Electromagnetic diffusion in anisotropic media

Authors


Abstract

[1] I present a plane-wave analysis of anisotropic electromagnetic media at the low-frequency range, where the displacement currents can be neglected and the field is diffusive. Anisotropy is due to the conductivity tensor and the magnetic permeability is a scalar quantity. The analysis includes the energy balance (Umov-Poynting theorem) and provides expressions of measurable quantities such as the phase and energy velocities, the attenuation factor, and the skin depth as a function of frequency and propagation direction. The balance of energy allows the identification of the stored and dissipated energy densities, which are related to the magnetic energy and the conductive part of the electric energy. For a real conductivity tensor, the stored energy equals the dissipated energy. I also establish fundamental relations, e.g., the scalar product between the slowness vector and the power-flow vector is equal to the energy density. For uniform plane waves, the phase velocity is the projection of the energy velocity vector onto the propagation direction and a similar relation is obtained by replacing the energy velocity with a velocity related to the dissipated energy. I have also obtained the Green function for an azimuthally isotropic medium (transverse isotropy), which is used to calculate transient fields.

1. Introduction

[2] Electromagnetic (EM) propagation at low frequencies (EM diffusion) is used in a number of applications, such as geothermal exploration [Pellerin et al., 1996], evaluation of hydrocarbon resources by mapping the subseafloor resistivity [Eidesmo et al., 2002], EM induction in boreholes and logging while drilling [Wang and Signorelli, 2004], magnetotelluric problems [Mackie et al., 1993; Yin and Maurer, 2001], and geoelectrical surveys for groundwater and mineral exploration [Oristaglio and Hohmann, 1984].

[3] The theory of EM diffusion in isotropic media is well established [see, e.g., Ward and Hohmann, 1988]. Anisotropy has been taken into account to model magnetotelluric fields, using a propagation matrix algorithm in 1-D layered models, where the conductivity is homogeneous both laterally and vertically within each layer [Mann, 1965; Loewenthal and Landisman, 1973; Abramovici, 1974; Kováčiková and Pek, 2002]. In all these works no analysis of the physics in 3-D space is performed. At most, the wave numbers along the vertical direction are obtained. This is because they only consider the z direction and then there is no dependence with the propagation angle in their equations. In fact, the study of anisotropic diffusion in three dimensions and from the point of view of the energy balance has not given much attention in the geophysical literature. An analysis has been performed by Carcione and Schoenberg [2000] and Carcione [2007], who considered the high-frequency range (waves), where the dielectric permittivity plays an important role. Analogies can be performed with the theory of elasticity to establish mathematical and physical formulations [Carcione and Cavallini, 1995; Carcione and Helbig, 2008].

[4] There are many material configurations in the subsurface that might lead to anisotropy [Negi and Saraf, 1989]. The geophysical motivation behind the use of an electrically anisotropic description of the Earth are given by Mann [1965], Carcione [1996], Weidelt [1998], Anderson et al. [2001], Carcione and Schoenberg [2000], Wang and Fang [2001], and Weiss and Newman [2002]. It might be that there are some preferred directions in the subsurface rocks, or some preferred orientation of grains in the sediments. Compaction, fine layering or a pronounced strike direction might lead to effective anisotropy. Alternations of sandstone and shales may give reservoir anisotropy [Carcione and Seriani, 2000; Davydycheva et al., 2003].

[5] The paper is organized as follows. First, Maxwell's equations are given, in the inhomogeneous and homogeneous cases. Then, I perform a plane-wave analysis and obtain the Kelvin-Christoffel eigensystem, whose eigenvalues yield the phase velocities and skin depth as a function of the conductivity components, frequency and propagation direction. The energy balance (Umov-Poynting theorem) is then established to obtain expressions of the energy densities and energy velocity. Finally, I obtain the Green function for a uniaxial medium.

2. Maxwell's Equations for General Anisotropic Media

[6] In vector notation, Maxwell's equations, neglecting the displacement currents, are [e.g., Carcione, 2007]

equation image

where the vectors E, H, B, JI, JS, and M are the electric field intensity, the magnetic field intensity, the magnetic flux density, the induced current density, the electric source current, and the magnetic source density, respectively. In general, they depend on the Cartesian coordinates (x1, x2, x3) = (x, y, z), and the time variable t. I have used the compact notation ∂t ≡ ∂/∂t.

