. Note that the ratios in
/~N o-kN~.QN
f//-1 ~N'I ~t~N -IJ (~N-~ ~~~
~~er~,r~
which becomes in the lirnit:
/~tJntJJ Q~N
... cv~ -1 '~N-~nN-~ ne,N~' ~...~~~
~zn2~z
R'~ = c~{~~n,h, + ca~-1 a'n' P1
P'oi the R'" function, ive have
'Txl o~Y of LLLCrt.zc:~L Sou~nz!~~c
Qo<~r,TraLr or •rli~: Co~.o~z:~no Sc~{oor_ or lIla~l:s
~
~
z~Pz
-i a , ~ ~
~~m1izh2 + ca~1
Let us review tl~e nature o[ the special Q and K functions. inasmuch as then
have been discussed in considerable detail in the literature I ~~ait. 1962:
NIeinardue, 1.96; Bodvarsson, 1965: lToone~ and others. 1966: I'lathe,
19~~: Slichter, 193:x: Onodei•a, 1960; Roman. 196:0.
It is important to note that in takin~~ a limit ~s frequ~i~cy i~a5 made small
the wave numbers dropped out o~ the 3i ftmction~. This results in the K(m)
function bein~~ composed ~~nly cif a real part. ~ti~hich co~~siderelbl~~ ~itnplifies its
evaluation. On the other hand. the Q function will normally consist of both
real and ima~inar}~ parts, because it is a function of tl~e wave nus~ibers for
the medium. Usual practice is to expi•e5s the Q function in terms of a ma~,~nitude and phase, rather than in tei~ins of real and ima~~inar~~ sizes.
'Che behavior of the K and Q functions with ch~n~~in~ m and <<~. respectively,
is shown in fi~~uz•es 62-66. In the first case, the independent variable is plotted
i~~ terms of the dimensionless product inh. Inasmuch as m has the dimensions
of inverse distance, and is related ro the ineerse spacin~~ bet~keen source and
receiver, the pi-~cluct mh may be vie~verl loosely as the ratio of layer thickness
Tlie Q and K ~ia~zctions
This function is one which is Snell known in electrical ~~~eoph}~sics also; bein.~~
the so-called kernel function for the di~•ect-current r•e~istivit} problem, ori,~inally developed by Stefanescu and others (1930).
These considerations of asymptotic beha~~ior for small m and small ~~> indicate the ~~eneraliCy of the R functions the~~ include as special cases the
theory for direct current methods as ~~-ell as the theorSr for the nia~~netotelluric
nleChod.
~N~Q'N
K = ~m R ~ = co~ ~ m ~ ,~, +
woo
(Note that the behavior of the h~~perbolic and inverse h~~~~erbolic 1'uuctions
which are of concern here teas summarized ,~raphicall}~ in fi;~;ure ~5). At
zero f~~ec~u~ency, the R ~unc(ivn becomes inclepenclenl o% the electrical j~roperties
of the sequence.
The corresponding; limit for the P'' funcCion is:
W -f0
,~,,,, R = co#~~mh~ +co~~1~co~Cm~z +co~~-'... ca~~l(~))...)~~'
14.6
~
10 2
10 ~
I
mh
147
10
pez
-spec ~
g2Cw~ = c~t~e~h~ ~- ~
-~ ~~ 2~ ]
K2Cm) = co~ Cm~,h, +- cv~ azpQ2
~2 2~
1231
function is indepenclei~t of m, but deto source-receiver separation. Tl~ie Q
The independent ~~ariable
pends on frequency throcnh the wave nur~~ber~, y.
l', ~~'hi~h ~~ in effect the
h"
/~„µpr
i
for the Q Junction plots is taken as ~~ h =
layer,'~[hus, in the plots
that
in
th
ratio of layer thickness to radian ~~~a4e ten
to the left rather
increase
len~~th
in fi~~ures 62-66, values for spacing or wave
than to the right.
functi<ms f~~r the pimple t«o-layer case
The expressions f<~r Che K and Q
with p./_p,>1 are:
function for the case of t~~~~~ la~•cr~.
Gictn:t: 62.— Aumericxl bcha~~iur of tlir K
~•
KEf~NEL FUNCTION, K(m)
TxLOF~Y of Ez.rc~rricnz. Sou~~ni~~c
IOG
N
O
~
~'SO
~O
4~~ ~°~
8
~~
~/~
~~~
i
~I
i ~
i
~~J
/G~h ~
~~2
2
5
'1i
i
Qti<~aTrrr.~ o~~ ~rxF: Co1.o~;.ano Scrroor. oE~ l~Il~~r:s
'l~he~e ttico <sE>rt dons are rciiiarkabl~ ~imi(ar ire foi~in_ but this similariCt~ is
cii~i~~i~ted i[ ice con~icicr Clie curnples nature of the ~ca~~e nwnber_ y,, iii
e~ uatioii Z::;Z. If ~~~e ~~ rite Y~ -- G i -'- ~G i~ evith G ~ = (~ic~~µ~ri
I'~- e~{nation
~~
'? 2 in~i~~ he~ lir~~k~~n iiit~~ ~r~al and ima~~inai~~ ~~arts:
I'ie~ r,ic 6:'>. --~ lIa_nitude of tLe Q"- luny lion I~~r the cu<e of 1~~u layers. C, is tht~ real
~~.n~t ..f tlic ~~a~i- number iu th~~ ;w~facc I<i~~rr, and hi i t}i~~ Iliicl:nc~;
~~I that I,~~~v~.
l - I•o
i
t
i
~:
a
Q
Z
v
~.
(~
C.~
~Z{
-1 /~P~z\'~a"1 -V- ~
2G~~1~
LG`~,1~ -4- ~K
`~ f ~
~~~
Zr>:>
~t~li~i<~I:n~l,~rarl
i /6~hi
I.~~<<~ an~~l~~~i~~ iC~~tlu
I~heihc ~~~a~v~+~numli~ii~~~~in~ t~h~~ tluri~u~~
that lari~r.
~~~
2
~
1~l)
rnh~~ K 2- 1
iZ>-1 i
zn~dc lar~E in equation Z~l or G~}~~
IC nay lie peen readily that a~ rnl~~ i~
and
lid peibolic cot ~ns~;ent become l u ~e_
in equation Z 3Z, the ar,:~ument of the
as~~~~ptotic value of wail} to the ri~,ht:
Che K~ or Q~ lunction~ aE~proach an
92,~w) _
i'~ct~.i:i~. 6L _
-3~
_4R
- 6C
-75
1'1ii~:orl or E~.i.c~rrzc~1. Sot~i~r
-90
).i
I
0
In
p2~y,=W
i/G~h~
I/100
I/50
/10
I/5
I/2
2
5
10
50
ioo
IC ❑~
QuAr,T~azY or z~x~ Cozo~zn~o Scxooz or 1l~IiN~s
The asymptotes for small mh~~or G~h~ are also easy- to find if the ar ument
of the inverse hyperbolic cotangent is greater than unity. 'The inverse
hyperbolic cotangent is clot defined for arguments less than unity; indicatinb
that if such arguments arise, the expression for the R functions in terms of
~h ~~
Ficoar; 65. —Real part of tl~e Q function for a ;i~~nlc ]aver resting on a uniforiti
substratum.
ip
1JO
•
•
00
0
~_
/G~h~
I /50
i/20
i/10
I/5
i/2
5
10
50
Pi~Po =ioo
~
••
Zjl
(236)
( 237)
mh,~+o
G,.~o C~z ~w) _ ~~Qi /
~e2)'2
hyperbolic tanbents must be used. "Chas the limits ~foi• small n1h~ vi' Gihi ere:
a. with p~ >p_
uniFicuiti 66.— Ima~inary pari ui tlic Q tur.ction for a ~in~lc la~cr resting on a
form .~ul>stratun~.
C~
Q
Q
z
a
r
H
Q
O
2~
10
~.CH~ORY OF ELP:CTRIC 1L SOU1`DIVG
~ZCm) _ ~z~Q2/~~.e~
~
'/z
~~~ QzCc~) _ ~~Pz ~'ei~
m h,-~o
.~um
I2.91
(2~~>
(1u,~r,Tl:r,T_Y or ~r~tr Co1.or,~no Scrzool~~ or llr~r~s
~?i~ i
~
P~P~~°°
e1 ~
ri'~~h~
i 21.2
The problem of the ensiti~ its of sn electrical sounding method to a thin
1a~er ~1~ith electrical properties different than Phi, liecls shove it and helotiv it is
!i cuicl Q fiuic/ions for thin layers
last one.
In both ~•ases. the equation is that of a -trai~ht line o❑ a Io~~~arilhmic ~~lot, ~a ith
a slope of -f-1, iniers~ctin~> the unite 1s}mptote as tuh~ =1 or yihi=l.
with d1e case of a buried conductor, the ii~t~ise is found. 7~he asymptotic
behavior at sma11 m or y, is that of a ~tia~~~ht Line on a 1o,~arithrnic plot, ~r~ith
a slope of —].; and ti~it~h an in~ercel~t ~~'ith tl~e unite aa~ mptoCe at ur~it~~. The
ts~o-la~~er eiivelopin~; kernel functions are sho~~~n,~raphicall~ in fi~>uee 6i.
_~(( ts~~o-la~et• Q and K functions it~ust Cie bellceen these limitiri~ curves an~~
the units Lis. In fact, the K and Q functions can be ~hc~~vn l0 1ie betstieen
t~~o limitin~~ curves, onE. p i ~sin~~~ thruu;~h the point mH or GH=1 ~4~ith a s1opE
uI i-1. and the oC~~er pi.sin~~~ tlirou,~h this point ~~ith a slope of —1. For anv
number of f~~er~, ~1 ith H beui~~ the total thickness of the I i~~ers. hbove the
w-ro J~~P,~°o
M~~
"Chese cur~~es provide a lii~~iting envelope for Che tw-o la~~e~r curt-es plotted as a
function of the contrast i❑ resistiviij~. I'or large m, or for lame y, the curves
approach the as~'~nptote 1 as specified in ecluatiuns 23-1 and 23~. I,or small m,
or for small y,, the asyinproCic form of 2-10 or 2-11 is:
~4z~w) _ ~~eth~
(2~0)
.~n Kz Cm) _ ~m ~ ~i
~ ~
1~~~ -~°°
A dil~erenC sequence musC be used in takin~ limits if the contrast in re~istivities bet~veeu the layers is very lai~~e. Chat is, if p~ /p ~ approaches either
zero or infinity. In the first case, that of a buried insulaCor. the inverse h5-perbolic ectan~~ent iu equations 2~1 and '232 is eseenCially zero and ina}' be discardecl in comparison with the fit~~t term. the oi~e containin~~~ m or y~
l~. ~~ ith p,<p_
1 ~2
mH or GH
1~3
of cuu~ideeable interest in the petroleum pru~~~ectiu~ a~~~plications. inasi~~uch a
most oil ficld~ are relativelt~ Chin eom~~arecl t~~ the <leE~th of burial.
The prol>lein mad be f~iri~~ulate<! bti a~suiniu~; [hat a iesislant Ia~er is
emliedded in an oCher~~~ise unifor~~~ halEspace ~aitfi a iesisti~ie}~ p,. T1~e re~isde~~tl~ C~~ the thin
tivity of the thin lati~er is taken as p_. ~+'ith p_»p~. Thy
of
these pv €imetecs
clPfinitic~ns
The
L
thi~l:ness
as
la}'er is t€ikezn a~ H. anal the
6~,.
Fi~ure
on
~Iescribin~> the' section <~re indicated
'I~hc ]h and Q functions I~<,r a Ll~re~e laver see{uen~~< are ri~eri b~
-1 ?~~
L~''~,H + C~~6C -~~a ~co~ ~ m~2t+
K3(rn) _
~3~s~~~
p
121-l~)
ealue
P~ctitt~: 6;. -~- l.imitin ~°~ilue; fur tl~e K ur O lunctiun I~~r i~v~~ Ia~~ci "I~lie actual
for t6c K ~~r (? limc~ti~~n inu~t iic Ixt~~~~~cn ~lie~r cui~~c~ anal the wtity
maonitucle• :axis. f[ i~ Ih~ tliii~knc~<.; ..f ~lir~ tu~~ lacer quid G is tk~c itia~c
ntunl~er in that I.~}'c~r.
MAGNITUDE OF
K or Q FUNCTION
(00
'['En~:ort~ ~r I:rr.c•rrrcu. ~oi~»z~c
~h-~0
Qt;_~rTLrzY or ~rxr Cor.or_~vo ScFiooL or l-lznEs
2-L~
~ ~~i H t-
_
—t}
~iY
~1(~Z e~
~z rnT2~
(2-1~r)
(2-1b)
~S,''mT2
L + ~~ ~
mN
(2'181
Tlie~e equations n is be rewritten usin~ the formulas for [he sums of t1r~u-~ /~F~
inc nt5 for h~~perl~ol~c tangents:
~ rrt ~~ H +
K3 ~'m~ ti
Let ns no~v consider the limiti~~~~ behavior ~~f these t~~-o e~~>ressions as ~~~e
IeC the ratio t,H hecume sivalL At the saute Ciine, ~~~e ~vil1 let the ratio p.~~p~
become lame in suds a wap' thaCthe ratio p~H~p_t remains constant I~~ot~ that
the product of [hc ic~isti~ ity and the thickness of a layer, such as p~II or p ~t, is
a parameter ti~hich ~r~i,ht he used to cliaracteu~e that Ii~er, and which is
called the iraus~ei5e iesi~tance; 'CI. With the laser rftiiativiCy much greater
than the iESistivit~~ of tl~e undet-lvin~~ i7iatei•ial, the inverse h~-•~~erbolic tln~,ents
in equaCions 21-I. <lnd '2-I~ becoui<~ ne,~li,~ible in comparison with the term mt
or y>t, as the case ma~~ be, rind the equations ma~~ be sir~iplified to the forms:
Q3(w) _
~ ~~ H +~-i(1~er 1~2 ~co~ ~ X ~ + c,~-i /1~Pz 1
z~~
Ftco~ttt fi8. — llc[i~~itiun of the mode( u;ed in cumputin<= the eflcct of a thin, resistant
I~rd on elrc~ric and ma i~c~tic field I~cha~~iur.
j1
PL-,°°p
H
154
15j
~~t ~ ~1- x~t~X,N - 1
1 ~ ~~t .~.~,w~ X H
(2~l}
Hit — ,
1 + ~ /t
~~ H
—•' ~~N ~ 1 — H~
~
~
ti - ~` ~r2
L t T,~TZ m H
(253j
(252)
These last Cwo equations show Chat the asympCotic behavior at sma11 m or
strai~he
~, is such that on the usual bilobarithn~ic plot, the curves tivill a~~proach
axis aC
lines with a slope o~f —1 to the left, with ~n interce~~t on the unity
It is interesting to note that for the K function, the values clepencl only on the
transverse resistance, ~I'~~, of the thin layer, and on none of its oCher eharacteristics. For the Q function, the ealues depend only on the thickness of the
thin layer, and not on the resistivity, so long as the resistivity contrast is
high enough that the approximations used above are valid.
~,,..0
~s C~v)-1 I
m -~ o
K3 Cm)-1,
Here, foi- convenience, the layers have been assumed to he isotropic. Let us
now consider the behavior of these two functions—the kernel function differences which define the sensitiviCy of various electrical sounding; r~iethods to Che
presence of a thin, resisCanl layer: The asymptotic behaviar~ for small m or• ~~
is found by ieplacinn the hyperbolic tangent by ies argument:
Qa ~m~-1 -
mT~
Note that as ins ~, K;; ~ m i ~ 1, and as o,~ ~, Q;;(w) ~ l; as is the ~~roper behavior in the limit. However, the K and Q functions would be uniquely
unity- for the case in which the thin Layer was not present, inasmuch as the
earth would be <1 uniform halEspace in that case. "Chun, in order to discuss
the anomaly caused by the ~~resence of the thin layer, we need to consider the
difference K,, i m~) —1 or Q;;(~~>) —l:
i
Q3C~) = 1 } g
1
~H + ~,~
THEOPY OI' ELECTRICAL SOUNDI:~'G
Qv;ar~rr.ri.r or •rEir. CoLO~t~vo S~xoo1, or ~It~~:s
—zG,~a
ZJJ)
(251)
= .I~2 t
~i ~
The R `, R, K and Q functions provide on1~ a partial solution to the probIem of clescribin~; electrotua~~neCie field behavior, except for the one caee of
the ma~~netotellui•ic method [ in which the earth resistivity i~ determined directle from measured Q functionsl. In all other case, the R-, R, ~i anc? Q
functions must be incorporated iii an inte~~ra1 transform before the field
quantity actually z7ieasured caii be determined. Conversel~~, in order to derive
the electrical properties of the earth from measured field data, these data may
1'IIF. HA1fiL:L. 13iANSP'Olill
it is apparent that the anomaly in the K function caused b}~ a thin resistant
bed ~~i11 alr~~ays be much lamer than the anornaly~ caused by the same bed in
the Q function.
Tz ~T
`I~he behavior of these three:-lager K and Q functions i~ shotivn ~ra~~hicall~r in
fi,ures 69-r I.; for various re~istivitti~ and thickness ~~arameters for Che thin
Layer. The K function passes through a minimtun tivhicb deepens as the ratio
~C /T~ I~ecoi~~es lamer. while the Q function has a mar;nitude tivhich passes
through a minimum ~ti~hich deepens as the ratio t/H }~econ~e~ larger. The
phase of the Q ftiinction di~ffers ~fi•om -1~5° only in the range o{ fiequencies
around yrI3=1. It is interestin~~ Co noCe Chat the behavior of the t~~~ro funcCions
is so similar. One difference which is important is that the anoinaly~ caused ley
the presence of the thin resistant Layer is proportional to the transverse resisCance for the K function. bul to the relative thickness of the thin bed in
the case of the Q function. Inasmuch as
W -~ cu
-z~,~t
1~3 Cm)-1 ~
m.~~
2vnFi
_ — 2e
1'or a large zn or hi,~l~ ~,~, the as}'m1~Cotic behavior can be found b~~ replacing
the hyperbolic tangent iri equatio~is 250 and '251 by 1-2e`'""~ or }~y
1-2e~--')~~ fl, as the case mag be:
'C~ jT,
— or on the unite axis ~'or y~I[ at (l — t/H.).
1 -r T~i"T,
15G
!_....
I
T. / T
____I C.14
IC
0.16
of KT~ /T~
ximum
0.18
!
DEPTH TO WAVELENGTH RATIO
I
n
appa~ciit resistivity
F~cuttF: 70.--Whin-bed anomaly in the Q Iunctiun exprc yea as xn
[/~a=(~~Qz)• ~I~h~r da~he~l part ref tht~ curve i5 ne,ati~~~~ with re°~l~ect to
the solid parr.
C~~■
I%
PERCENTAGE ANOMALY IN
APPARENT RESISTIVITY, per h/H =1%
limiting thinness. The
Frco~tF 69.— Behavior of the K function for a resistant bed of
by a bed of zero
function
K
the
contrihutio❑
to
the
i~
curve
upper
thukness Bind infinite r~e~i=tivit~. c6~u'uctr~rizcd Ly a i~~tns~e~r c ~c~-istance
The t~ln~ver~~ ~c~i~tancc of the overl~in~ bed i~ "T~. the ~nsct zit
7
fi'om
tkic lower ~inl~t shuw5 huw Elie uiaximmn poi~~t on th« curie ~lr.parts
T~.
i~~speet
to
with
lar,~cr
mode
is
T,
as
0,
its position for "I'.. =
Q3 L----1----~
0.1
O.Ci
m --
'~
~, y
QliART1:PLV OI' TH1: COLOP„1D0 SCHOOL OF ~I'~I:S
~~
In equaCion 135:
Lz —
x
n,, Ro+ ~~ Jo cmr) ~fm
n,~a
m
I2 -f ~o ~ ~ S Cmr) dm
(257)
(256)
have to be inverse tr•ansforrnecl to provide tl~e R. K and (~ functions for interpretation.
l~xain~les of the integral transforms which mush be z7~ade ire contained in
equations 13~~--136, for field components abouC a horizontal current dipole,
and in equations 1~6-158. for field coi~~ponents about a vertical ~na~netic
dipole. The particular forms of inCe~~ral transforms which must he e~raluated
in these expressions are as folloti~vs:
In equations 13-~, 1~6. 156 and 15l:
Ficisitt: 71. — Ph isc of tl~e reduced elnomalous impedance contribution, ~~-1 for the ceise
of a
thin, resi_tant be~1 imbedded in yin utl~el-~vise uniform medium in which the
real
part of thi, wa~~e number is G~.
45°
eo°
PHASE OF THE
REIXICED Qg FUNCTION
IJti
6
I -
15=
—
o
J
°
z
fi
~
}
Jo(mr)drn
n t~
Si(mr) dm
~n
Ric- ~J1(mr)dm
fin,
~,~
n` ~ --n'— Socmr) dm
~
no} R
Yen
~Q~Zno
~p
>.
x
~uZno
1
(2611
~ 260j
12 91
1258?
form.
note what hapLeC us first consider the integral I~ in equaCion 2~fi. First
F3essel function
pens if we consider spacims, r, which tend to be infinite. "The
sinusoid for
which is the kernel of tl~ie inCegral transform acts as a dainpea
dominated b~ its belarge values of its argument. "lhei-eEore, the inte~~ral is
integral can
havior at m-~0 when r-~ ~. In the limit for infinite spacin~, the
the ratiu function R
be evaluated by taking m=0~-e, defining t(~e value for
under Chew circumstances as R,,:
~Z
X262)
j2631
7'he follotvin~~ limiting
retaining only terms in m and ml a~ beinb significant.
to hold ~Tikhono~, l~)~9)
values for the integral transforms have been shown
m ~Z
m
m ~ ~. m+~
- rr~~Ro - ~ z + ...
-n~
-
transform in equation 2~6,
If we ex~~and the object function of Che intebral
assuming small values for m, eve have:'
m-so
-1 ~, r ~(~2h2~.~1 ...~s ~N -~ ~ ...)~~
'2 = ttirn l~ = c~{~,'n, + Cn-~t~
~N
°
4
I
~
' Paz
m~In
x 2~
1J~
trans[orins,
There are a variety of approaches t~> evaluating such inle,~~ral
to resource
from
s~~acir~~;
but let us first consider the case in which the
be
can
transforms
ceiver, r, is larbe. Iii Chis case, the object functions in the
closed
in
evaluated
approximated in such a ~~a~ t6at the. in~e,~rals ma~~ be
In equation 153:
In equation 13Ci:
In equation 136:
I3 =
In equation 135:
~IHLORY OF I.LlCl'IiICAf. 50U\llI\G
^
~
r3
~~ f m
2n
J ~mr)dm
~
p~ ~Y~mznITa~mr)dm = D~
r-,ao 0
Y'--s co
~~ fYmJ~Cmr)dm = p
r-,~ ~
n- o i, z ...
n = 1,2 _ ..
(267)
(266)
(265j
(264)
QU9RTERLY OF THE COLOPADO SCHOOL OF MIVP:S
~
R2 (
G
Rz
t
(268)
2tYr~
— 3.M~~ R z
(270)
~°a = P.R~
(271)
liIultiplication of the tangential electric field strength by the geometric factor
for aloop-to-wire coupling system, based on the field behavior over a homo~eneous earth, leads to:
~ ~Y-~cn
E ~`
(269j
I~valuations of the other• expressions may be found in several monobraphs by
Vanyan (1966, 1967).
It is of considerable interest to note what happens if only the .first order
term in series expansions such as these is considered to be significant. The
substitution of the first term from equation 26o in the expression for the
tangential electric field about avertical-axis magnetic dipole source (equation
161 provides the result:
~~
"'R°
n/R ~~ Jocmr)cam ~ ~~!
m ~n~}
~ m (1—
~
~ + ~o
~ z — m3Ro
~3 +.... ~ ~'oCmr) dm ~'
each of the intebrals may be evaluated in the same way. For example, I_
ma~~ he expanded for large r I or fir small m} as:
— Rz;~
2
That is, ~a-hen the zero-order Bessel kernel is multiplied by an odd power of m;
the infinite integral has zero value.
"I'he inte~;raL I~, minht he evaluated as follows:
IEO
161
(272,)
127j
~1{RCm),r~ = f WCmr)~"Fcm)~lcmr)drn
0
~'jo{ FCm),r~} =f (mr)'/z F(m).ToCmr)~Qm
(2~jj
(27~},j
For our purposes, only Hankel transforms of order zero and unity- are needed:
0
FCm) =f~ (mr)~zc~(r) 3y (mr)dm = ~v {S'Cm)~r}
The Hankel transform has the useful characteristic of beinn self-reciprocal;
that is:
c~ <r) = f ~(mr)v~ FCm~ J~,(mr~ c~m = ~y{F~m),r~'
0
The expansion of the Hankel transform integrals for large source-receiver
separations is useful in illustrating the nature of apparent resistivities measured
with the various induction methods, but in fact, spacinbs larbe enough that
this is a food approximation cannot always be used in the field. Therefore,
it is essential that the integrals listed above be evaluated in a more general
way. According to 3~rdelyi, the Hankel transform is benerally written as:
Gen-eral properties of the Hankel transforms
method.
Similar results may be found to each cif the methods, and it is obvious that
if only larbe spacings from source to receiver are considered, the apparent
resistivity curves obtained will reduce Co one of two forms: one specified by
the R„ function and the other by the R~,~^ function. If the spacinb is not as
large in terms or wave len~Chs as is required for this approximation to be
ood, it is not apparent that all the many source-receiver combinations will
behave similarly. There is considerable room for research on the relative
merits of the various source-receiver combinations for moderate source-receiver separations.
Thus if tlae spacing from source to receiver is large enough, an induction
sounding technique usin~y a vertical-axis loop source and a grounded wire to
detect the tangential electric field will provide the same curves for apparent
resistivity as a fzcnetion of frequency as tlae ~nagrzetotelluric method. The
distance, r, is normalized in terms of radian wave lenbths in the surface layer.
y~, and so the distance to which ehe ~pproYimation for larbe values of r is
valid will depend on frequency. As the frequency is lowered, the higher-order
terms in the series approximation for the integrals will become significant,
and the apparent resistivity will depart from that for the mabnetotelluric
~lHGOPY OF ELECTRICAL SOliNDING
Quar~r~xLY oF~ ~rxE Cozox~~DO Sc~zooL or MrN~s
(.276)
where
where
o
o
m
k k
k!•k~
z
kl• kt dm
m ~no+ ~~
~ _ "i/R
k-~
~z~ ` 4 ~ dm
~= n } n,
o -~-
k-b
~1•~ —r~
~
dm
o
k=o
~ ` m4r2~k
Z3 =
f~~3~
IZ =
I1
(2(9)
1278]
(27r)
Replacing the Bessel functions in equations 256-261 leads to the forms:
k=a
Jy(mr =
~~
~z~~~~ ki~ f(v+k+1)
~
k
One a~~proach to the evaluation of the transforms in equations 256-261 is
the use of series expansions which will eliminate the need for evaluatinb the
integral expressions. The case for small values of the separation, r, hzs been
treated by Meinardus (1967) as follows. The Bessel function Jv (mr) is replaced b5~ an ascending power series (Abramowitz and Stebun, 1965, p. 360)
Evaluation. by series e;epansion
A number of transform pairs of Uoth t}~pes are listed and shown braphically
in figures 72 and !3. The symmetry typical of transform pairs is readily
evident here: right is exchanbed for left and top is exchanged for bottom in
doing through a Hankel transformation, bul in addition, a rotation of the
transformed function takes place.
In determininb the I~~ehavior of the actual field quantities from the Hankel
transforms indicated in equations 256-261, computations are performed separately on the real and imaginary parts in the object function of the integrand.
Typical sets of source functions and transformed functions are shown in
fibures 7'1~-76.
