Ch3-EM_theory

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40

3.

CHAPTER

ELECTROMAGNETIC THEO
RY

3.1

INTRODUCTION

This chapter presents a very brief overview of relevant electromagnetic theory
necessary to collect and analyse electromagnetic geophysical measurements. Special
attention is given to principles related to the

ground
-
based transient electromagnetic
(TEM) method and its intended statistical analysis. Physical rock properties influencing
TEM measurements are also explained.

3.2

BASIC ELECTROMAGNETI
C PRINCIPLES

3.2.1

Introduction

The electromagnetic geophysical method util
izes electromagnetic induction
processes to gain information about the geo
-
electrical structure of the sub
-
surface (Ward
and Hohmann 1989). The most important principles of electromagnetic field behaviour,
laws of electromagnetic induction, and the relati
onship between electromagnetic fields
and electrical conductivity properties of materials are presented here. This chapter is a
summary of knowledge published in numerous textbooks (Keller and Frischknecht,
1966; Telford at al., 1976; Dobrin, 1981; Nabigh
ian, 1989; Strack, 1992, etc).

3.2.2

Maxwell’s equations

Maxwell’s four equations provide the starting point to obtain an understanding
of how electromagnetic fields can be used to determine the Earths electric or magnetic
properties. They can be written in man
y different, equally valid forms. Here the
equations are presented using the simplest possible notation.


One equation is used to represent the fact that magnetic fields are caused by
conductive and displacement currents:




41



x H = J +

䐠⼠

t





Eq慴楯

㌮3
-
1


where






is the ‘del’ operator (
z
k
y
j
x
i










),





x H

is the curl H (or rot H) =
3
2
1
H
H
H
z
y
x
k
j
i






,



H

is the magnetic field intensity (in ampere/m),



J

is the current density (in am
pere),



D

is the electric displacement (in coulomb/m
2
), and



t

is time (in second).


A second equation represents the physical law that electric fields result from
time
-
varying magnetic induction fields:




x E =
-


B 







Eq慴楯a
㌮3
-
2


where





is the ‘del’ operator (
z
k
y
j
x
i










),





x E

is the curl E (or rot E) =
3
2
1
E
E
E
z
y
x
k
j
i






,



E

is the electric field (in volt/m),



B

is the magnetic induction (in weber/m
2
), and



t

is time (in second)
.


The third and fourth equations describe that the source of electrical field is the
electrical charge, and there is no magnetic monopole:




E = q







Equation
3.2
-
3



䠠㴠〠







Eq慴楯a
㌮3
-



42


where






is the ‘del’ operator (
z
k
y
j
x
i










),





E

is the div E = (
3
2
1
E
z
E
y
E
x








),





H

is the div H = (
3
2
1
H
z
H
y
H
x








),



E

is the electric field (in volt/m),



B

is the magnetic i
nduction (in weber/m
2
), and



q

is electric charge (in coulomb).

3.2.3

Ampere’s law

Ampere’s law describes the electromagnetic force between two current elements.
The following equation describes the force acting at a point P
1

(see
Figure
3.2
-
1
)
:


12
3
12
2
1
2
1
1
)
(
r
r
l
l
I
kI
F










Equation
3.2
-
5


and k =


4
0


where




F is the force between two current elements (in newton),




0

is the magnetic permeability of the free space (
7
10
4




weber / ampere m),



I
1

and I
2

are currents ( in ampere),




l
1

and

l
2

are the length of the two current elements (in m), and



r is the distance between the two current elements (in m).


43


Figure
3.2
-
1
. Two current elements I
1

and I
2
, with lengths

l
1

and

l
2
, at a distance
r from each other. The electromagnetic force acting at
P
1

point is described by
Ampere’s law (see Equation 3.2
-
5).

3.2.4

Biot
-
Savart law

The Biot
-
Savart law is based on Ampere’s

law, describing the magnetic
induction at a point P due to an electrical current element (see
Figure
3.2
-
2
):


3
r
r
l
kI
B










Equation
3.2
-
6

or

2
7
sin
10
r
l
I
B











Equation
3.2
-
7


where





B

is the magnetic flux density (in teslas or weber/m
2
),



I

is the current (in ampere),




l

is the length of current element (in m),





is the angle between
P


l

and

l

(in degree), and



r is the distance between P and

l
(in m).


