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The Field of the Vertical Electric Dipole in the Layered Medium
APPENDIX FOUR
The Field of the Vertical Electric Dipole in the Layered Medium INTRODUCTION Here we consider the electromagnetic field generated in the Earth by a vertical electric dipole. The material complements Chapter 13, where we described the field generated by a horizontal electric dipole located at the Earth’s surface. In electrical prospecting on land, a horizontal electric dipole, grounded at its ends, is the most natural electric-field source for use in exploration geophysics, except in the case where the field is being measured at relatively high frequencies (20 kHz or more), far away from the source. In marine electrical prospecting, however, it is easy to use vertical dipoles both as the source of the field and as receivers. This configuration has potential applications in low-frequency electromagnetic exploration for offshore oil fields. O. Nazarenko first suggested the vertical-dipole configuration for electromagnetic sounding in 1961 (Russia); the method was revived through work by Nigel Edwards and others starting around 1981 (Canada). A brief history of electrical methods in marine exploration and in exploration for oil follows. The first application of electrical methods in a marine setting probably took place in Russia. Around 1930, large-scale electrical resistivity surveys for shallow oil fields were carried out both on land and offshore in the Caucasus region by teams of scientists working for a company owned by the Schlumberger brothers in a joint venture with the Soviet oil trusts. More than 70 years ago, profiling and sounding with electric dipoles were introduced for oil and mining exploration by L. Alpin, who developed the theory and interpretation of this method for direct (steady or time invariant) current flow. In contrast to the sounding methods introduced in the early 1900s by Conrad Schlumberger (later called “Schlumberger soundings”), dipole soundings do not require the use of long wires. For this reason, the dipole soundings were easily adapted to marine conditions, especially in shallow waters with depth less than 200 m. Marine dipole soundings were been used in the Azov Sea and Caspian Sea, as well as in the Sea of Okhotsk and other marine settings to study the basement surface, as well as the resistivity Methods in Geochemistry and Geophysics, Volume 45 ISSN 0076-6895, http://dx.doi.org/10.1016/B978-0-444-53829-1.19004-5
Ó 2014 Elsevier B.V. All rights reserved.
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structure of sediments. Of course, dipole soundings using direct current (DC) can be used to probe deep inside the Earth only when very resistive layers are not present within sediments, since such layers will form a barrier to the penetration of the current, screening deeper layers. Marine electric dipole measurements, like land measurements, were initially carried out with arrays arranged with different distances between the current dipole source and the dipole receivers. Later, at the beginning of the 1960s, transient soundings with measurements of the vertical component of the magnetic field were used in Baltic and Okhotsk Seas to provide sensitivity to deep layers even in the presence of very resistive layers. Dipole soundings were usually carried out with two shipsda source (current) ship and a receiver ship. The surveys generally used axial dipole arrays (recall that an axial dipole array is arranged so that the dipole direction for a given electrode pair lies along the same line as the profile direction). There were two main surveying methods: a two-sided method and a symmetrical (midpoint) method. In the first configuration, the ship towing the receiver array is held steady with the measurement line MN at a point along the profile where the natural noise level is low. The ship towing the current array then moves continuously toward the receiver array, until it passes by the receiver array and continues on a straight path away from it on the other side. In the second arrangement, the two ships start at a large distance apart and move toward each other at the same speed, so that their midpoint remains fixed. Marine induced polarization surveys have been used for more than 30 years to detect ore bodies beneath the sea bottom. Induced polarization methods also had a brief history in oil exploration. During the 1960s it was discovered that there could sometimes be accumulations, above oil fields, of pyrites that generate an induced polarization effect. But attempts to use induced polarization to locate oil reservoirs did not show consistent correlations between shallow pyrite halos and deep oil fields, and the method has more or less been abandoned. Standard electric and electromagnetic soundings have been used for many years as secondary techniques in oil exploration to delineate large structures beneath of the surface or sea bottom. But until recently these methods had rarely been deployed directly to detect the reservoir itself. Possibly the first successful applications of electromagnetic sounding for oil were transient soundings on land in the near zone that were carried out in the early 1970s to locate the depth of the oil–water contact in reservoirs. As shown in Chapter 9, when conditions are favorable, it is possible to find the
The Field of the Vertical Electric Dipole in the Layered Medium
729
boundary between the portion of a porous reservoir layer that is saturated with gas or oildand is therefore highly resistivedand the conductive part that is saturated with saltwater. When the medium surrounding the reservoir is itself resistive, the presence of an oil–water contact can be detected by inducing currents in the zone saturated with saltwater, which becomes the source of the secondary electromagnetic field. In 1981, Len Srnka (Exxon) suggested using marine electromagnetic surveys to study the distribution of resistivity in sedimentary basins that may contain petroleum systems. Sedimentary basins are generally composed of sandstone, limestone, and shale layers that, aside from the reservoir layer itself, are saturated with saltwater. The resistive oil-saturated reservoir layer cannot easily be detected by induction, because the induced currents will flow mainly in the surrounding conductive layers that can extend for large distances above and below the reservoir zone. For this reason, it was suggested that a horizontal grounded source near the sea or ocean bottom could be used to map the presence of an oil layer. A highly resistive layer in an otherwise conductive sedimentary column will impede the flow of vertical current from a grounded electrical dipole source and change the rate of decay of the electric field with distance from the source. This method, called marine controlled-source electromagnetic surveying, is now a standard technique in offshore oil exploration. Measurements of the electric and magnetic field are mainly performed with arrays having horizontal current and receiver dipoles, in configurations similar to the ones first used by Alpin. In the future, arrays with the vertical electric dipoles will undoubtedly find more application in oil exploration, so it is interesting to study the field of such a current source in the presence of a resistive reservoir layer. But before we solve the problem for a layer of finite thickness, we examine the case where the thickness of the resistive part of reservoir is much smaller than the other dimensions, so that it can be replaced by a plane with transversal resistivity T. We derive boundary conditions for the field at such a surface, which enables full solution of the boundary value problem for a thin resistive layer without having to solve for the field in the layer itself.
A4.1. BOUNDARY CONDITIONS AT THE SURFACE OF A PLANE T Suppose that a thin resistive layer is situated at a distance H from the origin and surrounded by a conducting medium as is shown in Figure A4.1(a).
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(a)
(b)
x ρ1
z
(1)
hi
(2)
(1)
(i)
(i)
Ez (a1)
ρi
Ez (a2)
L1
(2)
ρ1
h1
Ex , Bx
(1)
ρ0
H
O
Idz
x
ρ2
T
(2)
Ex , Bx
ρ2
L2
ρ2
z
Figure A4.1 (a) Illustration used in deriving boundary conditions at plane T. (b) Twolayer structure with plane T embedded in a basement layer.
The transversal resistance of this layer T is defined as T ¼ hi ri ;
(A4.1)
where hi and ri are the thickness and resistivity of this layer, respectively. If we assume that the layer is sufficiently thin, we can make use of the approximate boundary conditions at its surface. Applying the second Maxwell’s equation in the integral form: ð þ B$dl ¼ m0 j$dS L
S
to the path L1 in Figure A4.1(a), we have ð2Þ Bð1Þ x dx Bx dx ¼ m0 jy hi dx;
or
hi ð2Þ Bð1Þ x Bx ¼ m0 jy hi ¼ m0 Ey : ri (A4.2)
Similarly, along path L2 hi ð2Þ Bð1Þ y By ¼ m0 jx hi ¼ m0 Ex : ri
(A4.3)
Because we assume that the resistivity ri is very high, it is proper to neglect any currents flowing along the thin layer, and so we have ð2Þ Bð1Þ x ¼ Bx ;
ð2Þ Bð1Þ y ¼ By :
(A4.4)
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The Field of the Vertical Electric Dipole in the Layered Medium
Thus, the horizontal components of the magnetic field have the same values at both surface of the thin layer. Using the first Maxwell equation þ vF E$dl ¼ vt L
and applying it along the path L1, we obtain Exð1Þ dx EzðiÞ ða2 Þhi Exð2Þ dx þ EzðiÞ ða1 Þhi ¼
vBy hi dx; vt
where Ezi ða2 Þ and Ezi ða1 Þ are values of the normal component of the electric field inside the layer at points a2 and a1, respectively. As the thickness hi tends to zero, the flux caused by By also tends to zero, and we have for the complex amplitudes of the electric field Exð1Þ Exð2Þ dx þ Ezi ða2 Þ Ezi ða1 Þ hi ¼ 0: It is obvious that
Ezi ða2 Þ Ezi ða1 Þ ¼
vEzi dx vx
and hence
Exð2Þ Exð1Þ ¼ hi
vEzi : vx
(A4.5)
By analogy, applying the same equation to the path L2, we have Eyð2Þ
Eyð1Þ
vEi ¼ hi z : vy
(A4.6)
The last two equations can be rewritten in the following form
Exð2Þ Exð1Þ
ðiÞ
ðiÞ
vEz vg Ez ¼ hi ri gi ¼T i vx vx
¼T
vjz vx
and
Eyð2Þ Eyð1Þ
ðiÞ
ðiÞ
vEz vg Ez ¼T i ¼ h i ri gi vy vy
¼T
vjz ; vy
(A4.7)
where gi ¼ 1/ri, jz is the vertical component of current density inside the thin layer, and T is the transversal resistance. Inasmuch as the horizontal component of the current inside a thin resistive layer is negligible, its vertical
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component is a continuous function; that is it has the same values at both sides of the plate T:
gi EzðiÞ ¼ g1 Ezð1Þ ¼ g2 Ezð2Þ : ð1Þ
ð2Þ
Here g1 ; Ez and g2 ; Ez are conductivity and the vertical component of the electric field outside the plate T, but at its vicinity. Correspondingly, the boundary conditions at the plane T assume the following form: ð1Þ
Bx
ð2Þ
Ex
ð2Þ
Ey
ð2Þ
¼ Bx ;
ð1Þ
By
ð2Þ
¼ By ; ð1Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
¼ T g1 vEvxz ¼ T g2 vEvxz ;
ð1Þ
¼ T g1 vEvyz ¼ T g2 vEvyz :
Ex Ey
(A4.8)
These boundary conditions do not require information about the behavior of the field components inside the layer, a feature that markedly simplifies a solution of the boundary value problem. Unlike the tangential components of the magnetic field, the horizontal components of the electric field at the surface of the thin resistive layer are discontinuous functions. From the theory for the stationary field it is known that the tangential components of the electric field can be discontinuous only across a surface of a double layer with the changing surface density of dipole moments: ð2Þ
Et
ð1Þ
Et
1 vh ¼ ; ε0 vt
(A4.9)
where ε0 is the dielectric constant and “t” indicates a tangential direction. Comparing Eqs (A4.8) and (A4.9) we can say that there are electric charges with opposite signs on either side of the thin layer. For this reason, each surface element can be considered as being a dipole with a moment h given by h ¼ T g1 ε0 Ezð1Þ ¼ T g2 ε0 Ezð2Þ :
(A4.10)
Since we usually assume that conductivities above and beneath plane T are equal, the normal components coincide top, that is the simple layer of charges is absent. Therefore in place of Eq. (A4.10) we have hðpÞ ¼ Tgε0 Ez ðpÞ:
The Field of the Vertical Electric Dipole in the Layered Medium
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Note that the electric charges on the surface of the horizontal plane T create an electromagnetic field, which is equivalent to that caused by a system of vertical electric dipoles. Each elementary dipole is the source of a horizontal magnetic field, while the vertical component of this field is absent.