[7] Additional constitutive equations are needed. These are JI = σ · E and B = μH, where σ, the conductivity tensor, is a real, symmetric and positive definite tensor, and μ, the magnetic permeability, is a scalar quantity, and the dot denotes ordinary matrix multiplication. Substituting the constitutive equations into equations (1) gives

equation image

which are a system of six scalar equations in six scalar unknowns.

[8] In this work, I am concerned with triclinic media, where at each point of the space the conductivity tensor is nondiagonal. However, the tensor can always be rotated to obtain its expression in its principal coordinate system

equation image

Although the magnetic permeability is commonly assumed to be that of free space, some soils have a significantly higher value [Olhoeft and Capron, 1994], hence, μ can vary arbitrarily in space. In some cases, such as the CSEM problems μ is assumed to be spatially constant [Eidesmo et al., 2002].

[9] From equation (1) I obtain a vector equation for the electric field

equation image

Note that

equation image

where I have denoted ∂/∂xi by ∂i for brevity.

[10] Following Badea et al. [2001] and Stalnaker [2004], I consider the case

equation image

which is valid for ∇ ·JS = 0, according to equation (1), i.e., if the source current density is divergence free [Ward and Hohmann, 1988, equation 1.7]. This condition is satisfied by inductively coupled magnetic dipole controlled sources of the form

equation image

where A is a vector. In 2-D (x, z) space, an example is Js = (∂3F, −∂1F)h, where F(x) is the spatial distribution and h(t) is the time history.

[11] In homogeneous media with a diagonal conductivity tensor, equation (4) becomes

equation image

where γ = (μσ)−1. The explicit form of equation (8) is

equation image

[12] Consider the 2D anisotropic case, ignoring the y dimension. The corresponding equations are

equation image

From (6) one has σ11E1 + σ33E3 = 0. Combining this equation with (10) yields

equation image

where

equation image

is a modified Laplacian. Note that the field components have decoupled.

[13] In 3D space, the E3 component can be decoupled if one considers a transversely isotropic medium with σ1 = σ2. In this case, a similar procedure leads to

equation image

where

equation image

and a corresponding analytical solution is given below.

3. Plane-Wave Theory

[14] I assume nonuniform harmonic plane waves with a phase factor

equation image

where ξ, the complex slowness vector, is equivalent to k/ω, with k and ω being the wave number vector and frequency, respectively. The dot denotes the scalar product and equation image = equation image. Note the following correspondences between time and frequency domain:

equation image

where × denotes the vector product.

[15] Substituting the plane wave (15) into Maxwell's equations (2) in the absence of sources, and using (16) gives

equation image

and

equation image

For convenience, the medium properties are denoted by the same symbols, in both the time and frequency domains.

[16] Taking the vector product of equation (17) with ξ, gives

equation image

which, with equation (18), becomes

equation image

for three equations for the components of E. Alternatively, the vector product of equation (18) with ξ and use of (17) yields

equation image

for three equations for the components of H.

[17] The equivalent of the 3 × 3 viscoelastic Kelvin-Christoffel equations, for the electric field vector components, are

equation image

where the subindices take the values 1, 2 and 3, and eijk are the elements of the Levi-Civita tensor.

[18] Similarly, the equations for the magnetic field vector components are

equation image

Rotating the conductivity tensor to the principal system of coordinates, equation (3) is obtained. There is no loss of generality in this operation. Then, the equivalent of the viscoelastic Kelvin-Christoffel equation for the electric field vector is

equation image

where the EM Kelvin-Christoffel matrix is

equation image

After defining

equation image

the dispersion relation (i.e., the vanishing of the determinant of the Kelvin-Christoffel matrix) becomes

equation image

There are only quartic and quadratic terms of the slowness components in the dispersion relation.

3.1. Slowness, Kinematic Velocities, Attenuation, and Skin Depth

[19] The slowness vector can be split into real and imaginary vectors such that ωRe(tξ · x) is the phase and ωIm(ξ·x) is the attenuation. Assume that propagation and attenuation directions coincide to produce a uniform plane wave, which is equivalent to a homogeneous plane wave in viscoelasticity. The slowness vector can be expressed as

equation image

where ξ is the complex slowness and equation image = (l1, l2, l3) is a real unit vector, with li the direction cosines (equation imageli2 = 1). I obtain the real wave number vector and the real attenuation vector as

equation image

respectively.

[20] Substituting equation (28) into the dispersion relation (27) yields

equation image

It can be shown that the roots have the form

equation image

where s is a real slowness. Note the property

equation image

[21] In terms of the complex velocity

equation image

the phase velocity and attenuation are

equation image

respectively.