1
I
r
P
I
P
1
1
1
1
2
2
2

44


B
r
I

l
P


Figure
3.2
-
2
. The magnetic field at a point P due to an electrica
l current element
(after Sheriff 1984).


3.2.5

Magnetic field of a current line

Following from above, the magnetic field of a current line can be simply
expressed (see
Figure
3.2
-
3
):




2
I
H








Equation
3.2
-
8


where




H


is the azimuthal magnetic field intensity (in ampere/m),



I is the electric current (in ampere), and





is the radius in a cylindrical co
-
ordinate system (in m).


H
I
z

Figure
3.2
-
3
. Magnetic field of a current line. After West and Macnae (1991).


45

3.2.6

Magnetic field of a current loop

This description follows West and Macnae (1991). The magnetic field of a loop
current (such as an EM transmitter), at a distance r fr
om the loop center, can be
calculated from the Biot
-
Savart law (see Section
3.2.3
). For the case where a<<r, in
other words the loop is small compared to the distances measured and can be
approximated by a dipole, the magnetic

field strength can be expressed in the following
way, using spherical coordinates (see
Figure
3.2
-
4
):


3
cos
2
4
r
m
H
r










Equation
3.2
-
9


3
sin
4
r
m
H










E
quation
3.2
-
10


where


m =


a
2
I


and




H
r
is the radial magnetic field intensity (in ampere/m),



H


is the directional magnetic field intensity (in ampere/m),



r is the distance from the center of the loop
(in m),





is the angle between the loop axis and r (in radian),



a is the radius of the loop (in m), and



I is the electric current in the loop (in ampere).


46


Figure
3.2
-
4
. Magnetic field around a current loop
. After West and Macnae
(1991).


From Equation 3.2
-
9 and Equation 3.2
-
10, it follows that the magnetic field
intensity around a wire loop at a location far enough (where a<<r) is proportional to the
inverse cube distance from the centre of the source loop
. While this is true for large
distances, at distances close to the loop (r<<a), following from Equation 3.2
-
9, the fall
-
off in the magnetic field is proportional to the inverse linear distance from the wire.

3.2.7

Constitutive relations

Maxwell’s equations des
cribe the general behaviour of electromagnetic fields,
and, more specifically, Ampere’s and Biot
-
Savart’s laws describe the induction
processes between current elements and electromagnetic fields. However, in these
equations, no relationships are defined
between the electromagnetic field and sub
-
surface electrical conductivity properties, which could be linked to geology and used for
geological mapping. Those dependencies are expressed in constitutive equations.

3.2.7.1

Electrical conductivity

The most important
constitutive equation is Ohm’s law, relating current density
to electric field intensity. For most conductors, the current density is proportional to the
electric field intensity:


H
I
z
r
a

47

J =


E







Equation
3.2
-
11


where



J

is current density (in ampere/m
2
),





is the property of the medium, and is called the electrical conductivity of the
medium (in siemens/m), and



E

is electric field intensity (in volt/m).


Electrical conductivity is the most important propert
y of Earth materials that
influences electromagnetic measurements. Electrical conductivity properties, because of
their great importance in transient electromagnetic exploration, are discussed in detail in
Section 3.
3
.

3.2.7.2

Dielect
ric permittivity

Dielectric permittivity is a measure of the capacity of a specific material to store
charge when an electric field is applied. In other words, it is the measure of the
polarizability, or static electrical response, of the material in an e
lectric field. Dielectric
permittivity governs, in part, the response to high frequency alternating currents induced
into the sub
-
surface by conductive or inductive means. It is described by the relative
permittivity, formerly called dielectric constant
(

r
), which is the dimensionless ratio of
the dielectric permittivity of the material to that of free space (

0
). The dielectric
constant of free space is unity (Sheriff, 1984; Dobrin, 1981).


D =

r


0
E

=

E






Equation
3.2
-
12


where



D

is the electric displacement (in coulomb/m
2
),




r

is the relative electric permittivity, formerly called the dielectric constant
(dimensionless),




0

is the electric permittivity of free space, ( = 8.854187817


10
-
12

farad/
m),






is electric permittivity (dimensionless), and



E

is the applied electric field intensity (in volt/m).



48

As this property mostly affects high frequency (>10 kHz) survey methods, dielectric
permittivity has little effect on low frequency (<10 kHz) TEM m
easurements.