162
1
1
I
f
f
j
_r
`!Y
_
'I
_ /,IOC
__r~
r
a=0.1
__r 4~
ar
~~I~O.
a=10 a=1
~
r ~D
c~(r)_ ear
D =10
I~
__
a=1
a=10
,
a =1
u-.v
m
a =0.1
m
~
i
m
~~ Fcm~ = i -e
m
i
0=0.1 ~
a=10
a=1
Q=0.1
a=iv
a -v.~ v- ~
r ~
1 ~ \~o
Y
a=10 a=1 a- J
9cY>
~
fY
~
.rmFCm)=1
am
tion as a kernel.
FicuttF: 72. — I-Iankel transform pairs with the zero-order Pessel fi.u~c-
a=l
a=0
`~
g Cr) _ e or
~
Y
yCr)
rI'HIOPY OI' T~LLCTRIC 1L SOli\DItiG
163
„-~~
~~
a =~
a=i
a=0.1
i
.a2)~
a =0,1
a=1~
a
a =0
"
l0
a =10
'r
r
a
1 — Cazt~)'~z
r
a=0.1
r
0
D
-r2/a
1 -e
- ar
a =~
cl(r)
a=1
a =0.1
9cr)
1'ir,~~iir: r3.— f[~inkEl transform pairs with tl~e first-order Ressel
function as a %crnel
a =1
o=o.~
a=0
a=iU
m
-amt/4
,rm ~'~,-.,~ - o
j
1
{f
~
,o
~_~~k
o
1
z
~4C=~
k
6
m
dm
~
~ n°+ ~~.
~ _ n,
k=o
kt kt
~ (-m;~r)k
~b ~
dm
~Qno +R.
~z
+ ~ n.
5 _ n ~~
o ~
~~o
rrn ~ ~`
9
~~x
XiZno + ~~i,
1
~ ~q ~~ k! Ckt 1)t
k_b
o
r6=
o
R~
Yn~ h, ~S,z
(283)
(282)
(221_)
(280j
~qdm
+
18f
11'l5 ~~ c1M - -• ^ ~2$'~)
These series converge quickly only if the product ylr (the ratio of separation to skin depth in the surface layer) is considerably smaller than one.
jq = zf Yl'1 (~~c~f%1 - gf Y1'l3
and one of the unity order transforms; we have:
I1 =f ~1dm - 4 f m2~ldm + ~f rn4 ~ldm - ...
I9
m
3
~ ~
165
In order that these expressions he useful, it is essential that the series converae to an approximate value for the expression within a few terms. Writing
out the series represented by one of the zero-order transforms, we have:
°°
Z d,
,~
where
and
where
where
where
rrFIEORY OF ELECTRICAL SOUNDItiG
~
0.01
ice/
/i
0.1
KERNEL
~
~ i'
,~•'
~~
~
li
,~1
~
I
%~ ,ioo
~r ~~— ~
~
~
~~
~
~t1
~~
RESISTIVITY~~ i~ -~~
/i
~
~~
100
~
~
~
~
~~
~
10
~
Q~
io
100
is
ioo
G~h~
~o°
~
'-SPACING
RESISTIVITY
1000
Ftcuizr: 75. — Transformation of a Q-function for three layers (resistivity sequence,
1; 1/~:co, and tl~ickne~s sequence 1:5) to an apparent resistivity
curve for avertical-axis coil source and tangential electric field receiver.
RELATNE
MAGNITUDE
10
o.i
000
~
--'~
/T~ =10
10
F~ct:ite 74.-- Transformation of a K fttnctiun to the resistivity function fora Schlumber,~er ~u~ray. "I~lie case is that of three layers, with tl~e middle one being
thin and resistant, and tl~e luwest one bcin~ perfectly conduciin~;.
"1'.,~"T~ is the ratio in trai~svei-sc resista~c~^s between the surface layer
,ind the middle layer.
KERNEL
m -~
0.01
0.001
0.1
b ~
I
~~
~T
`
~
~~
,~
~
ii
::,~y,
~~.
~/t✓~
~~ dig
~i~/~
~~/
i~~~iii
4~
6
`~
~
i~/~ 3.4 \`~\
~~
,y.,
~i
~S
~
~~ `
/
3.4
4.8
r/h =8
6.7
--_ —
-
/
2rrGih
----_°--_
~ ~` ~
/
Q- MAGNITUDE -~/
~ ~(`~ ~
~.
~ ~
10
~
~~ ~
~
i,./4.8~~~ ~ ~~
~
~ ~ ~
.0 ~
~
/ifX6.7 ~\
Q - PHASE'S--
i
/
100
-~0°
~~
PHASE
900
167
These asymptotic expressions are useful only in evaluating the field components at very short spacinbs or low frequencies.
As pointed out by Baranov and Kunetz (1950), Bodvarsson (1966) and
Meinardus (1967j, an alternate approach to the series representation of the
Hankel transform can he made which provides insibht into the meaninb of
the transform process. Consider that any one of the object functions ~~'1 to ~~~;,
is the unilaternal Laplace transform of some ~funcCion, q(~)
I'i~ursE 76.— Example of the transfarmation of a Q function to the apparenC resistivity curves observed with electric dipole tangential electric dipole
system. The case is that of a single uniform layer wiCli a wave number
G covering an insulatin~~ substratum.
MAGNITUDE
Tx~:or.Y or E~rcTric.~L Sou~nl~c
~1
-mz
~
~~z)e"
~jz
JD
(285)
QliARTP;I:LY OT' THI's COTORADO SCHOOI. OI' MINES
mz
JoCmr) c~z dm
(286)
0
—~? d
Cry +zz)~~a Z
(2nt1
1+ 14emho _
~
~C
p P~.
=~ Az
(288)
(290)
(239j
Thus, q(z) consists of an infinite train of impulses along the z axis. These
can be thought of as being optical images of the source reflected about the
plane of sepaz-ation between the layer.
J=I
c~ Cz) _ ~Cz-o) + Z~ C-k)~ ~ Cz± 2~h~)
The inverse Laplace transform of this function is:
J=1
~oCm~ = 1 + 2~(-k)~ g2smh,
Inasmuch as ~ke~"'''~ G 1 (except for ,p_> = 0 or ~), the object function
may be expanded in the series:
~oCm) =
The quantity (r`~-I-z-̀')'~ can be t~iou~ht of as the distance from apoint displaced up or down the z axis and a point located radially outward from the
source at a distance r. The quantity q(z) can he thoubht of as a Green's
function, or a distribution of sources of field components along the z axis.
The integral in equation 2F>7 then represents the accumulation n£ effects at
the observation point from a fictitious distribution of sources all along; the z
axis.
In the zero-frequency limit, the distribution along the z axis has the si~;nificance of images, as discussed by many authors in the development of the
theory for direct-current flow (see Reiland, 1940; Roman, 1963; Meinardus,
1967; many oChers). For example, the object function to be transformed in
the zero-frequency case for two layers is simply:
I1
If the order of interration is changed, we have the Hankel transform of an ex~
ponential. which is we11 known (~rdelyi. 19 31
Il = ~~ ~ Cam) e
0
~
~e substituCe this integral expression in the transform inte~r~l (eq. 252 for
example),with the resulC:
16ii
169
k _ n2-n.
nz+n,
+ kezn,h
,i=t
~, !+ kez^~h
f 291)
(292)
qyN)e2vm
(293}
y =o
/z
In ~~~ -~ C4vz+r'')'
Ay~K)
(2990
The coefficients in the series approxirnation are evaluaCed with simple techniques described by Onodera (19631 or Lee (1959). 7'he transformation of
the exponential series provides the result:
v=o
(~n C~ x ~
N
It has been suggested by Onodera (196 1 that the Hankel lransforn~ for
the zero-frequency cage may be evaluated by replacing the object function
with a series of orthogonal polynomials of such a form that each term in the
polynomial may he transformed easily. 'The simplest function which might be
used in this way is air exponential, in view of the Lipschitz inte~raL The object function to be transformed is approximated with a series:
Evali~,ation, using ortlao~~orial polyro~ni.crl a2~~~roxima[~io~a
However. the intensity of the iina~~es decreases t~=ith distance along the x
axis exponentially, representing the effects of attenuation, and there is a
phase shift associated with each image.
Little has been done with the evaluation of the I~ankel Cransforms us~n~;
such an iina~;e analogy, buC it may provide a useful approach in the future.
~~Z~ M1 ezG,Z S
2. ~Cz-o) + 2~
J_~ (- k)s~(z ±2~h)~
The inverse Laplace tr~lnsforms of the terms in the series also consCitute a
sequence of imane strengths:
and
where
~1 I1'-s o0 ~ ~~
-zn,h
The same approach Ynay be used in evaluating the other Hankel transforms, though the imabe approach is used much less commonly for electromagnetic theor}~ than for direct-current theory. Let us consider the same
two-layer case, anc~ evaluate the integral I,. ThF object Iunction may be written as follows, if we assume the spacing, r, is relatively large:
Tx~o~Y of EL~cTricAL Sou~vnr~vc
Qli~1IiTIsI:LY OF' 'CIIT' COLOPADO SCHOOL OF NII11'liS
m~
m~
~oC''''~~ =f J-oCmr) dm
0
(295)
mJ r~
~°~ "'s
(297)
/Zaa.Q {rlCr)~ _ ~f i2eU-P {~1Cm)}Jo cmr)dm + r — ~o~m~)
y
(2)6)
o
~
The. finite range of integration is then divided into panels, so that equations
295 and `296 may be rewritten as:
q~imo~ Zli~r)~ =f ~~x-aq {~1Cm)~ SoCmr) c~m
for the real part of the object function and as:
f
m~
where
Rein {IlCr)~ =f Raa.Q~~1Cm)-}JoCmr)dm +r — ~oCrn~)
may be written as:
Meinardus (1961 1 has discussed in detail the evaluation of the Hankel transform for the zero-frequency case. He notes that there ire tHro basic approaches
to numerical ~uadratui•e: approximaLioii of segments of the object function,
such as <<'~ to ~~',; (taking their real and ima inary parts separately) with a
col}normal which provides an integral which can he evaluated in closed form
(the major difference between this and the preceding meChod is the fact that
only se.;~ments of the object function are approximated at an}~ one tune, rather
than the full fus~ctionl ; and methods in ~vhic}~ the complete inte6ral, both
object i'unction and B~ssel kernel, are approximated.
In Che first approach, it is argued that the ranee of inte~~ration for the
CranSform inte~;r~l need not actually he taken infinite as indicaC~cl in the definin~ equations 256-261, but may he terminated at some upper limit, m,..
The reason is tli~t as m becomes lar~~e, an5~ of the object functions approach
unity as the value of their real portions, while the imaginary portions become
small. I'or the real part, the contribution to the inte~;ra1 Irom the value m~,
at which the object function becomes sensibly constant tom = ~ can be
written as the value of a Bessel integral, which has been tabulated (Chistova,
1958) or• which may be compuied using; standard function subroutines
(Abramowitz and Ste~un, 1961). Thus, in practice, the integral transforms
Evalt~atio~z zcsin~ numerical quctdratz~re
It is interesting, to note that Che coefficients Av could be considered to he imabe
sources located at points along the z axis determined by the values selected for
v in sn~tchin~ the object function is~ the Hankel transform. These images are
then an abbreviaeed approxiiriation to the infinite series of optical images, an
approach resembling that used by van Darn f 1965, 1967) in evaluating the
zero-frequency Cransforrn expression.
I7O
~
m~~
a= t
~"a
~~x~ {IlCr)~ _~f d~m.oc~~~1(m)~ 3oCmr) cam
v
(298)
171
~
i=o
~~lCm)~ ^~
''O
R
bimi
X300)
X299)
^~o+~
n
m~}~
^~o
s=~ i =t rri~
tlmo.~~TiCr)~=~~f b,m~,ToCmr) c.~m
v
j=1 i•l
(302)
(301)
as those given by Chistova (1958j. The first procedure is described in detail
by Meinardus(1967).
This approach has been used by several anchors in computing zero-i'requency transforms, with food results (see l`Iooney and Wetzel, 1957, Galbraith and others, 1964).
Meinardus (1967) reports that even better results may be obtained in making apolynomial approximation for the entire rote;rand in the transform expressions. In numerical quadrature formulas, the integral as a whole is approximated by a finiCe sum of weighted ordinates of Che function, with the
wei;hts bein; obtained from a polynomial which matches the function exactly
at a series of sample points, There are a number of techniques for ap~~r•oximatina the inCe;rand, but Meinardus had most success with a Gaussian formula, where the samples ire determined by the zeros of Le~endre polynomials.
The technique is described in detail in ilVIeinardus (1967).
A number of compilations of values for the various integral transforms
used in describing electromagnetic field components are available in ehe literature. The most extensive catalons of such values for the non-zerafrequency
cases are those by Vanyan (196r), Vanyan and others 11961), Zhogolev and
These definite intebrals may be evaluated by computational al;orithms as
given by Abramowitz and Stenun (1964), or may be looked up in tables such
and
n-,
}=~~f a;m 3o~mr)a(m
Ra~{IlCr)
v
Combining these last two expressions, we obtain the quadrature formulas
used in actual computations:
and
Over each individual panel, the object function is approximated with a
polynominal of degree n, which passes precisely through n-.-1 points over the
ran;e of that panel:
~
~
and
TFZ~ORY OF ELECTRICAL SOUNDING
QU!1RTL;PLY OF THE COLORADO SCHOOL OF ~~TINES
where j is the current density vector,
J =~ E
(303)
If adirect-current method for measuring resistivity is to he used, one does
not normally compute the current field behavior by starting with Maxwell's
equations, as has been done so far in this approach. One usually considers
only the divergence condition for current flow, and solves a differential equation based on conservation_ of current flowing from a single-pole source.
Exact solutions for the single-pole potential function are restricted to those
cases in which the boundaries between areas of different resistivities can he
represented by simple beometric surfaces. Kraev (1951) has outlined a general
solution for the single-pole potential function in the presence of a disturbing
body of arbitrary shape, and this approach has been used by Alfano (199)
to compute the apparent resistivity for• a variety of prismatic models in a
single-pole current field. Vozoff (1960) has suggested a method of approxiinations for calculating apparent resistivities usiu~ Kraeds approach.
The method outlined by these authors requires extensive computations to
obtain hood results. A less exact form of Kraev's solution ma}' be expressed in
the form of a chart, similar• to the dot-charts used in coinputin~; the field effects
in Gravity and mabnetic methods. This chart permits a rapid, if not precise,
braphieal computation of the apparent resistivity anomaly clue to an arbitrarily shaped body.
The physical statement of the problem to be solved is indicated in figure
77a. A boundary with arbitrary shape sep~i•ates a volume with resistivity, p_>,
fi•om asemi-infinite medium with resistivity, pi. A sin~~le current pole is
located at point A on the surface of the semi-infinite medium. In order to
eliminate the effect of the earth's surface on current flow, an image of the
boundary is placed in the upper halfspace, and the earth's surface is removecl. The problem is to determine the single-pole potential function, U,
at a point M,on the plane z=0.
A scalar potential function can be defined using Ohm's law and the divergence condition. Ohn1's law states that:
AN ALTERNATE APPROACH TO TI3~ ZlRO FRI3QUENCY CASE
others (1962), anti Frischknecht (1 67). There ire also numerous catalogs
of curves and values for the more special case of the Q function, as liven by
Wait (1962), Jackson, Walters and Wait (1962) and Yunbul (1961). The
greatest number of curves and values have been published for the zero-frequency or direct-current application (CGG, 1957; 1VIooneS~ and Wetzel, 1957;
Al'pin, 1966; Orellano and Mooney, 1967).
172
N
173
1.30-1.)
or
02l = —~ ~ 0'S ~ 0'll- 0(P~ ~
~'~ _ — ~ O 02t — D'U- D(~~
Equation 304 expressed in terms of the potential function, U,is:
E _ — c"t~l
(307)
(306)
(305)
The potential function, by convention, is defined as a function whose
gradient is the electric field, E
0'J =p0-E +- E• D(p~
E is the electric field vector, and
p is the resistivity of the medium, assumed to be scalar, but not necessarily
constant.
The divergence of the current density vector must he zero every place except
at the current source (point AI. The divernence is:
FicUxr. 77. ~— Definition of the problem of measuring apparent resistivity over a
body of arbitrary shape.
Pi
a
THEORY OF ELECTRICAL SOliNDING
(~UA[iTTPLY OI~ 7'lIl~: COLOR:IDO SCHOOL OI' 1`II!NF.S
~~3~0)
W ~ 4m,1
r~
~
JE'°ou•o(p) do d5
s
~~here do is the nr~nna( direction to the surface. S.
res~ ritten :
(310)
~
~,
~
where the inte~~ration r carried out veer alb space.
Inasmuch as the tei~ui p U • Q t l ~~p) ie non-7~io only on sui f ices where re,i_ti~~it} changes, the ~~olu~ne inte~~r~1 c~los~l~ ieseinbles a ~uiface integral.
Consider that the entire volume under cou~ideration 3na~ be divided into t~vo
parts: one part coilsistin~~ of thi~i shells ~~~hich c~stc~nd 1 distancE~, E. ~~~hich mad
be arbitraril}~ sm~111, ~Srom each side of each surface of discontiuuit}~ iii re~istivity, and one part which includes all the rest of space. and ~~-hich does not
include any surface of discontinuiCy in icsisCivit~~. The integral iri equation
:i0~) for this second part will be zero. 'I,he inte~~ral for the first part may be
i
'
~
~
'
~,
rz
~
,
~
~
~
~
~
~
~
r
(309)
4n
~
~/~/ _ ~ ~~~~~ ~ ~1'~ ~(~/
Equation X03 can be ~~ie~ved a~ expic~~~n~ the total potential in terms oi' a
normal putetitial L~,,, ~.;izen b~' the first ~nl~ rat_ and a cli~turbii~, potcntiiL W,
liven b~~ tl~e se~~~~nd into.rat. `I he first ~nt~.~,ral i~ the potential clue to a jingle~olc currenC source in ~ uniform halfspace with re.isti~%ity p,. "I~he second
integral can }~e thou~~ht of as repre~~entin~; the effect of acurrent-source disti-ibution pp'~s p (I~/p I located on the ~arface ~~f di~coutinui<<~ in is i~tivit~,
inasinucli as tl~e tie are the on1~ place ~~~here the quantity Q l 1 jp) ~s nonzero. The inte.~ral erpicssion fur the disturl~in~~ potential is:
where z:~ i~ the di~[ance from a 1 point in the medium to the 1>oint lI at which
the potential function is bein r;valuated.
~~~~ ~U
- ~ ~~~ ~ dV- ~~~~,u,
r
2tt
4n
ri
z
z
!
= 4tYf~to~~ ~-r ou- o ~~)1 d~
u
,.
i?qu~tion 07 has the form ~~f Poisson's equation, so that tl~e terms to the
ri~~ht of the equalit}~ si~~n mi,,~ht be identified ~ti~ith current sources or charge
accumulations in the i~~eclium. "I'he ~~~luti~~n Co Poisson's equation is ~~iven by
Poisson's inte~~ral:
11-~~
sus cx,n~~
1 ~~
do dS
~;~11 ~
ssd max, y)
yz dx dy
L(X-a) +y~+z2]
~~_
~~'~
'
~
(31'21
_
p~
I
~x
x-.X,
~ a21y~
(31'~~1
J~z \ az Z ~=Q
i 3151
normal to the <. direction, the current ~le~nsity normal to the surface is pr~por-
Consider the evaluation of this boundary condition at a plane, S~ which
separates 7•eaions with resistivities p, and p.~. If the surface Si is oriented
P, \ 2z /Zazo
where pi and Ur are the resistivit}' and the potential on one side. and p_ and
U~ are the correspondin~~ quantities on the other side of the element. For an
element oriented normal to the 4 direction. the corresponding; equation ~~ould
]~e:
~(ay,(,)
_ 1 (aZlzl
p~ ~ ~X X~xo
l(aZl,~
surface element must be continuous. Tor a surfacF element normal to the x
axis, this is:
for a surface element nor~~~al to the >. axis.
Alfano 1.195)1 has shown how the charge dei~sitti funcCion, sl~.z1 or
s(x,y), may be evaluated by applS~n~~ varic~u~ 1_~oundary conditions. One of
these conditions states the normal component of current density through a
4n
W~ = t
fur a surface element normal to the a- axis_ ~~r:
S~C~'Y)
~ ~z
V~/ _ ~
X 4t1'~~[(x-a)z -t y~-i-z'']%z ~/
"l~he discussion mad be simplified b~ con~iderin~~ elements of surface
oriented normal to the 2 of - axes and ~~tendin~~ infinite l} far in both -fand -y direcCions. This limas u~ to con~iderin~ structures elongate iii a
clirection uorinal to the transverse alon, ~tihich j~otentials are bein,~~ obser~-ed,
but ordinarily, a re~istivit~ contrast n~u~t lx ~1on,ate in at least ot~e ditnensiou
in order that the potential field will lie mod~ficd suflicientl} [hat the ~modificaCron can be oh~eraed. "fhe~ surface inte~~ral llirn has the fortis:
S -E
4n~~~cX-adz+yZ +Z2~~~z
W _ ti
This inter=i-al may be evaluatec( i~} assuming the existence of an arbitrlry
current-source distribution s l x.~~.z 1 at the surface S. Desi~,~natin~; the coordinaces of the measuring point ~'I as t~.0.01. the integral for the distui~bin,~ potential becomes:
E
Tf~I:OIiY OF I'.LL''C'CPICAL SOLV'llIVG
~,
'~
Qo~~TErLi or TF~L CoLOha~o Scxooz, or Vlin~s
Pi
~~
+~
qtT
fJ 5az
a ~m,~
~
dxdy
~
_ ~ au i ('
Pz az~~P~ 1'2 az~~ 4np~.J~~az~ms~~`~/
~ au
pi
- P. az
P2
pi
(316)
Pz aZ Pz
t
pp~ az + 4n
= PzP~ a Z(o
~ a
\
~ a ~
5 ~P.~z ~ m,~ pz az ~mz1~ dxdy ~;~1<>>
~,
5r ~ a
i
i s
4tr,~,f Lp~ az ~ m~~ - ~ ~z ~m2~~ ~xcl~y
+4tr,~,~s~p, ~ ~m,~ - ~~z~mz'1~~°~y
ppz az
o= Pz:P~ a2(o }i
(319)
The remaining integral te~•in may% be evaluated by considering two ai-ou~s of
m~ and n~~ distances; namely, the distances frarn points Pi and P_ to all
points on the same surface at which the chat•~e density is beiii~~ evaluated will
be designated as mi' and m>' while distances from Pi and P_ to points on
other char~~ed surfaces will he designated as mi" and m=". The remaining
integral tern is then broken into two inte~~rals to separate these two types of
distances:
~
When the points P~ and P~ are ver}' close to the surface, the term involving
LJ„ simplifies:
1
r (
a?~o
Pl ~"` Pi pz az~lp }" `1n',1~5 LPi ~ ~ "~,~ – pz z~ mz~~ `~xdY
z
(317)
_ I
pi
o = Pi
'— aZ'' - ~ au~
~chere ins and ins are the distances froi~i the points P~ ~ncl P_ to the charged
surfaces, respectively.
As the points P~ end P, are brou~~ht into the surface S,, these two expressions for current densiCy become equal:
and
I~~ az
~ au,I _ ~ auo~
tional to a derivative of the potential taken in that direction. The component
of current den~ity in the z direction at points P, and 1'.~ located on opposite
sides of a surface element are
1r6
177
If r ~ ~dv~
(320j
1322)
4n,~,~ az~n~~~
The quantity in front of. the brackets is the reflection coefficient, K, for the
resistivity contrast between p~ and p~. Similar expressions can reac3il}' be
obtained for surface elemenCs oriented normal to the x and ~r directions.
Equations of the type in 323 may be used to calculate the potential about a
sinble-pole currenC source: for an arbitrary arrangement of boundaries between areas with different resistivities. The current-source, density, s, on the
various surfaces is found by numerical solution of equation <~23. Once the
charges are evaluated, the potential of an~~ pout, and at the measurin~~ point,
M,in particular-, is found from Poisson's ec~uati~~n
.Pz+p. ~ az
I)ividin~ this ec{uation by 1 p~ -1-p ~) and multi~lyin ~ by p~ p.., we have:
Pz-P. auo
pz~ aZ~m~~~ dxdy +- ~pz az
az ~ ~;~ = az\ mz = az ~ m"l
o = 2ns ~P,+.~~ + 4tr~J 5 ~p~
Equation X21 becomes:
Inasmuch as points P, and 1'. are very close together, the distances ~ni"
and m_" are nearly the same:
(32l)
j ~
(
( ~
i a o
O = ztfs ~Pt + ~~ }
41t ,~L pi ~ \m ~~ ~ az\ma'l~ c~ xcQy +~ z
(In the equations f~>r U~ and U~, it is assui~~ec~ the tEii7~ rel~resentin~; the
potential due to the e,har~e on the ,urfacc near ~~~hich the observation points
are located is much larger than the terms representia~~ the nor~~~al potential and
the potential due to charges on other surfaces.) Beneath the cl~ar~~ed sheet,
the direction of the electric field vector is opposite to that abo~>e the sheet, sv
equation 19 becomes:
~.
~ ((
E~ = 2~s = 2z ~ 4tY J 1 m;~xdY~
EL = 2Trs = az•_a(
— az L4tt~
When the points P~ and P~ are close to the surface S,, the surface appears to
be an infinite charged sheet. end Che electric field outside such a sheet i~:
T~~I~or,Y of ELLCrizrcaL SoL~~~llrnc
uM - u~ + 4n
L
z
~ r dU
(32~~j
COL.OP._1ll0 SCFIOOL OI~ ~'IINI:S
~ 5
QU91iTi~PLP OF T[if
i ~2~
xK dxdy
y' zZ]ya
z ~,2fZ
~
Q~~' -1- t
z~3/Z~1"~X-
X i-
13261
Z K dzdy
2]~
)z yz
Z,•JtXZ+yz+Zz~3/z C'
+z
•~(x-a ~
t~zl~'°°
1'he inte~~ration in y need he carried out only once, inasmuch as limits of
integration are fixed. 7'he intebration in :~ or z may be carried out ~~raphically
for each problem, since the limits of integration in x or z depend on the
geometry of a particular probiem. ~Che inte,~~rals in y were evaluated for
parametric values of x and r., using the facilities of the ~Xrestern Data Processin~; Center ~t the University of Califo:.~ia in Los Angeles. Contour maps indicating the contributio~i to the disturbin~~ function, W, form each element
of surface (and its ima~~e I are shown in fi~~ure 78.
4tTz~ x, _oo
rfx~~
=
w!
The disturbing potenCial contributed by an infinitely lon.;~ prism iiorinal to
the x-= plane ivas found by inserting, the ap~ropri ite charge densities in
Poison's inte,ral.
where for convenience it has been assumed that the surface S can be resole ed
into ~ series of x-and a-oriented plane elements.
S= = 2n as
zcr ax
sX = K auo
from the current source A. 'l~he derivative, ~af ~-)/an is small when the
~„
curvature of the surface, S. is slight (no corners 1. If an5~ of these conditions
holds, equation 323 simplifies lo:
Equation X23 would be simple to evaluate iI it were ~~ossible to i~~nore the
surface integral conCained in it. This would be possible if either s or
1
a I ~„
—)
' %~n were relatively small. The charge density. s, is small when the
reflection coefficient, K is small, or when the surface, S, is a lar~~e distance
Equation 32;i is not an explicit solution for s, and numerical values must
be determined by successive approyiinations (see Vozot~, 19601 , a procedure
which is tedious even with a hi<~h-speed computer, if a high de ree oF preeision is clesirecl.