[22] The skin depth is the distance d for which exp(−α·d) = 1/e, where e is Napier's number, i.e., the effective distance of penetration of the signal, where α = αequation image. Using equation (29) yields

equation image

since di = dli.

[23] Assume, for instance, propagation in the (1, 2) plane. Then, l3 = 0 and the dispersion relation (30) is factorizable, giving

equation image

These factors give the TM and TE modes with complex velocities

equation image

In the TM (TE) case the magnetic (electric) field vector is perpendicular to the propagation plane. For obtaining the slowness and complex velocities for the other planes, simply make the following subindex substitutions:

equation image

The analysis of all three planes of symmetry gives the curves represented in Figure 1b. There exists a single conical point given by the intersection of the TE and TM modes, as can be seen in the (1,2) plane of symmetry. The location of the conical point depends on the values of the material properties. The slowness curves of the TE modes intersecting the three orthogonal planes are circles (isotropy).

Figure 1.

(a) Phase velocity, (b) energy velocity, and (c) skin depth at the Cartesian planes. The frequency is f = 1 Hz. The modes are indicated in Figure 1b, together with the conductivity components associated with the TE mode; veTE = vpTE ∝ 1/equation image (see equations (39) and (58)).

[24] Consider the TE mode. Then, the phase velocity is

equation image

where f = ω/(2π) is the frequency. On the other hand, the skin depth and attenuation factor are given by

equation image

The velocity increases and the skin depth decreases with frequency. In the anisotropic case, these quantities depend on the propagation angle.

3.2. Umov-Poynting's Theorem and Energy Velocity

[25] The scalar product of the complex conjugate of equation (18) with E, use of the relation 2equation imageIm(ξ) · (E × H*) = (ξ × E) · H* + E · (ξ × H)* and substitution of equation (17), gives Umov-Poynting's theorem for plane waves

equation image

where

equation image

is the complex Umov-Poynting vector, and

equation image

are the electric (dissipated) and magnetic (stored) energy densities, respectively. The superscript “*” denotes complex conjugate. The imaginary part of equation (41) gives the balance of stored energy and the real part gives the balance of dissipated energy. This equation hold for a complex conductivity tensor, i.e., including induced polarization effects [Zhdanov, 2008]. In this work, σ is real and um and ue are therefore real quantities. Because of this fact, there is no electric stored energy in the diffusion process. Furthermore, it is shown in the appendix that

equation image

Therefore, the energy balance (41) is

equation image

The energy velocity vector, ve, is given by the energy power flow Re(P) divided by the total stored energy density, um. Thus,

equation image

3.3. Fundamental Relations

[26] The scalar product of equation (17) with H* and the scalar product of the complex conjugate of equation (18) with E yield

equation image

Using the property A · B × C = C × A · B, we obtain

equation image

Adding these two equations gives

equation image

For a real conductivity tensor, the real and imaginary parts of this equation yields

equation image

For homogeneous waves, it is

equation image

where I have used equations (33), (34), and (46), and

equation image

is a velocity associated with the dissipated energy. Similar relations are obtained in viscoelasticity and poroviscoelasticity [e.g., Carcione, 2007]. The first equation (51) indicates that the slowness and energy velocity surfaces are reciprocal.

3.4. The TE and TM Modes

[27] As an example, I consider the TE mode propagating in the (1,2) plane. Then,

equation image

where E0 is a complex amplitude. By equation (17),

equation image

where I have assumed uniform plane waves. Substituting the electric and magnetic fields into the energy density (43) (see equation (44)) yields

equation image

where vTE is given in (37), and the attenuation α is given by equation (34). Alternatively,

equation image

where vp is the phase velocity (39). The TE power flow vector is

equation image

where equation image = equation image1l1 + equation image2l2 defines the wave number vector direction. From equations (46), (56), and (57) I obtain the energy velocity for TE waves propagating in the (1,2) plane as

equation image

As in the viscoelastic case, the energy velocity equals the phase velocity when there is isotropy [e.g., Carcione, 2007].

[28] Performing similar calculations, the energy density and energy velocity for TM waves propagating in the (1,2) plane are

equation image

and

equation image

respectively, where vTM is given in (37). The expressions corresponding to the (1,3) and (2,3) planes can be obtained by making the substitutions (38).