3.2.7.3

Magnetic permeability

Magnetic permeability is the property of a material relating inducing magnetic
field strength to magnetic induction:


B =

r


0

H

=


H






Equation
3.2
-
13


and



r

= 1 +


m







Equation
3.2
-
14


where





B

is the magnetic induction (in weber/m
2
),




r
is the relative


magnetic permeability (dimensionless),




0

is the magnetic permeability of the free space (4


10
-
7

weber/ampe
re m),





is magnetic permeability (in weber/ampere m),




m

is magnetic susceptibility (dimensionless),



H

is the magnetic field intensity (in ampere/m).


Magnetic susceptibility is often measured on drill core samples during magnetic
surveys. The results ar
e used in conjunction with electromagnetic survey results to help
geological interpretation.

3.3

ELECTRICAL CONDUCTIV
ITY PROPERTIES OF RO
CK MATERIALS THAT
INFLUENCE ELECTROMAG
NETIC INDUCTION

3.3.1

Introduction

Understanding physical factors governing bulk electrical

conductivity is one of
the major topics of contemporary research in airborne and ground
-
based electromagnetic

49

exploration. Palacky (1986) gives a good summary of electrical conductivity properties
of bedrock conductors, clays and water. Strack (1992) de
scribes rock conductivity
dependencies on anisotropy, temperature and pore fluid chemistry. Emerson (1997)
provides data based on detailed laboratory measurements, investigating the effects of
water salinity on resistivity of clays. TEM methods can also
provide additional
information on structural elements, diapirism, volcanic cover, delineation of porosity
variations and water content, and some TEM methods are suitable for deep crustal
studies (Strack, 1992). However, due to the complex relationships be
tween individual
mineral conductivities and bulk
-
rock conductivity properties, TEM data give much more

reliable information on the location of conductive earth materials than on the factors
influencing conductivities of those rock units.

In this section,

the definitions of two important physical concepts, bulk electrical
rock conductivity and apparent resistivity are presented. Brief descriptions of the most
important physical and chemical factors influencing bulk electrical conductivity
properties are g
iven, and conductivity values for some rock materials are tabulated,
summarizing published data. The purpose of these descriptions is to demonstrate the
difficulties that EM interpretation techniques face.

3.3.2

Bulk electrical conductivity

3.3.2.1

Definition

Rocks are

made up of individual minerals with voids, which may or may not be
filled with fluids such as ground water. Electrical conductivities of individual minerals
are well known, measured, tabulated and published in the literature (Telford et al, 1976.
pp.451;

Nabighian, 1989. pp.23
-
24, etc.). However, where a rock comprises an
assembly of packed minerals, the bulk electrical conductivity is not a simple volume
average of the individual mineral conductivities. Bulk electrical conductivity is the
conductivity
value of a rock volume derived from
in
-
situ

measurements, with the
combined effect of all influencing physical and chemical factors, such as temperature,
pressure, pore fluids and structural elements. It can be extremely difficult to undertake
appropriate

in
-
situ

measurements. Bulk electrical conductivity is usually anisotropic,
highly variable, and poorly quantified for most rocks in mineral exploration (Telford et

50

al, 1976. pp.454
-
455; Dobrin, 1981. pp.570
-
571; Nabighian, 1989. pp.55; Strack, 1992.
pp.1
3
-
14).

3.3.2.2

Factors governing bulk electrical conductivity

TEM methodology is mostly used for investigations within the upper few
hundred m of the sub
-
surface. At such a depth, the most important factors governing
bulk electrical conductivity are the fracturin
g and porosity of the rocks, and the chemical
content and salinity of pore fluids (Keller and Frischknecht, 1966. pp.16). Electrical
conductivity of mineral grains will determine bulk conductivity only in special cases: for
fresh rocks with little or no p
ore space (such as massive sulphides), for very shallow
lying porous rocks, above the water table, and for rocks situated at such a great depth
that all pore spaces are closed by ambient pressure. Laboratory conductivity
measurements of rock hand
-
specimen
s, which have lost pore fluid, could be misleading.
Even individual mineral samples, because of impurities and changes in crystalline form,
show variations within a factor of ten or more in laboratory measured electrical
conductivity. The grain size of t
he minerals making up the rock is a minor factor.
Connectivity of pores and mineral grains is a more important factor influencing bulk
electrical conductivity.

