Geophysical Field Theory and Method, Part C, Volume 49: Electromagnetic Fields II (International Geophysics)

-

iWJ-L3M ikz 2rrk 2r 4 e '

(1.64)

Using the same approach with Eq. (1.24) we have

B

=

z

9Mp eikz 2rriwr5

(1.65)

Thus, for large values of r /8 the electromagnetic field observed beneath the surface of the earth decays exponentially in the vertical direction, just

1.1 Vertical Magnetic Dipole on the Surface of a Uniform Half Space

19

a M

h

o

'Y

z

b

0.50

0.30

0.70

.r,

s

1.10 +--..---r-r-T""T"T"1'"TT""--r--r-'"T""T--r'T"TTT---,----r--.,-"T'""T"~ .1

10

100

Fig. 1.1 (a) Vertical magnetic dipole above uniform half space; (b) quadrature and in-phase components of field bz ; (c) quadrature and in-phase components of field b.: (d) quadrature and in-phase components of field e'l" (Figure continues.)

as in the case of a plane wave propagating downward. An important feature of this behavior is that the phase of the field does not change in a horizontal plane. In other words, the horizontal planes are surfaces of equal phase. In contrast to the case with plane wave propagation into the earth, surfaces of equal amplitude do not form horizontal planes, since the field strength decreases with the distance from the dipole. As is well known, in carrying out magnetotelluric soundings the natural electromagnetic fields of the earth are used. In most cases these fields are

20

I The Quasistationary Field in a Horizontally Layered Medium

measured in the range of large parameters regardless of the type of generator of the primary field. For this reason, the ratio of tangential components of the field is independent of the distance from these generators. Thus, we have investigated in detail the behavior of the electromagnetic field over two asymptotic ranges of parameter p, In particular, for given C

br 0.40

-0.10

-0.60

-1.10

r

T

-1.60 .1

10

100

d

e.

0.90

0.70

0.50

0.30

0.10

.L S

-0.10 .1

Fig. l.l

10

(Continued)

100

I.1

Vertical Magnetic Dipole on the Surface of a Uniform Half Space

21

a

Ib~1 1.20

0.80

0.40

r

T

0.00

100

10

.1

b

Ibrl 1.20

0.80

0.40 r

T O.OO+--O;==;:....,.-rrT"T"T-r--..,......--,-.--r-rrT'T.--.----,....-'T""""T..,.....~

.1

10

100

Fig. 1.2 (a) Amplitude response of field b;; (b) amplitude response of field br ; (c) phase response of field b;; (d) phase response of field br • (Figure continues.)

distance r and conductivity 'Y, they correspond to the low- and highfrequency spectrum. The behavior of the quadrature and in-phase components of the field, as well as their amplitudes and phases, for arbitrary values of p are shown in Figures I.1b through I.3a. Until now we have considered the behavior of the field in the frequency domain. Next let us study the transient responses of the field of the vertical magnetic dipole on the earth's surface, assuming that the dipole

22

The Quasistationary Field in a Horizontally Layered Medium

c <j)(bi)

3.00

2.00

1.00

r 0

0.00 0.1 -1.00

-2.00

d 0.90 ell (b r) 0.40

"0.10

-0.60

-1.10

r

T

-1.60 .1 Fig. 1.2 (Continued)

10

I.1

Vertical Magnetic Dipole on the Surface of a Uniform Half Space

23

a 1.50

1.10

0.70

0.30

..L

s

0.1O+--,--,,.,...-r-rr-,.,.,..,r----,----,---.-;=;,TTT---.---r-"T'""T-rri~ .1

10

100

b trn

(0

. - - - - - - - - - f - - - - - - - . , Re (0

Fig. 1.3 (a) Phase response of field e
moment changes as a step function:

M(t) =

{~

t
(1.66)

In deriving the equations describing the transient field, we proceed from Fourier's transform, which in the case of the step function excitation has

24

I

The Quasistationary Field in a Horizontally Layered Medium

c 1.0 -+--=----~

0.5

Fig. 1.3 (Continued)

the form

B(t)

B( w) J __ ew 1 - - - J ---e Ttri w 1

= -

E( t ) -

00

-.

2rrl

iw t

dw

-iwt

dw

-00

00

E(w)

(1.67)

-00

where B(w) and E(w) are the fields in the frequency domain.

I.1

25

Vertical Magnetic Dipole on the Surface of a Uniform Half Space

First, we obtain expressions for the electric field Ecp and the vertical component of the magnetic field B z . With this purpose in mind, the following integrals will be used: 1 -f 27T

00

-00

e- i Wf 0 -dw={
tO ( 1.68)

and

t 0 where r/J(u) is the probability integral

and

27Tr u=-- , T

Differentiating the second equality in Eqs. (1.68) successively with respect to r, we obtain three useful identities: 1 -f 27T

1 -f 27T

00

.

e -iwt

ikre 1k r -

-00

1 -f 27T

00

.-

dco

~ =

.:

- u 3e- u

-IW

2

(1.70)

/2

7T

3 . e(ikr) e 1k r _ . _ doi =

-

~

113(U 7T

-IW

-00

(1.69)

/2

7T

2 . e -iwf (ikr) e 1k r _ . _ dw =

-00

2

ue?"

-

-IW

iw t

00

-

2

l)e- li

-

2

/2

(1.71)

Taking into account Eq. 0.34) and making the appropriate substitutions in the expression for the electric field in Eq. 0.67), we obtain

E

= cp

-

3Mp -

27Tr 4

[ r/J(U) -

~-

2]

1 2) e- li /2 u ( 1 + -u

3

7T

(1.72)

Applying the same approach to Eqs. (1.35) and 0.67), it follows that aB, - = -9Mp - - [ r/J(ll)-

at

27Tr 5

~-e7T

2 u2

/

4

2u ( 1+-+u u )]

3

9

(1.73 )

26

I

The Quasistationary Field in a Horizontally Layered Medium

It is clear that

(1.74) Considering that

(2

u

jo 4>(x)dx=u4>(u) + V;: (e-

U

2

/2-1)

and integrating each term in Eq. 0.73) by parts, we obtain

In order to determine the horizontal component of the magnetic field, we first consider the function aA'i;az. As was shown earlier, it can be expressed in terms of the function (1.76) where a 2 = 1/(YIL). To begin with we carry out a Fourier transform on the second derivative of the function F with respect to time. Doing this, we have a2F -2 =

at

MIL - - - 2-2

87Ta

iwt

(iw)2 e f-00(m+m 1) 00

2

dto

j

00

mlo(mr) dm

(1.77)

0

Consider for a moment the integral C defined by (1.78) where

The integrand in Eq. 0.78) has a branch point when m 1 = 0, that is

27

1.1 Vertical Magnetic Dipole on the Surface of a Uniform Half Space

Integrating along the negative imaginary axis instead of along the real values of w (Fig. I.3b) and considering that w changes within the interval:

and that on either side of the path the radical m 1 has opposite sign, we obtain

or (1.79)

Defining a new variable m 1 = -iv, we have

and

or

{II}

4

a i 00 C=--2] (m 2 + 471"

v

2 2 2 2 ) 2 e-al(m+v)v

0 2

.

(m + IV)

2-

2

(m - iv)

2

a 4 me- a m I 00 2 2 - - - - - ] v 2e- a t v dv 71"

dv

(1.80)

0

The integral on the right-hand side of Eq. (1.80) is known:

Thus (1.81 )

28

I

The Quasistationary Field in a Horizontally Layered Medium

Substituting this last expression into Eq. (I.77), we obtain

a2F -a 2 - t

Mf-L{;a 4 2 3/2 tr t

fme 00

2

2

2 -a 1m] (

0

0

mr )i d am

(1.82)

or

Making use of the tabulated integral

(r -2 2

fa

oo

2 e:" 2 1m] (mr) dm

0

=

{; r 2 I(Sa 2 III ---e-

2atl/2

0

)

8a t

we have (1.83)

The argument of the Bessel function can also be written as 27fr

where u = - T

Then, after the indicated differentiation and integration, we arrive at the result: (1.84)

and

where (1.85)

In summary, let us write down expressions for the quasistationary transient

1.1 Vertical Magnetic Dipole on the Surface of a Uniform Half Space

29

field observed on the earth's surface of a uniform half space.

where

(1.86)

br=4e-U2/4[(2+

a:" ~ _ ~:;,

[~(") _ ~

:2)/1(:2)_ :2 10(:2)]

e-"I'"( + ~' + ~')] 1

(1.87) Equations (1.86) describe the field components when the dipole current is turned on at the instant t = O. In order to obtain expressions for the field when the current is turned off, it is necessary to subtract the function b, from unity, while the functions e

cP(u)

~

1,

if u ~

00

we have

E and

3Mp
=---

21Tr 4 '

9Mp

B,= - - -5t •

21Tr

(1.88)

As follows from Eqs. (1.88), the dependence of the field on the resistivity

30

I

The Quasistationary Field in a Horizontally Layered Medium

of the medium and the separation r is the same as that in the frequency domain for large induction numbers. For example, on the earth's surface the electric field does not depend on time at the first instant, and the magnetic field is zero. With an increase of time both components of the magnetic field grow, but at the early stage

With greater conductivity in the medium or distance from the dipole, the early stage behavior is observed at later times. Comparison with the exact solution shows that Eqs. 0.88) are valid with a reasonable accuracy for the electric field E", and the component B, when 71r < 2, but for the horizontal component B, only if 71r < 1. It is clear that at the initial moment, induction currents concentrate near the dipole and with increasing time their magnitude becomes larger at greater distances from the surface z = O. At relatively short separations r, the early stage is observed when currents are located quite close to the earth's surface. However, at larger separations the field behavior described by Eqs. 0.88) still holds when the induced currents are located at relatively great depths. From Eqs. (1.88) it follows that at the initial instant, when the dipole current is turned on, the magnetic field is zero at every point on the earth's surface, but it begins to grow immediately. In other words, regardless of the separation for any small time t, the field is nonzero. This contradicts the fact that electromagnetic energy propagates with a finite velocity in the medium. This means that the equations describing the quasistationary field can only be used when a measurement time is several times greater than the time which is necessary for propagation of the field from the dipole to an observation point. It is proper to note that in the wave zone when either r 17» 1 or r18» 1, the Poynting vector has very interesting features. It is mainly directed downward and this fact indicates that the electromagnetic energy arrives at the wave zone through free space. Also, there is a small horizontal component of the Poynting vector, showing that some part of the energy travels along the earth's surface. Thus, in the wave zone the electromagnetic energy mainly penetrates into a conducting medium where it transforms into heat. At the same time, in the near and intermediate zones the energy can arrive at the observation point along different paths located in free space and the conducting medium. Next, let us consider the late stage of the transient response. Expanding the probability integral and the modified Bessel functions for small values

I.1

Vertical Magnetic Dipole on the Surface of a Uniform Half Space

31

of the parameter u:

cf>(u)"'"

{f ( -

7T"

5

U3

u 40

u - - + - - ... 6

)

we arrive at approximate expressions for the field components when the ratio 'T jr» 1. They are J-L5/2 r3/2M

E "'" r '" 407T";';t 5 / 2

B:"", Br

"'"

B - "'" z

±

J-LM [ 47T"r 3

1+

2

J-LM r 2 47T" 32t 2 (YJ-L) J-LM

(YJ-L )3/2]

15 r 3 ;';t 3 / 2

(1.89)

(YJ-L )3/2

-----:== -----::---,;:-307T";'; t 3 / 2

where B: and B; are the vertical components of the magnetic field during the "time on" and "time off," respectively. From Eqs. 0.89) it follows, in contrast to the early stage, that the vertical component of the field is stronger than the horizontal one (B z > B r ) , and it is independent of the separation r. Moreover, at the late stage the electromagnetic field is characterized by a relatively high sensitivity to changes in conductivity of the medium: E «p - y 3 / 2 '

Equations (1.89) describe the field with reasonable accuracy when the parameter 'T jr > 16. The transient responses for functions bz- , b., and e", are shown in Figure I,3c. At the initial moment when the current is turned off, the vertical component of the magnetic field is equal to the stationary field caused only by the current of the dipole, B~. Then, with an increase of time the field B, decreases, passes through zero at 'T jr "'" 2.3, and during

32

I

The Quasistationary Field in a Horizontally Layered Medium

the late stage reverses sign and is inversely proportional to t 3 / 2 • The horizontal component has a maximum around T Ir = 2 and its magnitude exceeds that of the vertical component B~ .

1.2 Equations for the Field on the Surface of a Layered Medium

Suppose that a magnetic dipole with moment M is located at the earth's surface (Fig. 1.3d) at the origin of a cylindrical system of coordinates. Because the primary electric field has only a tangential component E~ which does not cross interfaces between media with different conductivities, no electric charges appear. For this reason, the induced currents are situated in horizontal planes and in view of the cylindrical symmetry of the electromagnetic field, just as in the case of a uniform half space, there are only three components:

E={O,E'P'O},

B = {B r ,0, Bzl

(1.90)

We again describe the field with help of the vector potential of the magnetic type A*, which has only a z-component A*:

A*={O,O,A*}

(1.91)

As was shown earlier, the vector potential A* satisfies Helmholtz's equation in each medium, indicated by the subscript i: (1.92)

A;

where kl = iYi/.LW and is the z-component of the vector potential in the ith medium. From the continuity of the tangential components of the electromagnetic field at interfaces, it follows that at each boundary between layers we have

aA;

aA;+1

az

az

if

Z

=h i

(1.93)

where hi is the depth from the earth's surface to the boundary between the i and i + 1 medium. Equations (1.92) and (1.93), along with conditions near the dipole and at infinity, constitute a boundary value problem, which we solve in the

1.2 Equations for the Field on a Surface of the Layered Medium

33

frequency domain. First, let us consider a two-layered medium. Taking into account Eqs. (1.12)-0.15), the vector potential A* can be written in the following form for each medium:

where m 1 = /m 2 - ki, m 2 = / m 2 - k~, and k 1 and k 2 are wave numbers for the first and second layers, respectively. From the boundary conditions we obtain a system of algebraic equations for determining the unknown coefficients Do, C 1 , D 1 , and C 2 • In fact, making use of the orthogonality of Bessel functions, we have

(1.95)

where HI is the thickness of the first layer. Inasmuch as our main interest is to study the field on the earth's surface, we determine only the coefficient Do. Eliminating C2 , C 1 , and D 1 from the system (1.95), we obtain

(1.96)

where ( 1.97)

Now taking into account Eqs. 0.3), the components of the electromagnetic

34

I

The Quasistationary Field in a Horizontally Layered Medium

field at the earth's surface can be written as

(1.98)

where E~ and B~ comprise the primary electromagnetic field. It is obvious that using the same approach it is possible to find an expression for Do when the medium consists of N layers. In order to perform this generalization it is convenient to present function Do in a different form. With this purpose in mind, we take the ratio of the last two equations in the set of Eqs. (1.95):

or 1 + (Dj/Cj)e2mlHl

mj

1 - (Dt/Ct)e2mlHl

m2

Since

we have 1 + (Dj/Ct)e2mjHl

1 + e 2[ m

j

H j + (1/2j{n(DIiC jj]

1 - (DIICt)e2mjHj

1 - e 2[ m

j

H j +(1/Zj{n(DIiC j )]

Thus 1 coth m H + -1 en -D ] Cj [ t 1 2

=

-

mt mz

-

1.2 Equations for the Field on a Surface of the Layered Medium

35

Whence (1.99) From the first two equations of the set (1.95), it follows that I+D o

C\+D\

m

I-Do

C\ -D\

m1

---=

or 1 +D o

--- =

I-Do

mIDI --coth-tnm1 2 C1

Applying Eq. (1.99) we have

and I+D o =

2m -

-

-

-

(LlOO)

-

m +m 1/R 2*

where

R 2* = coth[ m,n, + coth-\ ::]

(LlOI)

Thus, the following expression holds for the vector potential on the surface of a two-layered medium:

A*

=

iWJLM [,mJo(mr) dm 27T

0

(Ll02)

m + m\/R 2*

Now, applying the method of mathematical induction, the expression for function R n * of the corresponding n-layered medium can be written as

s.: = coth[ m\H\ + coth- 1 : : R n - 1,*]

*

1

2

R n - 1 , = coth[m 2H2 + coth- m m3Rn R 1* = I

2'

*]

(Ll03)

36

I

The Quasistationary Field in a Horizontally Layered Medium

Correspondingly, for the vector potential A* we have (1.104) Let us note that Eq. (I.l04) also remains valid in the general case when the influence of displacement currents cannot be neglected, provided that m is replaced by m o (m o = -1m 2 - kJ). Next, we introduce a new variable x = mr in Eqs. (1.98). Then we have B,

=

(1.105)

B~br'

where ecp = 1 + {~)XDo(X)J\(x)

dx

o

b, = b,

=

[~x2Do(X)J\(X) o

dx

(I.106)

1 + {'x 2D o(x)Jo(x) dx

o

and

Xl -X

x lO = - - ,

Xl +X

(1.107)

and

It can be seen from Eqs. (I.l05)-(I.l07) that the electromagnetic field

expressions for a two-layered medium, just as in the case of a uniform half space, can be written as the product of two functions. One of them is the primary field, which depends on the dipole moment M, on the distance between the dipole and the observation site r, and, in the case of the electric field, on the frequency w. The second function (ecp' br , and b) describes the influence of induced currents on the electromagnetic field and depends on such dimensionless parameters as r / 8\, r / HI' and P2/P\, where 8\ is the skin depth in the first layer. These functions, ecp' b.; and bz ' are a measure of how much the corresponding components of the electromagnetic field differ from the

37

1.3 Behavior of Field when Interaction between Induced Currents Is Negligible

primary field. In order to evaluate them, the integrals in Eqs. 0.106) must be evaluated numerically. Of course, the representation of the field as a product of two functions also holds in the general case of an n-Iayered medium, and this representation follows from Eq, 0.104).

1.3 Behavior of the Field when Interaction between Induced Currents Is Negligible Now we investigate the behavior of the field on the surface of a layered medium assuming that the values of the parameters r /0; and H;/o; tend to zero. Let us begin with the case of a two-layered medium. Then expanding the functions x 10 and x 12 [Eqs. (1.107)] in a power series in kl/x 2 and discarding all terms but the first, which is proportional to w, we obtain

x -x(l- (kfr2/x2))1/2 x+x(l- (kf r2/x 2))1/2::::: -X2 12 x = Xl +X 2

1 kfr 2

-47

(1.108)

1 (kf-kDr 2

Xl

::::: -

4

x2

Substituting Eqs. (1.108) into Eqs. (1.107), we arrive at an approximation to the function Do: (1.109)

Replacing the function Do(x) in Eq. (1.106) by the right-hand side of Eq. (1.109), we obtain an asymptotic expression for the electromagnetic field for small induction numbers r /0. Because

(1.110)

38

I The Quasistationary Field in a Horizontally Layered Medium

we have the following expressions for the quadrature component of the magnetic field and for the in-phase component of the electric field:

Before analyzing the meaning of the expressions in Eqs. (Ll l l), it would be proper to make one comment. The amazing simplicity of Eqs. (1.111) is a consequence of the fact that the radical expressions Xl and X z have been expanded in a series of powers of klr z/x 2 • This is equivalent to the assumption that over the range of integration, the inequality Iklr 2 1/x2 < 1 holds. This is not strictly correct because there are always some small values of x for which the ratio Iklr 2 Vx2 is greater than unity. In other words, this approach cannot be used over the range in x corresponding to the initial part of the integration: O::;;x:::;;lkr! It should be noted that as Ik;rl decreases, this range becomes narrower and, in accord with Eqs. (LlO?), the integrands for small values of x are at least proportional to x 2. It turns out that an incorrect representation of the function Do(X) over the initial part of the range of integration does not result in an error in determining the leading term of the series for the quadrature component of the magnetic field when the parameters r/8; and H;/8; are small. It can readily be seen that an attempt to take successive terms of an expansion of radicals Xl and X 2 into account leads to divergent integrals. As follows from Eqs. (Ll l I), these asymptotic expressions describe only the quadrature component of the magnetic field which is proportional to w; they do not contain any information about the in-phase component of this field. That is, this asymptotic approach does not take into account the interaction of induced currents. Therefore, the in-phase component of the secondary magnetic field is assumed to be negligible with respect to the quadrature component over this particular range of parameters r/8;. To obtain asymptotic expressions for the in-phase component In BS and improve the representation of the quadrature component, we describe

1.3 Behavior of Field when Interaction between Induced Currents Is Negligible

39

in the next section another approach which is based on the correct expression for function Do for small parameters r18; over the entire range of the variable x. Referring again to Eqs. (1.111), it C8n readily be seen that these equations can be written in a more compact form: iWJLM. E E In £'1' = --IWJL{YIQI + 'Y2Q2} 167T

Q s, =

~~r

iWJL{

'Y)Q~

+ 'Y2Qn

(I.l12)

JLM Q B, = - 167Tr iWJL{ 'YIQf + 'Y2QD

where

Q~

=

../1 + 4hi ,

1 + 2h) - .

1 QZ = 1 - -p-======-

v1+ 4h

I

QZ _ 2 -

r

1

(1.113)

-,====,"

VI + 4hr VI + 4hr Qr _

2h)

-'-----;===~_

2-

V1+4hi

where h) = Hl/r. These functions Q are usually called geometric factors because they depend only on the distance r and the thickness of the layers. Equations (1.112)vividly demonstrate the remarkable simplicity of the behavior of the quadrature component of the magnetic field when the parameters r18 1 and r 18 2 are very small. In fact, this component of the field B and the in-phase component of the electric field are the sums of two terms, and each of them is a function of the geometric parameters and the conductivity of the appropriate layer. This means that in the present approximation the interaction between induced currents is neglected, that is, induced currents arise only as a consequence of the primary electric field:

£0= 'I'

iWJLM sin e 41TR2

In other words, it is assumed that the magnetic field of these currents is significantly less than the primary field B o ' and for this reason it has no appreciable effect on the induced currents. Thus, the density of induced

40

I

The Quasistationary Field in a Horizontally Layered Medium

currents at any point in the medium is independent of the intensity of currents in nearby parts of the medium. This independence is clearly reflected in terms of geometric factors that describe the field. Now let us consider some features of function Q. First, as follows from Eqs. (1.113), their sum is always equal to unity: Q~ + Qi = Qf + Q~ = Qr + Q~ = I

(1.114)

It is appropriate to point out that this result remains valid for an n-layered medium, too. In the case of a two-layered medium the geometric factors depend on the parameter h I only. For instance, if the separation r is very small with respect to the layer thickness HI' its geometric factor for each field component tends to unity, while the geometric factor of the lowermost medium (basement) becomes very small. This means that performing measurements of the quadrature component of the magnetic field or the in-phase component of the electric field near the dipole usually yields information about the conductivity of the upper layer only. In contrast, with an increase of the separation r, the geometric factor of the medium beneath the upper layer approaches unity. Therefore, the influence of induced currents in the layer with thickness HI becomes very small and the field on the earth's surface is practically defined by conductivity 'Yz. This consideration shows that measurements of the field Q B or In E in the range of small parameters with different separations allow us to obtain information about the geoelectric section. This procedure is usually called geometric sounding. At the same time, measurements with fixed separation r, that is, the profiling, permit us to trace lateral changes of conductivity, and this approach is widely used in geophysical methods. In accordance with Eqs. (1.113), geometric factors for different components of the field differ from each other. This means that in the range of small parameter r /8 and at a given distance from the dipole, these field components have a different depth of investigation. Let us note that this unusual simplicity of the expressions for the electromagnetic field greatly facilitates the solution of the forward and inverse problems. In contrast to electromagnetic methods based on the use of direct or alternating currents, we observe a unique situation where the field caused by induced currents in a given layer is independent of the conductivity of all other layers. Taking into account this fact, it is a simple matter to generalize our approximate solution to an N-Iayered medium, and with this purpose in mind it is sufficient to consider the function Do(x) for a three-layered medium. For instance, for the vertical component of the magnetic field we

1.4 Field of Vertical Magnetic Dipole in Range of Small Parameters r /

I)

41

have

j.LM N Q s, = --iwj.L L YiQi 167Tr i~l where the geometric factor of the ith layer is r Qi=

(4h~i+rZ)'/z

(1.115)

r

(1.116)

- (4hii+rz)'/z

and hI; and h Zi are the distances from the earth's surface to the top and bottom of the layer, respectively.