1%ii
~.~.
j
i
~`
-}-
+
_3
+
-p
+
-I
~
0
m
CONTOUR MAP OF
y az
/'~
+
x az
+
+
-~ ~x~,Y=~==~~=~~x-o,~.Y~,==~4z
+
o,\
MAP OF
+
CONTOUR
179
Ftct~itF. 78.— Contour map of the potential per unit surE~ce area and iu~it reflccti~~ii
coefficient contributed Uy y-oriented or x-oriented faces with infinite
length in tl~e c direction Iy is tl~e direction downwards. Interaction
terms between surfaces ar<~ ne~~l~cted.
~—.o~
F
_q
THEORY OF ELliCTffiGAL SOUNDING
Quai;T~r,1.Y or TFiI~, Cor,oFa~o ScxooL or 1~~Iz~vF,s
~.~J~
`Z.~)
I~
__
0.14b
0.1.46
0.078
~.~~~
0.1-(~
0.10-1~
0.0r7
~.~JCi
~).~)6~)
_
W/K
01-1-9
0.10r
0.0i9
K
__
It appears Chat the graphical computation ti=ill provide values for disturbing
~x~tential glue t~~ ail anomalous bid}~ buried at a depth comparable with or
2~ercenta,~e error increases with increa~ir ~ resistivit5r contrast.
"I~he d~eoz~~ outlined in lliis paper predicts that the disturbin~~ potential will
Jae prof~ortional to the reflection coef~'icient, ~~~hile the exact solution indicates
this is not true. 7~able 1-1~ lists the disturbing; potentials riven by the ~~raphical
~l~ethod as a function of reflection coefficient The correspondin~~ values for
disturbin.;~ potential taken from Roman's tables are also listed for comparison.
Tl~e clisturbin,~~ 2~otential determined bti ~~raphical means ranges from z
value about l~ percent too lar~~~e if the substratum is a perfect conductor to a
~~alue about ZO percent tco small if the substratum i5 a perfect insulator. Z'he
h = Q5a.
1,or coinpari~on. values :for the ratio of disturbing; potential to reflection
"K, ~cere taken from Roman's tables (19601 for reflection eo~~oef~icient_ W;
effic~iei~ts of 0.1 and -0.1. anc3 avera~~ed to determine the approximate value
of tkiis ratio for zero tc,istivity contrast. Except for the ease in ~~°hich the
electrode spacing i~ t«ice the layer thickness, the distui•bin,~ potential deteruiined kith the ~raphical method is ~~-ithin about 5 percent of the correct
~~lue. .~ relatirel~ lame error. about 20 percent< <ti~s obtai~ied For the case
0.1r6
0100
0.031
~
0.5
1.0
1.-1~
K
~I';ati~.r ]~.- Conzj~m~ison of ~raplaiccal tr~id e_Lact co~ra~~iitations for the dzsturbin~ potential clue to a single uniform Inyer of clifferea[ thicknesses
__
From Koman's tables 11960)
~V
W
W
, K=0.1 -, K= -01
Average
Graphical 1
h/a
The qualit}~ of the approximation made in neglectin~,~ the integral term
representing the interaction effect between ch~r:;ed surfaces ~n~y be deinonsCr•ated by~ applying the graphical method to a pi~oblPm ~rhose solution is well
knoFVn. Such a problem is Chat of a uniform lay ei of ogle resistivity co~~erin~~
a uniform substratum of infinite exient having a different resistivity (Roman,
1960). The ~rap}~ica1 method may be applied Co this prol,~lem u~in C~~e coutour map ehown in figure 78, inie~ratin~~ along a horizontal line at a depth
equal to the thickness of the assumed layer. Perfuri~~ing this iiite~raCion for
thicknesses ran,~iu~; From 1/'2a to 2a. the ~~esu(ts listed in table 13 are obtained.
180
1~l
0.0~>6
0.0~)r
0.103
O.Ocio~
C).0!7
0.0970
011~0
0.1300
-0.010 I
0.0107
0.021(
V.l)i)Jl
0.0150
O.O~(Z
0.0701
--0.011
t).Ul 1
i).0`~2
O.O.)2
0.O.~ >
~).0~.~
t).06.-;
-U.~Z~j
-O.O~iJ-I~
-0.0766
-0.0673
-0.05~u
-O.O 19J
--0.0 ~O1
--0.030-1
-0.09(
--0.08(i
-O.U7~
--l).U6~
~-O.OJ I
-0.0-~3
-O.032
---~).~)ZZ
Dis~urbin~ potential
(exactl
W
Di~turbin~ ~x>tential
I gr~~~hical l
~/
__
tY
s
rz
~,
(x,27)
structure, eve assume a
If, rzlther than cousideriii~~ a comE~letel~~ arl~~itrai~~~
an oeher~~~ise uniform
in
ei»bedded
i•e~i~ti~~it~~
high
model of a thi❑ layer ~~ith
1'or this model.
pi~o~pectin~,~.
petroleum
in
earth. ice obtain results of interest
of the la}~er
faces
opposite
on
sources
~~e can assume that the fictitious current
by an
replaced
be
may
zone
resi~ti4e
are equal but opposite in si.~ii. Tl~e
eve
hich
=
ti~
intensities
t~aith~
dipoles,
assemb(a~.ze of ~erticall~° oriented currenC
condiCions.
houndar~~
~zzti~~t still c3etermii~~ c>n the bads of
"the anomalous potential is
f~~
s~ eon 6
W =g
dS
lai~,'e iesisiirit~~ contrasC.
+1~itliin 20 percent for a very
percent for mod~ratc cesisti~~it~~ contr~a~t~. end
accurate within a few
sonie~~ hat ~reater th<l~~ the electr~>de spacin~~~ which is
---
0.~,
O.9
1.0
0.l
O.7
0.(
-O.1
O.l
0.2
O.)
0..~
--~).`7
--0.r
--0.6
-O.~
-0.1
-0.3
-O.~)
-0.~,
Reflection
coe(licient
h
for the clisT:~t3r.r: 11- Cona~~~ozson o% gra~liical cvul e~_~act cona~nctt~tions
a,
thickness,
of
layer
zuiifor~n
sin~~le
a
tia~bin~ ~~o~enlic~l c[ece to
ficicnts
for ~i;cn~ious reflection coef
'C~ir:orY or ELl:c~rrzc:~[. Soti~nrnc
Qu,u;TE:r,>>>~ or ~rui~: Coi.or,:~no Sc:lioo~. or lilsl:s
- p~ aZ
bard,
bound
~ au, ~ f
~ ~ ~ !a:~r„z)~ds =o i:~2~j
PZ~ aL + 4n ls S~n~ ~Z~M~~ -.~. a
F'u,i i;r: -9.
Potent iai probe
U~~fini~i~,n ~>I Ilu _~~nmrtr~~ used in evahiatina the resistivity anomal}'
e.:iu,~•rl lie a thin r~~>i~!ant layer with limited zu'eal extent.
Current source
L~•t u~ consider that the distances in are of t~~~o t}pes: 11, from the test points
to the c!u~r~ surface. designated as m'; and 2), from the test points to the ~lar
?urf~~~•~~. ale°~i,nat~~~l a~ ~1i". "l~h~n ec~uati~>n r>29 hec~mes
~P~
~
~chere m~ <i~id rr~., aie the distances from t~vo points, one above and one below
the u}~per buundar~. to any point on either surface which is contributing; a
fictitious current. In ev~luatin~~ the boundary condition at the upper boundary,
~~t• IE~C th~~~e t~~~~~ puiuts approach Chat boundary_ so that m,-stn ~:
(tiZc j
P. Lea~ + ~~ ,~5 az ~m~~ d 5~ - p2 ~ a~~ + 4tY,~s ~z ~mz~ ~ S~
~n~_ sul>~tituti~~,~ equation 309 for the potential as:
.Pi az
~chere s is the fictitiuu~ Curren[ dE.n~it~ pei~ unit of ~urtace area. dad~~. on t}ie
ir~i~tant bed_ tend ~t i~ t{ie current diF~ule mun~ent leer unit area. 'I~he fictitious
c~n~tc~nt den~it~ is deler~ninecl h~ a~>~~l~~i~~~~~ a condition re~~uiriu,~ continuit~~ of
the ~e~r~~cal c~~u~lx~nent of current ~hruu,~t~ the top cud Iwtt~~n~ surfaces of the
layer (fi,~. ~')1 . At [he top boui5dar~~_ [his cun[inuit~~ ~~~>nditiun is ~critteu as:
1~,2
az~ r,f-~~"~ -~ az~m;~~
SLp~ ~
~
ds
+`fin'
is
pz az ~mz~~ d5
ris
~
SLp~ az ~ m;~
i
az + 9rT
~:;:>t)~
~,~'
1 ~,.>
m, d5~ =o
Ez- az ~4tT ~ m~ d5~ _ - 4tTs
E2 - ~~~4Tr
t :3;-? ~
i,>:~li
4Trs
I
~
l a
~
i s
i
2z } f'~ + 4Tt 5 g ~ p, az ~m,~~ - p2 aZ ~mi~~ dS
~ :~:>:> >
~nJJo S~p,-P2~' CR2+tf)~~Rded2 = z~~~~ ~:>:~s~
°°/'s~'
_
~iIH
41Y f~~
S = ~ . P_2
z
rH
z
4n H3~~z ,- Nz+~~~
and so. the solution for Che fictii iou~ current tleusil~ i
aZ
f~,
4~(xz+ y 2+H~)~2
az
auo _
~,
~2
i'he normal ~,~otential term is
~ ;:,~~ ~
~~~~I .>.
i ;:~(~ t
where R. 8 is a circular courclivate system on the l~>~~ei~ buundai~~~ +pith i~
origin at the point iinmediatel~ belo~~ the ~~uint on the u}>pei• houndar~~ ~cl~er~°
the fictitious current c~ensity is l~ein~ eralu~lted. "I~hu
and
a
In evaluatia~ the second intE r~iL cun~i<I~~r ihzit ins„gym.,” ~ch~~n thf~ t~•~t ~~~>inCs
are infinitely close t~> the up~~er siirfa~~~°. 'I~h~~~i:
pj~
i°zPr 8Z~
These conditions allow u~ lu evaluate t}~e fiat irte~~~ral in e~~~uati~~n .3>U:
and
Inismuc}i a~ tl~e tr°~t ~~uint~ arer in[initel~ c I„~r~ t~~ th~~ up~>cr l~uundar~~. anv
continu<~u~ c{i~trihution ~~f futitiuu: c~urrcnt ~lr~n~~tir~ kill t~~~pcar a~ thuu~~~h
they ~~ere uniform and in~uite in e~ten~. 'Ihc ele~~tiic field frui>> such an infinite ~hect of ~ uerent ~lensit~ kill I~c~ I~:i -O al,~,~e~ tl~t~ Ix~un~lar~ anal 1.., ~- Ir;s
belut~~ the b~~u>>dar~ I l~et~ceen tl~e hr~~ ~6c~r~t.~1 . 'I~I~u
p,~Z,
I'z~~ a2lo
"Cii~~:orl~ ~~r E~.rc~riuc_v_ So~~~i>Ixc
QIIAtiT~1iLY OI' TFIE COLOIi9D0 SCFIOOL OF !VIINF.,S
c~XaQy
(~~39)
~l'pin. L. 1I., Ii~~rdichev~6i~. At. A., ~'ccirintse~°, C. A. and Gagarmi~tr, ~. 1~I., 1966,
Dipole uu~thnds f~,r m~~xstn~in~ rarih ~~nn~lucti~~it}~: A7e~+~ York, Consultants Bureau,
:302 ~ ~.
1lfanu, L.. 19i>9, Introduction to the interpretation of i~'is[ivit5~ measurements for
complicated structural conditions: C< i~h~s. Yiospectin_, c. 7, p. 311-366.
~bramowitr, Aiilton and Ste~un. Irene t1. (~dsJ, J96~~. HandLuok of niatl~ematical
fti~<,tiuns: A'xtl. lour. Standeird~, ~1~~ plied ~'Iaihcmatics series no. ~~, W i~1~in;~ton,
D. C.. II.S. Govt. Printin, 01~rce.
P1Ll;ERI:\CES
If the area dxdy oven ti~hich the inte~~ration is performed is allowed to become infinite. thie expression should reduce to that cleaved earlier For a layered
medium. I'or a resistant zone of limited dimensions, the integration process
may be thou~~ht of as detezminin~; the portion of the total anomaly for a
bed of infinite extent which is applicable for a bed of finite extent. Tl~ien, the
results obtained in inte~raein~~~ equaC~on 339 ma}' be used to form apoint-bypoint correction factor for convertin~~ the anomalt for a bed of infinite extent
into the anoi~ialy for a bed wiih specific finite dimensions. These correction
cu1-ves will depend on the size and shape of the resistant zone. and on the
array used in measuring resistivity, as well as on Che position of Che arra}r
relative to the resistant zone. As an example, the correction curve for a
case in ~r°hich the electrode array i~ centered svmme~rically over an nil zone,
with a circular shape, and ~ubtenc3in~ an ankle o~f one steradian is shown in
fi~ui~e o0. I'or shoz~t sp~cin~;s_ the correction is quite lar~~e, but there is almost
no anomaly to which the correction is to be applied. As the apacing is increasecl. the correctiorf passes throu~.;h a miP~imuin at ~pacin~~s shorter than that
required to see the maximum anomaly for beds of infinite extent, ~ncl then
increases a,ain at large spacings.
This expression holds For oil fields of quite arbitrary shape, and in the
general case. it must be evaluated numerically'. It shows the same properties
as the earlier, less ~~eneral solution for a resisti~-e bed o~l infinite extent: the
anon~al~~
in resistivity c2used by the oil zone is proportional to the ratio
~, ~,
~
H5
+1~C\H}2}CTS z+l 3
CC N ~z+~H~z
Tz
~rliere an approximation has been made on the basis of the ratio p_/pi
being; very lar~~e. Substituting; this expression in Che integral solution for
Poisson's equation, we have for Che anomalous potential•
1~~~
0.I
I
I0
l00
SPACING TO DEPTH RATIO
Anoma/y fob
finite extent
Anomaly for
finite extent
Correction factor for
~ finite extent
1£>>
(~ompa:;nic (;enrrale de Ceuphysiyu~~ ICGG~, 1)ii3, Ataster curre> for clech'ica]
sotmdin!t. ~~~~~und ~•diliuii: fur~,~x~an -1~~~~~ . }~;a~~loraPiim C~•uplp~cici<ts.
Cliistu~ i E. ~., 1958 [ il~les r~~ f c.~~~ l functiun~ ..I i ~ ail cn uuu~nt; and their in-tr~i~tl 11u•cuw. U..~d. Acaul. S~Sh. ~2$ p.
C~i~niard. Louis, 19 3, liu~ic t(i~~nr~- nl t(ic n~~i~neto-telluric uicthnd of _~e~~~pliyciceil
~iruspectin
Ci•u~~hy.:ic~. ~~. 1~, nn. 3, ~~. 60~-635.
Iiiiilvcu~rsi,n. (;.. 1966. I)irr~~~ inc~h~~~l. in ~i~~~~lird _~~~~~ili~~:i~~;: Gcu~~zpluratiun, v. 4,
no. 3. ~~. 113-13£3.
Iicrdiclievsl:i~. 11. \' ~., 19G:i, Electrical ~~ru<~pi~ctiu~ »~it[i the telluric cw'rc~~t nicthocL•
(~nluradu tichuul ~Iincs Outu~t., 4~. 60. no. 1. 216 p.
Rarunov. Vladimir. anc( Kun[ci (;c~za. (9:i8. Di~lrihutiun du ~w~cntirl clans tan milieu
stratifies: ..cad. Sci. Paris. (:~~mp~c> Rcndus, v. 2=67, nu. 23, p. 21 0-2171.
F~~;i;itF: 80.-- Lxaiupl~ u[ the ~orrectiun tivliich iiu=t lie muck fui the limited extent of
~i thin, is ist.mt Iiycr. cum~~utcci for a ~chluuibrr~,ei tnraiy ren[er~~d u~-cr
a circal~ii rc~istant rnn~~ which ~ubt~~nd~ iu~ an~lc~ of 1 ~teradian tit the
midpoint .,I ~hi~ arra~~.
•••
••
I%
PERCENTAGE f~ESISTIVITY
ANOMALY per T2/T~ = I%
'I'xcorY or ELLC~r1,zc:aL 5ou~ni~c,
QU~RTPPLY OF 'CHE COLOP,ADO SCHOOL OI' VIIN~S
$wide, ]~. ll., 1949, Lzu~tli conduction e~(lects in transiiiission s)~tems: New York,
D. Van Nostranci Company, 373 p.
Stcfanescu, S., Schlumbu~er, C., and Scfilunibe~;er, ~bT., 1930, Sur la distribution
elec[riyue putentiellr autour dune prise de tore ponctuelle clans u❑ terrain a
couches horizontules lionio;;enes: Juu~. 1'f~ysiyue et Radium, sec. 7, v. 1, p. 1321~1~0.
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--- 1963, Tl~e kernel function in the surface ~~otcntiat for a horizontally stratified
eai-t6: Geophysics. ~~. 28, no. 2, P. 232-2<1~9.
Roman. Irwin, 1910, _lpptii'ent rc_i~tivity of a single tmiform overbw~den: U.S. Ceol.
Sw~~ey Pr~,~f. Paper 365, Washin ton, D. L., I.S. Copt. Printing Office.
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1963, Numerical anal~~sis of relative resistivity for tt horizonttilly layered
earth: Geopfrysics, v. 28, no. 2, p. 223-231.
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layered ev~tl~: ~~Ii~~neapolis, Dniv. ~~Iin~esota YrESS.
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horizontally stratified media: Geophys. Prospecting, v. 3, no. 3, p. 268294• .
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earth, and application to ~'roand and airborne electroma7netie surveys Colorado
School Dunes Quart., v. 62, no. 1.
Galbraith, 7. N.> Simpson, J. !11., rind Cantwell, "I'., 1964, Computer• a~~plications in
geophysical ~nodelin~: Colorado School Nlines Quart., v. 59, no. 4, p. 67-80.
Reiland, C. A., 194Q Gr.,~pli}~sical exploration: New York, Prentice-Hall.
7ackso~, C. M., Waii, 7. R., and Walters, L. C., 1962, Numerical results for the surface impedance for a stratified conductor: L?.S. Nat. Bur. Standards, Tech. Note
143, Wushin~ton, U. C., Dept. of Commerce, Oi~ice "Tech. Services.
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ara~uments and their i~te~rals: ~1'To~cow, Al.ad, Nattk SSSK, 328 p.
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ll.Sc 1 hcsis, Colorado School 4lines.
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Erdelyi, t1. (Ed.l, 1953, Fli~her transcendental functions, v. 2: ~'ew York, D'IcGraw-
Ic~~'~
IH7
7,honolev, L. A., "Crif~~nov, iV. P., and Sliakhsuvxrov, I). N., 1962, Computation of the
electroma~~netic field in a layered medium (in Russian) in Numerical methods
Moscow State tiniv., Pub., p. 203-231. (English U'ans., CSi'VI
and progran~iuin
RL-4~, A u ~. 10, 1967).
Geophysics, v. `L6, no. 4~, P. '~6$-4.?3.
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"Pikhonov, A, N., aid S6ak6s~ivarov, J-). N., 1956, ?VIethod of ca]culating the eleeu~om~ynetic field generated by varying currents in a layered medium (in Russiaiil :Irv. Akad. Nauk SSSK, ser. ~~~eofiz., no. 3.
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used in Leo-electrical prospectin;: Geopf~ys. Prospecting, ~~. 13, no. 1, p. 37-65.
1967, Mathematical denotation of standard-graphs for rc~isti~ity prospecting
in vi<•w of their- calculation by means of a di~i[al computer: Geopl~ys. Prospecting,
v. 15, no. 1, p. 57-70.
Van Nosirand, R. G., and Cook, K. L., 1966, Interpretation of resistivity data: U. S.
(col. Si~rvPy Pruf. Paper 499 Wash~n rton, D. C., U.5 Govt. Printing Office, 310 p.
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- 1967, Electromagnetic depth soundings: Drew York, Consu]tanCS Bureau, 312 p.
Vanyan, L. L., Shi~~ul'skaya, "C. A., and Omet'cheko, 0. k., 19fi4~, Tables for co~nptttin~
theoretical curves for frequency sounding in the far zone: Geolo~ii i (;eofiziki, no.
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THEORY OI' ELF.CTRICAI. SOUNDING
189
Three basic arrays are considered: the Wenner, the Schlutnber~;er, and the
dipole array, f fi 53). In comparing Che various arrays, we must first
establish comparable conditions for each. "I'he lowest signal level which can be
detected depends on recordinb equipment used. In practice, the lowest signal
level which can be detected may be limited either by the sensitivity of the recording equipment or by the level of telluric noise between the measurinb
electrodes. Let us designate the threshold ~~oltage which can he measured in
the absence of noise as V,,. Similarly, let us designate the avera~,~e telluric
noise delivered from the measuring electrodes as V-r• The telluric noise level
varies widely with time, location and local resistivity, but these facCors wi11 be
common for all electrode arrays. However, telluric noise will also be proportional to the separaCion between zneasurin~ electrodes, so leC us clesi~;nate a
}primary telluric field strength,E-~,, such that:
A great many techniques have been used in measuring earth resistivity,
variations being primarily in the way the various electrodes are placed relative to one another. Since high power, and accordingly, heavy equipment, is
required in makinb deep resistivity surveys, both efficiency and safety are
factors to he considered more seriously than in normal shallow penetration
surveys. The first step, then, in planning a deep penetration survey usinb
zero-frequency i7ieChods is an analysis of the advantages and disadvantages of
tl~e vat-ious electrode arrays.
~.'O]~IPr1RIS01~ OIF 1~RRAYS USL:D IN ~IPECT-CUFRLNT ~~~SISTIVITY sURV~YS
So far we have considered the electrical characteristics of oil fields and the
rocks in which they occur, alonb with a generalized theor}~ for alternating current flow in the ground. The questions still remain: I3ow might one use electrical probing methods in search for oil? Ghat technique or techniques will
provide the best chances for success? What spacing is required between source
and receiver? ~X~hat frequencies must be used? Hoiv much potiver is needed
to achieve these spacings and frequencies, and sti11 have si~~nals at the receiver
which may be recognized above Che ambient electrical noise? What are the
chances o~ recognizing, the response one anticipates $•om an oil field in the
presence of many other responses to extraneous resistivity contrasts, or
"~;eolo,~ic noise"? We are now in a position to answer some of these questions, but in truth, complete answers to all of therl~ cannot be obtained without
more experience in electrical prospecting; in potentially petroliferous regions.
PART 3 — DESIGN CRITERIA P'C1R ELECTRICAL SURVEYING
VT = Mrt Er
~3~0}
QL~<>>~T~:r.~.Y or •rr~E~: Co~.ora~o Sc1-Loo[, ot~ l~Ti~~Ls
~.;_~l ~
(3=12
z'n,Q~E
= 6clp
1 '~;~'I:i
W~ _ ~ LA
(315)
to ~~E~erate. Thf weight, ~~,., ~~f the cable is proportional to its length, L, and
cress sectional area, A
The amount of current supplied t~ the ~~~round, I, can be considered to be
limited l~ti~ two factors: the weight of cable~W~., and batteries, W~, which one
~~ i~hes to transport_ and the maximum safe volta~ e, V~~ at which one chooses
bcI
P
1~. i~,~~r the Ex~lar di~~~~1e array. the measured ~rolta,~e is:
~~~l~ere b and c are the diFx,le lend>th~. and tt i~r, is the distance between dipole
centers.
~ ~E
V
`3. T~or the equat~n-ia) di~u>le arra~~. the. measured volta~~e is:
~~~6ere b is the spacinz; het~~~een inea~ur~n~ electrodes and cr, is half the spacing
l~et~~~een the current electrodes.
iT"4s
VS 5 = b~
~. F'or the Schluml~e.r~er ~~r reciF~roca( Sch(uml>er,,~er array, the n~ieasured
~~~~Itae i~:
~~ {iere I is the current supplied to the ,.;round. p is the rE~istieitti~ of tl~e
,round, and a,,. i~ the s~~acin~~ bet~~'een an~~ ts~o adjacent electrodes.
W
VS W = ~
1. F'~~r the ~~/e~iner <irra~~. the rneasurecl ~ol[a~e i~:
"~~he: ~- oltare ~~I~ich ~~e wish to compare tcith Chew Iimi[in,~ values is the
~i.;~na~ ~olta~~~e, V,. developed betwee❑ tl~e ins ~sui-in~; electrodes by the current supE>liecl ~~~ith the current electrode. `Phis ~ olta~;e may be computed for
an~~ of the arras for a uniform earth.
~~hc~re \'I\ is the ~pacin~ bet~~~een the ri~easurin~~~ electrodes. Berdiclievskiy
l~)6~) has shown That for a uniform earth ~~~ith a rfsi~tiviC}~> p, the electric
field i~ proportio~ial to the square root of iesi~ti~-itc_ E-i~ = k•i,p''.
190
I
t
1
t
a
i
4.j
191
~3~~.67
(347)
Wg
kiVg
(3=18)
Va
~~Lz ~ kiVa + R~
W~
Wg
(3 ~~`~
Wg = Ct —a)W
W~=aW
VB
A~b~z+ kiVa ~ R
~
<i-a)W
aW
(301
A maximum current can he provided to the around when the denominator
in equati~m 350 is a minimum. "fhe extreme vall~es ~f~r the denominator as
I
we have from equation 3-1-9:
~t,~3
total weinhC:
Generally, we will be concerned about Che total weight limitation,
W = W~. ~- W~. F..xpressin~~ the cable and battery weights as fractions of the
I_
where k~ is .02~ to 0.75 ohm-~ouncls per volt.
Using equations 31~, 316 anc~ X48 to re~no~~e some o~ the inter-related
parameters from Pquati~n 3=1-7. ~~~e have:
Ra
The internal resistance of a lead storage battery, which is probably the most
economical type batCery to use in deep resistivity surveys, is approximately inversely proportional to the weight of haCteries; and directly proporCional to
the hattf~~ry voltage:
UB
Z = ~c+Rg} R$
where p,. is the resistivity~of the material from which the cable is constructed.