Boundary layer considerations are important, since many minerals have
anomalous surface propert
ies. For example, galena and magnetite, which in themselves
are extremely conductive, usually have a thin and quite resistive surface coating over the
crystals. This means that even rocks consisting almost exclusively of one of these
minerals can be quit
e resistive.

Structural properties, the fabric of the assembly of minerals and voids, and
fracturing all play a part in determining bulk electrical conductivity. The fabric is
generally a function of mineral crystal form and of the shearing and folding hi
story of
the rock. The composition and mineral crystal form is controlled by the chemical,
thermal and burial history of the rock.

Keller and Frischknecht (1966) give a detailed summary of the electrical
conductivity properties of metals, semiconductors,
electrolytes and rocks, including
relationships between conductivity and porosity, texture, pore
-
fluid salinity, temperature
and anisotropy.


51

3.3.2.3

Changes in electrical conductivity

The electrical conductivity of rock materials is not constant, neither spatially

nor
temporally. The electrical conductivity of rocks can change as a non
-
linear function of
time, temperature, pressure, weathering state, age of rock and various environmental
factors (Nabighian, 1989).

An important cause of changing rock conductivities

is weathering. Weathering
is a complex chemical process, which results in decomposition of primary rock
-
forming
minerals and their substitution by clay minerals. Water
-
saturated clay minerals make
the alteration zone significantly more conductive than t
he parent rock (Palacky, 1987).
Different lithologies weather differently, producing different conductivities, and
consequently different EM responses. Weathered overburden has the effect of reducing
signals from buried targets (Doyle and Lindeman, 1985)
.

3.3.2.4

Anisotropy of rock conductivities

In isotropic materials, the three principal values of conductivities (

xx,

yy
,

zz

)
are the same. Most minerals and rocks, however, do not have a completely isotropic
structure or composition. In those materials, the

measured electrical conductivity is
influenced by the geometry of measuring equipment in relation to the target. This
property of materials is called “anisotropy of conductivities”. The origin of anisotropy
can be at atomic or molecular level, or could
be a directional aggregation of isotropic
material, for example layered sediments or volcanic sequences (Telford et al, 1976).
During inversion and interpretation, unless tensor measurements were made, isotropy is
assumed.


52

3.3.3

Electrical conductivity values
for some rock materials

The range of conductivities among rocks and rock materials is very large,
extending across about 25 orders of magnitude from 10
-
15

to 10
10

S/m. Rocks and
minerals with conductivities from 10
10

to 10 S/m are regarded as good conduc
tors; those
from 1 to 10
-
7

S/m intermediate conductors; and materials from 10
-
8

to 10
-
15

S/m poor
conductors (Dobrin, 1981). Bulk conductivities of different rock types overlap, making
identification of lithologies, based purely on conductivities, impossi
ble (see
Figure
3.3
-
1
). Gold deposits, because of their variable host rocks and low grades, do not have
consistent conductivities.


Figure
3.3
-
1
. Typical conductivities of rock ma
terials. From Palacky (1987).


There is a general relationship between conductivity and silica content.
Classification of igneous rocks based on silica content and conductivity is shown in
Tab
le
3.3
-
1
.

3.3.4

Conductivity properties of
regolith units

The susceptibility of rocks to weathering is dependent on both their mineral
composition and their texture. Consequently, a fully developed weathering profile will
define individual horizons that exhibit varying electrical properties.




53

Tab
le
3.3
-
1
. Classification of igneous rocks, based on silica content and
conductivity. Note the inverse relationship of conductivity to silica content. From
Palacky (1987).


Quartz Content

Plutonic

Vulcanic

C
onductivity

High

Granite

Rhyolite

Low


Syenite

Trachite



Qtz Diorite

Dacite



Diorite

Andesite



Gabbro

Basalt


Low

Peridotite

Ultramafitite

High

3.3.4.1

Transported material

Conductivity of alluvium and colluvium units is dependent on clay content,
poros
ity, salinity and water content. Highly conductive zones can be created by the
influx of saline waters. Consequently, large electromagnetic anomalies could be
associated with saline alluvium, which can hide underlying conductors (Palacky and
Kadekaru, 19
79).