1.4 The Field of a Vertical Magnetic Dipole in the Range of Small

Parameters r / 8

Now we study in detail the behavior of the field in the range of small parameters r/ 0, that is, in the low-frequency part of the spectrum. This topic is of considerable practical interest since this portion of the spectrum contains information mainly about the deepest layers which are being explored. In contrast to the previous section, the interaction between induced currents is taken into account. To obtain an analytic representation of the field in the range of small parameters, we first make use of the expression for the vector potential on the surface of a two-layered medium, caused by induction currents. From Eqs. (1.94)-0.107) we have

A:

A: =

iWj.LM r ) --1 Do x, -, s Jo(x) dx 47Tr 0 00

(

(1.117)

0

where (1.118)

and Xl -X

x lO = - X,

+x

, Yz

s="I, and

42

I

The Quasistationary Field in a Horizontally Layered Medium

A:

Next we seek an expression for the vector potential in the form of a power series in terms of the parameter a. This can be accomplished inasmuch as the skin depth in both media is much greater than the distance from the dipole to the observation site. We consider the integral in Eq. (1.117) as the sum of two integrals: (1.119) where lal is much less than unity, and we look for a power series expansion for each of integrals. It should be clear that (1.120) First consider the internal integral and at the beginning assume that the lower medium has a finite resistivity. This means that the magnitude of the term xlOx21e-2x,hj in Eq. (1.118) is less than unity over the entire range of integration:

Then the function DO<x) can be written as a series:

(1.121 )

The series on the right-hand side of this equation alternates and converges rapidly for any value of x. Assuming that over the range of integration, Osxslal, the value of 12x 1 t h 11 is less than unity, we expand the functions e-2xjhl(f-l) and e- 2 x\h\{ in power series. Then, making use of the identity 1

if x < 1

l+x

and performing simple algebraic transformations, we obtain 00

Do(x)

=

L p~l

Fp(x)

(1.122)

1.4 Field of Vertical Magnetic Dipole in Range of Small Parameters r /1)

43

where, for instance,

sa 4

F1 = -

2

(x 2 +x)

(1.123)

(s-1)a 4 F 2 = 2xh 1

2

(x 2+x)

and so on. Then, the first integral on the right-hand side of Eq. (I.119) can be written as ;. (Ial

c: I: FAx)fo(x) dx

p=l

(1.124)

0

Because the variable x is very small, the function fix) can be represented as a rapidly converging series

where Ii are known coefficients. After substitution of this series into Eq. (I.124) and multiplying the denominators of the functions Fp by their complex conjugates, the integrals in Eq. (I.124) reduce to a tabulated one of the form I n = l,(Ialx n( x 2 + sa 4) 1/2 dx

o

Having expressed these integrals in terms of elementary functions, we can write their expansion as a series in a. As an example, the series describing integrals 10 and I[ are given below: 10 =

ll } 4 I 2 { 2"a + "4 s +I 2"s fn 2 - "4s fns a 1

- -sa 4 fna

2

+

I

= 1

5

1

+ -S2a 6 16

7

1

-S3a 8

64

S5a12 + ... 64 X 32 1 1 _s3/ 2a 6 + -s2 a 7 3 8 1 1 - -S3a9 + - S 4all - _ S 5a13 + ... 48 128 256

S4alO _

32 X 24 3 a 1 + -sa 5 3 2 1

(1.125)

44

I

The Quasistationary Field in a Horizontally Layered Medium

Performing a summation of the series corresponding to different functions Fp [Eq. 0.124)], we obtain an expansion of the integral

It is essential to emphasize that terms of the series can contain integer and fractional powers of w as well as its logarithm tn w. Next, consider the expansion in a series in a of the second integral on the right-hand side of Eq. (1.119): ["DO(X)fo(x) dx lal

The method used above, based on a series expansion of the term e-2xjhj( and fix), is not effective for evaluating the outer integral since the variable x increases without limit. Therefore, it is necessary to seek an expansion in a series in a of integrals in which the integrand is a product of each term of Eq. 0.121) by the factor fo(x). For instance, for the first three integrals we have the following expressions: No =

-1

ooX

-x

_ I-fo(x)

lalxl

+x

dx

(1.126)

It turns out that the first term in the series expansion of each subsequent integral N, begins with higher power of a. This does not occur if the expansion of the integrals N, is considered over only the initial range of integration (O.:s;x.:s; laD. In this case, each series starts with a term having the same power in a but with decreasing coefficients. It is obvious that for the outer range of integration the condition

(1.127)

holds. Therefore, the radicals (x 2 + 0'4)1/2 and (x 2 + s( 4 ) 1/ 2 , which are present in integrands of ~, can be expanded in a reasonably rapidly convergent series of powers of a 4/x 2 • For instance, using such a series we

1.4 Field of Vertical Magnetic Dipole in Range of Small Parameters r /

45

l)

arrive at the following expansion:

where the a j are coefficients depending on s only. Correspondingly, the integrals N, can be expressed in terms of integrals which do not contain the parameter a in their integrands. These integrals have the form (1.128)

where f30

=

0,

f31

=

2h,

Now we introduce coefficients f~l)

f32

= 4h,

from the expansion 00

e -(3j x J o( x)

=

L

(1.129)

f~i)xn

n=O

Let us represent the integrals L k as series: ( 1.130)

where n can be negative. Taking a derivative of both sides of Eq. (LBO) with respect to a and setting the coefficients of the same powers of a equal, we readily find the coefficients a~ in terms of an expansion for L k • For instance, we have

L

a

=

1 1 2 - -f(i)a 3 B - f(i)a - -f(i)a a a 2 1 3 2

-

1 4 -f(i)a 4 3

-

1 -fO)a 5 54

fJi) CO) (0) 1 CO) 2 1 CO) 3 L 2 =B 2 + --f'a a + f1' t'na-f'a--f'a 2 2 3 3 4 and so on.

..•

(1.131)

46

I

The Quasistationary Field in a Horizontally Layered Medium

In these formulas the coefficients B k remain unknown. To determine them, the following relationships are used:

aL n

-=-L

af3

JiO(a)

n-I

oo Jo( U)

=

jlad - - d u II

Jil(a) =

(1.132)

oo JI( U) - - du

jlal

II

where the latter two are known functions. For the integral La we have La = jOOe-f3iXJo( X) dx lal

Letting 0'. be zero we obtain the Lipschitz integral:

Thus, Bo =

(1 + f3l) -1/2

The same approach allows us to determine other constants from Eq. 0.132). Thus, making use of the procedures described above we obtain the expansion of the integral

as a series in 0'.. Collecting the coefficients with the same power in 0'. in each of the series representing the inner and outer ranges of integration, we arrive at the following expression for the vector potential over the range of small parameters for r /8:

A:

A*s

iWJ.LM = ---

417"r

{'P a 0'.2

+ 'P I 0'.4

+'P4a8 t'na

t'na

+ 'P 2 0'.4 + 'P 3 0'.6

+ 'P5a8 + 'P6alO + " . }

(1.133)

where 'PI are coefficients depending on electrical and geometrical parameters of the medium.

1.4 Field of Vertical Magnetic Dipole in Range of Small Parameters r /

l)

47

It is appropriate to note that though this method of deriving asymptotic expressions is rather cumbersome, it is perhaps the only way to find, in an explicit form, relationships between the parameters of the medium and the field components in the range of small parameters. Let us choose an arbitrary small value of the variable of integration x =x o so that xo« 1. Then, Eq. 0.119) can be rewritten in the form

It is clear that for the outer range of integration, x ;:::xo, we have

x> la21 and

x> Isa 2 [

This means that the second integral on the right-hand side of Eq. (1.134) can be expanded in a series in a with powers 4n. In other words, this expansion contains only integer powers of w. It means that fractional powers of w, as well as logarithmic terms tn w, arise in the series (1.133) due to integration over very small values of x. Now taking corresponding derivatives of the vector potential [Eq. (1.133)] and discarding all terms except the leading ones, we obtain expressions for the field of a magnetic dipole on the surface of a twolayered medium, when the basement has a finite resistivity:

(1.135)

(1.136)

(1.137)

48

I

The Quasistationary Field in a Horizontally Layered Medium

It should be obvious that by including more terms from Eq. (I.l33) it is

possible to provide a significantly better asymptotic representation of the field. As is seen from Eqs. (I.l35)-O.137), different terms of the series are related to the parameters of the medium in various ways. The leading terms of these series for the quadrature component of the magnetic field and the in-phase component of the electric field, which are proportional to conductivity, were considered in the previous section. The analysis of other terms of the series permits us to establish some fundamental features of the field behavior within this range of parameters r l / 0 1 . First of all, the in-phase component of the secondary magnetic field as well as the quadrature component of the electric field approach the values for a half space with the conductivity of the basement, when the frequency decreases. This occurs regardless of the distance from the dipole to the receiver, that is, HI

if -

s

~O

and

r -~O

s

(1.138)

In other words, with a decrease of the frequency the first layer becomes transparent to the electromagnetic field and this phenomenon is independent of the conductivity of the layer and is observed at any distance from the dipole. Suppose that the separation r is greater than the thickness HI of the first layer. Then, the coincidence of functions In B, and Q E, ' observed on the surface of a two-layered medium, with corresponding components for a uniform half space with conductivity "Yz, occurs when the parameter r /0 is small in both media. If the separation is less than the thickness of the upper layer, the same coincidence is observed, providing that the parameter HI/o is small in both media. It is proper to note that this behavior of In B, and Q E, takes place regardless of the ratio of conductivities. In particular, the conductivity of the upper layer can be many times greater than the conductivity of the underlying medium. This phenomenon was, in essence, explained in Chapter II of Part B, where it was shown that with a decrease of frequency the maximum of the in-phase component of currents in the medium shifts to greater distances from the dipole. Therefore, the influence of induced currents in the vicinity of the dipole becomes smaller. This means that the depth of investigation when measuring the functions Q E, and In B, increases as frequency decreases, no matter what the separation r. For example, the distance r can be much smaller than the thickness HI' Thus,

1.4 Field of Vertical Magnetic Dipole in Range of Small Parameters r 18

49

with decreasing frequency there is always some lower frequency for which both functions Q E, and In B, practically coincide with the corresponding components, which would be measured at the surface of a uniform half space with conductivity 'Yz. A further decrease in frequency does not change the relationship between the field and conductivity. The succeeding terms in the series for the in-phase component of Band the quadrature component of E contain information about the conductivity 'Yl and the thickness HI' as well as the basement conductivity 'Yz. These terms reflect the interaction between the induced currents in both media. With decreasing frequency the in-phase component of the currents concentrates principally in the basement, but because of the interaction some currents arise in the upper layer. However, over the range of very small values for rio, their effect becomes negligible and the in-phase component of the magnetic field In B, depends only on 'Yz, Now let us compare the behavior of the quadrature and in-phase components of the field for small values of parameters rio and Hlo. We can draw several obvious conclusions from Eqs. (1.135)-(1.137). 1. The quadrature component of B z is directly proportional to the frequency W, but the in-phase component In is proportional to w 3/ Z • 2. The quadrature component of B, is directly proportional to the conductivity 'Y, while the in-phase component In is proportional to

B:

B:

'Y 3/Z. 3. The quadrature component of B, is larger than the in-phase component of B:, and with decreasing frequency the ratio In B:IQ B,

becomes smaller. 4. When the separation between the dipole and receiver is much less than the thickness of the upper layer, the component Q B, is mainly determined by its conductivity 'Y I' But the in-phase component of the magnetic field In is equal to that for a uniform half space with the conductivity of the basement 'Yz regardless of the separation. In other words, the influence of the separation r on the depth of investigation is different for these two components. In the case of the quadrature component, the depth of investigation decreases as the separation r is reduced, no matter how low the frequency. When the in-phase component of the secondary magnetic field or Q E~ is measured, the depth of investigation does not depend on the separation (r 1o ~ 0), and it is a maximum, since both functions In B: and Q E~ are practically defined by the conductivity of the basement. 5. With an increase in the separation r, the depth of investigation increases, when the quadrature component of B is measured. In the limit it becomes a function of the conductivity 1z only, along with the in-phase

B:

50

I

The Quasistationary field in a Horizontally Layered Medium

component of the secondary magnetic field. However, as was mentioned earlier, the in-phase component is more sensitive to a change in conductivity than is the quadrature component. 6. The depth of investigation over the range of small parameters depends primarily on the distance between the dipole and the observation point when the quadrature component of the magnetic field is measured. 7. When the parameters rio and H 1/0 decrease, the secondary magnetic field is mainly determined by the quadrature component. Therefore, it is proper to say that frequency soundings, made with relatively short separations (r 1HI < 1) and based on observations of the amplitude and phase or the quadrature and in-phase components of the total field, are limited in their depth of investigation. Correspondingly, they usually provide only information about the thickness and conductivity of the upper layer, unless 1'1/1'2 « 1. 8. Measurement of the in-phase component of the secondary magnetic field or the quadrature component of the secondary electric field usually represents a difficult task, when the parameter rio1 is small. This is related to the fact that this part of the field is much smaller than the functions Q B, and In E", and even more so than the primary field generated by the current in the dipole. In this light we consider the series for the quadrature component of the magnetic field [Eq, (1.135)]. The second term in the expansion for the quadrature component of B z ' like the first term for the in-phase component, does not depend on the conductivity of the first layer 1'1' Thus, in order to obtain the same depth of investigation as that for the in-phase component at relatively short separations (r IH I < 1), the first term in the series for the quadrature component has to be eliminated. This might be done by making measurements at two frequencies. For instance, the difference (1.139)

permits us to eliminate the influence of the first term of the series and it depends on only 1'2' if r 1° 1 « 1 and Hiiol « 1. It is interesting to note that this result remains valid even in the presence of lateral changes of conductivity within the upper layer. 9. As also was the case for the vertical component B z ' the leading term of the series (1.136), which describes the in-phase component of B r , is controlled by the conductivity 1'2' In contrast to In B, , this component of B, is more closely related with the conductivity (I'i tn 1'2)' However, as

1.4 Field of Vertical Magnetic Dipole in Range of Small Parameters r /8

51

follows from Eqs. (1.135) and 0.136), we have r

if - «1 8[

and

(1.140)

10. In accordance with Eqs. (1.136), the second term in the expansion for the quadrature component B, is proportional to w Z and does not depend on the conductivity of the upper layer. As was previously noted, the terms in the various series [Eqs. (1.135)-0.137)] which do not contain the conductivity 'Yl coincide with the corresponding terms of the series, representing the case of a uniform half space with conductivity 'Yz. From Eqs. (1.137) it follows that the main features of the behavior of the electric field E", for small values of the parameters r /8[, r /8 z , and H /8[ are practically the same as those for the vertical component of the magnetic field B z • Perhaps it is appropriate to point out that this analysis of asymptotic expressions allowed us to realize that in principle it is possible to reach any depth of investigation regardless of the separation between the dipole and an observation site. Thus, this study [Eq. (1.133)] shows that the electromagnetic field on the surface of a layered medium can be represented in terms of a generalized Maclauren series when small values for the parameters r/8; and H /8; are considered. These series contain integer and fractional powers of the frequency w, as well as logarithmic terms Unw), and each field component has the form (1.141)

For example, the series for the quadrature component of the magnetic field always starts with a term proportional to w, while the corresponding leading term of the series representing the in-phase component of the electric field is proportional to w Z• As was shown in the previous section, this behavior occurs when interaction between induced currents can be neglected. From Eqs. (1.135) and (1.136) it follows that the second term in the series for the quadrature component of the magnetic field, along with the leading term of the series for the in-phase component of the secondary field, are different for B; and B r • They contain either a fractional power of w or t'nw. For instance, we have

52

The Quasistationary Field in a Horizontally Layered Medium

Derivation of the various asymptotic formulas has shown that only the initial portion of the integral in Eq. (1.134) is responsible for the fractional power in wand the logarithmic terms. Making use of this fact, let us consider the asymptotic behavior of the electromagnetic field on the surface of an N-layered medium, provided that the basement is conductive, that is, "IN =1= O. In accordance with Eqs. (1.102) and 0.103), we have (1.142) where

Thus, in order to determine terms in the series containing fractional powers of w or logarithmic terms, we must define the limit for the integrand of Eq. (1.142) when k ~ 0 and m ~ 0, that is, m

lim m+mI!R * ' N

if m

~

0 and

k.,

0

~

From Eq. (1.103) we have

and finally if m

~

0 and

k, ~ 0

(1.143)

where (1.144) Substituting Eq. (1.143) into the integrand for A*, we see m

m

if m

~

0 and

k,

~

0

(1.145)

This limit corresponds to the case of a uniform half space with conductivity "IN'

1.4 Field of Vertical Magnetic Dipole in Range of Small Parameters r / l\

53

This means that the leading terms of the series which contain fractional powers of w or tn w coincide with the same terms for a uniform half space with conductivity of the lowermost layer. It should be clear that this result is invalid for the terms proportional to integer powers of w, because all spatial harmonics m in Eq. (1.142) contribute to this part of the series. Therefore, taking into account the results obtained in this section, we arrive at the following asymptotic expressions for the field of the magnetic dipole on the surface of an N-layered medium, when parameters r /0; and H;/o; are small and "IN =1= 0:

Thus, the in-phase component of the secondary magnetic field, in contrast to the quadrature component, tends to the in-phase component, corresponding to a uniform half space with conductivity "IN regardless of the distance between the dipole and the observation point; the same result applies for the quadrature component of the secondary electric field, that is, In

s, ~

In E r ( "IN)' r

if -

0;

~

0 and

(1.147)

In other words, over the low-frequency portion of the spectrum, when the in-phase component of the magnetic field or the quadrature component of

54

I

The Quasistationary Field in a Horizontally Layered Medium

the electric field are measured, all of the conducting layers become transparent, no matter what the separation r. This conclusion remains valid when the second term in the series expansion of components Q B and In E are measured. As was pointed out earlier, the successive terms in the asymptotic series represent the interaction of induced currents in various layers and, therefore, they usually depend on their thickness and conductivity. So far we have only discussed the behavior of the field for small values of the parameters r /0; and H;/o; when the underlying medium has a finite resistivity. Next let us consider the field at the surface of a layered medium when the basement is insulating. The method of expanding the integrand in Eq. (1.119) in a series which has been used previously cannot be applied in this case because the function x lOx 21e- 2 x ,h in Eq. (1.118) tends to unity when x ~ 0 if k 2 = 0, and therefore the corresponding series diverges. In order to avoid this problem we describe another approach, which is illustrated for the case of a two-layered medium. The vector potential [Eq. (1.142)] can be written as the sum of two integrals: (1.148) where

As we know, the second integral can be represented as a series containing only even powers of k l , that is, integer powers of w, since m6» Ik~l. Because the range of integration of the inner integral is small, the function ml/Ri can be expanded in a Maclauren series: (1.149) since

Here

Ri (0) = coth ik: I HI , because ml

Iirncoth " ' - ~ 0, m

if m

~O

1.4 Field of Vertical Magnetic Dipole in Range of Small Parameters r /8

It should be clear that

aR!

ax

1

am

sinh x am

where

x

=mlHl

+ coth- l

ml -

m

and

ax

mn,

am

= --;;;;

mn,

1

+

=--+ ml

a

ml

1 - mi/m 2 am -;;; 1

a

mu,

ml

m2 a

and

a

ml

m

ml

m 2-mi

am m

ml

m2

m lm 2

- - = - m - l--=----.".--

or

ax

mHl

1

am

ml

ml

-=--+Inasmuch as

sinh" x

1 = -----

cotlr' x-I

we have

and

ml ik , m - -- + Ri - RHO) R~*(O)

and

ml

--=--+--1 - mi/m 2 am m ml ki am m

[R

2 2

*(0) - 1]

55

56

I

The Quasistationary Field in a Horizontally Layered Medium

Therefore, the inner range of integration can be written in the form tn "

1o

mJ o( m r )

----dm a +bm

(1.150)

where a and b are coefficients which depend on the parameters of the medium and the frequency. It should be clear that this representation does not change for an N-Iayered medium. Now, expanding the Bessel function Jo(mr) in a power series in mr, we represent the integral in Eq. (1.150) as a sum of tabulated integrals, which have the form

Tn

=1 In" a +m"bm dm 0

Having taken these integrals, it is a simple matter to expand them in a series with respect to the parameter p = r18, which contains only the integer powers of wand logarithmic terms of w. For this reason, the electromagnetic field observed on the surface of a horizontally layered medium, when the basement is an insulator, is represented in terms of series without fractional powers of w, as follows: f-tM

InBs=--{m p4+ m p6+ m p8tnp+m p8+ ... } z 47Tr 3 1 2 3 4 P 6 8 f-tM 43 { ru2 p +n 2P 6 ,np+n Q B z= 3P +n 4P

1fr

f-tM

In B = - - {t p 4 + t p8 tnp r 47Tr3 1 2

+ ...

}

+ ... } (1.151 )

if P

~

a

where p = r181 , 8 1 = J2/"lif-tw is the skin depth in the upper layer, and m, n, t, f, C, and d are coefficients which depend on the electric properties of the medium and the distance.

1.5 Vertical Magnetic Dipole on Surface of Layered Medium

57

For instance, for the quadrature component of the magnetic field Q B, and the in-phase component of the electric field on the surface of a two-layered medium we have 1

Ylj.LWr 2

4

..jr 2 + 4H 12 r

4

+~(Wj.LSlr)3 2

(1.152)

tn wj.LSlr _ ... ] 2

2

' r

if 'Y2 = 0,

01

and

~O '

where S 1 = Y1 HI is the conductance of the upper layer. It is not difficult to generalize this result for an N-Iayered medium, introducing geometric factors of layers and replacing the conductance S 1 the total conductance S, equal to

1.5 Vertical Magnetic Dipole on the Surface of a Layered Medium when Parameter r / I) Is Large Now we suppose that the separation between the vertical magnetic dipole and the observation point r is significantly greater than the skin depth 0:

r -»1

s

(1.153)

First of all, let us assume that the layers and the basement are characterized by finite values of resistivity.

58

I

The Quasistationary Field in a Horizontally Layered Medium

In deriving asymptotic formulas we proceed from Eqs, (1.104) and 0.103). Then we have A*

=

B, =

iW/-LM --1 27T

0

/-LM -1 27T

m

m

* Jo(mr) dm

+ mjlR N m3

00

0

m

00

* Jo(mr) dm

+ mjlR N

(1.154)

In accordance with these expressions, the field can be considered to be the sum of cylindrical harmonics with a continuous spatial spectrum, where m is the spatial frequency. It is clear that the larger the value of m, the more rapidly the corresponding harmonic of the field will change. Defining a new variable in the expression for the vector potential, we have

where

As the value for the parameter r18j increases, the integral is mainly determined by small values of x because of the oscillating nature of the Bessel function, and this corresponds to small spatial frequencies m. In other words, for large values of the parameter r18, as well as in the zone which is far away from the dipole, the field changes relatively slowly. Taking this fact into account, we expand the integrand in Eqs. (1.154) in a power series in m I k and keep only the first term in this expansion:

mj ik, m+-;:::m+Rjy RN

(1.155)

where RN

=

lim Rjy,

as m

~O

1.5 Vertical Magnetic Dipole on Surface of Layered Medium

59

For example,

and

and so on. Whence

Substituting this series into Eqs. (1.154) we obtain the following expression for the vertical component of the magnetic field:

From Eqs. (1.15) and (1.31) we have

(1.158)

and

J.LMR N J 1

= - 27Tik]

Jr

-;s =

3J.LMR N 27Tik1 r 4

(1.159)

60

I

The Quasistationary Field in a Horizontally Layered Medium

By analogy for the electric field we have

or E.p=

iosu.M ---2 2rrk l

a -J m Jo(mr)dm= ar 00

3Mpi

2

- - - 4 R ;"

2rrr

0

(1.160)

Thus, the electromagnetic fiell observed at the surface of a layered medium for large values of the parameter r/ 0 is described by the expressions E

.p

= -

3M R 2 PI N 2rrr 4

(1.161)

where R N is a function of the geoelectric parameters of the medium and the frequency. It should be obvious that for a uniform half space (HI ~ (0) the function R N is equal to unity and Eqs, (I.16l) are the same as those for a uniform half space with resistivity PI' The function R N represents the influence of the layers, and correspondingly it can be described in terms of the ratio of the field in a horizontally layered medium to that in a uniform half space. (1.162)

We can see that the horizontal component of the field B, is less sensitive to a change in the geoelectric parameters than either of components (E.p, B z ) when the parameter r/o is large. However, it should be noted that this conclusion may not hold for other forms of the field excitation or for other models of the medium. The derivation of the asymptotic formulas shows that the field for large values of r /0 1 can be written as a series in powers of the parameter l/(k I r). In essence, the right-hand side of each equation of the set (I.16l) is the first term in such a series. As was also the case for a uniform half space, the electromagnetic field is similar to a plane wave field which only changes along the z-axis. In particular, its behavior as a function of the depth in the earth is characterized by the exponential terms e ik z and e- ik z • Moreover, the ratio of the tangential components of the electric and magnetic fields is equal up to a

1.5 Vertical Magnetic Dipole on Surface of Layered Medium

61

constant to the impedance of a plane wave field propagating vertically into an n-Iayered medium. In accordance with Eqs. 0.161), the components B, and E

ZN

B,

f.L

-=-=

iklPI

ZIRN

---R f.L

= --N

f.L

(1.163 )

where

and ZI = -iklPI is the impedance at the surface of a uniform half space. In contrast to the behavior of a plane wave propagating vertically into a conducting medium along the z-axis, the field generated by a dipole changes with the separation r. Also, the vertical component of the magnetic field is not zero, though IBzl < IB,I and their ratio

decreases with an increase of r. The behavior of the quasistationary field over this range of distances is characterized by some essential features of the propagation of the field in free space as well as in a conducting medium. When the parameters rio are large for each layer of the medium, absorption of the field within them is very strong. Therefore, the signal arrives at the observation point mainly by propagating through free space. At distances from the dipole to the receiver which are not large enough to meet the conditions for asymptotic behavior, the earth's surface serves to guide the propagation of the electromagnetic field. Here the Poynting vector has mainly a tangential component. But in the range of large parameters, rio» 1, the normal component of the Poynting vector becomes dominant. Now we return to the quasistationary field as given by Eqs. 0.161). These equations are remarkably simple. In fact, the right-hand side of each expression can be represented as the product of two functions. One of these functions depends on the dipole moment M, the distance r, the frequency W, and the resistivity of the uppermost layer PI' This function describes the corresponding components of the field for a uniform half space when the parameter r1°1 is much greater than unity. The second term in this product, R N or R'Fv, is a function only of the parameters of

62

I

The Quasistationary Field in a Horizontally Layered Medium

the medium and the frequency but does not depend on the dipole moment or the distance r, As long as the asymptotic behavior of the field holds, a change in the separation from the dipole to the receiver does not change the relationship of the field components to the electrical parameters for the medium. It should be clear that the asymptotic behavior for large values of the parameter r / 8 can be observed at any distance between the dipole and the observation site. Taking into account the fact that magnetotelluric soundings are based on measuring the field in the range of large parameters, let us consider in some detail the behavior of the function R N • From Eq. CU64) we see that when the parameter H l/8 1 increases, the function R N tends to unity: lim

RN~

(1.165)

1,

This indicates that as the skin depth 8 1 decreases, currents concentrate in the uppermost layer and the field corresponds to that of a uniform half space having resistivity PI' Now suppose that the skin depth in each layer increases and therefore the magnitude of k .H, goes to zero. Then from Eq. (1.164) we have

and in general (1.166)

Substituting this last equation into Eqs. 0.161) we have

3JLM B, = 2 7ft'k Nr 4

'

63

1.5 Vertical Magnetic Dipole on Surface of Layered Medium

Thus, when the skin depth in each of the layers increases, the electromagnetic field approaches that for a uniform half space with the resistivity of the lowermost medium PN. In this case all layers above the basement become transparent. Next we investigate the behavior of R N when the skin depth in every layer above the basement is significantly greater than its thickness. First, assuming that the two-layered medium is considered and the basement is more resistive (pz» PI)' we have

1 + (k1/kz)(1/iktH1) (1/ikIH 1) + (k 1/k 2 ) ( 1.168)

where 51 = YIHI is the conductance of the upper layer. Using the same approach we obtain the following expression for R J :

where

This result can easily be generalized to the case of an N-Iayered medium. Then we have (1.170)

if Hjo; < 1 and the basement is relatively resistive. Substituting this last result into Eqs. (1.161) we obtain asymptotic expressions for the field, describing the low-frequency part of the spec-

64

I

The Quasistationary Field in a Horizontally Layered Medium

trum within the wave zone: 1

9j.LM

Bz

= --5

2

27Tr (iwj.LS+k N )

3ij.LM

1

B = -4 - - - - - r 27Tr (iwj.LS+k N ) 3iwj.LM

E",

=

-

27Tr

4

(1.171)

1

H

(iwj.LS+k N )

if --.!... < 1 and 0;

2'

and the basement is relatively resistive. Here N-I

L

S=

;~I

N-I

S;=

L

;=1

H --.!... Pi

and k N is the wave number for the basement:

As follows from Eqs. (1.171), the magnetic field is defined by two parameters, namely, the total conductance of the section S and the conductivity of the underlying medium 'YN' It is clear that within this zone one can distinguish three intervals, each with a characteristic behavior. If the frequency is not sufficiently low that Wj.LS » Ik NI, then the field depends on the longitudinal conductance S, and 3j.LM

B

1

= ----r

27Tr 4 wj.LS '

E

3iM = ----,--_____=_

'"

27Tr 4Wj.L S2 (1.172)

This range of frequencies is usually called the S-zone. With decreasing frequency both terms in the denominators of Eqs. (1.171) become comparable, and correspondingly the second interval is controlled by both parameters Sand PN' With a further decrease in frequency, the term icop.S can be neglected and the field approaches the value for a uniform half space with resistivity PN' Now we study the case when the lowermost medium is relatively conductive. Starting with a two-layered medium and making use of Eq.

