Multi-Coil Differential Induction Probes

Chapter 7

MULTI-COIL DIFFERENTIAL INDUCTION PROBES

Analysis of the field in media with horizontal as well as cylindrical interfaces, performed in previous chapters, has shown that a two-coil induction probe does not possess noticeable advantages with respect to electric logging probes. Of course, such conditions as a nonconducting borehole surrounded by a relatively conductive medium with thin highly resistive layers can be considered as an exception. The influence of currents induced in a borehole and in an invasion zone is usually suflficiently great, and for this reason, in order to determine a formation resistivity, it is necessary to apply two-coil induction probes with practically the same length as in electric logging. Let us notice that at the range of frequencies and resistivities, where the skin effect manifests itself relatively weakly the influence of the surrounding medium is even greater than in electric logging. But with an increase of frequency, the conductivity of a medium, as well as formation thickness, absorption of the electromagnetic energy essentially improves the vertical response of the probe. This brief comparison with normal or lateral electric probes shows that application of two-coil induction probes is hardly reasonable. Moreover, it requires the use of more complicated equipment. For all these reasons H. Doll suggested, in 1946, multi-coil differential probes and also developed an approach allowing us to determine parameters of these systems. At the beginning we will consider multi-coil induction probes, proceeding from the theory of small parameters, when we can neglect the interaction between currents. In other words, currents in any part of the medium, regardless of how far from the probe it is located, create a signal which is only defined by the conductivity and the geometric factor of this part. It is obvious that the role of various parts of the medium in forming a signal essentially depends on the probe length. For instance, with an increase of the probe length the influence of more remote parts of the medium increases, and consequently, the contribution of currents induced near the probe becomes smaller. Applying probes of various lengths with different numbers of turns in coils, connected in series in the same or opposite directions, we can significantly reduce the signal caused by currents in any element of the medium independently of the distance to a probe. However, for improving the characteristics of a two-coil probe it is not sufficient to decrease a signal from some element of the medium. For improvement of the radial response of a two-coil probe in order to determine a formation resistivity it is necessary to decrease the contribution of currents induced in the borehole and in the invasion zone with respect to a signal caused by currents in the formation. In other words, it is essential to decrease the influence of parts of the medium located relatively close to the probe. At the same time the signal generated by currents 385

386

in remote parts of the medium should not be very smah, otherwise serious measuring problems would arise. For improving the vertical response of a two-coil induction probe it is necessary to decrease the relative contribution from the surrounding medium with respect to the signal caused by currents in the formation against which a two-coil induction probe is located. In other words, in this case the influence of more remote parts of the medium has to be reduced providing a significant signal from currents induced in that part of the medium which is located relatively close to the probe. Thus, improvement of radial and vertical responses of a two-coil induction probe is related to the development of multi-coil probes which have to satisfy opposite requirements. As will be shown later, under certain conditions we can improve both responses simultaneously. However, in a general case improvement of one response with respect to that of a two-coil induction probe is related with deterioration of another and vice versa. Arbitrarily, a multi-coil induction probe can be presented as a sum of two-coil induction probes. Currently, there are known symmetrical and non-symmetrical multi-coil induction probes, and their characteristics will be considered in detail in the next sections. As a rule in induction logging using one frequency the electromotive force, caused by the current in the transmitter coil or coils, is significantly greater than that generated by induced currents in the medium. For this reason an additional compensating coil to increase the accuracy of measurement is installed. Due to this the electromotive force caused by the primary field is practically equal to zero. It is also appropriate to notice that some differential probes do not require a compensation coil.

7.1. Methods of Determination of Probe Parameters Methods of choosing probe parameters are based on the use of differential and integral responses of two-coil induction probes. The differential radial response defines a signal from a thin cyhndrical shell, expressed in units of the signal, caused by currents in a uniform conducting medium. In accord with Doll's theory, described above, we have: oo

Gr=drfqdz

(7.1)

— OO

where q dr dz is the geometric factor of the ring with cross section equal to dr dz and q = {Ll2){r^/R\Rl)R, = [r^ + {z - L/2f]''^, R2 = [r^ -f (z + L/2)2]V'; r and z are cylindrical coordinates of a point, and the origin of coordinates system coincides with the probe middle; L is the probe length. Making use of results developed in Chapter 3 let us consider in detail the behavior of function Gr- As was shown in this chapter the magnetic field on the borehole axis, generated by currents in a thin cylindrical shell and expressed in units of the primary field, can be presented in the form: 00

m\^Kl{\r) •cosALdA + mh{\r)Ki{\r

(7.2)

387 where r is the shell radius; n = uj/iarAr; uj is the frequency; ji is the magnetic permeability equal to An x 10~^ H / m ; a and A r are conductivity and thickness of the shell; /i(Ar) and Ki{\r) are modified Bessel functions. As is well known, we have: A(Ar)^^

Ki{\r)^^

as Ar ^ 0 AT

Z

and hiXr)

-^

,

i^i(Ar) -^

A/TT

as Ar ^^ OO

Let us notice t h a t the product /i(Ar)i^i(Ar) does not exceed 0.5 when A varies from zero to infinity. For this reason at the range of small parameters we have: oo

r3

Qh,c^

r

in—

/ X^KliXr)

cos XL dX

(7.3)

0

Taking into account the relation between the quadrature component of the field, Q/i^, and the apparent conductivity: 2 ^a =

p^ Q hz jiujL'^

we obtain: Q.3 /err A r 2a^ — / rn?Kl{m)

TT

cos ma dm

C^-^)

J

where: a = L/r

or

Oa = c^Gr

(7.5)

oo

rAr2a^

f

o^.o. . . Ar 1 m'^K?(m) cos ma dm =^ 7;C(a) TT J r a"^

and oo

C(Q;) =

/ m^i^^(m) COS a m d m TT

J 0

It is obvious t h a t : a/ia;L2 A r 1

(7.6)

388 and correspondingly, for the electromotive force in the receiver of a two-coil induction probe we have: afiuL^ Ar 1

MrMRLOfi

^I^UJ'^MTMRI

Now we will consider the behavior of C(a) for large and small values of a. Suppose that a ^ oc. Introducing the notation 0(m) = im?Kl{m) and applying integration by parts we obtain an asymptotic presentation: OO

CO

f 1 l°° 1 \^ I 0(m) cosma dm = — (/)(77i) sinam H—r0'(m)cosma J a \Q a^ lo 0

I f i (/)" {m) cos ma dm Q:^ J 0

Taking into account that 0(0) = 1 and that the function along with its derivatives is zero as m. tends to infinity, we have: OO

OO

(/)(m) cos m a dm =

/

o^'W a^

o / ^"{'^) cos a m dm a^ J

0

0

If m -^ 0 we have: KAm) :^ m

h— m m 2

Therefore: (j)(rn) = m'^K^{m) c^ 1 -h m^ In m (/)'(m) ~ 2mlnm 0''(m) c^ 21nm ^ -2Ko{m) Whence: OO

/

OO

dim) cos m a dm ^ - / /^o(^) cos m a dm c::^ - ^ TTj

0

a^

0

smce OO

m) cos a m dm

(1 -h a2)i/2 0

Thus, we have: 2a^^7r C(a) ^ -;;^—-- = 2 TT a^ 2

if a ^ OO

389 TABLE 7.1 Values of function C{a) a

C{a)

a

C{a)

a

C{a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.58620x10-3 0.46248x10-2

2 3 4 5 6 7 8 9 10 11

1.3110 1.8546 2.0724

12 13 14 15 16 17 18 19 20

2.1056 2.0962

0.15254x10-^ 0.35029x10-^ 0.65736x10"^ 0.10830 0.16280 0.22858 0.30437 0.38845

2.1473 2.1551 2.1634 2.1530 2.1404 2.1278 2.1161

2.0879 2.0806 2.0740 2.0683 2.0632 2.0586 2.0545

In the opposite case, as a -^ 0, C{a) decreases directly proportional to a^, inasmuch as integral f^ m'^Kf{m) cos ma dm tends to the constant when a goes to zero. Values of this function are given in Table 7.1. As follows from this table C{a) can be presented as: C{a) c^ 0.586 a^

if a < 1

The differential radial response, G^, depends on two parameters, namely: the ratio of the probe length to the shell radius, L/r, and the ratio of its thickness to the radius, Ar/r. First, we will assume that for all shells A r / r = const. In this case let us present Gr in the form: Ar

if r -> 0, a -^ oo and C{a) -^ 2. Thus: n 2Ar 2 Gr -^ r In the opposite case, when r -^ oo, a -^ 0 and C{a) -^ ka^ {k = 0.586), we have: Gr -^

Ar 1 k—Lr r

Therefore, the geometric factor of cylindrical shells, the thickness of which is directly proportional to their radius, changes in the following way. For small values of r the geometric factor is also small, then it increases directly proportional to the square of the radius, reaches a maximum and with further increase of the radius it decreases inversely proportional to r (Fig. 7.1a).

390

0.06Ar/r=0.1

0.04-

Ar/L = 0.1

0.020

I

1

1

2

— 1 —

3

1

1

1

4

5

6

1

r/

Figure 7.1. Function GrNow we will present the medium as a system of coaxial cylindrical layers with the same thickness: Ar = const, and correspondingly we can write Gr in the form: Gr

Ar 1 C{a) L a

If r -^ 0 we have: (jr^

Ar 2 L L

while for large values of radius we obtain: KJr

kArL

1

Therefore, at the beginning Gr increases directly proportional to r, then it grows slower, reaching a maximum and for large values of r it decreases inversely proportionally to r^ (Fig. 7.1b). Correspondingly, the maxima of the geometric factor Gr for uniform and nonuniform presentation of the medium by shells are shifted from each other. First, we will consider the graphical method of determination of the parameters of the multi-coil differential probe. Let us notice that this was the main approach applied in developing the first induction probes. Making use of transparent paper and simultaneously putting on several differential responses of various two-coil induction probes, the parameters of the multi-coil probe are chosen so to reduce the influence of the shells surrounding the probe as much as possible. Correspondingly the depth of investigation of a multi-coil probe increases. If the differential response of the multi-coil induction probe within some interval is equal to zero, the geometric factor of any thin cylindrical shell of this interval is also equal to zero. In other words, the magnetic field of induced currents in such a shell creates a positive electromotive force in one group of receivers while in other groups of coils it generates the same signal by magnitude and different by sign. It is obvious that the radial differential response of the multi-coil induction probe does not depend on the distribution of conductivity in the radial direction if within every cylindrical layer the resistivity remains constant.

391 In the determination of the differential response there is an element of uncertainty, inasmuch as the dimensions of a shell are not defined. For this reason it is appropriate to find out the maximal thickness of the shell, when its geometric factor with sufficient accuracy is described by Gr- It is obvious that the geometric factor of the shell with thickness Ar can be presented as a difference of geometric factors of two cylinders with external and internal radius Ve and r^, respectively. For a sufficiently thin shell, in accord with eq. 7.6 we have: Gr{a) = ~\c{a) r a^

= Gi(ae) - Gi(a,)

(7.8)

where ae = L/r, ai = L/ri, a = L/r, r = {ri-\- re)/2 (re = r ^ Ar/2, Vi = r — Ar/2) and L is the probe length. It is clear that: a 1 + Ar/2r

a 1 - Ar/2r

Let us consider two limited cases. Case 1

If r ^> 0, a ^ cxD, C{a) -^ 2, Gi[a) —^ Xjo? and therefore: GrKOi) =

r a^ It means that if the external radius of the shell is many times smaller than the probe length, the geometric factor of such a thin shell is practically equal to that of the infinitely thin one. Case 2

If r ^ (X), a —> 0, Gi{a) ^^ 1 — ka, C{a) -^ ka^. Therefore: Ar — k a = k{ai — a^) or Ar -a

1 - (Ar/2r)2j

0

This relation becomes more accurate with a decrease of A r / r . Calculations show that if A r / r < 0.2, eq. 7.8 is valid practically for all values of a, i.e. the geometric factor of the shell is described by GrThe second radial response of the two-coil induction probe is its integral response (direct and inverse ones). The direct integral radial response defines the signal caused by currents in the cylinder referred to that from currents in a uniform medium as a function of

392 10 1--Gi(a)

0.1

0.01 G^{a) 0.001 0.01

0.1

1 a

10

100

Figure 7.2. Direct and inverse integral responses.

the distance from the axis where the probe is located. As is well known the geometric factor of the borehole is defined by Gi(a), where a = L/a. If instead of the borehole radius a we introduce radius r and the probe length L is considered to be constant, then G\{r/L) represents the direct integral radial response of a two-coil induction probe. If r ^ 0, Q; ^ cxD then G\{a) -^ Xjo? — r^jl?', i.e. the geometric factor of the cylinder at the beginning increases directly proportional to the square of its radius and inversely proportional to l?. For large values of r, a tends to zero, G\[OL) -^ \ — ka— 1 — kL/r, and correspondingly the response magnitude approaches to unity. It is obvious that coefficient k is equal to 0.586. The inverse integral radial response characterizes a contribution caused by currents in a medium outside the cylinder with radius r, and it is related with Gi{a) as: G'^'^l-Giia) Direct and inverse integral responses are presented in Fig. 7.2. Applying the graphical method we can choose corresponding characteristics of multi-coil induction probes. From the point of the depth of investigation the direct integral response of a multi-coil induction probe has to satisfy two conditions: • Its value near a probe has to be minimal. • It should approach relatively slowly to its asymptote, which is equal to unity for all induction probes. It is appropriate to notice that the graphical method of obtaining the radial responses of multi-coil induction probes has some shortcomings. They are:

393

• The process of choosing probe parameters is sufficiently cumbersome. • The accuracy of determination of small values of the geometric factor at the initial part of the response is usually very low. It becomes specially noticeable in the case of a nonuniform medium, for example, when the borehole is much more conductive than the formation. For these conditions the focusing abilities of a probe, chosen by the graphical method, can be insufficient in order to eliminate the influence of the borehole or the invasion zone especially, if the latter is more conductive than the formation. For this reason focusing features of a multi-coil induction probe, chosen by the graphical method, manifests themselves as a rule only for relatively small ratios P2/Pirn With an increase in the number of probe coils this process becomes more and more complicated. • Making use of the graphical method it is practically impossible to chose optimal parameters of the probe, when simultaneously with a decrease of the signal from the borehole and the invasion zone we can provide a maximal signal from more remote parts of the formation. • This method of superposition of radial responses of two-coil induction probes does not practically allow us to find parameters of probes which are sensitive only to certain parts of medium, similar to special probes in electrical logging as micrologs. In order to overcome these difficulties let us consider an analytical method of determination of probe parameters. As is well known the quadrature component of the electromotive force arising at the receiver of a two-coil induction probe due to induced currents in a medium is: Q ^ - Q/i, • 4 = ^ ^ a

•4

(7.9)

where ^0 = -Tr~7T~^l^

^^

Q^ =

;;

y^a

(7.10)

Here cr^ = ^^=1 (^iGi and Gi and ai are the conductivity and the geometric factor of the i-th part of a medium. Suppose that the induction probe consists of s transmitter and t receiver coils. Then it can be presented as a system of st two-coil induction probes, and the quadrature component of the electromotive force induced in measuring coils of this probe can be written as: s

t

^ = EE^"/5 a^l 13=1

(7.11)

394

{a is transmitter coil, f3 is receiver one). In accord with eqs. 7.10 and 7.11 we have: 2 2 ^

*

(7.12) a = l 3=1

La^ ,=1

Thus, the procedure of choosing parameters of multi-coil induction probes allowing us to eliminate the influence of k arbitrary parts of the medium is reduced to a solution of a system of k equations: ±±±'^G.iL.,)

=0

(7.13)

Let us remember that unknown parameters of the multi-coil differential probe are lengths of the two-coil probes and moments of coils. It is obvious that the number of equations can be taken either equal to the unknown parameters or greater. In the latter the least squares method can be apphed if the system is linear or in a more general case gradient methods can be used. Function Gi{La(3) can be either the geometric factor or a very thin cylindrical shell (the differential response) or that of a relatively thin cylindrical layer. In such a case this function is expressed through the integral response of a two-coil induction probe G\{La(3). The analytical approach is also convenient to analyze radial responses of known differential probes, specially when the probe consists of only one transmitter coil, while others are receiver ones or vice versa. Unlike the graphical approach this method allows in principle to solve several problems such as: • Choice of optimal parameters of a differential probe which provides a certain depth of investigation or a sensitivity to specific parts of a medium as well as a maximal signal from these parts of the medium. • Development of interpretation of soundings based on use of two- or three-coil induction probes with different lengths. Let us notice that if Gi{Laf3) in eq. 7.13 represents the differential radial response, the equation can be rewritten in the form: ttt^^CM^O

(7.4)

where values of C{ai) are given in Table 7.1. In accord with the definition of the integral radial response its expression for multi-coil probes has the following form:

G = ^=^^=^

1.1. 1 3=1

(7.15) ^"/^

395 where Gi{La(3/p) is the geometric factor of the cyhnder with radius r for a two-coil induction probe with length L^/?. The denominator in eq. 7.15 is proportional to the electromotive force in a multi-coil probe caused by currents in a uniform medium. It is appropriate to notice that eq. 7.15 can be used for calculation of radial responses of differential probes in a medium which is not uniform with respect to magnetic permeability. Later in this chapter we will perform analyses of radial responses of several multi-coil probes which will illustrate the efficiency of this approach.