3.3.4.2

Duricrust

The most resistive unit within the weathering profile is the duricrust. Palacky
(1987) found that the upper layers of duricrust and the mottled zone were typically more
resistive than other units within the weathered profile. The conductiv
ity of lateritic
horizons depends on the type of electrolyte present, and the pore geometry (Peric, 1981).

3.3.4.3

Saprolite

The most conductive and recognisable unit is saprolite, which has typical
conductivities between approximately 5 and 500 mS/m (Palacky, 198
7). Saprolite
conductivity is highly variable, and is dependent on the types of clays present and the
water content. The conducting minerals within a saprolite are hydrosilicates, which
exhibit high surface conductivity, typical of clay minerals.

It has
been demonstrated that saprolites, developed over mafic and ultramafic
rocks, are more conductive than that formed over felsic rocks (see
Tab
le
3.3
-
1
). The

54

main reason for the lower conductivities within the felsic rock types is t
he higher
content of resistive minerals such as quartz and muscovite (Palacky, 1987). An
important consideration for exploration geophysics is that saprolite developed over
mafic rocks is generally thicker and more conductive than saprolite over felsic ro
cks.

3.3.4.4

Saprock

Above the fresh rock is a relatively thin weathering boundary or saprock zone.
The conductivity is somewhere between fresh rock (low) and saprolite (high), which
makes resolution difficult, using electrical methods.

3.3.5

Archie’s empirical formula

Strack (1992. pp.11.) gives a good summary of changes in electrical properties
of different rock types with changing temperature, depth, salinity and porosity. All of
these factors, and possibly many more, need to be considered when estimating true bulk
conductivities. As Archie’s empirically derived formula for the resistivity of clean
compacted sand demonstrates, this task is extremely difficult:


n
w
S
F
1









Equation
3.3
-
1


where


m
a
F



is the Humble formation factor,


15
.
2
62
.
0


F

for sand,


2
1


F

for compacted formations (Archie formation factor),






is the resistivity of the formation,




w

is the resistivity of the pore fluid,




a is an em
pirically determined constant,





is the porosity of the formation,


55



S is the fraction of the pore volume occupied by formation water,



n is an empirical constant, recommended by Schlumberger (1987) as n=2 for
basin sediments, and



m is an empirically determin
ed constant, called the cementation factor.


For different formations, further corrections are needed, in the form of different
constants to be substituted into Archie’s formula. The constants vary widely for basin
sediments, and their application for gre
enstone belts is extremely difficult.

A theoretically defined quantity, apparent resistivity, provides a more viable and
practical approach.

3.4

APPARENT RESISTIVITY

3.4.1

Introduction

“Experience with transient coupling curves indicates that while in principle such

curves contain a great deal of information about the characteristics of the geoelectric
section, usually the curve shape is only weakly sensitive to changes in the profile of
resistivity with depth” (Kaufman and Keller, 1983, pp. 451).

Furthermore, in ord
er to characterise a given interval of the geoelectric section,
“the spacing between the transmitter and receiver must be many times greater than the
thickness of the section being investigated” (Kaufman and Keller 1983, pp. 451).

These two observations ex
plain why it is so difficult to carry out accurate
conductivity
-
depth inversion. Even though some of the following chapters of the
present work explore and describe the changes with depth in modelled and field
-
derived
TEM variables, lateral changes are co
nsidered as most important for the final
interpretation of the data.



56

3.4.2

The general definition of apparent resistivity

Apparent resistivity is the resistivity of a theoretical, homogeneous, isotropic
half
-
space, which would give the same voltage
-
current rela
tionship as the measured
value (Sheriff, 1984). The equation is derived from Ohm’s law for direct currents:


I
V
f
k
a
)
(








Equation
3.4
-
1


where




a

is apparent resistivity (in ohm
-
m),



k is
a geometric constant, which depends on the electrode array (in m),



f(V) is the measured voltage at a given frequency and time (in volt), and



I is the applied current (in ampere).

3.4.3

The definition of apparent resistivity for TEM measurements

The apparent resi
stivity equations in this section follow Kaufman and Keller
(1983). The following expression is based on the definition that the observed voltage
needs to be normalised to the voltage which would be recorded on the surface of a
uniform half space having t
he resistivity of the uppermost layer. This value will be
applicable at the early stage of electromagnetic diffusion:











)
(
)
,
,
(
1
1
1
t
E
r
t
E
un










Equation
3.4
-
2


where



1


is apparent resistivity, norma
lised to the uppermost layer,


1


is the resistivity of the uppermost layer,

un
E



is the electric field which would have been measured at the same instant
on the surface of a uniform half space with resistivity

1
,


E


is the electric field observed on a horizontally stratified medium,

t

is the time instant of the measurement, and

r

is the T
x
-
R
x

separation.

