Table Of ContentTSIL FO SLOBMYS
A,B current electrode positions
A vector potential of the electric type
A (cid:12)9 vector potential of the magnetic type
a,b,c,d,s,u,v,w constants in various expansions
B magnetic induction vector
fi, B time derivative of magnetic induction
C,D constants
C Euler's constant
dl length of current dipole source
Es-P-(pi) electric field measured on the surface of a uniform half-space
with resistivity pi
E electric field vector
tangential component of electric field
% tangential component of electric field, normalized
auxiliary functions used in the expansion of certain expressions
So(), ){oN Struve and Neuman functions of order 0
H,h thickness of a layer
H magnetic field vector, B = ~H
RH
radial component of magnetic field
zH vertical component of magnetic field
,R ,Z ~ components of magnetic field, normalized
h R , h ,z h o
Im operator notation for imaginary component of operand
i
j,J current density vector
iiv
wave number
/~,/2
definite integrals defined in equations 3.129 and 3.130
I current
In operator notation for inphase component
integrals defined in equation 2.253
Jo(mr), J1(mr) Bessel functions of first kind of argument (mr) and of order 0 or 1
as indicated
/o(), ~o() modified Bessel functions, order 0, of the first and second kinds,
respectively
i,,(), Kn() modified Bessel functions, order n, of the first and second kinds,
respectively
path of integration
L o definite integral defined in equation 1.346
definite integrals defined in equation 1.219, also in equation 1.341
L 1 , L 2 , L k
L length
P distance measured in units of skin depth, p = L/6
Lon, Mon definite integrals defined in equations equation 3.126 and
equation 3.127
M,N measuring electrode positions
M magnetic or electric moment
separation constant, inverse to distance
m
m i (m 2 _ k~) 1/2 modified wave number
-
n unit vector normal
n, i counter
n - (m 2 - kF) 1/2
modified wave number in the longitudinal and normal directions for
5 = (m 2- k~) 1/2
an anisotropic medium
reflection factor
operator notation for quadrature component of operand;
also, collection of variables in certain developments
viii
0 geometric factor in frequency sounding (equation 1.432)
Re operator notation for real part of operand
R,O,r spherical coordinates
r,O,Z cylindrical coordinates
radius vector in cylindrical components,
distance from source to receiver
Fresne's integrals
Rg grounding resistance
R. modified plane-wave correction factor for magnetic vector potential
Rn modified plane-wave correction factor for electric vector
potential -i~ Sr/ 2
lio saturation
S o
s,s
integral conductance rof layer, S = H/p
sign signum operator
t time
integral resistance for layer, T = Hp
x,y,z Cartesian coordinates
x = ikr distance normalized by wave number
U 2 = m2+ 2nk modified wave number
normalized distance in time domain
u = 2~R/~
u--mr representation rof argument in Bessel's equations
U Scalar potential
X,Z, gW auxiliary potential functions used in solving Maxwel's equations
Z impedance, Z - Eo/H r in plane wave
step function in time
Ah z difference of normalized vertical magnetic field at two frequencies
Ab z difference of normalized vertical induction at two frequencies
6O loss tangent
O6,~ constants
-io~G ,21//1 square of normalized thickness
xi
dielectric constant, arbitrarily small number
electromotive force
delta function, skin depth
conductivity
gamma function
ratio of thicknesses of two successive layers, v i = H i + I/Hi
q surface density of induced electric charges
r probability integral function of operand u
% wavelength, or ridge regression parameter
coefficient of anisotropy
r flux density
:I = ,j~R normalization factor for scaling time
resistivity
P
apparent resistivity, often with superscript indicating
aP
field component used in computing it
magnetic permeabiUty
ratio of resistivity of two successive layers, ~i = Pi + I / Pi
frequency, rps
co
s~ = 2 t/l~Sr time normalized to conductance of layer
longitudinal resistivity (parallel to bedding)
Pl
normal resistivity (across bedding)
nP
time-domain apparent resistivity computed by normalizing
the observed field component to that same field observed
over a half-space with the resistivity of the first layer
~P time-domain apparent resistivity based on late part
of transient curve
2~P time-domain apparent resistivity based on early part
of transient curve
porosity
NOITCUDORTNI
Almost twenty years ago one of the authors together with Professor .G .V Keller published
ycneuqerF" and Transient Soundings," Elsevier, 1983. This monograph described physical
principles, theory, and interpretation of different electromagnetic soundings. It also
contained examples of the application of transient soundings for petroleum and
geothermal exploration. Certainly, this information was not novel but rather summarized
achievements of many geophysicists during a long and successful history of the
development and application of soundings. Let us briefly outline these very important results:
.1 From the beginning it was realized that, as in the case of Schlumberger soundings, a
horizontally layered medium is the most suitable for the use of electromagnetic
soundings.