The current which can be supplied to the ground is limited by the combined internal resistance of the batteries, R73, the groundinn resistance at the
current electrodes, R.... and the cable resistance:
Rte= f~~L/A
where S is the density of the material from which the cable i~ constructed. On
the other hand, the cable resistance, R,., is proportional to the cross-sectional
area:
Dcszc~ C~iTEFia roe ELrcTRICAL Snxv~Yinc
a~x-
'~~'
~~~,, - 1
k~~s
-1 ± C~~~~
kv
~~ b Lz
(3521
ZnVT
~ —
p Yz
so:
~0
~ aW
(i-a)
Ua
AYz
l
a~n
~i-a)W
~~
as - ~4~ba~2~ k,~g +R ~ rrkT
Z
S
but M~~ = b and L = 2a
a 2 T b?p _
bpi2
2Tr ~kT
~h
,c~ ~,2, }<~~
k,vg +R~~~ 2n kT
C~~-
vg
vg
/~~V
is +
~ aW ~-,~jw
2. F'or the Schlumher~;er array
~'w
z
but MN = a,~. and L = 3a,,., so
Q"'
1. Fos• the We~nnei• array:
(3~-1)
r 3~3i
We are now in a position to compare the maximum spacings which nay
be obtained with different electrode, arra}s under comparable conditions. `I~his
is done by solving equations X41 through ~-~-1- for the spacing, a, a=ii~~ the
minimum detectable voltage venerated by telluric noise. V-~,, and C1~e maximum
possible current for maximize~~ values of p,., 8, V~~, any] W a~mm~~n t~, all
arrays.
which are:
kiVB
i s>S 11
Qv~1.Tr~LY or ~riir•: CoLOi.allo SciiooL or lIt~~rs
the parameter a is varied are given by thE~ solutions to the equ<ltion:
192
~
;
i
zfT"~/.r
Va
C~-«)W
~
Vs
~ olW
(~-a)W
(p~~~2+ k. a + ~J~
~
t1'kT
cp'/z
~R~\ 2nkT
~~'/z
(356)
(355)
19~
2•
C~-«)W
ocW
~ U-a)W
~~b a z k,VQ
aW
+ R~
9p~~uW+ k~v~ }
(357)
3
k
aW
k3 cS aDE
-~
k~Va
C~-a~1
~
'f R
~w + k,V~ + R
`3
2W z <l-a)W
'/z
(358}
~`"'
3 k3,~a~ ~ k,Va ~ R
d
(~-a)W
aW
(3~9
Similarl}_ the mat~mum spacing attai~~able with a polar dipole :~~ -ay, in proportion to the maximum spacing; attainable with a Wenner array
~w
aoe
The ratio of maximum spacing attainable with an equatorial dipole array to
the maximum spacing attainable with a Wenner array is then:
The dipole arrays may also be compared readily with the Wenner array,
providing; we express the source dipole length as a fixed fraction of the
maximum attainable sp~cinb:
c = kia
Qw =
as
In comparinb arrays, we might consider the ratios of maximum spacings
attainable. For example, the ratio of masinmm attainable Schlumber~er
spacing to the maximum attainable ~X~enner spacing ran he obtained from the
equation
app
3
4. ~'or the polar dipole array:
Vg
aDE-~~.aW~2+k,vs
1-a)
s
~
b
(/c Wiz+ 4C~u6 .}R) :ZA'h~l ~tT
but MN — U and we may write that:
a~E
a _ bclp _
3. For the equatorial dipole array
~~SIGN CRITlPIA FOP ~LFCTPIC9L SLItVEYING
Qun~TZr~LY of Txr CoLOxa~o Scxoor_ oz yIz:~~~.s
~~x
-1 ± qo~L
160 _ 1
L~
(~60)
~L
(361)
+ 3a,~,
9
(~\
z
+ cis
20
\k~ug~(l- zo~as
~,z
t/z
(362)
2. For the equatorial ~lip~le az•r1y i❑ comparieon with the Werner array,
eve have:
If the spacing a is sufficiently large (much lamer than -1.0 feet), the second
term in both the numerator aild denominitor is ne~~li~;ible in comparison tivith
the first Ce~m:
t
aw _ ~ 2~/4
=
1.9~
(363)
W
1'~ b
This equation implies that best penetration is obtained through the use of a
lieavy~ cable. rather than through the use of hinh-capacityr batteries.
,,..
Substituting; numerical ~~aluPS for the fixed parametei~~ in equations » ~,
3~8 and 3~9, ice have:
L For the Schlumberg~er arra~~ in comparison ~~~ith the Wenner array, and
assumin~~~ that the maxirnu~i~ attai~~able s~~~acing iii either case is much greater
than -10 feel ti4e have
G~eX ^ 1 .~
In deep electrical surveys, L i+'ill be much larger thin -10 feet, so this expression can be approximated as:
The extreme values for a are:
p~s
1.600 squire feet (used in equation 3521
= S~0 pounds per cubic foot (copper)
= h x 10-'ohm-feet
$
~
k~V~~ _
= 500 volts
= 0.0001. ohm-pounds per volt
V~
ki
At this point, it is desirable to assign numerical values to some of the fixed
parameters in these equations:
194
D
~~
I
t
l k3~
(~6~~)
195
36:i)
H
~ ~~f>)
i367i
~m
~36H)
For example, if we wish to detect the presence of a resistivity contrast beneath
a series of lavers in which the maximum rP~sistivity contrast is 1000:1, under
~"""`~' 2~Pm~~
~
where pi and .p~, are the maximum and minimum resistivities which occur in
the section above the contrast which is the Car~:eC horizon in soundin~~~. If the
range in resistivities is very large, this is approximately:
~"~X- zCp,~ }'/~
pi}pz
where p is the resistivity in the layered sequence as a function of def~th, z,
and II are the total c3ep[h to the basement.
The coefficient has a maximum value when the layered medium is divided
equally between two specific resistivities, and if the resistivity contrast is
lar~~e, the maximum value is
H
~ _ fpd~' J 1~dz
k
In equations >6~1• and ~i65, Ic:; must }>e much smaller t}iari unity i~~ order that
the equations for dipole resistivity lie correct.
The maximum spirin~; attainable for a riven array is not the only factor to
be considered in coinparin~; the utilitr~ of arra}'s for deep soundings. The
effective probing depth—tl~e depth at which a boundary may be detected with
a given array spacing;—must also be considered. The probing depth in resistivity suiveyin~ cannot be specified. As an example, consider one case of a
sequence of layers with alternaCin~ high and lo~v resistivities resting upon
an insulating basement in comparison and a second case of a uniform medium
of the same total thickness coverin, the basement rock. "I~he spacing between
electrodes required t~~ detect the E~resence of basement under the layered
sequence is greater by the factor ~, than the spacing required to detect basement under the rock with uniform resistivity. The factor A is the coefficient of
pseudoanis~~trr~E~y f~~r the layered sequence, ~~iven by
aD,,v - l ~ka J 1/~
v. F'or the polat~ dipole ai rav in comparison with the Wenner array, we
have:
aw
QpE _ (C~ \~~'4
llLSicN C~iT•i;rzra ror, Lr.rcTfuc:ar. St1avFYZ~vc
QUARTERLY OF THE COLORADO SCHOOL OI' R~IIN~S
2trrh
(370)
r
Ps =S
~~72)
where p is the resistivity of the layer and S is the longitudinal conductance,
h/p.
The apparent resistivity measured with the Schlumberber array is defined in
terms of the electric field intensity. With two current electrodes, as are
actually used with the Schlumberber array, the observed electric field intensity will be twice the value ~~~iven in equation ~~0 for a si~~~le-pole source. The
apparent resistivity is:
ps = '~'rzI
(371)
E =jf~ = 2rh — 2tYrS
where I is the total current, r is the distance from the source at which current density, j, is being calculated, and la is the Chickness of the layer. The
electric field intensity may be determined by applying Ohm's law:
~
(369)
the worst possible conditions. it may he necessary to use an array spacing
about 16 times larber than Chat which would be required if the same target
horizon were covered by a uniform medium.
It is possible to define a depth of probing; for a liven electrode array, providing the definition is ~~iven in terms of a specific set of conditions. Such a
definition is useful in coinparin~ array, but may not he used to esCimate the
actual spacing required iii a field survey unless tl~e correction for anisotropy
given in equation ~6l or 368 is applied. Probing depth (or' array spacing
factor) may be defined for measurements made at the surface of a single uniform layer with finiCe resistivity restinb on asemi-infinite insulating medium.
Consider the apparent resistivity which is observed with a Schlumberger
array under such circumstances. For array spacings considerably less than the
tklickness of the layer, the observed resistiviC~~ is nearly- equal to the actual
resisCivity of the layer, while for spacings lar,er than se~~eral times the layer
thickness, the observed resistivity is directly proportional to the array spacin~,~.
This proportionality may be shown as follows. Consider a point source of
current at the surface of the layer. 1V~o c~uri•e~it can leave the layer through
either the upper or lower planes, and so the current must spread uniformly
radially outward in two dimensions from the current source. The total current into the layer at the source can be used to calculate the current density
at a distance from the source which is large compared to the layer thickness:
I~(
1J
0.5
IO '
I
Current dipole length /Dipole separation (c /r,)
10
Ftcoits 81. -- Effective >liacin~ tactur as a functio>> of dipole len~t6s (I~ anel c ~ and
separation r~ for x p~ilar di~iole array (from Keller, 1966; rc}~riut~~d
by permission from the Society of ~sploralion Gcophysici~t i.
W
a
a
0
5
~i
In con~iderin~; the effective array spacing for such an electrode arra~~~~ement, c+~e assume the spacin~~ betF~-een dipoles is large compared to the la~~er
Of particultii• interest i~ t6~ polar-dipole array in ~~hich the dipole 1en~~tl~s
are not ~inall compered to di~~ole separation. The parameters used i❑ clefining~
such an arra~~ are shown in fi ure 81.
~Ve ma~~ define iht effective probin, df.~pth for the Schlwnber~;Er arra~~ ~s
r, t}~e halfspacin~; bet~ceen current electrodes. "This is the spacin_~ at which
the horizontal as~~ml~tute fui re~isti~~itie~ ~~~ea~ured with small spacings inCer~ecCs a ling rf~pre~entin, tt~e linear relation ~ iven in equation ~r2. "Che
existence of the i~~sulatin~, basement is little appar~~rnt for measurements made
~~ ith ~l~ ~cin~s r ~~ualler than the la~~er thickness_ /i, while the ineulatin~~ basement exerts a pronounced effect on mea~ureme~~ts inac3e kith large spacings.
Tl~e ef~ecti~~e probin~~ depths fur other arra~~s ma~~ be defined ei~nilarl~.
flOt78~I~F' I~ 1/S.
The observed iesisti~-it~~ i~ ~~rupurtiunal to s~~acin~~ and the constant uF prul~or-
D~;slc~~ C~,tTr:rt:~ roe; 1~:~.r:~:•rrtc:u. St~r.v~:~~inc
QUAIiTliPLY OP' THG ~OLORADO SCHOOL OI' 1'IINES
r
2srr S `~
r,,b 1
Edr
1 ~(r~+ b\
215 ~ r~ )
Y~
r
X373)
X37-1~I
r,
r~rb
~l+b
1
r+c
r+c
r~tb +c
r~ t~+c
~21a/
rr~+bl ( r~+c
(0'flA +
,~~
t, ~~ l~)
5
r,
tc
r
— c-~}b + r~+b~C
~,_ i~CC~~r,b)~Cr~
+~~~
~
i
t
'f~he et~ective s~~aci~~ lactor in eyuatiun 376 is then:
p~
(~77)
1376j
I❑ accordance with equation 372, the et~ective spacing factor, a', is defined
such thaC
Q'
t
r~
2tY
"The apparent resistivih' computed for the arr~n~~ement of electrodes shown in
figure ~l is:
~
( r,+ c
2n5 ~" \r,+b+~
r+c
L~21$ =f Edr
r,+b+c
~fhe difference ii1 potential observed bet~~een ~~oints il~I and N for a current
—I into the lad er at point B is:
_
~ 2tA —
— ~~+b
thickness. "Che difference in potential observed bets~een points NI and 1V for
a current I info the la}'er at point A is:
190
I
},
1
~,
199
i
1{b
~}r
~
~
I— ~}~ — ~ +b +i+e +~
e»
(ilia)
13~>1
A series of curves showing the ef~'PCtive sf~acing factor for various values of
b/c and c/r~ are given in fibure c>2. These curves provide reasonable spacin~~
facCors for- those cases in which the dipole separation, r, is larger than the source
dipole length, c. However, if the spacinb, r, is small compared to the dipole
length, c, the probinb depth given by equation 378 is larger thin we intuitively
expect. This is a consequence of the way the definition was set up, inasmuch as
it was required that the electrode separations be larbe in c~~>n~parison with the
thickness o~f the surfa;;e layer. The probinn depths given by equation 378 are
valid if the substratum is truly an insulator, but not when the resistivity of
the substratum is finite. When the resistivity of the substratum is finite and the
spacing, r, is much less than the dipole ten th, c, the array behaves a~~proximately as athree-terminal array with the contribution fi•om the distant current
electrode being insignificant. The potential OUR in equation 3r5 is arbitrarily
set equal to zero, with the result:
Unfortunately, we cannot examine the behavior of equation 3~~ for sma11
values of r, (r~ —~0) because in the model, ri was required to remain brae
compared to the layer thickness.
This is the spacing factor for the polar dipole array Given by A1'pin, (1966]
i
2'
t
This equation must apply for an ideal dipole array in which dipole lengths
are short. as we11 as for the nou-ideal dipole array shown in figure 81. In an
ideal dipole array, Che spaci~~g r is large in comparisan ~~-ith the ~pacin~;s b
and c, so that both the logarithm in the numerator and the fractions in the
denominator may he repla~~e~l hj. sh~~rt prnv<~r series:
r
~
For con~~e~~ience, this etjuatiun may he exf~ressed in terms of dimensionless
ratios:
Dr•.sic~~ Cr~rri;:~~ia ~~oi; I~:~_~~crric:~r. St~rv~:Yi~~c
~
~
~C~
.(
~~8 y
c/z
p
~/z
~
~
t0~
~
~
~
~
~~
~
~ ~
P
~0/ ~ ~
/
~
~
i
~
~/'~
' ~ ~
~ ~~~
~ ~~ ~ ~
QC',U~TI:RLy pF TE11: ~,OI.OiiADO SCISOOI. OP ~~jItiF:S
~
S
~
~
r, ~ r~ }b
~
~ A
~
r.
~Y,+ b~
(380)
harp:
~ _ _ . ~.
R.
~.
_
I)efining~ the effective spaciii~; factor in accordance with equation 376;~ -we
~a ~'
2 t1'
~ ~ +
r
`r, o b
~~t I~:Y~~I~,ra~ inn Gc~~pin~<i~~ist~~.
~1;~,~~1~~ ~tr~~m K~~il~~r. ]~~G(i: ~r~~riutcd b~~ pr~~~u;~sion Ir~m thf Society
c~•i~in,~ di~~ul~~ and tf~c radio. sec;or fru~n tlu, midEioint of [kic source
~h~ uzin,utl~ an_~I~~ I~~~t~~~~cn th~~ X115 dipole atis au~i the radius sector to
th~~ ~~~~int I'. ,u,d ~ i; tl~c 6~~a~inn an~~lc bet~~crn the direction of tl~e re-
F'ieiiir: 82. _Ui~finiti~~n of E,.n~imet~~i~ u~~~~i in di«u~<in= a ~u~eral dipole u~r~iy. 1
:tnd I, .u~c cw'rcrt cl~~i~u'ud~~>, f' i~ tlic ccnlct uF t ~ncasurin~ dipole, B is
B~
ZOO
-_
~
~
~
r
{
;
~
,~
Itr
~
.en (1~- ~
r~
r,+b
(Sc,2
181)
~~l
z
1383)
A
2~-rAH
E _ ~JJ = ~I
(~84~)
The magnitude of the electric field at the r•eceivinn dipole due to a current
I driven into Che ground at point A is:
to be located on the surface of a layer with resistivity, p~, with a thickness
much sr~iailer than tl~e di~pole se~~aration, r,., and resting on an insulating substratum. The source dipole is considered to have an appreciable leu~;th, c,
while Che i~~easurin~; dipole is considered to he so short that it can be heated
as an idealized dipole. "Che line from the midpoint of the source di~~ole to the
measuring point makes an angle B with the polar axis of the source dipole,
while the receiv~n~ dipole axis forms an an~~le /3 with the line connecting the
dipole centers (fib,. ice).
use of a nonlinear array. "Che effective probing; depth of such an array may be
defined in the same way as that for a colinear array. ~l'he array is considered
The probing depth curves in figure n2 should approach a limit of ]..0 as r~
becomes small, rather than become large without limit. Curves which behave
properly for afour-electrode array at large separations, and properly for a
three-electrode array- at small separations are shown as dashed lines on figure
82. The curves in figure 82 are valid when all electrodes lie along a common
line. In practice, this requirement is not usually rnet for the lar~~er separations.
With spacins;s
over a few miles, to~~o~ra]~h Y and road j>atterns ma}' dictate the
~
=1
b
Normall5r, the ratio b/r~ is quite small so that both the numerator and denominator in equation 382 may be expanded in series:
~i _
oz', using dimensionless ratios:
+,
~~SIGN CRITERIA FOR EL},CTFIC9L SliRVIiYIA'G
•
60
Azimuth angle (A), in degrees
30
Para~lel array
Maximum array
/~—Radiol
/
—array
Equatorial
array,
_—
90
2+r►z L
rA
rB
~ .1
(387j
(3a6)
as:
"Che an~~les a ~ and a~3 are very ne~rl5~ equal, so equation 307 may be written
~a+a,~
.~S (o~ ~(~-aA) _ coy.C(~ -cK )1
The total electric field observed alonb the receivinb dipole is:
Egg =-2~Bh e.~C~-~ag)
Similarly, the component o~ the electric field vector developed by the current
—I entering; t11e earth aC point B, projected along the receivin dipole, is:
.Pr
°°5
L.ffecti~~e sliaci~i;~ factor as a function of azimuth angle foi• various
dipole arrzi~'s (from Keller, ]966; reprinted by permission from the
Society of I:~ploration Cco~ili~ ~icistsl.
Polar
array
~~
Perpendicular
array\
--
---
the component of this vector directed along Che receiviiia dipole is:
F't~,t!itt•: 8.3.
w
v
'
0.5 i
'i
N
i0
d
U
O
0
0
n
~'
a~
0
a
o
~
o.
0
Parallel array —
__
~li:~l',1'I:RLY OP 'CH1 COLORADO SC1tO0I. OI~ ~~II\I~:S
i
~_
,°
i
~
7
~
g
~
(3<~d 1
2O3
2~rt-h ~. r- 2cc-ie
r~ z coy6
~p
rh~Z~`~'~i'°"`'na
~r
(391)
cm.a= 1-z a2
and the Sine and cosine may be approximated as:
~wi,a= a
+rce3F7co~coa.a~
~392~
We may simplif}' this further if we recognize that the an_~~le a is very small,
E~
This reduces to:
— co~~ c~a +,~,N ~3.a~n GY -E 2r cam.e(c~~ c~a a-.o~~,~ ~)~
CV1oC -t-.at.K ~s o~;.~a +fir Cap p ~ cos~3 COQcX +.a~.ri fS .c.c~c oC~
2r~'t[~~
cow C(~ —a 1= caa~3 coy a +.,o.c~~.n-<%x cx
cv~(~+a) = co~~3 cod a —.a,e~x ~3.n,c:~e a
7'he second term in the denominator is small compared with the first and
ma~~ he ne~lect~cl. Let us now expand the tr•i,ono~netric functions:
(3901
cvs«-«)~-zr cviec~CP-a)- ccrjC(~ta)-~ cozoco~(C3-w~)l
_ pI
E/3- 2~tv rh[
.l
1 - 4 ~r~z~z9
or_ making; a sink lei ~fi•acti~~n
~
E ^ pr(~ ~ -a) _ ~C +a)~
where c is the separation l~ehveen the twc~ current electrodes and B i~ the
angle between the axis of the current dipole and the radius vector r. 1?quation
~,
~,<, ecomes:
you
r~ — ~' + 2 'C.~'i 8
Y'A = r- z Ca18
The hvo distances r ~ and r13 from tl~e receiving; dipole to each of the current
electrodes may be expressed in terms of the central distance, r, i~F Che length
of the current dipole is small com}~ared to the distance r:
2crr h
DLSIGA` CP.ITrfiI.~ I~OR ELI;:CTRICAL SURVP;YING
QUARTERLY OF THE COLORADO SCHOOL OF MINES
(39-1)
~J~J~
~u
f~~
L ~,8~a
_ pr ara e ca1~s + .a.c.n O.u~(3l
+z,c.vi~ Ba.ui ~S S
2h
_ r. r.~,K 9~~,~r /3 + 11
~c~— SLXaxC~~~ +2J
(39S)
(39c)
(396)
These last equations indicate that the effective spacing facCor for a dipole
array is not only a function of the separation between dipole centers, but also
of the azimuth annle from the current dipole to the receiving dipole, and the
bearing an~1e of the receivinn dipole. Let us now consider a few special cases.
and:
~+1~
Q0.— 2hLLeo
Alternate ways of writing; the same equation are:
or, simply:
(~95j
'(1'Y3
1~T~x [,iut 8nt,,~ /3 + Cvi(3 Cai~3~
pa — IL~(cai6co~~ +'~.a~,;+ E3~.:,(3 ~ 2tYrhz
we have:
nr3
equation for apparent resistiviCy measured with a dipole array:
Rememberinb that the observed volta;e is DU = bE, and usinb the defining
E~ ~ 2rrhr2
~,a"~n f3~.e:+, B + ca~~ co~8~
The last term is very small and may be neglected:
E~
n.~"`~'°'`'" o + r ca1~ c.o1Gr — g(r~3~.~;►, e caie coi~~
2
mrh ~*
Thus. equation 392 may be rewritten approximately as:
1l~Ioreover, since a is a small anble, the product of the angle with a radius, such
as r, should be approximately equal to the length of the chord at the radius,
c sin B
tar = c,oc;,,e
2a!}.
(3991
2~5
-.~xzEJ + 1
(400
~r
J3 = — — B
2
H i~ arbitrary,
It is interesting to note that when the angle B has a tanbent, equal to -!- 2
(8 = 53° 44~'j, the effective spacing for a parallel dipole array becomes
larbe without limit. This is the same value for B for which the normal
electric field (that observed over a homogeneous earth), has a zero component
in the direction parallel to a sout•ce dipole. Accordingly, the geometric factor
for the parallel dipole array when tangy B = 2 is infinitely larbe. The presence
of a layer in the otherwise homogeneous earth will distort the dipole field so
that where previously no parallel component of electric field was observed
for B = 53° 4~~', some very slight electric field component is observed. Since
this small anomalous field value is multiplied by an infinitely larbe geometric
factor, the measurement becomes infinitely sensitive to the presence of layers at
depth, at least theoretically.
The effective spacing factor ~,oes to zero when the azimuth angle is such that
tan 8 = 1, or B = =~5°. with this aziinuCh annle Che observed resistivity becomes completely independent of the presence of a resistive layer at depth.
I~or the perpendicular array
and the effective spacinb factors:
B is arbitrary
/3 =~r—B
For the parallel dipole array:
~`ODE — ~
for the equatorial dipole array. both angles are 7r/2. and the effective spacing
factor i~:
a~P — 2
and the effective spacing factor is
B =0
/3 =0
With the polar dipole array, we have:
DESIGN CRIT~RI9 P'OR ELIsC'PRICAL SURVEYING
oR- z
j103)
ca~~ = z c~-~ Q
This ratio of field components is the coean~;ent of the angle:
Ero~~E+a~~ = 2 co~0
tangential compoi~enCs of electric field sti•en~Ch:
(40~)
(=b04)
The wide range over which the effective spacin~~ factor for the various
dipole arrays can vary sun~ests that care should be taken in orienting the two
dipoles. Arbitrary orientations caii lead to ~~idely scattered d~~ta. Therefoz~e,
if electric field observations are to be made over the full ran;e of azimuth
ankles, the measuring; dipole should preferably be oriented perpendicular to
the current dipole, or in the direction of greatest electric field strength (or the
direction which would be that of breatest field strength in a homogeneous
medium). A perpendicular array provides a fairly simple surveying problem,
but has the disadvanta,e that the amplitude of the electric field component
becomes quite sma11 for bearin~~ oi• azimuth angles near 0 or 7;/2. }_VIaximumfield orientation is a more difficult surveying; problem, but provides a much
lamer signal to be measured in many cases, and si,nal level is frequently a
critical problam in surveys made with the dipole methods.
The bearing angle for the direction of maximum field strength for a dipole
over a homogeneous earth may be found by taking the ratio of radial anc~
to ~r/2.
In view of the s}~mmetry of the electric field about a dipole source, we are
concerned only r~rith definitions applying to a single quadrant. Thus, while
tan B ct~uld have a value of —2, ~eneratin~; a singularity in the expression for
the effective spacing; factor for the radial dipole array, we have restricted our
considerations to a range of values for the azimuth an~1e running from 0
~, _ r
and the effective spacing factor is:
/3 =0
FI is arbitrary.
I'or the perpendicular array, [he effective s~~acing factor is independent of the
azimuth anble.
For the radial array:
(4.021
QUARTERLY OF THE COLORADO SCHOOL OF 1VIINES
and Che effective spacing factor is:
ZOO
I
2~zB + 2 r
_ z~a~cz8 + 1
l.000
i.000
.540
.538
.553
.609
.667
.727
.830
.865
.6b7
.667
.66r
.667
.667
.667
.667
Perpendicular
array
a'~r
(406)
207
= n—,5~2
~
2a 2'R'rSdr
ou=-zJ4~~r _ f~`z
over the limits 2a. to a, n being the Wenner spacing:
~4~~7~
electric field cti•en~th. The variation of sampling depth with bearing an~1e
indicates only that it is preferable. to have the receiving dipole directed in the
direction of maximum electric field strength.
The effective spacing factor for the standard type electrode arrays may be
used in conjunction with equations 363, 364 and 365 to determine the maximum depths at which resistivity contrasts may be detected for a given limit~tion to the weight of equipment and safe operating voltabe. In equations 363
to 365, the Schlumber~er and dipole arrays were compared with the Wenner
array. The effective spacii~b factor for a Wenner array may be found by
integrating Che expression for elecCric field intensity given in equation 370
For any biven azimuth an;1e, 8, there is Borne bearing angle, /3, which will
lead to an apparently infinite sampling depth. This infinite sampling depth
is of no praceical intereet, since it is associated with the direction of zero
l.00
.500
.~lJ7
.493
.440
0
2.00
1.08
1.02
a'~r
a'~r
a'~r
a'~r
.500..___.
a'~r
____
.500
.503
.SOr
.5~4,
.600
.710
.930
.968
Parallel
array
Radial
array
~.quatorial
arraq
Polar
array
~'Iaximum
an~ay
— _---These relationships are shown graphically in figure 24.
ao°
~o
15°
30°
4.5°
60°
t5°
io°
0°
Azimuth
angle, B
TABLE Ij.
Values for the ratio a'/r for various azimuth ankles are as follows:
a ~,"u`u
The effective spacing factor for such an array is:
DESIGN CRITERIA FOR ELECTRICAL SURVEYItiG
10-2
~~-I
'
~
\
—1
N~
N
~
,
~
p
N~
—{
.Wj ~
cP
W
cP cP ~ —1
U` —1 ~
~
10
N~\
~
'S
N
5
~ N N
N~~
~
N` = tJ
~
~~
\N
102
~
O
^~
—1
O
p
N
Effec4ive spacing /First-layer Thickness (a /h,)
N
O
\\
~
N; s,
~1
\
~
~ ~~ ~
~
`
`~\
.\
''
'/'
~~
—~
i~
~/
-~.
_~
/Maximum
resistivity
O
O
-0
~~
N~
103
QUARTERLY OF THE COLOR9D0 SCHOOL OF VIINES
FtcuaE 84. — Apparent resistivity curves obtained with a Schluinberger array over
a sequence of three layers in which the transverse resistance of the
middle layer is lame. Curves are taken from sets computed by Compagnie Generale de Geophysique (CGG?. The dashed curves are
taken from a set for a resistivity contrast of 39:1 between the second and
first Izyers (from Keller, 1966; reprinted by F~ermission from the Society
of Exploration Geophysicists).
a
0
a
C
N
N
T
i~
a
O
N
N
102
ZUcg
t
l408
209
,. ,n.i.
~ ~ ~~>
n,a~
_
~$
2C~W ~n 2
(ll~ 1
— 1.38 `21t J = 0.625 k3
(il ~)
i 11.2)
resistivity p, covering; an insulating; subsh~aturr~ are sho~vri in fi~,ure 85. All
the curves are plotted with the Lotal ~ii~tance between outermost electrodes as
the characteristic ~listan~~e, rather than the mm-P~ ~~~,mmr~nly ~lPfine~3 s}~acinrs.
The parameter, k:;, which is the ratio of receiving dipole lennth to dipole
separation, is quite sma1L If it is aesinned a value of 0.7., the maximum attainable "reach" with a dipole array is somewhat betCer than half the reach for
a 5chlumbei•~er ai•ray. with the same weight limitations.