At an early stage, the apparent resistivity tends to

1
:


57

1
1





as


t






Equation
3.4
-
3


The lower half
-
space departs from this response, caused by horizontal
stratification or lateral inhomogeneities. At the late stage

E

approaches the value of a
unif
orm half
-
space with the resistivity of the basement, and is proportional to
2
/
3
N

.
Because of that, at late stage


2
/
3
1
2
/
3
2
/
3
1
)
(
)
(
















N
N
un
t
E
t
E

as


t



Equation
3.4
-
4


To

derive an apparent resistivity curve demonstrating how at any moment the
measured field differs from a uniform half space, equally valid at early and late time, we
can write:


3
/
2
1
1
1
)
(
)
,
,
(









t
E
r
t
E
un









Equation
3.4
-
5


and from there




)
(
1
)
(
1
)
,
,
(
3
/
2
1
3
/
2
3
/
2
1
1
1
1
t
E
k
t
E
r
t
E
E
un











Equation
3.4
-
6


where



3
/
2
1
1
1
)
,
,
(
r
t
E
k
un





is the geometric factor for this array.

3.4.4

Apparent resistivity calculation for the Sirotem instrument

Duri
ng geophysical field surveys, apparent resistivities can be calculated using
empirical formulas specific to measuring instruments, and then the results can be
inverted using master curves or inversion routines, to build subsurface models.

Apparent resistiv
ity calculations produced by the Sirotem Mk3 instrument are
based on the following asymptotic formula (Buselli et al 1985), derived from Equation
3.4
-
6:


58


3
5
3
2
3
8
12
)
/
(
10
32
.
6




t
I
V
b
x
a





Equation
3.4
-
7


where



a


is the apparent resistivity (in ohm
-
m),



b is the loop length (in m),



V is the measured receiver voltage (in volt),



I is current in the transmitter loop (in ampere), and



t is time (in second).

3.5

TIME DOMAIN ELECTROM
AGNETIC THEORY IN EX
PLORAT
ION

3.5.1

Introduction

In time
-
domain EM methods, the response is expressed as a function of time, as
opposed to a function of frequency in frequency
-
domain methods. In time domain, the
transmitted signal is a low frequency, repetitive step function, the second
ary signal is
measured after the primary signal has stopped changing, and the primary field is not
present during conventional measurements.

In Australia, time
-
domain EM methods are preferred over frequency
-
domain
methods, as a large proportion of the cont
inent is subject to deep weathering. Time
domain methods give better penetration through the conductive cover, leading to better
quality measurements, and better definition of the top 50
-
100 m.

In this section, Maxwell’s equations are described in time do
main. The
diffusion characteristics of a plane electromagnetic wave and a current loop are shown
in a homogeneous half space, and Nabighian’s (1979) concept of “smoke ring” is
introduced. The important difference between smoke ring and eddy current diffu
sion is
shown, leading to interpretation principles of TEM anomalies that enable differentiation
between homogeneous half space, layered
-
earth and confined conductors.


59

3.5.2

Maxwell’s equations in time domain, in harmonic regime

Maxwell’s general equations are d
escribed in Equations 3.2
-
1 to 3.2
-
4. The
following is the summary of time domain relationships and is presented after Telford et
al (1976):


The following abbreviations are used:




E is the electric field intensity (in volt/m),



H is the magnetic field int
ensity (in ampere/m),



ω is the radian frequency (dimensionless),



μ is magnetic permeability (in weber/ampere m),



σ is the electrical conductivity of the medium (in siemens/m), and



ε is electric permittivity (dimensionless).


