The paradox of anisotropy in electric logging: A simple proof and extensions to other physics domainsAnisotropy paradox proof and other physics | Geophysics

ABSTRACT

Resistivity measurements in vertical wells through horizontally laminated formations suffer the paradox of anisotropy. In a borehole with negligible diameter, the measurement will only read the horizontal resistivity parallel to the laminae: It will be completely blind to the vertical resistivity perpendicular to the laminae, even though the source and sensor electrodes are vertically aligned. Coulomb’s law in anisotropic media explains this counterintuitive phenomenon. The anisotropy changes the Pythagorean distance to a new, anisotropic distance, which includes the inverse conductivity tensor. The mixed units of this anisotropic distance are reconciled in Coulomb’s law, whose normalization replaces the electric conductivity by the square root of the conductivity-tensor determinant. The special case of horizontal laminae and vertically aligned source and observation points simplifies Coulomb’s law in anisotropic media. The vertical conductivity can be extracted from the anisotropic distance as multiplicative factor, which then cancels a corresponding term in the normalization determinant. Any electrode-resistivity measurement can then be described as superposition of point sources and sensors. The analysis of Coulomb’s law in anisotropic media carries over to other physics domains with similar, close-form solutions: Compressional waves with anisotropic bulk modulus, thermal flow with anisotropic thermal conductivity, and fluid flow in porous media with anisotropic hydraulic permeability all are generalizations of the field equation in the anisotropic medium. Each physics domain introduces its own anisotropic distance.

anisotropy, electromagnetics, log analysis, resistivity log, stratigraphy

INTRODUCTION

Schlumberger introduced Wireline electric logging in 1927. In the following years, they explored repeat observations and recurring phenomena to gain a better understanding of their measurement. From their earlier surface electric surveys, Schlumberger understood that the electric measurements were related to the conductivity of the rocks, notably the fluids in the pore space.

In those years, all wells were drilled mostly vertical. Many reservoirs were sedimentary rocks showing horizontal stratification. This simple, axisymmetric configuration gave rise to an interpretation that allowed for lamination anisotropy in the electric conductivity, distinguishing a horizontal conductivity $σ_{h}$ parallel to the laminae and a vertical conductivity $σ_{v}$ perpendicular to the laminae (Figure 1).

The electric logs injected a current from a source electrode $A$ and measured the potential difference between two electrodes $M$ and $N$ some distance above the injection electrode. The vertically flowing current was expected to provide an apparent conductivity $σ_{a}$ that was close to the vertical conductivity $σ_{v}$ ⁠. However, time and again the logs were reading close to the horizontal conductivity $σ_{h}$ ⁠, apparently blind to the vertical conductivity.

This unexpected behavior became known as the “paradox of anisotropy.” It took five years before Henri G. Doll and his mathematician adviser Raymond Maillet gave a rigorous mathematical proof confirming this observation. Maillet and Doll (1932) proved that any vertically aligned electric measurement in a horizontally oriented lamination would only be sensitive to the horizontal conductivity and completely blind to the vertical conductivity, provided the borehole has negligible diameter. Since this early work, several authors have independently derived this result (van der Pauw, 1961; Wasscher, 1961; Bhattacharya and Patra, 1968; Hagiwara, 1994; Al-Garni and Everett, 2003), providing additional insights.

Many years later, Moran and Gianzero (1979) derived the electromagnetic fields for point dipole radiators in such laminated formations as close-form expressions. Their solution enabled the quantitative interpretation (Hagiwara, 1994; Lüling et al., 1994) of electromagnetic surveys in deviated wells through laminated formations. The Maillet and Doll (1932) and the Moran and Gianzero (1979) analyses relied heavily on the axisymmetry of the laminae to simplify the derivation of their solution.

The lamination anisotropy manifests itself in other physics domains as well. Seismic surveys must account for anisotropy in bulk and shear moduli.

Thermal flow in solids is controlled by the thermal conductivity, which may be anisotropic. Carslaw and Jaeger (1959) give the close-form solution in the eigensystem of the thermal-conductivity tensor for a $δ$ -pulsed thermal point source and a thermal point source with finite duration.

Fluid flow in porous media is controlled by the hydraulic permeability according to Darcy’s law. Testing measures this permeability, but is equally limited by some variation of the paradox of anisotropy. Goode and Thambynayagam (1990, 1992), Kuchuk and Habashy (1995), Kuchuk (1996), and Kuchuk et al. (2010) state the general solution for this problem in the eigensystem of the permeability tensor for homogeneous, anisotropic media and for layered media with anisotropy.

COULOMB’S LAW IN ANISOTROPIC MEDIA

The original Maillet and Doll (1932) proof for the paradox of anisotropy and the Moran and Gianzero (1979) solution for radiating dipoles in lamination-anisotropic media rely on a detailed analysis of the underlying geometry, exploiting the remaining circular symmetry in the plane parallel to the laminae. The present analysis abandons this simplification and builds on the fully anisotropic description with a symmetric $3 \times 3$ tensor.