7.2. Physical Principles of Multi-coil Differential Probes Until now it was assumed that interaction between currents is absent, i.e. all currents induced in a conducting nonuniform medium, regardless of the distance from the source, are shifted in phase by 90°. For this reason electromotive forces induced in measuring coils of a probe are in phase with each other, and they are added and subtracted in the same way as scalars. If only one component of the electromotive force, for instance the quadrature component, is measured it is subjected to the same operations as scalars, regardless of whether currents are shifted in phase by 90°, or the internal skin effect manifests itself and due to it at every point of a medium there are both quadrature and inphase components of the induced current. However, in the latter the magnitude of current density does not depend on the primary magnetic flux only but also on the intensity of currents in the neighboring parts, i.e. in essence on the distribution of conductivity in a medium. Correspondingly, a value of geometric factor becomes different from that which follows from the theory of small parameters when the skin effect is neglected. For this reason focusing features of the probe, the parameters of which were calculated assuming the absence of the skin effect, can be seriously deteriorated under real conditions. The character of the influence of the skin effect essentially depends on frequency, distribution of conductivities, and length of probes. In order to take into account the influence of the skin effect on the radial responses of the probe it is necessary to solve the direct problem for a given distribution of conductivities. It is obvious that the practical value of such multi-coil probes, where parameters are chosen proceeding from knowledge of a geoelectrical section, is negligible. For this reason we can say that a necessary condition for the application of differential probes is the absence of interaction between currents in those parts of a medium the influence of which should be significantly reduced. In other words, those parts of the medium have to correspond to DolVs area where the current density is defined by the primary magnetic flux and the conductivity at a given point. It is natural that in choosing parameters of probes in order to increase the depth of investigation in the radial direction, it is important to eliminate the influence of parts of the medium located relatively close to the source. As calculations show, this condition usually takes place even for sufficiently high frequencies. However, absence of the skin effect in the area, the influence of which it is necessary to reduce, is not sufficient for the efficiency of multi-coil induction probes. Let us present a signal measured by the probe as a sum of two signals, namely, one which is caused by currents in the area where there is not practically interaction between currents and a

396 second one caused by currents in the external area, for example in a formation. As was shown in previous chapters with an increase of the distance from the source the skin effect is more strongly manifested, and in a general case the distribution of currents in the external area (the formation) depends on the magnitude of currents in the first area, which is closer to the probe. For this reason even in those cases when it is possible to eliminate the signal from currents in the parts of a medium, the influence of which has to be reduced, their effect can be noticed indirectly, changing the distribution of currents in the formation. In this case the signal from the formation is not only a function of its conductivity, but it depends also on the conductivity and dimensions of the internal area, in particular on the borehole and the invasion zone. Therefore, the second requirement, in order to provide an efficient working of the multicoil induction probe, is the absence of the influence of currents in the near area on the distribution of currents in the formation. In other words, the skin effect in the formation has to manifest itself in the same manner as in a uniform medium with the resistivity of the formation. Inasmuch as the relation between the signal and the conductivity of a uniform medium is known, the correction of function a a due to the skin effect is often performed directly during calibration of the probe either with the help of conducting rings, the parameters of which are calculated taking into account the skin effect, or in measuring in models of a medium when its resistivity corresponds to t h a t under real conditions. Thus, for efficiency of differential induction probes designed to measure a formation conductivity, two conditions have to be met, namely: • The range of a medium which includes the borehole and an invasion zone must correspond to DolVs area, i.e. currents induced in this area are shifted in phase by 90°, and they are defined by a change of the primary magnetic flux and the conductivity at a given point. • Outside of this range, for example in a formation, the skin effect has to manifest itself as in a uniform medium with the resistivity of the formation, i.e. interaction between currents in these two regions has to be negligible. Both conditions formulated above coincide with conditions of application of the approximate theory, described in Chapter 3, and therefore we can expect t h a t the focusing features of probes will be preserved even at higher frequencies t h a n as follows from Doll's theory. As will be shown later, analysis of radial responses of some multi-coil probes confirms this fact. Also, a comparison of results of calculations, making use of the exact and this approximate solution, allows us to establish maximal frequencies for a given distribution of conductivities when both of these conditions are met. Analysis of physical principles of differential probes permits us to discuss one aspect related to obtaining radial responses of these probes. As is well known, physical modeling or the use of conducting rings permits us to define a signal in a probe located on the cylinder axis as a function of its radius for a given frequency and conductivity of a medium. At the same time, a space surrounded by the cylinder has an infinitely high resistivity. Results of measurements are usually presented

397 in the following form: the ratio of the cylinder radius to the probe length is plotted along the abscissa, while the ratio of the electromotive force induced by currents in the cylinder with radius r to that caused by currents in uniform space with the same resistivity is plotted along the ordinate axis. It is natural that the following question arises whether this characteristic allows us to obtain information on the focusing features of the probe in a medium when the resistivity changes in a radial direction. It is obvious that until the signal from cylindrical conductor does not depend on currents induced in the external medium, regardless of their resistivity, this function can be used for evaluating the efficiency of a differential probe in a uniform as well as in a nonuniform medium. The lower the frequency and the higher the resistivity, the greater the cylinder radius for which the radial response describes with sufficient accuracy the behavior of the field in a nonuniform medium. However, if the frequency and conductivity of the cylinder are relatively great and the distribution of currents in the cylinder is subjected to the skin effect then such response does not reflect the actual behavior of the field under real conditions. Correspondingly we can say that the radial response obtained by physical or numerical modeling can be used for understanding the focusing features of the probe if the skin effect is absent within the cylinder, and we can neglect the interaction between currents induced in internal (cylinder) and external areas. As an example, direct integral radial responses of a two-coil induction probe are presented in Fig. 7.3. The curve index is the product a/jtuj. As is seen from this figure the higher the frequency and the lower the resistivity of a medium the stronger the influence of an area relatively closer located to the probe. It is easy to explain, inasmuch as with an increase of this product, that the influence of the skin effect becomes stronger, and correspondingly, that the signal caused by currents outside the cylinder increases relatively slower than that generated by currents inside of it. For this reason a deviation from the radial response calculated with the help of geometric factor is observed, regardless of the cylinder radius. This consideration shows that the application of direct radial responses obtained from either physical modeling or making use of the exact solution is hardly useful for the determination of parameters of differential probes.

7.3. Radial and Vertical Responses of the Differential Probe l.L-1.2 As is well known, coil induction probes used in induction logging have various arrangements of coils. It is appropriate to distinguish in every differential probe the basic (main) two-coil probe having a maximal product of transmitter (T) and receiver (R) coil moments. Other coils are considered to be focusing coils and, they form several additional coil probes which provide focusing features of the induction tool. From the point of view of the location of the focusing coils with respect to the center of the basic probe, the multi-coil probes can be divided in symmetrical and nonsymmetrical ones. In symmetrical probes focusing coils are located in such a way that for every pair: transmitter-receiver coils, displayed with respect to the center, there is another pair with

398

CO

0.01 rIL

Figure 7.3. Direct integral responses of a two-coil induction probe. Curve index a/j.uj; L=lm.

the same product of moments displayed at the same distance on the opposite side. Signals measured by symmetrical pairs have the same sign. It is also obvious that symmetrical probes have symmetrical profiling curves with respect to the formation center provided that the resistivity of the medium from both sides of the formation is the same. From the point of view of the position of focusing coils with respect to basic ones multi-coil probes are classified as probes with internal, external and mixed focusing. For internal focusing additional coils are placed between basic ones; for external focusing they are located outside the main probe, and finally for mixed focusing additional coils are placed inside as well as outside the basic probe. The configuration of symmetric diflPerential probes is shown in Fig. 7.4. In addition it is appropriate to notice that in symmetrical multi-coil probes moments of transmitter and receiver coils of corresponding pairs for example, basic probe, are the same, i.e. Mr = MR, MT^ = MRJ,. We will consider several diflPerential probes and will start with a symmetrical four-coil probe with additional internal coils. This type of probe is defined by three parameters, namely: the distance between basic coils, L, the ratio of the length oi focusing probe RTp to that of the basic probe, which is denoted by p, and the ratio of moments of focusing coils to the moment of basic ones (parameter c). Let us consider a symmetrical four-coil probe with internal focusing l.L-1.2 (Fig. 7.4a) (p = 0.4, c = 0.05, L = 1.2 m) Comparison with the results of calculation with coils having finite dimensions has shown that we can neglect their length and diameter in calculating radial and vertical responses.

399

R T Rp

RF

R

T

TF

TF

Figure 7.4. Symmetrical differential probes: (a) internal focusing; (c) mixed focusing.

(b) external

focusing;

For this reason a signal in symmetrical four-coil probe can be written in the form: S' = STR — 2STRF

IJ.'^UJ^MTMR

47rL

+

^TFRF

fv-^

r

, ,

2c^ ,

,

c^

^ ..

. 1 1

- {H^i l^ii^) - ^G,{pa) + YZ-^^^[(^ - 2^)^]] }

(7.16)

where L is the length of the basic probe, TR; pL is the length of probe, Rp T; and (1 — 2p)L is the length of probe, Rp Tp. In a uniform medium we have: ^^

II^U'^MTMR

a

1

2c p ^

c^ l-2p

inasmuch as geometric factors of the whole medium for a two-coil induction probe: Gi{a), Gi{ap) and Gi[{l — 2p)a] are equal to unity {a is the ratio of the length of the basic probe to the cylinder radius on the axis of which the probe is located). Therefore, the coefficient of the probe is: ATTL

(7.17)

K.= uj^fimTMn

1- - + . ' p 1

„ -2p

and the geometric factor of the whole space is equal to unity as is the case for two-coil probes. For this reason the expression for the apparent conductivity a^ has the form: (7.18) where

Gi{a) - -Giipa) G* =

P

+ rr^Gi[{l L — Zp

1

2c c' " p " ^ 1 - 2p

- 2p)a] (7.19)

400 This function is the geometric factor of the cyhnder for a four-coil induction probe with internal focusing and parameters p and c, i.e. it is the integral radial response. Let us notice t h a t normalization of geometric factor allows us to compare radial responses of various probes. Now we will consider the behavior of function G* as a function of the cylinder radius r ( a = L/r). If r ^ 0, (a —> oc) we have: G^{a)-^\

G,{pa)^-^

Gi[{l - 2p)a

^

Substituting these values into eq. 7.19 we have: ,

2c

c2

2c

c'

p ^

l-2p

^

- ^^^

^^-^^^

where ,

2c

c2

1-^ p« + ( l - 2 p ) 3 p

l-2p

Therefore, the smaller the value of coefficient Ki the lesser the value of the geometric factor of the area directly surrounding the probe. On the other hand, the electromotive force due to the primary field induced in receivers of the probe by currents in the transmitter coils is: _

^IWMTMR

1 - p3 ^ + (1 - 2p)3

(7.21)

Thus the rate of compensation of the electromotive force of the primary field defines a value of coefficient Ki. For a probe with parameters p = 0.4 and c = 0.05, coefficient Ki is 0.32. Introducing a fifth compensating coil, coefficient Ki becomes equal to zero, and correspondingly the radial response somewhat improves at its initial part. From this consideration follows t h a t with an increase of the length of two-coil induction probes, forming a differential four-coil probe, the cylinder radius, characterized by small values of geometric factor, increases, provided t h a t the primary electromotive force is compensated. In the opposite case, as r —> oo (a ^ 0) we have: G i ( r ) -^1-Ka

Gi{pa) = 1 -

and G i [ ( l - 2p)a] ^ l - K { l -

2p)a

Kpa

401 10

0.004 G;

/

0.1

/

0.002

/°

0.1 '

^-v,^

0.2

/

i0.002

0.01

0.3 r/L

1'

i

1-0 004 1-0.006 -0.008 -0.01

0.001

10 r/L

0.1

100

1000

Figure 7.5. The integral radial response of a four-coil probe with parameters p • 0.4 and c = 0.05.

where K c^ 0.586. Substituting these expressions into eq. 7.19 we obtain: l-2c

Gl-^l-K-

2c

~^~^

+ c^ __i

= 1-

KK2a

(7.22)

l-2p

where Ko =

1 - 2c + c^ p ~^

(7.23)

l-2p

Coefficient K2 exceeds unity, and therefore the radial response approaches its asymptote slower than that of a two-coil probe, i.e. the four-coil induction probe possesses a greater depth of investigations with respect to a two-coil induction probe of the same length. Figure 7.5 presents the integral radial response of a four-coil probe with parameters p = OA and c = 0.05. Unlike the radial response of a two-coil induction probe, in this case function G\(r/L) at the beginning has small but negative values, near r/L = 0.27 it changes sign and monotonically approaches unity.

402

Now let us consider the difTerential radial response of this probe. In accord with eq. 7.13 the electromotive force in receivers, caused by currents in a thin cylindrical shell, can be written in the form: ^ = ^ ^ ^ V ^ ^ l ^ ^ ( ^ ) _ ?£ ^^(^^) ^ _ ^

^^((^ _ 2p).]}

(7.24)

where Gr = (Ar/r)(l/a^)C(a). Values of C{a) are given in Table 7.1. We will present eq. 7.24 as

where Gr{a) - - Gripa) + - ^ G: = \ 2c ^ p ^

^ [ ( 1 - 2p)a] (^-2^)

l-2p

G* is the differential response of a four-coil induction probe and defines the ratio of signals from a thin cylindrical shell to that from a uniform medium. From eq. 7.25 we have:

2c

C^

oP' r

^ ~ 7 "^ 1 - 2p Assuming that a medium is presented as a system of shells with the same thickness Ar it is convenient to write down G* as:

^^C(.)-g^C(.a).^q(l-2,H

As has been shown, if o; —> 0 then C(a) -^ K^, and for a —^ CXD C{a) —> 2. For this reason, if r ^ 0 (a ^ CXD), we have: G* ^

2Ar 1 ^ K LQ a

i.e. again compensation of the electromotive force of the primary field provides minimum of geometric factor of cylindrical layers located close to the probe.

403

0.06Ar//. = 0.1

0.040.02-

0 V

1

1

1

1

1

1

2

3

4

5

6

rIL

Figure 7.6. Differential response of a four-coil probe with parameters p = 0.4 and c = 0.05.