If E and H fields are varying at an arbitrary frequen
cy

, than


t
j
e
E
t
E

0
)
(









Equation
3.5
-
1

and

t
j
e
H
t
H

0
)
(








Equation
3.5
-
2



By taking the curl of equations 3.5
-
1 and 3.5
-
2
, and using the identity;


A
A
A
2
)
(
)
(














Equation
3.5
-
3


the equations can be written the following way:


E
E
i
E



2
2









Equation
3.5
-
4

H
H
i
H



2
2









Equation
3.5
-
5


These are the “Telegraphers’ equations”. It is apparent from the above equations
that E and H fields propagate through space as a wave. During geophysic
al exploration,

60

when the ground is conductive, and frequencies are low, the real part of the expressions
becomes negligible, so that the equations reduce to the following form:


E
i
E



2







Equation
3.5
-
6

H
i
H



2







Equation
3.5
-
7


These are the well
-
known diffusion equations. The field at low frequencies is
independent of the dielectric constant and depends mainly on the

conductivity of the
ground. The permeability is almost always assumed to be that of free space.

The above equations indicate that, during time domain geophysical exploration
over conductive terrain, which is typical of Western Australia, the electromagne
tic field
is best described using diffusion properties rather than wave properties. This is termed
the quasi
-
stationary approximation to electromagnetic fields. It has great importance for
the selection of suitable statistical techniques for the new inte
rpretation methodology
presented in subsequent chapters.

3.5.3

The diffusion characteristics of a plane electromagnetic wave

The diffusion of a plane electromagnetic wave into a conductor, such as the
ground, is shown in
Figure
3.5
-
1
:



Figure
3.5
-
1
. Plane electromagnetic wave diffusion into a conductor. From Grant
and West (1965).



61



















x
x
t
H
H
exp
cos
0




Equation
3.5
-
8


where





2


and




H is magnetic field intensity during diffusion (in ampere/m),



H
o

is initial magnetic field intensity (in ampere/m),





is angular frequency (dimensionless),



t is time (in second),



x is the depth of penetrati
on (in m),





is the skin depth, the distance where the amplitude falls by a factor of 1/e (in
m),





is magnetic permeability (in henry/m), and





is electrical conductivity (in siemens/m).


The “skin depth”,

, has a great importance in electromagnetic meth
ods,
determining the possible depth of exploration. Because of the great attenuation within
the distance of one wavelength at low frequencies, time
-
domain electromagnetic
methods work within distances less than a wavelength.

3.5.4

The diffusion characteristics

of a smoke ring

Nabighian (1979) showed that the secondary current
in a homogeneous half
space,

induced by a transmitter loop, is analogous to a diffusing “smoke ring” of current
(see
Figure
3.5
-
2
).


62


Figure
3.5
-
2
. A smoke ring moving downward and outward in a homogeneous
half
-
space as a function of electrical conductivity and time. From Nabighian
(1979).


As the secondary magnetic field decays, the smoke ring disperses downward and

outward away from the center of the transmitter loop, giving information about deeper
regions at later time. The smoke ring is defined by the current density maximum.
Nabighian (1979) describes the current density maximum in terms of velocity, radius
an
d diffusion distance:


t
V
0
2









Equation
3.5
-
9




0
37
.
4
t
R








Equation
3.5
-
10




0
2
2
t
d








Equation
3.5
-
11


where



V is the velocity of the smoke ring (in m/second),



R is the radius of the smoke ring (in m),



d is the diffusion distance of the smoke ring downwards (in m),




0

is the magnetic permeability of free space
(4




10
-
7

weber / ampere m),





is electric conductivity (in siemens / m), and


t = t
t = t
t = t
I
R
z
Loo p
1
2
3

63




t is time since turn
-
off (in microsecond).


From these equations, it is apparent that the velocity, radius and diffusion
distance of the smoke ring are all an inverse square
-
root functions of the electrical
conductivity of a homogeneous half
-
space. The velocity is also an inverse square
-
root
function of the time elapsed since turn
-
off, while the radius and diffusion distance are
square
-
root functions of time. The smoke ring
reaches a horizontal distance r from the
centre of the transmitter loop at a t time given by:


2
23
.
0
R
t









Equation
3.5
-
12


where



t is time since turn
-
off (in second),





is the magnetic pe
rmeability of the material (in weber/ampere m),





is conductivity (in siemens/m), and



R is the horizontal distance from the centre of the transmitter loop (in m).