A4.2. MECHANISM OF APPEARANCE OF THE SECONDARY FIELD In this light let us notice that the presence of the thin resistive layer may be detected only due to charges arising at both its sides. They have opposite signs and equal magnitudes if resistivity of a medium above and beneath is the same. In fact, as is well known from the theory of a quasi-stationary field the surface charge density on the boundary between two media (i) and (j) is defined from the same equation as in the case of the time invariant field: sðpÞ ¼ 2ε0 Kij Enav ðpÞ: Here Lij ¼
rj ri ; rj þ ri
where ri and rj are resistivity of a medium, but Enav is the normal component of the total field at the same point p caused by all charges, except charge at the point p. Also this field includes the vortex part generated by a change of the magnetic field with time. It is essential that the normal n is directed from medium (i) to medium (j) that gives charges of different sign at the upper and lower surfaces of the thin layer. By definition, we may assume that Kij ¼ 1. As was pointed out it is natural to treat the field Enav as a sum of the normal and secondary fields, where the latter field is due to the presence of the plane T. It is obvious that if the normal electric field does not have component perpendicular to this layer the charges are absent, and the secondary field is equal to zero. The classical example of such field is the vertical magnetic dipole, since its vortex electric field is tangential to this horizontal layer, and correspondingly the secondary dipole moment is zero. Therefore, the necessary condition for detecting the thin resistive layer is the presence of the relatively strong component of the normal electric field perpendicular to this layer. For simplicity we assume that the thin layer has an infinite extension in horizontal direction, which will allow us in a simple manner to solve
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boundary value problem and obtain an expression for the electric field in the explicit form. In spite of such approximation we will be able to obtain some useful information about an influence of frequency or time, as well as a separation between the current and receiver electrodes on the depth of investigation for different resistivity structure parameters. In our case an array consists of two parallel vertical lines and length of which may be comparable with the distance between these current and receiver lines, again for simplicity we replace them by dipoles. At the same time transition to real system can be done by summation of the fields of dipoles along these lines. In such case the main attention will be paid to the vertical component of the electric field. At the beginning we consider the normal field in two-layer medium, shown in Figure A4.1(b).
A4.3. EXPRESSIONS FOR THE NORMAL FIELD AT THE SEA BOTTOM Let us choose the Cartesian system coordinates with the origin 0, located on the sea bottom and suppose that the vertical electric dipole is placed beneath the sea surface at the point 0 and directed along the z-axis, Figure A4.1(b). As usual we seek a solution in the frequency domain in terms of the vector potential, A. Its complex amplitude satisfies the Helmholtz equation in each layer: V2 A þ k2 A ¼ 0: As is known, B ¼ curl A ;
E ¼ iuA þ
1 grad div A : gm0
(A4.11)
A4.3.1. Boundary Conditions for the Vector Potential Taking into account the axial symmetry of the field and the model of a medium, we will try to solve the boundary value problem with help of one component of the vector potential Az. Then, as follows from Eq. (A4.11), we have Bx ¼
vAz ; vy
By ¼
vAz ; vx
Bz ¼ 0
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The Field of the Vertical Electric Dipole in the Layered Medium
and Ex ¼
1 v2 Az ; gm0 vxvz
Ey ¼
1 v2 Az ; gm0 vyvz
Ez ¼ iuAz þ
1 v2 Az : (A4.12) gm0 vz2
From continuity of the tangential components of the magnetic and electric fields at sea surface and sea bottom, we have ð0Þ
Az
ð1Þ Az
ð1Þ
¼ Az ¼
ð2Þ Az
ð0Þ
ð1Þ
and
r0 vAvzz ¼ r1 vAvzz ;
and
r1 vAvzz
ð1Þ
¼
ð2Þ
r2 vAvzz
;
if z ¼ h1 :
(A4.13)
if z ¼ h:
Here r0, r1, and r2 are resistivity of the upper space (air), the seawater, and a medium beneath, respectively, and h1 is the thickness of the seawater, Figure A4.1(b). At the surface T, we have ð1Þ
Az
ð2Þ
vAz vz
ð2Þ
¼ Az ; ð1Þ
vAvzz
¼ Tg
ð2Þ k22 Az
ð2Þ
2 Az þ v vz 2
(A4.14) ;
if z ¼ H:
Note that in deriving Eqs (A4.13) and (A4.14), we used the fact that from continuity of function’s tangential derivatives at the boundary follows the continuity of the function itself.
A4.3.2. Expressions for the Vector Potential Az of the Normal Field As was shown in the Chapter 13 the vector potential in a uniform medium is written as Az
m Idz expðik1 RÞ m0 Idz ¼ 0 ¼ 4p R 4p
N ð 0
m expðm1 jzjÞJ0 ðmrÞdm: m1
(A4.15)
Correspondingly, we have the following expressions for the complex amplitudes of the vector potential Að0Þ z
m Idz ¼ 0 4p
N ð
C0 expðm0 zÞJ0 ðmrÞdm; 0
if z < h1 ;
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Að1Þ z
m Idz ¼ 0 4p
N ð 0
m expðm1 jzjÞ þ C1 expðm1 zÞ þ D1 expðm1 zÞ m1
J0 ðmrÞdm if h1 < z < 0; (A4.16) Að2Þ z
m Idz ¼ 0 4p
N ð
D2 expðm2 zÞJ0 ðmrÞdm;
if z > 0:
0
Here 1=2 1=2 1=2 m0 ¼ m2 k20 ; m1 ¼ m2 k21 m2 ¼ m2 k22 : Substituting the latter into boundary conditions, we obtain the system of equations for determination of unknowns m C0 expðm0 h1 Þ ¼ expðm1 h1 Þ þ C1 expðm1 h1 Þ þ D1 expðm1 h1 Þ; m1 r0 m0 C0 expðm0 h1 Þ ¼ r1 ½m expðm1 h1 Þ þ m1 C1 expðm1 h1 Þ m1 D1 expðm1 h1 Þ; (A4.17) D2 ¼
m þ C1 þ D1 ; m1
r2 m2 D2 ¼ r1 ð m þ m1 C1 m1 D1 Þ: In deriving this system we assume at the beginning that the dipole was located slightly above the boundary z ¼ 0. Eliminating C0 and D2, we have: m m01 expð2m1 h1 Þ ¼ m01 expð2m1 h1 ÞC1 þ D1 ; m1 m m12 ¼ C1 m12 D1 ; m1 0 r1 m1 1 r2 m2 ; m12 ¼ rr1 m . where m01 ¼ rro m o m0 þr1 m1 1 m1 þr2 m2 This gives
C1 ¼
m 1 m01 expð2m1 h1 Þ m12 m1 1 þ m01 m12 expð2m1 h1 Þ
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The Field of the Vertical Electric Dipole in the Layered Medium
and m 1 þ m12 : D1 ¼ m01 expð2m1 h1 Þ 1 þ m01 m12 expð2m1 h1 Þ m1
(A4.18)
Further we will be mainly interested by the case when the upper medium is an insulator, that is m01 ¼ 1. Then we have C1 ¼ and D1 ¼
m 1 expð2m1 h1 Þ m12 m1 1 þ m12 expð2m1 h1 Þ
m 1 þ m12 expð2m1 h1 Þ : 1 þ m12 expð2m1 h1 Þ m1
(A4.19)
A4.3.3. Expressions for the Normal Field Beneath Sea Bottom Taking into account that at the cylindrical system of coordinates with the origin 0, we have 1r r14 1z 1 v v v B ¼ vr v4 vz r 0 0 Az Therefore B4 ¼
vAz : vr
As follows from Eq. (A4.16) B4
¼
m0 Idz B0 4 ðk1 RÞ þ 4p
N ð
m½C1 expðm1 zÞ þ D1 expðm1 zÞ J1 ðmrÞdm 0
(A4.20) Here B0 4 ðk1 RÞ is the magnetic field of the electric dipole in a uniform medium with resistivity r1. In particular, in the plane z ¼ 0, we have B4
¼
B0 4 ðk1 rÞ
m Idz þ 0 4p
N ð
mðC1 þ D1 ÞJ1 ðmrÞdm; 0
if z ¼ 0: (A4.21)
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For the radial and vertical components of the electric field, we have ¼ E1r
1 v2 A1z g1 m0 vrvz
and E1z ¼ iuA1z þ
Then Eq. (A4.16) gives E1r
¼
r Idz 0 E1r ðk1 RÞ 1 4p
1 v2 A1z : g1 m0 vz2
N ð
mm1 ½C1 expðm1 zÞ 0
D1 expðm1 zÞ J1 ðmrÞdm and E1z
¼
r Idz 0 E1z ðk1 RÞ þ 1
N ð
4p
m2 ½C1 expðm1 zÞ
0
þ D1 expðm1 zÞ J0 ðmrÞdm:
(A4.22)
At the points of the plane z ¼ 0 the latter gives E1r
¼
r Idz 0 ðk1 rÞ 1 E1r 4p
and E1z
¼
r Idz 0 E1z ðk1 rÞ þ 1 4p
N ð
N ð
mm1 ðC1 D1 ÞJ1 ðmrÞdm 0
m2 ðC1 þ D1 ÞJ0 ðmrÞdm
(A4.23)
0
Note, that we will consider only the vertical component of this field. In general, determination of the field requires numerical integration, except one case when r1 ¼ r2, and it can be expressed by elementary functions.