Electrostatics commonly describes conductive media by an electric conductivity

\hat{σ} [A / V m]

and current point injector

I_{0} [A]

as source; the injection point is described as

δ

-function distribution with the source current density

j_{0} = \partial_{t} q [A / m^{3}]

⁠;

j_{0} (\vec{r}) = \partial_{t} q (\vec{r}) = I_{0} δ^{3} (\vec{r}) .

(1)

The isotropic case is solved by the known electrostatic Coulomb potential

Φ [V]

Φ (\vec{r}) = \frac{I_{0}}{4 π σ r} .

(2)

The gradient of this potential is the electric field

\vec{E} [V / m]

\vec{E} (\vec{r}) = - \vec{\nabla} Φ (\vec{r}) .

(3)

The electric field

\vec{E}

relates to the electric current density

\vec{j} [A / m^{2}]

\vec{j} (\vec{r}) = \hat{σ} \cdot \vec{E} (\vec{r}) .

(4)

Maxwell’s equation for the displacement current

\vec{D}

provides the charge density

q

\vec{\nabla} \cdot \vec{D} (\vec{r}) = q (\vec{r}) .

(5)

The divergence of Maxwell-Ampère’s law together with equation ensures charge conservation with the injection-current as local source

\vec{\nabla} \cdot (\vec{\nabla} \times \vec{H}) = 0 = \vec{\nabla} \cdot \partial_{t} \vec{D} (\vec{r}) + \vec{\nabla} \cdot \vec{j} (\vec{r}) \Rightarrow \vec{\nabla} \cdot \vec{j} (\vec{r}) = - \partial_{t} q (\vec{r}) .

(6)

The field equation combines equations

\vec{\nabla} \cdot \hat{σ} \cdot \vec{\nabla} Φ (\vec{r}) = \partial_{t} q (\vec{r}) = I_{0} δ^{3} (\vec{r}) .

(7)

The conductivity

\hat{σ}

is a symmetric, second-rank tensor because it couples to the gradient vector on both sides; any skew-symmetric part in the tensor would cancel. The conductivity is real-valued, so the symmetric conductivity tensor will have real eigenvalues and orthogonal eigenvectors.

Appendix A derives the solution to the field equation , which introduces a new distance concept — the anisotropic distance

r_{{\hat{σ}}^{- 1}} \equiv \sqrt{\vec{r} \cdot {\hat{σ}}^{- 1} \cdot \vec{r}} .

(8)

This distance is distinct from the Pythagorean distance

r = \sqrt{\vec{r} \cdot \vec{r}}

⁠; its units do not even combine to

m

⁠. This distance describes the apparent distortion of space in electric measurements due to the formation anisotropy.

With this anisotropic distance, Appendix A shows that Coulomb’s electrostatic potential in anisotropic media has the close-form solution

Φ (\vec{r}) = \frac{I_{0}}{4 π \sqrt{| \hat{σ} |} \sqrt{\vec{r} \cdot {\hat{σ}}^{- 1} \cdot \vec{r}}} .

(9)

The potential normalization

σ

in equation is replaced by the square root of the tensor determinant, which thus reconcile the units of the anisotropic distance with the isotropic case.

ELECTRIC FIELD AND ELECTRIC-CURRENT DENSITY WITH SIMPLE EXAMPLES

The Coulomb potential in equation gives the electric field

\vec{E}

\vec{E} (\vec{r}) = - \vec{\nabla} Φ (\vec{r}) = \frac{I_{0}}{4 π \sqrt{| \hat{σ} |} {\sqrt{\vec{r} \cdot {\hat{σ}}^{- 1} \cdot \vec{r}}}^{3}} {\hat{σ}}^{- 1} \cdot \vec{r} .

(10)

This electric field does not point in a radial direction — it is distorted by the inverse conductivity tensor

{\hat{σ}}^{- 1}

⁠. Thus, one fundamental insight of Coulomb’s law must be amended and generalized in the presence of anisotropy.

The electric-current density

\vec{j}

relates to the electric field

\vec{E}

with the conductivity tensor

\hat{σ}

\vec{j} (\vec{r}) = \hat{σ} \cdot \vec{E} (\vec{r}) = \frac{I_{0}}{4 π \sqrt{| \hat{σ} |} {\sqrt{\vec{r} \cdot {\hat{σ}}^{- 1} \cdot \vec{r}}}^{3}} \vec{r} .

(11)

The electric-current density is purely radially oriented; however, its magnitude varies as a function of direction because of the anisotropic distance in the denominator. This anisotropic distance weighs the different directions in space according to the inverse conductivity tensor

{\hat{σ}}^{- 1}

⁠.