'Che relative capacities cif various arrays for detecting; boundaries at depth
may perhaps he better seen by comparing; theoretical sounding; curves over
simple layer sequences. Curves for the very sirrople case of a single layer ltiith
a'
_ ~.os k3
a~„, — i.3$ ~ k3)
and for the polar dipole aria}'
Thus, for a even total ~vei~~ht limitation, resistivit}' co~~tcast~ ma}' }~e found
at 5 percent <~~reater deaths with the Schlumber~;er arra~~ than with the Wenner
array.
Similarly-, for the equatorial dipole arra}~:
°s _ 1.96
c`;,~
1.3g = 1.05
(.ill}
~s the ratio of rnaximuai attainable spacings arrived
~`w
Q`
5
at in equation 35~. harin~~~ the ~~alue I ':~ i '^ = 1.-lb
where a
ZS.
The ratio of effective spacinT factors fog- the Scl~lumber~;er and Wenner arrays
Defining the e{~ective spacin~~ factor as in equation 7~i:
~w = S
"' L„ 2
oz•, substituting from equation -~07:
~u
PW = 2rra W Z
The equation defining apparent resistivity for the l~enner array is:
DFSlc~ Cai~•Lrr.a ror, }r_rcT~tic_aL Survi.Yln~c
~
I
1'a~Pi
1
dipo%
be~~er
QUARTERLY OF 'THE COLORADO SCHOOL OF NIIN~S
dipole array.
"Chi•ee-la}'ei• sequences are considered in figures 86 and 87, the first being a
sandwich of lavers ~a-ith the middle one being relatively more conduceive than
the outer layers, and the second being a sandwich of layers with the middle one
being relatively more resistive ehan the outer layers. It should be noted that
the introduction of the third layer does not affecC the i•elaCive positions of the
rising= asymptote for the various arrays. However, the capacity for• an array
to see t6rou~;h a resistive layer into an underlyinn conductive layer does not
varti~ in the same wa5~ as does the capacity to see Yhrou~;h a conductive layer
into a resistive layer. T'or example, it is necessary to use spacings twice as
great wily ~~olar dipole array as with an equatorial dipole array to see a buried
insulator. However, the polar dipole ~i~acing required Co see a buried conductor is only slinhtl}' greater than the spacing; required with the equatorial
It is interesting to note, that with one exceE~tion, all curves approach a rising
as}'mptote for long spacings, with the position of the asymptote depending on
the array considered. The one exception consists of an array of parallel dipoles
arranged so that the receivin~~ dipole lies almost at the position where no
electric field ~~~ould be detected for a uniform ezrth. In this one case, a buried
insulator looks like a buried conductor.
Fico~tt: 85. ilpparent re_isticitg' curves for se~~eral electrode arrags for the case of a
eingle uniform layer cu~~erin~, Dui insulating substratum.
210
~
1
I
~
~
~
~I~
i
~~
—_
_-
ACTUAL RESlS7/V/TY
PROF/LE
-----
\
~
-------~~-'P
~
RELATIVE SPACING
90
/
O/~
10
i
i //
i
r-
211
100
We will i~ow consider Che requirements i'or frequencies and source-ieceiv~r
separations in oi•cier that a layered earCh inay be i~ivesti~;ated to a desired depth.
Such requirements must be considered in determining how much power is
required with the various source and receiver sensor configurations to reach
a givers depth, if the relative merits of the methods are to be investigated.
We inusC first agree on how the "depth of prohinn" or the "depth of penetration" ~f a method is to he measured. No simple definition of depth of penetration is available for any of the electrical geophysical methods because the
ability to detect a bed at a given depth depends on the electrical properties of
all the, beds lying; above that depth, and no single, universal measure of depth
of penetration ran he specified. Rather, since we a~-e interested onl}' in comparative measures of depth of penetration, we wi11 pustulate a simple model,
and compare requirements to see throunh such a simple model with each of
the tec}~niques. 'i~he simplest m~~~lel which has mPaninr is a two-la~~ei• model,
1)ri~~r~i o~~ ProE3rnc ~vrrt~ E~i.i:cT«<>tit,~~n~°r1~. YIi~:•r~lons
FicuaE 86.— Various dipole re~istivi[y curves for a common set of ]ayers with a sequence of
resisti~'ities of 1:1~4~:1 and a seyucnce of thickness of 1:5. 1~he da~l~ed curve is
valid for the perpendiculeir dipole array, while the solid curves are valid for p~ireillF~l
dipole arran~emenis with the xrimuth ankles as indicated.
0.1
RELATIVE
RESISTIVITY
10 ~
DESIGN CRITERIA FOH ELECTRICAL SURVEYIA~G
0I
I
10
RELATIVE SPACING
QL_~ltri.r~~r o~ z~}rr Cor.or„»o Scilooz_ or \Ti!vr:s
ICO
(~,1~)
A~surnin~ either that p~/p,-~0 or ~~, we have t~vo asymptotic expressions:
~ + co~3'L,
chiiacterized by a single overburden layer with a resistivity, p~, and a thicknc , h_ re~ti~~, on n halfspace with a resistivity,,p~, (fig. ~>81 .
The model could be investigated for an arbitrary conti~a~t in resistivitiee,
p_:"p,. but in view of the complexity of the expressions for coupling between
~ arious sources and receivers, this would have to be done numerically, so
that generalizations would be difficult to make. Fortunatelq, the various expre~~ions can be simplified by considering extreme contrasts in resic~ivity; that
i~. p yip,~0 or P_/A? ~ a~
~Je 1vi11 first consider the expression for tan~;entiai electric field about a
~~ertical-axis coil source.
T'oi• the t~vo-layer case, R~, is:
~
for the pe~peridieular dipolf. irieu~~ement, ~~l~ile the. solid curves are valid for
~~eu~zillel dipole arrangements with tl~e azimuth un~~Ir pis indicated.
l~tr,t rt: 8~. -- b'ariuus dipole resistivity curves for a common set o1 lagers with a sequcnc<~ of re~i~ti~ities of 1:19:L and a sequence of tl~ickn~ =es of 1 5. 1'he dished cuirc is valid
10
RELATIVE
RESISTIVITY
21Z
`
'
213
fz-moo
~
cvt~X~h
~o~ = ~~~
=
~~z-~
R
(416)
(415)
bl. For o~-~0
a2. T'or w-~ ~
al. For o,~0
R ~
_ i
= 2
I
z-~ o
R
Igo'
_ ~~1
P2+o
(41~~
(41!)
Let us now examine the low-frequency and hibh-frequency asymptoCes for
these two expressions. F'or small arguments, the by>perbolic cotangent can be
replaced with the reciprocal for its ar~uinent, while for large arguments, its
value approaches l:
b. I'or p,/p,-~ ~
a. For p~/p,~0:
Ftcutt~: 88.— Formulation of the two-la}er problem for determining, the depth of
investis~ation of various electromagnetic coupling techniques.
DESic~ CrzT~:Fia Foa E~rcTrrcaL Survl:Yin~c
~,
~a~i~= 1
(420)
QUAF.TrRLY OF TIIB COLORADO SCHOOL OF MINAS
i
z
f.~o~ ~'1
(421)
We are also concerned with the requirement for the spacing necessary to
see to a liven depth, inasmuch as the amount of power required will vary
rapidly with spacing. We may forecast the nature of the results we are after
by considering some couplinb curves which have already been computed by
Vanyan (196'71. The curves shown in figure 90a are for the case of a twola} er sequence with the lover medium a perfect conductor. With progressively
shorter spacinbs in comparison with the thickness of the overburden, the
curves for finite separation depart from the curve for• infinite separation at
hi~~her and higher frequencies, until at very short spacings, the presence of
the lower halfspace cannot be detected easily.
A similar situation is seen if the lower medium is an insulator (fib. 90b)
thou~~h the curves differ in detail If the separation is too short in comparison
Frith the thickness of the surface layer, the effect of the lower halfspace becomes
di(licult to see.
In the various integral e?cpressions, distance from source to receiver is contained only in the argument of the Bessel function, and so these integrals
i~~ust be evaluated in order to determine for what spacings the curve for R2
first cle~~arts a given amount from the curve for R„'-'. The separation parameter
W<
depth ; or
Radian wave lengths somewhat longer than the wave length
in the surface layer (or layers) must be used
to see through a sequence to a riven
frequency requirement:
of penetration: the depth of penetration is the intersection of the low-frequency
and high-frequency asymptotes. For frequencies hither than that intersection
frequency, the lower medium has little effect on the field components,
while for frequencies lower than the intersection frequency, the lower medium
has a progressively larger effect on the field components. Therefore, we have a
These asymptotic conditions have simple forms which plot as straight lines
on the usual logarithmic coordinates 1 fib,. 89j. IJsin~ the appropriate definition for apparent resistivity, we find that the ratio p„/p~ is constant at unity
for the two low-frequency asymptotes. At high frequencies, the apparent resistivity increases directly as yr2h if the lower medium is a perfect insulator,
and decreased directly as 1/yl2h, if the lower medium is a perfect conductor.
These two asymptotic conditions pirovide a fairly simple definition for depth
b2. For ~„~ ~o
ZIT
~
~ih
ZZJ
= Ro d R.
(422)
~(n,h + ca~E -1 n')
z
~ ~
~ i ~
~ ~
ca~n,h ~ c~ X,h + z~(1- co~2X,L,~
z
2 %2 ti
,1
-.r 112(1 +z~"'t
nz= Cm2+~Z)
--,,j
~
For small values of in, we may make the followinb 1pproximations:
FZ
~`~2%i
j::1251
(423j
Because the case is somewhat simpler, let us first consider the requirement
for spacing when p_=0. The ratio function is
Al<f
R~
m has roubhly the significance in the integrals of the inverse of the separation.
Thus, if we are concerned with relaxing our limit condition on r from infinity to merely lame, it is equivalent to relaxing the condition on m from
zero Co merely sma1L In place of R~„ we can use a better approximation:
I'icaex: 89.—Use of asymptutes for asyruptutic coupling function to define depth of
investi~;ati~in as a function of frequency.
log Ro
DESIGN CRITERIA T'OR ELI:CCIiIC9L SUR~'I;YING
r/h =
IOg f~ 2
G~h
Couplin bE°twecn i ~°erti~il ixis loop and a tan~,fntial wnc, a~ a function of Source-rcc~ n~er s~ p u~ation for the t lse of i -in~le 1 iper covering
a perfect conductor or a perfect insulator.
B. Coupling over a buried insulator
log R 2
ugh
QU.ARTIsPLY OF THl? COLORADO SCHOOL OF fVIINES
A. Coupling over a buried conductor
Ftcuae 90.
216
'R~
~'~
t
21.~r
L
~~
X, + ~ X,1~
~
z
~~1
co~ X h
~ZL ~; c~ — ~Z~~h>
i ~ l.;i l 1
~~
a'f2
Considering equation -1~1, it is obvious that tl~c firsC term ~~~ill contribute
nothing on integration because it leads to an nr' weinht in intes~~ration. $o,
we need ~>nly consider inte,~rati~~n ~>f the terms:
YR+ a—~'
~Je have achieved our foal—that of ex~,>ressin~; the behavior fur m nut
quite zero as an additive c~ri•eetion to R,,. }~ernember that the tei~~r~ cc~i~tainiii~;
R in the inte~~rand of the Sc~mmerfiefd integral can Jae expanded a~ foll~~~cs fur
small values of m:
m
~ yylR
m~R~
= Ro + L1R
I
~ 4n <E R° ~ 1 -
~ i:~~~ ~
1- co~ ?I h
` +~z 2Y,
KeeE~in~; terms ~>nl~' ~~f ~,rder n~`-' and takinn y,jy., O. ice hare:
_ ~, 1} ~
~~ ~ ~Li + ga z
equation. ti~ ith thc~ result:
"l~he terms ~shich represP~~l the ratio, fin- ni=0 can he ~li~~ided ~~ul ~~f lhis last
~ i•?~~,
~ h +2
~(1-coil( Y,~)+
~~.1~
zz~~z -~x)~~ ►~~ t 1
The terms in the deuuminatur can be reai~raii~;ed s~~ that all ~~~6ich are. i~~ultiplied b}~ m'-' can 1>e treated as a small r{uantit~~ ~~ hr~n ~~~~n~E~zu~ecl ~~ith tl~e rest
of the terms in ihr~ d~nr~minat~~r•:
(=128 j
~~1 +z~~~~z- 7z~~'~~X,h+~(1- coC~X,h)~ ~- 1
Substitutin.t~ i~hese nppr~>ximatio~~~ in e~juali~>n lZ .gee have for ~inall m:
Drsicn~ Ci,i~ri,:rtn roi, l:i.rcri,ic ~i. St~r,~~r;~ i~c
ESP
R~' = CRo +~fZ)2= Cho + 2 RL1R + ~FZz
('132)
QUARTL:IiLY OF THF, COLOR9D0 SCHOOL OP' ~VIINLS
~oCMr)clrn
'
+-,2 ~~h
~~
-~Xh~ ~
z
.~
1434)
23oCmr)dm~
(433)
~ `1~~6)
We are now in a position to ask the question: Having chosen a spacing and
frequency, and knowing the approximate conductivity of the earth we wish to
zneasure, what source strenbth do we need and what receiver sensitivity is
required? If we examine the expressions for an}' of the source and receiver
types listed in the firet section, we note an extremely important facC:
For a~ay sourcea~eceiver co~nbinaLion., the apj~m~ent resistivity is
c~ function of the ratio of meccsurecl feld slrengt~C to source strength.
As a result, there is a direct trade-off beCween source strerinth and receiver
Norse Co~vsri>~haTio~s nn~~ PowLi~ REQui~~vrLrrTs
Our result indicates that the spacing must be comparable to the thickness, and
be greater than tjae thickness by tlae same ratio as the spacing' i-s greater than
a soave le~agth irz the surface layer (by 2~rJ.
Similar analyses should be made of the spacing requirements for other
combinations of sources and receivers, but on the basis of this analysis, we
may infer that the spacing between source and receiver must probably be at
least five times the depth to be explored.
C X,r~~ - ~r~~{Q
where Q is the fractional accuracy required, presumably of the order of unity
for realistic requirements. Because y~h<1, we might make the approximation:
The second term in the brackets represents the fractional correction to the
value of tangential electric field arising from finite separation:
Eq~ — Zn~'[1 -~r~• Xr'
{2
Perforn~inb the indicated integrations and differentiation:
°
~~ztt'~ ar~- R2f m
The first of these three terms provides the answer obtained previously for
infinite source-receiver separation. The last term leads to ~n intebral with
weight m'', which we will neblect. With the first two terms:
ZI$
21~
With both an upper limit on source power and an upper limit on receiver
sensitivity, there is an ultimate capability for electromagnetic soundin~~ methods
within a given set of constraints. This ultimate capability is not the same for
all coir~binations of source and receiver, and so, it will appear that for measurements which strain the constraints of a riven problem, some systems will be
beeYer than others.
I?lectric field components are measured quite simply, using a pair of
electrodes in contact with the mound. Various considerations enter into the
choice of electrodes: if very low frequencies (less than 1.0 cycles per second)
are being; used, the old non-polarizin~~ type of electroc(es consisting of a
metal rod immersed iii a saturated solution of one of its salts, all carried in a
porous cup, must be used to avoid diET~iculties with electrode drift after it is
placed in t~~e ground. At hi~~her frequencies, the choice oI electrodes is less
important, and the only ehin.; which trust be considered is the development of
a resistant surface corrosion la}'er nn many metals. Oxidized lead electrodes
and cast iron electrodes are.;~c;nera(ly r•ecominended; stainless steel electrodes
or copper-clad steel electrodes should he avoided.
Electric field is not measured directly. Rather, the voltage drop is measured
between electrodes with 1n appreciable separation, and the electric field is
assumed to bE~ about equal to the ratio of voltage drop to electrode separation.
Ho~~ever, the electric field is usually determined more precisely, the loner the
electrode separation is made. Near-surface rocks nearly alti~~ays have inhomo~;eneitie~ in resistivity which distorC the current flow locally. With a long
electrode separation, these irre~~ularities in current 1-low tend to average out.
Also, with loner sepai•ltions, the bearinn of tl~e electrode lair can be established with more precision and less trouble. Usuall}r, it is wise Yo use as
larbe an electrode separation as is compatible with the assumption of dipole
sources and receivers; that is, receiver dipole len~~ths of approximately onefifth the spacinb. At very lame spacings, in excess of 10 kilarneters, this is
no longer so important because the precision with which the electric field is
measured becomes limited by the ambient noise field.
ceiver sensitiviCy.
2. The maximum receiver sensitivity is almost always limited by the ambient noise field. Cenerallq, weight is not a factor in determining re-
strength.
~l. The maximum source stren~[h is almost always limited bj~ the amount
of weight which is allowed in the source. For all except small source
stren~tl~s. the weight of a source is very nearly proportional to the source
sensitivity. so long as the do not encounter any nonlinear constraints not biven
in these expressions. The tsvo n~~ost important nonlinear constraints we will
be coiice~rned with are:
DL:SIGN CRITERIA FOR ~LEC1'P,ICAL SUP.VlYING
QUARTERLY OI' THE COLOPADO SCHOOL OF MINAS
likely to be concerned with in electromagnetic sounding.
Most of the distant spherics noise originates in the tropical areas where
electrical storms are common. During the course of a day, the center of
storm activity shifts westward with the sun, providing a surprisingly consistent level of spherics activity. The least active period of the day corresponds
to midafternoon in the Inid-Pacific where the sCorm activity is least, but the
diurnal pattern to noise level is difficult to reco~~nize.
There is a moderaCe seasonal variation in distant spherics noise level at a
given station, as the center of storm activity shifts north and south with the
climate. In northern latitudes, the spherics noise level is lower during the
winter months than durin~~ the summer months. There is also a pronounced
Which of the three noise sources is most iir~poi•tant depends on the frequency ranbe under consideration. ~~Iicropulsations of the earth's magnetic
field ire important only below frequencies of about 1 cps. Po~~~er distribution
noise is a problem whenever measurements are made within several miles of
power line or transformer. The worst difficulties are presented by low-volea~e
hi~~h-current distribution systems, and ~~articularly from 11.00/220 volt transformers which are common on poles near homes, irrigation wells and so on.
Where noise from AC distribution sysCems is a problem, the problem may
be minimized with filterin~~. At the higher frequencies, noise from atmospheric
electric discharges contributes to the received electric field at all frequencies,
and normally constitutes the background against which a signal must be
recobnized at frequencies above 1 cps.
During local summer seasons, spherics noise from nearby showers may be
so intense as to make electrical prospecting; infeasible, not only because of high
noise levels, but also because of high li,~htnin~ hazards. Even during nonstorm periods, spherics noise is contributed by lightning discharges around the
world, which excite a cavity resonance between the earth and the ionosphere at
low frequencies, or ttihich propagate as a trapped wave in the earth-ionosphere
wave guide at hi~~her frequencies. The lowest resonance frequency, the socalled Schumann resonance, is observed at about o-1/2 cps, with other resonances ~t intervals of about 4~ cis goin upward in frequency• Above about
20 cps, noise contribuCed by spherics forms a smooth spectrum up to frequencies in excess of 1.0 kilocycles per second. the highest frequency we ire
Noise above I cps
The ambient noise field is contributed lar~~ely by Chree sources:
1. Rapid variations of the earth's magnetic field (micx•opulsations)
2. Induction from power distribution systems, primarily at 60 cps and
harmonics.
3. Energy from atmospheric electric c~ischar~~es.
220
221
(~1~37)
io-
io~
--
.._
frequency, cps
io'
~c2
io
iaz
i62
142
122
density, db below I a/m
,., v2
-~
Derived magnetic field
1~'[cuas 91.--0bscrved spectral density of the verticu] electric field during the suntmer season i❑ Colorado. Horirortal magnetic field density is derived by
as~umin~~ ~ilan~°-wave imprdancr. 1 Frr~m Maxwr~ll. 19671.
~v
14C
12c
ioc
eo
Observed electric field
density, db below I v/m
per cps ~Z
60
The scale to the right of figure ~1 was obtained using; this relationship.
These noise spectra show some characterisCics which are common to noise
spectra observed the world o~~er. The noise power density decreases gradually
with increasinn frequency, except that a minimum power density is observed
at frequencies of 7 to 5 kcps, an~i a local maximum power density is observed
IEI _
— 377
latitude dependence to the amplitude of spherics noise, with noise levels
dropping ofd markedl5r at auroral latitudes.
Maxwell (1J67 I has summarized observations of the noise environment
above 20 cps for a variety of locations in the northern and southern hemispheres. Data are primaril}' for the vertical electric field intensity above the
surface of the earth. "I~hese epectra provide a basis for rough estimates of
noise field intensities over the world.
Average spectre for the vertical electric field for summer in Colorado are
shown in figure 91 (from Maxwell, 1'~6l). ~Chese spectra, as are most spec
tra, are ex~>ressed as ~owe~- density spectra. fl scale for power density in the
horizontal magnetic field is included on fi~;ui•e 91, although the field quantity
measured was the vertical electric field. In free space, and assuming large
distances from the source, oi•tho~;onal electric and ma~~~netic fields are related
to one another by the f~~eespace impedance:
~L;SIGiV C[iI1'P•.hIA FOR ~Li:CTRICAI. SITI2VF~:YING
QUARTERLY OF THF. COLORADO SCHOOL OF MINES
~~
~~
FREQUENCY, cps
~..
~Ex~ = ~(,~~c,~~~
/"
`i-}r~
(439)
where the parameters k~, and n will depend on season and latitude.
intensity is
In methods in which the horizontal component of electric field
the
estimate
to
used
measured, the harizontal magnetic noise density can be
relationthe
of
amount of noise detected with the receiver. We may make use
above the earth
ship between Che horizontal component of the magnetic field
Cagniard:
and the orthononal component of electric field in the earth liven by
estimating the best
As a crude estimate of noise density suitable for use in
these spectra beapproximate
can
we
used,
receiver sensitiviCy which may be
the logarithmic
on
line
strai~~ht
with
a
tween frequencies of 50 Co 5,000 cps
plot, which has an equation:
z
z s
~
component of magnetic inFtctntr: 92. —Power density spectrum for the horizontal
region (from
tensity, summarized for the summer season of 1963, Alaska
\4~xwell, 19671.
io
I/2
HORIZONTAL MAGNETIC
S,r.. .,,. ....~,.... mini .,e. ~~t
FIE!D DEN
1967).
the source of
in the neighborhood of 5 to 10 kcps. As one does further from
increases, and
the noise. the rate of decrease of power density with frequency
illustrated by noise
the minimum at 1 to 5 kcps bets deeper. This feature is
J2 1 from Maxwell,
figure
in
shown
Alaska,
in
season
spectra for the summer
2Z2
223
tensity lasting only a fe~~ti c~~cles is termed lra~isie~il naicro~~ulsation ac•ti~z~ily,
sometimes designated by the symbol Pt. Pt pulsations consist of several series
of oscillations, each series bein~~ a set of heavily dail~ped oscillations persistin~~
for a few minutes,. with 1-1/2 to S cycles. The }>eriod of pulsations lies in the
range from 20 seconds to several minutes.
NIicropulsatiun activity which ~ on~ists of oscillations iu in~gnetic field iri-
activity.
"The lo~v-frequency natural electroma~>netic field is varied in chaz-acter, but
it dies show characteristic structures tivhich have led to the classification o~
numerous types of micropuleation activitti~. Classification may be on the
basis of the frequency content or the duration in tii7ie of the micrupulsation
Frequencies above 1 cps provide relatively- shallo~~ penetrations and require relatively small separations between source end receiver, ~o that obtaining sul~~icient signal strength to provide recognizable signals in the presence
of noise is usually not a serious problem. Durin~~ the periods of Local thundershower activity, the noise environment tray make ineasui-einents infeasible because of high-frequency impulsive noise, but generallti•_ the Iar~~est problem
is with noise at frequencies below 1 cps. ror deep soundings in conductive
sedimentary basins, the required frequencies mad be as lo~~~ as 10 ~~ c~~s, and
the source-receiver spacinb a~ large as 2U miles.
i1'oise in the frequency range below 1 cps has been de5cribecl in a large number of scientific papers published in recent years in journals such as the
Journal of Geophysical Research. For oui- purpose. useful descriptions of
low-frequency noise have been ~:iven by Santirocco end Parker (19631. Davidson (196-1.1, Troitskaya t 196-1- I , Heirtzler (196-1 1. Lokken 1 196 l 1, Pritchard
(196-1~} ,Herron (196r I .Hoffman and f-Iorton (19661. Balachandran 1 196r 1
and Welch (1968).
At frequencies belo~~~ 1. cj~s, the natural noise e~ivironmerit at the earth's
surface is dominated by~ sma1L-amplitude fluctuations in the ma~;neCic field,
having amplitudes up to Lens of gammas and a wide range of perio~l~, kno~ti~l
as micropulsaCions. The micropulsations obserred at the earth's surface are
the consequence of electroma netic wave disturbances propa~~~ati~~ to the surface of the earth from the outer atmosphere. "l~he frequency content of these
waves aboae 1 cps is strongly aCtenuated in the ionosphere before reaching the
surface of the earth, ~ccountin~ for the se~araCion in frequency as ~>>el1 ~~ in
origin beri~een micropulsations and pub-audio frequenc~~ noise I \a~rrocki and
Papa, 1963; fib;. i I. The ultimate source of [he electromagnetic ~~a~~ec called
micropulsations is the interaction of solar particles with the boundarti~ of the
earth's ma~~~netic field.
Noise below 1 cps
D~szc~~ Crz~~Leza roi. IlLi:c~rr~zcnL Su~,vrYZ~c
QUARTERLY OI' TFIE COLORADO SCHOOL OF MIA'ES
the past three cycles of sunspot activity is given in figure 9%~.
aml~~litudes during the night hours.
The structure of the pulsation nurse field also shows an approximately
monthly pattern related to the period of rotation of the sun, and the occurrence o£ magnetic storms. Magnetic stories are periods when increased
amounts of solar plasma are emitted by the sun.. leading; to turbulent and
chaotic interactions with the earth's tna~;netic field. Some types of micropulsation activity are enhanced during magnetic storms (Pt, Po-I1, while other
t}~pes of micropulsation activity are inhibited 1 Pc-III). The periods of Pc-I
activity are shortened during; magnetic storms: during the peak of storm
actiait5~. Pc-I periods may be as short as 15 seconds, lengthening; ~zaduaIly to
about 50 seconds after the storm is over.
Long-period cycles in ina~~netic stori~i actiaity are also observed, with the
11-~~ear c~~cle beir.~; the best known. During the peaks of these c}~cles, the
essentially monthly occurrence of magnetic storms continues, but the probability of a particular storm being intense increases. Inasmuch as mannetic
storms appear to be closely associated with sun spots, it is common to measure
solar activity in terms of the number of sunspots which occur. Data on sunspot numbers are available over about the past t~~~o and a half centuries, as
indicated iii figure 93. It is readily apparent that the 11-year cycles of activity
~ho~v marked long-period variations in intensity, and the current cycle of
acti~~ity, which should reach a peak during 1969, may be one of the most
intense in the past t~ao and a half centuries. t~ more detailed comparison of
sufficient amplitude to be obset-vable.
The amplitude of micropulsation noise is by no means completely random,
but sho~~~s manv consistencies which may be used in predictinn noiee amplitudes. ti'Tan~ t}~pes of micropulsation noise show a diurnal pattern of intensity. The Pc-I and Pc-II t~:pes of continuous inicropulsation acxivity are
normally most intense during the da}`time hours and least intense during; the
ni~~ht hours. PoIII pulsations show the opposite behavior, having the largest
VIicropulsation activity which lasts for lone periods with fairly constant frequenc5r of oscillation of the magnetic field is terrned continicous micro~ulsation activity, sometimes designated by the symbol Pc. Pc pulsations have a
number of •characteristic frequencies, and are di~~ided into three types, depending on the characteristic frequenc}': Pc I, Pc II, and Pc III, having
periods in the ranges 10-~0, 60-150 and 1~0-600 seconds, respectively.