4. Transient Analytical Solution

[29] A transient solution can be obtained to describe the amplitude of the diffusion field and test simulation codes. Consider equations (13), denoting E3 by E. Redefining the source term, I have

equation image

Defining x′ = x/equation image, y′ = y/equation image and z′ = z/equation image, I obtain

equation image

The solution for the Green function is obtained for JS = −δ(x′, y′, z′)δ(t). I get [Carslaw and Jaeger, 1984]

equation image

where

equation image

If JS = −δ(x, y, z)h(t), the solution is given by

equation image

where “*” denotes time convolution here.

[30] The solution corresponds to transverse isotropy, for which there are two eigen directions, and only two, for which all three tensors have equal eigenvalues. This electromagnetic symmetry includes that of hexagonal, tetragonal and trigonal crystals. These are said to be optically uniaxial.

5. Examples

[31] I consider σ1 = 0.2 S/m, σ2 = 0.03 S/m and σ3 = 0.05 S/m, and μ = μ0 = 4 π 10−7 H/m. The phase velocity, energy velocity and skin depth at the Cartesian planes are shown in Figure 1. The frequency is f = 1 Hz. The electric (magnetic) field is perpendicular to the plane for the TE (TM) mode (see Figure 1b). The first are isotropic while the TM modes show anisotropy. Both the velocity and skin depth decrease for increasing conductivity. This means, for instance, that the diffusion process is slower in salt water than in fresh water, or slower in brine-saturated sediments than in oil-saturated reservoirs. On the other hand, the velocity increases and the skin depth decreases with the frequency.

[32] Let us consider a transient source, whose time history is

equation image

where tp is the period of the wave (the distance between the side peaks is equation imagetp/π) and I take ts = 1.4tp. I consider σ1 = σ2 = 0.1 S/m and σ3 = 0.05 S/m and a central frequency fp = 3 Hz. Figure 2 shows the time history at two different distances from the source location, corresponding to the anisotropic and isotropic cases (solid and dashed lines, respectively). The electric field is normalized with respect to maximum amplitude. Isotropy corresponds to σ1 = σ3 = 0.1 S/m. The amplitude and phase variations with offset are shown in Figure 3. As can be appreciated, the differences can be substantial.

Figure 2.

Electric field as a function of propagation time at (a) (x, y, z) = (2, 1, 1) km and (b) (x, y, z) = (3, 1, 1) km from the source location. The solid and dashed lines correspond to the anisotropic and isotropic cases, respectively.

Figure 3.

(a) AVO and (b) PVO responses along the x direction (solid lines) compared to the isotropic case (dashed lines). The frequency is 2 Hz and (y, z) = (1, 1) km.

6. Conclusions

[33] The study of electromagnetic propagation in anisotropic media requires a detailed plane-wave analysis and the establishment of the energy balance to obtain the expression of measurable quantities such as the energy velocity as a function of frequency and propagation direction. In the case of uniform plane waves, the stored and dissipated energies have the same value. Fundamental relations are obtained, e.g., the energy density can be obtained as the scalar product of the slowness and the power-flow vectors. For uniform waves, one of the relations indicates that the energy velocity and slowness surfaces are reciprocal.

[34] The theory can be generalized to the case of induced polarization, i.e., a complex and frequency-dependent conductivity tensor. The generalization is straightforward. In this case, the stored electric energy is not zero and the magnetic stored energy does not have the same value of the dissipated energy. In terms of mechanical models, it can be shown that the conductivity components can be represented by Kelvin-Voigt elements.

[35] A closed form solution is obtained in the time domain for a uniaxial (transversely isotropic) medium. The solution is useful to obtain amplitude and phase variations as a function of the propagation distance (offset) in the frequency domain.

[36] The examples illustrate the differences between the anisotropic and isotropic cases, which can be significant.

Appendix A:: Equivalence Between the Stored and Dissipated Energy Densities

[37] I show here that, for uniform plane waves and a real conductivity tensor, the stored energy equals the dissipated energy over a cycle. The magnetic energy (43) is

equation image

where I have used equation (17) and the property (A × B) · (C × D) = (A · C)(B · D) − (A · D)(B · C). On the other hand, the electric energy (46) is

equation image

where I have used equation (20), the fact that σ is real and the property A × (B × C) = (A · C)B − (A · B)C.

[38] From equations (31) and (32), it is ξ = (1 − equation image)sequation image and ∣ξ2 = equation imageξ · ξ. Then, the last equation and ∣ξ · E*∣2 = equation image(ξ·E)(ξ · E*) imply

equation image

Acknowledgments

[39] This work was funded by EMGS ASA.

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