A4.3.4. The Normal Field When r1 [ r2 (Uniform Half-Space) As follows from Eq. (A4.19), we have C1 ¼ 0
and
m D1 ¼ expð2m1 h1 Þ; m1
(A4.24)
since m12 ¼ 0. Therefore, taking into account Somerfield integral, we have r1 I expðik1 RÞ expðik1 R1 Þ Að1Þ ; if z < 0: (A4.25) ¼ z 4p R R1
The Field of the Vertical Electric Dipole in the Layered Medium
739
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here R ¼ x2 þ y2 þ z2 ; R1 ¼ x2 þ y2 þ ðz þ 2h1 Þ2 . This means that as in the case of the time invariant field the electromagnetic field is represented as the field of two electric dipoles, located symmetrically with respect to the boundary. Their moments have the same magnitude but opposite directions. It may be proper to note that this is the rare case when the electromagnetic field of the dipole located beneath the boundary is expressed in terms of elementary functions. It is obvious that the normal component of the electric field is equal to zero at the conducting side of the sea surface as well as the magnetic field. B4 ðr; h1 Þ ¼ 0:
(A4.26)
The latter follows from the fact that near the surface, currents have only tangential component and the field is independent on the angle 4. Correspondingly, in the quasi-stationary approximation the magnetic field is absent in the upper nonconducting half-space. Note that the same is true in the case when instead of a conducting uniform half-space there is a horizontally layered medium. Absence of the magnetic field indicates that electromagnetic energy does not propagate through the upper space, if displacement currents are neglected in the upper halfspace. From Eq. (A4.25) we see that the normal field beneath the sea surface consists of the direct and reflected wave, and the latter is equivalent to the field of the electric dipole in a uniform medium, as if it is located above the boundary at the point, which is mirror reflection of the origin where the source is placed. As follows from Eq. (A4.25) the amplitude of the reflected wave does not exceed that of the direct one and with an increase of the distance both waves begin to decay as exponential functions. In particular, at relatively large distances, R [ h1, these waves almost cancel each other. In a fact we have 1 0 2h21 exp ik 1R r I expðik1 RÞ @ Az Að1Þ ¼ 1 1 z 2h2 4p R 1 þ 21 R
i2k1 h21
r1 I expðik1 RÞ ; 4p R2
if R/N:
(A4.27)
In the Chapter 13 it was shown that in a uniform medium and in the spherical system of coordinates the complex amplitudes of the field components ER and Eq are
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2r1 Idz expðik1 RÞð1 ik1 RÞcos q and Eq 4pR3 r Idz ¼ 1 3 expðik1 RÞ 1 ik1 R k21 R2 sin q: 4pR
ER ¼
Also B4 ¼ By ¼ Then
(A4.28)
m0 Idz expðikRÞð1 ikRÞsin q 4pr 2
Ez ¼ ER cos q Eq sin q;
and therefore the field in the presence of the upper boundary is:
r Idz Ez ¼ 1 3 expðik1 RÞ 2ð1 ik1 RÞcos2 q 4pR r Idz 1 ik1 R k21 R2 1 cos2 q þ 1 3 expðik1 R1 Þ 4pR1
2ð1 ik1 R1 Þcos2 q1 1 ik1 R1 k21 R12 1 cos2 q1 or Ez
r1 I expðik1 RÞ z2 2 2 1 þ ik1 R þ k1 R þ ½3 ik1 Rð3 ik1 RÞ 2 ¼ R 4p R3 ( r I expðik1 R1 Þ 1 1 þ ik1 R1 þ k21 R12 3 4p R1 ) ð2h1 þ zÞ2 þ ½3 ik1 R1 ð3 ik1 R1 Þ ; R12 (A4.29)
since cos q ¼
z R
and cos q1 ¼
2h1 þ z R1
For the magnetic field, we have m0 Idz 1 1 expðik1 RÞð1 ik1 RÞsin q 2 expðik1 R1 Þ B4 ¼ 4p R2 R1
ð1 ik1 R1 Þsin q1
(A4.30)
The Field of the Vertical Electric Dipole in the Layered Medium
741
These equations, describing the field in a uniform half-space, can be used to introduce the apparent resistivity, and correspondingly, it is natural to study this field in some detail. First, consider the asymptotic behavior of the field in the frequency domain. Expanding Eqs (A4.29) and (A4.30) in the series by small parameters k1R and k1R1, we have 1 2 2 2 3 3 1 2 2 z2 r1 I 1 Ez z 1 þ k1 R þ ik1 R þ 3 þ k1 R R2 4p R3 2 3 2 " # r1 I 1 1 2 2 2 3 3 1 2 2 ð2h1 þ zÞ2 1 þ k1 R1 þ ik1 R1 þ 3 þ k1 R1 4p R13 2 3 2 R12 (A4.31) and m Idz B4 z 0 r 4p
1 k21 R2 1 3 3 1 k21 R12 1 3 3 1þ þ ik1 R 3 1 þ þ ik1 R1 ; 2 2 R3 3 3 R1
if k1 R 1 (A4.32) First of all from these equations it follows that the series, describing the low-frequency part of the spectrum, does not contain the term in the power u3/2. In other words, after the term with u the next two terms of the series are proportional to u2 and u5/2, correspondingly. And, as we know (Chapter 9) this means that at the late stage both fields decay with time as t5/2. Let us discuss Eqs (A4.31) and (A4.32), which are valid regardless of the frequency, as soon as the wavelength is much greater that the distance from the dipole and its mirror reflection, (R/l1 1), that is they characterize the field at the near zone. The leading term coincides with that of the time invariant field: #) " ( r1 I 1 1 z2 ð2h1 þ zÞ2 ; þ3 5 Ez1 ¼ R 4p R15 R13 R3 and B41
m0 Idz 1 1 : ¼ r 4p R3 R13
(A4.33)
It is clear that this field synchronously changes with the dipole current and by definition has a pure galvanic origin, since its electric field is caused
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Principles of Electromagnetic Methods in Surface Geophysics
by charges on the surface of the dipole electrodes and charges distributed on the boundary z ¼ h1. The field of these charges gives rise to currents, which are the sources of the magnetic field B14 . As follows from Eqs (A4.31) and (A4.32), the next term for both fields is ( " #) r1 Idz 2 1 z2 1 ð2h1 þ zÞ2 1þ Ez2 ¼ k 1þ 2 R 8p 1 R R1 R12 and B42
m0 Idz 2 1 1 : ¼ k r 8p 1 R R1
(A4.34)
The latter arises in the following way. A change of the magnetic field B41 with time generates the vortex electric field Ez2, as well Er2, and in accordance with Ohm’s law, we have j2 ¼ g1 E2 ; that causes the magnetic field B42. In the same manner the field E3 and B3, associated with the third term of the series and proportional to k41 , arises. It turns out that the first three terms of the series describe portion of the fields, which is hardly related to diffusion. In contrast, the next terms of the series, starting from the term with power of k51 ðu5=2 Þ, contains some information about a diffusion of the field. Note that considering separately the series for the in-phase and quadrature components of the field we can see that in the first case the term k51 is only the second one, while for the in-phase component it is the third. Correspondingly, we can expect that the 1Þ 2Þ quantity QEuz ðu QEuz ðu depends on resistivity of layers practically in the 1 2 same manner as the transient field does. Until now we focused on the lowfrequency spectrum. As concerns the opposite case when the wavelength becomes comparable or smaller than the distance between the dipole and the boundary h, its influence vanishes, and happens due to attenuation of the dipole field. Now making use of Eqs (A4.31) and (A4.32), consider the dependence of the field, normalized by the field ðEzun ; Bun 4 Þ, on distance from the origin, Figure A4.2(a)and (b). Here r Idz m0 Idz Ezun ¼ 1 3 and Bun y ¼ 4px 4px2 are the fields of the dipole in a uniform medium when frequency is equal to zero and an observation point is located at an equatorial plane, z ¼ 0. Index of
The Field of the Vertical Electric Dipole in the Layered Medium
743
Figure A4.2 Dependence of normalized fields: Ez =Ezun and B4 =Bun 4 on the distance.
curves is the ratio l1/h1. Along the vertical axis we plot either the value of the ratio Ez =E un or B4 =Bun . First of all, it is clear that the ratio pf the electric field tends to unity near the dipole. Then at the beginning with an increase of the distance x we first observe a maximum, and it happens due to opposite directions of the real and fictitious dipoles, but and after it the ratio starts to decrease. Behavior of normalized magnetic field, Figure A4.2(b), is sufficiently simpler and in the vicinity of the dipole, x < l1 is described by Eq. (A4.33). Next consider the transient electric and magnetic fields when the dipole current is turned off. Unlike the frequency domain here we deal with the diffusion process only. First of it is clear that at the first instance when the current vanishes, the field cannot disappear instantly, and at each point of a medium it is equal to the field, caused by the constant current. Then with increasing of time the field starts to decay and at sufficiently large times (late stage) it decreases as t5/2. The latter follows from the fact that the leading term of the series with fractional power of u, describing the low-frequency spectrum, has the power 5/2. Now we derive expressions for the transient field and with this purpose in mind, use again (Chapter 9) the equalities: 8 N ð < 0; t < 0 expðiutÞ 1 1 2p du ¼ : iu 1; t > 0 N (A4.35) 8 N ð 0; t < 0 < expðiutÞ 1 expðikRÞ du ¼ ; 2p : iu 1 FðuÞ; t > 0 N
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where F(u) is the probability integral: 1=2 ðu 2 FðuÞ ¼ exp x2 =2 dx p
(A4.36)
0
and u¼
1=2 2pR : ; s ¼ 2prt107 s
(A4.37)
Differentiating Eq. (A4.35) successively with respect to R, we obtain 1 2p
N ð N
1 2p
N ð N
2 1=2 expðiutÞ 2 u ikR u exp expðikRÞdu ¼ 2 iu p 2 1=2 2 u 2 expðiutÞ 3 ðikRÞ u exp : expðikRÞdu ¼ 2 iu p
(A4.38)
Taking Fourier’s integral from expressions for the electric and magnetic fields, Eq. (A4.28), we obtain h 2 i 2 1=2 1 Idz FðuÞ u exp u2 cos q; ER ¼ 2r 3 p 4pR Eq ¼ and
r1 Idz 4pR3
h 2 i 1=2 FðuÞ p2 u 1 þ u2 exp u2 sin q
" 1=2 2 # m0 Idz 2 u FðuÞ u exp sin q: B4 ¼ 2 2 4pR p
(A4.39)
(A4.40)
Note that with accuracy of a constant, expressions for the electric field, Eq. (A4.39), coincide with those of the magnetic field, caused by the magnetic dipole situated in a uniform medium. Of course, as in the frequency domain, the electric field and the current are located in the vertical planes Z0R, but the vector lines of the magnetic field are horizontal circles with the centers on the z-axis. As usual the study of the transient responses of the field we start from its asymptotic behavior. Bearing in mind that FðuÞ/1; at the initial stage, we have
if u/N
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The Field of the Vertical Electric Dipole in the Layered Medium
ER ¼
2r1 Idz r Idz m Idz cos q; Eq ¼ 1 3 sin q; B4 ¼ 0 2 sin q; 3 4pR 4pR 4pR
if t/0; (A4.41)
that is due to inertia of the magnetic field, which cannot change instantly, at the moment when the dipole current is switched off, the electromagnetic field remains the same as before. Correspondingly, for the vertical component of the electric field, we have 2 Idz z Ez ¼ 3 2 1 ; if t/0: (A4.42) R 4pR3 To derive the formulas for the late stage the following expansion is used 1=2 2 u u2 u4 2 u3 u5 u þ / z1 þ /; FðuÞz exp 2 2 8 6 40 p Their substitution into Eqs (A4.39) and (A4.40) gives r Idz 2 1=2 u3 u5 cos q; ER z 1 3 2 3 10 4pR p r Idz 2 1=2 2 2 Eq z 1 3 u3 u5 sin q 4pR p 3 5 and
m0 Idz 2 1=2 u3 u5 B4 z sin q; 3 10 4pR2 p
if u < 1:
(A4.43)
(A4.44)
Correspondingly, in the same manner as in the frequency domain we have for the vertical component of the electric field caused by the dipole r1 Idz 3 5 z2 2 3 2 5 u 2þ u þ u Ez z 4pR3 5 R 3 5 and, therefore, the field in the presence of the boundary is " r1 Idz 2 1=2 1 2 3 2 5 3 z2 5 u u þ u Ez z 4p p R3 3 5 5 R2 1 2 2 3 ð2h1 þ zÞ2 5 u1 3 u31 þ u51 5 5 R12 R1 3
!#
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Principles of Electromagnetic Methods in Surface Geophysics
and m0 Idz 2 1=2 1 1 3 1 5 r 1 1 3 1 5 r B4 z u u u u 4p p R2 3 10 R R12 3 1 10 1 R1 (A4.45) After simple algebra, we have 3=2 5=2
g m h ðh þ zÞ 1 5=20 Ez z 1=2 1 1 t 20pðpÞ Idz
and 5=2 5=2
g m m0 Idz B4 z h ðh þ zÞr 1 5=20 : 1=2 1 1 t 40pðpÞ
(A4.46)
The latter is valid when
r1 t m0
1=2
[r:
(A4.47)
Applying the same approach we have also for the radial component of the electric field at the late stage, we have Idz 3=2 5=2 Er z g m0 h1 r: 1=2 5=2 1 40pðpÞ t
(A4.48)
As we see the field at the late stage decays relatively quickly, as t5/2, while the components Ez and B4, caused by the single dipole decrease more slowly, as t3/2. This means that the field, associated with the surface charges, cancels at the late stage the term proportional to t3/2. At the same time the radial component Er due to the single dipole also decays as t5/2. From Eq. (A4.46) it follows that between the dipole and a boundary, components Ez and By decrease toward the boundary, but beneath the dipole they linearly increase. Unlike them the radial component of the electric field is independent on z. In particular, at the plane z ¼ 0, we have 3=2 5=2
g m Idz Ez z h2 1 0 ; 1=2 1 t 5=2 20pðpÞ
Idz 3=2 5=2 Er z g m0 h1 r 1=2 5=2 1 40pðpÞ t
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The Field of the Vertical Electric Dipole in the Layered Medium
un Figure A4.3 Normalized un transient responses of the electric field jEz j=Ez (a) and magnetic field B4 =B4 (b). Curve index is x/h1.
and
5=2 5=2
m0 Idz 2 g1 m0 h1 r : B4 z t 5=2 40pðpÞ1=2
(A4.49)
Correspondingly Er 1 r z Ez 2 h1 and with increasing the distance r the contribution of the horizontal component becomes stronger. Behavior of transient responses of the electric and magnetic fields is shown in Figure A4.3(a) and (b).