Pearl-type pulsations ai•e re~~~ular amplitude-modulated sinusoidal oscillation~ with periods in the ran,e 0.3 to ~ seconds. They occur in the form of
separate bursts, ,.;~raduall~~ developing into a eei•ies of ~~ulsations persisting; from
tens of minutes to tens of hour. Pearl pulsations ina~~ have quite large amplitudes in the auroral latitudes. but in the inid-latitudes,. they rarely are of
22~
Z
c .
g 5
A,
_
3
_
_....,
-I
1
~
2
3
1
d
5
5
6
I
]
8
9
9
10
11
II
IS
I]
is
Ib
13
15
17
Moving 22-Year Means
12
Moving 11 -Year Means
13
Annual Means
IB
19
19
vo_
T~T.
...
•,
MHO iq
.....
~
.. _.""
1971
1972
No .s
zoa
SUNSPOT No.
R
........T-....-....._,.,.
-...~. 110
.
.~.-•-•-^---^-""'-
~o ,~
zoo
30K
50 ~
60
~o
h
o
p
~ o
p
_
~o
d w th Ih
r
b q
ng
q,
_~
nc d ng
t~s
~-
'
""'._..,...J
"i___
1966 ~ 7967
1968 _ 1969 r 1970
_a1
s
--
--{- 197_
1973._1
~
eo
1967, p. 2).
1icur~. )t. — C:omparisun of activity patterns for sunsputs over recent c}~cles (from
<:The current try°nd in solar activity." in TQSY Notes No. 20, April,
~
'CYCIe No '.
___'-19?~
~~
.zo
30
~un~pot Cycles No'S,17, 1$. 19 and 24 Zunch Smoothed Relative Sunspot Numbers (Ri
~— ~
,_
.. ...
Cyde
?~
"'..
1963 ~ I964~1965
~~/fg.,
50
60
~ ~o
~ 80
~ 90
80
~ wo
~ ,zo
no
~ 130
Iao
iso
90
7,
?.
vo
- 160
ioo
i zo .
no -
130 -
l ao
isoi
I60-
CY<le No.€
N 1
ti,
r~
'....
190
.....
180
...
180.
._..
190.
zoo -
SUNSPOT No.
R
210
I'rct~r.~•: 93. — Various measures of stt~ispot activity over the period from 1700 [o
1968 (from Handbook vi Gr~ophysr.rs, "Che 1~1acNlillan Company, New
Rork, 1961).
:o
eo
so
30
ioo
:o
~o
~o
io
00
IO
00
22S
ac
~0 20 30 <0 1750 ao ~o so vo 1800 io zo 3o ao 850 ao ~o eo 9a 9~ io 20 3o ao
X950
1700
YEAR
0
zo
a
g
f ~o
`o
N
a
0 6
E g
~o
m
D~SIGi~` CRITERIA FOR ELECPRICAL SURVI:YIN(;
QuartTerr.Y or Tr3~ CoLOra~o Scxoor. of ~Tz~vrs
from t}ie amplitude of fuzzing of the trace. It is apparent that in almost all
cases, the Po-I oscillations are much larger during the daylight hours than
durin~~ the ni~~ht huur~. Hoti~~ever, the amplitude is ~uuch less for the days of
Jule 26 and 2c than on the days o~l July 29 and 30. The sudden increase in
le~~el is associated ~~ ith the occurrence of a small magnetic sCorm which started
just after midnir;ht of July 2Sj29.
On records of electric field recorded ~~ith a wide-band recorder (flat frequencv response u}> to a frequency of 1 cps oi- sol, the Pc-I oscillations are
apparent about 21.00 hours FIST.
The record in fi~~~ure J5 shows the fairl}~ typical increase in the level of
Pc-I activity durir.~ the dayliht hours. The increase in this activity is not
uniform from day to day, but highly irrenular. This is shown by a typical
set of records for a period o~f 9 dav~st shown in fi~~ure 96 (covering lh~ period
from the mornin~~ of July 2 l~, 1964, to the morning; of August 2; 196-1).
F..ach scan line on this illustration represents a duration of approxiinatel}' '1-8
hours (note that the length of each trace is from [he time on one morning when
the ma;~~netic tape on a recorder ~~~as changed to the time on the morning two
day°s later ~~hen the tape vas changed, aiicl not necessarily exactly '18 hours).
On this time kale, the Pc-I activit~~ is observable on15r as a fuzzing; of the
trace: individual oscillations clnnot be discerned on the reproduction of these
records. Ho~ce~~er, the relative amplitudes of the Pc-I noise can be recognized
tural noise field.
Let us no~v examine the Cypical character of natural electromagnetic noise
belotia 1 cps by examinin~~,~ records obtained at the Cecil Green Geoph}'sical
Observatory, operated by Che Colorado School of l2ines. The Observatory is
located near Berrien Park, Colorado, at ~9° -1~2' North. 105" 22' West. The
site is on metamorphic rocks ~~=ith a resisti~it~r of about 1000 ohm-meters, in
the front Ran,e ~~eat of Denver. Electric field effects are detected with ortho~~oi~al electrode arras, with electrode separltions of approximately 0.3 kilometer. Electrodes are lead plates buried at a depth of six feet to minimize
the effect of diurnal Yeinperature ranges on electrode potential.
A record showing electric-field variations over a t~~~ent}r-four hour period
fairl~~ tS~pical of a non-storm period is shown iu figure 9~. On this record,
each sca~i line repre~enCs three hours of time. The lar<.~,~e-amplitude ra~icl
fluctuations apparent on ever~~ scan lice except the ones ~tartin~; at 2200 and
0100 hours Il~Iountain Standard "Cimel are Pc-I oscillations havin~~ about a
twenty -fig e second apparent period. I:lcellent ex<~inples of Pt oscillations are
'l~Iicro~~ulsation activity is usually considered to be variation in m~~~netic
field intensity, but such variations also induce current Ilow in the earth, giving
rise to electric-field noise which is essentially identical in character, and so, no
separate terminolo~~}~ ~s needed to describe the electric component of the na-
226
227
FicuttE 95.— Electric field variations recorded over a 24-hour period at the Green
Ceoplrysical Observatory at 1'>erp~en Park, Colorado. Each scan line
represents three horn's of recording.
usually the most obvious of the inicropulsaCion types. The inefficiency of
incluctiari at lotia-er frequency discriminites a~~ainst the recognition of Pc-II
and Pc-III oscillations, as well as long-period events oil records made in this
way. The existence of Pc-III oscillations may better be r~coanized on recoids
where the higher-frec~ue~~cy contributions have been rejected by filtering.
Typical 2l~-hour records of electric field made at Bei•~en Park usinb a recording system which had a decrease in response o£ --1.81 db/octave above .01
cps are shown in finures 9t-99. These records have a scan duration of one
hour. Tht: three records are typical of periods of low magnetic activity (fig.
J7), moderate activity I fig. 93) and high activity l fin. 99). It is of interest
to noCe that the T'c-III oscillations a~~e hest developed at nighC clurin~; periods of
low ma~tletic activity.
F'or design of a recorclin~; system to detect signals in the presence of lowfrequency noise, it i~ desirable to summarize information on the noise in the
foam of power density spectra, as has been clone by Pritchard (1964~~1, Welch
(1968) and others. Spectra for electric field intensity recoreled at three
locations f Hawaiian Volcano Observatory, China Lake Naval Ordnance Test
SCation and Ber~~en Park 1 are shotim in fiaui•e 100 for a magnetically quiet 2~hour period, and fi~ui•e 101 for a ma~~,~netically active period. These spectra
sho~~r chaT•actFri~tics which are typical ~~f mant~ ~~hich haee been published:
ll~sicn~ Crz~rLrr~ ror, Ei.rcTxic.ai. Su~;vrYZ~c
5
t
}
4
s
~~
a
o
''
3~
~3
223
h
h
~.
t~
c
t~
}
~~
,,
N
QLiART~RLY OI' THI: COLORADO SCHOOL OF MINES
`
`Z29
(440)
KN
2.4
2.6
2.7
Santirocco and Parker,
1)60
Davidson, 1961
Hoffmai7 and Horton,
1966
Herron, 196r
Source
where the new constant k,~ is now a function only of the source stren~;Ch of the
noise field, and not of the electrical properties of the earth.
Tt~~enty-four hour aeeraaes of power density spectra do not provide insibht into the hom•-by-hour variation in noise levels which we know is present.
KN =k~I'
—50db%ly=/cps
1~-10,000 sec
h
2.5
"l~he amplitude of the noise spectrum computed fi-orn electric field observations is proportional to the effective resistivity of the earth, measured at the
frequency under consideration:
—53db/ly=/cps
—35db/ly-/cps
— <Zdl>jly';'cps
10— 1,000 sec
1Q0-10.000 sec
1— 2;000 sec
Periods
7'ABLt; 16. — Parameters describing ~~ower density spectra
where aP~,/al.~ is the ~~ower density, f is the frequency cycle per second, h
is a constanC with a value of 1.5 to 2, and K~ represents the control of source
and local ,~eolo~;y on noise power. Values of b and K~ taken from published
spectra are listed in table 16.
aaF
p" = K nF -b
1. The spectral density increases nearly linearly with increasinb period, if
the spectral peaks are noe considered;
2. The spectrum consists of sharp peaks (almost linesj correspondinb to the
named varieties of micropulsations 1 Pc-I, T'c-II, Pc-IIIj superimposed on the
linear background spech~um;
3. The effect of a magnetic storm is to "fill in" the spectral density between
the peaks or lines, indicatin~~ that the increased activity clurin magnetic storms
is much more random than the activity associated with the normal quiet-period
pulsations.
Summaries of spectra reported by Parker (1963j, Davidson (196=1~),
Hoffman and Horton 1.1966) and Herron (196r) are given in figure 102.
Each shows the linear increase in potiver density with increasinb period, though
levels of noise differ because of Che differing earth resistivity at the various
recording sites. If the ignore the specYt-al lines,. the noise density may be expressed as a function of frequency as:
DESIGN CRIT~P,IA FOR EL~CTKICAL SURVEYING
QUARTERLY OF THE COLORADO SCHOOL OF MINE5
Jan. 28: 19-}Jan. 29: 9
Jan. 30: 5
Jan. 31: 4
Feb. 1: 9—
Feb. 2: 4
Feb. 3: 4
activity, Kp2, for these days were:
FtcuaE 97. — Electric fields recorded at the Green Geophysical Observatory, during
a relatively quiet period frwn 17:02, Jan. 28 to 18:32 GM"1', I'eb. 3,
1967. The daily sums of the three-hour range indices of magnetic
Havinb specified the frequencies which must he used for direct-current of
zero-frequency behavior, and having, specified the noise density as a function
of frequency, if not of time, we are in a position to estimate the current required in making surveys over sedimentary basins of specified depth. In the
exploration of sedimentary basins, the lowermost layer is usually very much
more resistant than any of the overlying rocks. and can he considered to be
an insulator. When this is the case, the extreme ribht-hand portion of the
resistivity-spacing curve will approach asymptotically a line rising with a
slope of 1 on logarithmic coordinates, no matter which of the basic arrays is
used. The ratio of spacing to apparent resistivity for an}~ point along this
line will be some constant multiplied times S, the longitudinal conductance for
Current regzeire~nents for aero-frequ-e~acy methods
Eve, 1966.
The noise density map in figure 105 is representative of a disturbed magnetic interval on December 1-1,, 1966. The record from which this noise
density map was taken was characterized by the occurrence of numerous Pt
pulsations. As a consequence, the noise density map is less structured than
during the quiet period of December 24-25, and shows a more uniform noise
density as a function both of time and frequency.
I'or example, "runninb" spectra for electric .field data avera;ed over onehour time intervals are shown in finure 103 for the time interval from 0500
hours to 1700 hours, December 7, 1966. These spectra show the shift between
dominant frequencies from hour to hour which may take place.
An effective approach to description of noise density is a pz~esentation as a
contour map in which the independent variables are frequency and clock time.
Two examples of such noise density maps are shown in fi~,ures 104 and 105.
The map in figure 10<1• is representative of a quiet interval durinD the times
from noon on December 24•, 1966, to noon on December 25. The consistency
of the Pc-III pulsations is apparent from the smooth contours of noise
density at frequencies of about .00~ cps throubh the nibht hours of Christmas
23O
~
-
~`;~~
-~ —_
~ `~
~„'-'`
-~-~f -=
_
-
_
_
---
y ,~
_.__
_.
_
-
~~
~
-.
_~=
_,~ ,v~-_-"_-
(~4.2~
(4'i~3)
Jan. 1: 24~—
Jem. 2: 14—
Ficuee 98.— Electric fields recorded at the Green Geophysical Observatory durinnr
a period of muderute activity from 17:12 Gn'IT, Dec. 28, 1966 to 17:24
CHIT, Jan. 2, 1967. Tl~e daily sums of tl~e three-hour range indices of
magnetic activit}', Kph, for these days were:
De.c. 28: 22
Dec. 29: 15—
Dec. 30: 11—
Dec. 31: 6—
0.~ >~ C H
where p~ is the geometric average of the transverse and longitudinal resistivities. Therefore, the maximum spacinn required to probe Chrough a uniformly
anisotropic series of layers and establish the r•inht-hand asymptote for the
resistivit}~ sounding curve is:
~s _ ~~~
.I~4 = ~~t~ P)
where the value for the coilsCant, C depends only on the type of array used.
It has a value of unity for the Schlumberger array, a value of 2 In 2 for the
Wenner array and of l/2 for the polar dipole array.
If the rocks above the insulating; zone are perfectly uniform and isotropic,
the left-hand asymptote is a horizontal line with p;, = p,. "The intercept of the
two asymptotes defines the leasC electrode spacinb required to first detect the
presence of insulaCin~; basement rock, and so, is of considerable importance in
designing the requiremenCs for electrical resistivity surveys to see through a
column of sedimentary rocks.
The problem is not so simple if the conducting layers are anisotropic or have
variable resistivities. To a first approximation. a layered medium in which the
layers have different resistivities znay be treated as an anisotropic medium,
provided the basement resistivit}' is large compared to the layer resistivities.
Dakhnov 1.1960) 17as biven an analysis of the measurement of resistivity in
an anisotropic medium which is uniformly anisotropic—that is, the whole
medium can be characterized by a single value for the coefficient of anisotropy.
This mathematical treatment indicates that in the case of an anisotropic
medium restin~~ nn an insulator, the interpreted thickness of the conductive
beds will he too large by the zatio of anisotropy, and the interpreted resistivity
for the anisotropic medium will be:
U/p;, = CS
(441)
Quar•r~~LY or Txr CoLO~~DO ScxooL of Mines
the rocks above the insulator.
232
~
~,
,_
~G. r
~
-~/
Div---~-~~,~.~
~~
_
r^_ :.
~
~
1
...~
~
w ~-v ~
,
~,"'~,~~
~ _~
Qu~l~T~rLY or z~xr Coi,orano Scxooz or A~Il~vcs
R -~ -p0.
al Observatory durinn
Ficu~t~ 99. -- Electric fields recorded at the Green Geophysic
G~~1T, llec. 12 to
16:19
from
activity
ncra~e
than
a period of hither
range indices
17:08, llea 16, 1966. Tl~e daily sums of the three-hour
of magnetic activity, Kph, for these days were:
Deg 12: 3—
Dec. 13: 26
Dec. 14: 35
Deg 15: 25—
AFC. 16: 17
in
The mutual resistance is a function of both Che resistivity of the material
the
relawhich the array is located and the array geometry. Curves showing
are liven in
tionship between array beor~~etry, spacing; and mutal resistance
y likely
figure 106 for two extreme values of apparent resistivit
ohm100
and
er
ohm-met
1
on,
explorati
m
in
petroleu
to be encountered
the
for
that
ly
arbitrari
assumed
was
it
curves
meters. In calculating these
must
It
n.
separatio
dipole
Che
tenth
oneare
dipole arra~~s, the dipole lengths
each
be recobnized, however, that these s~acii:~s are not usually increased
the
when
less
is
e
resistanc
mutual
Chat
and
time the array spacing is increased,
curves
the
in~;
Consicler
spacing.
Che
to
dipole lengths are shorter in proportion
ohms must
in figure 106, it is apparent that mutual resistances as small as 10—~
ohms, if
10—~
as
lar~~e
as
and
be measured if the dipole arrays are to be used,
current electrodes:
array.
a lar~~e
Resistivity soundings made to spacin~~~s of tens of kilometers require
Surveys
tine.
the
present
at
y
done
commonl
quite
are
but
amount of effort,
reported in
made with array dimensions as large as 100 kilometers have been
(1966),
Keller
Ind
Anderson
,
1966)
Keller
f
by
crustal-scale soundings
,
Cant~~ell
by
and
,
(19G3)
'1VIa~we11
and
Ililathews
Jackson (1966), Watt,
Galbraith and Nelson (1964).
are best
The sensitivity requirements for azero-frequency resistivity sin~vey
and the
s
electrode
current
the
expressed in teams of rniitual resistance between
of the
ratio
the
as
defined
is
e
measurinb electrodes. The mutual resistanc
to the
supplied
current
the
to
voltabe detected aC the measuring electrodes
The anisotropy coefficient for a sedimentary column may range from l.2
for asandstone-shale sequence to l .5 for a section with many lirnestone layers,
and larger if the section contains a high fraction of ev~porites. In order to
penetrate a column of sediments 6 kilometers thick with an anisotropy° coefficient of 1.3, a maximum spacing; of nearly 16 kilometers would he required
with the Schlumber~;er array, or• of nearly 32 kilometers with the polar di~~ole
234
~,rr_~_~,a
~~
FIGURE
99V~.
;_,~..
—,
~_r`
'~-=~~ mac-=-~_--~~
~-:w L_~~_.~~
„'
_
~Vii
~"--tip
_
~,
~.~.~'
`"~
~
_.--
~~
_
/10
~p
r
100
P2fIOCj~ S2C0(iCjS
1000
n N/S
Chino Loke, Ca/if.
.~~
Calif.
E/IN
N/S
Bergen Pork, Colo.
-r
Quar~rrr,LY or Tt~zr Cor.or,_~vo ScxooL or iVIinrs
rc~lati~'ely ~~uiet period ~ Guru Pritchard, 7964).
l~ict:itr: 100. — _lmplitude ;pcctr,il density c~ui~es toi~ natural electric Fields a~c~ti~cd
over a common 2<L-lwur pe~riud at sc~eral recurdiu~ stations for a
0.001
O.I
10
236
Gmplitude Density,
my/km/cps
g
10
1
/
X1.1 ~ /r "1
i
M'
100
i
i
Perini, seconds
1000
Chi.~a Lake, Calif.
N/S component, E- field
V
/
/
Chi,~o Lake, Colif.
E/W component, E-field
/
Bergen Pork, Colo.
E/W compo~n; E-field
i
i
237
over a conunon 24-l~ow~ period at several rccorrling locations for a
n~a netically active period Ifi~oui Pritch~n~d. 196~1~1.
Ficutte lOL— Amplitude spectral density curves for nattu~al electric fields averaged
0.001
••
0.i
l
Amplitude Density,
my/km/cps
10
U~SIGN CRI'I'ERLA FOR ELI:CTP,IC9L ~LiP,VI;YING
a~
/~\7
I
100
-------- J
IO,OC~J
PERIOD, seconds per cycle
t
1000
-~
-
for mutual resistance
the Scl~lur7~ber,er array is to be used. "I~he upper limit
for ineasurrequirement
a
but
on,
pre~euts no special problems in instrumentati
or better
percent
few
a
of
in~ 10 milliri~icro~~olts with an~ absolute precision
challenge.
a
~~rovide
does
for each ampere oI current provided to the ground
magnetic vzn~i~itions from the
I~rcur~~; 102.--- "Cypical spectral density cttrvcs for
literature.
Curve l: IIeri'on, 1967.
Curve 2: Hoffman and Horton, 1966.
Curve 3: Da~~idsun, 1)69.
Curve 4~: S<u~tirocco and Parker, 1963.
10 T L
~p
102
104 ~--
.,
Qu_nr,TLi;LY or TI~iL CoLOrn~o ScxooL of l~Iin~~s
SPECTRAL CENSITY, `j2/cps
233
m
a
O.OI
0.03
FREQUENCY, cps
0.02
i
23)
clic~tecl by the curves in fi~~ure Fl.
c~~~~d~~
~~~e~ss~~
Achieving a sensitivity approaching 10-8 colts per ampere requires
the use of as lane currents as can be obtained. and advanced sianalprocessin,~ procedures to lower the detectable voltage level to as low a value
as possible. The minimum detectable voltage level is determined primarily b}~
natural noise in the frequency ran~.~~e at which measurements must be made,
and is itself a function of the frequency. The frequency musC he made su{~iciently low Chat the resistivity measured ie strictly astatic-fielc1 value, as in-
P~icutt~ 103.-- Running spech~al density curves for electric field observed at the
Green Ceupfiysical Observatory, I3ernen Park, Colorado, for the
interval 0600-1300 VIS'T, Dece~uber 7,~ 1966. Eacf~ spectral density
curve: rcpresent~ one hour of recor~lin~.
0
D~slc~ C~zrr~ria roe L~LC°rricaL Surv~Yln~c
io
,5000
500
t
N
8
~
O
~
~
O
o
DECEMBER 25,
O
p
N
N
i
i
i
0 o g oc0
I
~
~
/pp
~
—
O
~
5~
g~, o
~~
`
/00
io
place of the resistivity for a uniform medium.
for these
The period correspondin to the maximum permissible frequency
three working models is shown as a function of dipole s~acinn by the curves
While the estimate of frequency obtained from these curves is strictly valid
set of
only for a uniform earth, it is a reasonable approximation. A simple
A~~parent
107.
figure
working models for sedimentary basins is illustrated in
to
resistivity curves for three models are shown; each model is assumed
values
resistivity
represent a sedimentary thickness of 10 kilometers. Three
corare considered; 1ohm-meter, 10 ohm-meters, and 100 ohm-meters, with
103
mhos,
respondin~ values for the longitudinal conductance being 10~ mhos,
the
and 10`-' mhos. The basement is assumed to he a perfect insulator, so that
rise
to
assumed
right-hand asymptote for t ie apparent resistivity curve can be
he
with a slope of ~-l. An approximation to the highest frequency which may
in
used can he obtained by using the apparent resistivity from such a model
noon, December 25, 1966. SE~ectra are computed for one hour intervals
(from Welch, 1968).
FtcottF 104. — Contour map of spectral density for electric field intensity recorded
at the Green Geophysical Observatory between noon, December 24 and
24
~
c9
~Y
~
DECEMBER
I
I
~ ~
I
/00
0.0011
0.002
/000
~5~
~,~ -50 ----'
~~
~o
QUARTERLY OF THE ~.'OLOh'.ADO .SCHOOL OF .MINES
/~/
0.004~
_/
icn
!oo
0.003
0.005 ~--
0.006
0.01 — ~
0.008
0.007
FREQUENCY, cps
`Z~~
O
50
/O
1000
SGI~
500
O
O
O
O
O
~
—
N
~
~
~
o 0
500
ioo
50
~o
io
DECEMBER 14, 196G
~
o~ 0 8 0 0 0 0 o S
/00
ioo
5~ —~—
io
resisti~-its° of rune o}gym-tnetF°r.
l;valuation ~~f the none detected at the receiver i~ must simple if the output
of the clei~ector I the electrode pair in this cire J is first filtered sharpl}~, as is
advantageous ~~~hert measueeil~ents are Ix:in~~ ~nad~_~ ~t ~in,~le frequencies. With
narro~e-band filter, ~~hich i~ one ~+ith ti bni~d~~~idtl~, :~f, conciderabl~~ less
than the mid-frequent}~, f,,, the encrr~~ which persists after filterin~~ is ~~er~
nearly th~~ ~n-~xlu~~t of han~l~~iclth anal the none clen~it}~ al the ft~equenc}' f .
in figm-e 10~,. "I~he curves nip, straight lines iisin,~~ ~ti~ith a slope ~2 for spacings over ~~ hich the apf~arc i~t ie~is[ivity is constant, and change to strai~l~t lines
rising with a slope ~>f ~1 at spncin~;s over a~liich the apparent i•e~istivit} i~lc~a~eas~s linearl~~ ~~~ith ~pacin~. A~ one mi~~l~t expect; lon~~~er periods are require<l at the lar,er dipole ~c~l>ara~ion~ and ~~~ith the lover resistivity models.
If an upper practical limit of une hour is Enlaced on the wave period which is
to lie used. a ~e~>aratiou ~~f ~10 kilometers may he obtained in a medium with a
I`tct iti, 105. - - Cnn~uur snap i~~ .:pe~ctral ~lcti~it~~ fur ~•I~•ctric fielel int~~n~ity recorded
ut tl~c Green Geup6}sicul Obse~n~a~ur~- h~~twecn 0 00 and 1E00. CIS"I~,
1)ecenil~er 14. 1966. S~u~riru ar~~ ~v~ni~~uti~d Gar ~~ne h~~ur intervals (from
W cl~•h. 19fift ~ .
O.OI
I
0.02 f--
0.03
•••
FREQUENCY, cps
0.05 r
~)I~SIG_1 (~.RITI:IiIA l~ OI: ~LI~:CT1iICAI, .~liIiVIiYINGZ!j.]
p~F
S
io9
10
\~
~
\
10
\
IO 4
10 3
102
\\
103
\
~
~
j
m
W~
104
Q
o
°
J
a
W
~
ao
\
ohm-meter
100
100
00 ohm-m
10'
SPACING, meters
~
Y}1C 8~)dCIR~".
measin'ed in investigating
I~tccr~: l0fi. — ~'Iutual re~i<t~uuc ~alucs ~~-hich must Ise
ohm meters.
sequen~c~ of iurl.s ~vitli ~~eistnitir~ in tht ran c 1 to 100
while the
Uipul~ Icn!!th_ arc a uiued to he one tenth the zpa~ing,
one -fifth
be
to
soloed
i
is
iy
ui
cr
Scl~lumb~i
tl~t
fur'
HIV cEiu ttion
~
Q
~--
CI)
~
a
W
v
z
0
a.
~.
10 ~
I00
1000
SPACI NG, meters
IO,aZ7fl'
24
~ 11.;
a F2 — L F
a
p Z ~,.c,,c z
1 1(o}
f~
~•~2
~Che signal power for a transinittecl sinusoid depends c>n tl~ie number of
cycles transmitted. Accorc~in to F~harkevich 11960), the running spectral
density foi• n cycles of a sinusoidal signal is:
a
E2 ~ OF a~2
F
1''u,t ~tF: 10i.-- Idealized zipparent resistivity sounding curves which would be obtained user ei sedimenteiry sequc:~ice 10 kilometers thick ~estin~ ou insulati~~; ba~~~m~~nt, with asi~mrd r~-~sistivitic~s of 1, 10 and 100 olimnieters.
••
APF'Af~ENT f~ESISTIVITY,
Ohm-meters
1000
D~srcV CriTExzn roe k:Lrc•rl,zcn~ SL~avrYZnc
—
_
_
~/
/
Equatorial array
Polar array
Schlumberger array
~
/
//
~/
/j
//
~
/
///
ICz
SPACING, meters
1
~
_ 10
—=i
~
~
~
~ 103
~
~~4
PERIOC, sec.
f~ =10~ ohm n
///
~
~~
/
//
~//
/
—
//~ = I ohm-m
//
~
/
//
— —
— — — — —
— __T-- __..._._-_--__
MINBS
QU9P.TPFLY OF THE COLORADO SCHOOL OI'
aaF ~ A2p~~4F2
,;1~,;~
shown in figure 109. It should
A three-dimensional sketch of this spectrum is
frequency of the transmitted
be noted that the height of the spectrum at the
for an infinite number
signal groti~s with the number of cycles. Ultitnitely,
large at the frequency f,,.
of cycles, the spectral density becomes infinitely
density is related to the
At the frequency, f,,, the amplitude of the spectral
number of cycles as:
z
for• direct-current soundings for tl~e
Frcu~te 108.--- I'requcneies which are required
107.
figure
in
viven
model cases
102
10 3
10
_4
FREQUENCY
Z~,
a
fo
Frequency
Q`
~~o
0
245
(-1 ~~3 i
n — OF
~o
If a nreater number of cycles is transmitCecl_ the width of the spectral peak
diminishes as tl~ie height increases, and the amount of power transferred
through the filter remains nearly constant, for any greater number of c~'cles.