2. Measurements of either frequency or transient responses of electromagnetic fields
allow us, in principle, to study a geoelectric section.
Because of the skin effect at the high frequency end of the spectrum and at the early
0
stage of the transient response, depth of investigation is minimal.
In contrast, at the late stage of the transient response depth of investigation
Q
increases. In other words, the upper layers of a horizontally layered medium become
transparent.
It is essential to our understanding that such behavior is observed regardless of
separation between transmitter and receiver. This fact permitted development of
transient soundings in the near zone, which became a conventional widely used
method.
.5 At the low frequency part fo the spectrum, depth fo investigation yllaitnesse depends
on the component fo the magnetic dleif measured. roF ,ecnatsni fi a reviecer si located
ylevitaler close ot the source and the quadrature component fo the magnetic dleif si
observed, this depth si usually very ,llams sseldrager how large skin depth .si
.6 At the same time, when the in-phase component fo the secondary magnetic dleif si
measured, the depth fo investigation becomes greater with decreasing frequency.
This tcaf remains true even when separation tends ot .orez
.7 At the low frequency end fo the spectrum, the quadrature component fo the magnetic
dleif si defined by the primary cirtcele dleif and the distribution fo .ytivitcudnoc
In other words, noitcaretni fo currents can be neglected. This tcaf yllacitsard seifilpmis
the theory fo the electromagnetic gniliforp and the noitaterpretni fo .atad
xi
8. At each frequency the quadrature and in-phase components of the magnetic field are
related to the geoeLectric section in a different manner. However, their frequency
responses contain the same information.
9. With increasing separation between transmitter and receiver, depth of investigation
also increases when the quadrature component of the magnetic field is measured,
provided that frequencies are sufficiently low. roF this reason an equivalence of
frequency and transient soundings in the far zone can be easily observed.
10. However, when separation is small, an advantage of the transient soundings becomes
obvious. It is related to the fact that at the woL frequency part of the spectrum, the
secondary in-phase component is much smaller than the primary field, as well as the
quadrature component. Correspondingly, measurements of the in-phase component are
a very difficult task in this frequency range. Thus, in spite of the fact that in theory
(Fourier's Integral) transient and frequency soundings are equivalent, realization of
this phenomena is hardly possible at small separations.
At the same time, the use of relatively small arrays is very attractive because lateral
resolution becomes better.
11. Measurements of electrical fields greatly increase the influence of geologic noise and,
therefore, observations of magnetic fields are much more preferable.
lLA this information about electromagnetic fields in a conducting medium was known for
many years and became the foundation for different electromagnetic soundings and
profiling methods. Among them:
1. Frequency soundings in the far zone, such as magnetoteLLuric (MT) and controlled
source audio-magnetoteLluric soundings .)TMASC(
2. Transient soundings with relatively short separations.
3. Geometric soundings at the woL frequency end of the spectrum, provided that
magnetic fields are measured.
4. E(ectromagnetic profi(ing.
Without any doubt this comprehensive analysis, as LLew as field experience, had a strong
influence on the application of these methods. In this Light it is proper to emphasize the
outstanding role of the Canadian company Geonics Ltd., which developed several widely
used systems for transient soundings and electromagnetic profiling. For many years
.rM Duncan LLieNcM successfuLLy deL this company.
During the Last twenty years the area of application of electromagnetic methods was
strongly enlarged. They are now widely applied in ground water investigations and
geotechnicaL and environmental engineering, along with their use in hydrocarbon
exploration, mainly in Russia and China.
This fact motivated us to write this monograph, in which we systematicaLLy explain the
physical principles, theory, and interpretations of the different electromagnetic soundings
and profiling and illustrate their applications in new areas.
iix
CHAPTER I NIAMOD-YCNEUQERF ROIVAHEB FO ECHITTENGAMORTCELE
DLEIF DETARENEG YB A CITENGAM ELOPID
Introduction
In developing the theory for electromagnetic sounding, we will be concerned mainly with
the behavior of electromagnetic fields observed on the surface of a horizontally stratified
Earth; for the most part, we will not be concerned with the behavior of the fields within
the Earth. However, in order to understand completely the behavior of the electromagnetic
field, it is useful first to investigate in detail the behavior of the electromagnetic field of a
magnetic dipole situated within a uniform full-space. This approach will permit us to
obtain insight into the physical principles on which electromagnetic soundings are based
even though the effect of boundaries between various media cannot be discussed.