A4.3.5. The Normal Field at the Sea Bottom When r1 s r2 Unlike the previous case there are two boundaries, and at both of them electric charges arise, and naturally there is an interaction between them. Before we begin to discuss results of the field calculations let us briefly consider its behavior at the vicinity of the dipole and at the wave zone. As well known, the electric and magnetic fields change almost synchronously with the dipole current, when the distance from an observation point to the dipole is much smaller than the wavelength. In other words, it behaves as the time invariant field. For this reason, in order to obtain the asymptotic expressions at the near zone we can assume that k / 0. Then Eq. (A4.19) gives 1 expð2mh1 Þ C1 ¼ K12 ; 1 þ K12 expð2mh1 Þ D1 ¼ expð2mh1 Þ
1 þ K12 1 þ K12 expð2mh1 Þ
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and K12 ¼
r1 r2 r1 þ r2
Functions C1 and D1 contain information about an influence of both boundaries. To study their effect it is convenient to expand the fractions in the last equations in the series. Considering only the leading term, which does not contain h1, we have C1 zK12 and D1 ¼ 0 and therefore from Eq. (A4.21) it follows that at the point z ¼ 0 3 2 N ð m Idz 4 1 m Idz r1 B4 z 0 þ K12 mJ1 ðmrÞdm5 ¼ 0 2 ; (A4.50) 4p r 2 2pr r1 þ r2 0
since
N ð 0
v mJ1 ðmrÞdm ¼ vr
N ð
J0 ðmrÞdm ¼ 0
1 : r2
In the same manner the leading terms of the electric field (Eq. (A4.22)) give r Idz r Idz r1 ¼ 0 and E1z z 1 3 ð1 þ K12 Þ ¼ 1 3 (A4.51) E1r 4pr 2pr r1 þ r2 because N ð
2
N ð
m J0 ðmrÞdm ¼ 0
0
m2 expð mzÞdm ¼
v2 2 2 1=2 r þ z ; vz2
if z/0:
Thus in the near zone and plane z ¼ 0, we have E1z z
r1 Idz r Idz r1 m Idz r1 ð1 þ K12 Þ ¼ 1 3 ; B4 z 0 2 : 4pr 3 2pr r1 þ r2 2pr r1 þ r2 (A4.52)
It is obvious, that the latter describes the field, provided that an influence of the upper boundary is neglected, while its contribution is taken into account by the following terms of the series describing functions C1 and D1. Note that the same result can be obtained using the residual theorem, since these functions have poles on the complex plane of variable integration m.
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The Field of the Vertical Electric Dipole in the Layered Medium
As concerns the wave zone when a separation exceeds the wavelength we may expect that the field arrives propagating mainly through a more resistive medium. Dependence of the field magnitude on the distance from the dipole is shown in Figure A4.4(a) and (b). At the same time the values of normalized field near the origin are well defined by Eqs (A4.50) and (A4.51). Of course, with an increase of the frequency an influence of the upper boundary becomes smaller and due to attenuation it can be neglected when l1/h1 < 10. Next consider the transient responses of the electric and magnetic fields when the dipole current is turned off, which have the same features as in the case of a uniform half-space. For instance at the early stage when time tends to zero the field coincides with that of the time invariant field and, therefore, near the dipole it is almost described by Eqs (A4.50) and (A4.51). With an increase of time there is a moment when the vertical component of the electric field changes sign and at the late stage it is described as 3=2 5=2
g m Ez z h ðh þ zÞ 2 5=20 : 1=2 1 1 t 20pðpÞ Idz
(A4.53)
It is essential that their behavior at relatively large times is described by Eq. (A4.46), when r1 is replaced by r2, provided that inequality Eq. (A4.47) is held. In other words, due to diffusion an influence of resistivity of the upper medium is negligible at the late stage, and the component Ez coincides with that in a half-space with the resistivity of the lowest medium. In general, in the case of a layered medium we also have
Figure A4.4 Normalized electric and magnetic fields as functions of spacing in twolayer structure, when r2=r ¼ 3:33. Curve index is l1/h1. 1
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Principles of Electromagnetic Methods in Surface Geophysics
Figure A4.5 Apparent resistivity curves for the electric and magnetic fields in two-layer structure (r1 ¼ 0.3 Ohm m, r2 ¼ 1 Ohm m). Index of curves is x=h . 1
3=2 5=2
g m Ez z h ðh þ zÞ N 5=20 : 1=2 1 1 t 20pðpÞ Idz
(A4.54)
As results of calculations show that at the late stage the magnetic field is 5=2 5=2
g m g1 m0 Idz h ðh þ zÞr 2 5=20 B4 z 1=2 1 1 g2 t 40pðpÞ
(A4.55)
and, correspondingly, an influence of the upper layer remains at all times. It turns out that in a general of the N-layered medium, we have 5=2 5=2
g m g1 h ðh þ zÞr N 5=20 : B4 z 1=2 1 1 gN t 40pðpÞ m0 Idz
(A4.56)
Thus, measuring each of these fields in principle it is possible to carry out soundings, regardless of separation between the dipole and observation point. Note that expressions for the electric and magnetic fields, Eqs (A4.54) and (A4.56), at points of the upper layer obey the second Maxwell equation: curl B ¼ m0g1E. To illustrate a behavior of the fields in the two-layered medium, we represent them in the form of the apparent resistivity, Figure A4.5, making use the expression for the field at the late stage for uniform half-space.
A4.4. INFLUENCE OF THE PLANE T Now we study the influence of the thin resistive layer and for illustration consider a medium when r1 ¼ r2. Substituting Eq. (A4.16) into
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The Field of the Vertical Electric Dipole in the Layered Medium
boundary conditions, including Eq. (A4.14), we obtain the system of equations with respect to unknown coefficients: C0 expðm0 h1 Þ ¼
m expðm1 h1 Þ þ C1 expðm1 h1 Þ þ D1 expðm1 h1 Þ; m1
r0 m0 C0 expðm0 h1 Þ ¼ r1 ½m expðm1 h1 Þ þ m1 C1 expðm1 h1 Þ m1 D1 expðm1 h1 Þ;
(A4.57)
m expðm1 HÞ þ C1 expðm1 HÞ þ D1 expðm1 HÞ ¼ D2 expðm1 HÞ; m1 m expðm1 HÞ m1 C1 expðm1 HÞ þ m1 D1 expðm1 HÞ ¼ m1 þ T g1 m2 D2 expðm1 HÞ: Eliminating from the first two equations the unknown C0, we have m01 C1 expð2m1 h1 Þ þ D1 ¼
m m01 expð2m1 h1 Þ: m1
(A4.58)
The last two equations of the set Eq. (A4.57) give T g1 m2
m expð2m1 HÞ ¼ 2m1 þ T g1 m2 C1 m1 þ Tg1 D1 m2 expð2m1 HÞ
(A4.59)
From Eqs (A4.58) and (A4.59) we find coefficients, characterizing the field between the sea surface and plate T C1 ¼ T g1 m2
m expð2m1 HÞ m1
2m1 þ T g1
m2
1 þ m01 expð2m1 h1 Þ T g1 m2 m01 expð2m1 HÞexpð2m1 h1 Þ
and D1 ¼
m m01 expð2m1 h1 Þ m1
2m1 þ Tg1 m2 ½1 expð2m1 HÞ 2m1 þ Tg1 m2 Tg1 m2 m01 expð2m1 HÞexpð2m1 h1 Þ
(A4.60)
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Principles of Electromagnetic Methods in Surface Geophysics
A4.4.1. DC Soundings Suppose that we perform DC geometrical soundings with an array consisting of vertical electric dipole and number of receivers measuring the components of the field Ez and B4. Inasmuch as signals are measured on the sea bottom we may expect that at small separations the field mainly depends on the resistivity r1. Then an influence of the plane T becomes stronger and after it with further increase of the offset the field tends again to that in a uniform halfspace. For illustration consider dependence of the ratio of the total electric and magnetic fields to their normal values (plane T is absent) on the array offset x, Figure A4.6. Index of curves is the parameter T/T0, where T0 ¼ r1H and the ratio x/h1 is plotted along the horizontal axis. All curves are calculated for the same ratio H/h1 ¼ 1, and the apparent resistivity is introduced as ra Ez ra B4 ¼ N or ¼ N r1 Ez r1 B4 where EzN and BN 4 are normal fields given by Eq. (A4.33). As we see behavior of the apparent resistivity curves for the electric and magnetic field is almost similar and for illustration let us consider the case of the electric field. As we know, the field in presence of T-plane is created by source dipole, its mirror reflection with respect to seawater/air boundary and the secondary dipoles is distributed over the T-plane. By definition the vertical component of the field generated by the single dipole is proportional to # " 2 1 3z 1 1 EN ; ¼ 4p ðx2 þ z2 Þ5=2 ðx2 þ z2 Þ3=2 and its behavior is shown in Figure A4.7.
Figure A4.6 Apparent resistivity curves for the electric field Ez (a) and magnetic field B4 (b).
The Field of the Vertical Electric Dipole in the Layered Medium
753
Figure A4.7 Illustration for explanation of DC sounding apparent resistivity behavior. EN1 as function of offset x. Curve index is depth z.
It is essential that at some point, where x ¼ 21=2 z the component Ez changes its sign. Now, as the first approximation, assume that the dipoles at the plane T arise due to the dipole, located at the origin of coordinates and its mirror reflection with respect the upper boundary. Note that moments of these dipoles have opposite direction. At the beginning consider the secondary field due to the first dipole with the moment directed downwards. In 1 the internal area, where x < 2 /2 z, it generates the secondary dipoles directed opposite to the z-axis, while in the external area of the plane T dipoles have opposite direction. It is almost obvious that, if the observation point at the plane z ¼ 0 is located relatively close to the z-axis the dipoles of the internal area give the main contribution and the normal and secondary fields Ez have the same direction. Correspondingly, the ratio ra=r exceeds 1 unity. In contrast, far away from the z-axis the influence of dipoles of the external area is dominant. This means that the direction of the normal and secondary fields is different, and as a result the ratio ra=r is less than unity. 1 Such behavior of the apparent resistivity can be observed, if the upper boundary would be absent. In our case we have to take into account the effect of the vertical component, caused by the second dipole, which is a mirror reflection of the first one. Since this dipole is located at greater distance from the plane T and its moment has the opposite direction, it causes a different distribution of the secondary dipoles. For this reason, we observe two points where ra ¼ r1, and correspondingly more complicated behavior of the curves of the apparent resistivity.
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Principles of Electromagnetic Methods in Surface Geophysics
A4.4.2. Transient Soundings Next, assume that transient soundings are performed at different distances from the dipole and for illustration consider the case when T =T0 ¼ 3 and H/h1 ¼ 1. Apparent resistivity curves are shown in Figure A4.8, and their behavior in the quasi-stationary approximation is almost obvious. At the first instance, when the dipole current is turned off, electrical charges arise around electrodes, as well as the current between them, and they preserve the same electric and magnetic field as before. Because of transformation of the electromagnetic field into heat, intensity of these charges and the field becomes smaller, and we begin to observe diffusion. It is natural that at the early stage the left asymptote of the curves corresponds to the stationary field, and it contains some information about plane T. Its influence is very small, when observation point is close to the dipole and also it vanishes at relatively large distances. At the late stage, that corresponds to the right asymptote of the curves, the apparent resistivity approaches to that of the underlying medium, and the effect from the plane T becomes negligible. Because of diffusion there is a time interval when the magnitude of the dipole moments at this plane becomes maximal and, therefore, the secondary field becomes relatively large. It may be proper to notice that a change of a sign of the transient responses does not have any relation to the presence of the plate T. This is the feature of the field of the electric dipole, located beneath the upper boundary. As is seen from the curves, the influence of the thin resistive layer is more noticeable at relatively small distances from the primary dipole.
Figure A4.8 Apparent resistivity curves the electric field Ez (a), and the magnetic field B4 (b) in presence of T-plane Index of curves is the ratio x=h . 1
The Field of the Vertical Electric Dipole in the Layered Medium
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REFERENCES AND FURTHER READING [1] P.O. Barsukov, E.B. Fainberg, A mobile time domain sounding system for shallow water, First Break 31 (10) (2013) 53–63. [2] A. Chave, C. Cox, Controlled electromagnetic sources for measuring electric conductivity beneath the oceans. Forward problem and model study, J. Geophys. Res. 87 (1982) 5327–5338. [3] B.S. Svetov, Foundation of Geoelectrics, LKI, Moscow, 2008.