65

1.5 Vertical Magnetic Dipole on Surface of Layered Medium

(1.156), we have (1.173)

since

A similar expression is obtained for the function R 3 :

H2

-<1

82

P3

'

- « 1, P2

and

P3

-

«1

PI

By analogy, for an N-layered medium we have (1.174)

where H = 'LH; is the total thickness of all the layers. Substituting the last expression for R N into Eqs. (1.161) we arrive at equations for the field in the range of large parameter r/8, when the skin depth in each layer is greater than its thickness and the basement is relatively conductive: B

=

_

z

9f.-LM

(_1_ + iH)2

21Tr 5 k N

(1.175)

Ecp= -

3iWf.-LM( 1

21Tr

4

-k +iH

)2

N

Thus, over this range, the field is defined by two parameters of the medium, namely, the basement resistivity PN and the total thickness of the layers above basement H.

66

I

The Quasistationary Field in a Horizontally Layered Medium

In particular, when IkN < H I the field is determined by the total thickness of the layers only. This frequency range is often called the H-zone. Next we investigate the behavior of the electromagnetic field over the range of large values of the parameter r /8 1 , when the underlying medium is an insulator. As an example, consider a two-layered medium. From Eqs. 0.148) and (1.156) it follows that

Whence

and

Repeating the procedure for deriving the field components that was carried out above, we obtain

(1.176) r

if - » 1

81

At high frequencies (H I/8 1 > 0, as a result of the skin effect, the

67

I.S Vertical Magnetic Dipole on Surface of Layered Medium

electromagnetic field is the same as for a uniform half space with resistivity Pi- In the opposite case, for low frequencies, (r /0, > 1 and H,/o, < 1), replacing coth x by the first term in its expansion l/x we have

(1.177) 3Me- i 7T / 2 = ----::---,-

E ip

1rw/-LS?r 4

These equations imply that the quasistationary field, described by Eqs. (1.177), results from the propagation of the field principally along two paths: (1) in free space along the earth's surface and (2) vertically downward into the conducting medium along the z-axis, then horizontally through the insulating basement and vertically upward to the observation point. In accordance with Eqs. (1.177), at the low frequencies, the components E

°

holds. It is clear that with increasing separation r due to attenuation of the field in the resistive basement, the field observed at the earth's surface approaches that described by Eqs. (1.161). In conclusion, we demonstrate the behavior of the magnetic field, normalized by the primary field, which was calculated using the exact solution [Eqs. (1.154)] (Figs. L4a and b), when it is caused by the magnetic dipole and the current loop.

68

The Quasistationary Field in a Horizontally Layered Medium

a 1.4

Ibzl 1.2 1.0 0.8 0.6

'~~""'' 'i P2

0.4 0.2 0.0

b

Fig. 1.4 (a) Spectrum of field Ibzl caused by magnetic dipole; (b) spectrum of field Ibzl at the loop center.

1.6 The Early Stage of the Transient Field on the Surface of a Layered Medium

69

1.6 The Early Stage of the Transient Field on the Surface of a Layered Medium Suppose that the primary field B o ' caused by a current in the vertical magnetic dipole, varies as B o( t)

=

t 0

BO {0

(1.178)

Then, as in the case of a uniform half space, we can distinguish three stages of the transient response caused by induced currents in the medium -namely, the early, intermediate, and late stages. In order to study the asymptotic behavior we proceed from the Fourier transform

F(t)

=

1 -f 27T

00

-00

F(w) -.-e-iw1dw

(1.179)

-lW

where F(w) is the spectrum of the field component. In this section we consider the early stage and with this purpose in mind let us compare the frequency and transient responses at the surface of a uniform half space for the range of large parameters rI 8 1 and rI 7"1 . As follows from Eqs, (1.59) and (1.88),

E (w)

3Mp . 27Tr4'

= _ _ e'"

'P

3Mp 27Tr4

E (t)=-'P

(1.180)

3wMe- i 37r / 4 Br ( w)

=

r::-::-::

27TVYJ.LW r

4 '

As may be seen from these expressions, the behavior of the field for large induction numbers r18 and at the early stage is almost identical. In particular, the equations for the electric field and the vertical component of the magnetic field at the early stage can be obtained by replacing w by lit in the corresponding expressions in the frequency domain. It suggests that by applying Fourier's transform and making use of expressions for the field in the range of large parameters r I 8 we can derive equations that correspond to the early stage behavior.

70

I

The Quasistationary Field in a Horizontally Layered Medium

For example, in the case of the electric field we have 3Mp 1 E",(t) = - 2'lTr4 2'lT

e- iwt

3/J-p -iw do: = - 2'lTr4

00

1_

00

Of course, this transition from the frequency to the time domain is valid only over a limited range of time, since a field at the small and intermediate values of the parameter r/01 is not taken into account in this relationship. Now, applying this approach for a layered medium and taking into account Eqs. 0.161), we obtain 9/J-M 1 R~e-iwt BzCt) = 2'lTr5 2'lTI_ 00 -iwkf dto 00

Br(t)

=

3/J-M 1 2'lTr4 2'lT

E",(t)

=

-

00

1_

3MpI 1 2'lTr4 2'lT

00

RNe- iwt wk] dw 00

1_

(1.181)

R~e-iwt

-iw

00

dto

These equations are valid when the condition

( I.l82) is met, where T i = V2'lTp;t X 107 and Pi is the resistivity in the ith layer. Each expression of the set (1.181) can be represented as being the product of two terms; one term depends on the dipole moment and the separation r, while the second is a function of the time and the geoelectric parameters of the medium. Thus, during the early stage of the transient response, the relationship between the field components and the distribution of conductivity in a horizontally layered medium is independent of the separation r, Therefore, there is a strong resemblance in the behavior of the field for large parameters r /Oi in the frequency domain and at the early stage of the transient field. The integrals on the right-hand side of Eqs. (1.181) generally cannot be expressed in terms of known functions, and they must be evaluated numerically. However, in the case of a two-layered medium these expressions can be simplified. In accordance with Eq. (1.164), we have R 2 = coth[iklH I + coth-

I

~]

71

1.6 The Early Stage of the Transient Field on the Surface of a Layered Medium

Inasmuch as coth(a + coth" ' b)

1 + b coth a = ----

b

+ coth a

1 + I( b - l)/(b + 1)] e- Za 1- [(b - l)/(b

+ l)]e- Za

we have

where

Assuming that the basement is not a perfect insulator (Q =1= 1), we can expand the function 1

in a power series. Then after some algebraic operations we have RZ= 1+ 2

L:

Qne-Zik1H1n

(1.183)

n=l

and R~

= 1+4

L: nQn e

- 2i k 1H 1n

n=l

Substituting expressions (1.183) into Eqs. (1.181), we can represent the transient field during the early stage on the surface of a two-layered medium in terms of the probability integral cjJ(x). As an example, we have for the electric field

72

I

The Quasistationary Field in a Horizontally Layered Medium

Using Eq. (1.68) we have

Ecp(t) = - -3MPI[ - 4 1 + 4 L nQ" 1 - ¢ (47Tn)}] -27Tr , ~l Tl/H l 00

{

(1.184)

Let us examine the behavior of the electric field as a function of the parameter T1/H] . During the very early stage, when the parameter T1/H I tends to zero, the probability integral approaches unity, and therefore

3Mpl 27Tr

E(t)--+--4 ip

ift--+O '

That is, the field is the same as that for a uniform half space with resistivity Pl. In contrast, when the parameter TI/Hl increases, the probability integral tends to zero and we have

Ecp( t)

=

-

3Mp [ 1 + 4 L nQ" , --~ 27Tr n=1 00

]

Inasmuch as Q

LnQn

=

(1 _ Q)z

we obtain

3MpZ 27Tr

E(t)--+---4

cp

Thus, even during the early stage when T I is significantly greater than the thickness of the first layer, it becomes transparent to the transient field, which becomes the same as that for a uniform half space having resistivity pz (r /H l » 1). The same behavior is shown by the magnetic field and is valid for a medium with any number of layers. As was pointed out in the previous section, the term early stage does not always mean that induction currents are located only near the earth's surface. Suppose that the separation r is less than the thickness HI' that IS,

Then, in accordance with the inequality (1.182), during the early stage we

1.7 The Late Stage of the Transient Field on the Surface of a Layered Medium

73

also have

Therefore, the field is defined by the resistivity of the upper layer since induction currents are mainly located in this layer. This means that if the separation r is less than or comparable to the thickness HI' the transient field observed during the early stage does not contain information about a medium beneath the layer. However, the transient field at the early stage can be used in principle to investigate the geoelectric section, provided that the separation r is much greater than the total thickness of layers. In this case, the condition 0.182) is met even when parameter Ti/H[ exceeds unity in every layer. Accordingly, induction currents can be situated at greater distances from the dipole beneath the earth's surface, even though the early stage behavior is observed at the receiver. As follows from inequality 0.182), with an increase of the distance r or with a decrease in the resistivity of the medium, the early stage persists to greater times.

1.7 The Late Stage of the Transient Field on the Surface of a Layered Medium In this section we study the field behavior on the earth's surface when the parameter TJr in every layer is much greater than unity: T;

-» 1 r

(I.l85)

With this purpose in mind, as in the case of the early stage, we proceed from the Fourier integral. For instance, for the magnetic field we have

2 -1 'P[(w)coswtdw 00

Bs(t) =

1T 0

and

2 --1 'P2(w)sinwtdw 00

Bs(t) =

1T 0

(I.l86)

74

I

The Quasistationary Field in a Horizontally Layered Medium

where

QB(w)
=

Re B( w)
, W

(1.187)

and B; is any component of the secondary field caused by currents induced in a horizontally layered medium. Assuming that time t increases without limit, and then integrating Eq. 0.186) by parts, we obtain 2 [
-1

=

~ 17"

I'" +
[
(1.188)

0

or

Bs(t)

I'"

-1

1 2 (
= -

17"

00

)

00

=

2 [
1 + 2" 1 0

1
"

•

]

to

Thus, we have obtained a series expansion in powers inversely proportional to t, which can be used in determining the late stage of the transient field. As follows from Eqs. (1.188), this stage is controlled by the lowfrequency spectrum along with its derivatives with respect to frequency, such as

d
75

1.7 The Late Stage of the Transient Field on the Surface of a Layered Medium

In accordance with Eq. 0.141), the low-frequency part of the spectrum for any component of the field can be written as a sum: 00

2n 2n 1 '\' '-' C In k + '\' '-' C 2n k + + '\' '-' C 3n entnk n=1

n~1

(I.189)

n=1

where k = (iYILw)I/2 and C; are coefficients depending on the geoelectric parameters, the separation r, and the dipole moment. First, we demonstrate that the sum 2n '\' '-' C Ink

(I.190)

n~1

has no effect on the late stage of the transient response. Let us write Eq. (1.190) as the sum of the in-phase and quadrature components, respectively:

E C lnk 2n = E a 1nw 2n + i E blnw2n-l n=l

n=1

(1.191)

n~l

Substituting this into the Fourier transform, we obtain two types of integrals, namely,

L;

= l"'w 2n- 1 sinwtdw

o

(1.192)

and M n = jOOw 2n- 2coswtdw

o

They can be considered as being the limiting cases for large t of more general integrals:

L n = lim jOOe-f3ww2n-1 sin cot dco o M

= n

lim rooe-f3Ww2n-2 cos cat dto Jro '

(I.193) as f3

--+ 0

and

t --+ 00

This approach is valid because the introduction of the exponential term

e- f3 w does not change the initial part of the integration which defines the

values for the integrals in Eqs. (1.193) when the parameter t tends to

76

I

The Quasistationary Field in a Horizontally Layered Medium

infinity. The integrals

and

["'e-l3wW2n-z cos wtdw o are very well known, and they are expressed in terms of elementary functions. It is essential that their values vanish as the parameter f3 tends to zero. Therefore, we can conclude that the first sum in Eq. (1.189), which contains only integer powers of w, makes no contribution to the late stage of the transient field. Correspondingly, only the fractional powers of w and logarithmic terms are important in determining the late stage behavior. This fact plays a fundamental role in understanding the relationship between the frequency domain and time domain responses of the electromagnetic field in a conducting medium. For example, the quadrature component of the magnetic field at low frequencies is principally controlled by the leading term in its series representation, being directly proportional to w. But the following terms, which contain fractional powers of wand tn w, have a relatively negligible effect. However, in accordance with what was shown above, these less important terms in the frequency domain define or control the behavior of the transient field at the late stage. Usually the relationship of the first term in the series for the quadrature component Q B, which is directly proportional to w, with the geoelectric parameters of the medium and with the separation r differs essentially from that of the rest of terms. Therefore, we can readily understand the fundamental difference between the behavior of this component of the magnetic field at low frequencies and the transient response during its late stage. On the other hand, if the lower part of the medium is characterized by a finite resistivity, then, in accordance with Eqs. (1.146), the leading term in the series expansion for the in-phase component of the magnetic field contains either a fractional power of w or tn w. Correspondingly, one might expect that the behavior of this component of the magnetic field at low frequencies will be practically the same as the behavior of the transient field at the late stage. As we know, the complex amplitudes of the electromagnetic field in the frequency domain are expressed in terms of an integral of the type

77

1.7 The Late Stage of the Transient Field on the Surface of a Layered Medium

However, it has been demonstrated that only the initial part of the range of integration, where m ~ 0, is responsible for the existence of fractional powers of wand logarithmic terms. Therefore, one can say that the late stage of the transient field is controlled by the long-period spatial harmonics, characterized by very small values of m. Now we describe a method of deriving asymptotic formulas for the transient field during the late stage. After separation of the real and imaginary parts, the second sum in Eq. (1.189) can be written as '\' C2n k 2n + 1 = '\' a W(2n+ 1)/2 + i '\' b W(2n+I)/2 LJ LJ 2n LJ 2n n=1

n=l

n~l

As an example, let us use the in-phase component of the sum 00

L

a 2n W n + 1/2 = a 2 w 3 / 2

+ a 4 w 5 / 2 + ...

n=1

Substituting this sum into the Fourier transform [Eq. (1.186)], we obtain 2

00

1a 00

-- L

a 2n

7T n~I

W

n- I/2

sin wtdw

(1.194)

Inasmuch as we are concerned with the behavior of the integrals when t increases without limit, we pay attention only to the initial part of the range of integration. Letting n = 1, we have

Integrating II by parts and taking into account the fact that the quasistationary field at high frequencies tends to zero, we obtain II = -

1 -1 w to 00

l 2d /

1[ cos cot = - - W I/ l cos wtl~ t

1

= ~ cos wt dco = _1_ 2t a w l / 2 2t 3 / 2 00

1 cosIXx dx 00

0

This last integral is well known: 00

1a

cos x dx

IX

=

rrr V2

-

1 cos t ] -1 --dto 2 V;;; 00

0

to

78

I The Quasistationary Field in a Horizontally Layered Medium

Thus,

iii

I =-=--

For

11 =

(1.195)

2/2 t3/2

I

2 we have 12 = ["w 3/ 2 sin tat dto

o

Integrating 12 twice by parts, we obtain

12 = 3

=-

t1[ w

3/ 2

00

{

2t10

cos wtlo -

wI/2COS

3

'2 fa

00

wl/

3 tot dto = { 2t 2 10

2

00

cos tot dio

wl/2d

]

sin wt

3 [ l/ 2 sin tot w sin wtl'O - -1 ( - dto ] 2t 2 210 .j;;; 00

= -

3 = - 4t 5/ 2

fa

00

sin x

..;x

dx

Inasmuch as sin x 1o --dx= IX 00

rr -

2

we have (1.196)

Making use of a similar approach, one can calculate any integral in the sum (1.194), and we can see that any term proportional to to" + 1/2 generates a term in the time domain proportional to t- n - I / 2 • Therefore, the portion of the spectrum described by the sum

is responsible for the appearance of a sum of the type (1.197)

in the expression for the late stage of the transient response.

79

1.7 The Late Stage of the Transient Field on the Surface of a Layered Medium

The third sum in Eq. (1.189) can be written as '\' i... n=l

C3n k 2n tnk =tnk i... '\' C3n k 2n n=l

=tn (VY/-Lwe iTr/ 4

f

)

C 3n(Y/-Lw)n e i(Tr/2)n

n=l

7T = (tnVY/-Lw +i

)[

4

f

C3n(Y/-LW)ncos

n=l

+i

L 00

7T

2

n

7T ] C 3n( YILW) n sin -n

2

n=l

Letting n = 2p and n = 2p - 1 in the first and second sums, respectively, and taking into account the fact that cos TTP = (-l)P and sin[(2p - 1)/2]7T = ( -1)P - 1, we have the following expressions for the real and imaginary parts of the third sum in Eq. 0.189):

Substituting the real part of this last equation into the Fourier transform [Eq. (1.186)], we obtain two types of integrals:

1 w2p- 2 sin tat di» 1o W2p- tnw sin wtdw 00

A = P

0

(1.199)

00

Bp =

1

For example, when p = 1 we have

1o sin wtdw = lim 1 e-{3w sin wtdw = -,1t 00

Al =

00

0

if {3

~

a

and

t

~

OCJ

80

I

The Quasistationary Field in a Horizontally Layered Medium

and

where F(w)

B1 =

w tnw. Integrating by parts we obtain

I

00

-- (

t Jo

= -

=

=

-

F(w)dcoswt

+{ +

~ faooF' ( w) cos wt d w}

F ( w) cos wt I -

[F( w )cos

wtl~

-

faoo F'( w)d sin cot]

1 1 { roo } = - (F( w) cos wtl o + fi F'( w )sin wtl o - J F"( w )sin «it dt» o 00

00

Since

F'(w)

=

F"(w)

1 +tnw,

1 =-

w

we have

B

= _ 1

2- roo sin wt dw [2J O

W

=

_

~ 2t 2

Applying the same method, one can derive integrals A p and Bp when p is not unity. It is readily seen that the portion of the low-frequency spectrum described by the last sum in Eq, (1.189) gives rise to terms in the representation of the late stage of the transient field, proportional to l/t n • Thus, the following sum appears in the expression for the transient field: 00 1 '\' a* - n 1..J 3n t

(1.200)

n~l

Therefore, in accordance with Eqs. (1.197) and (1.200), the late stage of the transient electric and magnetic field in a horizontally layered medium can be represented as follows: 00 1 '\' a* - 1..J 2n n + 1/2 t n~l

1

00

+ 1..J '\' a*3n[n n~

I

(1.201 )

1.7 The Late Stage of the Transient Field on the Surface of a Layered Medium

81

where ain and a!n are related to the coefficients describing the lowfrequency part of the spectrum. They are functions of the geoelectric parameters of the medium and the separation between the dipole and observation site. Under certain conditions some coefficients are equal to zero. For instance, as will be shown, the late stage of the transient field, when the basement is an insulator, is described by a sum containing only integer powers of t, that is, ain = O. Having completed a general analysis of the field at the late stage, we should now obtain the leading term of this asymptote for each component of the electromagnetic field. From Eqs. (1.146) and (1.186) we have

Bz(t)

f-LM = --Z-3

27Tr

[1 L (Yi/-UZ)Qtj 00

-

4 i =1

00

coswtdw

0

(1.202)

As was shown earlier, the first integral in Eq. (1.202) is zero, while the second is

thus, (1.203)

Of course, the same result follows if the in-phase component of Bz(w) is used. In the case of the horizontal component Br(t) we use the quadrature component as follows:

82

I

The Quasistationary Field in a Horizontally Layered Medium

The first integral is again zero, but the second integral can be written as 00

1o

a

1

00

wcoswtdw=-j e- f3W sin wt dw = 2 ' at

t

0

if f3

~

0

and

t

~

00

Thus, (1.204) Finally, for the transient electric field, we have (1.205) In summary, the late stage of the transient field when the vertical magnetic dipole is located at the surface of a horizontally layered medium is B ( ) z

y 3/ 2

M

t "'" 30Tr{; f.L

5/2

N

t 312

Mrf.L3 y~

(1.206)

Br(t) "'" 128Tr (2 f.L5/2MryJ/2

E
'

Therefore, at sufficiently late times induction currents are primarily located in the lowermost medium, and for this reason expressions (1.206) are exactly the same as those for a uniform half space having conductivity YN' In other words, all the underlying layers, regardless of their conductivity, become transparent during the late stage of the transient response. This means that the depth of investigation, obtained with transient soundings, is in principle controlled only by time, and the separation has no practical significance. As follows from Eqs. (1.206), the vertical component of the magnetic field during the late stage does not depend on the distance between the dipole and the point of measurement. This feature of the behavior of B, is related to the fact that induced currents, generating this component, are mainly located far away from the earth's surface. This study also shows that during the late stage the component B)t) is more sensitive to a change of conductivity than the quadrature component, observed at small induction numbers, or the total field when the parameter r //5 is large. Although the horizontal component B, is less

1.7 The Late Stage of the Transient Field on the Surface of a Layered Medium

83

than B, at the late stage, it possesses a higher sensitivity to a change of conductivity. Of course, the conclusion concerning the depth of investigation during the late stage is also valid for the in-phase component of the magnetic field over the range of small values of induction numbers. It is readily seen that at the late stage the following relationship holds: (1.207) Until now we have considered only the leading term of the asymptote for each component of the field at the late stage. These expressions can be markedly improved by applying the results obtained in Section 104. For instance, taking into account the terms of the series expansion for the vector potential A* in the case of a two-layered medium [Eq. 0.133)] and making use of the Fourier transform, after simple algebra we obtain

(1.208)

where 'Yz PI 0 s=-=-=I= 'Yt

pz

It is clear that the second and following terms in the asymptotic series (1.208) reflect the influence of induced currents in the upper layer, and

84

I

The Quasistationary Field in a Horizontally Layered Medium

these expressions can be used only if the corresponding terms decrease rapidly. Now, applying the same approach we arrive at the asymptotic formulas for the field at the late stage for an N-layered medium, when the basement is an insulator:

where S is the total conductance of layers. Thus, the vertical component of the magnetic field during the late stage does not depend on the separation r and is directly proportional to the cube of the longitudinal conductance S. For this reason, relatively small changes in the thickness of a sedimentary sequence, resting on the basement, can be observed by measuring the transient field, regardless of the separation between the dipole and an observation point. As follows from Eqs. (1.209), the horizontal component B, is even more sensitive to a change in the conductance S, though it is less in magnitude than the vertical component. It is a simple matter to improve the expressions for the late stage. In particular, in the case of a two-layered medium we have

(1.210)

In conclusion, examples of the transient responses of field i; are shown in Figure I.5a and b, where t, is the ratio of function e, to that at the early stage. They are calculated using the exact solution for the field in the frequency domain and the Fourier transform.