3.5.5

Eddy currents in confined conductors

If there are confined conductors underground, the changi
ng primary field on the
surface results in circulating eddy currents within the conductor. The strength of the
eddy currents depends on the
rate of change

of the primary field, as well as on the
electrical conductivity contrast

between the anomalous body
and the host rock, and not
on the electrical conductivity of the body (West and Macnae, 1991):


J
a

= (

b

-


h
) (
E
p

+
E
s

)





Equation
3.5
-
13


where



J
a

is the anomalous (eddy) current density vector inside

the conductor (in
ampere/m
2
),





b

is the electrical conductivity of the body (in siemens/m),





h

is the electrical conductivity of the host rock (in siemens/m),




E
p

is the primary electric field vector (in volt/m), and


64





E
s

is the secondary electric field

vector (in volt/m).

Eddy currents decay more slowly than smoke rings. The rate of decay is
described using the decay constant (

) of the secondary signal at the receiver, which is
the time for the decaying voltage to decrease by 1/e from its initial value

(Sheriff, 1984).


E(t) = E
o
* e
-
t/









Equation
3.5
-
14

and



= t
2
-

t
1
/ ln (E
1
/E
2
) = L / R






Equation
3.5
-
15


where




E(t) is the measured EM respo
nse as a function of time (in volt/m),



E
o

is the initial value of E (in volt/m),



t is time (in second),





is the decay constant or time constant (dimesionless),



L is self inductance (in weber
-
turns/ampere), and



R is loop resistance (in ohm).



Figure
3.5
-
3
. Eddy currents in a sheet conductor. From Coggon (1995).

3.5.6

The secondary EM signal in time domain

The ratio of the secondary voltage to the primary voltage, as measured at the
receiver, according to Nabighi
an (1988) is:






















2
2
1



i
L
M
M
M
V
V
TR
CR
TC
p
s




Equation
3.5
-
16


65

where



Vs is the secondary TEM response,



Vp is the primary TEM signal,










L
M
M
M
TR
CR
TC

is the geometrical factor or coupling coefficient, where th
e mutual
inductances are:



M
TC

between transmitter and conductor,



M
CR

between conductor and receiver,



M
TR

between transmitter and receiver,





2
2
1








i

is the response function,




L
R
,





is the angular frequency,



R is the resistance

of the secondary loop, and



L is the self
-
inductance of the secondary loop.


Following from above, the duration and amplitude of the secondary TEM signal
in time domain depends on:



the electrical conductivity of the sub
-
surface (the background conductivit
y),
which is a function of numerous physical and chemical factors, as described
in Section
3.3
,



the shape and size of the confined conductor (the anomaly), and



the position of the conductors with respect to the receiver loop,

such as depth
and attitude.

3.6

SUMMARY

According to Maxwell’s first equation, an electrical current flow produces a
magnetic field, and, according to the second equation, a time
-
varying magnetic field
produces an electrical field.


66

In a time
-
domain electromag
netic geophysical survey, electric current flows in a
transmitter wire loop layed on the ground. According to Maxwell’s first equation, this
primary current generates a primary magnetic field, which penetrates into the ground,
diffuses downwards and outwa
rds. When the primary current is switched on and off in
the loop, the changing primary magnetic field induces secondary currents in the sub
-
surface conductors. Constitutive equations describe how physical properties of
materials influence electromagnetic

field behaviour. The most important physical rock
property for electromagnetic exploration is electrical conductivity, described by Ohm’s
law.

EM receivers can measure either the primary and secondary fields, or only the
secondary response during off
-
tim
es of the primary field. The electromagnetic response
depends upon the frequency, the electrical conductivity structure, and the geometric
coupling between the transmitter and the receiver. Electrical conductivity is the most
important property of Earth
materials influencing electromagnetic measurements. The
secondary or the total (primary plus secondary) field response is interpreted and the
electrical conductivity structure of the ground is determined.

For low frequencies used in time domain exploratio
n, there is a great attenuation
within the distance of one wavelength. Therefore, diffusion processes are more
appropriate to describe EM field behaviour than wave processes. Structures of interest
are small with respect to the free
-
space wavelength; con
sequently, the diffusion
processes can be described as quasi
-
stationary.

In order to characterise a given interval of the geo
-
electric section, the spacing
between the transmitter and receiver must be many times greater than the thickness of
the section be
ing investigated.
The shape of transient coupling curves is only weakly
sensitive to changes of resistivity with depth, in the geological profile.

Kaufman and
Keller’s (1983) observations explain why it is so difficult to carry out accurate
conductivity
-
depth inversion.