The Field of the Vertical Electric Dipole in the Layered Medium INTRODUCTION Here we consider the electromagnetic field generated in the Earth by a vertical electric dipole. The material complements Chapter 13, where we described the field generated by a horizontal electric dipole located at the Earth’s surface. In electrical prospecting on land, a horizontal electric dipole, grounded at its ends, is the most natural electric-field source for use in exploration geophysics, except in the case where the field is being measured at relatively high frequencies (20 kHz or more), far away from the source. In marine electrical prospecting, however, it is easy to use vertical dipoles both as the source of the field and as receivers. This configuration has potential applications in low-frequency electromagnetic exploration for offshore oil fields. O. Nazarenko first suggested the vertical-dipole configuration for electromagnetic sounding in 1961 (Russia); the method was revived through work by Nigel Edwards and others starting around 1981 (Canada). A brief history of electrical methods in marine exploration and in exploration for oil follows. The first application of electrical methods in a marine setting probably took place in Russia. Around 1930, large-scale electrical resistivity surveys for shallow oil fields were carried out both on land and offshore in the Caucasus region by teams of scientists working for a company owned by the Schlumberger brothers in a joint venture with the Soviet oil trusts. More than 70 years ago, profiling and sounding with electric dipoles were introduced for oil and mining exploration by L. Alpin, who developed the theory and interpretation of this method for direct (steady or time invariant) current flow. In contrast to the sounding methods introduced in the early 1900s by Conrad Schlumberger (later called “Schlumberger soundings”), dipole soundings do not require the use of long wires. For this reason, the dipole soundings were easily adapted to marine conditions, especially in shallow waters with depth less than 200 m. Marine dipole soundings were been used in the Azov Sea and Caspian Sea, as well as in the Sea of Okhotsk and other marine settings to study the basement surface, as well as the resistivity Methods in Geochemistry and Geophysics, Volume 45 ISSN 0076-6895, http://dx.doi.org/10.1016/B978-0-444-53829-1.19004-5
Ó 2014 Elsevier B.V. All rights reserved.
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j
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Principles of Electromagnetic Methods in Surface Geophysics
structure of sediments. Of course, dipole soundings using direct current (DC) can be used to probe deep inside the Earth only when very resistive layers are not present within sediments, since such layers will form a barrier to the penetration of the current, screening deeper layers. Marine electric dipole measurements, like land measurements, were initially carried out with arrays arranged with different distances between the current dipole source and the dipole receivers. Later, at the beginning of the 1960s, transient soundings with measurements of the vertical component of the magnetic field were used in Baltic and Okhotsk Seas to provide sensitivity to deep layers even in the presence of very resistive layers. Dipole soundings were usually carried out with two shipsda source (current) ship and a receiver ship. The surveys generally used axial dipole arrays (recall that an axial dipole array is arranged so that the dipole direction for a given electrode pair lies along the same line as the profile direction). There were two main surveying methods: a two-sided method and a symmetrical (midpoint) method. In the first configuration, the ship towing the receiver array is held steady with the measurement line MN at a point along the profile where the natural noise level is low. The ship towing the current array then moves continuously toward the receiver array, until it passes by the receiver array and continues on a straight path away from it on the other side. In the second arrangement, the two ships start at a large distance apart and move toward each other at the same speed, so that their midpoint remains fixed. Marine induced polarization surveys have been used for more than 30 years to detect ore bodies beneath the sea bottom. Induced polarization methods also had a brief history in oil exploration. During the 1960s it was discovered that there could sometimes be accumulations, above oil fields, of pyrites that generate an induced polarization effect. But attempts to use induced polarization to locate oil reservoirs did not show consistent correlations between shallow pyrite halos and deep oil fields, and the method has more or less been abandoned. Standard electric and electromagnetic soundings have been used for many years as secondary techniques in oil exploration to delineate large structures beneath of the surface or sea bottom. But until recently these methods had rarely been deployed directly to detect the reservoir itself. Possibly the first successful applications of electromagnetic sounding for oil were transient soundings on land in the near zone that were carried out in the early 1970s to locate the depth of the oil–water contact in reservoirs. As shown in Chapter 9, when conditions are favorable, it is possible to find the
The Field of the Vertical Electric Dipole in the Layered Medium
729
boundary between the portion of a porous reservoir layer that is saturated with gas or oildand is therefore highly resistivedand the conductive part that is saturated with saltwater. When the medium surrounding the reservoir is itself resistive, the presence of an oil–water contact can be detected by inducing currents in the zone saturated with saltwater, which becomes the source of the secondary electromagnetic field. In 1981, Len Srnka (Exxon) suggested using marine electromagnetic surveys to study the distribution of resistivity in sedimentary basins that may contain petroleum systems. Sedimentary basins are generally composed of sandstone, limestone, and shale layers that, aside from the reservoir layer itself, are saturated with saltwater. The resistive oil-saturated reservoir layer cannot easily be detected by induction, because the induced currents will flow mainly in the surrounding conductive layers that can extend for large distances above and below the reservoir zone. For this reason, it was suggested that a horizontal grounded source near the sea or ocean bottom could be used to map the presence of an oil layer. A highly resistive layer in an otherwise conductive sedimentary column will impede the flow of vertical current from a grounded electrical dipole source and change the rate of decay of the electric field with distance from the source. This method, called marine controlled-source electromagnetic surveying, is now a standard technique in offshore oil exploration. Measurements of the electric and magnetic field are mainly performed with arrays having horizontal current and receiver dipoles, in configurations similar to the ones first used by Alpin. In the future, arrays with the vertical electric dipoles will undoubtedly find more application in oil exploration, so it is interesting to study the field of such a current source in the presence of a resistive reservoir layer. But before we solve the problem for a layer of finite thickness, we examine the case where the thickness of the resistive part of reservoir is much smaller than the other dimensions, so that it can be replaced by a plane with transversal resistivity T. We derive boundary conditions for the field at such a surface, which enables full solution of the boundary value problem for a thin resistive layer without having to solve for the field in the layer itself.
A4.1. BOUNDARY CONDITIONS AT THE SURFACE OF A PLANE T Suppose that a thin resistive layer is situated at a distance H from the origin and surrounded by a conducting medium as is shown in Figure A4.1(a).
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Principles of Electromagnetic Methods in Surface Geophysics
(a)
(b)
x ρ1
z
(1)
hi
(2)
(1)
(i)
(i)
Ez (a1)
ρi
Ez (a2)
L1
(2)
ρ1
h1
Ex , Bx
(1)
ρ0
H
O
Idz
x
ρ2
T
(2)
Ex , Bx
ρ2
L2
ρ2
z
Figure A4.1 (a) Illustration used in deriving boundary conditions at plane T. (b) Twolayer structure with plane T embedded in a basement layer.
The transversal resistance of this layer T is defined as T ¼ hi ri ;
(A4.1)
where hi and ri are the thickness and resistivity of this layer, respectively. If we assume that the layer is sufficiently thin, we can make use of the approximate boundary conditions at its surface. Applying the second Maxwell’s equation in the integral form: ð þ B$dl ¼ m0 j$dS L
S
to the path L1 in Figure A4.1(a), we have ð2Þ Bð1Þ x dx Bx dx ¼ m0 jy hi dx;
or
hi ð2Þ Bð1Þ x Bx ¼ m0 jy hi ¼ m0 Ey : ri (A4.2)
Similarly, along path L2 hi ð2Þ Bð1Þ y By ¼ m0 jx hi ¼ m0 Ex : ri
(A4.3)
Because we assume that the resistivity ri is very high, it is proper to neglect any currents flowing along the thin layer, and so we have ð2Þ Bð1Þ x ¼ Bx ;
ð2Þ Bð1Þ y ¼ By :
(A4.4)
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The Field of the Vertical Electric Dipole in the Layered Medium
Thus, the horizontal components of the magnetic field have the same values at both surface of the thin layer. Using the first Maxwell equation þ vF E$dl ¼ vt L
and applying it along the path L1, we obtain Exð1Þ dx EzðiÞ ða2 Þhi Exð2Þ dx þ EzðiÞ ða1 Þhi ¼
vBy hi dx; vt
where Ezi ða2 Þ and Ezi ða1 Þ are values of the normal component of the electric field inside the layer at points a2 and a1, respectively. As the thickness hi tends to zero, the flux caused by By also tends to zero, and we have for the complex amplitudes of the electric field Exð1Þ Exð2Þ dx þ Ezi ða2 Þ Ezi ða1 Þ hi ¼ 0: It is obvious that
Ezi ða2 Þ Ezi ða1 Þ ¼
vEzi dx vx
and hence
Exð2Þ Exð1Þ ¼ hi
vEzi : vx
(A4.5)
By analogy, applying the same equation to the path L2, we have Eyð2Þ
Eyð1Þ
vEi ¼ hi z : vy
(A4.6)
The last two equations can be rewritten in the following form
Exð2Þ Exð1Þ
ðiÞ
ðiÞ
vEz vg Ez ¼ hi ri gi ¼T i vx vx
¼T
vjz vx
and
Eyð2Þ Eyð1Þ
ðiÞ
ðiÞ
vEz vg Ez ¼T i ¼ h i ri gi vy vy
¼T
vjz ; vy
(A4.7)
where gi ¼ 1/ri, jz is the vertical component of current density inside the thin layer, and T is the transversal resistance. Inasmuch as the horizontal component of the current inside a thin resistive layer is negligible, its vertical
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Principles of Electromagnetic Methods in Surface Geophysics
component is a continuous function; that is it has the same values at both sides of the plate T:
gi EzðiÞ ¼ g1 Ezð1Þ ¼ g2 Ezð2Þ : ð1Þ
ð2Þ
Here g1 ; Ez and g2 ; Ez are conductivity and the vertical component of the electric field outside the plate T, but at its vicinity. Correspondingly, the boundary conditions at the plane T assume the following form: ð1Þ
Bx
ð2Þ
Ex
ð2Þ
Ey
ð2Þ
¼ Bx ;
ð1Þ
By
ð2Þ
¼ By ; ð1Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
¼ T g1 vEvxz ¼ T g2 vEvxz ;
ð1Þ
¼ T g1 vEvyz ¼ T g2 vEvyz :
Ex Ey
(A4.8)
These boundary conditions do not require information about the behavior of the field components inside the layer, a feature that markedly simplifies a solution of the boundary value problem. Unlike the tangential components of the magnetic field, the horizontal components of the electric field at the surface of the thin resistive layer are discontinuous functions. From the theory for the stationary field it is known that the tangential components of the electric field can be discontinuous only across a surface of a double layer with the changing surface density of dipole moments: ð2Þ
Et
ð1Þ
Et
1 vh ¼ ; ε0 vt
(A4.9)
where ε0 is the dielectric constant and “t” indicates a tangential direction. Comparing Eqs (A4.8) and (A4.9) we can say that there are electric charges with opposite signs on either side of the thin layer. For this reason, each surface element can be considered as being a dipole with a moment h given by h ¼ T g1 ε0 Ezð1Þ ¼ T g2 ε0 Ezð2Þ :
(A4.10)
Since we usually assume that conductivities above and beneath plane T are equal, the normal components coincide top, that is the simple layer of charges is absent. Therefore in place of Eq. (A4.10) we have hðpÞ ¼ Tgε0 Ez ðpÞ:
The Field of the Vertical Electric Dipole in the Layered Medium
733
Note that the electric charges on the surface of the horizontal plane T create an electromagnetic field, which is equivalent to that caused by a system of vertical electric dipoles. Each elementary dipole is the source of a horizontal magnetic field, while the vertical component of this field is absent.