1.7 The Late Stage of the Transient Field on the Surface of a Layered Medium

85

a

10-8

b

Fig. 1.5 (a) Transient responses of field responses of field i, at the loop center.

bz

caused by magnetic dipole; (b) transient

86

I

The Quasistationary Field in a Horizontally Layered Medium

1.8 Magnetic Field of a Vertical Magnetic Dipole Located inside a Layer

Suppose that there are two parallel interfaces which divide a medium into three parts, as shown in Figure 1.6a. The vertical magnetic dipole is placed at the origin of the cylindrical system of coordinates. Its moment is oriented along the z-axis and changes with time as function e- i w i • It is obvious that as in previous cases the electric charges do not arise and the electric field has only component Ecp. Correspondingly, the electromagnetic field can be expressed in terms of the vertical component of the vector potential A* only. Making use of results obtained in Section 1.2, we can represent the function A* in every medium in the following way: A* = 1

---J D iWfLM 47T

00

0

---1

A*2 -- iWfLM 4 7T

I

emJzJ

OO[ m

0

0

(mr) dm

_e-mllzl

m2

'

] J (mr) dm + D2 e'"?" +3 D e- m1Z 0 (1.211)

where

and H = hI + h 2 is thickness of the layer, while k~ = iY2fLW, and YI and Y2 are conductivities of the layer and surrounding medium, respectively. Now, by making use of the boundary conditions [Eqs. 0.93)], we obtain the system of equations to determine unknown coefficients:

(1.212)

Inasmuch as our main interest is a study of the field inside the layer, we

87

1.8 Magnetic Field of a Vertical Magnetic Dipole Located inside a Layer

z

a

M

In b~

b

.1 H -=2

L

.01

.001

-l-~'-r----'.:"""""---l...---rL-1,---r----,---r----,----r--.----.-.J

.01

.1

10

Fig. 1.6 (a) Magnetic dipole inside layer; (b) frequency responses of in-phase component of magnetic field; (c) frequency responses of in-phase component of magnetic field; (d) frequency responses of quadrature component of magnetic field. (Figure continues.)

88

The Quasistationary Field in a Horizontally Layered Medium

c

.1

.01

L

~

.001 +-......,--'--.,.-L-+--'--,-----r--,---.---.,.--.,.--.-------r--..I .01 .1 10 100

Fig. 1.6 (Continued)

calculate only coefficients D 2 and D 3 • Solving the system we have =

2

mk e-2m2hl(1 + k e-2m2h2) 12 12 m (1 - k 2 e-2m2H)

=

3

mk e-2m2h2(1 + k e-2m2hl) . 12 12 m (1 - k 2 e-2m2H)

D

D

2

12

2

(1.213)

12

Substituting these expressions into Eq, 0.212) we obtain

A*

=

iwp.,M col m -4-1T-fa -m-

e-m2lzl

2

mk 12 {e -m2(2h

j

-z)

+ e -m2(2h 2+ z ) + 2k 12 e - 2m2 H cosh m 2 z}J0 (mr) dm

+-_....:..-_---------,----------=-----=------:,.,.,----------=-----m 2(1

-

k~2r2m2H)

(1.214)

1.8 Magnetic Field of a Vertical Magnetic Dipole Located inside a Layer

89

d o s,

.1

.01

-r--,---.....-,..~,.---r--r--,--r--.,r-----r--,-:.+-I

.001 .01

.1

10

100

Fig. 1.6 (Continued)

where

m 2-m j k 12 = - - - m 2 +m j Making use of relationships between the vector potential and the electromagnetic field [Eqs, (I. 1)], it is a simple matter to derive expressions for the electric and magnetic fields. This study is of great practical interest for the induction and dielectric logging, where the vertical component of the magnetic field B; is measured along the z-axis. It is customary to call the system, which consists of the current and received coils, the two coil probe, and the distance between these coils is the probe length. Let us restrict ourselves to the case when the probe is located symmetrically inside the layer. Then, applying Eqs, (1.212) and (1.213), we obtain the expression for the vertical component of the magnetic field on the z-axis, related to the

90

I

The Quasistationary Field in a Horizontally Layered Medium

primary field:

(1.215) where a = H / L, L is the probe length, and b~ is the field in a uniform medium with conductivity 'Yl, normalized by the primary field caused by the dipole current, and

k 12 =

X z -XI --Xz

+x I

Before we continue it is appropriate to note that Eq, (1.215) remains valid in the general case when both the conduction and displacement currents are present. Now proceeding from Eq. 0.215), we begin to study the frequency and transient responses of the quasistationary field B z . First, let us consider the low-frequency spectrum of the quadrature component of the field B z , assuming that parameters IklLI and IkzLI are very small. Then, representing radicals X I and X z as 1 k;L z

x ::::::x---1 2 x '

1 kiLz x ::::::x---Z

2

x

we can expand the integrand in Eq. 0.215) in the series by small parameters Ik1 LI and IkzLI. Considering only the first term of this series, we obtain for the integrand

Correspondingly, the integral becomes equal to

Thus, the quadrature component of the field b, at the range of small parameter is (1.216)

1.8 Magnetic Field of a Vertical Magnetic Dipole Located inside a Layer

91

or

1)

2

1]

[y ( 1--+yQ biwp.,L =-z 2 1 2a 2 2a '

if a

~

1

Introducing notations and

1 Q2 = 1- 2a

we finally have (1.217) It is clear that at the low-frequency part of the spectrum the quadrature

component of the magnetic field is mainly caused by induced currents, which appear due to the primary electric field, E~:

iwp.,Mr

EO

= -------,,...,-;:-

'I'

47T(r2+z 2 ) 3/ 2

In other words, in this range of frequencies, the component Q B, is not practically subjected to an influence of interaction of induced currents. Functions Q1 and Q2 are called geometric factors of the layer and the surrounding medium, respectively. It is proper to note that Eq. 0.217) allows us to evaluate the vertical resolution of the induction logging in a relatively resistive medium. Now we improve the representation of the field at the low-frequency spectrum [Eq. (1.217)]. With this purpose in mind, we apply the method described in Section 1.4. Then the field B, can be represented as

where

1 1 C =-+-(s-l) 1 2 4a '

Y2

s=-

as(s-l)

C3 = - - 4 - -

Y1

(1.219)

if a

~

1

92

I

The Quasistationary Field in a Horizontally Layered Medium

It is obvious that the first term of this series describes the quadrature component of the field at the low-frequency spectrum when interaction between currents is negligible. At the same time, the second term of the series gives a contribution to both the quadrature and in-phase components of the field, and the latter coincides with that in a uniform space with conductivity of the surrounding medium 'Yz, that is,

(1.220) It means that the in-phase component at the low-frequency spectrum, measured within the layer, is practically independent on its resistivity PI' regardless of the ratio of the conductivities 'Y2/'YI and the probe length. At the same time, the quadrature component Q B; essentially depends on the layer resistivity, especially when the layer is more conductive. Thus, at the low-frequency spectrum the components of the magnetic field Q B; and In B, have completely different vertical resolution. In the opposite case, that is, at the high-frequency spectrum, due to the internal skin effect, induced currents concentrate mainly near the dipole. Correspondingly, the influence of the surrounding medium decreases (a > 1) when either the quadrature or in-phase components are measured. It means that with an increase of a frequency it is possible to improve the vertical resolution of the induction logging. This is vividly seen from frequency responses of the magnetic field shown in Figure 1.6b through 1.7a. The ratio Liol is plotted along the axis of abscissa, where 0, is the skin depth in the layer. The index of curves is the ratio 'Yl/yz and components Q b, and In b: are plotted along the axis of ordinate. These curves clearly illustrate the fact that with an increase of frequency the influence of the surrounding medium becomes smaller and the field approaches that of a uniform medium with the resistivity of the layer. Next we study the transient responses of the field, assuming again that the dipole current changes as a step function. It is obvious that at the early stage, due to the skin effect, the currents are mainly located in the vicinity of the dipole. For this reason it is natural to expect that the field coincides with that in a uniform medium with the layer resistivity, if t ~ O. In fact, as calculations show, this asymptotic behavior takes place if both the dipole and the receiver are located within the layer (a > 1). Now let us consider the opposite case, that is, the late stage of the transient response. For instance, performing Fourier transform and taking

1.8 Magnetic Field of a Vertical Magnetic Dipole Located inside a Layer

93

a

Fig. 1.7 (a) Frequency responses of quadrature component magnetic field; (b) transient responses of magnetic field. (Figure continues.)

into account Eq, (1.218), we obtain

aB

Mp 'TTL5

__ z = __ I

at

(2) _

7T"

1/2

ui { S3/2 -

2

-auI(s -l)s +

(1.221)

7T"

where

This equation describes the field at the late stage when the surrounding medium has a finite resistivity. In accordance with Eq. 0.221), at suffi-

94

The Quasistationary Field in a Horizontally Layered Medium

b

32

10

.1

10

100

1000

Fig. 1.7 (Continued)

ciently large times the field tends to that in a uniform medium with the resistivity of the surrounding medium (pz =1= 00), as takes place in the case of the in-phase component In B; at the low-frequency spectrum. Applying the same approach, it is not difficult to show that if the surrounding medium is an insulator, we have at the late stage (1.222)

In this case induced currents are distributed uniformly within the layer,

1.8 Magnetic Field of a Vertical Magnetic Dipole Located inside a Layer

95

c 32 16

~=4

10

8 4

2

112 1/4

118 .1

1/16 1/32

'tYL

10

100

1000

Fig. 1.7 (Continued)

and the field B, is directly proportional to the cube of the longitudinal conductance S (S =Hlpl). It is convenient to represent results of calculations of transient field in terms of the apparent resistivity PT as a function of parameter TIlL. Examples of curves PTIPI are shown in Figures I.7b, c, and d. The index of curves is Pzlpl. The apparent resistivity is related to the change of the field with time as

(1.223)

96

The Quasistationary Field in a Horizontally Layered Medium

d

.1

10 Fig.1.7

100

1000

(Continued)

where Bun(PI) is the function aBzunlat in the uniform medium with resistivity of the layer PI. Such form of representation clearly illustrates an influence of induced currents in the layer and surrounding medium at different stages of the transient response. Considering these curves we can conclude that 1. At the early stage when TIlH < 5, currents are mainly concentrated in the layer and therefore the field depends only on the layer resistivity. For this reason the left-hand asymptote of curves PTI PI approaches unity (a> I),

2. For relatively large values of parameters T1IH, when currents are mainly located in the surrounding medium, curves of PTIP I approach their right-hand asymptote, equal P21PI.

1.9 Vertical Magnetic Dipole in Presence of Horizontal Conducting Plane

97

Inasmuch as induced currents at the late stage are located at distances which essentially exceed the probe length, the field is practically independent on L. Usually, the influence of the surrounding medium becomes significant at sufficiently small times, and this fact is a serious shortcoming of the transient induction logging when the layer has a relatively small thickness.

1.9 Field of a Vertical Magnetic Dipole in the Presence of a Horizontal Conducting Plane

In studying the behavior of the electromagnetic field of a vertical magnetic dipole located on the surface of a layered medium, we paid some attention to one special case when the underlying medium (basement) is an insulator. This analysis showed that under certain conditions the field is defined by the total conductance of layers S: N-l

L

S=

YiHi

i~l

but it is independent of the distribution of the conductivity. Such behavior of the field occurs if we consider the following: 1. The quadrature component of the magnetic field in the range of small parameters: r

H; - «1 0;

-« 1

° ' l

and the separation r is much greater than the total thickness of layers N-l

r

ze-

L

Hi

i~l

2. The in-phase component of the magnetic field InBS, or the quadrature component of the electric field Q caused by induced currents in the range of small parameters, regardless of the distance between the dipole and the observation point. 3. The field in the high-frequency part of the spectrum, when

E; ,

r - » 1 and 8]

H

-<1 °min

98

I

The Quasistationary Field in a Horizontally Layered Medium

where N

H= LH; i~l

4. The transient field at the early stage if

r - »1 T;

and

H

-- < 1 Tmin

5. The transient field at the late stage:

regardless of the distance r. Assuming that one of these conditions is met, we can say that a system of layers is equivalent to a thin conducting plane with the same conductance S. Taking into account this fact, it is natural to investigate the field generated by conduction currents in the plane. With this purpose in mind, suppose that a vertical magnetic dipole with the moment M is located at the origin of a cylindrical system of coordinates at distance h from a plane with the conductance S (Fig. 1.8a). It is obvious that the currents, induced in the plane, possess axial symmetry and have only a cp-component. Correspondingly, the electromagnetic field above and beneath the plane has the following components of the field:

As before, we describe the field with the vertical component of the vector potential A*, which is related to the field by

E

aA*

'"

=--

ar '

a2A*

iwB

=-r

araz

(1.223)

In accordance with Eqs. (1.94), and taking into account the fact that the medium surrounding the plane is an insulator, we have if z
p,M A*=iw--j 2 47T 0 Dm e-mzJ0 (mr)dm ,

(1.224)

00

if z > h

a

M

(I)

T

b

0

Ih

I

(2)

r

S

E(l)

h

8(1)

~

i

'E($)

8(~)

z

z

c bz

d

0.60

br 0.0

0.90

-0.40

0.40

-0.90

0.0 ~Sr

~Sr

-1.40

I .1

Iii

""

Iii 1

I "

"

iii i 10

~

-2 .-, -0.60 I 100·1

'i i "

i ;-;( 1

i

I

""

i i i " i .. I 10 100

Fig. 1.8 (a) Vertical magnetic dipole over conducting plane; (b) tangential components of electromagnetic field near plane S; (c) behavior of quadrature and in-phase components of field b, ; (d) behavior of quadrature and in-phase components of field b,.

100

I

The Quasistationary Field in a Horizontally Layered Medium

Because the conducting sheet is vanishingly thin, we can make use of approximate boundary conditions which do not require determination of the field inside the conducting layer. In fact, writing Maxwell's equations in the integral form, we have

¢. B . d ~ L

=

JL

1j . dS S

(1.225)

¢. E . d ~ = ico J(sB . dS L

Evaluating the first integral of the set 0.225) along the path shown in Figure 1.8b, we have (1.226) where B 2 r and B 1r are the tangential components of the magnetic field at each side of the conducting plane, and i

(1.229)

That is, the electric field is continuous across the conducting sheet, but the tangential component of the magnetic field is discontinuous and this discontinuity is defined by the current density at a given point. It is essential that the boundary conditions (1.229) contain only tangential components of the field for free space; this facilitates its determination. The components of the magnetic field on either side of the conducting sheet B 1r and Bz, are caused by the dipole current and by induced currents in the plane S. The primary magnetic field B o is continuous across the sheet. Therefore, the first condition in Eq. (1.229) can be

1.9 Vertical Magnetic Dipole in Presence of Horizontal Conducting Plane

101

written as (1.230) where E

Now, taking into account Eqs, 0.223), the boundary conditions for the vector potential have the form

A;

Ai =Ai aA*

aA*

__ 2

1

az

= -iwJLSA* = -iWJLSA*

az

1

(1.231)

2

Substituting Eqs. 0.224) into Eqs. (1.231) we arrive at a set of two equations with two unknowns: (1.232)

-mDm e- m h +me- m h -mCm e m h

=

mh -iwIISD r m e-

whence

2m

D

=----m

2m - iWJLS

(1.233)

and

e- 2 m h Cm

=

iwJLS----2m - iWJLS

(1.234)

102

I

The Quasistationary Field in a Horizoutally Layered Medium

Thus,

/-LM [ 1 e-Zmhemz Ai=iw-- -+iW/-LSj . Ja(mr)dm 47T R a 2m -IW/-LS 00

1

(1.235)

In the case when both the magnetic dipole and the observation point are situated on the conducting plane, Eq. 0.235) is slightly simplified and we have

/-LM mJa(mr) AHp) =iw-j . dm 27T a 2m - lW/-LS 00

(1.236)

since 1 -

r

1a Ja(mr) dm 00

=

Correspondingly, (1.237) Thus, iWIISA* r-:

=

aA* 2 __1

ifz=h=O

az'

1

Then from Eqs. (1.223) we obtain

E

=

iW/-L~1°omzJI(mr) 27T a

sp

dm 2m - iW/-LS

(1.238)

1

. M mZJ1( mr) dm -IW/-L-/-LS 47T a 2m - iW/-LS 00

B

= r

(1.239)

and

Bz

_ /-LM -

3J

a(mr) dm 27T 100m a 2m - iW/-LS

,

if z

=

h

=

0

(1.240)

Inasmuch as the horizontal component of the primary magnetic field is zero on the conducting plane (h = 0), the tangential components B, and E'f' must satisfy the condition

/-LS B Zr

=

TE'f'

and this also follows from Eqs. (1.238) and (1.239).

(1.241)

1.9 Vertical Magnetic Dipole in Presence of Horizontal Conducting Plane

It is convenient to introduce a new variable x (1.238)-(1.240). Then we have

=

mr

103 in Eqs.

(1.242) where (1.243) and (1.244) while (1.245) and iosp.Sr q= - - 2 -

=

-ips

The parameter

WJkSr

PS=-2plays the same role as the ratio between the distance r and the skin depth 8 for a conducting half space. First we study the range of small parameters Ps. Applying the method of deriving asymptotic formulas described in Section lA, we obtain if Iql < 1 or

WJkSr)3 t nWJkSr (WJkSr)3 - - - a - - + ... 2 2 0 2

Qb "" (- z

(1.246) s

Inb "" z

(JkWSr)2 (WJkSr)3 (WJkSr)4 - - +- - - - - - + ... 2 2 2 2 7T

104

I

The Quasistationary Field in a Horizontally Layered Medium

where

a o = t'n 2 - C + 2,

C = 0.57721566

Therefore, in contrast to the case of a layered medium, the in-phase component of the secondary field B, dominates over the quadrature component, which is proportional to w 3 t'n wand the cube of longitudinal conductance S3. It can be readily seen that in the low-frequency range (P s < 1) the component Q B, is practically independent of the distance from the dipole to the receiver. In fact, as follows from Eqs. (1.244) and (1.246), we have QB

J-tM ( wJ-tS )3 -t'nw 47T 2 '

0::::-

z

if

w ~

0

(1.247)

It is proper to note that the vertical component of the transient magnetic field B/O is also proportional to S3 and is independent on the distance r at the late stage. This particular behavior of the quadrature component Q B, is observed only when both the dipole and the observation site are located on the conducting plane. For example, if the dipole and the observation point are placed at a height h above the plate S, the secondary magnetic field BL can be written in the form

In the limit when the parameter Ps tends to zero, we have if P«

~

0

(1.248)

that is, the leading term of the quadrature component Q B, is directly proportional to w, and it is not equal to zero. In defining the vanishing thin conducting sheet, it was assumed that the model would be equivalent to the real case of a relatively thin layer with a finite thickness. However, as this analysis shows, these models are not equivalent when the quadrature component of B, is considered in the range of small parameter Ps and the dipole and observation point are located on the plane S. At the same time, the first term of the expression for the quadrature component and the second term for the in-phase component [Eq. 0.246)]

1.9 Vertical Magnetic Dipole in Presence of Horizontal Conducting Plane

105

are the same as corresponding terms in the series representing the field at the surface of a layered medium lying above an insulating basement. Applying the same approach for deriving asymptotic formulas, we have the following formulas for the electric field in the range of small parameters Ps:

where C

b

tn2

=--1

4

4

The first term in the expression for the quadrature component is the primary electric field E~. For small values of Ps this field nearly defines the current in the conducting plane: w/LM i =--S

(1.250)

This current gives rise to the magnetic field which is also directly proportional to wand S. But, as has been shown previously, the vertical component of this magnetic field is zero, if z = h = O. The first term of the expression for the in-phase component, [Eq. (1.249)] is caused by a change of the quadrature component of the magnetic field with time, which is proportional to w 2 . The second and following terms in Eqs. 0.249) also reflect the interaction between currents. Finally, from Eq. (1.241) we have for the horizontal component B,: W/LS [ oiu.Sr ( cop.Sr InB z--/LM --+ - r 8'1Tr 2 2 2

)3 tn--+b W/LSr ( W/LSr )3 - - + ,., ] 2

1

2

(1.251) and W/LS [ -1+ ( W/LSr QBrz--/LM 8'1Tr 2 2

)2 -

W/LSr -'1T ( -

4

2

)3 + ... ]

106

I

The Quasistationary Field in a Horizontally Layered Medium

Comparing Eqs. (1.246) and (1.251), it can be seen that in contrast t-o the case of B z ' the quadrature component of B contains a term which is directly proportional to w. Moreover, the second term of the in-phase component and the third term of the expression for the quadrature component are more sensitive to changes in conductance S than corresponding terms in the series for the quadrature component of the magnetic field B z . This behavior holds equally well in the case of a horizontally layered medium when the basement is insulating. Next, let us consider the behavior of the field when parameter Ps is large. Representing the fraction 1

x+q in Eqs. (1.243)-0.245) as a series:

and performing integration, we obtain the following expressions for the leading terms of the series which describe the field in the range of large parameters: 36

(wJ.LSr) 2

12 '

ecp : : : -

(wJ.LSr)

2 '

if

v.> 1

(1.252)

Thus, in this range the in-phase component of B, as well as the quadrature component of the electric field are dominant and they are inversely proportional to S2. For instance, if u.> 1

(1.253)

Taking into account Eq. (1.241), it can be shown that the quadrature component of the horizontal magnetic field B, is less sensitive to a change in the longitudinal conductance than the in-phase component of B z . In fact, we have if Ps» I Graphs of frequency. responses of the quadrature and in-phase components of functions b, and b, are shown in Figures 1.8c and d.

1.9 Vertical Magnetic Dipole in Presence of Horizontal Conducting Plane

107

Next we investigate the transient responses of the electromagnetic field, caused by currents in a conducting plane. Applying a Fourier transform to the spectrum of the vector potential A* of the second field [Eq. (1.67)], we have

p,SM

00

Ai( t) = -iw--p,j 2 16'7T

2h )J

0

o( mr ) dm .

iw t

ef-oom-/w/b . dto 00

em ( z -

(1.254)

where 2 b=-

(1.255)

p,S

First consider the integration over frequency: 00

e -iwt

(1.256)

L=f dto -oom - ito /b The integrand has a pole when Wo=

-imb

but there are no branch points. Placing the path of integration as shown in Figure I.3b and applying the residue theorem,

where

f(w) = 'PI(W) 'P2( w) In our case,

Thus, and correspondingly if z
108

I

The Quasistationary Field in a Horizontally Layered Medium

or

Aie t ) =

f-LM -

-

1

417 [r 2 + (bt + 2h - z

)2]

1/2

(1.257)

Therefore, for the electromagnetic field we have

f-LM

r 2-2(bt+2h-z)2

s, = 4 17 [r 2 + (bt + 2h _Z)2 ]5/2 n, =

3f-LM r(bt+2h-z) -4- [ ]5/2 17 r2+(bt+2h-z)2 3M

r(bt + 2h -z)

Eop = - 217S [r2 + (bt + 2h _ z )2f /2

(1.258)

(1.259)

(1.260)

These expressions are remarkably simple and they often play an important role in the interpretation of transient soundings. It can be seen from Eqs. (1.259) and (1.260) that in all cases there is a single relationship between the electric field and the horizontal component of the magnetic field: (1.261) Now we consider several specific cases:

Case 1 First, let us assume that the dipole and the observation site are situated on the same axis, r = O. Then we have

f-LM 1 Bz(t)=-3 217 (bt + 2h -z) and

(1.262)

1.9 Vertical Magnetic Dipole in Presence of Horizontal Conducting Plane

109

It is clear that during the early stage of the transient response

B (t) z

p,M --+ - ---~

21r(2h-z)3'

if bt-e: 2h - z

This expression also describes the field of a fictitious magnetic dipole located on the z-axis at the point z = 2h. This analogy can be generalized to a more common case when neither t nor r are zero. As follows from Eq. (1.258), the late stage begins when 2t

-.-» 2h-z

(1.263)

Mp,4S 3 B:::=--z 161rt3

(1.264)

p,S

and then we have

We see that during the late stage the field B, is independent of distance (either z or h), and it is characterized by a very high sensitivity to the parameter S: .

s, ~S3 From the physical point of view, condition (1.263) indicates that induction currents are situated relatively far from the z-axis. As follows from Eq. (1.262), in order to determine the conductance S and the height h, it is sufficient in principle to make measurements at only two times, corresponding, for example, to the intermediate portion of the transient response. If the receiver and transmitter coils are coincident, then from Eq, (1.262) we have

p,M B=-----__;:_ z

21r(bt+2h)3

and therefore a single coil system can be used in determining both parameters Sand h.

Case 2 Now suppose that the dipole and an observation point are situated on the plane S, that is, z = h = O. Then the field components are described by the

110

I The Quasistationary Field in a Horizontally Layered Medium

following expressions:

IJ-M r 2 - 2b 2t 2 B =-------z 47T (r2 + b'1(2)5/2 B

=

3IJ-M bt 47T (r2+b2t2)5/2

-·--r-----=-=

r

(1.265)

3M bt E = - --r-----=-= 2 cp 2rrS (r + b 2 t 2 ) 5 / 2 Let us represent these equations in the form

IJ-M B =--bz z 47Tr3 and M E =---e cp 2rrSr 3 cp

where (1.266)

and 7" S

During the early stage when

7"5

bt

2t

r

IJ-Sr

=-=--

(1.267)

tends to zero and we have if

7"s

« 1

(1.268)

Thus, when the dipole current is turned off, induction currents arise immediately in the vicinity of the dipole, creating a vertical component of the magnetic field which is equal to the primary magnetic field (r ~ 0). At that instant the horizontal component of the magnetic field, along with the electric field, at the plane S is zero. This means that at the moment t = 0 induced currents are absent everywhere except the lateral surface of the current source.

1.9 Vertical Magnetic Dipole in Presence of Horizontal Conducting Plane

III

In the other extreme during the late stage we obtain 3

2 b~--

3 ' 75

z

b~r

4

75

or

(1.269)

Thus, at the late stage of the transient response the vertical component of the magnetic field is independent of the separation r . Such behavior has already been observed in the case of a horizontally layered medium. At the same time, the horizontal component B, is less than the vertical component, but it is more sensitive to a change in the conductance:

In accordance with Eqs. (1.265), the current density at any point in the plate S is

It is clear that regardless of the distance r (r -=1= 0), the surface density i

M

(im) max :::: 0.37Tr 3 ~

Case 3 Next we suppose that both the dipole and the observation point are situated at the same height h above the plate S. Then from Eqs.