In view of the low frequencies required in sleep electrical prospectin;~,
efficient field operations require that as fe~v cycles as possible be used. Let us
assume that i~he number of cycles of si,>~nal transmitted will be the inaerse of
the band~~>idth-frequenc~~ ratio:
P ~ ~P
where p is the numberi of. cycles transmitted, and A is the amplitude of the
sibnal at the receiver cpmputed foi• the noise-free condition, before filtering.
If the bandwidth ~f the fi1Cer is less than the width of the spectral peak associated with tl~e transmitted signal, the noise po«er transfel•red through tl~e
filter is merely the pz~ocluct of the spectral densiC} <~iven b} equation ~-1-~ r and
the filter bandwidth. ~Chis condition ~vil1 halt] so l~n ~ as Che number of c~~cles
transmitted is less than the ratio o~f the center• frequency to Che band~~-idth:
Ficta~tti 109. — Running ~:pectr~i fur p cycles of a sinusoid ifrom Kharkevicli, 1960)
Power
DLSZC~ C~iT~:~zrA nor ELrcTRrcaL Su~zv~Yln~c
QL'ART~RLY OF THL COLORADO SCHOOL OF ~IIN~S
Pn
ps
F°
qk Fo~ ~ OF
ES
(1.5p)
'149)
(4511
2
~2~
~° ~y
~° ~G-7.0
(452)
of the
The quantities w°hich are contained in the first term to the ri,hY
for
0.002
and
array
equatorial
For
the
0.00~1~
of
order
of
the
are
equalit}' sign
Park
Bergen
the
surveys
at
density
power
noise
on
the polar array, based
Observatorj- of the Colorado School of l7ines.
Assuming; a desired signal/noise ratio of 100 (20 decibelsl acid a bandfor
width-frequency ratio of 01, the currents required for a liven spacing, a,
110.
fi~~ure
in
graphically
shown
t~esi~tivities of 1 and 100 ohm-meters are
H/lVI and Q is a quality factor; deCermined by the precision with tivhich the
direct-current behavior must be approximated. 1or an approach within 5
percent of the static behavior, Q should be approximately 1. Combining
equations -1~~1 and =~~2 with equation X50, we have:
X 10—~
where f„ is the operating frequenc}' aC maximum spacing, µ„ is ~7r
~~
Q
The highest frequency ~shich may he used is also the most efficient one.
At the maximum required spacing, Che frequency required to penetrate a
section with unifori2~ resistivity, p> is
arrangement.
depending
where a is the spacing, between dipole centers, and c is a parameter
on the arz~ay used, being 1 for the polar arrangement and 2 for the equatorial
I
ES = io~2
The signal level, E„ can he evaluated from the appropriate expression for
apparent resistivity for the electrode array beinb used. For the equatorial and
polar dipole arrays, assuming the source dipole length is one-tenth the spaoina between dipole cenCers, the signal level is:
The signal-to-noise ratio is:
4~o
PS - S
_ EZ
sinusoid
The amplitude of the peak in the signal spectrum for a terminated
is:
filtering
after
cycles
n
for
meter
with an amplitude E, volts per
Zcj,(
~, ~
0.I`
__~
--
_ Polar
-.
'
100
-- -----
___--
— -
~
~
-~
I OOC
,10,EX00
Spocing, mete~s~
--1--
_._
Let us use this as the spacing for ~vhic}i the porgy-er requirements for various
receivei••source. co~nbinationc are to be com~~ared.
(~~I}
With AC meChod~, the ~i,~~~al po~~-er incident to a filter at the receiver is 1
function of the source ~trengtn, the distance from the source to the receiver.
and in some case, the frequency and the resi~tiviCy of the earth. "Che source
strength is a ~~x~cified parameter. which can be chan,ed almost at will to obtain
aciclitional se~~~iti~ its. and so_ i~ uoC of primary concern iii considering design
criCeiia in tf ims of earth piopertiES.
1 or simplicity iii com~>it~c~>~~ of various source-receiver combinations, let
us consider onle x uniform earth. We have no real rec~uire~~iei~t on the separation between source and re~cciver_ c~LCept that iinpo4ecl by the need to have a
spacing comparable to a ~~~ave len~>~tl~
in the
Q~
~ earth:
Current regt~ireiraenGs in AC nzetlzods
Currents in excess of L000 amperes may he required for spacings of tens of
kilometers. Cur•i•ent requirements can be reduced in several ways: the length
of the ~oui•ce dipole may he increased with a proportionate decrease in the
required current, or the bandwidth-frequenc~~ ratio may be reduced, with a
reduction in required current i~l proportion to tl~e square root of the reduction
in fractional bandwidth. These factors. combined with operation during
quieC periods at large spacin s can reduce the required current to several
handreds of amperes, which is well within i•eaaon. Jackson 1 1966) reports the
use of currents up to 300 amperes in crustal resistivity surveys, where all the
power ~~~as provided by batteries.
Fictntr 110. — Current reyuircd to ~~ruvide x 20-dccil~cl signal to noise }ia~ through
a (iltcr with a relzlti~-~~ heu~dwith of 01, when t}u direct~m~renC method
is considered.
0.01 '
10
-
247
Equatoriai
~ ~'
-- —0
-"
ohm - meters
--=t_
_
~
~_
f ~ ~-
~ — --~- ~
10
ioo
Required current, amperes
Drsrc~ C>>rTi;F;r:~ Fors I:i.rcTr~ical, Survrl~in~c
QuAi~T~r~LY o~ Txr CoLOFa~o Scxoor. o~ ~IinFs
Q12nQ5M
(4551
6~fi6~o~~~/~~
-9t At
l'}cJ7~
z
(fol2
81Ar ~7/-~oo"~t~'l2
(~~~9}
(4,58)
(~~61)
hhere AS is the effective area of the source coil.
the conductivity, reflecting; the
The current required depends very ,neatly on
of 2 PS
2 _ ~z8 Try K$Z
I - 8~ a;o~~o a-' ~ Fo ~ ~ ~N~
ps
PN
(x,60)
81 ~k'a'~ Mz ~~~~z~
128 n K~
operate at the Lower limit of the
Or, converselj, the current required to
asymptotic frequency ranee is:
m~~;netic induction noise at
where K1~, is the spectral density of thc: vertical_
ehe frequency under consideration.
output of the filter is:
The ratio of signal power to noise power at the
°~~
9~n'~'F~'Ar Kam(
Pte- p~
0 aF~uN~ =
The noise density which penetrates the filter will be:
OF 2
filter, we have:
product of area and the numwhere A,. is the effective area of the receiver, the
ber of turns.
of signal transmitted is
If ti=e take as a standard that the number of cycles
the filter, then usinb
of
ratio
the inverse of the bandwidCh-center frequency
through the
transferred
equation 4-17 to determine the si~~nal po~~rer density
V~HZ~p,m~nl-
with an induction coil, the
If the magnetic field at Che receiver is detected
be:
will
~rolta~;e output of the induction coil
for which equation 455 is valid, we have:
the least spacing
Using condition -154- to find the ma~~neCic field componenC at
HZ
strength from a
Vertical magnetic cli~ole soicrce: Consider first the signal
component at
field
magnetic
vertical-axis ma~~netic dipole source. The vertical
a receiver tivill be:
Zq.g
1
2-=1.9
(4~2)
g(T3
~'~G3~
vE~,~w~in~
/
CC~Chin ` '1>°`m~b
12~ tt-s
~F
C{ C2M2 F ~f.Co
~ ~ ~~
(~1~65)
~~'~~~)
256 try KE
~~Pl
(467)
(-1.6 1
Considering; fl~~t the source tT~omenl is still IA„ the required current is:
Pri
PS _ gMza-3F2,u~
Fo \2
E ~ of ~
aF ~VK ~ - pµ~
The rati~~ of si~nal power lc> >ioi~~ pc~~ver is:
p~ _ ~F
"Che voltage detecCed with a pair of elech•odes is proportional to the spacing
bet~~veeii Che electrodes, and to the square root of the resistivity of the earth.
T~~~refore. the noise power which penetrates anarrow-band filter will b~:
~5
(2 il'~s/2
3C M ~~'~~a/z~'/2
~l'he signal E,x~wer density which passes the filter is:
~/
The electric field is usually detecCecf by i~~ea~urin,~ the voltage drop between
a pair of electrodes ~~ho~e separation is a small fraction of the spacing, the
Fraction being; G:
E7~amin
_ 3M~Koo'
Using condition -1~5-1~ to find the electric field e•om~onene at the least spacin for
which equation -162 is valid. tine ha~~e:
E = 3M
`~ 2tT0.4~..
Let u5 nest con~idei- the tan~entia) electric fielc{ from a vertica~~xis mabnetic dif>o1e source s~~ that we ma}' compare the effectiveness of the two types
of re~ceiaer for a c~~rnin~~n sr~urce. The signal strennth is:
on spacin~~.
fact tk~at as the conductivity varies, the spacin~~ varies and the amplitude of
the si~;na1 varies 1s the inverse fifth po~~er of the spacinn. A slight decrease in
concluciivity. which causes a slight increase in the required ~pacin~;, will require alarge inca•ease in ehe ieguired current. "Chic is offset in part by the
fact that the requir~ement~ on frequency, f,,, are inverse to the requirements
D~:slc~ CriT~ri_~ ror. Er.~~;cT~;ic:~~ SLrvrn~~c
Iz _
gAsZ63 F2 Ko
256 n6 K~
(~F z
\ ~o ~ PN
1-1.68)
QuAi~TLr,LY or Tx~ CoLOF~a~o Scxooz of ~Vlrrr~s
E
(K0z 1%z
3Fo Q"z ~2~No~/2 \
1
~~
TY~c3
_ r~
(=170)
E~ —
~----~~~~j~
spacing
Using condiCion ~5=-l~ to find the electric field coz~~~~onent at the least
for Svhich equation 4~~0 is valid, we base:
/z
_ I cQn F3~,uo26-~
(~~7~)
moment Ids is:
of 0.1
As an example, consider a case in which the earth has a conductivity
the
for
61,
figure
to
Referring
Colorado.
mhos per meter. as in eastern
100
he
would
f„
frequency
the
behavior,
minimum conditions for far-field
would be:
cps fora 1220 meter spacing. The raCio of currents in this example
(l~C RZ\%z
lO ~
KE
noise for vertical
The ratio of currents rec~uirecl depends on the statistics of the
on
induction receivers and E-fie13 receivers. Little information is available
probably
is
/K~,
such statistics_ but at Ber~~en Park, Colorado, the ratio KF3
efficient
of the order of 10—~}. In this case, the E-field detector is much more
lower
much
he
could
ratio
noise
the
However,
receiver.
coil
than the induction
infrequency
transition
the
or
as
increases
conductivity
in other areas. As
with
comparison
in
decreases
detector
E-field
ereases, the advantage of the
at short
the induction coil receiver. "I'Ile induction receiver will be preferable
of
spa^ing
a
bein
transition
the
wiCh
earth,
spacinbs over a highly conductive
i~leter.
jeer
mho
1
of
conductivity
with
a
a fe~v hundred meters over an earth
electric
We might also consider the relative merits of a coil source and an
array
dipole
equltorial
au
~~ith
seen
dipole source. Consider the electric field
displaced
receiver
dipole
(an elecCric dipole source ~~ith a parallel electric
along the equatorial axis of the source).
ti~ith a
The signal str•en~th at a distance a fror~i a current-dipole source
Iel~ctric field de-tec-for
Tver~~-t axis ind~tioR coy l
taken:
induction and EWe are now in a position to compare the relative merits of
current required
the
field receivers for a mabnetic dipole source. The ratio of
an electric field
with
for an induction coil receiver to the current required
the square root
with
detector is the ratio of the expressions in ~~61 and ~•6S
250
251
2mz
CF ko Ic~
(4727
~o1
C~tlo2 ~Ic~n~z
C ~F J
i6 t7'~
(=1.73
\~Fl
~,Uo 6`2C~li)z \~~ )z
s
32tY5KE
N
Ps
~oµo~2CI~)~'( fo 1z
~ 1X55 ~
j1.7(j
SouKC
gn
~ga'~'~t,(o
_ /
%z
Ag
ds
(
'
lib
Y1C~5~'
2fl~Y~ C.
_ (2~n-Poµo~)~z
2
I
z — 3n C
ftCC,~min
I
(.4.7d j
l`1((~
where n is [he number of turns in the source coil. IC is interesting to note that
the electrical properties of the earth do nut enter, and that amulti-turn coil is
always preferable [o an electric dipole source. "I~his result is deceptive; hoti~ever,
because the amount of current which can be driven into the ~;rounci depends to
a very ~~.reat de,ree on the resistivity of that around, particul~i•ly near the surface ~~-here the electrode c~nt~icts must be made. Although it~ wi11 not he shown
here, it is reasonable to expect that th<~~ coil source wi11 be preferable Co the
current dipole source in regions where the sui•ficial resistance is high.
and equation -176 becomes:
~5
ds _ IQs
However, consider that the maximum dimension of a coil source is the same
as the len,~~th of an electric dipole source. limited by the size Ca,,,;,,. Then:
Iel~ctric d~~o1~ so~vice
I verti~q~ cAxis m~icJnet~c
source to that rec~uirec3 tivit}i an electric ~~~uree is:
1~ow, let us compare a ina<~~netic source and an electric source, usin an
electric receiver ~tiith both. ~'1~he .ratio of current r~c~uired with a mannetic
I2
Piv
and the current required is:
p
The noise dower is the same as that liven by equation -166 for~an E-field
recei~rer from a mailetic source. 1~he ratio of ~i~;nal power to noise ~~o~ver is:
PS —
The signal }~o~~er i~-hich pastes through the filter. from equation ~1=1<<, is:
VS =
1f the ltn~;th of the receiver electrode ~E~read is Ca,,,,,,, the voltage sensed at
Che rec~ieEC is:
I)rsi~n~ C~~rri:rra r•or, I?Lf~:c.Trzc,~i. St~i~v;~ri~c
QUARTERLY OF TIII: COLORADO SCI~OOL OF ~'TIN~S
G~:ozocic NoisE a~~ P~ECZSiov or l~Irnsurrvi~~rTs
surveying
Other factors to be considered in desining a galvanic resistivity
Usually we may
program are those iirvolved in the precision of measurement.
of accuracies of
assume that instrumental errors will be insignificant, in terms
being measured
a few percent. However, even with the observed quantities
considerable
show
will
values
resistivity
of
plot
a
quite precisely, commonly
crustal-scale
a
of
one
is
'The
example
lll.
figure
scatter, as in the example in
quite well.
nozse
~eolo~~ic
of
problem
the
resistivity sounding, but it illustrates
imensmall-d
by
caused
resistivity
Geologic noise is the variation in apparent
elecvarious
the
near
and
Dyer
surface
sioned variations in resistivity in the
sounddistinctive
to
lead
variations
trode locations. Large dimensioned lateral
theoretical curves which
in~-curve character and inay~ be interpreted using=
Keller and T'risch1957;
have been given in the literature (see Kalenov,
We shall not
1966j.
Cook.
knecht, 1966; Al'pin, 1966; Van Nostrand and
sedimentary
in
soundings
be concerned with these in a discussion of electrical
if major
However,
basins, where usually, lateral uniformity can he assumed.
available.
lateral changes are to be located, interpretation methods are
can be quite
Small-dimensioned resistivity variations in the surface layer
discussion of the ef~'eet
arbitrary in shape, and thus, a detailed, quantitative
our purposes we
they have on observed resistivity is tedious. However, for
a consideration of the
need only asemi-quantitative discussion, and for this,
resistivity is useful. If
effect that hemispherical pods znay have on measured
array', the behavior
dipole
a
in
spacin
the
with
the pods are small compared
pod will be apthe
of
vicinity
the
in
dipole
source
of Che current from the
been computed
have
curves
Many
sheet.
proximately that of a planar current
structures
resistivity
various
by
sheets
for the distortion of planar current
i~iduc3ing
1965),
r,
Berdichevski}
1956;
(see Kunetz and Chastenet de Gery,
subsurface
a
on
resting
layer
surface
the hemispherical pod embedded in a
field curves over relayer restinb with difFerent resistivity. Typical electric
112. The important
figure
in
sistane pods and conductive pods are shown
pod is never
resisCive
a
inside
feature of these curves is that the electric field
if the pod
observed
be
more than about twice the electric field which ~>>ould
by
decreased
is
pod
were not there, while the electric field inside a conductive
expensive, time or source power?
ineasurina reCorrelation filter techniques appear to provide a means for
of measureexpense
the
at
but
,
requirements
sistivity with relatively low power
for obtainmeads
a
provide
to
appear
tnent time. Pulse filter techniques also
required
being;
sources
power
ing measurements, but probably with higherapproaches
t~vo
the
between
than for correlation filter methods. The choice
which is more
must be based on economic considerations for a given problem:
252
a
x
x
i
~o
o0
I
_
—
-
o
10
x
o I
~
r
_ ..~
-~—~.
°
00
°
o
Resistivity
25.3
o
100
I0
100
the ratio of the resistivity contrast between the pod and Che surrounding
medium. 'The efTect is diminished by about 1.0 ~~ercent if the resistivity decreases with depth, and is enhanced by about 10 percent if the resistivity increases with depth below the surface layer.
These results pertain only to hemispherical pods, and the results znay be
expected to he much more com~~lic~ted for less symmetrical pods. However,
it is a meaningful ~eneralil~ to say that conductive iudusions in a surface layer
will lead to ~i•eater amplitude geologic noise than wi11 resistive inclusions. If
we were to assume a uniform distribution density of pods with resistivity both
treater than a~~d less than the gross resistivity of a surface layer, the errors
in observed electric field intensities caused by these pods would be skewed
towards the low side as indicated in figure 113a. The probability density for
errors on the low side will he the same as the probability density for the occurrence of pods with low resistivity, while the density of errors on the hibh
side will show a peak for err~t• ratios r~~f about 2. Since we have assumed an
I~trurtF: 111.-- ~xa~uple of typical scatter of llC resistivity determinations caused by
lateral inhomo,eneities in resistivity. 'The surveys were made using two
source dipole lucations, with electric field measurements about etch
being difFerentiated by the symbol used in plotting the apparent resistivity value. The survegs were made on the Columbia River Plateau
i~t the state of Washington.
~
1000
r-+ 10 000 ohm-m
o o ~ ~,
_1 }
-~
--+
~-
oo p
o
t`
o
_
- .T ~
~
x~x —
~
X x
~;_ _-
--
__.
-
-
-
Spacing - kilometers
x
_
—
o00x
0. I
x
0.01
x
_.
_
-- _____
° o0 oX
x
-
-- _—
_— —..
~ ~
__
'~
DESIGN CPIT7?RI9 FOR ELECTRICAL SURVEYING
OBSERVED ELECTRIC FIELD
FIELD WITN NO POD
OL OF MINAS
QUARTERLY OF THL COLORADO SCHO
large, higher than the
If the resistivity of the surface layer is relatively
y curves, there is a
median resistivity indicated in these distribution densit
tive than the layer than
greater probability that pods will be more conduc
the avera;e error further
shifts
turn
in
This
that they will be more resistive.
of pod resistivities above
from zero than in the case of an equal distribution
of the surface layer is
vity
resisti
the
If
and below the layer resistivity.
ted in these distribution
indica
vity
resisti
n
relatively low, lower than the media
will be more resistive
hods
that
density curves, there is a greater probability
I, Section 2).
pods; the errors in the
equal probability for resistive pods as for conductive
in the high direction.
low direction will be much more important than errors
rm distribution of
unifo
a
be
to
there
In the beneral case we do not expect
in a surface layer
vities
resisti
of
ence
pod resistivities. Some idea of the occurr
determined from
vities
resisti
earth
may be obtained from the compilation of
stations (Part
ast
broadc
rd
radio•tivave intensity measurements about standa
by a hemispherical pod at the surFtcoae 112. — Anomalies in electric field caused
z~ planar current sheet.
of
case
the
for
d
earth,
layere
of
a
face
2S'~~
pod=~'lay~r
°pod>ployer
ed cicch~ic field intensity wliicli ~ni~i~t
1`re[iaE 11'3.— Distributiun of errors iii measm~
vities embedded in a uniform
be caused by pods of different iE.sisti
surface layer.
.
B. Distribution of errors caused by pods
Observed ~
p of medium
A. Uniform distribution of pod resistivities.
pod <J°,byer
QUAKTERLY OI' THE COLORADO SCHOOL OF ~~IINES
than the layer, than that they will he more conductive than the layer. This
shifts the: average error towards zero, or in extreme cases, slinhtly above zero.
This non-uniform occurrence of errors is demonstrated by the hypothetical
probability curves in figure 11 ~. Although this discussion is not quanCitative,
two things should be noted:
1. Geologic noise does not necessarily contribute random errors in resistivity, so siinj~le averaging will not always reduce errors. However, if the
surface layer is relatively conductive, aeoloaic noise is more likely to have
zero average than if the surface layer is relatively resistant.
2. Geologic noise should pzovide loti~er variance (averane of squared
errors) when the surface layer resistivity is low than when it is high.
Since the surface layer covering a sedimentary basin is usually less resistive
than the average surface layer, it appeals that electrical exploration in such
uasins is favored over exploration in other areas. In fact, from the noise point
of view, it appears that the optimum surface layer for electrical exploration
would be a thin sheet of sea water. Because of its uniformity in electrical
character. the scatter to observed resistivities would be virtually zero.
The variance contributed by geologic none is of concern in specifying the
number of resistivity observations which must he made to determine the shape
of a resistivity sounding curve wiChin a required precision. If the variance is
zero, m• at least less than the precision required for a resistivity sounclin~
curve, the number of points needed to specify a sounding curve may be estimated using; the Nyquist sampling theorem. The sharpese curvature and thus
the l~i~;hest required sampling; density, is observed for Che case of a thin resistant layer embedded in an otherwise homogeneous medium (Keller, 1966).
The soundino~ curves for the various arra}'s which would be obtained for such
a sequence of layers are shown in figure ll5. The horizontal scale shown in
fia~ure 115 as log a/hi is equated with a time scale in sampling theory. while
the vertical scale shown in figure 115 as lob; p;,/p,,,,,;,, is equated with the amplitude scale in samplinb theory. In this manner, resistivity sounding cm~ves
be
may be thought of as time funcCions. The power density spectrum may
obtained from the Fourier transform of the equivalent time function. The
power density spectra for the two curves in figure 115 are shown in figure
cycles per
ll6. Most of the power density is found at frequencies under O.o
at least
have
must
we
theorem,
sampling
Nyqui~t
the
decade. According Co
in
a refrequencies
these
of
presence
the
deCerznine
ttivo eamples per cycle to
of
rates
sampling
to
corresponds
requirement
This
sistivity sounding curve.
~t least 1-1/2 points per decade.
In Clue absence of noise, a single sample can be assumed to be a sinble
measurertient. However, if there are random measurement errors such that a
standard
group of iz~easurements within a '~yquist sampling interval has a
25E1
`L57
deviation (-square root of variance I of p percent, and if we wish to have our
sample accurate within q perceuC 1 assumed to be smaller than p percent]
their, for a normal distribution of en-ors with zero averabe, the number of
measurements rec~uii-ed per sample is I p/cj1 ~ -I- 1. 7`hus, if the standard
nu~re resistive.
F'~;~ rr: 11'1-. — Di~h~ibutions u[ errors in clect'ric field intensity v,~hich mi~~ht be caused
by a disiribiuiun of pods ~vit}i dit~erent resistivities in ~ surface la}~er
which is e~ith~~r more c„nductivc than the acera~c pod resistivity or
C. Error distribution for conductive surface Iayer.
Average resistivity of surface /oyer
Averoge error
B. Error distribution for resistive surfoce layer.
Averoge resistivity
of surface layer
Average error
A. Assumed resistivity in surface layer.
DI'sSIGN CPIT~P,IA FOR ELFCTFICAL SUPV~YIATG
__.
~~
~
Equalarial dipo/e and
Schlumberge~ arrays
„~
Polar dip~/e array
~/
//
i
~~
a /amax
1
\
`
NIr~Fs
QuArtT~r,LY or ~riir~. CoLO~~~»o Scxool oF~
i(
0
~ r,
5
:;~
Frequency, cycles per decade
i~
~~
curves for thin
sPecti'a" c~ilculated for resistivity
I~tcinzi: 116.-- "Pu~~er dcneit)'
earth.
uniform
otl~er~visc
an
in
r~~si,tant 1<ivers emheddcd
0
2
curvatw'e, and
curees which exi~ibit maximum
Fzct?itr: 115.-- Direct-current iesi~tivity
curves have
"I~he
arrays.
~evcral
therefore, hi h "frequency,, content, for
ap~~ai'ent resi.tivity and the
observed
maximum
the
to
d
been normalize
recorded.
spacing at which that mttxintum is
••
•~•
Pa/pma
ZJii
25~)
~
It is perhaps important to point nut here that the scatter caused 1~~~ random
inhozno~;eneiCies of the surface layer is most important in the af~plicatio~~s of
determined.
where E; is the error in interpretation and ~' is the number of ~aroples per
decade range in spacing, or the ranee over which the shale of the anomalF is
/z '/z
~- _ ~~ ~~N-1'
The precision with which a contrast in transverse resistance <~r a contrast
in lun,~itudinal conductance can be determined is:
anomal~~ on the soundiu~,~ curve is the same as the contrast in tt~e la>~er parametei~~. If the conLra~t in laver parameters i~ 10 E~er~rent the an~~~litude ~>f Lhe
an~»naly~ on the sounding; curve is about 5 percent Sli,htl~ lamer anomalies
are obtained with the }polar dipole gray than with the Schlumber~;er or
equatr~rial dipole arra}~~.
In ~~i~~~c of Lhe naturr~ of a ~~ ~~ical oil field a~ ~~i~~e°n itt Part 1. ice should
cuu~i<3i~r the resuluti~in kith ~chicli a thin (aver can be detected. Lack of
thxkne~s i~ u(Iset 1~~~ the I~r~;er contrast in ~rFSistitiit~, if a thin la~~er is
c~~t~bcdded in a hu~uureneuus uieclium. thf ~e~i~ti~~it~~ cure shuns a bum~> or a
deprc~ssi~~n_ as indicated in fi ore l l "r. The amplitude of the )pump i~ a function only ~~1~ they contrast in trap ~~erse iE.~i~tance of the ]a}~er to that of the
o~-erl~~in~ me~diuin 17=,j'I~i I if [he la}~er is ii~orE ie~istant than the rest of the
mediwn, ur ~~nlv of the cu~~trast in lun~~itudinal conductance of the Iti~er to
that of the u~-erlvin~> nieclium, if the lavci is less resistant than the rest of the
medium. F~~r small contrasCs in 'i~.~j~l'~ or S../S,. the amplitude of the
1(1 ~ierc~~nt.
~~ related cunsi~leratiun i~ the mattr~r of resululiuu—ho~~- s~»~ill a than e
in ir~i~ti~it~~ ~~r hu~~~ thin a la~cr can be rr~ol~ed al depth. It see~tr rea~onal~le
t~~ r~°quire ~am~>lin~ t~~ pro~~ide a precision of ttico f,>ercent on the a~~E>ra~e.