A4.2. MECHANISM OF APPEARANCE OF THE SECONDARY FIELD In this light let us notice that the presence of the thin resistive layer may be detected only due to charges arising at both its sides. They have opposite signs and equal magnitudes if resistivity of a medium above and beneath is the same. In fact, as is well known from the theory of a quasi-stationary field the surface charge density on the boundary between two media (i) and (j) is defined from the same equation as in the case of the time invariant field: sðpÞ ¼ 2ε0 Kij Enav ðpÞ: Here Lij ¼
rj ri ; rj þ ri
where ri and rj are resistivity of a medium, but Enav is the normal component of the total field at the same point p caused by all charges, except charge at the point p. Also this field includes the vortex part generated by a change of the magnetic field with time. It is essential that the normal n is directed from medium (i) to medium (j) that gives charges of different sign at the upper and lower surfaces of the thin layer. By definition, we may assume that Kij ¼ 1. As was pointed out it is natural to treat the field Enav as a sum of the normal and secondary fields, where the latter field is due to the presence of the plane T. It is obvious that if the normal electric field does not have component perpendicular to this layer the charges are absent, and the secondary field is equal to zero. The classical example of such field is the vertical magnetic dipole, since its vortex electric field is tangential to this horizontal layer, and correspondingly the secondary dipole moment is zero. Therefore, the necessary condition for detecting the thin resistive layer is the presence of the relatively strong component of the normal electric field perpendicular to this layer. For simplicity we assume that the thin layer has an infinite extension in horizontal direction, which will allow us in a simple manner to solve
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Principles of Electromagnetic Methods in Surface Geophysics
boundary value problem and obtain an expression for the electric field in the explicit form. In spite of such approximation we will be able to obtain some useful information about an influence of frequency or time, as well as a separation between the current and receiver electrodes on the depth of investigation for different resistivity structure parameters. In our case an array consists of two parallel vertical lines and length of which may be comparable with the distance between these current and receiver lines, again for simplicity we replace them by dipoles. At the same time transition to real system can be done by summation of the fields of dipoles along these lines. In such case the main attention will be paid to the vertical component of the electric field. At the beginning we consider the normal field in two-layer medium, shown in Figure A4.1(b).
A4.3. EXPRESSIONS FOR THE NORMAL FIELD AT THE SEA BOTTOM Let us choose the Cartesian system coordinates with the origin 0, located on the sea bottom and suppose that the vertical electric dipole is placed beneath the sea surface at the point 0 and directed along the z-axis, Figure A4.1(b). As usual we seek a solution in the frequency domain in terms of the vector potential, A. Its complex amplitude satisfies the Helmholtz equation in each layer: V2 A þ k2 A ¼ 0: As is known, B ¼ curl A ;
E ¼ iuA þ
1 grad div A : gm0
(A4.11)
A4.3.1. Boundary Conditions for the Vector Potential Taking into account the axial symmetry of the field and the model of a medium, we will try to solve the boundary value problem with help of one component of the vector potential Az. Then, as follows from Eq. (A4.11), we have Bx ¼
vAz ; vy
By ¼
vAz ; vx
Bz ¼ 0
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The Field of the Vertical Electric Dipole in the Layered Medium
and Ex ¼
1 v2 Az ; gm0 vxvz
Ey ¼
1 v2 Az ; gm0 vyvz
Ez ¼ iuAz þ
1 v2 Az : (A4.12) gm0 vz2
From continuity of the tangential components of the magnetic and electric fields at sea surface and sea bottom, we have ð0Þ
Az
ð1Þ Az
ð1Þ
¼ Az ¼
ð2Þ Az
ð0Þ
ð1Þ
and
r0 vAvzz ¼ r1 vAvzz ;
and
r1 vAvzz
ð1Þ
¼
ð2Þ
r2 vAvzz
;
if z ¼ h1 :
(A4.13)
if z ¼ h:
Here r0, r1, and r2 are resistivity of the upper space (air), the seawater, and a medium beneath, respectively, and h1 is the thickness of the seawater, Figure A4.1(b). At the surface T, we have ð1Þ
Az
ð2Þ
vAz vz
ð2Þ
¼ Az ; ð1Þ
vAvzz
¼ Tg
ð2Þ k22 Az
ð2Þ
2 Az þ v vz 2
(A4.14) ;
if z ¼ H:
Note that in deriving Eqs (A4.13) and (A4.14), we used the fact that from continuity of function’s tangential derivatives at the boundary follows the continuity of the function itself.
A4.3.2. Expressions for the Vector Potential Az of the Normal Field As was shown in the Chapter 13 the vector potential in a uniform medium is written as Az
m Idz expðik1 RÞ m0 Idz ¼ 0 ¼ 4p R 4p
N ð 0
m expðm1 jzjÞJ0 ðmrÞdm: m1
(A4.15)
Correspondingly, we have the following expressions for the complex amplitudes of the vector potential Að0Þ z
m Idz ¼ 0 4p
N ð
C0 expðm0 zÞJ0 ðmrÞdm; 0
if z < h1 ;
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Principles of Electromagnetic Methods in Surface Geophysics
Að1Þ z
m Idz ¼ 0 4p
N ð 0
m expðm1 jzjÞ þ C1 expðm1 zÞ þ D1 expðm1 zÞ m1
J0 ðmrÞdm if h1 < z < 0; (A4.16) Að2Þ z
m Idz ¼ 0 4p
N ð
D2 expðm2 zÞJ0 ðmrÞdm;
if z > 0:
0
Here 1=2 1=2 1=2 m0 ¼ m2 k20 ; m1 ¼ m2 k21 m2 ¼ m2 k22 : Substituting the latter into boundary conditions, we obtain the system of equations for determination of unknowns m C0 expðm0 h1 Þ ¼ expðm1 h1 Þ þ C1 expðm1 h1 Þ þ D1 expðm1 h1 Þ; m1 r0 m0 C0 expðm0 h1 Þ ¼ r1 ½m expðm1 h1 Þ þ m1 C1 expðm1 h1 Þ m1 D1 expðm1 h1 Þ; (A4.17) D2 ¼
m þ C1 þ D1 ; m1
r2 m2 D2 ¼ r1 ð m þ m1 C1 m1 D1 Þ: In deriving this system we assume at the beginning that the dipole was located slightly above the boundary z ¼ 0. Eliminating C0 and D2, we have: m m01 expð2m1 h1 Þ ¼ m01 expð2m1 h1 ÞC1 þ D1 ; m1 m m12 ¼ C1 m12 D1 ; m1 0 r1 m1 1 r2 m2 ; m12 ¼ rr1 m . where m01 ¼ rro m o m0 þr1 m1 1 m1 þr2 m2 This gives
C1 ¼
m 1 m01 expð2m1 h1 Þ m12 m1 1 þ m01 m12 expð2m1 h1 Þ
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The Field of the Vertical Electric Dipole in the Layered Medium
and m 1 þ m12 : D1 ¼ m01 expð2m1 h1 Þ 1 þ m01 m12 expð2m1 h1 Þ m1
(A4.18)
Further we will be mainly interested by the case when the upper medium is an insulator, that is m01 ¼ 1. Then we have C1 ¼ and D1 ¼
m 1 expð2m1 h1 Þ m12 m1 1 þ m12 expð2m1 h1 Þ
m 1 þ m12 expð2m1 h1 Þ : 1 þ m12 expð2m1 h1 Þ m1
(A4.19)
A4.3.3. Expressions for the Normal Field Beneath Sea Bottom Taking into account that at the cylindrical system of coordinates with the origin 0, we have 1r r14 1z 1 v v v B ¼ vr v4 vz r 0 0 Az Therefore B4 ¼
vAz : vr
As follows from Eq. (A4.16) B4
¼
m0 Idz B0 4 ðk1 RÞ þ 4p
N ð
m½C1 expðm1 zÞ þ D1 expðm1 zÞ J1 ðmrÞdm 0
(A4.20) Here B0 4 ðk1 RÞ is the magnetic field of the electric dipole in a uniform medium with resistivity r1. In particular, in the plane z ¼ 0, we have B4
¼
B0 4 ðk1 rÞ
m Idz þ 0 4p
N ð
mðC1 þ D1 ÞJ1 ðmrÞdm; 0
if z ¼ 0: (A4.21)
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Principles of Electromagnetic Methods in Surface Geophysics
For the radial and vertical components of the electric field, we have ¼ E1r
1 v2 A1z g1 m0 vrvz
and E1z ¼ iuA1z þ
Then Eq. (A4.16) gives E1r
¼
r Idz 0 E1r ðk1 RÞ 1 4p
1 v2 A1z : g1 m0 vz2
N ð
mm1 ½C1 expðm1 zÞ 0
D1 expðm1 zÞ J1 ðmrÞdm and E1z
¼
r Idz 0 E1z ðk1 RÞ þ 1
N ð
4p
m2 ½C1 expðm1 zÞ
0
þ D1 expðm1 zÞ J0 ðmrÞdm:
(A4.22)
At the points of the plane z ¼ 0 the latter gives E1r
¼
r Idz 0 ðk1 rÞ 1 E1r 4p
and E1z
¼
r Idz 0 E1z ðk1 rÞ þ 1 4p
N ð
N ð
mm1 ðC1 D1 ÞJ1 ðmrÞdm 0
m2 ðC1 þ D1 ÞJ0 ðmrÞdm
(A4.23)
0
Note, that we will consider only the vertical component of this field. In general, determination of the field requires numerical integration, except one case when r1 ¼ r2, and it can be expressed by elementary functions.
A4.3.4. The Normal Field When r1 [ r2 (Uniform Half-Space) As follows from Eq. (A4.19), we have C1 ¼ 0
and
m D1 ¼ expð2m1 h1 Þ; m1
(A4.24)
since m12 ¼ 0. Therefore, taking into account Somerfield integral, we have r1 I expðik1 RÞ expðik1 R1 Þ Að1Þ ; if z < 0: (A4.25) ¼ z 4p R R1
The Field of the Vertical Electric Dipole in the Layered Medium
739
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here R ¼ x2 þ y2 þ z2 ; R1 ¼ x2 þ y2 þ ðz þ 2h1 Þ2 . This means that as in the case of the time invariant field the electromagnetic field is represented as the field of two electric dipoles, located symmetrically with respect to the boundary. Their moments have the same magnitude but opposite directions. It may be proper to note that this is the rare case when the electromagnetic field of the dipole located beneath the boundary is expressed in terms of elementary functions. It is obvious that the normal component of the electric field is equal to zero at the conducting side of the sea surface as well as the magnetic field. B4 ðr; h1 Þ ¼ 0:
(A4.26)
The latter follows from the fact that near the surface, currents have only tangential component and the field is independent on the angle 4. Correspondingly, in the quasi-stationary approximation the magnetic field is absent in the upper nonconducting half-space. Note that the same is true in the case when instead of a conducting uniform half-space there is a horizontally layered medium. Absence of the magnetic field indicates that electromagnetic energy does not propagate through the upper space, if displacement currents are neglected in the upper halfspace. From Eq. (A4.25) we see that the normal field beneath the sea surface consists of the direct and reflected wave, and the latter is equivalent to the field of the electric dipole in a uniform medium, as if it is located above the boundary at the point, which is mirror reflection of the origin where the source is placed. As follows from Eq. (A4.25) the amplitude of the reflected wave does not exceed that of the direct one and with an increase of the distance both waves begin to decay as exponential functions. In particular, at relatively large distances, R [ h1, these waves almost cancel each other. In a fact we have 1 0 2h21 exp ik 1R r I expðik1 RÞ @ Az Að1Þ ¼ 1 1 z 2h2 4p R 1 þ 21 R
i2k1 h21
r1 I expðik1 RÞ ; 4p R2
if R/N:
(A4.27)
In the Chapter 13 it was shown that in a uniform medium and in the spherical system of coordinates the complex amplitudes of the field components ER and Eq are
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Principles of Electromagnetic Methods in Surface Geophysics
2r1 Idz expðik1 RÞð1 ik1 RÞcos q and Eq 4pR3 r Idz ¼ 1 3 expðik1 RÞ 1 ik1 R k21 R2 sin q: 4pR
ER ¼
Also B4 ¼ By ¼ Then
(A4.28)
m0 Idz expðikRÞð1 ikRÞsin q 4pr 2
Ez ¼ ER cos q Eq sin q;
and therefore the field in the presence of the upper boundary is:
r Idz Ez ¼ 1 3 expðik1 RÞ 2ð1 ik1 RÞcos2 q 4pR r Idz 1 ik1 R k21 R2 1 cos2 q þ 1 3 expðik1 R1 Þ 4pR1
2ð1 ik1 R1 Þcos2 q1 1 ik1 R1 k21 R12 1 cos2 q1 or Ez
r1 I expðik1 RÞ z2 2 2 1 þ ik1 R þ k1 R þ ½3 ik1 Rð3 ik1 RÞ 2 ¼ R 4p R3 ( r I expðik1 R1 Þ 1 1 þ ik1 R1 þ k21 R12 3 4p R1 ) ð2h1 þ zÞ2 þ ½3 ik1 R1 ð3 ik1 R1 Þ ; R12 (A4.29)
since cos q ¼
z R
and cos q1 ¼
2h1 þ z R1
For the magnetic field, we have m0 Idz 1 1 expðik1 RÞð1 ik1 RÞsin q 2 expðik1 R1 Þ B4 ¼ 4p R2 R1
ð1 ik1 R1 Þsin q1
(A4.30)
The Field of the Vertical Electric Dipole in the Layered Medium
741
These equations, describing the field in a uniform half-space, can be used to introduce the apparent resistivity, and correspondingly, it is natural to study this field in some detail. First, consider the asymptotic behavior of the field in the frequency domain. Expanding Eqs (A4.29) and (A4.30) in the series by small parameters k1R and k1R1, we have 1 2 2 2 3 3 1 2 2 z2 r1 I 1 Ez z 1 þ k1 R þ ik1 R þ 3 þ k1 R R2 4p R3 2 3 2 " # r1 I 1 1 2 2 2 3 3 1 2 2 ð2h1 þ zÞ2 1 þ k1 R1 þ ik1 R1 þ 3 þ k1 R1 4p R13 2 3 2 R12 (A4.31) and m Idz B4 z 0 r 4p
1 k21 R2 1 3 3 1 k21 R12 1 3 3 1þ þ ik1 R 3 1 þ þ ik1 R1 ; 2 2 R3 3 3 R1
if k1 R 1 (A4.32) First of all from these equations it follows that the series, describing the low-frequency part of the spectrum, does not contain the term in the power u3/2. In other words, after the term with u the next two terms of the series are proportional to u2 and u5/2, correspondingly. And, as we know (Chapter 9) this means that at the late stage both fields decay with time as t5/2. Let us discuss Eqs (A4.31) and (A4.32), which are valid regardless of the frequency, as soon as the wavelength is much greater that the distance from the dipole and its mirror reflection, (R/l1 1), that is they characterize the field at the near zone. The leading term coincides with that of the time invariant field: #) " ( r1 I 1 1 z2 ð2h1 þ zÞ2 ; þ3 5 Ez1 ¼ R 4p R15 R13 R3 and B41
m0 Idz 1 1 : ¼ r 4p R3 R13
(A4.33)
It is clear that this field synchronously changes with the dipole current and by definition has a pure galvanic origin, since its electric field is caused
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Principles of Electromagnetic Methods in Surface Geophysics
by charges on the surface of the dipole electrodes and charges distributed on the boundary z ¼ h1. The field of these charges gives rise to currents, which are the sources of the magnetic field B14 . As follows from Eqs (A4.31) and (A4.32), the next term for both fields is ( " #) r1 Idz 2 1 z2 1 ð2h1 þ zÞ2 1þ Ez2 ¼ k 1þ 2 R 8p 1 R R1 R12 and B42
m0 Idz 2 1 1 : ¼ k r 8p 1 R R1
(A4.34)
The latter arises in the following way. A change of the magnetic field B41 with time generates the vortex electric field Ez2, as well Er2, and in accordance with Ohm’s law, we have j2 ¼ g1 E2 ; that causes the magnetic field B42. In the same manner the field E3 and B3, associated with the third term of the series and proportional to k41 , arises. It turns out that the first three terms of the series describe portion of the fields, which is hardly related to diffusion. In contrast, the next terms of the series, starting from the term with power of k51 ðu5=2 Þ, contains some information about a diffusion of the field. Note that considering separately the series for the in-phase and quadrature components of the field we can see that in the first case the term k51 is only the second one, while for the in-phase component it is the third. Correspondingly, we can expect that the 1Þ 2Þ quantity QEuz ðu QEuz ðu depends on resistivity of layers practically in the 1 2 same manner as the transient field does. Until now we focused on the lowfrequency spectrum. As concerns the opposite case when the wavelength becomes comparable or smaller than the distance between the dipole and the boundary h, its influence vanishes, and happens due to attenuation of the dipole field. Now making use of Eqs (A4.31) and (A4.32), consider the dependence of the field, normalized by the field ðEzun ; Bun 4 Þ, on distance from the origin, Figure A4.2(a)and (b). Here r Idz m0 Idz Ezun ¼ 1 3 and Bun y ¼ 4px 4px2 are the fields of the dipole in a uniform medium when frequency is equal to zero and an observation point is located at an equatorial plane, z ¼ 0. Index of
The Field of the Vertical Electric Dipole in the Layered Medium
743
Figure A4.2 Dependence of normalized fields: Ez =Ezun and B4 =Bun 4 on the distance.
curves is the ratio l1/h1. Along the vertical axis we plot either the value of the ratio Ez =E un or B4 =Bun . First of all, it is clear that the ratio pf the electric field tends to unity near the dipole. Then at the beginning with an increase of the distance x we first observe a maximum, and it happens due to opposite directions of the real and fictitious dipoles, but and after it the ratio starts to decrease. Behavior of normalized magnetic field, Figure A4.2(b), is sufficiently simpler and in the vicinity of the dipole, x < l1 is described by Eq. (A4.33). Next consider the transient electric and magnetic fields when the dipole current is turned off. Unlike the frequency domain here we deal with the diffusion process only. First of it is clear that at the first instance when the current vanishes, the field cannot disappear instantly, and at each point of a medium it is equal to the field, caused by the constant current. Then with increasing of time the field starts to decay and at sufficiently large times (late stage) it decreases as t5/2. The latter follows from the fact that the leading term of the series with fractional power of u, describing the low-frequency spectrum, has the power 5/2. Now we derive expressions for the transient field and with this purpose in mind, use again (Chapter 9) the equalities: 8 N ð < 0; t < 0 expðiutÞ 1 1 2p du ¼ : iu 1; t > 0 N (A4.35) 8 N ð 0; t < 0 < expðiutÞ 1 expðikRÞ du ¼ ; 2p : iu 1 FðuÞ; t > 0 N
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Principles of Electromagnetic Methods in Surface Geophysics
where F(u) is the probability integral: 1=2 ðu 2 FðuÞ ¼ exp x2 =2 dx p
(A4.36)
0
and u¼
1=2 2pR : ; s ¼ 2prt107 s
(A4.37)
Differentiating Eq. (A4.35) successively with respect to R, we obtain 1 2p
N ð N
1 2p
N ð N
2 1=2 expðiutÞ 2 u ikR u exp expðikRÞdu ¼ 2 iu p 2 1=2 2 u 2 expðiutÞ 3 ðikRÞ u exp : expðikRÞdu ¼ 2 iu p
(A4.38)
Taking Fourier’s integral from expressions for the electric and magnetic fields, Eq. (A4.28), we obtain h 2 i 2 1=2 1 Idz FðuÞ u exp u2 cos q; ER ¼ 2r 3 p 4pR Eq ¼ and
r1 Idz 4pR3
h 2 i 1=2 FðuÞ p2 u 1 þ u2 exp u2 sin q
" 1=2 2 # m0 Idz 2 u FðuÞ u exp sin q: B4 ¼ 2 2 4pR p
(A4.39)
(A4.40)
Note that with accuracy of a constant, expressions for the electric field, Eq. (A4.39), coincide with those of the magnetic field, caused by the magnetic dipole situated in a uniform medium. Of course, as in the frequency domain, the electric field and the current are located in the vertical planes Z0R, but the vector lines of the magnetic field are horizontal circles with the centers on the z-axis. As usual the study of the transient responses of the field we start from its asymptotic behavior. Bearing in mind that FðuÞ/1; at the initial stage, we have
if u/N
745
The Field of the Vertical Electric Dipole in the Layered Medium
ER ¼
2r1 Idz r Idz m Idz cos q; Eq ¼ 1 3 sin q; B4 ¼ 0 2 sin q; 3 4pR 4pR 4pR
if t/0; (A4.41)
that is due to inertia of the magnetic field, which cannot change instantly, at the moment when the dipole current is switched off, the electromagnetic field remains the same as before. Correspondingly, for the vertical component of the electric field, we have 2 Idz z Ez ¼ 3 2 1 ; if t/0: (A4.42) R 4pR3 To derive the formulas for the late stage the following expansion is used 1=2 2 u u2 u4 2 u3 u5 u þ / z1 þ /; FðuÞz exp 2 2 8 6 40 p Their substitution into Eqs (A4.39) and (A4.40) gives r Idz 2 1=2 u3 u5 cos q; ER z 1 3 2 3 10 4pR p r Idz 2 1=2 2 2 Eq z 1 3 u3 u5 sin q 4pR p 3 5 and
m0 Idz 2 1=2 u3 u5 B4 z sin q; 3 10 4pR2 p
if u < 1:
(A4.43)
(A4.44)
Correspondingly, in the same manner as in the frequency domain we have for the vertical component of the electric field caused by the dipole r1 Idz 3 5 z2 2 3 2 5 u 2þ u þ u Ez z 4pR3 5 R 3 5 and, therefore, the field in the presence of the boundary is " r1 Idz 2 1=2 1 2 3 2 5 3 z2 5 u u þ u Ez z 4p p R3 3 5 5 R2 1 2 2 3 ð2h1 þ zÞ2 5 u1 3 u31 þ u51 5 5 R12 R1 3
!#
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Principles of Electromagnetic Methods in Surface Geophysics
and m0 Idz 2 1=2 1 1 3 1 5 r 1 1 3 1 5 r B4 z u u u u 4p p R2 3 10 R R12 3 1 10 1 R1 (A4.45) After simple algebra, we have 3=2 5=2
g m h ðh þ zÞ 1 5=20 Ez z 1=2 1 1 t 20pðpÞ Idz
and 5=2 5=2
g m m0 Idz B4 z h ðh þ zÞr 1 5=20 : 1=2 1 1 t 40pðpÞ
(A4.46)
The latter is valid when
r1 t m0
1=2
[r:
(A4.47)
Applying the same approach we have also for the radial component of the electric field at the late stage, we have Idz 3=2 5=2 Er z g m0 h1 r: 1=2 5=2 1 40pðpÞ t
(A4.48)
As we see the field at the late stage decays relatively quickly, as t5/2, while the components Ez and B4, caused by the single dipole decrease more slowly, as t3/2. This means that the field, associated with the surface charges, cancels at the late stage the term proportional to t3/2. At the same time the radial component Er due to the single dipole also decays as t5/2. From Eq. (A4.46) it follows that between the dipole and a boundary, components Ez and By decrease toward the boundary, but beneath the dipole they linearly increase. Unlike them the radial component of the electric field is independent on z. In particular, at the plane z ¼ 0, we have 3=2 5=2
g m Idz Ez z h2 1 0 ; 1=2 1 t 5=2 20pðpÞ
Idz 3=2 5=2 Er z g m0 h1 r 1=2 5=2 1 40pðpÞ t
747
The Field of the Vertical Electric Dipole in the Layered Medium
un Figure A4.3 Normalized un transient responses of the electric field jEz j=Ez (a) and magnetic field B4 =B4 (b). Curve index is x/h1.
and
5=2 5=2
m0 Idz 2 g1 m0 h1 r : B4 z t 5=2 40pðpÞ1=2
(A4.49)
Correspondingly Er 1 r z Ez 2 h1 and with increasing the distance r the contribution of the horizontal component becomes stronger. Behavior of transient responses of the electric and magnetic fields is shown in Figure A4.3(a) and (b).