112

I

The Quasistationary Field in a Horizontally Layered Medium

(1.258)-(1.260) we have

and E

M

= 'P

----e 27TSr3 'P

where bz =

1 - 2( T s + 2h o)2 [1

+ (Ts + 2h o)2r /

2

3( T s + 2h o)

(1.270)

and

In this case the early stage occurs if t h»-

(1.271)

p,S

If so, we have 1- 8h6

b r

=

6h o -----=-= (1 + 4h6)5/2

Comparing Eqs, (1.270) and (1.266) we see that they coincide when the inequality (1.272) is met. In other words, the influence of the height practically vanishes if the time of observation satisfies the condition

t> p,Sh

(1.273)

It is also obvious that the late stage of the transient field contains no

information about the height h, and it is described by Eqs, (1.269). Examples of transient responses of functions b, and b, are given in Figures I.9a and b.

1.9 Vertical Magnetic Dipole in Presence of Horizontal Condncting Plane

113

a 0.27

bz

0.19

0.11

0.03

-0.05

-0.13 't s

-0.12 .01

.1

10

b 1.20

br

1.00

0.80

0.60

0.40

0.20

0.00 .01

.1

Fig. 1.9 (a) Transient responses of field bz ; (b) transient responses of field b.: (c) horizontal dipole over layered medium; (d) profiling array with horizontal dipoles. (Figure continues.)

114

I The Quasistationary Field in a Horizontally Layered Medium

c M fr-~-----~X

z

d

M

/~/

Fig.1.9

Rl

R2

R3

£ /79////T/7TQ

/ /////7

(Continued)

///7

1.10 Horizontal Magnetic Dipole on Surface of Layered Medium

115

1.10 A Horizontal Magnetic Dipole on the Surface of a Layered Medium

Suppose that a horizontal magnetic dipole is located above the earth's surface at height hI' and that its moment M is directed along the x-axis as shown in Figure 1.9c. First we derive expressions for the electromagnetic field in the frequency domain and then investigate the behavior of the quadrature component of the magnetic field in the low-frequency part of the spectrum. It is convenient to introduce both Cartesian and cylindrical systems of coordinates with a common origin where the dipole is situated. As before we make use of the vector potential of the magnetic type A*, which is related to the electric and magnetic fields by E = curIA*,

iwB

=

k 2A*

+ graddivA*

(1.274)

and at regular points satisfies the Helmholtz equation: (1.275) Inasmuch as the primary electric field intersects the earth's surface, we have to expect appearance of electric charges. This means that unlike the previous cases, the electromagnetic field is caused by both induction currents in the conducting medium and surface charges. Correspondingly, the geometry of the field becomes more complicated and for this reason we use two components of the vector potential A~ and A~ , assuming that

At =0. As follows from Eqs, (1.274) and (1.275), each component of A* obeys the Helmholtz equation: (1.276) and for the components of the electromagnetic field we have

aA*

E = __ Z x

aA*x E= __

ay'

az

Y

aA~ _

ax'

E = z

aA*x

ay

and

iwB

a

x

=

k 2A* + -

=

-

=

k 2A*Z

x

a

iwB Y

iwB Z

ay

ax div A*

divA*

a

+ -az

divA*

(1.277)

116

I

The Quasistationary Field in a Horizontally Layered Medium

Therefore, in order to provide continuity of the tangential components of the electric and magnetic fields at the ith interface, the vector potential should satisfy the following conditions: aA~,i

aA~,i+!

ay

ay

aA~,i aA~,i -----

az

aA~,i+!

ax

aA~,i+l ax

az

a

klA~

,

i + - divAj ax

=

kl+!A~

a

- divAj ay

(1.278)

a

a = -

ay

'

i+! + - divAj+! ax

divAj+!

Certainly, this system is rather complicated, but it can be drastically simplified if we represent Eqs. 0.278) as two groups of equations-namely, the first group

az

az

(1.279)

and the second group A~,i

=A~,i+l'

div A'~

=

divAj+ I

(1.280)

where aA* div A* = __x ax

aA*

+ __z az

(1.281)

It is obvious that if Eqs. 0.279) and 0.280) are satisfied, then the components of the vector potential also obey Eqs. 0.278). It is essential that the conditions for the component A~ at interfaces, [Eqs. (1.279)] do not contain the other component A~ . This fact allows us to determine first of all the component A~ and then find out the expression for the vertical component A~ . With this purpose in mind, let us formulate the boundary value problem for the component A~. In accordance with Eqs. 0.276) and (1.279), at regular points and at interfaces the component A~ satisfies the following conditions:

117

1.10 Horizontal Magnetic Dipole on Surface of Layered Medium

and aA~

2 k 12A*x,I -- k i+1 A*x,i+l'

i

aA~,i+l

aZ

aZ

respectively. Near the dipole we have

ieu M 47TR

A*

iiou M 47T

CQ

--+ - - = - - ]

x,o

0

e,-m1zlJ (mr) dm 0

if R At the same time, at infinity A~,i

A~

,

--+ 0

(1.282)

tends to zero: if R

--+ 0,

(1.283)

--+ 00

Taking into account Eq. (1.94), the component A~,i

can be written as

Thus, we have iosu M A*x,o = -47T - ]0

A*x,l

ioiu M =--] 47T

A*x,2

= --] 47T 0

CQ

dm ,

00

(Ce-mlz+DemlZ)J(mr)dm 1 I 0 ,

0

iosu M

(e- mizi + D 0 emz)J0 (mr)

CQ

(C 2 e- m2z + D 2 e m2Z)J 0 (mr) dm '

ifh 2
if z > h N

where

As was shown earlier, the function each layer.

A~,i

obeys the Helmholtz equation in

118

I

The Quasistationary Field in a Horizontally Layered Medium

Now, applying conditions 0.279) at each surface we obtain a system of equations for determination of the unknown coefficients:

Performing the standard procedure for solving this system we find that

(1.285)

and

Therefore, in the conducting medium the horizontal component of the vector potential A~ is equal to zero: A~

(1.286)

=0,

In accordance with Eq. 0.277), we have E

= Z

JA*x Jy

Thus, beneath the earth's surface the vertical component of the electric field is equal to zero: (1.287) that is, induced currents are located in horizontal planes.

1.10

119

Horizontal Magnetic Dipole on Surface of Layered Medium

At the same time, above the earth's surface the component of two parts. In fact, from Eqs. (1.284) and 0.285) it follows

A~

consists

or

A*

=

iWf.LM (~+~)

xO

R

47T

RI

(1.288)

where R

=

( r 2 +z 2) 1/2 ,

In particular, when the dipole is located on the earth's surface, hI have

=

0, we

(1.289) It is proper to note that this component does not contain any information about the geoelectric parameters of the medium. In the upper half space, unlike the conducting medium, the normal component of the electric field is not equal to zero, and in accordance with Eqs. (1.277) we obtain

(1 1)

E -iWf.LM - - -3+ z 47T y R Ri

(1.290)

Correspondingly, as follows from Eq. (1.56) in Part B, the surface density of charges is

or

I

iWf.LM = E O - - 2-

27TR

sin cp,

(1.291)

As is seen from Eqs. (1.291), charges of opposite sign appear at the same instant at points where y > 0 and y < 0, respectively, and their magnitude decreases rapidly with distance. Of course, in the vicinity of the x-axis, as well as at interfaces beneath the earth's surface, charges are absent.

120

I

The Quasistationary Field in a Horizontally Layered Medium

Next we derive an expression for the vertical component A~. First of all, in accordance with Eq. (1.286), the expression for div A* in the conducting medium is drastically simplified and we have aA*

divA* = __Z

az '

Therefore, the conditions (1.280) have the form aA~,i A~,i

and

=A~,i+1

aA~,i+j

az

(1.292)

az

and at the earth's surface

+ aA~.o

aA~,o

A~.o

_ aA~,

j

----a;- a;- - a;-'

=A~.j,

ifz=h 1

(1.293)

In order to satisfy the last condition in Eqs. 0.293), all terms must have the same dependence on coordinates of a point. It is clear that aA~.o

ar

aA~,o

-- = -- -

ax

ar

ax

aA~,o

= cos cp - ar

iWfLM

=

()()

---coscpj m(e-mlzl+Doemz)Jj(mr)dm (1.294) 4rr

0

since

Correspondingly, we represent A* . = Z,l

iWfLM cos tp 4rr

--

A~.

1

as

i

00

0

[C*e-m,z I

+ D*em'Z]J 1(mr) I

dm

(1.295)

and this function obeys Helmholtz's equation:

In particular, above the earth's surface, we have A* = z,o

iWfLM JOO coscp 0 D*emzJ (mr) dm 4rr 0 l

---

(1.296)

Now, applying the conditions 0.292) and (1.293) at the interfaces, we can and For our purposes it is determine the unknown coefficients

Cr

Dr.

1.10

121

Horizontal Magnetic Dipole on Surface of Layered Medium

sufficient to consider only a three-layered medium. Then we obtain

(1.297)

Eliminating all coefficients except Dri, we obtain

2me- 2 m h , Dri=---m +m 1 P3

(1.298)

where

P2 =

m l -m 2 P I

-

-

-

-

m l +m 2 P I

and

k 23

m 2-m 3

= ----

m 2+m 3

Therefore, the vertical component of the vector potential in the upper half space is

Usually, in performing electromagnetic profiling the horizontal component B; is measured in the equatorial plane of the dipole (cp = 1T/2). Taking into account this fact, we derive an expression for B x .

122

I The Quasistationary Field in a Horizontally Layered Medium

As follows from Eqs. (1.277),

a ax

iwB x = -

div A*o,

Inasmuch as

aA*.e.o aA*Z,o diIV A*0 = --+--

ax

az

we have

Correspondingly, at the y-axis when x

=

a we obtain

Next, suppose that both the dipole and the receiver are located on the earth's surface. Then we obtain (1.299) Now we pay attention only to the low-frequency part of the spectrum. Taking into account that 1

kf

m·zm - - 12m'

if

k~O

k

n

we have 1

kn -

= -4m-2(q- -

1.10 Horizontal Magnetic Dipole on Surface of Layered Medium

and

Whence

m I P3

=(m-~)(l-~)2m

2m2

where

and

Substituting the latter into Eq. (1.299), we obtain Ex =

f.LM [

-

21T

-

1

1

00

+ - j mf l( mr) dm r 2r 0

3"

k?joofl(mr) Q_k?jOOe- 2mHt dm + 8 f l ( mr) dm 8r 0 m rom

+-

+ k'J - k~ JOO e -2m(H, +Hz ) f ( mr) dm] l 8r 0 m Inasmuch as ooe- m1zl

Vr 2+z 2 _ z --fl(mr)dm=---jo m r we have

oo f l( mr ) ---dm=l jo m'

OO e- 2mHI dm jo ---fl(mr) m

-/r2+4HI2 -2H I

= -------

r

123

124

I

The Quasistationary Field in a Horizontally Layered Medium

and co

1o

VrZ + 4( HI + Hz)z - 2( HI + Hz)

e-Zm(H +H2 ) j

Jl(mr) dm

m

Thus, the secondary field

E~

= ------------

r

is

(1.300)

This equation clearly demonstrates that in the low-frequency part of the spectrum the quadrature component of the magnetic field, Q Ex , as in the case of the vertical magnetic dipole, can be described with the help of geometric factors. In fact, the field Q Ex can be represented as (1.301)

where G

I

G, ~

=

r 1_ (

Z + 4H Z)I/Z - 2H 1

1

r

(rz+4Hnl/Z-2HI =

r

-

[r Z+4(H I + Hz)Zf/Z -2(H] +H z)

-=-----------=--------r

(1.302)

are geometric factors for the upper intermediate layers and lowermost medium, respectively. The function G z is called the geometric factor of a layer of finite thickness, and it is obvious that the functions G 1 and G 3 can be derived

1.10 Horizontal Magnetic Dipole on Surface of Layered Medium

125

from G z . As follows from Eq. (1.302),

This analysis shows that though the distribution of the quadrature component of induced currents is different from that when the field is caused by a vertical magnetic dipole, we can still use the concept of the geometric factor. This is related to the fact that surface charges are absent beneath the earth's surface and in the low-frequency part of the spectrum the interaction of the quadrature component of induced currents is negligible. Therefore, we can generalize Eq, 0.301) for an N-layered medium and then obtain

WJ.1,zM Q Ex = -16 1Tr

N

L y;Gi i=l

(1.303)

The latter is used for interpretation of electromagnetic profiling with different separations between the dipole and the receiver (Fig. L9d). In conclusion, let us briefly study the behavior of the in-phase component of the secondary magnetic field in the low-frequency part of the spectrum. With this purpose in mind, we represent the coefficient D'6 in a different form. First of all, it is clear that systems (1.95) and 0.297) are similar. Applying the same approach as in the case of the vertical magnetic dipole, the field Ex can be written as

(1.304) where (1.305)

As was demonstrated earlier, in the low-frequency part of the spectrum the in-phase component of the magnetic field In BS is determined by the small values of m. In accordance with Eq. (1.305), we have if m

~

0

and

k

~

0

126

I

The Quasistationary Field in a Horizontally Layered Medium

and therefore m

m

------ ~ ----1 + (m/m])R 3 that is, the integrand in Eq. (1.304) approaches that for a uniform half space with resistivity P3. In other words, if k

~

0

and

P3 =1=

00

This result is easily generalized for an N-Iayered medium. Also taking into account the relationship between the low-frequency spectrum and the late stage of the transient response, we can conclude that with an increase of time the field approaches that in a uniform half space with the resistivity of the lower medium.

1.11 A Vertical Electric Dipole on the Earth's Surface

Now we study the field of a vertical electric dipole located near the earth's surface and measured in the far zone, where the distance from the dipole is much greater than the wavelength in free space. This condition is usually observed in geophysical methods based on measuring fields which are generated by low-frequency radio stations. Suppose that a dipole with moment (1.306) is situated above the earth's surface as is shown in Figure I. lOa. Here (1.307) where eo is the free charge and dt is the dipole length. It is obvious that the field possesses axial symmetry. Correspondingly, we introduce a cylindrical system of coordinates so that the dipole is located at its origin and the moment M is directed along the z-axis. In order to determine the field, it is convenient to make use of the vector potential of magnetic type A. In accordance with Eqs, (1.392) from Part B, we have B = curl A

J.L( y

-

iWE)E = k 2A + grad divA

(1.308)

127

1.11 Vertical Electric Dipole on Earth's Surface

b

a z

z

M

Fig. 1.10 (a) Electric dipole over uniform half space; (b) electric dipole in uniform medium.

and function A satisfies Helmholtz's equation: (1.309) Also, we use the relationship between the scalar and vector potentials: iw U= - -divA k2

(1.310)

First, let us assume that the dipole is located in a uniform medium with parameters 'Y and E. Then, by analogy with the magnetic dipole, we find the field using only the single component of the vector potential A z , which depends on coordinate R. In other words, in the spherical system of coordinates oA, _4

oA

z =-= 0

oep

00

and Eq. (1.309) has the form

-

1

R2

d

oA z

dR

oR

- R2 -

+ k 2A =0 z

128

1 The Quasistationary Field in a Horizontally Layered Medium

As is well known, the solution of this equation satisfying the condition at infinity is eik R

A z = CR-

(1.311)

where

R

=

( r 2 +z 2)1/2

To determine the constant C we study the field in the vicinity of the dipole. It is convenient to represent the dipole as a system of two small electrodes connected by a wire. Due to the current in the wire, electric charges arise on the electrode surfaces. Thus, from the physical point of view the dipole can be described as a combination of the current in the wire and two surface charges of opposite sign. Both the current and charges vary as sinusoidal functions, and in general there is a phase shift between them. It is proper to note that this model of the electric dipole implies that measurements are performed at distances which significantly exceed the dipole length, and that the current along the wire has the same magnitude and phase. Of course, there are also charges on the lateral surface of the wire, but their influence on the field is usually very small. In accordance with Coulomb's law, in the vicinity of the dipole the electric field of the charges coincides at each instant with the static field Eo, and its potential Va is defined as (1.312) On the other hand, as follows from Eq. (1.310), the potential of the electromagnetic field V is

ito ilA z V= - -2- k

oz

=

-

ieo ilA z --2 k

ilR

cos 8

Then, taking into account Eq, (1.311), we obtain

V=

u«: 0

e i k R ( 1 - ikR) R2 cos 8

Inasmuch as in the vicinity of the dipole V---')

ii;

(1.313)

1.11

Vertical Electric Dipole on Earth's Surface

129

we obtain

M

iw

= -k c 2 47TE or

c=--47TEiw Therefore, the vertical component of the vector potential A z is k 2M e i k R A=--z 47TEiw R

(1.314)

In deriving expressions for the field components we use a spherical system of coordinates, R, e, and ip, where r

sin e = R

and

z cos e = R

As is seen from Figure 1.10b,

In accordance with Eqs, (1.308), we have

and k2 - ER

=

iw

a2A z k 2A R + __ aeaz

2

k -.-Eo uo

=

k 2A o +

1

a2A z

R aRaz

E
130

I

The Quasistationary Field in a Horizontally Layered Medium

we obtain expressions for the field components of the electromagnetic field in a uniform medium:

»»

k 2M

Bcp

= 47TEiw R 2 M

ER

=

(1 - ikR) sin ()

eik R

27TE ~

(1 - ikR)cos (}

(1.315)

M e ik R E e = - - --3-(1 - ikR - k 2R 2)sin (} 47TE R First, consider the relationship between charges and the current in the dipole on one hand and the parameters of the medium on the other hand. As was pointed out earlier, near the dipole the electric field is practically caused only by charges, and therefore we have

edt ER

~

Ee ~

27TE OR

3

(1.316)

edt 47TE OR

cos (}

3

sin (}

where e is the amplitude of the total charge on the electrode surface and is the dielectric constant. Comparing Eqs, (1.315) and (1.316), we see that the total and free charges are related to each other in the following way:

EO

(1.317)

Since the free and bound charges have opposite signs, the total charge becomes smaller than the free charge and, correspondingly, the electric field decreases. Taking into account the continuity of the normal component of the total current density on the electrode surface and neglecting displacement currents in the wire, we have

or (1.318)

where En is the normal component of the electric field on the external side of the electrode surface.

Vertical Electric Dipole on Earth's Surface

1.11

131

Taking the integral of both sides of Eq. 0.318), we obtain

where S is the electrode surface except in the vicinity of the connection point of the wire with the electrode. Thus, the dipole current is e

1= (y-iWE)-

eo

=

(y-iWE)-

EO

(1.319)

E

since

Therefore, Eqs. (1.315) can be rewritten as

ILl dt . 47TR2

= _ _ e ' k R(l

B 'P

- ikR)sin 8

I dte i k R ER

=

(

.

)

27T Y -lWE R

3

(1 - ikR) cos 8

(1.320)

I dte i k R E o= ( . ) 3(1-ikR-k 2R 2)sin8 47T y-lWE R Now, proceeding from Eq. 0.319), let us consider the linkage between charges and the current in two special cases when either the conduction or the displacement current can be neglected. For instance, in an insulator we have 1= -iwe o

(1.321)

that is, the dipole current is equal to the rate of change of the surface charge, and they are shifted in phase by 90°. If the frequency is equal to zero, displacement currents disappear and, unlike the electric field, the magnetic field vanishes. In a conducting medium, when wE/y ~ 0, the phase shift between the current I and the surface charges is equal to zero and, correspondingly, they change synchronously. It is essential that in this case the total charge e is independent of the dielectric permittivity of the medium and (I.322) In the general case, when both the conduction and displacement currents

132

I

The Quasistationary Field in a Horizontally Layered Medium

are present, the relationship between the current and surface charges, including the phase shift between them, depends on w, 'Y, and E. Next, we study the main features of the field behavior. As in the case of the magnetic dipole, it is appropriate to distinguish the near, intermediate, and wave or far zone. Inasmuch as the dipole current I can be measured, we make use of Eqs. (1.320). When the parameter IkRI is very small, we have for complex amplitudes of the spectrum

Bop =

j.Lldt --2

417R

sin e

1dt' cos e ER=-----~

217( 'Y - iWE)R 3

(1.323)

I dtsin ()

Ee =

-

-

-

-

-

----,,-

417( 'Y - iWE)R 3

Assuming that the current I is given, we conclude that, even in the near zone, the electric field, unlike the magnetic one, contains information about the conductivity and dielectric permittivity of the medium. In the opposite case of the wave zone, when IkRI» 1, we have B

op

=

E = R

E

u ] dt . -ik _ _ e1k R sin ()

417 R

wj..tldt

217kR 2

e i k R cos ()

(1.324)

iosu I dt .

e

=

-

417 R

e 1k R sin ()

It is clear that in this zone the azimuthal component of the electric field prevails:

and the electromagnetic field resembles a plane wave. The difference in the behavior of the field in the near and far zones is especially seen if the medium is not conductive and therefore the transformation of the electromagnetic energy into heat is absent. Now we derive expressions for the field of a vertical electric dipole located on the surface of a uniform half space, (Fig. I. IOa). Taking into account the axial symmetry of the field, we still look for a solution with the help of one component of vector potential A z : Inasmuch as the compo-

1.11 Vertical Electric Dipole on Earth's Surface

133

nent A z in a uniform medium [Eq. (1.311)] can be represented as (1.325) we can write expressions for the vector potential A z in each medium in the following form:

A zo = C jOO[~e-molzl rno

+ Fe moz] Jo(rnr) dm ,

°

if z

<0 (1.326)

if z > 0

A Z1 = C jOODe-m1ZJo(rnr) dm, o where 2M

k __ C=_o 47TiwE

(1.327)

As follows from Eqs. 0.308), B r = 0,

B

cp

aA z =--

ar ' 0.328)

and

Ecp =0

Therefore, in order to provide continuity of tangential components of the field at the interface z = 0, the component A z has to satisfy two equalities:

A zo =A z) 1 aA zo

---

k5 oz

1 oA z )

----

kr

if z

oz '

=

0

(1.329)

Substituting Eqs. (1.326) into Eqs. 0.329), we obtain two linear equations with two unknowns: m

-+F=D rno

kr( -rn + rnoF)

=

-k5 rn)D

134

I

The Quasistationary Field in a Horizontally Layered Medium

Solving this system we have

and

(1.330)

Respectively, the vector potential A z and its derivative JAjJz at both sides of the interface z = 0 are

(1.331)

We assume that the upper space is a free space, that is,

and in the conducting medium the influence of displacement currents is negligible:

k[=(iY/-Lw)

1/2

1+i 8

=--

Also, in studying the field we restrict ourselves to the case when the skin depth 8 is much smaller than the distance r:

r -» 1 8

(1.332)

Let us assume that the dipole is located on the surface of the ideal conductor, that is, k 1 ~ 00. Then, as follows from Eqs. 0.331), A zo =

/-Lo! dt '" m --1 -Jo(rnr) drn 277" rno 0

1.11 Vertical Electric Dipole on Earth's Surface

135

or

A

/.Lo! dt eikor 21T r

=--zO

(1.333)

and

dA zo --=0

dz

since

Thus, on the surface of a perfect conductor the function A z is two times greater than that in a uniform medium. Making use of Eqs. (1.328), we obtain for the field components E, = 0

Ez

= -

eo dt 2 1TE 3 or

(1 -

" ik 0 r - k2r2)e'kor 0

(1.334)

since !

=

-iwe o

Of course, the components of the field also double with respect to the case of a uniform medium. It is obvious that the electromagnetic energy does not penetrate into an ideal conductor and, correspondingly, it propagates along its surface. The same conclusion follows from the fact that the tangential component of the electric field E; is equal to zero. In particular, if the parameter kor is large, we have B =
and

iko/.Lo 1 dt "k e' or 21Tr (1.335)

It is essential that these equations also describe the main part of this field in the far zone, even when the conductivity Y1 has a finite value.

136

I The Quasistationary Field in a Horizontally Layered Medium

As follows from Eqs. 0.335),

_ ~ (/La /La E a

1 )1/2 --z /La

where Z is the impedance of the plane wave in free space. Since the electric field is equal to zero inside the ideal conductor, the surface density of charges can be represented as (1.336) Thus, at every point of the surface there are charges which vary with time and distance in the same way as the vertical component of the electric field e..

References Bursian, V. R. (1972). "Theory of Electromagnetic Fields Applied in Electroprospecting." Nedra, Leningrad. Fok, V. A., and Bursian, V. R. (1926). Electromagnetic field of a current in a circuit with two grounds. Phys. Chern. Soc. 58(2). Kaufman, A. A., and Keller, G. V. (1983) "Frequency and Transient Soundings." Elsevier, Amsterdam. Sheinman, S. M. (1947). On transient electromagnetic fields in the earth. Appl, Geophys. 3. Tikhonov, A. N. (1950). On transient electric currents in an inhomogeneous layered medium. Jzv. Akad. Nauk SSSR, Ser Geogr. Geojiz. 14(3). Vanyan, L. L. (1965). "Electromagnetic Soundings." Nedra, Moscow. Wait, J. R. (1962). "Electromagnetic Waves in Stratified Media." MacMillan, New York.

Chapter II

The Behavior of the Field in a Medium with Cylindrical Interfaces

11.1 The Field of the Vertical Magnetic Dipole on the Borehole Axis 11.2 The Quadrature Component Q b, in the Range of Small Parameters: Ik i a il « 1 and IkiLI« 1 II,3 The Behavior of the Field in the Range of Small Parameters ajD i II.4 The Magnetic field on the Borehole Axis in the Far Zone II,5 Behavior of a Nonstationary Field on the Borehole Axis II,6 Magnetic Dipole on the Borehole Axis when the Formation Has a Finite Thickness 11.7 The Field of a Current Loop around a Cylindrical Conductor 11.8 Integral Equation for the Field Caused by Induced Currents References

Electromagnetic fields have found broad application in borehole geophysics for the study of the resistivity and dielectric permittivity of the formations surrounding a borehole. For instance, in induction and dielectric logging an alternating magnetic field is usually measured when a primary electromagnetic field is caused by the vertical magnetic dipole located on the borehole axis. Taking this fact into account, we consider in detail the frequency and transient responses of the magnetic field B, in the borehole.