In the opposite case, when r ^ oc (a —^ 0) we have: Gl -^ ^KK^a^

(7.28)

A differential response of a four-coil probe with p == 0.4 and c — 0.05 is presented in Fig. 7.6, provided that the shell thickness is constant. In accord with eq. 7.18 the expression for the apparent conductivity in a three-layered medium (borehole, invasion zone, formation) has the form: ^a --=

aiGl H- cr2Gl+

(7.29)

C7SG;

where: G,{a) -

Gl

2c V

Giipa) + —

- 2p)a]

C2

P G2{a) G*2

2c P

1 -2p

G2ipa) + Y:- ^

^^t^^-

2p)a]

C2

p

G,{a) G*s

^='«'

2c P

1 -2p

Gaipa) + —^ ' ' • K ' - 2p)a] c2

1 - - +1 -2p P

It is obvious that:

G* + G; + G* = 1

1

404

TABLE 7.2 Values of function a,2/cri ^ " ^ ^ ^ - ^ ^ ^ ^ 0-2/0-1

Probes

^^^^-^^^

Two-coil probe Four-coil probe l.L-1.2

0.500

0.250

0.125

0.0625

0.0313

0.0156

0.505 0.498

0.258 0.246

0.133 0.121

0.0724

0.0414

0.0580

0.0268

0.0259 0.0111

and

here /3 is the ratio of radius of the invasion zone to that of the borehole {P = a2/ai). Proceeding from equations derived above we will consider a behavior of the apparent conductivity in a medium with cylindrical interfaces. First, let us assume that the invasion zone is absent. It is simple to show that for the four-coil probe with internal focusing and L — 1.2 m we have: Gl 2=^ -0.0045

Gl = 1.0045

Table 7.2 contains values of (Ja/ai for both two- and four-coil induction probes. For small values of P2/P1 (P2/P1 < 1) the difference between the radial responses of differential and two-coil probes is insignificant. If the conductivity of the borehole does not exceed more than 30 times the formation conductivity, the probe l.L-1.2 ehminates the influence of the borehole almost completely. Focusing features of this probe in the presence of the invasion zone are illustrated by values of apparent conductivity, CTa/cri, given in Table 7.3. Regardless of the resistivity of the invasion zone (4 ^ P2/P1 ^ 64), if Ps/pi ^ 30 the influence of induced currents in the borehole and the invasion zone on the signal, measured by the probe, is small (Table 7.3). The influence of the invasion zone on the two-coil probe with length L = 1.2 m is also insignificant if ps/pi ^ 10. For this reason when penetration of the borehole filtrate is not deep and ps/pi is relatively small the value of apparent conductivity, aa, measured by a two-coil induction probe L = 1.2 m, is close to the formation conductivity, and correspondingly in such conditions the role of focusing features of the probe, regardless of its type, is insignificant. With an increase of the radius of the invasion zone focusing features of the probe manifest themselves stronger. Practically, values of cr^ are close to as if the ratio of conductivities (as/ai) is not less than 1/20. At the same time the character of the penetration P3/P2 > 1 or P3/P2 < 1 does not have a strong effect on the value of the apparent conductivity, aaFor deep penetration of the borehole filtrate {a2/ai ^ 8) the focusing features of this probe do not allow us to obtain the formation conductivity even in such cases when the borehole and the invasion zone have greater resistivity than the formation. It is explained by the fact that the geometric factor of the formation is about 0.7 (a2/ai = 8) and for this reason, even if the conductivity of the borehole and the invasion zone is equal to zero, the apparent conductivity differs from that of the formation by 30%.

405

TABLE 7.3 Values of function a 2/cri Probe type

^2/cri

(^3/^1 1

0.500

0.250

0.125

0.0625

0.0313

0.0156

a2/ai = 2 2-coil 4-coil

1/4 1/4

0.975 1.003

0.500 0.499

0.255 0.247

0.140 0.121

0.0800 0.0575

0.0500 0.0260

0.0330 0.0102

2-coil 4-coil

1/8 1/8

0.980 1.004

0.500 0.499

0.250 0.247

0.132 0.121

0.0750 0.0580

0.0450 0.0265

0.0300 0.0107

2-coil 4-coil

1/16 1/16

0.980 1.004

0.500 0.500

0.250 0.247

0.130 0.121

0.0720 0.583

0.0430 0.0268

0.0280 0.0110

2-coil 4-coil

1/32 1/32

1.000 1.004

0.500 0.500

0.250 0.248

0.129 0.122

0.0710 0.584

0.0420 0.0269

0.0265 0.0111

2-coil 4-coil

1/64 1/64

1.000 1.004

0.500 0.500

0.250 0.248

0.129 0.122

0.0710 0.585

0.0420 0.0270

0.0260 0.0112

Probe type

CT2/(Tl

1.00

0.500

0.250

0.125

0.625

0.3130

0.0156

2-coil 4-coil

1/4

0.900 0.955

0.480 0.483

0.250 0.248

0.150 0.128

0.1000 0.6960

0.0750 0.0401

0.0640 0.0253

2-coil 4-coil

1/8

0.986 0.947

0.475 0.476

0.240 0.239

0.135 0.121

0.0830 0.0621

0.0570 0.0326

0.0430

1/8

2-coil 4-coil

1/32 1/32

0.986 0.943

0.440 0.471

0.230 0.235

0.125 0.117

0.0730 0.0583

0.0485 0.0288

0.0330 0.0140

2-coil 4-coil

1/64 1/64

0.986 0.940

0.440 0.468

0.220 0.232

0.120 0.114

0.0650 0.0554

0.4000 0.0259

0.0250 0.0112

Probe type

CT2lcri

1.000

0.500

0.250

0.125

0.0625

0.3130

0.0156

2-coil

1/4

0.405

0.260

0.172

0.1500 0.1150

0.1280 0.0931

0.120 0.0283

02/^1 = 4 1/4

0.0178

02/^1 = 8 4-coil

1/4

0.700 0.764

0.418

0.248

0.158

2-coil 4-coil

1/8 1/8

0.650 0.726

0.360 0.380

0.205 0.207

0.135 0.120

0.0750 0.0767

0.0800 0.0551

0.0650 0.0443

2-coil 4-coil

1/16 1/16

0.620 0.707

0.318 0.361

0.180 0.188

0.110 0.101

0.0720 0.0580

0.0530 0.0361

0.0450 0.0253

2-coil 4-coil

1/32 1/32

0.600 0.697

0.320 0.351

0.170 0.178

0.0960 0.0915

0.0600 0.0482

0.0420 0.0266

0.0322 0.0158

2-coil 4-coil

1/64 1/64

0.600 0.692

0.320 0.346

0.168 0.173

0.0920 0.0868

0.0550 0.0435

0.0360 0.0219

0.0250 0.0111

406

Until now it has been assumed that at all points of a medium induced currents are shifted in phase by 90°, i.e. the skin effect is absent. Now we will investigate radial responses of the probe, making use of results of the exact solutions in a medium with two cylindrical interfaces when the four-coil induction probe is located on the borehole axis. The electromotive force in receivers of the four-coil symmetrical probe with internal focusing can be presented as:

On the other hand: ^ = ^ohz, here hz is the quadrature component of the magnetic field expressed in units of the primary field and obtained from the exact solution. Therefore: '(2)

(1) (On

S

/i(l)_2/i(2)!fO_

z.(3) •

,

^ ( 3 ) ^

0O

(Of)

where: (2)

(JJ/IMTMRF

)

(1)

I

(JJ^IMTMR

L^

^TRF

(7.30) (3)

UJIIMTF^RF

) (1)

I

^IIMTMYI L3

^TFRF

(l-2p)3

The electromotive force measured in receiver coils of the probe referred to that in a free space for two-coil basic probe is defined as:

/,, = 4 , = h^) - ?£/^(^) +

''

(1 - 2p)

(1)

Sn

/il^'

and correspondingly the expression for the apparent conductivity, aa, is: 1)

1 1

2c

C

.(3)

P

CTl

1

2p ax

(7.31)

2p

where Ga /CTI, CTQ, /cf\, cri /<7i are functions corresponding to the basic two-coil probe (T, R), the differential one (T, Rp) and another differential probe (T/r, RF), respectively. It is appropriate to consider simultaneously field, electromotive force and apparent conductivity, calculated from the approximate theory assuming that in the borehole and the invasion zone, regardless of their conductivity and dimensions, currents are shifted in phase by 90° but in the formation the skin effect manifests itself as in a uniform medium with the resistivity of the formation. Then, according to results derived in Chapter 3 we have: CJa = (^1 - CTs)Gl + (^2 - CTS)G; +

af

407

or

^=(,_^)GI+(^)G;

+ ^ ^

(7.32)

Results of calculation of function Oajoi by exact equation, Oal^x-, and the approximate formula 7.31, CF^/CFI, are given in Table 7.4. For illustration, values of cr^^/as are presented in Table 7.5 (A^3 = a^iiijja\). Comparison with results of calculations for very small parameters shows t h a t the influence of the skin effect in a relatively conductive medium can significantly change the value of CTa/ai. For instance, for P2/P1 = 32, ps/pi = 16 and a 2 / a i = 4, the value of (Ta/cJi calculated by Doll's formula is 0.0564. If the frequency of the field is 60 kHz and pa = 1.1 ohm-m, the factual value Oajox is 0.0298, i.e. it is almost two times smaller. A decrease of the quadrature component of the field and correspondingly the apparent conductivity is mainly the result of the skin effect in the formation for geoelectric parameters considered here. It follows from the coincidence of the calculated results based on exact and approximate solutions, because one of the main assumptions of the latter is t h a t the skin effect is absent within the borehole and the intermediate zone. In this case the skin effect does not change the focusing features of the probe since the change of apparent conductivity, a^, is the same as in a uniform medium with the resistivity of the formation. Numerical d a t a show t h a t within the borehole and the invasion zone conditions of small parameters are preserved for a sufficiently large range of resistivities and dimensions. This fact allows us to choose properly a frequency for a given probe. As is well known, with an increase of frequency the vertical response of the probe becomes better, and we can measure higher resistivities of a formation. However, if the frequency is chosen too high at least two problems arise, namely nonuniqueness in determination of resistivity by measuring the quadrature component of the field, and the radial response can be significantly worse t h a n t h a t calculated with the assumption t h a t the skin effect is negligible. Comparison of calculated results, based on exact and approximate solutions, defines the maximal frequency for which the skin effect is still absent in the borehole and the invasion zone but in the formation it manifests itself in the same manner as in a uniform medium with the same resistivity. We can think t h a t the maximal frequency for this probe, derived from this comparison and taking into account its radial response, is defined from the relation: / ^ (2.0 - 2.2)p^inl0^ Hz

(7.33)

For example, if the minimal resistivity of a medium is about 1 ohm-m, the frequency can be increased up to 200-220 kHz. This analysis of the focusing features of the probe l.L-1.2 with p — 0.4 and c = 0.05 in media with cylindrical interfaces allows us to establish the range of frequencies as well as parameters of borehole and invasion zone when induced currents within t h e m do not have an influence on the measured signal. If the resistivity of the invasion zone exceeds t h a t of the formation, P2 > Ps, and a2/ai ^ 4, the apparent conductivity practically coincides

408 TABLE 7.4 Values of functions (Ja/cri and o^jox asfiujal X 10^

Ps/pi =: 1 ^a/^l

Ps/Pi == 4


(y a 1(^1 aijax

1 2 4 8 16 32 64



0-a/(Ti


0.233

0.233

0.0545

0.0546

0.0247

0.0248

0.924

0.228

0.227

0.0532

0.0533

0.0240

0.0240

0.892

0.894

0.219

0.220

0.0510

0.0510

0.0229

0.0230

0.848

0.849

0.208

0.208

0.0483

0.0484

0.0215

0.0216

0.785

0.785

0.192

0.193

0.0441

0.0442

0.0194

0.0196

0.700

0.698

0.171

0.171

0.0383

0.0385

0.0164

0.0169

0.585

0.583

0.141

0.142

0.0304

0.0307

0.0119

0.0133

P2/P1 == 16

0.950

0.950

0.233

0.234

0.0549

0.0550

0.0250

0.0250

0.924

0.925

0.227

0.228

0.0533

0.0534

0.0243

0.0244

0.892

0.893

0.220

0.221

0.0514

0.0515

0.0233

0.0234

0.847

0.848

0.208

0.209

0.0483

0.0484

0.0218

0.0219

0.786

0.788

0.192

0.193

0.0445

0.0446

0.0197

0.0200

0.700

0.701

0.171

0.172

0.0387

0.0388

0.0167

0.0168

0.585

0.586

0.142

0.143

0.0309

0.0310

0.0125

0.0126

0.0550

0.0552

0.0251

0.0252

0.0536

0.0536

0.0244

0.0244 0.0235

0.947 0.924

0.947 0.924

0.234

-32 /92/pi -0.235

0.228

0.229

0.893

0.894

0.220

0.221

0.0515

0.0516

0.0234

0.847

0.848

0.209

0.210

0.0485

0.0486

0.0219

0.0220

0.0446

0.0446

0.0199

0.0200

0.785

0.785

0.193

0.194

0.700

0.702

0.171

0.172

0.0388

0.0389

0.0169

0.0170

0.585

0.142

0.143

0.0312

0.0313

0.0127

0.0130

0.583 0.947 0.925 0.892 0.847 0.785 0.700 0.583 0.890 0.865 0.834 0.788 0.728 0.643 0.538

/92/pi •-= 64

0.234

0.0550

0.0551

0.0252

0.0253

0.229

0.0537

0.0537

0.0245

0.0245

0.220

0.0514

0.0515

0.0234

0.0235

0.209

0.210

0.0485

0.0486

0.0220

0.0221

0.193

0.194

0.0446

0.0447

0.0199

0.0201

0.698

0.171

0.172

0.0390

0.0394

0.0170

0.0174

0.583

0.142

0.143

0.0313

0.0318

0.0128

0.0138

0.0590

0.0311

0.0310

0.0572

0.0574

0.0303

0.0302

0.213

0.0552

0.0555

0.0292

0.0292

0.202

0.0522

0.0524

0.0277

0.0278

0.184

0.185

0.0482

0.0483

0.0256

0.0258

0.163

0.164

0.0422

0.0429

0.0219

0.0231

0.135

0.137

0.0340

0.0358

0.0164

0.0195

0.949 0.924 0.894 0.849 0.785

0.234 0.228 0.219

a2/ai -•=

1 2 4 8 16 32 64

32

0.949

^2/^1 == 2

1 2 4 8 16 32 64

P3/P1 =

0.940 0.924

a2/ai -= 2

1 2 4 8 16 32 64

^a/^1

16

^"8 = 2 P2/P1 =

a2/ai =- 2

1 2 4 8 16 32 64

PZIP\ =

0.891 0.866 0.836 0.791 0.727 0.640 0.535

0.225 0.219 0.211 0.200

4

p2lp\ = 8 0.226 0.220

0.0589

409

TABLE 7.4 (Continued) as/jLual

X 10^

Ps/Pi =-- 1 CTa/cri

CTa/f^l


32


(^a/cri


0.886

0.887

0.221

0.223

0.0548

0.0549

0.0272

0.0273

0.860

0.860

0.215

0.217

0.0525

0.0525

0.0264

0.0265

0.830

0.831

0.207

0.210

0.0514

0.0515

0.0253

0.0255

0.785

0.787

0.196

0.198

0.0481

0.0483

0.0239

0.0240

0.722

0.725

0.180

0.182

0.0445

0.0446

0.0217

0.0220

0.640

0.643

0.158

0.160

0.0388

0.0390

0.0187

0.0193

0.532

0.534

0.131

0.132

0.0309

0.0312

0.0141

0.0157

P2/P1- 3 2

0.882

0.882

0.219

0.220

0.0524

0.0526

0.0251

0.0252

0.860

0.862

0.213

0.215

0.0510

0.0510

0.0244

0.0246

0.827

0.205

0.200

0.0493

0.0496

0.0234

0.782

0.829 0.784

0.194

0.195

0.0465

0.0466

0.0219

0.0236 0.0221

0.722

0.725

0.179

0.180

0.0425

0.0427

0.0198

0.0200

0.635

0.637

0.157

0.158

0.0371

0.0375

0.0169

0.0173

0.528

0.520

0.130

0.132

0.0294

0.0298

0.0127

0.0137

TABLE 7.5 Values of function a"^ jo-^', a i^3 X 10^ Ps/pi

PZIPI =

P2/P1= 16

4

•• 4

1 2 4 8 16 32 64

CTa/o^l


a^jax =-1 2 4 8 16 32 64

Ps/pi = 16

Ps/pi == 4

10,L = 1 m

1

16

32

64

128

1

0.945

0.921

0.891

0.844

0.781

0.695

0.576

0.432

2

0.473

0.461

0.446

0.422

0.391

0.348

0.288

0.216

4

0.236

0.230

0.233

0.211

0.195

0.174

0.144

0.108

8

0.118

0.115

0.111

0.106

0.976

0.869

0.0720

0.054

16

0.0590

0.0575

0.0557

0.0528

0.0488

0.0434

0.0360

0.027

32

0.0295

0.0287

0.0278

0.0264

0.0244

0.0217

0.0180

0.013

64

0.0148

0.0143

0.0139

0.0132

0.0122

0.0108

0.0090

0.006

with that of the formation provided that pa/pi < 20. However, if the resistivity of the invasion zone becomes smaller than pa the depth of investigation of this probe decreases. Thus, for certain conditions the value of the apparent conductivity, cr^, in a formation with a very large thickness coincides with the apparent conductivity in a uniform medium having the formation conductivity. However, in more complicated cases, when measured electromotive forces are subjected to the influence of parameters of the borehole and the invasion zone, interpretation cannot be performed without additional information. Now let us consider vertical responses of the four-coil induction probe l.L-1.2. Calcula-

410 tions of apparent conductivity, aa, in formations with finite thickness have been performed proceeding from equation: ^a

-3

1 2c i_ ' p ^

c2

02

P (72

1 - 2p 0-2

l-2p

where a i 70-2 and a i 7 ^ 2 are values of Oalo2 for two-coil probes located symmetrically with respect to the formation boundaries while a a I02 is the value of Gal(y2 for the probe displaced with respect to the formation center. Curves of apparent conductivity Oajcf^ for the probe l.L-1.2 are presented in Figs. 7.77.10. Values of a^l^ujl? and ratio cra/^2 are plotted along axes of abscissa and ordinate, respectively. The curve index is Oxjo^ [ox and a^ are conductivities of the formation and the surrounding medium). Every set of curves is characterized by the ratio / / / L , here L is the length of the basic-two coil probe. For given frequency and probe length the abscissa represents the conductivity of the surrounding medium. For example, i f L = 1.2 m, f = 6 x 10^ Hz, then a2/xa;L^ c^ 0.7cr2 or ^2 = 1.44( p2, and its resistivity does not exceed 10 ohm-m the influence of the surrounding medium becomes negligible for the given probe when the formation thickness exceeds 3.0 3.5 times the probe length. In cases when the difference of the conductivities is relatively small, equality cr^ = cr^^ takes place even for sufficiently thin layers. W i t h an increase of the formation resistivity the influence of currents in the surrounding medium becomes more significant. A comparison with corresponding curves for a twocoil induction probe shows t h a t the influence of the surrounding medium on the four-coil induction probe is somewhat greater t h a n that on a two-coil probe of the same length when the thickness of the formation is equal to or greater than the probe length. However this difference does not exceed 10-15%. Here it is appropriate to make the following comment. Results of calculations, presented in Figs. 7.7-7.10, have been performed in a medium with only horizontal interfaces. Certainly, such an assumption would lead to significant errors if a two-coil induction probe

CM

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Jim oc

1

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o

^Dl'^D

'^Dl'^D

7

7 o

7o o

CM

O

.2

00

bJO

X

.2 "-4-3

x>bJO

411

412

o

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0

0

00 II

5

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5

4

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CN

^"

o

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1

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go

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CM CO

-^ CO

00 CM :^

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b"

b"

S ^

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^Dl^n

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B

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X

•^^l

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5-1

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413

TABLE 7.6 Values of function Oajoi', P2 = ^ ohm-m ^l/cr2

H/L 1

V2 2 2\/2 4

1

1/2

0.56 0.56 0.56 0.56 0.56 0.56 0.56 0.56 0.56

4^2 8 8^/2 16

TABLE 7.7 Values of function cri/cr2

(^a/(^2\ P2

1/4

1/8

0.42

0.34

0.38 0.35 0.34 0.34 0.34 0.34 0.34 0.34

0.28 0.23 0.20 0.18 0.18 0.18 0.18 0.18

031 0.22 0.16 0.12 0.110 0.105 0.105 0.105 0.105

1/16

1/32

1/128

-

-

0.21 0.13 0.085 0.065 0.056 0.054 0.053 0.053

0.20 0.12

0.270 0.170 0.094

0.070 0.042 0.032 0.029 0.028 0.028

0.049 0.026

-

= 2 ohm-m

2

1

1/2

1/4

1/8

1/16

16

1.10

0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68

0.51 0.45 0.41

8 8^/2

099 1.10 1.10 1.10 1.10 1.10 1.10 1.10

0.40 0.39 0.39 0.39 0.39 0.39

0.42 0.33 0.27 0.23 0.22 0.21 0.21 0.21 0.21

037 0.27 0.19 0.15 0.13 0.12 0.11 0.11 0.11

0.25 0.17 0.11 0.080 0.065 0.058 0.057 0.057

H/L 1

72 2

2V2 4 Ay/2

TABLE 7.8 Values of function
H/L 1 s/2 2 2V^ 4 4v/2 8 8^2 16

(^alcf2, P2

-

= 4 ohm-m

4

2

1

1/2

1/4

1/8

L80 2.10 2.20 2.30

1.18 1.25 1.35 1.40 1.40 1.40 1.40 1.40 1.40

0.77 0.77 0.77 0.77 0.77 0.77 0.77 0.77 0.77

058 0.50 0.46 0.44

048 0.31 0.31 0.27 0.24 0.22 0.22 0.22 0.22

0.420 0.220 0.230 0.180 0.145 0.125 0.120 0.115 0.115

2.25 2.20 2.20 2.20 2.20

0.42 0.41 0.41 0.41 0.41

414 TABLE 7.9 Values of function (Jalo2\ p2 = 8 ohm-m

1

72 2 2\/2 4

4v^ 8 8\/2 16

16

8

4

2

1

1/2

3.40 4.10 4.50 4.50 4.50 4.50 4.50 4.50 4.50

2.00 2.40 2.60 2.80 2.80 2.80 2.80 2.80 2.80

1.25 1.40 1.50 1.50 1.55 1.55 1.55 1.55 1.55

0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85

0.63 0.56 0.51 0.48 0.46 0.46 0.45 0.44 0.44

0.55 0.42 0.34 0.30 0.27 0.24 0.23 0.23 0.22

1/2

1/4 0.68 0.60 0.55 0.50 0.48 0.47 0.46 0.46 0.46

TABLE 7.10 Values of function aa/02', P2 = 16 ohm-m cri/a2

H/L 1

V2 2 2^2 4 4v/2 8 8v/2 16

6.6 8.2 9.0 9.0 9.0

3.8 4.6 5.2 5.5 5.5

2.3 2.6 2.9 3.0 3.0

1.3 1.5 1.6 1.6 1.7

0.90 0.90 0.90 0.90 0.90

9.0 9.0 9.0 9.0

5.5 5.5 5.5 5.5

3.0 3.0 3.0 3.0

1.7 1.7 1.7 1.7

0.90 0.90 0.90 0.90

is considered. However, for a differential probe the effect, caused by the influence of the borehole is usually very small. In fact, it is known t h a t the probe, within a certain range of change of pi and p2, eliminates the influence of the borehole and the invasion zone. Analysis of geometric factors of cylinders of finite thickness shows t h a t as soon as the height of the cyhnder exceeds the probe length its geometric factor is almost equal to t h a t of an infinitely long cyhnder, provided t h a t the probe and the cyhnder are symmetrically located. For this reason if the formation thickness is equal to or exceeds the probe length then the electromotive force is not subjected to the influence of t h a t part of the borehole which is located against the surrounding medium. Moreover, the geometric factor of the part of the borehole located against the formation practically coincides with the geometric factor of the borehole, and due to focusing this part of the borehole as well as the rest of it do not affect the signal measured by the probe. Analogous behavior is observed for the invasion zone. W i t h an increase of the formation thickness and resistivity of the surrounding medium errors in the determination of the apparent conductivity, a^, decrease.

415 TABLE 7.11 The position and parameters of coils Two-coil probes

Length, m

T-R

1.00 0.75 0.75 0.50 0.50

1.0000 0.2900 0.2900 0.0841 0.0200

0.50 2.00 0.75 0.75

0.0200 0.0004 0.0058 0.0058

MiMj/MrMR

Sign of Signal

Probe 6F1M

T F I - R T - R F I TFI

- Rpi

TF2

- R

T-RF2

- RF2 TFI - RF2 TF2 - RFI TF2

— + + + H-

Probe 4F1 T-R TF2-R TF3

- R

1.00 0.586 0.320

1.000 0.350 0.025

+ +

1.000 0.350 0.025

+ +

Probe 4F1.1 T-R ~ R TF3 - R

TF2

1.100 0.586 0.352

7.4. Radial and Vertical Responses of Probes 6 F 1 M , 4 F 1 and 4F1.1 The probe 6F1M has six coils with symmetrical internal and external differential probes (mixed focusing) based on the use of frequency 50 kHz. Probe 4F1 is a four-coil nonsymmetrical system with an internal differential probe and frequency of the current is 70 kHz. Finally, probe 4F1.1 is a four-coil nonsymmetrical probe with internal focusing and frequency 1 MHz. Table 7.11 describes the position and parameters of coils of these probes. Let us notice that all considered probes are systems where the electromotive force caused by currents in the transmitter coils, i.e. the primary electromotive force is compensated. In other words, the moments of coils satisfy the condition:

This fact turns out to be very essential in the investigation of the radial responses of multi-coil probes located on the borehole axis. Calibration curves of probes 6F1M, 4F1 and 4F1.1 are presented in Figs. 7.11-7.13. Ratio of Q S jS^"^^ and resistivity of the medium are plotted along axes of ordinate and abscissa, respectively. Here S^ — UOIIMTMR/2TTL^ is the primary electromotive force

416 of the basic probe. For illustration let us assume that the minimal value of the ratio Q^/4^^^ measured is 5 x 10^ Then the range of resistivities defined by these probes is: 6F1M 4F1 4F1.1

0.4-12 ohm-m 0.5-25 ohm-m 10-400 ohm-m

Due to the relatively high frequency applied in probe 4F1.1 the range of resistivities is shifted to larger values than those for probes 6F1M and 4F1. It is appropriate to notice that with an increase of the degree of compensation of the primary electromotive force the upper boundary of resistivities is shifted to larger values. In accord with the calibration curve for the probe 4F1.1 we have: pmax — 400 ohm-m. However, the determination of function p of such resistive layers can be complicated due to the radial response of the probe, if the borehole resistivity is sufficiently small (pi 2:^ 1 ohm-m). If the electromotive force is measured with an accuracy of about 5%, maximal errors in determination of resistivity near the low boundary of the range do not exceed 10-15% for these probes. Calibration curves for basic two-coil induction probes are also shown in Figs. 7.11-7.13. From comparison with calibration curves of corresponding differential probes it follows that a decrease of the signal with respect to that of the basic probe within the range of measured resistivities in average constitutes: 3.1-3.4 times for 6F1M 2.2-3.0 times for 4F1 2.3-2.8 times for 4F1.1 Radial responses of these probes are shown in Figs. 7.14-7.16. The initial part of these responses are presented on a larger scale. The cylinder diameter is plotted along the axis of the abscissa. Comparing the integral radial responses we can see that in a two-layered medium when the borehole radius changes from 0.1 to 0.15 m probe 6F1M provides more accurate values of the formation over a wider range of (J2/CF1 than probe 4F1. However, in the presence of an invasion zone (0.4 0.8 m) we can expect that the measurements by probes 4F1 and 6F1M are closer to each other. Concerning probe 4F1.1, we can notice the following: in sections where the geoelectric parameters correspond to the range of resistivities for probes 6F1M and 4F1, results of measuring with probe 4F1.1 are almost the same since their radial responses practically coincide. For a more accurate evaluation of parameters of a medium, when probes 6F1M, 4F1 and 4F1.1 allow us to determine the formation resistivity, it is necessary to calculate the apparent conductivity for these probes in media with cylindrical interfaces. Results of such analysis are described below. Let us notice that calculations have been performed proceeding from the approximate theory which takes into account the skin effect in the external area. First, consider the influence of the borehole (the invasion zone is absent) on the apparent conductivity, Ga. Curves of (Jal(J2 as a function of the borehole resistivity, pi are shown in Figs. 7.17-7.22. The curve index is formation resistivity, p2- For every probe there are two groups of curves, corresponding to different values of the borehole radius, ai = 0.10 m and ai — 0.15 m. An analysis of these curves allows us to make the following conclusions:

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T^ y

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/

i

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iiy\^I^K^

in

CM/

1

CM/

(0'',^/<^D

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in

in

CM

CM

lO

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o

0)

I

o

o

i

o

o

03

o

bJO.

417

418

\^^^

\7

2 10"

X

10>-2

10 2 p, ohmm

Figure 7.13. Calibration curves: (1) two-coil probe, L — 1.1 m; (2) probe 4F1.1.

• For a given value of the borehole resistivity the difference between Oa and (72 increases also with an increase of formation resistivity. It is explained by the fact that with a decrease of G2 currents induced in a formation decrease, and the contribution of the electromotive force caused by the magnetic field of currents within the borehole becomes more essential. Let us notice that radial responses of probes within considered borehole radii are not equal to zero, and therefore the influence of the borehole cannot generally be ignored. • W i t h a decrease of the borehole resistivity for a given value of the formation resistivity, the deviation of the apparent conductivity, Ga, from G2 becomes stronger since the density of currents induced in the borehole increases. • W i t h an increase of the borehole radius the difference between Oa and GI increases also, inasmuch as radial responses of all considered probes are worse for a == 0.15 m t h a n for the case when the borehole radius is equal to 0.1 m. If a = 0.1 m the difference between aa and G2 does not exceed 5% for the probe 6F1M when the borehole resistivity changes from 0.05 to 1 ohm-m. It permits us to perform measurements with probe 6F1M in boreholes with strongly minerahzed solutions. For

419

420

Figure 7.16. Radial response of probe 4F1.1.

example, if pi = 1 ohm-m and a^ = 0 . 1 m, the ratio aa/cr2 is equal to unity with an accuracy of 1% as 0.5 < p2 < 50 ohm-m. The probe 4F1 is more sensitive to currents induced in a relatively conductive borehole. However, for pi = I ohm-m and ai = 0.1 m the apparent conductivity aa with an accuracy of 5% coincides with cr2 within the whole range of formation resistivities measured by the probe. Curves of the aa/cr2 for the probe 4F1.1 given in Figs. 7.21-7.22 correspond to larger values of the borehole resistivity with respect to those for probes 6F1M and 4F1. It is related to the fact that, due to the high frequency used in this probe, the condition of focusing {ai/hi < 0.3) is not vahd anymore in boreholes with high mineralization. For instance, if borehole radii are 0.1 m and 0.15 m we have for the minimal resistivity, pi satisfying this inequahty, 0.5 ohm-m and 1 ohm-m, respectively. If pi = 1 ohm-m and ai = 0.1 m the difference between a'^{02 and cra/a2 does not exceed 2% provided that 0.6 < P2< 100 ohm-m. Now let us consider the influence of parameters of a three-layered medium (an invasion zone is present) on the apparent conductivity measured by probes 6F1M, 4F1 and 4F1.1. Values of (Ja/(^^ for various parameters of a three-layered medium along with data for a two-coil induction probe are given in Tables 7.12-7.22. The borehole resistivity is assumed to be 0.5 ohm-m. The behavior of (Ja/cT^ depends to a certain extent on the sign of geometric factor of G\{r) for r = ai and r — a2. Let us introduce notations for specific points of the radial responses; namely Tmin is the

421

1.0

0.95

0.9

10.4^-1.6 3.2 6.4 12.8 25.6

^

^

51.2 Gf-i = 0.2 m

10.-1

10^

p^

Figure 7.17. Apparent conductivity curves for probe 6F1M (/ = 50 kHz). Curve index P2.

1.0

:^aa——^^^g

0.2^0.4 0.8 1.6

0.95 3.2

b"

0.9 6.4

^ 0.85 C/i =

0.8

12.8 1

25.6

10"^

0.3 m

51.2

2

10"

2

Pi

Figure 7.18. Apparent conductivity curves for probe 6F1M (/ = 50 kHz). Curve index P2.

422

10

^^^

0.2 0.8 1.6 3.2

•_

6.4

b^

12.8 Cf-i =

/25.6

10"

3.2 m

/51.2

10"

10"

2

p^

Figure 7.19. Apparent conductivity curves for probe 4F1 (/ = 70 kHz). Curve index p2-

di = 0.3 m

1.0 ^

b"

0.8

^1

V^ A 5

lO"'' 2

6.4/

2

/25.6

/l2.8

5

/5I.2

10°

2

Pi

Figure 7.20. Apparent conductivity curves for probe 4F1 (/ = 70 kHz). Curve index p2-

423

1.5 d,= D.2m

1.0

1.6 3.2 25.6 51.2

b" ^ 0.5

102.4

10"

P^

Figure 7.21. Apparent conductivity curves for probe 4F1.1 (/ = 10^ kHz). Curve index P2.

.6.4 .^-"^^^

12.8

^^^^^

25.6

0.95

b"

1

• 51.2

0.9

• 102.4

0.85

0.8 / 0.5

/

LI

1.0

d-i = 0.3 m

2.0

P\

Figure 7.22. Apparent conductivity curves for probe 4F1.1 (/ = 10^ kHz). Curve index P2-

424 point where Gl{r) reaches a minimum; TQ is the point when Gl{r) is equal to zero. Values of Tmin and To for probes considered here are: Probes

Rmin, m

6F1M

022

ro, m 0.28

4F1

0.28

0.39

4F1.1

0.31

0.43

We will notice the following features in the behavior of apparent conductivity for various parameters of a medium. Case 1 Consider the case when ps/pi > 1. An increase of the radius of the invasion zone results in an increase of (Ta/(Ji if a2 < Vmin^ and in a decrease of this function if (I2 >• TYuinFor example, for probe 6F1M when P2 = 16 ohm-m, ps = S ohm-m and ai = 0 . 1 m. a2, ni (Ta/(T3

0.28 0.998

0.40 0.986

0.56 0.948

0.80 0.977

An increase of resistivity p2 leads to an increase of (Ja/crs, if 0^2 < '^o, and to a decrease of aa/os when 02 > TQ. For instance, for probe 4F1 we have following values of (Ja/cTz (p3 = 1 ohm-m, a\ = 0.1 m): p2, o h m m

2

a2, m 0.28 0.56

4

8

16

1.014

1.021

0.929

0.893

1.025 0.876

1.027 0.867

In the case when 02 > TQ with an increase of formation resistivity, p3, (Ja/crs increases also. However, if a2 < ^0 an increase of ps leads to a decrease of aa/crs- For instance, for probe 4F1.1 we have following values of (Ja/crs (p2 = 128 ohm-m, ai = 0 . 1 m): p25 o h m m a2, m 0.28 0.56

8

16

32

64

1.035 0.867

1.019 0.911

1.006 0.933

0.991 0.948

Case 2 Let us consider the case when P3/P2 > 1. If a2 < Vmin the apparent conductivity decreases with growing a2 becomes greater with an increase of 02, if 0^2 > rmin- For instance, for probe 6F1M, when p2 = 2 ohm-m, Ps = 16 ohm-m and ai = 0.1 m we have: a2, m

0.28

0.40

0.56

0.80

aa/(J3

0.999

1.156

1.666

2.616

425

TABLE 7.12 Values of function (Jal<^?,\ ai = 0.1 m, p2 = 4 ohm-m Probe type

/03, ohm-• m 0.5

1

2

6F1M Two-coil

0.999 0.893

0.999 0.993

0.999 1.085

4F1 Two-coil 4F1.1 Two-coil

1.036 0.881

0.999 0.937 1.021 0.932

1.010 0.992

-

-

-

0.996 1.088 0.995 1.157

0.960 0.788 0.994 0.764

0.974 0.860 0.996 0.850

0.985 0.947 0.996 0.945

0.999 1.085 0.996 1.088

1.022 1.338 0.996 1.345

1.065 1.823 0.994

-

-

-

-

-

4

8

16

32

64

128

0.998 1.588 0.932

0.996 2.238 0.852 2.250 0.822 3.297

0.992 3.523 0.695 3.541

-

0.671 3.454

0.380 5.696

1.835

1.147 2.774 0.992 2.792

1.309 4.654 0.996 4.680

-

-

-

-

a^jax = 2 ^ 2 0.998 1.256 0.974 1.262 0.952 1.359

1.596 0.904 1.690

a2/ai = 4 6F1M Two-coil 4F1 Two-coil 4F1.1 Two-coil

TABLE 7.13 Values of function 0-^/(73; ai -= 0.1 m, P2 = 4 ohm-m Probe type

ps, ohm-m 0.5

1

2

4

8

16

32

64

2.341

4V2

a2/ai = 0.833 0.632 0.824 0.590

0.747 0.893 0.728

0.939 0.879 0.941 0.873

0.999 1.086 0.996 1.088

1.099 1.459 1.084 1.469

1.284 2.174 1.245 2.191

1.640 3.574 1.553 3.600

6.348 2.155 6.380

-

-

-

-

-

-

-

-

6F1M Two-coil 4F1 Two-coil

0.596 0.440 0.517 0.376

0.738 0.605 0.709 0.578

0.853 0.795 0.842

1.241 1.607 1.244

4.263 8.415 4.264

1.620

1.691 2.605 1.696 2.628

2.558 4.557 2.564

0.785

0.999 1.085 0.996 1.088

4.593

8.467

4F1.1 Two-coil

-

-

-

-

-

-

-

-

6F1M Two-coil 4F1 Two-coil 4F1.1 Two-coil

0.892

= 8

0 2 / ^ 1 ••

128

426

TABLE 7.14 Values of function (Ja/cy?,', ai = 0.1 m, p2 = 16 ohm-m Probe type

ps, ohm- m 0.5

1

2

6F1M

0.999

0.999

0.999

0.999

Two-coil

0.882

0.917

0.958

1.020

4

8

16

32

64

128

0.998

0.996

0.994

0.988

1.133

1.143

1.769

2.602 0.896

a 2 / a i = 2v/2

4F1

1.040

1.027

1.019

1.012

1.002

0.986

0.956

Two-coil

0.881

0.922

0.956

1.021

1.135

1.353

1.778

2.613

-

4F1.1

-

-

-

-

1.017

0.993

0.965

0.918

0.832

1.184

1.406

1.802

2.551

4.000

-

Two-coil a^jax - 4 6F1M

0.956

0.974

0.980

0.986

0.996

1.015

1.051

0.766

0.968 0.822

Two-coil

0.878

0.956

1.092

1.348

1.846

2.828

4F1

0.994

0.995

0.995

0.994

0.991

0.986

0.976

0.995

Two-coil

0.739

0.809

0.872

0.955

1.094

1.353

1.855

2.841

4F1.1

-

-

~

-

1.005

0.993

0.978

0.951

0.902

1.125

1.406

1.887

2.782

4.505

16

32

64

128

-

Two-coil

TABLE 7.15 Values of function CTO/CTS; a\ = 0 . 1 m, p2 = 16 ohm-m Probe type

/93, ohm- m 0.5

1

2

4

8

a 2 / a i == 2 ^ 2 6F1M

0.815

0.865

0.894

0.918

0.948

0.996

1.086

1.257

Two-coil

0.593

0.680

0.758

0.861

1.032

1.348

1.960

3.166

4F1

0.805

0.867

0.898

0.921

0.947

0.986

1.056

1.188

Two-coil

0.546

0.656

0.747

0.857

1.032

1.353

1.970

3.181

4F1.1

-

-

0.927

0.993

1.064

1.176

1.380

1.031

1.406

2.019

3.143

5.380

-

Two-coil a^lai = 8 6F1M

0.553

0.673

0.743

0.803

0.877

0.996

1.217

1.642

Two-coil

0.380

0.505

0.612

0.744

0.958

1.348

2.100

3.580 4.264

4F1

0.517

0.709

0.842

0.996

1.244

1.696

2.564

Two-coil

0.376

0.578

0.758

1.088

1.620

2.628

4.593

8.467

4F1.1

-

-

-

-

-

-

-

-

Two-coil

427

TABLE 7.16 Values of function (Ja/cJ^', ai = 0.1 m, p2 = 64 ohm-m Probe type

P3, ohm- m 0.5

1

2

0.999 0.879 1.042 0.865

0.999 0.912

0.998 1.004

1.028 0.906

0.999 0.949 1.021 0.947

-

-

-

-

6F1M Two-coil 4F1 Two-coil

0.955 0.760 0.994

0.966 0.812

0.975 0.924

0.773

0.995 0.799

0.972 0.860 0.995 0.854

4F1.1 Two-coil

-

-

-

4

8

16

32

64

128

0.996 1.288 0.999 1.292 1.015 1.336

0.993 1.562

0.986 2.372 0.947 2.381 0.980 2.325

-

0.986 2.372 0.947 2.381 0.980 2.325

-

a2/ai =:2V2 6F1M Two-coil 4F1 Two-coil 4F1.1 Two-coil

1.016 1.000

0.998 1.102 1.010 1.104 1.103 1.140

0.981 1.660 1.000 1.679

0.945 3.574

a2/ai = 4 0.982 1.614

0.993 0.922

0.977 1.031 0.990 1.031

0.979 1.229 0.984 1.232

-

1.013 1.036

1.000 1.263

0.971 1.620 0.994 1.637

16

32

64

128

0.910 1.031 0.913 0.924

0.925 1.229 0.992 1.144

0.986 2.372

-

0.876 0.873

0.923 1.149

0.947 1.614 0.932 1.563 0.951 1.571

0.881 1.487 0.859 1.492 0.861 1.485

0.986 2.372

0.955 3.646

TABLE 7.17 Values of function cJa/crs; di = 0.1 m, p2 — 64 ohm-m Probe type

p3, ohm- m 0.5

1

2

4

8

a2/ai =• 2^/2 0.898 0.924

0.860 0.638

0.882 0.860 0.888 0.715

-

-

-

-

0.542

0.716 0.566 0.695 0.545

0.754

0.365 0.452 0.293

0.657 0.479 0.618 0.441

0.659 0.740 0.648

-

-

-

-

6F1M Two-coil 4F1 Two-coil 4F1.1 Two-coil

0.810 0.760 0.800 0.535

0.858 0.812

6F1M Two-coil 4F1 Two-coil 4F1.1 Two-coil

0.903 0.799

0.947 2.381 0.980 2.325

1.023 3.760

a2/ai = 8 0.785 0.795 0.773 0.791 0.586 0.660

0.823 1.034 0.809 1.034 0.753 1.002

0.947 2.381 0.980 2.325

1.167 3.906

428

TABLE 7.18 Values of function Oalo'^', ai = 0.1 m, p2 = 4 ohm-m p3, ohm- m

Probe type

0.5

1

2

0.945 0.785 0.967 0.760

0.963 0.877 0.973 0.868

0.977 0.996 0.971 0.996

-

-

-

4

8

16

32

64

0.993 1.196 0.963 1.202

1.020 1.566 0.944 1.578

1.069 2.281 0.905 2.300

1.163 3.687 0.829 3.714

1.348 6.468 0.678 6.506

-

-

-

-

-

'

'

^2/^1 = 2\f2 6F1M Two-coil 4F1 Two-coil 4F1.1 Two-coil

^2/^1 = 4 6F1M Two-coil 4F1 Two-coil 4F1.1 Two-coil

0.799 0.621 0.772 0.577

0.868 0.757 0.855 0.739

0.924 0.925 0.908 0.921

0.993 1.200 0.963 1.202

1.108 1.693 1.045 1.708

1.320 2.650 1.193 2.674

1.730 4.527 1.473 4.563

2.534 8.240 2.021 8.292

-

-

-

-

-

-

-

-

TABLE 7.19 Values of function a^^ iM; «! = 0.15 m^, P2 Probe type

— 16 o h m - m

p3, ohm •m 0.5

1

2

4

8

16

32

64

128

-

0 2 / 0 1 =:2v/2

6F1M Two-coil 4F1 Two-coil 4F1.1 Two-coil

0.939 0.762

0.954 0.838

0.962 0.926

0.967 1.065

0.970 1.316

1.069 2.281

1.163 3.687

1.348 6.468

0.964

0.968 0.826

0.963 0.922

0.949 1.067

0.918 1.323 0.886 1.435

0.905 2.300 0.840 1.932

0.829 3.714 0.740 2.822

0.678 6.506 0.546 4.507

0.974 1.798 0.857 1.810

1.061 2.863 0.829 2.882

0.840 1.932

0.843 2.962

0.735

-

0.169 7.767

02/01 = 4

6F1M Two-coil 4F1 Two-coil

0.777 0.580 0.748 0.532

0.836 0.688 0.821

0.869 0.800 0.852

0.665

0.791

0.896 0.965 0.866 0.963

4F1.1 Two-coil

-

-

~

~

0.926 1.253 0.868 1.258 0.793 1.336

1.228 4.964 0.768 4.993 0.813 4.890

0.737 8.608

429

TABLE 7.20 Values of function (Ta/cj'i] ai — 0.15 m, p^ — 16 ohm-m P3, ohm- m

Probe type

0.5

1

2

0.500 0.364 0.388 0.292

0.632 0.510 0.577 0.475

0.709 0.652 0.688 0.635

-

-

-

4

8

16

32

64

0.773 0.846 0.728 0.841

0.850 1.117 0.783 1.181

0.974 1.798 0.857 1.810

1.201 3.005 0.982 3.026

1.637 5.386 1.213 5.416

-

-

-

-

-

0.759 1.103 0.686 1.105

0.974 1.798 0.857 1.810

1.370 3.146 1.158 3.168

2.131 5.802 1.729 5.836

-

-

-

-

fl2/ai = 2 v ^ 6F1M Two-coil 4F1 Two-coil 4F1.1 Two-coil

«2/ai = 8 6F1M Two-coil 4F1 Two-coil 4F1.1 Two-coil

0.516 0.504

0.161 0.151 0.027 0.054

0.385 0.335 0.295 0.286

0.455 0.481

0.625 0.728 0.568 0.720

-

-

-

-

TABLE 7.21 Values of function (Jal^'i] <^i = 0-15 m, p2 = 64 ohm-m Probe type

P3, ohm- m 0.5

1

2

6F1M Two-coil 4F1 Two-coil

0.938 0.756 0.963 0.728

0.952 0.828 0.967 0.815

0.958 0.908 0.961 0.904

0.960 1.032

4F1.1 Two-coil

-

-

-

-

6F1M Two-coil 4F1 Two-coil 4F1.1 Two-coil

0.772 0.570 0.741 0.521

0.828 0.671 0.812 0.647

0.855 0.769 0.838 0.758

0.871 0.907 0.841 0.904

-

-

-

-

4

8

16

32

64

128

0.951 1.678 0.846 1.688 0.838 1.785

0.935 2.507 0.714 2.522

0.901 4.146 0.454 4.168 0.544 4.040

-

0.888 1.586 0.774 1.594 0.743 1.664

0.894 2.447 0.668 2.462 0.688 2.500

0.901 4.146 0.454 4.168 0.544

-

a2/ai = 2^2

0.945 1.033

0.958 1.254 0.912 1.259 0.885 1.345

0.739 2.566

0.165 6.886

a2/ai = 4 0.881 1.143 0.823 1.146 0.722 1.171

4.039

0.246 7.006

430

TABLE 7.22 Values of function (Ta/(J2,] cti — 0-15 m, p2 = 64 ohm-m Probe type

p3, ohm- m 0.5

1

2

4

16

32

64

128

0.748 1.010 0.676 1.011

0.779 1.476 0.654

0.824 2.380 0.592

0.901 4.146 0.454

1.483

2.390

0.410 1.954

0.560 1.513

0.590 2.408

4.168 0.544 4.039

-

0.587 0.880 0.505 0.878 0.020

0.647 1.368 0.516 1.373 0.330

0.729

1.357

0.740 2.305 0.503 2.318 0.468 2.317

8

a2/ai = 2\/2 6F1M Two-coil 4F1 Two-coil

0.486 0.349 0.373 0.274

0.613 0.484 0.556 0.447

0.678 0.605 0.634

0.718 0.758 0.669

0.586

0.751

4F1.1 Two-coil

-

-

-

-

6F1M Two-coil 4F1 Two-coil 4F1.1

0.141 0.130 0.052 0.031

0.355 0.300 0.260 0.250

0.464 0.442 0.400 0.415

0.532 0.612 0.469 0.599

-

-

-

-

0.400 7.157

aijai = 8

Two-coil

0.901 4.146 0.454 4.168 0.544 4.040

0.594 7.313

With an increase of p2 ^al^^z increases if a^ < TQ. In the opposite case i.e. as a2 > TQ with growing resistivity of the invasion zone (Ta/crs decreases. For example, for probe 4F1, when p3 = 32 ohm-m and ai = 0.1 m we have following values of Oajo-^'. p2, ohmrQ ^2, m 0.28 0.56

2

4

8

16

0.714 2.215

0.852 1.553

0.921 1.221

0.956 1.056

If a2 < To then (Ja/(^?, decreases with an increase of formation resistivity. On the contrary when a2 > ro, CFal^z increases when p3 increases. For instance, for probe 4F1.1, when /92 = 16 ohm-m and ai = 0.1 m we have following values of Oajo-^'. P2, ohm-m a2, m 0.28 0.56

32

64

128

0.965 1.064

0.918 1.176

0.832 1.380

An increase of the borehole radius from 0.1 m to 0.15 m results in a decrease of Oaloz-, when p\lp2 < 1. In conclusion in Table 7.23 ranges of the change of formation resistivity are presented for which aa/cTs differs from unity less than 10%. Now we will investigate vertical response of these differential probes. As has been shown differential probes allow us to reduce significantly the influence of the borehole and the invasion zone, that is areas of a medium, directly surrounding the probe. On the other

431 hand, in order to provide satisfactory vertical response in layers with a finite thickness it is necessary that the main part of a measured signal is defined by currents in areas which are relatively close to the probe. Therefore as was pointed out earher, requirements for improvement of radial and vertical responses of a probe in essence contradict each other. Correspondingly, it is natural to expect that since multi-coil induction probes have better radial responses than the basic two-coil probe, they are more sensitive to the surrounding medium at least in those cases when the formation thickness exceeds the distance between the most remote coils of the probe. However, with help of external focusing when some coils are located outside the formation, we can improve the vertical response due to the fact that for such position a differential probe permits us to reduce the signal from the surrounding medium to a greater extent than from the formation. Proceeding from Doll's theory let us first consider vertical responses of the internal differential probes. Suppose the probe is located against the formation, and we will introduce notations: Gl and G2, which are the geometric factors of the formation and the surrounding medium (shoulders), respectively, while d and G2 are corresponding geometric factors for the basic two coil probe. Values of criGl/a2G2 and aiGi/(72G2 characterize a relation between signals caused by currents in a formation and in the surrounding medium for differential and two-coil induction probes, respectively. Let us show that in this case the following inequality takes place: a,Gl ^ aiGl 0-2 G 2

^2^2

or

l
<"^'

In accord with this relation we can consider a uniform medium in order to demonstrate that a multi-coil probe with internal focusing has higher sensitivity to the surrounding medium than the basic two-coil probe when the formation thickness exceeds the probe length. As was mentioned above every multi-coil induction probe can be considered as a sum of two-coil probes namely the basic induction probe and additional coil probes which provide improvement of the radial response. Electromotive forces induced in these probes can have the opposite sign to that in the basic probe as well as the same sign. However, the probes where the electromotive force has opposite sign play the most essential role and correspondingly, only they will be taken into consideration here. In accord with Doll's theory, described in detail in Chapter 3, it is appropriate to emphasize two main features of the geometric factors: • The geometric factor of the whole medium for every probe is equal to unity. • The part of the medium against which the two-coil probe is located has a geometric factor of 0.5.

432

TABLE 7.23 Ranges of the change of formation resistivity; pi = 0.5 ohm-m, ai = 0.1 m p2, ohm-m

a2/ai

Probe type

2V2

4

4V2

0.5-20 0.5-8

0.5-8 0.5-40

Ti

-

-

-

0.5-20 0.5-16 8-16

0.5-16 0.5-40 8-32

1-8 1-8

-

6F1M 4F1 4F1.1

0.5-20 0.5-40 8-32

0.5-32 0.5-40 8-64

2-16 2-16 8-16

6F1M 4F1 4F1.1

0.5-20 0.5-40 8-64

0.5-64

2-32

0.5-40 8-128

2-40 8-32

6F1M 4F1 4F1.1

32

0.5-20 0.5-40 8-128

0.5-64 0.5-40 8-64

4-64 4-40 4-40

6F1M 4F1 4F1.1

64

0.5-20 0.5-40 8-128

0.5-64 0.5-40 8-128

8-64 4-40 16 128

6F1M 4F1 4F1.1 6F1M 4F1 4F1.1

16

128

1-4

0.5-20 0.5-40

0.5-64

4-64

0.5-40

8-128

8-128

4-40 16-128

6F1M 4F1 4F1.1

From these two facts we can conclude that the differential two-coil probe reduces the signal from the formation to a greater extent than that from the surrounding medium, and therefore inequahty 7.35 is valid. Now we will briefly discuss vertical responses of probes with external focusing coils. It is obvious that in the case when the whole probe is located against the formation the influence of currents induced in the surrounding medium is stronger than that for the basic two-cofl probe (Fig. 7.23a). If additional probes, as shown in Fig. 7.23b, are mainly outside of the formation, the eff"ect caused by currents induced in shoulders will be reduced stronger than that in the formation, and correspondingly some improvement of the vertical response of the basic two-coil probe will be observed. In conclusion of this analysis of vertical responses of multi-coil probes with internal and external focusing it is appropriate to notice the following: • Focusing probes located against the formation possess higher sensitivity to the surrounding medium than the basic two-coil probe.

433

>RF

a i

A

r 'T

"I

^1

L

H

H

a-2

•T-F

^ •R

^r

1

r

T

>R a-2 ^Tf=

Figure 7.23. Various positions of probes with respect to formation boundaries.

• Application of probes with external focusing results in some improvement of the vertical response within the interval of formation thicknesses: L

where L and / are length of the basic and additional probe, respectively. However, the influence of shoulders for such thicknesses as pi > p2 is so large that the apparent conductivity, aa, essentially differs from <7i for probes with internal as well as external focusing. In the case when pi/p2 and H > L probes with both types oi focusing have practically the same vertical responses. As was demonstrated in Chapter 5, evaluation of vertical responses of induction probes, based on the theory of small parameters, very often has a qualitative character. For this reason let us consider the results of calculations based on the exact solution. Curves of aa/cri for various parameters of geoelectric section when differential probes 6F1M, 4F1 and 4F1.1 are located symmetrically against the formation are presented in Figs. 7.24-7.29. As is seen from these curves: • With an increase of frequency the influence of shoulders (surrounding medium) becomes smaller. This effect is more noticeable when Pi/p2 and the formation thickness is greater. For example, if pi = 32 ohm-m, p2 = 2 ohm-m, if = 2.5 m for probes 4F1 (/ = 70 kHz) and 4F1.1 (/ = 1 MHz) we have cfa/cri = 2.5 and (^a/(^i = 1-05, respectively. • An increase of shoulder conductivity deteriorates the vertical response of the probe, and it manifests itself stronger the lower the frequency and the smaller the thickness of the formation.

434

CO

CO

II

CM CO

E E 1 o

CO

CD

lO

'^

CO

/

CO

E E o _^

^^ 1

X
E E o II

CD

00 II

1[

I TT

^X>/%

CO

CsJ

CN

/ ^

1

l\

1

E

CO

vl

1

/

5 ^

Vl

E 5f

CM

I

1

T;

CO

CN

N

O

CO

o

o OH

;^

bJO OJU

o

CO

o

OH

O

03

;^

bJO

X! a;

1^

3

1 5

yI

CM

'^

CM

CD

E E o II

CNI

/ "* CM

\y Ico

1 T—

CO

CO

II

CO

E E 1 o ncvi

CD

m

uo

1 ^

I "r

E E o 00 11

"^

CO

CO

/ -*

/ ^

~1 1 E

>J

1

1

1

5 C

CM

CO

E a:

1 I

CM

/

/

CM

O

"^

O

O

03

> CNI

tbJU

X CD

-0

^ ^ 0

fo O ^ ffi o

rX

t-

"^

O

a;

s:^ O

a

C/5

1

^

QJ

crt (M

o

-HJ

CD

<«:L

^

a

> ^

CD

(M X3 ta bJO

^ CJ

435

436

CD

CO

E E o 00 CM

II

1 ^ CD

E E JO.

o 00 II

CM

^

lO

y Tt

[ ^5^^^^

X-

/

CO

1

CO

CO

/

^

E E SI

II

CNJ CO

1 \ CO

5 ^

1]1

'i-

i

1 V

CM

"^

/ CM

E

CO

CNJ

E 1

CO

CM

O

0)

Pi O

03

CM
1^

X

fl

i

g

X a;

bX) g

O

o

03

00

CM

$-1

bJO 3

437

• The probe with external focusing 6F1M is characterized by smaller values of aa/(Ji than the probe with the internal focusing 4F1, if 1.5 < H < 2.5 m and pi > p2- For instance both probes provide the same value of cTa/cri, equal to 1.75 when H = 2.5 m, Pi — 8 ohm-m and p2 = 0.5 ohm-m. However, ii H = 1.75 m values of da/cri for probes 4F1 and 6F1M are equal to 2.5 and 1.8, respectively. Within this interval of thicknesses and P2 == 2 ohm-m values of apparent conductivity for both probes are practically the same. Thus, the advantage of the external focusing manifest itself only for large values of pi/p2 when values of apparent conductivity essentially differ from the formation conductivity.

7.5. The Influence of Finite Height of the Invasion Zone on Radial Responses of Probes 6 F 1 M , 4 F 1 and 4F1.1 As is known, parameters of the multi-coil induction probe are chosen with the assumption that the invasion zone has an infinite extension along the borehole axis. In real conditions penetration of borehole filtrate into a formation and surrounding medium occurs in a different manner due to the difference of their physical properties. For this reason it is appropriate to consider focusing features of multi-coil induction probes when the invasion zone has limited dimension along the borehole axis. We will investigate the case where the borehole filtrate penetrates into the formation only (Fig. 7.30). Calculations of the magnetic field on the borehole axis in such model is performed on the base of the approximate theory which takes into account the skin effect in the external area. As was demonstrated in Chapter 3 for the quadrature component of the magnetic field of a two-coil induction probe we have: Q H , = Q F O + ^ [ ( a i - a4)Gi(a) + {a, - a3)Gf\a)

+ {a^ - ^ 3 ) 0 ^ (a)]

(7.36)

where Q H^ is the quadrature component of the magnetic field in a horizontally layered medium when the borehole and invasion zone are absent; G\ (a) and G2 (o^) are geometric factors of the part of the borehole located against the formation and the invasion zone, respectively. It is obvious that an influence of finite height of the invasion zone is defined by difference between geometric factors G^ {a) and G2 {a) and those corresponding to infinitely long cylinders. First of all we consider this question for a two-coil induction probe. We will assume that the probe is completely located within the cylinder interval (iJ/2, — iJ/2), and L is the probe length, but e is coordinate of the transmitter coil (Fig. 7.31). Then the expression for geometric factor of the cylinder in accord with eq. 3.104 can be written in the form:

0 -if/2

438

I k—•

^

('4

^3

a2

Figure 7.30. The model of the invasion zone with finite height.

1 'f

1

i[—^'^

H

+

1 I0 i

s r 1

i

1 /0

1 ^^ 12H Figure 7.31. Two-coil induction probe within the cyhnder with finite thickness, H.

439 where: Ri = x/r2 + (z - ey

R2 = V[L - {z - e)]^ + r^

It is obvious that:

-00

0

0 H/2

Integrals on the right-hand side of eq. 7.38 describe a difference between geometric factors of a cyhnder of finite height and that of an infinitely long one. Let us investigate these integrals. After a change of variables a sum of them, S, can be presented as: 1 H/2-£

a

S=-^

r'dr

[ "

*^^

(7.39)

H/2-e

It is clear that H/2 — e = /+ and H/2 + e = /^ are distances from the transmitter coil to upper and lower boundaries of the cylinder, respectively. Suppose that a/l^
Suppose also that:

1 - L//J > a//J

1 + L/IQ > a/l^

or /+ = /^ - L > a

/~ =

ZQ"

+ L> a

where /+ and /~ are distances from the receiver to upper and lower boundaries, respectively. Taking into account these inequalities, radicals in the integrand of eq. 7.40 can be expanded in series. Then the integral can be presented as:

440 fix) \

0.2

0.4

0.6

Figure 7.32. Behavior of function f{x).

After relatively simple transformations we have: ^4

s =-

a

/iSV/f!;

(7.42)

^0-

where f{x) = {x' - 1)

6x - (1 - x'^) 2x2

6 Inx

(7.43)

Asymptotical expansion 7.42 is valid when the distance from coils to the upper and lower boundaries exceeds the cylinder radius. A graph of f{x) is shown in Fig. 7.32. If L/lo tends to zero expansion of eq. 7.42 results in the following: 5 = -

1

(U)^ + (lori

(7.44)

Equations 7.42-7.44 show that the value of S has high order with respect to small parameter a/L or a//J, CL/IQ, and therefore the geometric factor of a cylinder of finite length for a two-coil induction probe {H > L) slightly differs from that of an infinitely long one, i.e. Gi. In the case of symmetric location of the probe {e = 0) this fact was demonstrated in Chapter 5.

441 Thus, we can assume that the influence of finite dimensions of the invasion zone along the borehole axis on focusing features of the multi-coil probe will be usually small. Calculations confirm this assumption. Values of EMF expressed in units of electromotive force in a horizontally layered medium are presented in Tables 7.24-7.26. These data show that probes 6F1M, 4F1 and 4F1.1 preserve focusing features even when the invasion zone has finite dimensions along the 2;-axis.

7.6. Three-coil Differential Probe Now we will investigate the simplest differential system, namely the three-coil probe which consists of two transmitter and one receiver coil or one transmitter and two receiver coils. The distance between the pair of transmitter or receiver coils is significantly smaller than that to the remote coil. Let us suppose that the probe is located on the borehole axis, and it consists of one receiver coil with moment A^i, and two transmitter coils with moments Mi, and M2. The latter are characterized by opposite direction of turns. Then, for electromotive force induced in the receiver by currents in the borehole and in the formation at the range of small parameters we have: ^ — 011 — 0\2 —

, Ml M2 cTi ( - — G i ( a i ) - - ^ G i ( a 2 ) L\ L2

;;

47r + ^2 ( ^ — G ' 2 ( Q ; I ) -

(7.45)

-j—G2[a2^

Li

L2

where Li and L2 are lengths of two-coil induction probes forming the differential probe ( i i > L2). Suppose that the electromotive force of the primary field is compensated, i.e. we have Mi/Ll = M2/LI. Then we obtain:

4ITLI

\"i

where t = L2/L1 < 1 and:

G, {a,)-t^G G*i =

G; =

1(^2)

l-i2

9li{ai)-t'G

(7.47) 2(0^2)

Let us assume that the lengths of both two-coil probes are much greater than the borehole radius, i.e. c^i > 1 and a2 > 1. Then, making use of the asymptotic expression for function Gi: 1

3 In a - 4.25

442

TABLE 7.24 Values of EMF for probe 6F1M; pi = 1 ohm-m, P4 = 4 ohm-m P2 = 8 ohm-m, p^ — 32 ohm-m H/L

a 2A /ai

6

1.015 1.025 1.007 0.995 0.994 0.993 0.992

1

V2 2 2\/2 4 4A/2

p2 = 32 ohm-m, ps = S ohm-m

1.041 1.073 1.047 1.035 1.044 1.053 1.059

6

1.096 1.169 1.169 1.184

0.997 0.996 0.999 1.000

1.232 1.277 1.309

1.000 1.000 1.000

0.978 0.967 0.978 0.983 0.982 0.981 0.981

0.938 0.909 0.915 0.916 0.912 0.910 0.909

TABLE 7.25 Values of EMF for probe 4F1; pi = I ohm-m, P4 = 4 ohm-m P2 = S ohm-m, p3 = 32 ohm-m

P2 = 32 ohm-m, p3 = 8 ohm-m

a2/ai H/L 0.973 0.984

1

V2

0.980 0.973 0.996 0.960 0.956

2 2v/2 4 4^2

0.957 0.986 0.995 0.996 0.996 0.995 0.995

TABLE 7.26 Values of EMF for probe 4F1.1; pi

0.995 1.062 1.118 1.171 1.220 1.260 1.286

1.007 1.005 1.005 1.005 1.005 1.005 1.005

1.020 1.003 0.997 0.995 0.995 0.994 0.994

0.989 0.952 0.929 0.918 0.913 0.911 0.911

1 ohm-m, P4 = 4 ohm-m

P2 — S ohm-m, ps = 32 ohm-m

P2 = 32 ohm-m, p3 = 8 ohm-m

a2/ai H/L 1 2 4

4V2

0.929 0.969 0.965 0.957 0.958 0.959

0.862 0.938 0.959 0.959 0.960 0.961

0.917 1.077 1.284 1.306 1.301 . 1.295

1.021 1.017 1.014 1.014 1.014 1.014

1.087 1.041 1.014 1.013 1.013 1.013

1.033 0.934 0.862 0.861 0.862 0.862

443

4.0 d, m

Figure 7.33. Radial responses of a three-coil probe (L = 1.4 m). Curve index L2/L1.

we have for the geometric factor of a three-coil probe, G^, the following expression: 1

Gl

2.17-^-31n., 1 — r^

(7.48)

Thus due to the compensation of the electromotive force of the primary field the value of the geometric factor Gl turns out to be much smaller than the values of the geometric factor for two-coil probes, Gi(ai) and Gi(a2). Radial responses of three-coil probes characterizing focusing features of these systems are presented in Figs. 7.33-7.35. At the initial part of the radial response, G^ has negative values which, due to compensation of the electromotive force of the primary field, are much smaller than those of two-coil induction probes. Near point QQ when 3 In

ao

2.17-

3lnt

geometric factor G^ is equal to zero and then monotonically increases, approaching asymptotically to unity. Combination of two factors, such as the compensation of the electromotive force of the primary field and the behavior of the function Gi{a) as 1/a^, provide

444

4.0 d, m

Figure 7.34. Radial responses of a three-coil probe (L = 1.8 m). Curve index L2/L1.

4.0 d, m

Figure 7.35. Radial responses of a three-coil probe (L = 2 m). Curve index L2/L1.

445 a significant reduction of the influence of currents induced in the borehole, as the probe length Li is several times greater than the radius a, and with an increase of the probe length the effect of focusing manifests itself stronger. This behavior of the radial response of a three-coil differential probe is of great practical interest for solution of various problems, in particular for the determination of a relatively high resistive formation when the borehole is filled by a strongly mineralized solution {s = (^2/0'! < 0.001). Inasmuch as for obtaining reliable measured signals created by currents in a slightly conductive medium it is necessary to apply relatively high frequencies, when the skin effect in the mineralized solution of the borehole can be noticeable, it is appropriate to use results of calculations by exact formulae. As an example Table 7.27 contains values of ratio of quadrature component of electromotive force when the probe is located on the borehole axis to that corresponding to a uniform medium with conductivity of the formation {t = 0.8). As is seen from the table the skin effect does not practically affect the radial responses of the three-coil probe provided that the thickness of the skin effect, /ii, is related with the borehole radius as: hi > 2V2ai

(7.49)

Values of apparent conductivity, Oajox-^ for a three-coil probe [L\ = 1.2 m, t = 0.833) demonstrating its focusing features are presented in Tables 7.28-7.31. Let us notice that calculations have been performed making use of the exact solution provided that ai = 0.1 m. As is seen from these tables, if penetration of borehole solution is not very strong (a2/ai < 4) the value of CFa/cfi does not practically differ from that corresponding to a uniform medium with the formation resistivity when parameter a^jiLxjai < 0.64 x 10"^. In this considered range of parameters the borehole and the invasion zone do not have an influence on the apparent conductivity which is defined by the formation conductivity only. With an increase of the invasion zone radius the difference between Oa and a^ becomes more noticeable. Comparison of (Ja/cr2 for a three-coil probe located symmetrically with respect to the formation boundaries with the same function for a two-coil probe shows that if the formation thickness exceeds the probe length the influence of the surrounding medium on the three-coil probe is somewhat greater than that on the two-coil induction probe of the same length (Figs. 7.36-7.38). In this case ai and 02 are conductivities of the formation and the surrounding medium. Also the curve of profiling for a three-coil induction probe is shown in Fig. 7.39 which demonstrates only slight asymmetrical behavior with respect to the center of the formation. Now let us consider the calibration curve for a three-coil probe (Fig. 7.40). For comparison values of quadrature components for both two- and three-coil probes are given in Table 7.32. It is clear that the range of resistivities measured by a three-coil probe is narrower than that for a two-coil probe. As was shown before, the better the focusing of a multi-coil induction probe the narrower the range of measured resistivities, and correspondingly the same effect is observed for other differential probes. We will characterize the range of measured resistivities by the

446

TABLE 7.27 Values of function Oal^^x a

14

10

ai/hi

18

10

14 cr2/o-i = 1 X 10-3

(J2/CJ1 = 2 X 1 0 - 3

I/V2 1/2 I/2V2 1/4 I/4V2 1/8

0.550 0.730 0.796 0.821 0.832

0.580 0.809 0.888 0.916 0.926

0.550 0.826 0.919 0.950 0.962

0.459 0.589 0.641 0.664 0.675

0.586 0.765 0.828 0.852 0.861

0.609 0.824 0.897 0.923 0.932

0.837

0.931

0.966

0.681

0.866

0.936

cF2lcr\ = 5 X lO""^ 0.522 0.652 0.702 0.722 0.732 0.738

0.231 0.294 0.329 0.349 0.362 0.369

1/V^ 1/2 I/2V2 1/4 l/4v/2 1/8

18

^^2/^1 = 2.5 X 10-4

0.614 0.782 0.842 0.864 0.872 0.876

-0.252 -0.301 -0.291 -0.275 -0.261 -0.252

0.345 0.413 0.447 0.465 0.475 0.482

0.554 0.679 0.726 0.744 0.752 0.657

TABLE 7.28 Values of function cTa/ai 0-2/0.1 = 4i, P2/P1 = 16

0 2 / ^ 1 = 4, P2IP1 = 8

Pz/pi CTl/LtU;

0.01 0.02 0.04 0.08 0.16 0.32 0.64

1

4

16

32

1

4

16

32

0.910 0.877 0.826 0.770 0.686 0.574 0.433

0.225 0.220 0.209 0.193 0.172 0.144

0.0587 0.0563 0.0536 0.0496 0.0440 0.0356 0.0260

0.0298 0.0283 0.0272

0.905 0.875 0.830 0.768 0.687 0.570 0.433

0.228 0.219 0.207 0.193 0.171 0.143 0.107

0.0572 0.0556 0.0524 0.0490 0.0427 0.0362 0.0260

0.0296 0.0282 0.0266 0.0250 0.0214

0.108

0.0258 0.0216 0.0182 0.0120

0.0180 0.0123

TABLE 7.29 Values of function aa/cri a2/ai = 4, P2/P1 = 64

a2/ai = 4, P2/P1 = 32 Ps/Pi (Jipuo 0.01 0.02 0.04 0.08 0.16 0.32 0.64

1 0.910 0.882 0.830 0.772 0.684 0.570 0.440

0.226 0.220 0.208 0.192 0.172 0.143 0.107

16

32

1

0.0570 0.0545 0.0520 0.0481 0.0428 0.0353 0.0260

0.0298 0.0275 0.0260 0.0242 0.0214 0.0174 0.0124

0.907 0.874 0.830 0.768 0.682 0.590 0.428

0.226 0.218 0.208 0.191 0.170 0.142 0.107

16

32

0.0569 0.0550 0.0520 0.0478 0.0427 0.0357 0.0255

0.028 0.027 0.025 0.024 0.021 0.017 0.012

447

TABLE 7.30 Values of function (Jajox a2/ai = 4, P2/P1 = 16

a2/ai = 4, P2/P1 = 8 Ps/Pi aifiLU

0.01 0.02 0.04 0.08 0.16 0.32 0.64

1

4

16

32

1

4

16

32

0.780 0.748 0.706 0.643 0.564 0.460 0.342

0.207 0.203 0.189 0.176 0.155 0.128 0.095

0.0670 0.0661 0.0628 0.0528 0.0530 0.0440 0.0310

0.0439 0.0429 0.0413 0.0390 0.0350 0.0280

0.769 0.738 0.694 0.634 0.554

0.200 0.192

0.0574 0.0055 0.0527 0.0488 0.0433 0.0358 0.0260

0.033 0.032 0.031 0.029 0.026 0.021

0.0157

0.453 0.335

0.181 0.166 0.145 0.119 0.088

0.014

TABLE 7.31 Values of function (Ja/cri a2/ai = 4, P2/P1 = 16

a2/ai = 4, P2/P1 = 8 Ps/pi aifiiJ

0.01 0.02 0.04 0.08 0.16 0.32 0.64

1

4

16

32

0.764 0.732

0.195 0.186 0.176 0.161 0.141 0.115 0.085

0.0523 0.0504

0.0286 0.0276 0.0262 0.0242 0.0214

0.690 0.630 0.550 0.448 0.328

0.0476 0.0438 0.0385 0.0315 0.0230

0.0175 0.0143

1

4

16

32

0.762

0.192

0.730 0.686

0.185 0.173 0.158 0.138 0.113 0.083

0.0498 0.0478 0.0452 0.0412 0.0361 0.0296 0.0213

0.02 0.02 0.02 0.02 0.01 0.01 0.01

0.627 0.547 0.447 0.330

TABLE 7.32 Values of quadrature components (xlO^) cr/iCjL^

0.01

0.02

0.04

0.08

0.16

0.32

0.64

1.28

2.56

5.12

Two-coil probe Three-coil probe

0.474

0.929 0.328

1.80 0.63

3.45 1.17

6.48 2.17

11.8 3.77

20.4

32.7 8.48

46.4

53.2 3.87

0.169

6.10

8.89

448

^nl^n

X-

T-

O

o

"^

O

7

1 O

CO

)

CN

^

b"-

O

Q (D

^

r ^ •+^

a o CO


J-i

^

b

XJ

o b X C/3

13 .S

t ^ ^-A

> o *•

00

CO CO

S o 5 II

PlH

O

C/2

b

C^

O

cp

rX;

o b C/2

> O CO ^ ^ CO CO I^ ^

bJO

449 10' 64^ 16

10'

8-

H/L = 8 1

1

4 2

^ 10°

1 1/2 1/4 1/8 • 1/16 _ 1/32 1/128—-

10"

10,-2 10"

10~

10"

10"

(72 jucoL^

Figure 7.38. Vertical responses of a three-coil probe {t = 0.833). Curve index cri/a2.

ratio: A.=

(7.50) Pmin

where Pmax and pmin are upper and lower boundary of this range, respectively. If we assume that Pmin — 0.07 and Pmax = 1.4 (P = tijK) then for a two-coil induction probe Ap :^ 400. For multi-coil induction probes Ap depends on parameters of the probe. For example, we have:

A.=

30 6F1M 50 for 4F1 40 4F1.1

Assuming that for three-coil induction probe Pmin = 0.13 and Pmax = 0.8 we obtain Ap = 40, i.e. it has a relatively broad range of measured resistivities. Of course, in practice this range is essentially wider. In conclusion, it is appropriate to notice that three-coil differential probes due to their simplicity can be used for lateral soundings. Examples of sounding curves, calculated for the range of small parameters, are given in Figs. 7.41-7.44. They demonstrate that such soundings can be performed with relatively short probes.

450

-^lisssIJ*"*

•

•

L2/H = 0.333 Li/H = 0.400 (J2lCT'\ - 8

R zlH

Figure 7.39. Profiling curves of induction probes: (la) three-coil probe S^L\ = (^2^\\ (lb) three-coil probe ^^ = S'^] (2) two-coil probe; (3) four-coil probe p = 0.4, c =^ 0.08.

f=0.8

Figure 7.40. Calibration curve for three-coil probes. Curve index t.

vj II

O CO

^II T—

o>

T—

CO

G^_ b o eg J ^ II W b CM CO

o

o>

b^ d

II

'^ ^ II ^

Ig o

H^

T-

V

00

'*

CM

'^

o o

•

^

^£>I^X)

CM

00

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CM

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r

s

00 CM

CO in CM

T-

II

^

o

7

lO

CM

O

CO

O

CO

o O

V

^

iin m 11

CO

L r^ l^ " r- Q

CM

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^ol^o

o

a;

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^

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3 b !>:

1^ a; bjO - o

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E .3

a; o

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o

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bJO

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451

452

^I?/%

l-X)/»D

r

^

o o o

-^

O

03

^

0)

o

-^ fl^

TJ P^ C/J

^ O 03 $-1

QJ 4^

01 KJ CO

^ CD ;-i

^

CD

CO

b

b X

•S

G) bJO - a PH

453

7.7. The Influence of Eccentricity on Focusing Features of Multi-coil Induction Probes In Chapter 4 we investigated an influence of displacement, ro, from the borehole axis on the radial responses of a two-coil induction probe. These results allow us t o consider the dependence of radial responses of multi-coil probes on the value of displacement roBy analogy with the case when a probe is located on the borehole axis the geometric factor of the probe, shifted from the axis, is defined in the following way:

cti

Gl{ai,e,s)

=

(7.51) MjNj

where £ = r g / a i , s = o^jox. If we suppose t h a t all two-coil induction probes forming a multi-coil system are sufficiently long and t h a t we can use the asymptotic expression for geometric factor G\, then instead of eq. 7.51 we have:

1+

(8-l)(2g-H) i=l

G i ( a i , £ , s ) :^

i=l

3=1

3=1

4

(7.52)

Lij

Inasmuch as in all considered probes moments of coils and distances between t h e m are chosen in such a manner t h a n the electromotive force of the primary field is compensated, the right part of eq. 7.52 turns out to be zero. Therefore, if the differential probe does not contain relatively short two-coil probes we can expect t h a t such a probe preserves focusing features even when it is shifted from the borehole axis within a wide range of parameter 02lo\. For illustration radial responses of probes 6F1M, 4F1 and 4F1.1, calculated for three values of borehole radius (ax = 0.1, 0.125, 0.15 m) and two values of parameter s {s — 1/32, 1/2), are presented in Figs. 7.45-7.48 as a function of the displacement e — r^/a. As is seen from the curves the radial response of probe 6F1M does not practically depend on s and £, if ai = 0.1 m. It is related with the fact t h a t the shortest two-coil probe forming 6F1M has in this case the relative length a = 5.0. If ai — 0.125 m and a\ — 0.15 m, due to a decrease of the relative length of probes, specially the shortest one, dependence of geometric factor G\ on parameters s and e begins to manifest itself. This behavior occurs sufficiently favorable: the geometric factor of probe 6F1M decreases by absolute value with an increase of £ for s = 1/32, as well as for s = 1/2. For example li s — 1/32 and a\ — 0.125 m function G\ has at initial part negative values then changes sign and begins to increase.

E

T-

^

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/ / (

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CD

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c o o ^. "-^ CM « - CO CM

It

5 •»-

II

T-

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o I

1/ . 1

1

-LL CO

454

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(X)

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2^ LL CO

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LL

LL

LL

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(30

^

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10 C\J -: O 11

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rri T1

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455

1

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a; ;-i

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03

m

C«

Tl

fe

456 TABLE 7.33 Values of function Gl{s) x 10^

1.2 1.4 1.6 1.8 2.0

0

0.1

0.2

0.3

0.4

0.5

-1.95 -1.26 -0.842 -0.582

-1.92 -1.25 -0.836 -0.578 -0.411

-1.85 -1.21 -0.818 -0.568 -0.405

-1.74 -1.16 -0.789 -0.552

-1.58 -1.08 -0.748 -0.529 -0.382

-1.39 -9.86 -0.696 -0.497 -0.367

-0.413

-0.396

The dependence of the geometric factor G^ of probes 4F1 and 4F1.1 on parameters ai, s and s is displaced in more complicated manner. For instance if ai = 0.1 m and s = 1/32 as well as s = 1/2, radial responses of both probes increase by absolute value with an increase of £, while for ai = 0.125 m and ai = 0.15 m as s = 1/32 they decrease. At the same time Gl for probes 4F1 and 4F1.1 does not practically depend on £, if s = 1/32 and tti = 0.125 m, or ai = 0.15 m. In accord with results obtained in Chapter 4, the geometric factor of a two-coil probe, displaced from the borehole axis, behaves practically as: ^

ifa»l

Taking into account that the electromotive force of the primary field is compensated in a three-coil probe we can expect that the displacement of this probe with respect to the borehole axis produces a relatively small efi'ect on the radial response. This fact is illustrated by values of geometric factor G*i{e) for s = 1/2, ai = 0.1 m, t = 0.8, presented in Table 7.33.

7.8. Choice of a Frequency for Differential Probes Considering the field of the magnetic dipole as well as induced currents in a uniform conducting medium it was established that with an increase of frequency the role of those parts of the medium which arc relatively close to the probe increases. For this reason the vertical response of a two-coil induction probe significantly improves, but simultaneously the influence of currents induced in the formation, with respect to those in the borehole and in the invasion zone, becomes lesser. Unlike of a two-coil induction probe the influence of the medium directly surrounding the multi-coil probe is very small, and an increase of frequency up to a certain limit does not practically change its radial response. Besides, improvement of the vertical response of a multi-coil induction probe due to measurements at higher frequencies allows us to apply induction logging in a more resistive medium, inasmuch as the ratio between the quadrature component of the electromotive force and that of the primary field increases. In order to define the upper limit of frequencies several factors have to be taken into account. First of all comparison of results of calculations based on the exact solution and the approximate theory, described in Chapter 3, permits us to establish the maximal

457 frequency when the radial response is not yet distorted. In fact two main assumptions form this theory, namely: • The skin effect is absent in the borehole and in the invasion zone. • In the formation the skin effect manifests itself in the same manner as in a uniform medium with conductivity a^. For this reason, coincidence of results of calculations by both methods establishes maximal frequencies for which, first of all, geometric factors of the borehole and of the invasion zone correctly describe the influence of these parts of the medium, and secondly, the electromotive force induced by the magnetic field of currents in the formation does not depend on currents induced in the borehole and in the formation. Also, a choice of a frequency is defined by focusing features of the probe. For example, for multi-coil probes having a greater depth of investigation in a relatively conductive medium the frequency should be smaller. In particular, it was established that for probe l.L-1.2 and three-coil probe of the same length the maximal frequency is defined from the relation: fma. ^ (2.0 - 2.2) X lOV. ^^min If we assume that Pmin = 1 ohm-m, the maximal frequency can be increased from 60 kHz to 220 kHz. An increase of frequency of almost four times results in a significant improvement of the vertical response of the probe, specially in a low-resistive medium, and it also allows us to perform reliable measurements in a more resistive medium. An additional limitation on the choice of the maximal frequency is related with the fact that the quadrature component of the electromotive force as a function of the formation conductivity has a maximum, and in order to avoid nonuniqueness it is necessary that the range of conductivities should correspond to the ascending branch of this response of EMF. Finally, it is appropriate to notice that the upper limit of measured resistivities is defined by instrumental problems as well as the possible influence of the dielectric constant.

7.9. Determination of the Coefficient of Differential Probes First, let us assume that a two-coil induction probe, which has coil dimensions that are much smaller than its length, is located in a uniform medium and the skin depth is much greater than the separation between coils. Then, as was shown in Chapter 2, for the quadrature component of the electromotive force we have: S = —--—a

(7.53)

ATTL

i.e. the EMF depends on conductivity, frequency, the probe length and coil moments, M and N.

458 For this reason in two probes characterized by various lengths or coil moments or both of them, different electromotive forces will be induced in the same medium. It is natural to eliminate the influence of these factors, depending on probe parameters, and introduce apparent conductivity which in a uniform medium coincides with its conductivity when parameter Ljh is sufficiently small. In fact this approach has been used widely in several chapters of this monograph. In particular, in accord with eq. 7.53 we have:

""

li'^uj'^MN

where 47rL

K is the probe coefficient and unlike corresponding coefficients of electric logging it depends not only on the probe length but also on coil moments. Equation 7.54 can be written as:

where <^o is the electromotive force of the primary field. As was shown above, in a nonuniform medium apparent conductivity, a a, depends on the distribution of resistivity and the probe length. For interpretation it is appropriate to present results of a solution of the forward problem as well as experimental data as a function of apparent conductivity, the probe length and parameters of a medium. Knowing the probe length and the electromotive force of the primary field, it is a simple matter to calculate the probe coefficient and transform the quadrature component of the electromotive force into the apparent conductivity. In accord with eq. 7.12 the coefficient of the coil probe can be written as: K =

47r ^^^^^^^

(7-56)

and it depends on distance between coils and values of their moments. Let us present the probe coefficient through the electromotive force of the primary field of one of two coil probes, for example, L n . Then, instead of eq. 7.56 we have: K =

^

(7.57)

where S'Q^ is the electromotive force of the primary field of a two-coil probe with length Liu Pij is the ratio of length, L^j, of a two-coil probe to L n ; Q and Cj are ratios of coil moments to the moment of one of the coils of the probe with length L n .

459 In accord with eq. 7.57 for determination of the probe coefficient it is necessary to measure the EMF of the primary field of one of the two-coil probes that requires switching oflF two coils of the system. Since this is impractical, calculation of the coefficient of multicoil probes can be performed by using eq. 7.56 only. In deriving the formula for the probe coefficient it was assumed that the radius and height of coils are many times smaller than the corresponding length of the two-coil probe. In other words, the EMF induced by currents in a medium depends on the distance between coil centers and their moments only. Under certain conditions the influence of finite dimensions of coils on the probe coefficient is insignificant, and it can be neglected. But in general, coil sizes have to be taken into account in calculating the probe coefficient and therefore the apparent conductivity. Let us consider this question in more detail. We will assume that coils of a two-coil probe present themselves as layered ones placed on a nonconducting base. Then, in accord with results obtained in Chapter 4 for the quadrature component of the electromotive force at the range of small parameters we have:

0

sin(A//2)sin(A6/2) ..^ . ^^ ^^ X —^— \n ——^Hi^^) COS ALo dA A"^ where rir and TIR are the number of turns per unit of length of transmitter and receiver coils; / is current; I and h are coil lengths; LQ is the distance between coil centers; r is the coil radius. Let us introduce the following notations: Ar = m, riTTrr^I and riRTrr'^ are moments of transmitter and receiver coils per unit of the length, M^, M^, respectively; l/2r = Si, b/2r = S2, Lo/r = a. After simple transformations we have: oo

^ ^ Suj'^a'^ ,.n,^n /" // ,sin5imsin52771/?(m) , QS' = a ^rM^M?. / (^(m) ^ ^ cos ma dm TT^ J m m m^ 0

Whence, for the coefficient of the probe having single layered coils we have: K = o 9 9 7»^0]i#o f j.r Nsmsimsms2m/i2(m) Suj^LL^rM^lMVf / 6(m) ^-^-^^ cos ma dm J m m m^ where 4>{m) = ^ [2Ka{m)KM)

- m{Kl - K^)]

/^ ro\ (7.58)

460

The expression for coefficient K, when coils are infinitely small, can be obtained assuming that the linear dimensions of the coils are much smaller than the probe length. In this case a — Lo/r -^ oo, and the integral in eq. 7.58 is defined by the behavior of the integrand for small values of m. If m —> 0 then: sinsim m

sins2m m

> si

> 52

If{m) :: m^

1 . ,. ^ r^ / x > - and 0(m) -^ Ko{m) 4

Therefore we have: 00

oo

f sin Sim sin S2mlf{m) siS2 f j . ^ . . 1 (pirn) r—cos ma dm ^ ^--— / ivo(^) cos m a dm J m m m^ A J 0

0

As is well known: 1 (l + a2)i/2

2 f — / Ko(^) cos m a dm TT J 0

Whence: oo

^. . sin5imsin52m/?(m) ^ SiS2 ^— cos ma dm = —, / (bim) v^v ; _ _ _2 8/rT^

7rsiS2 2± 8a

Substituting this value of the integral into eq. 7.58 we obtain the known expression K for point sensors:

UO'^^'^MTMRL

where Mr = M^ /, MR = M^ h. If the height of both coils is the same, then S\ = S2 = s, and we have: K =

,,

—^

^

(7.59)

Suj^fi^rM^M^ 10(m) ^^^^^^ ^ ^ ^ cos ma dm J m^ m^ 0

For a multi-coil probe located in a uniform medium with conductivity a, the quadrature component of the electromotive force in measuring coils is defined from relation:

Q^ = ^ Q 4 , = ^ ^ a = a5^-i

461 where Q Sji and K^ are electromotive forces and coefficients of n two-coil probes forming the differential system. Therefore, the coefficient of the multi-coil induction probe can be presented in the form:

and

where Q(^ is the electromotive force, induced in receiver coils of a focusing probe. As follows from eq. 7.58 determination of the probe coefficient, X, is related with calculation of the integral in the denominator of this equation. For the case when coil lengths are the same, values of the integral: oo

., .sin^ ms I?(m) . r—cos m a dm m^

/ 0(m)

. . (7.61)

and its asymptotic values are given in Table 7.34. With an increase of the probe length the value of the integral approaches its asymptote and, within certain limits, the influence of coil length is negligible. In known probes finite dimensions of coils do not practically influence the value of the probe coefficient. For example, if ai = 0.1 m, the four-coil differential probe l.L-1.2 has the following parameters: ai = 24, 0^2 = 9.6, as = 4.8, Smax = 1, p = 0.4 and c = 0.05. Taking into account the ratio between turns of coils we can see that the probe coefficient is the same as that for the probe with infinitely small coils. This is true to an even greater extent for three-coil differential probes with length exceeding, by at least four-five times, the borehole radius. If the probe length is sufficiently small and the dimension of coils has to be taken into account, the determination of coefficient K is defined by eq. 7.58. In conclusion of this chapter let us make several comments. • The use of differential multi-coil probes is the most conventional approach in application of induction logging. • Due to the use of these probes induction logging in most cases has the greatest depth of investigation among other logging tools. • Any multi-coil induction probe, regardless of the amount of transmitter and receiver coils, by no means performs focusing of the field as it takes place, for example, in optics. In effect, every induction probe, except a two-coil one, is a differential system measuring such a difference of signals in receivers that the influence of induced currents in the borehole and in the invasion zone is significantly reduced.

462

TABLE 7.34 Values of integral (7.61) a

s = 0.2 Accurate

Approx.

Accurate

Approx.

Accurate

Approx.

1 v/2 2 2^2 4 4^2

0.811x10-2

0.157x10-1 0.111 0.782x10-2 0.556 0.391 0.278 0.196 0.139

0.325x10-2

0.626x10-1 0.444 0.314 0.222

0.250 0.177 0.125 0.885x10-1 0.624

0.978x10-3

0.130 0.116 0.974x10-1 0.773 0.585 0.429 0.309 0.220 0.156

0.700 0.489

0.279 0.196

0.711 0.594 0.472 0.359 0.265 0.192 0.137

8.0 8v/2 16.0 16V^ 32

0.976x10-3 0.692

a

s = 1.6

1

V2 2 2V^ 4 Ay/2 8.0 8^2 16.0 16\/2 32

0.490

s = 0.4

0.285 0.239 0.190 0.144

s = 0.8

0.106 0.768x10-2

0.156 0.111 0.782x10-2

0.549

0.560

0.391

0.391 0.280 0.196

s = 2.0

0.111 0.784x10-2

0.443 0.312 0.221 0.156 0.111 0.780x10-2

5 = 4.0

Accurate

Approx.

Accurate

Approx.

Accurate

Approx.

0.483 0.454 0.404

0.718 0.684

6.28 4.44 3.14 2.22

0.247 0.177 0.126 0.890x10-^ 0.629 0.444 0.314

0.250 0.177 0.125 0.890 0.625x10-1 0.445 0.313

1.570 1.110 0.783 0.556 0.392

2.27 2.23 2.14

0.330

1.000 0.710 0.500 0.355

0.625 0.530 0.403 0.285 0.200 0.140 0.980x10-1 0.695 0.491

0.278 0.196 0.139 0.986x10-1 0.695 0.490

2.00 1.76 1.40 0.933 0.602 0.408 0.283 0.198

1.57 1.11 0.784 0.560 0.392 0.280 0.196

• Interpretation of induction logging data measured by such differential probes is mainly based on measuring the quadrature component of the electromotive force, shifted by 90° with respect to the primary electromotive force. • In such cases when the depth of investigation of the multi-coil probe is not sufficient in the radial direction, and correspondingly the apparent conductivity differs from the formation conductivity, in order to perform interpretation it is necessary to have additional information derived from either other induction probes or applying different logging methods. It is obvious that similar types of problems arise when the influence of the surrounding medium becomes essential.