~Iu<h higher preci~iun tntrht he i~btained in ma~me exploration ~~~rha~~ ~~ith
ayes t,e ~~irur.~ ~~f thr~ ~~rd<~r ~>f O.Z ~~ercent. EI~~~~f~~e~r, if the ar~fc i ~e. ~amplin;
error i~ two percent. ~cr may e~~~~ect to obtain rcrlial~le~ interE~retation; f~>i la}~er~
ti~~hich cuntrihute~ a ~ h~in~~~ in ~~l~ rr~~E~cl rf~;i~ii~ it~~ ~~f fi~~e dines a~ much. or of
de~~iation of mersuren~euts i~ 1O ~~ercent. and the desired precisio~~ for etich
\~~yuist ~<unple i:~ Z percent. 2(~ mcasure~>>ent. must 1>e ~~~ade in each sam~~lin
inter~~al. S~~n~e caution in thy° use ~~f' uch e~limattn mint be e~er~~i~e~d wile;
unt~ is quite ~ur~~ thtit th~~ ~~rrur~ ~~~in he ~i~~umetl to lur nurm~ll~~ cli~trihute<I
kith ce~r~~ i~~era~e 'I~hi~ ~~ a iu~ur~ rci~~>nal~le as~utn~~tiou for es~~loration
peu}~lein~ in t.hich thesu~fi~~r~ la~ei i~ t-elati~E~l~~ cuixlucti~e tha❑ iii exE~l~~ratiun
~~ruhlern.~ its ~chich the ~~irfacr~ layer i~ rel~tti~~~l~~ re~is[i~ e.
Ui~;stc~ (:rrri•:rl,~ t or I~:i.i~:crni<:.~~. Soi:v~~:l~i~~c
pa~Ps
QL1AFiT~RLY OF TIIF, COLOPADO SCHOOL OF NIINI:S
andLa~armistr, A. M., 1966,
APpin, L. ~'I., Berdiclievskiy, 1T. N., Vedrintsev, G. A.,
Consultants Bureau,
Dipole methods- for measuring earth conductivity: New York,
302 p.
probes in the
Anderson, L. A., and Keller, G. V., 1966, Txperimen[al deep iesistivity
central and eastern United States: Geophysics, v. 31, no. 6, p. 1105-11`22.
geomagnetic
I3alachxndran, K, 1967. A theory of the solar wind interaction with the
fields: D.Sc. Tlicsis, Colorado School Mines, 69 p.
method:
l3erdiclievs~i~-, _~L N., 196.1, Electrical prueptctin~ with tl~e telluric current
Colorado School alines Quart., v. 60, no. 1.
resistivity results from
Cantwell, T., Galbr~iitl~, J. V., ,)r., and nelson, P., 1964, Deep
20, p. 4367-4376.
no.
v.
69,
Research,
l~~ew York and Virginia: lour. Ceophys.
REF'ERF.NCES
be moved
dipole resistivity surveys, where one or the other of the dipoles must
the
about over the surface of the earth. In a system where the source and
penetrations,
receiver are fixed, and frequency is varied to obtain a range of
data
observed
the
of
shape
the
distort
to
inhomogeneities
is
the effecC of surface
done
been
has
nothing;
virtually
However,
randomly..
smoothly, rather than
surfacein the analysis of errors which may arise with such systems when the
la}'er resistivity varies.
Sel~lumber~er array by a
Ficui~s 117.- Anomaly in resistivity meeisured with a
in an otherwise uniform
embedded
layer
conductive
thin resistant or
spacing to
earth. The relative spacing is the ratio of tl~e Schlumber~er
the depth to the thin layer.
'Z(jp
261
Tarkhov, ~i. G., (EdJ, 1963, Spravochnik Gcofi~ika: Tom 3, Elektrorazvedka: ~VIoscow, Co~toptekirdat, 582 p.
TroiYekayR, V. A., 1964, Rapid v~n~iations of tl~e electromagnetic field of the earth
i~n. Research in Geophysics, v. 1, Carnbrid~,e. _'1~Iass. InsL Tech. Press, p. 485-532.
Vanyan, L. I,., 1967, El~ctromti~netic depth soundings: New York, Consultants Bureau,
312 p.
Watt, A. D., AZathews I S., rind ~1ax~r II, E. L., 1963, Some eleot~-ical characteristics
of the earth's cru_t. Inst. Elec. &' I.Icctronic engineers Trans., ~~. 51, no. 6, p.
897-910.
Welch, 'I'. D~1., 1968, Variation, in telliu~ic current power spectrums with local
time
and magnetic activity: NT.Sc, Thesis, Colorado School Mines, 113 p.
:~:~n.~-ss:~s.
D~khnov, V, iV., 1962, Geophysical wc11 logging: Colorado School Mines Quart., v.
57, nu. 2.
llavid~•~on, '1~L 1., 1964•, Average diurnal characteristics of geomagnetic power spectrums
in the period range ~.5~to 1000 seconds: .Tour. Geophys. Research, v. 69, no. 23,
p. 5ll6-5119.
Jacobs, J. ~., 1964, ~ticropulsations of the f.arth's electromagnetic field in the frequency range 0.1 to 10 cps in National electromagnetic phenomena below 30
KC/S: New York, Plenum Press, p. 319-350.
Heirtzler, 1., 1964, A summary of the observed char icteristics of ~em~iagnetic micropulsations in. Natural electromagnetic phenomena below 34 KC/S: New York,
Plenum Press, p. 351-372.
Herron, T. J., 1967, An avera < ;eomi~neti< power sp~,~ctrum for the period range
9,5 to 12,900 seconds: Jour C;eopl~ys. RESCarch, v. 72, no. 2, p. 759-761.
Hoffman, A. A. ,J., and Horton, C. ~/., 1966, An analysis of sonic magnetote]]uric recolds from Takhaya 13ay, U.S.S.R.: Jour. Geophys Research, v. 71, no. 16, p.
4047-4051.
Jackson, D. B., 1965, Deep resistivity probes in the southwestern United States: Geophysics, v. 31, nn. 6, p. 1123-1149.
Kalenov, E. N., 1957, Interpretatsii krivik6 vertikalnovo elecktricheskovo zondirovania:
D4oscow, Costoptekizdat, 473, p.
Keller, G. V., 1966, Dipole methods for deep resistivity studies: Geopl~vsics, v. 31,
no. h, p. 1088-ll04.
Keller, C. V., and I'rischknecht, I'. C., 19F>6, I.lectricxl n~e[hods in ~eop6ysical prospecting: Oxford, Per„anion Press, 527 p.
Kharkevich. A.A., 1960, Spectra and analysis: New York, Consultants I3urcau, 222 p.
Kunetz, G., and Chastener de (eery, 1., 1956, La representation confonne et divers
problems de potentiel dins les milieux de "pennexl~ilite" differente: Revue de
Inst 'Tech. I'rancais Pet., Octol~re, p. 1179, 1192.
Lokke~n, J. E., 1969, Instrumentation for receiving electromagnetic noise below 3000
cps in. Natural electromagnetic phenomenal below 30 KC/S: New York, Plenum
Press, p. 373-428.
Maxwell, E. L., 1967, Atmospheric noise from 20 H, to 30kHz: Radio Science, v. 2,
no. C.
Naidti, Prabakar, 1967, Random telluric. field and signal extraction: Jour. Geo~~hys.
Research, v. 72, no. 20, p. 509-~OC9~.
1V'ati~rocki, P. J., and Papa, Robert, 196.9, Atmospheric prrocesses: Englewood Cliffs,
New Jerseq, Prentice-Hall.
Pritchard, 1. L, 1964, Spectral analysis of twenty-four-hour time intervals of micropulsations observed at stations in mid latitudes: _1VI.Se. Thesis, Colorado School
Mines, 12~ p.
San[irocco, X. A., and Parker, ll. C., 1963, The polarization and power spectrums of
Pc micropulsations in Bermuda: Jour. Geophys. Research, v. 68, no. 19, p.
DESIGN CRITERI.A FOR ~LI:CTPICAL SURVrYING
<
263
With AC sounding methods the response caused at the surface by a resistivity
contrast p_/p, at depth is a function of the square root of the conerast in
Id%ith DC sounding methods, the depth of investigation cczn be controlled o~zly
by changing the positions of the electrodes on the su,rjace of the eartla. Surface irrega~larities in resistivity cage cause Dirge scatter in tlae ~neasure~nents
known as "geologic noise."
With AC sounding methods, the depth of investiation may be controlled
either by varying the spacinb between source and receiver or by chanting the
frequencies used. In the latter case, scatter caused by surface irregularities in
resistivity can be minimized.
T~ith DC sounding nzet7iods, a value for "a~~parent" resistivity which has
some significance can always be co~npicte~l from field data using; simple
formulas.
With AC methods, field observations can be converted to values of "apparent"
resistivity using simple formulas only for• a restricted ran,e of frequencies,
and the particular range for which computations ma~~ be made depends on the
resistivity, or a priori knowledge of it.
I~/ith DC sounding methods, Llaeory is bnse~l on the solutr;on of Laplace's
egzccztion. Such solutions are much si~n~ler tha~z solutions to Nlaxzuell's equations for comparable conditio~zs.ljaus, more e:~ tensive catalogs vf reference
curves are available for the DC ~n.ethods.
With AC sounding; methods, theory is based on the soluCion of Maxwell's
equaCions, usually with propa~~ation effects and displacement currents beinb
neglected. Such a solution is difficult for any but simple ~,eometries.
1. AC and DC methods:
We might first summarize the factors which would be involved in the
choice of an AC method or a DC n7eihod.
Considering the rnaterial presented in the precedin;> three sections, one mibht
use a variety of approaches to explurinn for oil with electrical prospecting
i~~ethods. Let us now consider what still needs to he clone in defininn the
capabilities of the electrical prospectinn methods, as well as their limitations,
and the strategies which might be employed in applying the methods in
exploration.
sL~l~~tn~Y n~ r~ c<~~cl.t sloes — s~rr~ATt~cY
Sc<<oot. ol~ till~l:s
Qo:~i:Tr~:rt.~~ o~~ ~r~it: Coc~r,.at>o
26.1,
to sinalL contrasts.
,ds are relati~ elf in~ensiti~~e
iesi~ti~ it~~. 'Chin. the AC metki<
methods.
kith lar,,~e eontia;t~ as du the DC
lout d~~ not saturate as raE~i<ll~
by n resis[ii~ity
response caused a1 !ke s~n~~ace
!~/iI/i I)C soundin~~ naei{io~ls, the
DC methods
77ius,
function of the r~rtio directly.
conh~~i,st (~_ dpi rat de~~th is ~r
st~turates
ii•ity
sensil
clian,~es in resis[ii~it~~. but this
m~e quite scr~si(ire to small
quich~ly pis the coritrtisl increases.
surface from a thin,
little efTrcl i~ seen at the
With ~1G inetkio<I~. relati~~cl~
its ~~resence does
la~cr i~ di(licult to detect. but
insulating layer at dearth. Such 1
deeper la}er~.
nut interfere ~~~ith detection of
t~ely prohibits the
n ~hrn insiilci[in~~~ l~rti~er efJecti
l~itlt DG soitndiiz~ methods_
ds the tlee~er
tl~c~ earth. rmtl completely scree
~~ene(ratio~i of cin~ren~ ~urihcr i~i[o
lnyers front detectio~i.
la}'et com~,letely
a tliiu, perfectl}~ conducting
~Vit}i AC soundin~~~ il~etliod~,
mation can be
infor
elic ficl~l. ~u l6at nu
reflects the incident electroiva~n
obtained zllxnil deel>ci la}erg.
will tray all
lliir~_ perfectly conduclin:; layer
!!'i~1a DC so~uiclin~ methods, a
r the AC
neithe
1'lius,
to d<;eper Icryers.
ei~rre~it, evict done ~te~i11 pe~ieh~nle
lnyer.
ning
scree
cti~i~e
cmi see tlu~oii~la ci co~iclu
met]io~ls Dior the DC ~netlaods
comparison
~~ rapidly ~vilh AC methods in
~ieasureinents ma}~ be made relati~°el
separations
r
eceive
e-r
only oile or a few sourc
~~ ith UC methods; inasi~iuch as
magnitude
of
order
an
s of interest are at Iea~C
need be used, and the frequencie
methods.
hi~xl~er than those used in the llC
uch cts many
the DC sozuulin~ nietlaods, inasm
Field worl~ is cunabersome with
low frevery
ing,
be icse~l. In deep sound
source-receiver se~~aratio~is niaist
tine
are
ts
u~emen
inclecction ef~ects, nncl ~neast
quencies must be icsecl to avoid
consz~minb.
soundin` method, in
lity in the design of AC
There is considerable Aesibi
depending on
ctive receivers may be used,
that either inductive or condu
h detects
~~hic~
ique
the choice of a techn
operlting conditions, allowin~,~
noise.
minimum electrical or ~;eoloaic
there is no
zise c~ concductive recei~uer, and
pith DC nzetliods; one Iaas to
st electrical or geolobic noise.
fle.~ibility in cliscriminatiaa~ a~;airi
earth is
the IongiCudinal resistivity of the
With some of the AC methods. only
aries
bound
to
s
are interpreted are true depth
detected and the depths ~vhic.i
rethe
ds,
metho
resistivitie~. with other AC
between layers with different
+
26
in~~ 1n ap"ChB target of exploration is ~Isu a maCter to consider in desi~n
be the
mi~;~hC
proach to oil exploration using elECtrical methods. "The tarnet
in
resise
a ehan~;
anomaly in resistiviCy caused b~ oil saturation directlt~. or
the
in
out
ed
tivity in areas favorlbLe for oil accumulation. l\s w1s point
oil
with
ly
direct
section on detection of the resistivity contrlst associated
st would cause the
saturation, the IeasC of oil fields w~~ich is of economic intere
~e by about onechan,~
to
ce
s
surfa
earth'
the
at
red
field components measu
or by about a
used,
are
ds
metho
ing
sound
DC
if
qu~rter to one-half percent,
nit entirely
is
it
,
"Thus
used.
d4
are
tent- ~s much, if AC sounding metho
from oil
al}
a
direct
of
anoin
ion
the
detect
unreasonable to base exploration on
ions of
precis
with
mace
only
comm
saturation. Conventional 1)C surveys are
of a
on
fracti
a
01'
al}~
anor~~
the order of a few perceuC, and detection of an
DC
of
ion
precis
ChB
in
C
percent would require only a moderate improvemen
ha~~e
%s
surve}
netic
romag
sounding inethoc3s. In the ~~asC decade, airborne elect
with a resolution of a fE~v
been developed to the point of detecting anomalies
would be neeclecl in
hunc{red parts per million, ~~~hich is the precision which
the scale of the airborne
oil exploration. It should he noted, of course, that
2. Tlae tarbet
an~i de~~l}is
spouse ~~ ill depend on loth vertical and luu~ituclinai rESistivity~,
a comwill be distorted by° anisoh~upy. In theory at least_ it is p~s~ible to use
s. and the effect of
bination of AC methods to find both longitudinal iesistivitie
anisc~Cropy.
roj~y. I7ae
With DC methods, the results cn~e alivays c~istortecl by aniso[
h~~ms~verse
l
mid
measured resistivity is th.e gu~aclrczlic avera~~e of the lon~:;it«clina
cierit o~
coe~fi
l{ce
vales; and interpreted depths are alauays scaled z~1~ by
hn~l
true de~~llcs.
a~n.isotropy. Auxiliary clata~ of some type must be proc~irlecl to
AC methods.
The consideraCions relating; to the ease of field operations with
would apility
Ilexib
and
,
noise
riic
their rapidity, lack of sensitivity to ~;<~~olo
_An exc1~es.
in
most
ds
pear to make AG methods prE~ferable to DC metho
may
ts
remen
measu
DC
~~
here
ception would be the case of marine opera[ion~,
.
vessel
~~
movin
a
from
be made Basil}' by traiLin~~ the mea~uremeot electrodes
asepar
er
e-receiv
of
sourc
This reduces the work invol~-ed in ,~etting~ a number
~~~ill reduce the problaver
~vaCer
sea
a
of
rmity
unifo
the
ion,
iu
addit
tions, and
con~idc~red seriouslems with ~eolonic noise to the ~~oint ti-here tk~P~ need not be
>ds in this case,
ly. Havin„ eliminated the ~~rincipal aclvanlagES c,f the AC ineth~
small-dimensioned
the advantage of the DC method in beins~, more ~en~iti~~e to
1io~vever. if I)C
anomalies in the subsurface resi~tivit~ I~f~ce>rnes d~,minanG
be provided to
methods are used at sea. some auxiliary inf~_~rmaeion must
correcC de~~ths f~~i• the ef~'e~~t r>f ani~c,leoE>y.
Su~t~r;~i~l ;~~i~ Co~c:t,usio~s-5~ri:n•rf•:c~
MINES
COLORADO SCHOOL OF
QUART~FLY OF Tft~
with standard methods wi-Eke
suc}i measurements may be made
oil exploration.
smaller than that needed for
electroma<~netic surveys is much
of oil fields,
ion
methods for the direct locat
In the application of electrical
may be
ents
measurem
whether the accm-acy of the
the question is not so much
rather,
but
detected,
the direct anomaly can be
improved to the point where
milper
500 parts
percent in DC resistivity or
whether an anomaly of one-half
is
It
.
saturation
reco~iiized as representing oil
lion in AC coupling caii be
field
e measured
ges in a secCioil will caus
readily- apparent that other chan
even when oil is
nts
amou
m
these minimu
quantities to vary by snore than
would seem to he
of oil with electrical ineChods
not present. Direct Location
field was known
nil
an
of
ap~~roximate location
feasible only in cases where the
known that oil
he
ma5~
it
in depth. 1'or example,
beforehand, both in }plan and
that othel-and
line,
d
voir hori~~n, alone a tren
pools occui in ~ g~ieen reser
d then
woul
One
er.
iomv~ries in a logical mann
~~ise, the character of the sect
depth.
fic
speci
thaC
resistivity associated with
prospect for sma11 differences in
s per
part
500
or
a half percent in DC resistivit~~
In such case_ an anomaly of
even
,
ation
satur
oil
be reco~,~ilized as re~~resentin~~
million in AC coupling might
er
lac
ably
ider
cons
of the section would cause
thou~~h changes in other parts
ch ~n~~es in observed resistivity.
exploration is in
of electrical methods in oil
Another approach to tl~e use
mn which affect the
tions in the stratigraphic colu
the study of litholo~ic varia
lo~;ic variations will
y, an understanding of litho
electrical properties. Hopefull
large-dimen~ionec3
of oil. However, detection o~
assist indirecNv in tl~e location
much simpler
with litholo~~ic h•ansitii~ns is a
ch antes iil iesisti~it~ associated
directly with oil
contrase in resistivity associated
problem than detection of the
mentary column,
in average resistivity ~f a sedi
saturation. !~~Ilpping~ of changes
first part of this
the eleceric lob studies in the
such as those apparent from
ties ~f electrical
capabilities of any ~~~f the varie
monograph, are ~~~ell within the
a formatiari or
of
y
tivit
resis
~ here. Average
prospectinn methods c3escribec
2 to ~ over a
of
rs
facto
appears to vary by
group of similar formations
of ehe section,
logy
litho
nce of the chan;e in
~'~edimenCary basin, as a conseque
within reason
even
is
It
of the connate water.
or of changes in the salinity
are run with
eys
surv
is Caking place, if electrical
to recogniae which change
A combinapy.
otro
gnition of the effect of anis
methods which allow the reco
y and a
tivit
resis
inal
itud
which ,gives boeh the long
tion of induction methods
the use
ple,
exam
tl~e effect of anisotropy (fir
resisti~Tity value di~iorted by
field
etic
magn
and
with electric field detectors
of an electric dipole soiu-ce,
an
out
with
y
resistivit
this. A change in lonbitudinal
in
detectors) would permit
ces
chan
d indicate
coefficient of anisotropy woul
associated chanbe in the
ind
woul
tropy
chan~.;~es in the coefficient of anis
connate water salinity, while
iderin that
Cons
.
ence
sequ
ary
ment
of the sedi
dicate changes in the litholog~y
adequate pre-
Z(j(j
Z6(
~. Problerras to be solved
in a petroleum e~ploratiun proBefore electrical methods can find a place
remain to }~e sol~~ed. 1`he least of
~ram, there are a namber of problems t~~t
Ntnent w~itli the }>t~ti~er capacity- r1nc1
these is the ~~t-ol~letn of develoE~in~* equi
accuracy required in oil exploration.
he evaluated ~~~ithou~ the expense
A ~~ei-~~ important prohlen~ which mati~
electrical surve}~s is the probof wldertakin~; field operations with lame kale
tires as~ociatecl ti~~ith oil occurrence.
lem as to th<~ size and dc~tectability of anom
oil ~aturatior~ can be estimated by
The size of the ano~~~aly caused direcll~ 1~~~
been obtained in oil fields The nlaranaly~zin~ the electii~. lobs which l~a~~e
cl ~ti~itli litholo~,~ic chan~~es wiChin a
nitude of chines in resistivity a~~oe•iate
i e re~i~tivities from lo~~s as ~tias
basin may be studied by compiling avei
~h. Perhaps even more iiYiportant,
dotle iri the fiat section ~>f this mono~~~ra~
usin~~ iE~istivities from electric lc~~~s
synthetic electrical ~une~s ma~~ be run by
~aundi~~~~ methods. This aprical
to compute the response to various elect
cor_~icleral>le experience in evaluati~i
proach ~~ould allow tl~e accumul~ltion o{
in field curve}'s.
electrical surv~}'s ti~ithout the cost in~~ol~~ed
of interpretaCion Present techniques
A second problem to be sul~-ed is that
date are arcl~aic~. consi~tin~ usuall}- of
used in interpreting electrical eur~~ev
n~ of cur~cs computed [or carious
e~°eball cuniparison of field data with albu~
~~~orkc 1~ell ens>u,~~h for field date i~l which tl~e
earth models. Such a procedure
ot
~~ ith more precise field data, one cann
precision is fig e percent or less, but
rlt
~imou
l
smal
the
by
es ~+Erich ~~ar~
cliacciminate ~i~ua1L}- anion,, curve shap
dais. Several approaches ma~~ be
the
i❑
s
rence
ditrc
t
fican
ti~'hich are signi
rved
h is the trial and error fitting of obse
available iii such a case, one of whic
o~>~v throu~~h combined measurecision to accomplish this, Che study of Iithol
be~the most powerful application
ments of re~sisti~ity and ani-.otrop}' may well
of electrical methods in oil prospecting.
methods in oil prospecting; is in
A third approach to t1~e use of electrical
. ~/ith electrical methods as they are
the tna~~pin~~ of elee~trical marker horizons
f of a marker hoiizoi~ ~~~ith a much
used tod~~~. it is feasible to map the relie
, sequence with a precision of the
hiahe~r or luti~er resisti~~it~- khan the o~~er1}~in
n is comparable to Chat obtainable
order of une or t~~o percent "[~hi~ resoluti~>
is less than can lie obtained ~+-iCh a
with the seismic refr<lction r~iethod, buC
survevin~; methods are riot inexpe~~seismic reflection ~ui-ve~. f)ee~~ electl-ical
~ur~ ev ~~ ith 1 similar covera~,e b~
sivf~, thous>h they m i~ curt leis than ~ei~inic
trical ~urve~~s to mapping; relief of
a factor of 5 to 10. "Che application of elc.,•
in cages ~~-here. the. marker cannot
a marker horizon is prol>abl~~ of value only
one reason or another.
he easil~~ ma}~pecl b~~ Seismic mcthc>ds for
9TI:GY
~li~I.11ARY ANll CO\CLliSIO\S-.~TR
o?. or• 111~~:s
Q~~:~itTr.i:i.~ or •ruF: Co~.oi;:~uo 5cuo
.1 more
cur~~e.~ in ur~lr~ [u ,,Et a best fit.
data ~citl~ thc~~reti~ ~ll~ cak•ulated
~~ed
ob~er
the
ul
n
iti~~
of direct cle~um~~os
e~citinT l~u_- ibilit~ is thr~ concept
r
~~er. that <Z ivaju
h function. It i~ certain. ho~cc
data int<~ a rep<i~ti~ its ~~.~. cle~>l
the full ca~~acit~~
re
bef~~
interl~retatiun capal~ilit~~
impru~ement ~~~u t be made in
~ratiui~.
espl~
ca~i 1>e rec~~~~~ize~d in ~~il
.,f electrical l~r~~~~~e~ tin metl~ud~
~~reci~i~m of
~Che
i~ that ~,f ~eul~>;~:ic none.
A third }~ruble~n to be s~>l~~e~l
the scatter
b~
zip~~ear~ to he limited F~rimaril~~
elt~~~trictil sur~~e~ in techni~~ue~
ods are
meth
in rt~i~ti~~it~. S~>n~e ~ur~e~
~ ~,ntrihute~l 1>~ ~urficial ~ ariati~~n~
c~,iL
li~>n
h induc
lh~in uther~: inr•~hu~ls in tchic
If.=, ~cn~iti~~~ to _c~~l~,sic n~,i~c
>enei
n~~~
in}~oi
are much lc.~; ~en~iti~~e to lu~•al
arr~ u~e~l a~ ~uur~~~~~ anti recci~er~
and
:
ts are u~~d
~ls in ~~~hi~~h ele~•tru~le cuiita~°
tir°; in rr~~i <ti~ its than tu~r methu
~cr ~eparatiun
rceci
e~uure
than
r
~arie<l rathe
n~c~thud~ in ~chick~ fre~~uen~~~ it
~U~<,. them ~trr~ tip~urfa<~e~ inh~~mu~E~ncitie~.
~~~hil~it I<~; ;~ attar ~ziu~~~<1 h~
rall~ the cage:
r i~ of lr°~ ~~uncern than i~ ;~~•ne
~~li~~utiun in ~chi~h ,~cul~~,~i<~ »ui.~
une t~~ pruhlem
r layer ~uf~ r;.~entiall~ inui~
~ur~e~~ ina~lc ~~~er a ~e<i- ~catc~
the surface ~~~il
e~
~~hcr
~ are ~,n -shore array
<~au~~~~l I,~ _e~~l~~~~ii~ n~~i~~~. and tht~rf
kin~ ~~nl~ in
~~ur
.
f~~er
}lu~c
~~r~~l~lem i~ inii~ur.
i> ~u~li~~i~~ntl~ unif„rni that t{ir
h arr~ lezist
~~hic
.~
~~r~,l~lc~in. ur unl~ kith tcchnit~uc
arca~ ~chcr~~ n~~i;~~ i< rn,t zt
lit~ ~~I
icahi
ap~~l
the
unnece•~ai~~ limitation un
~ul~je~~t t.> n~,i~<~ ina~ lac an
~~~hich
hocl~
m<~t
Gin=i~leratiun ~~( fic~l~l =uc~c~~~
t~l~~~~trii~~tl ~~r~„Eu~~~tin~, n~eth~xl<.
~~in~
pruee
data
~l h~ ~e~~l~,_ic noise. and of
kill n~ininiize thc~ r~rn,r~ eau~e
cial
~urfi
(ruin
al~
~e~~ar<ili~,n of ~Ic•e~>-~eate<l :i;~ii
t~~~~hniyuc~ ~chi~h gill all~~~c iht~
this E~ruhlr~in.
n~~i~r ~na~ ~~r~~~~i~l~~ a ~uluti~~n [~~
nt tech~~ic~ue~~lil7~i~~ull~ ~,f fin~lin~ ~~il ~citl~ prese
1~~ ~innn~ar~ _ ~~umi~lerii~~~~ thr~
to khe problem.
of c~lr~~trical ~ur~c~ in~~~ methods
anal thi~ u~~~~arc~nt a~~~~li~ al~ilit~
more ~~i<]el~' in
nx~tl~uds l~a~e nul been used
it i~ sin~~~ri~in~ that ~~Ic~~trical
E ~~~I~,rati~~n („r oil.
Z6~