11.1 The Field of the Vertical Magnetic Dipole on the Borehole Axis

Suppose that a vertical magnetic dipole is located on the borehole axis, and that its moment M changes as a sinusoidal function of time. Due to the change of the primary magnetic field with time, a primary electric field arises which has only an azimuthal component E ocp ' as shown in Figure

138

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

lI.la. Correspondingly, induced currents appear in the conducting medium which also have only an azimuthal component, since the interaction between currents does not change their direction. Therefore, electrical charges do not arise at interfaces. Correspondingly, the generators of the secondary field are induced currents located in horizontal planes which have only the component i; and their vector lines are circles with centers on the borehole axis. We see that the geometry of currents is the same as that for a uniform medium. To derive the expression for the field we again proceed from Maxwell's equations and formulate a boundary value problem. In accordance with Eqs, (1.391) in Part B, the electromagnetic field is related to the vector potential of magnetic type A* by E = curlA* iwB

=

k 2A*

+ graddiv A*

(II.1)

where (11.2)

and A* obeys Helmholtz's equation: (11.3)

Let us choose a cylindrical system of coordinates (r, tp; z ) and suppose that the vertical magnetic dipole is placed at the origin of this system (Figure II.1a). The moment of the magnetic dipole is oriented along the z-axis. Taking into account the geometry of current distribution, we look for a solution of the problem with the help of the single component of the vector potential, A~ . As follows from the behavior of the electromagnetic field, the vector potential must satisfy several conditions: 1. The function A~ is a solution of Helmholtz's equation in the borehole and in the formation

In a cylindrical system of coordinates this equation is written as (11.4)

II.1

The Field of the Vertical Magnetic Dipole on the Borehole Axis

139

a z

......

~

b

---

-a0.1

M.....

t.

v<,

0.01

'Y2

/11

/12

£1

£2

--

a. 0.001+----,.---,.....--~

10

0.1

100

....

./

C 100

i: 10

d

64 32

r,~

16

8

4

R2

2

Rl L2 L,

0.1

T 0.01

a 01

10

100

Fig. II.I (a) Vertical magnetic dipole on borehole axis; (b) behavior of geometric factor G l(a); (c) curves of apparent conductivity; (d) three-coil probe.

140

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

2. Near the origin of the system the function A~ tends to the vector potential of a magnetic dipole located in a uniform medium with parameters YI' £1' and ILl' which characterize the borehole. Thus we have if

(II .5)

R~O

3. At each interface r = G i the tangential components of the (electromagnetic) fields are continuous functions. The electric field has only the component E
B

= r

1 a2A*z ico araz

and

(II .6)

Therefore, the boundary conditions for the vector potential interface r = G i can be written in the form aA~.i ar

aA~.i+1 ar

A~

at the

(11.7)

where A~.i and A~.i+1 are vector potentials in the "ith and i + 1-th medium, respectively. 4. With an increase of the distance from the magnetic dipole, the function A~ tends to zero. The function A~ also satisfies the following conditions related to the geometry of the model and the position of the current source: (a) Due to the axial symmetry of the field, the vector potential and all components of the field do not depend on the (p-coordinate, that is, A~ =A~(r, z ), (b) The vector potential does not depend on the sign of the z-coordinate; that is, it is symmetrical with respect to the horizontal plane passing

II.1

The Field of the Vertical Magnetic Dipole on the Borehole Axis

141

through the dipole: A~(r,z)

=A~(r,

-z)

We start the determination of the field from a solution of Helmholtz's equation. At the beginning let us assume that there is only one cylindrical interface between the borehole and the formation. We look for a solution of Helmholtz's equation as the product of two functions: . A~

= T(r)cP(z)

Substituting this expression for A~ into Eq. (II A), we obtain instead of Helmholtz's equation two ordinary differential equations of the second order:

and

dZT(y) 1 dT(y) -d-y-Z- + dy - T( y) = 0

y

where

The solutions of the first equation are the functions sin mz and cos mz , while the solutions of the second equation are modified Bessel functions of zero order Io(Y) and Ko(Y). Taking into account the symmetry of the field with respect to the plane z = 0, the expression for the vector potential within the borehole can be written as

(II.8) since the function KO(m 1 r) tends to infinity as r

~

0, and

It is well known that the function e i k j R /R can be represented in the form

142

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

Thus,

r < a1 (11.9) Inasmuch as the function IO<mzr) increases unlimitedly when r ~ vector potential within the formation is

00,

the

(I1.10) where

rn.im

Substituting Eqs. (I1.9) and into Eqs, (11.7), we obtain a system of equations which determine the unknown coefficients C and D:

+ CI1(m1a 1)} =

m 1{ -K1(m1a,) p,zmHKo(m1a 1)

-mZK1(mZa1)D

+ CIO(m 1a 1)} = P,lm~K()(mZal)D,

since

IMx) K~(x)

=

dIo( x) dx

=

dKo(x) dx = -KI(x)

=

I,(x)

Solving this system, we have

and p,zm l

D=--;-----------,-----------,------,mZal{P,lmzKo(mzadll(mlal)

+ p,zmIIO(mlal)Kl(m2ad}

(11.12) According to Eqs, (II.6), the components of the electromagnetic field

11.1

The Field of the Vertical Magnetic Dipole on the Borehole Axis

143

within the borehole are

(II.13)

where E ocp ' B o z ' and B Or are components of the field in a uniform medium with parameters 'Yl' fJ.l' and B l· In particular, on the borehole axis we have:

e; =

Br=O,

°

and

B, = Boz -

M

fJ.I- 2 -

2'iT

1 mic cos mzdm 00

(11.14)

0

The primary magnetic field along the z-axis caused by the dipole current is

and, correspondingly, the expression for the vertical component of the magnetic field represented in terms of the primary field is (11.15) where function b oz was described in detail in Part B, Chapter II, and L is the distance between the dipole and the observation site, that is, the probe length. It is obvious that in the presence of several cylindrical and coaxial interfaces, we arrive at the same expression for the field as given by Eq. (I1.15) but with a different function C. For instance, in the case of two cylindrical interfaces and a nonmagnetic medium, the function C is

(11.16)

144

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

where ~l

=

[-m 2JO(m2a\)K t(m\ad -mlKo(m\a\)lj(m2al)] X

[m3K\(m2a2)Ko(m3a2) -m2Ko(m2a2)Kl(m3a2)]

+ [m2Ko(m2adKt(mtat) - mIKo(mtat)K\(m2ad] X

~=

[-m31j(m2a2)Ko(m3a2) - m21o(m2a2)Kj(m3a2)]

[-m 2JO(m2a\)It(m ta\) +mtJo(mtat)lt(m2aj)] X

[m 3K t( m 2a2) K O(m3a2) - m 2Ko( m 2a2) K t(m 3a2)]

+ [m 2Ko( m 2a t)Ij( mtal) + mIJO( mjat)K t ( m 2at)] X [

-m3Jt(m2a2)Ko(m3a2) - m2JO(m2a2)Kt(m3a2)]

and

and a 1 and a 2 are the radii of the borehole and invasion zone, respectively. Thus, the complex amplitude of the magnetic field on the borehole axis is expressed in terms of an improper integral, and its integrand represents the product of the complex function mrC and the oscillating multiplier cosmL. Now we are ready to study the field behavior, and first of all we consider the quasistationary field, measured in induction logging.

11.2 The Quadrature Component Q b, in the Range of Small Parameters: Ikiail « 1 and IkiLI « 1 As was mentioned earlier, we assume that the influence of displacement currents is negligible. Then by definition of the wave number, the range of small parameters means that the skin depth in every uniform part of the medium is much greater than the radius of the borehole and the invasion zone, as well as the probe length:

In the previous chapter we demonstrated that in this case the electromagnetic field plays the dominating role. In other words, neglect the interaction of induced currents and consider that the density at every point of the medium is defined by the primary

primary we can current electric

11.2 The Quadrature Component Q bz in the Range of Small Parameters

145

field only. Therefore, in accordance with Eq. (I1.11) in Part B, we have

iY!-LwrM Mr -k z 3 '" - 47TR 47TR 3

j -'1£0-

'" -

and, as follows from the Biot-Savart law, the secondary magnetic field in this approximation has to be directly proportional to k.', Correspondingly, in order to derive an expression for the field b, in the range of small parameters, it is necessary to expand the right-hand side of Eq. (IUS) in a series and then discard all terms but the first one, which is proportional to k z. In accordance with Eq. (I1.50) in Part E, the component b oz can be represented as L

if -

8j

«1

(11.17)

Now we determine the leading term of the expansion of the integral in Eq. (IUS). Assuming that the medium is nonmagnetic (!-Lj =!-Lz = !-Lo) and there is no invasion zone, we have z _ zmzKo(mzaj)Kj(mjaj) -mjKj(mzaj)Ko(mjaj) mj C - ml mzKo( mzal)Ij( mja l) + mlK j( mzal)Io( mja j) It is obvious that if

(11.18)

Ikjl -e.m and Ikzl« m,

(11.19)

and 1 kfa j IO(mla,) ;::::;Io(ma j) - ---Ib(ma t) 2 m

It(mja l )

;::::;

1 kfa j Ij(ma j) - - --I; (ma j) 2 m 1 kfa j

KO(mla j) ;::::;Ko(ma t) - ---K~(mal) 2 m 1 kZa Kj(mta j) ;: : ; Kj(ma l ) - - _1_K; (mal) 2 m

(II .20)

146

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

Substituting Eqs. (11.19) and (IIol0) into Eq. (11.18) and making use of the recurrence relations of Bessel's functions

2v

Ie-I( x) -Ic+ I( x) Ke_l(x) -KC+I(x)

(11.21)

-1,,( x)

=

x

21'

=

-

-Kl'(x) X

I"_I(x) +I"+I(x) =21;.(x) K,'_I(X) +K"+I(X)

=

-2K;,(x)

after simple algebra we obtain

mal 2 Z - k ) m 2I C = (k 1 22 X

[2K o(ma l)K I(ma l) -mal(Ktcmal) -K5(ma l))]

(11.22)

Thus, the quadrature component of the magnetic field, expressed in terms of the primary field, is T'I

L3

f.L(vLZ

Qb z=

00

+-(s-lhlf.Lwj mal 27T 0

2

(11.23)

X(2KoKI-ma(K~-K5))cosmLdm

where s = T'z/T'I' Let us introduce notations x = rna l

(Il.24 )

,

Then Eq. (II.23) can be rewritten as

f.LWLZ{ Q b,

=

-2-

2a

fa

ooX

2

T'I

+ (T'2 -

X

[2K o( X)K I( x) - x( K~ -

T'd --:;;:-

K(~)]

cos axdx}

or (11.25)

11.2 The Quadrature Component Q bz in the Range of Small Parameters

147

where

and (11.27) Functions G z and G 1 are called the geometric factors of the formation and the borehole, respectively. It is essential that G z and G j depend on the parameter a =Lja j only, which characterizes the length of the two-coil probe, and, in accord with Eq. (11.27), the sum of the geometric factors is equal to unity: (11.28) regardless of the parameter a. Therefore, due to the absence of interaction of induced currents, in the range of small parameters the quadrature component of the magnetic field bz can be represented as a sum of two terms so that each one of them depends on the conductivity and geometric factor of the corresponding part of the medium. Applying the same approach in the case where the invasion zone is present, we obtain (II.29) where 1'1' 'Yz, and 1'3 are conductivities of the borehole, invasion zone, and the formation, respectively, and G Z=G 1(

;

)

-Gj(a)

G 3 = 1 - G 1 - GZ

L a=aj ,

(11.30)

az aj

{3=-

where az is the radius of the invasion zone. Now it is obvious that Eq, (II.25) can be generalized for the medium which consists of N uniform coaxial cylindrical layers. Then we have J.LwLz Qbz

= -2-

N

E 'Y;G; ;= I

(II.31)

148

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

where G, is the geometric factor of the cylindrical layer. In particular, G 1 and G N are geometric factors of the borehole and the formation, respectively. Taking into account the fact that the interaction of the quadrature component of currents is negligible, we can represent the geometric factor of the cylindrical layer as the difference of geometric factors of cylinders. Then, we obtain

2a. coX --'1-[2K0 K 1 -x(K12 - K 02)]cosa.xdx I 71"

0

2

(11.32)

where L

ai- 1= - - , ai-I

and ai-I and a i are radii of the internal and external surfaces of the ith cylindrical layer. The first equation of the set 01.30), which defines the geometric factor of the invasion zone, vividly illustrates this relationship. As in the case of one cylindrical interface, the sum of geometric factors remains equal to unity: N

L, G i = 1

(11.33)

i=l

Inasmuch as all geometric factors are expressed in terms of the function G 1 , let us describe its behavior in detail. In accordance with Eqs. (11.26) and (11.27), we have

2« -1 A(x) cos ax dx co

G1(a) = 1-

17"

(11.34)

0

where

(II .35) First, consider the dependence of this function on x. For sufficiently large

Il.2

The Quadrature Component Q bz in the Range of Small Parameters

149

values of x we have

7T )1/2(

Ko(x)

=:;

e- x ( 2x

7T )J/2(

KJ(x)=:;e- x ( 2x

0.125 )

1 - -x-

0.375) l + -x

For this reason, when x -) 00

In the opposite case, as x -) 0 we have Ko(x)

--t

-tnx,

Substituting these values into Eq. (11.35), we obtain A(x) -)Ko(x) -) -tnx,

that is, the integrand has a logarithmic singularity when x tends to zero. In order to remove this singularity we make use of the following equation: 1 ----1/-2 =

(1+a 2 )

2 00 -1 Ko(x) cos ax dx 7T 0

Correspondingly, the function G 1 can be represented in the form 2a 00 GJ(a) = 1- - [ A(x) cos ax dx

7T

0

a =1-

(1 + a 2 )

2a

00

1/2+-[ {Ko(x)-A(x)}cosaxdx (11.36)

7T 0

The integrand in this equation does not have a singularity, and therefore calculation of function G 1 is a relatively simple task. First, we study the asymptotic behavior of the geometric factor G J and begin with the case when the parameter a is sufficiently large. Then, due to the oscillating character of the integrand in Eq. (11.36), the value of the integral [00( x) cos ax dx

o

150

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

where

is defined by the behavior of the function
00

[
Jo

w

-

00

[

a Jo


1 sin ax 1 - -1 [
=-
00

1 sin ax I'" + 2"
a

()

a

0

1 - 2" (
a Jo

(11.37)

In the limit when x ~ 00, function
1 1 [
00

For small values of x we obtain

x2 x2 K (x) "" -tnx - -tnx + - - C + Ii + ... o 4 4 1 K (x) "" 1

X

x

x

+ -tnx - 2 4'

if x

~

0

where C is the Euler constant. Substituting these expressions into
1

+ -x 2tnx 4

II.2

151

The Quadrature Component Q bz in the Range of Small Parameters

and

x

1 c/JI/(x) :::: "2fnx,

c/J'(x)::::"2 t nx,

if

x~

0

Thus 00

1o

c/J(x)cosaxdx~

1

00

--2 (

2a )0

fnxcosaxdx

100 :::: --Z (

2a )0

17T'

Ko(x)cosaxdx~

--2-'

2a 2a

if a

~

00

Substituting this expression for the integral into Eq. (11.36), we have if a »1

(11.39)

Therefore, for large values of a the geometric factor of the cylinder, in particular, of the borehole, decreases inversely proportional to a Z; that is, if a » 1

(11.40)

Comparison with the results of calculations using Eq. (11.36) shows that the asymptotic behavior [Eq. (11.40)] practically begins when the ratio L/a t >4. Making use of the same approach, we can obtain the following terms in the expansion for function G I . For example, a more accurate expression of the geometric factor Gt(a) for large values of a has the form 1 3tna-4.25 G 1( a ) :::: 2 + ----:--a a4

(11.41)

In the opposite case of small values of a, the function G 1( a ) approaches unity as

(II .42) Thus, for small values of parameter a the geometric factor of the borehole is close to unity and its decrease is directly proportional to a (a < l), The behavior of the geometric factor G 1( a ) is shown in Figure II.lb.

152

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

In accordance with Eq. (II.31), it is clear that the function G /0') allows us to investigate the radial responses of induction probes in a medium with cylindrical interfaces, provided that the quadrature component is measured and parameters Ik;a;1 and Ik;LI are small. With this purpose in mind, let us again introduce the concept of the apparent conductivity, Ya , which we define in the following way: or

Ya

(1I.43)

1'1

where 1'1 is the borehole conductivity and Q B, and Q g' are the magnetic field and the electromotive force measured on the borehole axis, respectively. At the same time Q B:n(YI) and Q g'un(YI) are the magnetic field and the electromotive force, respectively, measured by the same two-coil probe when the medium is uniform and has conductivity 1'1' As follows from Eqs. (1I.43), the ratio Ya/Y I shows how the field or the electromotive force measured on the borehole axis differs from the same quantity in a uniform medium with conductivity 1'1' Making use of Eqs, 01.31) and 01.43), we have the following expression for the apparent conductivity: N

Ya

=

(II.44)

Ly;G; ;=1

For instance, for a two-layered medium when there is no invasion zone we have (1I.45)

while in the presence of an invasion zone (1I.46)

In accordance with the behavior of the geometric factor GI(O'), a very short induction probe is mainly sensitive to the currents arising in the vicinity of the dipole, that is, Ya

~Yl'

if 0'

~

0

(1I.47)

On the other hand, with an increase of the probe length the geometric factor of every cylindrical layer of finite thickness decreases, while the geometric factor of the formation approaches unity, and therefore the depth of investigation in the radial direction increases. This means that in the limit for any function y(r), the apparent conductivity Ya approaches

I1.2 The Quadrature Component Q hz in the Range of Small Parameters

153

the formation conductivity "IN: if a -

IX)

(11,48)

and in this case the magnetic field B z coincides with that in a uniform medium having conductivity "IN' As an example, the behavior of the function "IaI"I\ is shown in Figure IL1c. The index of curves is ratio "12/"1\·

Thus, performing measurements with two-coil induction probes having different lengths, it is possible to investigate the distribution of conductivity in the radial direction. Such an approach is often called geometric or lateral sounding, and in spite of the fundamental difference of physical principles, it is very similar to Schlumberger soundings; in both methods the distance between the current source and the receiver is the single factor defining the depth of investigation. Now let us return again to the two-coil induction probe. It is obvious that the influence of the borehole and invasion zone becomes greater with an increase of their conductivities and radii. In fact, calculations based on Eq. 0I,46) clearly show that in many cases it is necessary to use a relatively long two-coil probe in order to reduce to a great extent the effect of currents induced in the borehole and in the invasion zone. However, with an increase of the probe length, its vertical resolution becomes worse. Because of this, many coil differential probes have been developed and they have been successfully used in induction logging for more than 40 years. The physical principle of these probes is based on the assumption that the behavior of the quadrature component of induced currents corresponds to the range of small parameters. In other words, it is assumed that the interaction between currents creating the field Q B, is negligible. To illustrate the concept of a differential probe, let us consider its simplest configuration, which includes one generator and two receiver coils so that the latter have opposite directions of their turns (Fig. ILld). In essence, such a three-coil probe can be treated as a combination of two-coil induction probes of different lengths L 1 and L 2 • In general, the moments of receiver coils M\ and M 2 are also different. As follows from Eqs, (ILl7) and (II,43), the electromotive force iff and the apparent conductivity "Ia for the two-coil probe are related to each other by (11.49)

where iffo(L) is the primary electromotive force.

154

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

First, suppose that there is one cylindrical interface between two uniform media. Then, in accordance with Eqs, (II.45) and (II.49), the electromotive force measured by the three-coil probe is ilg'=g'(L\) -1?'(L 2 )

=

2WJ.L

[ 1'1 { L\l?'o(Lt)G\(L\) 2

+Y2{L~l?'o(

-L 22I?'o(L2)G t(L 2) }

LdG 2( L\) - L~l?'o(

L 2)G 2( L 2)}] (11.50)

where the first term in the square brackets characterizes the influence of the borehole. In order to remove this influence we have to choose the moments of the receiver coils in such a way that the first term on the right-hand side of Eq. (11.50) vanishes, which happens if

(11.51) or

since

Thus, if receiver moments and distances satisfy the condition (11.51), the measured signal ill?' becomes a function only of the conductivity of the external medium. It is clear that currents induced in the cylinder (internal medium) generate the electromotive force in each receiver, but by measuring their difference it is possible to greatly reduce the influence of these currents. This consideration emphasizes the fact that the three-coil probe, as well as many other coil probes, is a system which measures a difference of signals but does not possess any focusing features. Until now we have assumed that the internal cylinder is uniform. If its resistivity changes with the distance from the z-axis, condition (11.51) is usually not sufficient to reduce the effect caused by currents in such a medium. In this case we have to mentally divide this internal part of the medium into several cylindrical layers so that the conductivity 1'( r ) of each one of them is practically constant. Then, applying Eq. (11.51) or a similar equation to each cylindrical layer, it is possible to reduce to a great extent the influence of currents induced in the internal cylinder. This result can be

II.2

The Quadrature Component Q bz in the Range of Small Parameters

155

practically achieved by using either a relatively long three-coil probe or probes with more coils. In conclusion let us make several comments:

1. The theory of induction logging based on the concept of the geometric factor was developed by H. Doll. 2. In many cases which are typical in the application of induction logging, the parameter LI8 is small. For example, if the length of a two-coil probe is 1 m, the frequency of the field is 20.10 3 Hz, and the resistivity of the medium is 5 ohm-m, we have L

p = -8 z 012 .

Of course, there are also exceptions, especially in those cases when the frequency is sufficiently high. 3. From the study of the field in a uniform medium (Part B, Chapter II), we know that with an increase of the distance from the dipole the interaction between currents begins to play an essential role, and this happens regardless of how small the frequency. At the same time, we assumed that within the range of small parameters the interaction of induced currents is negligible at any distance from the dipole. However, in spite of this approximation, the results of calculations based on the asymptotic formula [Eq. (11.31)] give sufficiently accurate values of the quadrature component Q bz , provided that the induction numbers are small. This apparent paradox is not difficult to resolve. In fact, with a decrease of frequency, the area around the dipole, where the quadrature component of induced currents is mainly defined by the primary field E~, increases, and, correspondingly, their interaction begins to display itself at greater distances from the dipole. Inasmuch as the depth of investigation depends to a great extent on the probe length, it is not difficult to imagine such cases where the magnetic field on the borehole axis is practically caused by induced currents which are not subject to the skin effect. In other words, the interaction between currents is always present, but at the range of small parameters it occurs at such great distances that its influence becomes very small. 4. Since within the range of small parameters we neglect the interaction of currents, the in-phase component of these currents and therefore the in-phase component of the secondary magnetic field is equal to zero. Next, we describe a more general case when only some of the induction numbers are small, and this allows us to partially take into account the influence of the skin effect.

156

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

II.3 The Behavior of the Field in the Range of Small Parameters aJ'b i The study of the field of a magnetic dipole in a uniform conducting medium (Part E, Chapter II) vividly demonstrated that with an increase of the distance from the current source the quadrature component of induced currents becomes smaller than that at the range of very small parameters. Moreover, comparison of the magnetic field on the borehole axis, calculated by the exact formula [Eq. (II.15)] and making use of Eq. (lIAS), confirms this feature of the current distribution. Now, proceeding from the physical point of view we describe the field behavior in the range of small parameters aJoi when the induction number Ljo; can be relatively large. The method of deriving the asymptotic formulas in this case is very simple. Let us represent all current space around the induction probe as a sum of two areas (Fig. II.2a), namely, (a) The internal area where the induction probe is located. (b) The external area. For simplicity we suppose that the conductivity of the external area is constant. Later, this restriction will be removed. We also assume that the distribution of induced currents is characterized by two features: 1. In the internal area induced currents are shifted in phase by 90° with respect to the dipole current, and their density depends on the geometric parameters and the conductivity in the vicinity of a point. In other words, interaction between currents induced within this area is practically absent and they arise only due to the primary electric field E~. 2. The density of currents in the external area does not depend on the distribution of resistivity within the internal area. In particular, the skin effect in the external medium does not change when both parts of the medium have the same resistivity. This means that the distribution of currents in the external medium with conductivity 'Ye coincides with that in a uniform medium which has the same conductivity. It is obvious that both assumptions imply that there is no interaction between currents located in different areas, and the skin effect manifests itself first of all at relatively large distances from the dipole. Now we are prepared to derive sufficiently simple expressions for the field in the borehole. Let us represent the quadrature components Q B, as a sum of magnetic fields caused by currents in the internal and external areas:

11.3 The Behavior of the Field in the Range of Small Parameters

157

a external area

internal area

b

Irn (0

-------t--~

1m (0

Re (0

---------H~Re

(0

Fig. 11.2 (a) Internal and external areas of current field; (b) path of integration in Eq, (Il.7l); (c) spectrum of quadrature component of field bz ; (d) spectrum of in-phase component of field bzP'1 = 27TC5 1) . (Figure continues.)

158

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

c 10,.--------------------------,

Qbz

1Q-4-+--..--"'T"""--.--..----,---r--..----,--..--..--"'T"""-+-I 10-4 Fig. 11.2 (Continued)

or

(II.52) where Q b, is the vertical component of the magnetic field related to the field in a free space B~ , but Q B; and Q B; are quadrature components of the magnetic field, generated by currents within the internal and external areas, respectively. Suppose that the internal area includes the borehole and the invasion zone. Then, the first assumption allows us to represent the magnetic field

11.3 The Behavior of the Field in the Range of Small Parameters

159

d 10,._------------------------,

L

-=8 81

lO-4'-+--~....L.~....L.,._..L-,...._L-__.J.-.,J_-,...._-"""T""-~__,,__-~.....,

10-4

Fig. 11.2 (Continued)

due to currents in this area as

(11.53) and correspondingly

(11.54)

160

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

where (11.55) where 'Y\, G\ and 'Y2' G 2 are the conductivity and geometric factor of the borehole and the invasion zone, respectively. Next, we assume that the conductivities of all parts of the medium are the same and that they are equal to 'Ye. Then the magnetic field Q b a/ 'Ye) in such a uniform medium can be represented in the form ( 11.56) since in accordance with the second condition the field of the currents in the external medium does not depend on the conductivity of the internal area. Here Q is the quadrature component due to currents in the internal area when its conductivity is equal to 'Ye, and it can be expressed in terms of geometric factors in the following way:

b;e

(1l.57) Therefore, for the quadrature component of the magnetic field caused by currents in the external area we have Q b:

i:

=

Q b oz( 'Y3) - Q

=

wp.,L2 Q b az( 'Y3) - -2-'Y3( G\

+ G z)

(II .58)

where 'Y 3 = 'Ye is the conductivity of the formation. Then, substituting Eqs. (II.53) and 01.58) into Eq. (II.52), we arrive at an expression for the quadrature component of the magnetic field caused by currents in both areas having different conductivities:

or

Equation (1l.46) has been derived directly from the exact expression for the field, assuming that all induction numbers are very small. In contrast, the derivation of Eq. (II.59) is based only on the two assumptions (listed

11.3 The Behavior of the Field in the Range of Small Parameters

161

at the beginning of the section) characterizing the distribution of induced currents. However, later we show that this expression for the field can be treated as the first approximation in a solution of the integral equation when the skin depth is much greater than the radius of the invasion zone. In accordance with Eq. (11.59), the quadrature component of the magnetic field on the borehole axis consists of two parts. The first one describes the magnetic field caused by currents in the borehole and invasion zone, provided that the interaction between these currents is absent. At the same time, the second part characterizes, the quadrature component of the magnetic field in a uniform medium with the conductivity of the formation 'Y3' It is essential that this part of the field reflects the interaction of currents, which occurs in the same manner as if the whole medium were uniform. Equation (11.59) clearly demonstrates that the skin effect occurs, first of all, at greater distances from the dipole while in its vicinity the density of induced currents is governed by the primary vortex field E~. As was pointed out earlier, with a decrease of the frequency the interaction between currents becomes noticeable at such great distances that the field measured by the induction probe is not practically subjected to its influence. Correspondingly, instead of Eq. (11.59), we again obtain Eq, 01.46), which describes the field Q b, in the range of very small parameters. In fact, since

we have

or

In the opposite case, when the frequency increases, the interaction between currents in the invasion zone and then in the borehole begins to influence the current distribution. For this reason the use of geometric factors becomes invalid. Moreover, the interaction between currents located in the internal and external areas also increases. In other words, the magnitude of the currents in the formation becomes a function of the conductivity of the borehole and the invasion zone. Thus, we cannot assume that the skin effect in the formation manifests itself in the same way as that in a uniform medium with conductivity 'Y3' This analysis shows

162

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

one more time that Eq, (11.59) describes the behavior of the quadrature component of the magnetic field only within a certain range of induction numbers Llo; and aJo;. Comparison with the results of calculations based on the exact solution allows us to define the limits of application of Eq. (II.59), For instance, with an error not exceeding 10% we can use Eq. (II.59) provided and

(II.60)

To illustrate these inequalities, suppose that the invasion zone is absent and determine the maximal frequency when Eq, (II.59) is still valid. Letting a

~ 0, since O2

=

=

0.3

and

03 ' we have

fm~x

' {50P2 2P1 } < 10 4 nun -2-' -2-

L

a1

(11.61)

It is obvious that frequencies which are used in conventional induction

logging (f < 60 X 104 Hz) usually satisfy these conditions, and correspondingly Eq. (II.59) correctly describes the quadrature component of the magnetic field. For this reason this asymptotic expression can in many cases be treated as a simple and reliable equation for calculation of the field measured in induction logging. Also, it is proper to emphasize that the inequalities in (II.60) are of great practical importance. In fact, they define the range of resistivities and frequencies where interaction between induced currents in the borehole and invasion zone is negligible, but the skin effect in the formation manifests itself in the same manner as in a uniform medium with conductivity Y3' At the same time, we know that this behavior of currents is vital for the application of differential induction probes. It is interesting to note that Eq. (II.59) suggests a method of eliminating the influence of induced currents in the borehole and in the invasion zone, Suppose that measurements of the quadrature component of the magnetic field are performed with a two-coil induction probe at two frequencies which satisfy conditions (II.60). Then, in accordance with Eq. 01.59), we

1I.3 The Behavior of the Field in the Range of Small Parameters

163

see that the difference

(II.62) is independent of Yl and Yz, but it is a function of the formation conductivity Y3' Until now we have discussed the asymptotic behavior of the quadrature component of the magnetic field. At the same time, this study allows us to obtain some information about the in-phase component of the field, too. We assumed that the skin effect manifests itself in the external area (formation), and the in-phase component of induced currents within the borehole and invasion zone is negligible. Therefore, it is natural to assume that the in-phase component of the magnetic field is caused by currents in the formation only. Proceeding from this consideration we can rewrite Eq. (II.59) in a more general form:

or

and In bz

=

In b oz ( Y3)

if conditions (II.60) are met. Thus, the in-phase component of the magnetic field is practically not subject to the influence of induced currents in the borehole and in the invasion zone. Moreover, it coincides with the in-phase component in a uniform medium with the formation conductivity Y3' This means that in this approximation induced currents which arise in the internal area contribute to the quadrature component Q b, only. This consideration clearly shows that the in-phase and quadrature components of the magnetic field have a different sensitivity to the geoelectric parameters and different depths of investigation. Certainly, understanding of this feature of the field behavior is of great practical interest for developing methods of induction logging. In fact, some of the induction probes presently used in practice are based on measuring both components of the magnetic field. From the physical point of view it is clear that Eq. 01.64) is valid regardless of the probe length and the distribution of conductivity in the

164

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

radial direction within the internal area. In particular, this result is applicable even when the probe consists of one coiL This behavior of the in-phase component In b, vividly emphasizes that the distribution of the in-phase component of induced currents has a diffusive origin. It is not difficult to realize that the asymptotic behavior of the field In b, [Eq. (11.64)] is still observed even when the conductivity changes arbitrarily within the internal area but conditions (11.60) are met. For instance, if the invasion zone contains a confined inhomogeneity, then electric charges arise on its surface. Since these charges appear due to the primary electric field EO, they only create a quadrature component of the electric field. Therefore, currents which arise under the action of this field generate a quadrature component of the magnetic field, while the in-phase component In b, is not subject to the influence of these currents. The same conclusion remains valid when the induction probe is not located on the borehole axis. In fact, due to a displacement of the probe, electric charges appear on the borehole surface as well as on the interface between the invasion zone and the formation. However, their density is proportional to the primary electric field, and, correspondingly, currents which are related to these charges do not contribute to the in-phase component of the magnetic field. In particular, this analysis shows that the micro-induction probe, which consists of one coil and measures the in-phase component of the field at a sufficiently high frequency, can be useful for determination of resistivity around the borehole. Finally, let us make two comments: 1. In accordance with Eqs. (11.64), at the range of small parameters

when

we have

and

It is obvious that the second term of the quadrature component and the

leading term of the in-phase component of the magnetic field B; do not depend on the probe length. Therefore, as was pointed out earlier,

11.4 The Magnetic Field on the Borehole Axis in the Far Zone

165

regardless of the separation between coils, measurements of these quantities can essentially increase the depth of investigation. 2. The approach which allowed us to derive Eq. (11.59) will be used later for a study of the field behavior for more complicated models of the medium.

U.4 The Magnetic Field on the Borehole Axis in the Far Zone

In the induction logging, the probe length usually exceeds the borehole radius, and often the ratio a = L/a l reaches ten and more. Correspondingly, it is appropriate to investigate the behavior of the field when the parameter a is large. As will be shown in this section, within this range of parameters

L -» 1 ' al

L

-» 1 az

(11.66)

the field behavior is characterized by some features which can be used for increasing the depth of investigation of the induction logging. The study of the field in the far zone will proceed from Eq, (11.15): (11.67) where

For obtaining the asymptotic formulas we take into account the distribution of singularities of the integrand m~C on the complex plane m. In accordance with Eq. (11.67), the variable of integration m has only real values (0 :::; m < 00), and a relative length of the probe a plays the role of a multiplier of the argument of the oscillating term cos am. Also, for sufficiently small values of the parameter Ik l a II, the function m ~ C decreases rapidly with an increase of m. Moreover, with an increase of a, due to the presence of the oscillating factor cos am, the contribution of the integrand, corresponding to large values of m, decreases. For this reason the integral

[Xlo miC cos am dm

(11.68)

166

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

is mainly defined by the behavior of the function mfC, , when the variable

m is very small. This consideration allowed us to derive the asymptotic formula for the geometric factor of the borehole G1(a). However, with an increase of the wave number Ikl, the integrand mfC begins to decrease more slowly with increasing m, and, for instance, if m < Ikal, it practically does not change. Therefore, the fact that the number of oscillations is large (a» 1) does not mean that the value of the integral [Eq. 0I.68)] is mainly defined by the initial part of integration. For this reason, in order to obtain an asymptotic expression for the field in the far zone it is necessary to perform preliminary transformations of Eq. (11.68). Inasmuch as the integrand of this equation is an even function with respect to m, we have 3

1=

3

a a -1 mfC cos am dm = -1 mfCe 2w 2w 00

0

00

0

im a

dm

(11.69)

Now we make use of Cauchy's theorem, according to which the integral of an analytical function does not change under deformation of an integration contour if it does not intersect singularities of this function in the complex plane of the variable m. It is clear that deforming the contour of integration in the upper half plane (Im m > 0), the exponent e im a tends to zero with an increase of Im m. First, we consider the case when there is no invasion zone. Taking into account the fact that the function mfC contains radicals m l and m 2 , the integrand has in the upper half plane two branch points k I a I and k 2 a I which are defined by the equalities (11.70)

As is well known, in order to apply Cauchy's theorem we have to deal with a single-valued analytic function. With this purpose in mind, let us draw branch cuts. From the condition Remj>O

and

Rem 2>0

which arises as a result of a solution of the boundary value problem, it follows that these branch cuts must distinguish areas where the real parts of the radicals m I and m 2 are positive. Thus, the equations of these lines are Re m 1 = 0

and

Re m 2 = 0

11.4 The Magnetic Field on the Borehole Axis in the Far Zone

167

Next we replace the contour of integration along the real axis m by that along both sides of two branch lines, where Re m 1 = 0 and Re m 2 = O. It is also assumed that, within the area surrounded by the real axis of m and these contours, singularities are absent (Fij, II.2b). Therefore

or (11.71 )

since the integral along the semicircle of infinitely large radius is equal to zero. Integrating along the branch line D 1 , where Re m 1 = 0, we introduce a new variable of integration m 1 = it. Here t is the parameter of the branch cut which ranges from 0 to 00 on the right side of the branch cut and from - 00 to 0 on the left side, because in passing around the branch point the radical m] changes sign. The variable of integration of m along contour D 1 can, represented as . 2)1/2 . 2)1/2 m = ( m 21 + m] = ( -t 2 + tn = 1.( t 2 1

. 2)I/Z

tn 1

and correspondingly

itdt dm = -(-2-'-.-2)71J=2 ' t -tnl where

Thus, for the integral along the cross section D I we have the following expression: [~ 2 (m 2K o(m 2) K 1( it ) -itK o(it)K 1(m2) o (-t) m 2K o(m 2)ll(it) +itK 1(m z)Io(it)

n:

_ m ZKO(m2)K]( -;ot) + ;,tK o( -it)K 1( z) ) mzKo( m z) ll( -it) -itK]( mz)Io( -it) (11.72)

168

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

Making use of the relations lo( -it)

=

ll( -it)

=

10 {it )

K o( - it)

=

Ko{it) + iTT 10 {it )

-ll(it)

K I( -it)

=

-K 1{it)

+ iTTll{it)

(11.73)

we can represent the second term in parentheses of Eq. (11.72) in the following form: m 2K o(m 2 ) [ -KJ{it) +iTTll{it)] +itK I(m 2)[K o{it) +irrlo{it)] -m 2K o( m 2)II(it) - itK l(m 2)Io{it) =

m 2K o(m 2)K 1(it) -itK 1(m 2)K o{it) m 2K o( m 2)Il(it)

+ u« I( m 2 ) 10 (it )

. -ITT

(II.74)

The first term on the right-hand side of Eq. (11.74) is equal to the first term in parentheses of Eq, (II.72). For this reason the integral along the branch cut D I is greatly simplified and we have 00

t3e-au2-inT)112

TTl

(t 2 -

o

inn

1/2

dt

This integral, multiplied by a 3 ITT, represents the field of the magnetic dipole b oz in a uniform medium with conductivity YI. Then, as follows from Eqs. (11.67) and (11.71), the field on the borehole axis is expressed only in terms of the integral along the branch cut D 2 (Re m 2 = 0). Making the replacement of variables m 2 = it we have _ "( 2 "2)1/2 m-It-11l 2

it dt dm=----...,.. "2 )1/2 2 ( t -m 2

'

Correspondingly, the integral along the path D 2 can be written as

ite _a(1 2 _in~)1/2 X (t

2

. 2 )1/2

-1ll2

dt

(II.75)

11.4

The Magnetic Field on the Borehole Axis in the Far Zone

169

Making use of Eqs. (11.73) and presenting the integrand as one fraction, we obtain for the numerator of the parentheses the following expression:

mjit{Io(mj)Kj(m j) +Ij(mj)Ko(m j)} X{K o( -it)Ki(it) +Ko(it)K j( -it)} Inasmuch as

the numerator is equal to itt, Correspondingly, after simple algebra the field b; on the borehole axis is expressed through the integral along the right-hand side of the branch cut D 2 :

Let us consider the integrand as a product of two functions:

At the initial point of the integration, the function F depends only slightly on the parameter t. The second function

(

t2 -

. 2) l/2

ln 2

is the integrand of the Summerfield integral, which describes the field in a uniform medium with resistivity of the formation. If the skin depth in the formation and the probe length are much greater than the borehole radius: (11.77) we can let t = 0 in the expression of Ft.m, , t} and take this function out of the integral. Thus, we obtain an asymptotic equation for the field in the far

170

II

The Behavior of the Field in a Medium with Cylindrical Interfaces

zone:

1

or

(II.78)

L if - » 1

and

at

Comparison of fields calculated by the exact and asymptotic formulas shows that if the skin depth in the borehole is greater than its radius and P2 > PI , the error in determination of the amplitude and phase of the field b, by Eq. 01.78) does not exceed 5%, provided that L

a=->4 at

(II. 79)

Now we represent the complex amplitude of the field in the following form: (II.80) where A * and 'P * are functions of both the borehole and formation conductivity but are independent of the distance L. Suppose that the field is measured at two distances from the dipole L, and L 2 which correspond to the far zone, and consider the ratio of amplitudes and difference of phases of fields bz(L t ) and b z(L 2 ) . Then, in accordance with Eq. (II.80), we obtain

Ibz ( L 2 ) I Ibz(Lt)1 and

A o(k 2L 2) A o(k 2L t ) (II.8D

Thus, measurements of either the ratio of amplitudes or the difference of phases allow us to eliminate the influence of the borehole; and this approach is used in high-frequency induction logging. In this light it is appropriate to note that the field behaves in the same way in the presence of the invasion zone if the observation point is located

11.4

171

The Magnetic Field on the Borehole Axis in the Far Zone

in the far zone. In fact, performing similar derivations we find that 1 b = z

2 _ k 2)1/2 }J 2{(k 2 _ k 2)1/2 } J2{(k o 2 1 a1 0 3 2 a2 if Ika2 1max < 1 and

b (1) Oz

(II .82)

3

L

->4 a2

Therefore, measuring the same parameters Ib z ( L 2)l!lb z ( L 1)1 and D.ip, it is also possible to reduce to a great extent the influence of the invasion zone. Now let us return again to Eq. (II.78). Since for small values of the argument

we can represent Eq. (II.78) as (II.83) which coincides with Eq. (II.59) if a » 1 and 12 = 13 . In the same manner, we obtain from Eq, (I1.82) (II.84) As follows from Eq. (II.78) in the range of large parameters (Ikal » 1), for instance in the high-frequency spectrum, the field b, tends to zero. This means that induced currents concentrate in the vicinity of the dipole, and the secondary field is almost equal in magnitude to the primary field, but it has the opposite sign. However, in the far zone, unlike the near one (L < 2a l ) , the influence of the medium surrounding the borehole remains regardless of the frequency. Finally, if the conductivity of the borehole is much greater than that of the formation, then, in accord with Eq. (I1.78), we have (II.85) We have considered the field behavior in the low- and high-frequency parts of the spectrum as well as in the near and far zone. Now several

172

II

Behavior of Field in Medium with Cylindrical Interfaces

examples of frequency responses of the field b, on the borehole axis, calculated by the exact formula, are shown in Figures II.2c and d. In conclusion, let us make the following comment. Equations derived in Section 11.1 remain valid in the presence of displacement currents when the wave number k is

k

=

2 )1/2 . ( lYJLW + W EJL

Correspondingly, Eqs. (11.78) and (II.82) describe the behavior of the field at the far zone in the general case when both the conductivity and dielectric permittivity influence the field. In fact, Eqs. (11.78) and (11.82) are the theoretical basis of dielectric logging, which measures the ratio of amplitudes and the phase difference of the magnetic field with a three-coil probe in order to increase the depth of investigation in the radial direction.

II.S Behavior of a Nonstationary Field on the Borehole Axis

Now we study the transient responses of the magnetic field on the borehole axis when the dipole moment changes as a step function: t.:::;;O

t> 0

(II.86)

As is well known, at the first instant (t = 0) induced currents are concentrated in the vicinity of the dipole. Therefore, the field at the early stage is strongly subject to the influence of the borehole. Then, with an increase of time, currents due to diffusion appear at greater distances from the borehole and we can expect that the depth of investigation will also increase. For this reason, we mainly pay attention to deriving asymptotic formulas for the late stage of the transient field. With this purpose in mind, we make use of Fourier transforms: sb _z =

at

-1 2

'TT

00

Qbz(w)sinwtdw

0

and Bb:

2

-" = --

at

'TT

'" 1 Inbz(w)coswtdw 0

where bz(w) is the spectrum of the magnetic field [Eq, (11.15)].

(11.87)

II.S

Behavior of Nonstationary Field on Borehole Axis

173

Integrating the first integral of Eq. (II.87) by parts, we obtain an expression for the field at the late stage as a series in powers of lit:

.

2 ({ cP I ( w) .cos w t cp'] ( w) sin w t - -------:-t t2

b (t) ::::; - Z

Tr

CPI cos tot -

/I

3

}

I'" + 3"1 1

tot

cc

a

cp'j ( w) cos w t d w )

(11.88)

where CPj(w) = Q bz(w). A similar expression can be derived from the second integral in Eq. (II.87). Therefore, derivation of asymptotic formulas for the late stage of the transient field consists of two steps. 1. The representation of the low-frequency spectrum as a series with respect to w. 2. The determination of coefficients of the asymptotic series by powers of l/t. At the beginning we study the late stage behavior, proceeding from approximate equations derived in the previous sections. First of all, in the range of very small parameters [Eq, (H.29)], we have

Inasmuch as this field is caused by induced currents, which arise only due to the primary field, diffusion is not taken into account. Correspondingly, this part of the field does not contain any information about the transient field. The same conclusion follows from the relationship between the low-frequency spectrum and the late stage [Eq. (H.88)]. In particular, we demonstrated that terms of the series describing the low-frequency spectrum of Q b., which contain only integer powers of w, do not contribute to the late stage asymptotic expansion. Next, consider the spectrum of the field when the interaction of currents in the formation is taken into account. In accordance with Eq. 01.59), iWf.LL 2

bz = b Oz ( Y3) + - 2 -

2

L

(Yi-Y3)G i

i~l

Since the second term does not contribute to the late stage behavior, a

174

II

Behavior of Field in Medium with Cylindrical Interfaces

Fourier transform yields if t

(II.89)

~ 00

Thus, we see that at the late stage both the borehole and the invasion zone become transparent, and this happens regardless of their conductivity or the probe length. At the same time, Eq. (II.89) correctly describes only the leading term of the series, characterizing the late stage. Therefore, instead of Eq. (11.89), we can write if t

~

00

(II.90)

but the following terms of the series remain unknown. This problem can easily be solved provided that the observation point is located in the far zone. Then, expanding Eq. (11.82) in a series we obtain

bz

=

1 + 'Pzk~

+ 'P3 k i + 'P4 k i + 'Pski + ...

where

S12 =

l-s z + (sz -SI)/3Z

Then, applying the relationships between the low-frequency spectrum and the late stage, from Chapter I, we obtain

II.S

Behavior of Nonstationary Field on Borehole Axis

175

Now let us discuss the general approach, which allows us to derive an asymptotic expression for the transient field at the late stage. In accordance with Eq, (I1.15), the vertical component of the magnetic field on the borehole axis is represented by a sum of cylindrical harmonics which are characterized by spatial frequency m. The greater m, the more rapidly a corresponding harmonic of the field changes with the distance from the dipole. This means that a nearly uniform field is formed by the low-frequency spatial harmonics, which constitute the initial part of integration over m. On the other hand, at the late stage induced currents are located relatively far away from the dipole, and for this reason they generate an almost uniform field B, near the dipole. Therefore, the function mrC(m1, m 2 , m 3 ) contains the information about the late stage if the variable m is very small. This conclusion can be confirmed in different ways. In fact, as we know from Chapter I the late stage of the transient field is defined by only those terms of the series which represent the low-frequency spectrum and contain either odd powers of the wave number k or logarithmic terms t'nk. Let us write down the integral

f'"o mr C cos am dm as a sum

r

J, mrCcosamdm o

=

r:mi Ccoe am dm + f'" mrCcosamdm

l,

mo

0

where m a has an arbitrary small value (m a « O. Within the external interval trn > m o) the radicals m j,m 2 , and m 3 can be expanded in series by powers k 2 j m 2 , and this allows us to represent the external integral

f'" mr C cos am dm mo

as a convergent series containing only even powers of wave number k:

f '"mrCcos am dm m"

=

( k LAI' I' m

)21'

Thus, we see again that the low-frequency part of the spectrum, which contains odd powers of k as well as logarithmic terms, can be obtained

176

II

Behavior of Field in Medium with Cylindrical Interfaces

only from the internal integral: (mO I;

o

mr C cos am dm

Performing some transformations, it is possible to see that in the general case, when the observation point is located in the far zone, the late stage behavior is still described by Eq. (H.9!). As follows from this equation, at sufficiently large times the field is independent of the parameters of the borehole and invasion zone, and it approaches the field at the late stage in a uniform medium with resistivity of the formation. Such behavior of the field, as in the case of the in-phase component, is observed even in the presence of inhomogeneities within the invasion zone. In fact, at the late stage the electric field changes with time as 1/t 5 / 2 [Part B, Eq. (H.59)]. Correspondingly, the electric charges arising on the inhomogeneity surface also vary as 1/ t 5/2. Therefore, the electric field of these charges generate currents which, along with their magnetic fields, change in the same manner. However, in the absence of the inhomogeneity the magnetic field decreases as 1/t 3 / 2 • Thus, with an increase of time the influence of an inhomogeneity becomes relatively small. It is obvious that this conclusion also applies to the case when the induction probe is not located on the borehole axis. This study clearly shows that in measuring the transient field in the borehole the depth of investigation increases with time regardless of the probe length and its position. In particular, the probe can consist of one coil only. Moreover, the field at the late stage has a relatively higher sensitivity to the conductivity than in the case when, for example, the quadrature component of the magnetic field is measured in the range of small parameters. Comparison of calculations of the transient field by the exact and asymptotic formulas allows us to establish the range of parameters when the late stage behavior takes place. For instance, if there is no invasion zone, Eq. (II.9l) provides accurate values of the field if

Until now we have considered the asymptotic behavior of the field. Next let us describe the transient responses of the field aBz/at calculated using

II.S

Behavior of Nonstatlonary Field on Borehole AxIs

a

10

5

2...L------r------.------r-------..-----l~

10

5

50

20

b

10

5

2...L----,,....--------.----"""""""T--------,---~,..

5

10

20

50

Fig. 1l.3 Transient responses of apparent resistivity. (Figure continues.)

177

178

II

Behavior of Field in Medinm with Cylindrical Interfaces

c

10

5

50

10

5

2..L-----..----------.-------.------..,.----~

10 Fig. II.3

(Continued)

20

50

100

11.6 Magnetic Dipole on Borehole Axis When Formation Has Finite Thickness

179

Eq. (I1.15) and Fourier's transform, which includes the early, intermediate, and late stages. It is convenient to introduce the apparent resistivity PT , which is related to aBz/at in the following way:

(11.92)

This transformation allows us to see more vividly the influence of the conductivity distribution on the field behavior. Examples of apparent resistivity curves are given in Figure II.3. The indices of the set of curves for two- and three-layered media are PZ/PI and PZ/Pl - aZ/a l - P3/Pl' respectively. Each curve is characterized by the parameter a. All curves correspond to the case when the probe length exceeds the borehole diameter. For this reason, at the early stage the field does not tend to that in a uniform medium with the resistivity of the borehole. With a decrease of time the value of PT increases unlimitedly, that is, the field at the borehole is much smaller at the early stage than that calculated by the formula for the late stage. The right asymptote of all curves is the formation resistivity. With the approach to this asymptote of the transient field, the influence of the probe length becomes smaller. As follows from the study of these curves, the field practically coincides with that in a uniform medium with the resistivity of the formation pz if

t

90ai

> - - f.Lsec 21TPl

provided that the invasion zone is absent.

U.6 Magnetic Dipole on the Borehole Axis when the Formation Has a Finite Thickness

Next we consider the field in a more complicated medium when the formation (bed) has a finite thickness and there is also an invasion zone (Fig. IIAa). At the beginning, let us assume that the skin depth in every part of the medium is much greater than the probe length L, the thickness H of the

b

a R

-..,

r

I'--.

..-/

R

14

t H

T 11

~

12

13 14

--

<,

I

a

----

c

d z p

1 100

12 £2

11 £1

R

z

y

-==t--=- -I

/-

<,

x (a) Model of medium with borehole, invasion zone, and layer of finite thickness; (b) illustration of geometric factor of elementary ring; (c) a medium with cylindrical interfaces; (d) current filament. Fig. 11.4

II.6

Magnetic Dipole on Borehole Axis When Formation Has Finite Thickness

181

bed, and the radius of the invasion zone a20 From a physical point of view this implies that the interaction between induced currents is negligible, that is, the current arises only due to the primary electric field. Then, for the quadrature component of the magnetic field on the borehole axis we have Qbz

W/-LL2 4

= -2- "'V.G. L..J i=1

I, ,

(II.93)

where 'Yi and G i are the conductivity and geometric factor of the ith part of the medium. At the same time, the in-phase component of the secondary field is equal to zero. As follows from Eq. (11.93), in the absence of an invasion zone we have (II.94) It is obvious that in this case the geometric factor of the formation be written as

Gi can (II.95)

where G 2 is the geometric factor of the bed when the borehole is absent and can be expressed in terms of elementary functions, while G 12 is the geometric factor of that part of the borehole which is located against the formation. As is seen from Figure II.4a, this part of the medium has the shape of a cylinder with the radius of the borehole and its height equal to the bed thickness H. By analogy, the geometric factor of the medium with resistivity P3 can be written as (II .96)

where G 3 is the geometric factor of the medium surrounding the formation when the borehole is absent and G 13 is the geometric factor of the part of the borehole which is located against this medium. It is clear that the sum of factors G 12 and G 13 is equal to (II.97) where G 1 is the geometric factor of the borehole.

182

II

Behavior of Field in Medium with Cylindrical Interfaces

Taking into account Eqs. (11.94)-(11.97), we obtain

=

Wj.LL 2 - 2 - (-yPl wj.LL2

=

-2-(( 1'1

-

+ 'Y2 G2 + 'Y3 G3 -

'Y2 G[2 - 'Y3( G[ - G l2 )

)

'Y3)G1 + (1'3 - 'Y2)Gl2 + 'Y2 G2 + 'Y3 G3) (11.98)

Thus, the quadrature component Q b, is expressed in terms of the geometric factors G 1 , G 2 , and G 3 , which are easily calculated and

At the same time, the function G 12 can be represented as

where L

r3

g=-2 R 3R 3 1

2

is the geometric factor of the elementary ring with a unit cross section (Part B, Chapter II). Coordinates r, ZI' and Z and distances R 1 and R 2 are shown in Figure IIAb. In accordance with Eq. (11.98), the apparent conductivity is (II.99)

and Eqs. (11.98) and (11.99) describe the field and the apparent conductivity regardless of the position of the two-coil induction probe with respect to the horizontal interfaces. Here it is appropriate to make the following comments: 1. The first term on the right-hand side of Eq. (II.99) does not depend on the probe position, since it characterizes the field caused by induction currents in a borehole with conductivity (I'I - 'Y3)G 1. 2. The last two terms of the equation

I1.6

Magnetic Dipole on Borehole Axis When Formation Has Finite Thickness

183

describe the field generated by induction currents in the formation and in the adjacent medium when the borehole is absent. 3. Equation (11.99) can be rewritten as

where 'Y~1) is the apparent conductivity, which corresponds to a horizontally layered medium. Next, we consider a more general case when the borehole solution penetrates into the formation (Fig. IIAa). Applying Eq. (11.93) we obtain the following expression for the quadrature component of the magnetic field: WJ-LL

Q bz =

2

-2-( 'Y4 G4 + ('Y1 + ('YJ -

=

'Y4)G I2 + 'Y3 G3

'Y3)G I2 + ('Y2 - 'Y3)Gz )

wJ.LL Z - 2 - ('Y3 G3 + 'Y4 G4 + ('Yl - 'Y4)G 1

+ ('Y4 -

'Yz)G 12 + ('Yz - 'Y3)G Z )

(II.lOO)

where 'Yz, G z, 'Y3' G 3 and 'Y4' G 4 are the conductivities and geometric factors of the invasion zone, the formation, and the adjacent medium when the borehole is absent, respectively. As before, G 12 is the geometric factor of the borehole, which is located against the invasion zone. Correspondingly, for the apparent conductivity we have

The simplicity of these equations is obvious. At the same time they are very useful for the study of the radial and vertical responses of induction probes, when it is possible to neglect the skin effect. Now we improve the approximate solution, given by Eq. (II.lOO), and consider the field behavior when the skin effect occurs only in the formation and in the adjacent medium, and it manifests itself in the same manner as in a horizontally layered medium. This means that the currents induced in the borehole and in the invasion zone are defined by the primary electric field and do not influence the currents in other parts of the medium. In deriving the expression for the field we apply the approach described in Section Ir.3. First, let us represent the field in a medium with

184

II

Behavior of Field in Medium with Cylindrical Interfaces

two horizontal interfaces, as a sum of two fields:

(11.102) where b, is the vertical component of the magnetic field, expressed in units of the primary field, caused by currents in the cylinder with the borehole radius; be is the field of currents induced outside the borehole. The field b, can be expressed in terms of geometric factors: iWj.LLz

b, = - 2 - ( Y ZG 12 + Y3 G13)

(II .103)

where G 12 and G 13 are the geometric factors of the borehole, located against the formation and the adjacent medium, respectively. Thus, the magnetic field of currents induced in the formation and in the surrounding medium when the borehole is not conductive and there is no invasion zone can be represented as

where G,z + G 13 = G,. On the other hand, the currents induced in the borehole, create only a quadrature component, which is equal to

Therefore, for the magnetic field on the borehole axis we have iWj.LLZ

iWf.L L z

bz = b oz - - 2 - ( y zG 12 + Y3 G13) + - 2 - y,G, iWf.L L z

=

b oz + - 2 - [( Y3 - yz)G 12 + (y, - Y3)G,]

(11.104)

As was demonstrated in the previous chapter, the magnetic field of the vertical magnetic dipole boz in a horizontalIy layered medium is expressed in an explicit form. For instance, when the probe is located symmetrical1y with respect to the bed interfaces, we have

oom3 e m 2H +K lZ coshm Z L b Oz= b z+L3 l,( - K 12 e -Zm2H Z dm, o mZ 1-K 12 e Zm2 H

if H;;::.L

I1.6

185

Magnetic Dipole on Borehole Axis When Formation Has Finite Thickness

and if H
bz = b oz + - 2 - [( 'Y2 - 13)G 23 + ('Y4 - 'Y3)G 12 + ('YI - 'Y4)Gd (I1.10S) where 'YI' 'Y2' 'Y3' and 'Y4 are the conductivities of the borehole, invasion zone, the formation, and the adjacent medium, respectively. G] and G l2 are the geometric factors of the borehole and its part, located against the formation, while G 23 is the geometric factor of the invasion zone, which can be expressed in terms of G 12' Thus, the determination of the field on the borehole axis in the presence of horizontal and cylindrical interfaces consists of two steps. The first one is a calculation of the field in a horizontally layered medium, when the borehole and invasion zone are absent. The second step is a determination of the geometric factors of cylinders of finite height, the axes of which coincide with the borehole axis. In accordance with Eq. (IUOS), we have the following expressions for the quadrature and in-phase components of the field: WJLL 2

Q bz

=

Q b oz

+ - 2 - [( 'Y2 -

'Y3)G 23

+ ('Y4 -

'Y3)G l2 + ('YI - 'Y4)G1]

(II.106) and

The latter clearly indicates that induced currents in the borehole and invasion zone do not have an influence on the in-phase component of the magnetic field, and this field coincides with that in a horizontally layered medium. As was pointed out earlier, this fact shows one more time that by measuring the in-phase component it is possible to achieve a greater depth of investigation than in the case of the quadrature component. In particular, we found in Chapter I that with a decrease of the frequency, the field

186

II

Behavior of Field in Medium with Cylindrical Interfaces

In b oz' measured in the bed of a finite thickness, tends to that in a uniform medium which surrounds this formation. Therefore, taking into account Eqs. (11.50) and (II.106), we have s

In bz ~

-

1 2 3/2 M (Y4J.LwL) , 3v2

L if--«l °i(min)

This behavior of the spectrum also allows us to conclude that the late stage of the transient field, measured on the borehole axis, is determined by conductivity of the medium 'Y4 surrounding the formation. Of course, with an increase of the thickness and conductivity of the bed, this asymptotic behavior will occur at greater times. Comparison with calculations, based on the use of numerical methods of solution of this boundary value problem, shows that with an error of 10% or less we can use Eqs. (lI.l06), if conditions (II.60) are met.

11.7 The Field of a Current Loop around a Cylindrical Conductor

Suppose that an infinitely long cylinder with radius a is surrounded by a medium with conductivity 'Y and dielectric permittivity E. The primary field is caused by a horizontal current filament with radius r 0 whose center is situated on the cylinder axis (Fig. IIAc). It is also assumed that the cylinder has an infinitely large conductivity. In order to determine the field we introduce a vector potential of the electric type A. Then, in accordance with Eqs. (1.392 in Part B), we have B = curIA J.L( 'Y - iWE)E = k 2A + grad divA

(11.107)

and

First, we find an expression for the vector potential A in a uniform medium in terms of cylindrical functions. With this purpose in mind, it is useful to imagine the current ring as a system of electric dipoles. Earlier we demonstrated that the vector potential d A of the electric dipole [Eq. (1.314)] is J.L/ dt 'k dA=--e' R 47TR

(11.108)

187

1I.7 Field of Current Loop around Cylindrical Conductor

We choose a cylindrical system of coordinates so that its origin coincides with the center of the current ring. It is clear that the vector potential A at every point p can be represented as a sum of vector potentials caused by all elements of the current loop. As can be seen from Figure IIAc, due to the axial symmetry, the radial component A r is equal to zero, and therefore the vector potential of the current ring has only the component A
(11.109)

where

d = ,; r~

+ r 2 - 2r or cos a

Next, we make use of the known integral representation of function eikRjR:

(Il.IlO) where m]

= Vm 2 - k 2

Substituting Eq. (I1.11O) into Eq. (11.109) and integrating along the ring, we obtain an expression for A
1 cos mzdm 1 Ko(mjd)cos ada 00

21T

0

(11.111)

0

In accordance with the addition theorem of modified Bessel functions, we have

Ko(mtr)Io(mtr o) + 2

L

Km(m]r)Im(mtfo)coS m a ,

if r ~ "o

m~j

(Il.112) 00

Ko(mjd) =Ko(m]ro)/o(mjr) +2

L

Km(mjro)/m(mjr)cosma,

m~j

Now we replace the function Ko(mjd) in Eq. (ILl1l) by the right-hand

188

II

Behavior of Field in Medium with Cylindrical Interfaces

side of Eq. (II.1l2) and apply orthogonality of trigonometric functions: 2 7T

1

a cos mo: cos a. da =

{O

7T

if m if m

* 11 =

Then, we obtain an integral representation of the vector potential of the electric type when the generator of the primary field is a current loop: if r ~ "o (11.113) and if r .:s; "o Inasmuch as the vector potential has only the component A!p' which is independent of the coordinate ip, we have divA

=

0

For this reason, the electric field of the current ring in a uniform medium can be written as if r ~ r o (11.114) and

Now we are prepared to determine the electric field in the presence of the cylinder. First of all, it is obvious that, due to the axial symmetry, charges are absent and the secondary electromagnetic field is caused by currents in the conducting medium and on the cylinder surface, since its resistivity is equal to zero. These currents have only a q;-component, and they are mainly distributed in that part of the cylinder which is close to the current ring. The solution of the boundary value problem is greatly simplified, since on the surface of the ideal conductor the total electric field E!p vanishes, that is, if r = a

(II.1l5)

IL7

189

Field of Current Loop around Cylindrical Conductor

where Es'P is the secondary electric field, caused by the change with time of the secondary magnetic field. As follows from Eq. (11.114), the secondary field Es'P can be represented as: (II.116)

since the electric field tends to zero at infinity. Ci m) is an unknown function. Substituting Eqs. (II.116) and (11.114) into Eq. (11.115), we obtain

Therefore, (II.11?)

Correspondingly, the secondary electric field is

Further, we consider the field behavior when the current and receiver coils have the same radius r = ro' Then, the total electric field at the distance z = L is if r = ro (II.119)

where (I1.120)

Correspondingly, the electromotive force arising in the receiver coil is (II.12l)

where n is number of turns of the receiver coil. I As usual the distance between the coils L is called the length of the two-coil probe.

190

II

Behavior of Field in Medium with Cylindrical Interfaces

The study of this field is of great practical interest. For instance, induction logging is often used during drilling for measuring the formation resistivity. In this case, the radius of the coils r 0 is only slightly greater than the cylinder radius a, that is,

ro - a !J.a --=-«1 a a

(1l.122)

For this reason, induced currents appear very close to the current loop and they have a direction which is opposite to I. Therefore, at distances essentially exceeding the loop radius, the field is practically equivalent to that of a vertical dipole placed at the origin of coordinates. This result can also be derived from Eqs. (1l.119) and (Il.120). In fact, with an increase of the probe length L, the integral on the right-hand side of Eq. (Il.119) is defined by small values of m. Substituting asymptotic values of Bessel functions into Eq. (1l.120), we have a2

F(m) -d - 2" ro

=

1-

a2

txa

(a+!J.a)

2 :::: 2--

a

Thus, in the far zone, when L » ro, Eq. (11.119) can be represented as if L » ro (II.123) Then,

~aking

into account Eq. (II.114), we have if L » r0

and

r = r0

(II.124)

At the same time, the field Eo'P coincides with that of a magnetic dipole in a uniform medium, if L» ro' Therefore, from Eq. (II.14) in Part B it follows that

iosu.M 2!J.a . E = -_ _ e' k R ( 1- ikR) sin () 'P 411'R 2 a or, since sin () = ralR when R:::: L, we have E

'P

iWJ.LM!J.a . - r e ' k L ( 1 -- ikL) 3 211'L a 0

= --

(11.125)

11.7 Field of Current Loop around Cylindrical Conductor

191

where M = hrrgn is the dipole moment. Correspondingly, the electromotive force in the receiver coil is (II.126)

where M R = 7Trgn R and n R is the number of turns of the receiver coil. Comparison of Eqs. (11.14) in Part Band (11.125) shows that the presence of a very conductive cylinder results in a decrease of the electromotive force in the far zone by approximately 2(!1ala) times with respect to the case of a uniform medium. As follows from Eq. (11.126), by measuring the amplitude and phase of fJJ it is possible to determine the resistivity and dielectric permittivity of the medium. Now let us suppose that around the metal conductor there is a cylindrical zone with parameters 'Y 1 and E I' which differ from corresponding values of the formation (Fig. IIAc). The internal and external radii of this zone are a and r, respectively (a < r 0 < r). Taking into account the fact that in the first approximation the presence of the cylinder results in only a decrease of the moment M, we can make use of Eqs. (11.78) and (11.126) and represent the electromotive force in the far zone as if L» r

(II.127)

where k 1 is the wave number of the medium surrounding the cylinder. Then it is clear that by measuring either the ratio of amplitudes or the difference of phases with two probes of different length it is possible to reduce the influence of currents induced in the intermediate zone. As is well known, this approach is used in measuring the resistivity of the formation during drilling of wells. If the contribution of displacement currents is negligible and observations are performed in the far zone (Llr» 0, measurement of either the ratio of amplitudes or the phase difference is sufficient to determine the formation resistivity. In those cases when displacement currents play an essential role, measurements of both parameters allow us to calculate 'Y and E. Also, this can be done by measuring one parameter, for instance, the phase difference, with three-coil probes having different length. It is appropriate to note that, in accordance with Eq. (11.127), measurement in the far zone of either the ratio of amplitudes or the phase difference provides the same depth of investigation in the radial direction.

192

II

Behavior of Field in Medium with Cylindrical Interfaces Induction Logging

Table II.1a

Probes

N

Measuring parameters

4

Two-coil probe Differential probes Lateral soundings One-coil probe

QW and QW and Q,if and In W

5

Three-coil probe

IW(L 2 ) 1 and

acp = cp(L2 ) - cp(L j )

6

Lateral soundings

IW(L 2 ) I

acp = cp(L2 ) - cp( L I)

7

Two-coil probe

tJ.{Q go) = Q W(w,) - (::) Q W(W2)

8

One- and two-coil probes

Transient response of W{t)

1 2

3

Table II.1b N

In W In W In W

W(LI)

W(LI)

and

2

Dielectric Logging Probes

Measuring parameters

Two-coil probe

Iwi and cp(W)

2

Three-coil probes

3

Two- and three-coil probes

W(L2) - and acp W(L,) First arrival and transient response

I

I

In this chapter we have described the field behavior and the principles of electromagnetic methods which are used for determination of conductivity and dielectric permittivity of the medium surrounding a borehole. As a summary, Tables n.1a and Il.Ib list these methods, which are, in essence, different modifications of the induction and dielectric logging methods.

II.S Integral Equation for the Field Caused by Induced Currents The study of electromagnetic fields in media with either horizontal, cylindrical, or spherical interfaces allowed us to lay a foundation for many geophysical methods. For such models, application of the method of separation of variables is the most natural approach, enabling one to represent the field in an explicit form by known functions. In more complicated models where, for instance, plane interfaces are not parallel to each other or there are both the horizontal and cylindrical interfaces, the method of separation of variables cannot be used. For this reason,

11.8 Integral Equation for Field Caused by Induced Currents

193

numerical methods, such as integral equations or finite differences are mainly applied to solve the boundary value problems. In this section we describe three modifications of integral equations, some of which have found broad application for solving forward problems of different electromagnetic methods. To illustrate this approach, we suppose that a vertical magnetic dipole is located on the borehole axis and the medium possesses axial symmetry (Fig. II.Sa). In accordance with the Biot-Savart law, the current of the magnetic dipole creates the primary magnetic field, and its change with time generates the primary vortex electric field. Due to the axial symmetry, the electric field does not intersect boundaries between media with different conductivities and therefore surface charges are absent. As a result of the existence of the vortex, electric field currents arise at every point of the conductive medium with a density given by (II.128) where Eg is the primary electric field, while E, is the secondary electric field (caused by a change with time of the secondary magnetic field), and '}' is the conductivity in the vicinity of some point. Inasmuch as electric charges are absent, the induced currents as well as the primary electric field have only an azimuthal component j cp in a cylindrical system of coordinates (r, tp, z ). It is clear that the interaction between currents does not change the direction of the current flow, and this single fact drastically simplifies the derivation of the integral equation. Thus, the total electric field is (II.129) As was shown in Part B, Chapter I, an elementary current tube passing the point q creates an electric field at point p (Fig. II.Sa) equal to

where dS is the cross section of the tube, G(p, q) is a function which depends only on geometrical parameters and can be represented in an explicit form, and j /q) is the current density at the point q, Now, applying the principle of superposition, the total electric field can be written as (II.130)

194

II

Behavior of Field in Medium with Cylindrical Interfaces

a

b

z

z

c/ 'Y2

'Y1

'Y1

'Y2

'Y1

'Y2

13

a1 .q

~

M

---- --

M

81

,...-

.......

S2 .......

d C

82

I 82

'Y3

I

13

I

83

I

81 I

81

12

I I

I

11

'Y2

'Y4 'Y3

I

'Y3

Mt

'Y1

Mt I

I Fig. 11.5 (a) A medium with one cylindrical interface; (b) a medium with two cylindrical interfaces; (c) a medium of borehole and layer with finite thickness; (d) a medium with borehole, invasion zone, and layer of finite thickness.

11.8

Integral Equation for Field Caused by Induced Currents

195

where S is the half-plane described by the conditions r

> 0 and

~

00

< z < 00

Making use of Ohm's law,

we obtain (11.131) This is an integral equation of the Fredholm type of the second kind with respect to an unknown function ElF" Since dS

=

drdz

we have

Taking into account Eq. 01.129), the integral equation with respect to the secondary field has the following form

where (11.134) is a known function which describes the secondary electric field when the interaction between induced currents is negligible. In other words, this first approximation corresponds to the range of small parameters when the field can be expressed in terms of geometric factors. We can conceptually replace the half plane with a system of small cells within each of which the electric field is practically constant. In doing so the integral equation (II.133) can approximately be represented as N

Esrp(p) =F(p) +iwJ.L

L: y(q)G(p,q)Esrp(q)!lS ;~

1

Having written this equation for every cell, we obtain a system of N linear equations with N unknowns.

196

II

Behavior of Field in Medium with Cylindrical Interfaces

However, Eq, (1I.133) is not convenient to use, since the infinite limit with respect to coordinates rand z creates serious numerical problems for determination of the electric field. Also, it does not allow one to derive relatively simple asymptotic formulas, except the very-low-frequency limit when the skin effect is completely ignored. In order to facilitate calculations of the field and obtain asymptotic formulas, we derive the integral equation when the area of integration with respect to distance r is limited. First of all, let us assume that the invasion zone is absent. Then, proceeding from Green's formula we obtain an integral equation for the component Ecp, in which the integration is performed over the cross section of the borehole only. At the beginning, suppose that the formation is uniform and that it has conductivity 'Y2' as shown in Figure lI.5a. As we know, the electric field E in every part of the medium satisfies Helmholtz's equation: (II.13S) Let us represent the electric field as a sum, (11.136) where Eo is the field in a uniform medium with the conductivity of the formation 'Y2' It consists of the field caused by a dipole current in free space and the field of currents induced in this medium. E 1 is the field due to the presence of the borehole with conductivity 'Yl and radius a. The field Eo satisfies the following equation: (11.137)

where •

k 22 = 1'Y21J. W In accordance with Eqs. (lI.l3S) and (II.136), the electric field E, is a solution of the equation

(11.138) Taking into account Eq, (II.l37), we have for the formation and the borehole, respectively, if r > a

(II.139) if r
(11.140)

II.S

Integral Equation for Field Caused by Induced Currents

197

where

It is obvious that

(II.141)

where I,!, is the unit vector directed along the cp-coordinate line. Next, let us introduce a vector function G = GI,!, which is continuous along with its first derivative and satisfies the equation (11.142)

except at the point p at which the field E 1 is determined. At this point, the function G = GI,!, has a logarithmic singularity. Consider the expression

Inasmuch as

and

we have G· V' 2 E I - E 1 . V' 2 G

=

Epl,!, V' 2 1,!, + G V' 2 E 1 - GEII,!, V' 2 1,!, - E 1 V' 2 G

= G V 2E 1 - E 1 V' 2 G Thus (II .143) It is also appropriate to make the following comment. In practice the magnetic dipole is a small loop. In approaching this loop the primary electric field tends to infinity as a logarithmic function, while the electric field caused by induced currents does not have a singularity. Correspondingly, the electric field E I is a continuous function everywhere. Taking into account the axial symmetry of the field and the model of a medium, we make use of the two-dimensional form of Green's formula. Let us assume that the point p is located inside the borehole. Then, we have for the

198

II

Behavior of Field in Medium with Cylindrical Interfaces

borehole

and for the formation (II.145) where n + is directed along the coordinate r, but n _ = - n +' t is the straight line on the borehole surface which is parallel to the z-axis, and to is a contour around point p (Fig. II.5a). From Eqs. (11.140)-(11.142) it follows that within the formation

and in the borehole

Near the point p the field E[ has a finite value, but the function Eo increases without limit as tnr, where r is the radius for the circumference to' Therefore, the value of the integral along the contour to tends to the value -2rrE[(p). Combining Eqs. (11.144) and (11.145) and making use of the continuity of tangential components of the electric and magnetic fields, we eliminate the integral along the line t. Respectively, we obtain an integral equation which includes a surface integral only over a restricted area in the radial direction, which corresponds to half of the cross section of the borehole:

(II.146) The function Gtk 2' p, q) describes with the accuracy of a constant the

11.8 Integral Equation Cor Field Caused by Induced Currents

199

electric field in a uniform medium with conductivity Y2' generated by a circular current filament. There are several known expressions for this function and, in particular, it can be represented as an integral of elementary functions. Now we suppose that there is an invasion zone, but the formation still has an infinite thickness, as shown in Figure II.5b. As before, the function G is chosen in such a way that it satisfies the equation (II.I47) and it has a logarithmic singularity at the point p. In accordance with Eqs. (11.139) and (II.I40), we have for the field E}

+ (kj - kDEo,

V 2E I

=

-kfE I

V2E I

=

-kiE} + (kj - knEo,

if 0 < r < a l (II.I48)

if r> a 2 Again applying Green's formula for the borehole, the invasion zone, and the formation, we have, respectively,

(II.I49)

(II.I50)

(II.I5I) where t I and t 2 are straight lines located at the boundaries between the borehole and the invasion zone and between the invasion zone and the formation, respectively, while 51' 52' and 53 are their cross sections. Now, taking into account Eqs. (II.I48) and performing a summation of Eqs. (II.l49)-(II.I5I), we obtain an integral equation which contains two surface integrals over half cross sections of the borehole and invasion

200

II

Behavior of Field in Medium with Cylindrical Interfaces

zone:

(II.152) It is obvious that the integral equations (11.146) and (II.152) coincide with

each other if k 1 = k 2 or k 2 = k 3 • We have illustrated the derivation of the integral equation in two cases when the solution of the boundary value problems can be obtained in the explicit form by making use of the method of separation of variables. In both cases the same Green's function has been used, which corresponds to a uniform medium with the conductivity of the formation. Next, we derive the integral equation for the case when the formation has a finite thickness (Fig. II.5c). With this purpose in mind, let us introduce a new Green's function, which obeys the following conditions: 1. It is a solution of equations

and

(II.153)

in a horizontally layered medium when the formation and adjacent medium are characterized by wave numbers k 2 and k 3 , respectively. 2. The function G = GI