1. niamoD-ycneuqerF citengamortcelE ffeld of a Magnetic Dipole in a Uniform
gnitcudnoC dna elbaziraloP ecapS-lluF
Let us define a magnetic dipole with the moment:
-icot
M = Moe z 0 )1.1(
where M o = nSoJ is the magnitude of the moment, J = oJ e -Rot is the current provided
to the dipole, oc is frequency in radians per second, n is the number of turns in a loop, S is
the area enclosed by one turn of the loop, and z o is a unit vector along the z-axis. The
behavior of the electromagnetic field is completely specified by MaxweU's equations:
E~
curl H = oE + ~-~e (1.2)
~B
curl E = -~-~ (1.3)
div H = 0 (1.4)
div E = 0 (1.5)
where H and E are the magnetic and electric field vectors respectively, o is the
conductivity, e is the dielectric constant, B = ~H and ~ is the magnetic permeability. The
most direct approach for obtaining a solution for MaxweWs equations is through the
RETPAHC 1
definition of a vector potential of the magnetic type based on the fact that the divergence
of the electric field is zero:
E = curl *A (1.6)
The vector potential *A is not uniquely defined by equation 1.6, inasmuch as the
divergence condition can be met for any additive scalar function. Substituting
equation 1.6 into equation 1.2, we can obtain an expression in which the scalar function
can be defined as a scalar potential, :U
*A~
H = *A~( + ~-~ - gradU )7.1(
Thus, in accordance with equations 1.6 and 1.7, the electromagnetic field can be described
using two potential functions, *A and .U Now, we can demonstrate that the field can also
be defined using only the vector potential. Expressing equation 1.3 in terms of the
potentials *A and U rather than the field components E and H, we have:
*A~ *A2~ ~U
curl curl A* - - ~ ~ - ~~ +
~t 7 ~ grad~-~
Taking into account that the field is oscillatory and making use of the vector identity:
curl curl A* = grad div *A - V 2 *A
we can rewrite this equation "sa
grad div A* - V2A * = (iapa) + co2sp)A * - Ro~gradU (1.8)
Because the relationship between the potential functions *A and U has not been specified
up to this point, it is possible to simplify equation 1.8 by selecting a gauge condition that
will permit us to eliminate the scalar potential from further equations. Consider a
relationship between the potentials U and *A as follows:
(1.9)
divA* = -Rop U
With this gauge condition, the wave equation for the vector potential A* becomes:
V2A * + k2A * = 0 (1.10)
where k 2 is the square of a wave number for electromagnetic waves traveling through the
medium and is defined in terms of the properties of the medium as follows:
k 2 = i(~o3 + (o2Ep (1.11)
Thus, using the gauge condition in equation 1.9, the electromagnetic field can be
expressed in terms of only the vector potential A*"
NIAMOD-YCNEUQERF ROIVAHEB FO CEIHTTENGAMORTCELE DLEIF DETARENEG YB A CITENGAM ELOPID
E - curt *A and i(opH - k2A * + grad div *A (1.12)
oS far, derivations of the equations describing the etectromagnetic field have been done
independently of the coordinate system. Let us now take a spherical coordinate system ,r
0, and r and a cylindrical coordinate system ,R 0, and r with a common origin. The dipole
is located at the origin with its moment directed along the z-axis of the cylindrical system
and at the angle sin0 = 0 of the spherical system (see figure 1.1 for a definition of these
directions).
Z
l~,I<
erugiF 1.1. Definition of a spherica coordinate system for esu in soMng s'ewxaM
.snoitauqe
Assume that the components A~ and A* r era ,O os that the fietd is comptetety described
by a singte component of the vector potential, A*, which is a function of the distance .R
Equation 1.10 takes the following form in the spherical coordinate system:
1dR( dA*~ k2 )3~.~(
2R R2---~) + %A = 0
A sotution for this equation is:
ikR
A*- C e --K 141.1(
where C is a constant. The divergence of the potential given by equation 1.1/+ is:
OA* e ikR
divA* = az = C -7 (ikR- 1) cos 0
so that from the definition in equation 1.9, we have the following expression for the scalar
potential:
RETPAHC 1
ikR
e
U - C (l-ikR)cosO (1.15)
i(ogR 2
It has been demonstrated that the static magnetic field, ,oU caused by a magnetic dipole is-
M
= cos0 (1.16)
oU 47rR 2
eW can utKize this as a limit condition near the source and determine the constant 6 by
evaluating equation 1.15 as U--) U ,o with oc --) 0:
C- ir (1.17)
4n
Therefore"
ikR
A* -- ia)~M e (1.18)
z 4n R
and
ikR
divA* - - i~opMe (ikR-1)cose )9~.1(
r74 R 2
It is now possible to obtain expressions for the electric and magnetic field components
using equation 1.12. From equation 1.6, the electric field vector lies in a plane
perpendicular to the dipole axis and has only the 0E component, which can be described
as:
E e = cufl~A* = } (RA~)- (1.20)
In the cylindrical system we have the following relationships between vector potential
components"
A~ = A*cosO, ~A = -A*sinO
Substituting these expressions into equation 1.20 and carrying out some relatively simple
algebraic operations, we obtain:
i~o~tMeikR( 1 - ikR) sin 0 )~2.1(
0E = 47rR 2
The magnetic field vector has two components:
