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Quasistationary Field of Magnetic Dipole in a Uniform Medium
CHAPTER FIVE
Quasistationary Field of Magnetic Dipole in a Uniform Medium Contents 5.1 Expressions for the Field 5.2 Low and High Frequency Asymptotic 5.3 Expression for Induced Currents Further Reading
163 166 168 172
In developing the theory of induction logging, we focus our attention on quasistationary fields observed in the borehole in the presence of cylindrical and horizontal boundaries. However, to gain understanding of peculiarities of fields in complicated formations, it is useful to study fields in a uniform medium excited by a vertical magnetic dipole and obtain insight into the physical principles that form the basis for induction logging.
5.1 EXPRESSIONS FOR THE FIELD When a magnetic dipole with a sinusoidal current is placed in a uniform conducting medium, a change of the primary magnetic field with time causes a primary vortex electric field, and the latter gives rise to the induced currents. These currents and their interaction cause an appearance of the secondary magnetic and electric fields. Due to the symmetry, the interaction does not change a current’s direction, and in the spherical system of coordinates they have only a ϕ-component. Because the system is linear, the secondary field is also a sinusoidal function of the same frequency as the primary field. In Chapter 4, we derived equations for the electromagnetic field of the magnetic dipole in a uniform medium when both conduction and displacement currents are present. Taking into account Eqs. (4.42), (4.45), we have for the complex amplitudes of the quasistationary field:
Basic Principles of Induction Logging http://dx.doi.org/10.1016/B978-0-12-802583-3.00005-8
© 2017 Elsevier Inc. All rights reserved.
163
164
Basic Principles of Induction Logging
iμ0 ωM0 ð1 ikRÞ exp ðikRÞ sin θ 4πR2 μ M0 B∗R ¼ 0 3 ð1 ikRÞ exp ðikRÞ cos θ 2πR μ M0 B∗θ ¼ 0 3 1 ikR k2 R2 exp ðikRÞ sin θ 4πR Eφ∗ ¼
(5.1)
Here the wave number is 1+i 2 1=2 103 ¼ ð10ρT Þ1=2 m k¼ , δ¼ 2π δ γμ0 ω
(5.2)
where T is the period of oscillation and, as before, δ is the skin depth. The dipole moment varies as M ¼ M0 cos ωt
(5.3)
and, in accordance with the Biot-Savart law, it generates primary magnetic (0) fields, B(0) R and Bθ : ð0Þ
BR ¼
μ0 M0 μ M0 ð0Þ cos θ cos ωt and Bθ ¼ 0 3 sin θ cos ωt 2πR3 4πR
(5.4)
This field is confined to meridian planes and synchronously changes with the dipole current. Earlier we called this field quasistationary. Its variation with time causes the vortex electric field (Chapter 3) with complex amplitude: ð0Þ∗
Eϕ
¼
iωμ0 M0 sin θ 4πR2
and for the field E(0) ϕ we have: iωμ0 M0 ωμ M0 ð0Þ Eφ ¼ Re exp ðiωtÞ ¼ 0 2 sin θ sin ωt 2 4πR 4πR
(5.5)
(5.6)
which is confined to horizontal planes and exists at any point in space regardless of whether the medium is conductive. The primary electric and magnetic fields are shifted in-phase with respect to each other by 90 degrees. As in the general case (Chapter 4), it is convenient to express the complex amplitudes of the field in a conducting medium in terms of the primary field, that is
Quasistationary Field of Magnetic Dipole in a Uniform Medium
b∗R ¼ ð1 ikRÞexp ðikRÞ b∗θ ¼ 1 ikR k2 R2 exp ðikRÞ
165
(5.7)
e∗φ ¼ ð1 ikRÞexp ðikRÞ Inasmuch as the right-hand sides in Eq. (5.7) are complex numbers, we can say that there is a phase shift between the field and the dipole current. For instance, in the case of the radial component we have: BR ¼
μ0 M0 cos θRe½ðcR + idR Þ exp ðiωt Þ 2πR3
or BR ¼
μ0 M0 cos θ½cR cos ωt + dR sin ωt 2πR3
(5.8)
where cR + idR ¼ b∗R : By analogy, Bθ ¼
μ0 M0 sin θ½cθ cos ωt + dθ sin ωt 4πR3
(5.9)
Here cθ + idθ ¼ b∗θ In essence, the field is a sinusoidal wave that relatively rapidly decays with the distance from the dipole. We can also interpret fields as a sum of two harmonic functions, called the in-phase and quadrature components: InbR ¼ cR cos ωt QbR ¼ dR sin ωt Inbθ ¼ cθ cos ωt Qbθ ¼ dθ sin ωt
(5.10)
By definition, the real and imaginary parts of the complex amplitude are the amplitudes of the in-phase and quadrature components, respectively. The in-phase component changes synchronously with the primary field, whereas the quadrature component is shifted in-phase by 90 degrees. In general, these oscillations have different amplitudes. Similarly, the electric field and the current density can be represented as the sum of the quadrature and in-phase components. According to the Biot-Savart law, the quadrature
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Basic Principles of Induction Logging
component of the secondary magnetic field arises due to currents that are shifted in-phase by 90 degrees with respect to the current in the dipole, whereas the in-phase component of the field is the algebraic sum of the primary field and the in-phase component of the secondary field. The latter is contributed by induced currents in the medium that are in-phase with the dipole current. This representation is useful for understanding the physical principles of induction logging, which is based on measurements of corresponding components of the field. It is natural to distinguish two special cases when either radial or equatorial components exist: θ ¼ 0 (bR 6¼ 0 and bθ ¼ 0) and θ ¼ π=2 (bR ¼ 0 and bθ 6¼ 0).
5.2 LOW AND HIGH FREQUENCY ASYMPTOTIC First, consider the low frequency spectrum (or limit) of the field. Expanding exp(ikR) in the series exp ðikRÞ ¼
∞ X ðikRÞn n¼0
n!
and substituting this into Eq. (5.7) after some simple algebra, we have: ∞ X 1 n n=2 n 3πn 2 p exp i (5.11) b∗R ¼ 1 + n! 4 n¼2 Here p¼
γμ ω 1=2 R 0 R¼ 2 δ
(5.12)
is the parameter characterizing the distance between the dipole and an observation point expressed in units of skin depth δ. Sometimes the parameter p is called the induction number. Taking into account Eq. (5.11), we see that the series describing the low frequency part of the spectrum contains whole and fractional powers of ω. As follows from this equation:
or
2 2 Imb∗R ¼ dR p2 p3 and Reb∗R ¼ cR 1 p3 3 3
(5.13)
" # 2 2 3=2 μ M γμ R ð γμ R Þ 0 0 ImB∗R 0 3 cos θ ω 0 1=2 ω3=2 2πR 2 3ð2Þ
(5.14)
Quasistationary Field of Magnetic Dipole in a Uniform Medium
167
and ReB∗R
" # 3=2 μ0 M0 ðγμ0 R2 Þ 3=2 cos θ 1 ω 2πR3 3ð2Þ1=2
(5.15)
Thus, within the range of small parameter p, the quadrature and in-phase components are related to the frequency, the conductivity, and the distance from the dipole in completely different manners. The first term on the righthand side of Eq. (5.15) characterizes the primary field, which is caused only by the dipole current. The next term describes the in-phase component of the secondary magnetic field, which arises due to the currents induced in the conductive medium. At the same time, all the terms describing the quadrature component correspond to the secondary field. Comparison of the last two equations shows that the in-phase component of the secondary field is more sensitive to changes in conductivity than the first term of the quadrature component, and in this low frequency limit the in-phase component is independent of the dipole-receiver distance. In fact, this interesting feature at p ≪ 1 indicates potentially large depth of penetration of the in-phase component compared to that of the quadrature component. In a similar manner, we obtain expressions for the azimuthal component of the field: 4 4 Imb∗θ p2 + p3 and Reb∗θ 1 + p3 3 3
(5.16)
In accordance with Eq. (5.7) at the high frequency range when p ≫ 1, the in-phase and quadrature components of the field approach zero: Reb* ! 0 or Rebs* ¼ b0 and Imb* ! 0 where bs* is the complex amplitude of the secondary magnetic field. At such frequencies the induced currents are concentrated in the vicinity of the dipole causing strong skin effect. Correspondingly, the secondary in-phase component differs from the primary field by sign only. Since the radial and azimuthal components behave similarly, we may focus on the radial component: Imb∗R ¼ exp ðpÞ½ð1 + pÞ sin p p cos p Reb∗R ¼ exp ðpÞ½ð1 + pÞ cos p + p sin p
(5.17)
The graphs, illustrating dependence of both quadrature and in-phase field components on the parameter p, are presented in Fig. 5.1A and B. With an increase in the induction number, the quadrature component (Imb∗R )
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Basic Principles of Induction Logging
In b*z
0.8
Q b*z
0.6
0.4
0.4
0.2
0.2
(A)
0
2
4
6
p
0
0.5 2
4
p
(B)
Fig. 5.1 (A) Quadrature and (B) in-phase components of the magnetic field.
increases, reaches maximum, and then tends to zero. By contrast, the in-phase component decreases and then, like the quadrature component, approaches zero in an oscillating manner. According to Eq. (5.13), at the low frequency limit, the amplitude of the quadrature component prevails over the secondary in-phase component InbsR, and we have: QBR ¼
μ0 M0 γμ ω cos θ sin ωt, p ≪ 1 4πR 0
(5.18)
Hence in the range of a small parameter, the quadrature component is directly proportional to the conductivity and the frequency, and inversely proportional to the distance from the magnetic dipole. As will be shown later, some of these features of the field also remain valid in a nonuniform conducting medium. From Eq. (5.17), we also see that at p ≪ 1 the in-phase component of the secondary field InBsz is much smaller than the primary field and the quadrature component of the secondary field QBR: ð0Þ
InBsz ≪ QBR ≪ BR
(5.19)
Because of this inequality (5.19) low frequency induction measurements require high-accuracy compensation of the primary field.
5.3 EXPRESSION FOR INDUCED CURRENTS Let us analyze the behavior of the field in terms of the distribution of induced currents. Applying Eq. (5.7) and Ohm’s law: j ¼ γE
Quasistationary Field of Magnetic Dipole in a Uniform Medium
169
we have the following expression for the current density at any point in a uniform medium: jϕ∗ ¼
iγμ0 ωM0 exp ðikRÞð1 ikRÞ sin θ 4πR2
(5.20)
As in the case of the magnetic field, we represent the current density as the sum of the quadrature and in-phase components, and, using Eq. (5.20), obtain: γμ0 ωM0 r exp ðpÞ½ð1 + pÞ cos p + p sin p 4πR3 γμ ωM0 r Rejφ ¼ 0 3 exp ðpÞ½ð1 + pÞ sin p p cos p 4πR
Imjφ ¼
(5.21)
The distribution of currents represents a system of rings located in planes perpendicular to this axis (Fig. 5.2A) and having a common axis with that of the dipole. According to Eq. (5.5), for the density of induced currents arising due to the primary electric field, we have: jφð0Þ∗ ¼ γEφð0Þ∗ ¼
iγμ0 ωM0 r 4πR3
(5.22)
and their phase is shifted by 90 degrees with respect to the dipole current. If interaction between induced currents is negligible, then Eq. (5.22) describes the actual distribution. In this case, the current density at any point in the medium is a product of two terms. The first term depends on the dipole moment, frequency, and conductivity at the observation point; the second is determined by coordinates of the point of observation. Finding current distribution and magnetic field of these currents is an elementary task when interaction between induced currents is negligible and the primary electric field does not intersect any boundaries. This last condition is critical because appearance of the electric charges changes the direction of the current density; the geometry of currents becomes unknown, making it impossible to apply the Biot-Savart law. In Chapter 6 we demonstrate that the approximation based on the use of Eq. (5.22) is the foundation of Doll’s “geometrical factor theory” in “low frequency” induction logging. The behavior of amplitude of the current j(0) ϕ in planes perpendicular to the dipole axis is shown in Fig. 5.2B. It also illustrates that increase of the distance from the dipole along z-direction z1 < z2 < z3 leads to the shift of the maximal density along the radial direction.
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Basic Principles of Induction Logging
–5
z3
z2
z1 < z2 < z3
1
z1
(A)
(B) 1 0.8 0.6 0.4 0.2 0 –0.2
(C)
(D)
Fig. 5.2 (A) Geometry of current tubes; (B) distribution of current density, j0ϕ, in planes perpendicular to the dipole axes; (C), (D) quadrature and in-phase components of the current density, respectively.
Introducing notation: j0 ¼
γμ0 ωM0 r 4π R3
we may rewrite Eq. (5.21) as Qjφ ¼ j0 exp ðpÞ½ð1 + pÞ cos p + p sin p Injφ ¼ j0 exp ðpÞ½ð1 + pÞ sin p p cos p
(5.23)
Analyzing functions Eq. (5.23) we can see how the actual current density, jϕ, differs from j0 for different values of p. The quadrature and in-phase components of jϕ normalized by j0 are shown in Fig. 5.2C and D. For small
Quasistationary Field of Magnetic Dipole in a Uniform Medium
171
values of the parameter p 0:7, the quadrature component of the current density is essentially the same as j0, indicating that interaction between induced currents is negligible. As the parameter p increases, the ratio Qjϕ/j0 decreases, passes through zero, and for large p, approaches zero in an oscillating manner. The ratio In(jϕ/j0) has a completely different character. At small p the ratio Injϕ/j0 approaches zero, then increases to a maximum value at p 1:5 and tends to zero again in an oscillating manner. The actual distribution of currents, in contrast to j0, is determined by both geometric factors and the interaction of currents. Although at small values of p the quadrature component of the current density is dominant (Fig. 5.2C and D), there is a range of p where the in-phase component is significantly larger. The main features of the magnetic field can be analyzed, proceeding from the distribution of the corresponding components of the current density. If the frequency is low enough and the medium has a relatively high resistivity, the range of distances for which the actual current density Qjϕ is almost equal to j0 becomes large and the magnetic field QB is defined entirely by currents in this area. In this frequency limit the depth of investigation cannot be increased by lowering frequency despite increased penetration of the field into the formation. Both the current density Qjϕ in this area and magnetic field caused by these currents are directly proportional to the frequency, Eq. (5.22). Within some range of the parameter p, the dimensions of this volume remain much greater than the distance from the dipole to an observation point. As the parameter p increases (e.g., by an increase of the frequency), the size of this volume becomes smaller, leading to decreased growth of QB with frequency. As frequency increases further, there is a rapid decrease of both ratio Qjϕ/j0 and the quadrature component of magnetic field. By analogy, the behavior of the in-phase component of the field can be explained by the in-phase component of the current. Unlike the quadrature component Qjϕ, which is not indicative of the diffusion in the medium, the in-phase component clearly shows a diffusion process. For instance, a maximum of Injϕ moves away from the dipole when the frequency decreases, indicating an increased sensitivity of magnetic field to the distant parts of a medium. The depth of penetration of the in-phase component gradually increases with a decrease of frequency, regardless of the distance between the dipole and an observation point. This feature of the in-phase component manifests itself primarily when the separation between the dipole and receiver is comparable to or less than the thickness of the skin depth (similar behavior is observed in the transient field discussed in
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Basic Principles of Induction Logging
Chapter 10). Inasmuch as the density of the current Injϕ around the dipole is small, the field component, InB, is defined by currents located relatively far from the probe. For this reason, a change of relatively small distance between the dipole and receiver practically has no effect on the field. However, with further increase of separation, the dipole-receiver distance has greater influence. These general features of the quadrature and in-phase components of the field remain valid for a nonuniform medium as well.
FURTHER READING [1] Kaufman AA. About the theory of induction logging. Moscow: Geology and Prospecting; 1960.
Quasistationary Field of Magnetic Dipole in a Uniform Medium Contents 5.1 Expressions for the Field 5.2 Low and High Frequency Asymptotic 5.3 Expression for Induced Currents Further Reading
163 166 168 172
In developing the theory of induction logging, we focus our attention on quasistationary fields observed in the borehole in the presence of cylindrical and horizontal boundaries. However, to gain understanding of peculiarities of fields in complicated formations, it is useful to study fields in a uniform medium excited by a vertical magnetic dipole and obtain insight into the physical principles that form the basis for induction logging.
5.1 EXPRESSIONS FOR THE FIELD When a magnetic dipole with a sinusoidal current is placed in a uniform conducting medium, a change of the primary magnetic field with time causes a primary vortex electric field, and the latter gives rise to the induced currents. These currents and their interaction cause an appearance of the secondary magnetic and electric fields. Due to the symmetry, the interaction does not change a current’s direction, and in the spherical system of coordinates they have only a ϕ-component. Because the system is linear, the secondary field is also a sinusoidal function of the same frequency as the primary field. In Chapter 4, we derived equations for the electromagnetic field of the magnetic dipole in a uniform medium when both conduction and displacement currents are present. Taking into account Eqs. (4.42), (4.45), we have for the complex amplitudes of the quasistationary field:
Basic Principles of Induction Logging http://dx.doi.org/10.1016/B978-0-12-802583-3.00005-8
© 2017 Elsevier Inc. All rights reserved.
163
164
Basic Principles of Induction Logging
iμ0 ωM0 ð1 ikRÞ exp ðikRÞ sin θ 4πR2 μ M0 B∗R ¼ 0 3 ð1 ikRÞ exp ðikRÞ cos θ 2πR μ M0 B∗θ ¼ 0 3 1 ikR k2 R2 exp ðikRÞ sin θ 4πR Eφ∗ ¼
(5.1)
Here the wave number is 1+i 2 1=2 103 ¼ ð10ρT Þ1=2 m k¼ , δ¼ 2π δ γμ0 ω
(5.2)
where T is the period of oscillation and, as before, δ is the skin depth. The dipole moment varies as M ¼ M0 cos ωt
(5.3)
and, in accordance with the Biot-Savart law, it generates primary magnetic (0) fields, B(0) R and Bθ : ð0Þ
BR ¼
μ0 M0 μ M0 ð0Þ cos θ cos ωt and Bθ ¼ 0 3 sin θ cos ωt 2πR3 4πR
(5.4)
This field is confined to meridian planes and synchronously changes with the dipole current. Earlier we called this field quasistationary. Its variation with time causes the vortex electric field (Chapter 3) with complex amplitude: ð0Þ∗
Eϕ
¼
iωμ0 M0 sin θ 4πR2
and for the field E(0) ϕ we have: iωμ0 M0 ωμ M0 ð0Þ Eφ ¼ Re exp ðiωtÞ ¼ 0 2 sin θ sin ωt 2 4πR 4πR
(5.5)
(5.6)
which is confined to horizontal planes and exists at any point in space regardless of whether the medium is conductive. The primary electric and magnetic fields are shifted in-phase with respect to each other by 90 degrees. As in the general case (Chapter 4), it is convenient to express the complex amplitudes of the field in a conducting medium in terms of the primary field, that is
Quasistationary Field of Magnetic Dipole in a Uniform Medium
b∗R ¼ ð1 ikRÞexp ðikRÞ b∗θ ¼ 1 ikR k2 R2 exp ðikRÞ
165
(5.7)
e∗φ ¼ ð1 ikRÞexp ðikRÞ Inasmuch as the right-hand sides in Eq. (5.7) are complex numbers, we can say that there is a phase shift between the field and the dipole current. For instance, in the case of the radial component we have: BR ¼
μ0 M0 cos θRe½ðcR + idR Þ exp ðiωt Þ 2πR3
or BR ¼
μ0 M0 cos θ½cR cos ωt + dR sin ωt 2πR3
(5.8)
where cR + idR ¼ b∗R : By analogy, Bθ ¼
μ0 M0 sin θ½cθ cos ωt + dθ sin ωt 4πR3
(5.9)
Here cθ + idθ ¼ b∗θ In essence, the field is a sinusoidal wave that relatively rapidly decays with the distance from the dipole. We can also interpret fields as a sum of two harmonic functions, called the in-phase and quadrature components: InbR ¼ cR cos ωt QbR ¼ dR sin ωt Inbθ ¼ cθ cos ωt Qbθ ¼ dθ sin ωt
(5.10)
By definition, the real and imaginary parts of the complex amplitude are the amplitudes of the in-phase and quadrature components, respectively. The in-phase component changes synchronously with the primary field, whereas the quadrature component is shifted in-phase by 90 degrees. In general, these oscillations have different amplitudes. Similarly, the electric field and the current density can be represented as the sum of the quadrature and in-phase components. According to the Biot-Savart law, the quadrature
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Basic Principles of Induction Logging
component of the secondary magnetic field arises due to currents that are shifted in-phase by 90 degrees with respect to the current in the dipole, whereas the in-phase component of the field is the algebraic sum of the primary field and the in-phase component of the secondary field. The latter is contributed by induced currents in the medium that are in-phase with the dipole current. This representation is useful for understanding the physical principles of induction logging, which is based on measurements of corresponding components of the field. It is natural to distinguish two special cases when either radial or equatorial components exist: θ ¼ 0 (bR 6¼ 0 and bθ ¼ 0) and θ ¼ π=2 (bR ¼ 0 and bθ 6¼ 0).
5.2 LOW AND HIGH FREQUENCY ASYMPTOTIC First, consider the low frequency spectrum (or limit) of the field. Expanding exp(ikR) in the series exp ðikRÞ ¼
∞ X ðikRÞn n¼0
n!
and substituting this into Eq. (5.7) after some simple algebra, we have: ∞ X 1 n n=2 n 3πn 2 p exp i (5.11) b∗R ¼ 1 + n! 4 n¼2 Here p¼
γμ ω 1=2 R 0 R¼ 2 δ
(5.12)
is the parameter characterizing the distance between the dipole and an observation point expressed in units of skin depth δ. Sometimes the parameter p is called the induction number. Taking into account Eq. (5.11), we see that the series describing the low frequency part of the spectrum contains whole and fractional powers of ω. As follows from this equation:
or
2 2 Imb∗R ¼ dR p2 p3 and Reb∗R ¼ cR 1 p3 3 3
(5.13)
" # 2 2 3=2 μ M γμ R ð γμ R Þ 0 0 ImB∗R 0 3 cos θ ω 0 1=2 ω3=2 2πR 2 3ð2Þ
(5.14)
Quasistationary Field of Magnetic Dipole in a Uniform Medium
167
and ReB∗R
" # 3=2 μ0 M0 ðγμ0 R2 Þ 3=2 cos θ 1 ω 2πR3 3ð2Þ1=2
(5.15)
Thus, within the range of small parameter p, the quadrature and in-phase components are related to the frequency, the conductivity, and the distance from the dipole in completely different manners. The first term on the righthand side of Eq. (5.15) characterizes the primary field, which is caused only by the dipole current. The next term describes the in-phase component of the secondary magnetic field, which arises due to the currents induced in the conductive medium. At the same time, all the terms describing the quadrature component correspond to the secondary field. Comparison of the last two equations shows that the in-phase component of the secondary field is more sensitive to changes in conductivity than the first term of the quadrature component, and in this low frequency limit the in-phase component is independent of the dipole-receiver distance. In fact, this interesting feature at p ≪ 1 indicates potentially large depth of penetration of the in-phase component compared to that of the quadrature component. In a similar manner, we obtain expressions for the azimuthal component of the field: 4 4 Imb∗θ p2 + p3 and Reb∗θ 1 + p3 3 3
(5.16)
In accordance with Eq. (5.7) at the high frequency range when p ≫ 1, the in-phase and quadrature components of the field approach zero: Reb* ! 0 or Rebs* ¼ b0 and Imb* ! 0 where bs* is the complex amplitude of the secondary magnetic field. At such frequencies the induced currents are concentrated in the vicinity of the dipole causing strong skin effect. Correspondingly, the secondary in-phase component differs from the primary field by sign only. Since the radial and azimuthal components behave similarly, we may focus on the radial component: Imb∗R ¼ exp ðpÞ½ð1 + pÞ sin p p cos p Reb∗R ¼ exp ðpÞ½ð1 + pÞ cos p + p sin p
(5.17)
The graphs, illustrating dependence of both quadrature and in-phase field components on the parameter p, are presented in Fig. 5.1A and B. With an increase in the induction number, the quadrature component (Imb∗R )
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Basic Principles of Induction Logging
In b*z
0.8
Q b*z
0.6
0.4
0.4
0.2
0.2
(A)
0
2
4
6
p
0
0.5 2
4
p
(B)
Fig. 5.1 (A) Quadrature and (B) in-phase components of the magnetic field.
increases, reaches maximum, and then tends to zero. By contrast, the in-phase component decreases and then, like the quadrature component, approaches zero in an oscillating manner. According to Eq. (5.13), at the low frequency limit, the amplitude of the quadrature component prevails over the secondary in-phase component InbsR, and we have: QBR ¼
μ0 M0 γμ ω cos θ sin ωt, p ≪ 1 4πR 0
(5.18)
Hence in the range of a small parameter, the quadrature component is directly proportional to the conductivity and the frequency, and inversely proportional to the distance from the magnetic dipole. As will be shown later, some of these features of the field also remain valid in a nonuniform conducting medium. From Eq. (5.17), we also see that at p ≪ 1 the in-phase component of the secondary field InBsz is much smaller than the primary field and the quadrature component of the secondary field QBR: ð0Þ
InBsz ≪ QBR ≪ BR
(5.19)
Because of this inequality (5.19) low frequency induction measurements require high-accuracy compensation of the primary field.
5.3 EXPRESSION FOR INDUCED CURRENTS Let us analyze the behavior of the field in terms of the distribution of induced currents. Applying Eq. (5.7) and Ohm’s law: j ¼ γE
Quasistationary Field of Magnetic Dipole in a Uniform Medium
169
we have the following expression for the current density at any point in a uniform medium: jϕ∗ ¼
iγμ0 ωM0 exp ðikRÞð1 ikRÞ sin θ 4πR2
(5.20)
As in the case of the magnetic field, we represent the current density as the sum of the quadrature and in-phase components, and, using Eq. (5.20), obtain: γμ0 ωM0 r exp ðpÞ½ð1 + pÞ cos p + p sin p 4πR3 γμ ωM0 r Rejφ ¼ 0 3 exp ðpÞ½ð1 + pÞ sin p p cos p 4πR
Imjφ ¼
(5.21)
The distribution of currents represents a system of rings located in planes perpendicular to this axis (Fig. 5.2A) and having a common axis with that of the dipole. According to Eq. (5.5), for the density of induced currents arising due to the primary electric field, we have: jφð0Þ∗ ¼ γEφð0Þ∗ ¼
iγμ0 ωM0 r 4πR3
(5.22)
and their phase is shifted by 90 degrees with respect to the dipole current. If interaction between induced currents is negligible, then Eq. (5.22) describes the actual distribution. In this case, the current density at any point in the medium is a product of two terms. The first term depends on the dipole moment, frequency, and conductivity at the observation point; the second is determined by coordinates of the point of observation. Finding current distribution and magnetic field of these currents is an elementary task when interaction between induced currents is negligible and the primary electric field does not intersect any boundaries. This last condition is critical because appearance of the electric charges changes the direction of the current density; the geometry of currents becomes unknown, making it impossible to apply the Biot-Savart law. In Chapter 6 we demonstrate that the approximation based on the use of Eq. (5.22) is the foundation of Doll’s “geometrical factor theory” in “low frequency” induction logging. The behavior of amplitude of the current j(0) ϕ in planes perpendicular to the dipole axis is shown in Fig. 5.2B. It also illustrates that increase of the distance from the dipole along z-direction z1 < z2 < z3 leads to the shift of the maximal density along the radial direction.
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Basic Principles of Induction Logging
–5
z3
z2
z1 < z2 < z3
1
z1
(A)
(B) 1 0.8 0.6 0.4 0.2 0 –0.2
(C)
(D)
Fig. 5.2 (A) Geometry of current tubes; (B) distribution of current density, j0ϕ, in planes perpendicular to the dipole axes; (C), (D) quadrature and in-phase components of the current density, respectively.
Introducing notation: j0 ¼
γμ0 ωM0 r 4π R3
we may rewrite Eq. (5.21) as Qjφ ¼ j0 exp ðpÞ½ð1 + pÞ cos p + p sin p Injφ ¼ j0 exp ðpÞ½ð1 + pÞ sin p p cos p
(5.23)
Analyzing functions Eq. (5.23) we can see how the actual current density, jϕ, differs from j0 for different values of p. The quadrature and in-phase components of jϕ normalized by j0 are shown in Fig. 5.2C and D. For small
Quasistationary Field of Magnetic Dipole in a Uniform Medium
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values of the parameter p 0:7, the quadrature component of the current density is essentially the same as j0, indicating that interaction between induced currents is negligible. As the parameter p increases, the ratio Qjϕ/j0 decreases, passes through zero, and for large p, approaches zero in an oscillating manner. The ratio In(jϕ/j0) has a completely different character. At small p the ratio Injϕ/j0 approaches zero, then increases to a maximum value at p 1:5 and tends to zero again in an oscillating manner. The actual distribution of currents, in contrast to j0, is determined by both geometric factors and the interaction of currents. Although at small values of p the quadrature component of the current density is dominant (Fig. 5.2C and D), there is a range of p where the in-phase component is significantly larger. The main features of the magnetic field can be analyzed, proceeding from the distribution of the corresponding components of the current density. If the frequency is low enough and the medium has a relatively high resistivity, the range of distances for which the actual current density Qjϕ is almost equal to j0 becomes large and the magnetic field QB is defined entirely by currents in this area. In this frequency limit the depth of investigation cannot be increased by lowering frequency despite increased penetration of the field into the formation. Both the current density Qjϕ in this area and magnetic field caused by these currents are directly proportional to the frequency, Eq. (5.22). Within some range of the parameter p, the dimensions of this volume remain much greater than the distance from the dipole to an observation point. As the parameter p increases (e.g., by an increase of the frequency), the size of this volume becomes smaller, leading to decreased growth of QB with frequency. As frequency increases further, there is a rapid decrease of both ratio Qjϕ/j0 and the quadrature component of magnetic field. By analogy, the behavior of the in-phase component of the field can be explained by the in-phase component of the current. Unlike the quadrature component Qjϕ, which is not indicative of the diffusion in the medium, the in-phase component clearly shows a diffusion process. For instance, a maximum of Injϕ moves away from the dipole when the frequency decreases, indicating an increased sensitivity of magnetic field to the distant parts of a medium. The depth of penetration of the in-phase component gradually increases with a decrease of frequency, regardless of the distance between the dipole and an observation point. This feature of the in-phase component manifests itself primarily when the separation between the dipole and receiver is comparable to or less than the thickness of the skin depth (similar behavior is observed in the transient field discussed in
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Basic Principles of Induction Logging
Chapter 10). Inasmuch as the density of the current Injϕ around the dipole is small, the field component, InB, is defined by currents located relatively far from the probe. For this reason, a change of relatively small distance between the dipole and receiver practically has no effect on the field. However, with further increase of separation, the dipole-receiver distance has greater influence. These general features of the quadrature and in-phase components of the field remain valid for a nonuniform medium as well.
FURTHER READING [1] Kaufman AA. About the theory of induction logging. Moscow: Geology and Prospecting; 1960.