A4.3.5. The Normal Field at the Sea Bottom When r1 s r2 Unlike the previous case there are two boundaries, and at both of them electric charges arise, and naturally there is an interaction between them. Before we begin to discuss results of the field calculations let us briefly consider its behavior at the vicinity of the dipole and at the wave zone. As well known, the electric and magnetic fields change almost synchronously with the dipole current, when the distance from an observation point to the dipole is much smaller than the wavelength. In other words, it behaves as the time invariant field. For this reason, in order to obtain the asymptotic expressions at the near zone we can assume that k / 0. Then Eq. (A4.19) gives 1 expð2mh1 Þ C1 ¼ K12 ; 1 þ K12 expð2mh1 Þ D1 ¼ expð2mh1 Þ
1 þ K12 1 þ K12 expð2mh1 Þ
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and K12 ¼
r1 r2 r1 þ r2
Functions C1 and D1 contain information about an influence of both boundaries. To study their effect it is convenient to expand the fractions in the last equations in the series. Considering only the leading term, which does not contain h1, we have C1 zK12 and D1 ¼ 0 and therefore from Eq. (A4.21) it follows that at the point z ¼ 0 3 2 N ð m Idz 4 1 m Idz r1 B4 z 0 þ K12 mJ1 ðmrÞdm5 ¼ 0 2 ; (A4.50) 4p r 2 2pr r1 þ r2 0
since
N ð 0
v mJ1 ðmrÞdm ¼ vr
N ð
J0 ðmrÞdm ¼ 0
1 : r2
In the same manner the leading terms of the electric field (Eq. (A4.22)) give r Idz r Idz r1 ¼ 0 and E1z z 1 3 ð1 þ K12 Þ ¼ 1 3 (A4.51) E1r 4pr 2pr r1 þ r2 because N ð
2
N ð
m J0 ðmrÞdm ¼ 0
0
m2 expð mzÞdm ¼
v2 2 2 1=2 r þ z ; vz2
if z/0:
Thus in the near zone and plane z ¼ 0, we have E1z z
r1 Idz r Idz r1 m Idz r1 ð1 þ K12 Þ ¼ 1 3 ; B4 z 0 2 : 4pr 3 2pr r1 þ r2 2pr r1 þ r2 (A4.52)
It is obvious, that the latter describes the field, provided that an influence of the upper boundary is neglected, while its contribution is taken into account by the following terms of the series describing functions C1 and D1. Note that the same result can be obtained using the residual theorem, since these functions have poles on the complex plane of variable integration m.
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The Field of the Vertical Electric Dipole in the Layered Medium
As concerns the wave zone when a separation exceeds the wavelength we may expect that the field arrives propagating mainly through a more resistive medium. Dependence of the field magnitude on the distance from the dipole is shown in Figure A4.4(a) and (b). At the same time the values of normalized field near the origin are well defined by Eqs (A4.50) and (A4.51). Of course, with an increase of the frequency an influence of the upper boundary becomes smaller and due to attenuation it can be neglected when l1/h1 < 10. Next consider the transient responses of the electric and magnetic fields when the dipole current is turned off, which have the same features as in the case of a uniform half-space. For instance at the early stage when time tends to zero the field coincides with that of the time invariant field and, therefore, near the dipole it is almost described by Eqs (A4.50) and (A4.51). With an increase of time there is a moment when the vertical component of the electric field changes sign and at the late stage it is described as 3=2 5=2
g m Ez z h ðh þ zÞ 2 5=20 : 1=2 1 1 t 20pðpÞ Idz
(A4.53)
It is essential that their behavior at relatively large times is described by Eq. (A4.46), when r1 is replaced by r2, provided that inequality Eq. (A4.47) is held. In other words, due to diffusion an influence of resistivity of the upper medium is negligible at the late stage, and the component Ez coincides with that in a half-space with the resistivity of the lowest medium. In general, in the case of a layered medium we also have
Figure A4.4 Normalized electric and magnetic fields as functions of spacing in twolayer structure, when r2=r ¼ 3:33. Curve index is l1/h1. 1
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Principles of Electromagnetic Methods in Surface Geophysics
Figure A4.5 Apparent resistivity curves for the electric and magnetic fields in two-layer structure (r1 ¼ 0.3 Ohm m, r2 ¼ 1 Ohm m). Index of curves is x=h . 1
3=2 5=2
g m Ez z h ðh þ zÞ N 5=20 : 1=2 1 1 t 20pðpÞ Idz
(A4.54)
As results of calculations show that at the late stage the magnetic field is 5=2 5=2
g m g1 m0 Idz h ðh þ zÞr 2 5=20 B4 z 1=2 1 1 g2 t 40pðpÞ
(A4.55)
and, correspondingly, an influence of the upper layer remains at all times. It turns out that in a general of the N-layered medium, we have 5=2 5=2
g m g1 h ðh þ zÞr N 5=20 : B4 z 1=2 1 1 gN t 40pðpÞ m0 Idz
(A4.56)
Thus, measuring each of these fields in principle it is possible to carry out soundings, regardless of separation between the dipole and observation point. Note that expressions for the electric and magnetic fields, Eqs (A4.54) and (A4.56), at points of the upper layer obey the second Maxwell equation: curl B ¼ m0g1E. To illustrate a behavior of the fields in the two-layered medium, we represent them in the form of the apparent resistivity, Figure A4.5, making use the expression for the field at the late stage for uniform half-space.
A4.4. INFLUENCE OF THE PLANE T Now we study the influence of the thin resistive layer and for illustration consider a medium when r1 ¼ r2. Substituting Eq. (A4.16) into
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The Field of the Vertical Electric Dipole in the Layered Medium
boundary conditions, including Eq. (A4.14), we obtain the system of equations with respect to unknown coefficients: C0 expðm0 h1 Þ ¼
m expðm1 h1 Þ þ C1 expðm1 h1 Þ þ D1 expðm1 h1 Þ; m1
r0 m0 C0 expðm0 h1 Þ ¼ r1 ½m expðm1 h1 Þ þ m1 C1 expðm1 h1 Þ m1 D1 expðm1 h1 Þ;
(A4.57)
m expðm1 HÞ þ C1 expðm1 HÞ þ D1 expðm1 HÞ ¼ D2 expðm1 HÞ; m1 m expðm1 HÞ m1 C1 expðm1 HÞ þ m1 D1 expðm1 HÞ ¼ m1 þ T g1 m2 D2 expðm1 HÞ: Eliminating from the first two equations the unknown C0, we have m01 C1 expð2m1 h1 Þ þ D1 ¼
m m01 expð2m1 h1 Þ: m1
(A4.58)
The last two equations of the set Eq. (A4.57) give T g1 m2
m expð2m1 HÞ ¼ 2m1 þ T g1 m2 C1 m1 þ Tg1 D1 m2 expð2m1 HÞ
(A4.59)
From Eqs (A4.58) and (A4.59) we find coefficients, characterizing the field between the sea surface and plate T C1 ¼ T g1 m2
m expð2m1 HÞ m1
2m1 þ T g1
m2
1 þ m01 expð2m1 h1 Þ T g1 m2 m01 expð2m1 HÞexpð2m1 h1 Þ
and D1 ¼
m m01 expð2m1 h1 Þ m1
2m1 þ Tg1 m2 ½1 expð2m1 HÞ 2m1 þ Tg1 m2 Tg1 m2 m01 expð2m1 HÞexpð2m1 h1 Þ
(A4.60)
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Principles of Electromagnetic Methods in Surface Geophysics
A4.4.1. DC Soundings Suppose that we perform DC geometrical soundings with an array consisting of vertical electric dipole and number of receivers measuring the components of the field Ez and B4. Inasmuch as signals are measured on the sea bottom we may expect that at small separations the field mainly depends on the resistivity r1. Then an influence of the plane T becomes stronger and after it with further increase of the offset the field tends again to that in a uniform halfspace. For illustration consider dependence of the ratio of the total electric and magnetic fields to their normal values (plane T is absent) on the array offset x, Figure A4.6. Index of curves is the parameter T/T0, where T0 ¼ r1H and the ratio x/h1 is plotted along the horizontal axis. All curves are calculated for the same ratio H/h1 ¼ 1, and the apparent resistivity is introduced as ra Ez ra B4 ¼ N or ¼ N r1 Ez r1 B4 where EzN and BN 4 are normal fields given by Eq. (A4.33). As we see behavior of the apparent resistivity curves for the electric and magnetic field is almost similar and for illustration let us consider the case of the electric field. As we know, the field in presence of T-plane is created by source dipole, its mirror reflection with respect to seawater/air boundary and the secondary dipoles is distributed over the T-plane. By definition the vertical component of the field generated by the single dipole is proportional to # " 2 1 3z 1 1 EN ; ¼ 4p ðx2 þ z2 Þ5=2 ðx2 þ z2 Þ3=2 and its behavior is shown in Figure A4.7.
Figure A4.6 Apparent resistivity curves for the electric field Ez (a) and magnetic field B4 (b).
The Field of the Vertical Electric Dipole in the Layered Medium
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Figure A4.7 Illustration for explanation of DC sounding apparent resistivity behavior. EN1 as function of offset x. Curve index is depth z.
It is essential that at some point, where x ¼ 21=2 z the component Ez changes its sign. Now, as the first approximation, assume that the dipoles at the plane T arise due to the dipole, located at the origin of coordinates and its mirror reflection with respect the upper boundary. Note that moments of these dipoles have opposite direction. At the beginning consider the secondary field due to the first dipole with the moment directed downwards. In 1 the internal area, where x < 2 /2 z, it generates the secondary dipoles directed opposite to the z-axis, while in the external area of the plane T dipoles have opposite direction. It is almost obvious that, if the observation point at the plane z ¼ 0 is located relatively close to the z-axis the dipoles of the internal area give the main contribution and the normal and secondary fields Ez have the same direction. Correspondingly, the ratio ra=r exceeds 1 unity. In contrast, far away from the z-axis the influence of dipoles of the external area is dominant. This means that the direction of the normal and secondary fields is different, and as a result the ratio ra=r is less than unity. 1 Such behavior of the apparent resistivity can be observed, if the upper boundary would be absent. In our case we have to take into account the effect of the vertical component, caused by the second dipole, which is a mirror reflection of the first one. Since this dipole is located at greater distance from the plane T and its moment has the opposite direction, it causes a different distribution of the secondary dipoles. For this reason, we observe two points where ra ¼ r1, and correspondingly more complicated behavior of the curves of the apparent resistivity.
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Principles of Electromagnetic Methods in Surface Geophysics
A4.4.2. Transient Soundings Next, assume that transient soundings are performed at different distances from the dipole and for illustration consider the case when T =T0 ¼ 3 and H/h1 ¼ 1. Apparent resistivity curves are shown in Figure A4.8, and their behavior in the quasi-stationary approximation is almost obvious. At the first instance, when the dipole current is turned off, electrical charges arise around electrodes, as well as the current between them, and they preserve the same electric and magnetic field as before. Because of transformation of the electromagnetic field into heat, intensity of these charges and the field becomes smaller, and we begin to observe diffusion. It is natural that at the early stage the left asymptote of the curves corresponds to the stationary field, and it contains some information about plane T. Its influence is very small, when observation point is close to the dipole and also it vanishes at relatively large distances. At the late stage, that corresponds to the right asymptote of the curves, the apparent resistivity approaches to that of the underlying medium, and the effect from the plane T becomes negligible. Because of diffusion there is a time interval when the magnitude of the dipole moments at this plane becomes maximal and, therefore, the secondary field becomes relatively large. It may be proper to notice that a change of a sign of the transient responses does not have any relation to the presence of the plate T. This is the feature of the field of the electric dipole, located beneath the upper boundary. As is seen from the curves, the influence of the thin resistive layer is more noticeable at relatively small distances from the primary dipole.
Figure A4.8 Apparent resistivity curves the electric field Ez (a), and the magnetic field B4 (b) in presence of T-plane Index of curves is the ratio x=h . 1
The Field of the Vertical Electric Dipole in the Layered Medium
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REFERENCES AND FURTHER READING [1] P.O. Barsukov, E.B. Fainberg, A mobile time domain sounding system for shallow water, First Break 31 (10) (2013) 53–63. [2] A. Chave, C. Cox, Controlled electromagnetic sources for measuring electric conductivity beneath the oceans. Forward problem and model study, J. Geophys. Res. 87 (1982) 5327–5338. [3] B.S. Svetov, Foundation of Geoelectrics, LKI, Moscow, 2008.