The divergence of the electric-current density is zero away from the origin; here, the purely radial direction of the current density is crucial

\vec{\nabla} \cdot \vec{j} (\vec{r}) = \frac{I_{0}}{4 π \sqrt{| \hat{σ} |}} (\frac{\vec{\nabla} \cdot \vec{r}}{{\sqrt{\vec{r} \cdot {\hat{σ}}^{- 1} \cdot \vec{r}}}^{3}} + \vec{r} \cdot \vec{\nabla} \frac{1}{{\sqrt{\vec{r} \cdot {\hat{σ}}^{- 1} \cdot \vec{r}}}^{3}}) = \frac{I_{0}}{4 π \sqrt{| \hat{σ} |}} (\frac{3}{{\sqrt{\vec{r} \cdot {\hat{σ}}^{- 1} \cdot \vec{r}}}^{3}} - \frac{3 \vec{r} \cdot {\hat{σ}}^{- 1} \cdot \vec{r}}{{\sqrt{\vec{r} \cdot {\hat{σ}}^{- 1} \cdot \vec{r}}}^{5}}) = 0 .

(12)

The normalization of the anisotropic Coulomb potential follows from the analysis in appendix A.

The equipotential surfaces are no longer concentric spheres, but ellipsoids that illustrate how the conductivity tensor weighs the different spatial directions. A simple example shall illustrate the ellipsoidal equipotential surfaces. The conductivity tensor shall be diagonal in the (⁠

x

⁠,

y

⁠,

z

⁠) coordinate system; furthermore, it shall be degenerate in the horizontal

x - y

–plane.

\hat{σ} = (\begin{matrix} σ_{x} & 0 & 0 \\ 0 & σ_{x} & 0 \\ 0 & 0 & σ_{z} \end{matrix}) .

(13)

A $2 ∶ 1$ anisotropic medium with $σ_{x} = 2 S / m$ and $σ_{z} = 1 S / m$ is compared to an isotropic medium with conductivity $σ = 1 S / m$ ⁠. In Figure 2, the equipotential surfaces are plotted for an $x - z$ –plane with a small distance $y = 0.1 m$ off the origin to avoid the singularity of the potential. The equipotential contours are displayed on a logarithmic scale, which is pleasing to the eye without any scientific value.

The anisotropic case shows that the equipotential contours are distorted from circles into ellipses. The principal axes of the ellipses obey a ratio $\sqrt{σ_{x} / σ_{z}} = \sqrt{2}$ ⁠, which corresponds to the ratio of the axes in the anisotropic distance 8 with the tensor 13.

In Figure 3, the electric field 10 is superposed as evenly distributed arrows with a length proportional to the square root of the field magnitude. This normalization is chosen for aesthetic reasons without any scientific significance.

The anisotropic figure illustrates the electric-field distortion from purely radial into $x$ -direction.

In Figure 4, the electric-current density 11 is superposed as blue arrows. Again, the arrow length is chosen as square root of the magnitude for aesthetic reasons without any scientific significance.

In the isotropic case, the current density overlays the electric field. On the other hand, in the anisotropic case, the radially oriented electric-current density deviates from the electric-field direction off the $x$ - and $z$ -axes, illustrating the misalignment caused by the anisotropy.

PROOF OF THE PARADOX OF ANISOTROPY

The close form of Coulomb’s potential in equation applies to the special case of a horizontally laminated formation. These formations are described by a diagonal conductivity tensor that retains rotational symmetry in the horizontal

x - y

–plane. The conductivity parallel to the laminae usually is called horizontal conductivity

σ_{h}

⁠, whereas the conductivity perpendicular to the laminae is called vertical conductivity

σ_{v}

⁠, similar to the tensor in the above examples

\hat{σ} = (\begin{matrix} σ_{h} & 0 & 0 \\ 0 & σ_{h} & 0 \\ 0 & 0 & σ_{v} \end{matrix}) .

(14)

This tensor has the determinant

| \hat{σ} | = σ_{h}^{2} σ_{v},

(15)

and the anisotropic distance

r_{{\hat{σ}}^{- 1}} \equiv \sqrt{\vec{r} \cdot {\hat{σ}}^{- 1} \cdot \vec{r}} = \sqrt{σ_{h}^{- 1} (x^{2} + y^{2}) + σ_{v}^{- 1} z^{2}} .

(16)

Coulomb’s potential in this anisotropic medium is then with equation

Φ (\vec{r}) = \frac{I_{0}}{4 π \sqrt{σ_{h}^{2} σ_{v}} \sqrt{σ_{h}^{- 1} (x^{2} + y^{2}) + σ_{v}^{- 1} z^{2}}} .

(17)

Special case of vertical alignment — the proof

The paradox of anisotropy describes vertical wells, where the current-injection point and any potential-observation point are vertically aligned with some distance

z

⁠:

\vec{r} = (\begin{matrix} 0 & 0 & z \end{matrix})

⁠. This vertical distance vector simplifies the anisotropic distance to

r_{{\hat{σ}}^{- 1}} = \sqrt{σ_{v}^{- 1} z^{2}} .

(18)

Coulomb’s potential cancels the inverse vertical conductivity

σ_{v}^{- 1}

from the anisotropic distance with the corresponding term in the determinant

Φ (\vec{r}) = \frac{I_{0}}{4 π \sqrt{σ_{h}^{2} σ_{v}} \sqrt{σ_{v}^{- 1} z^{2}}} = \frac{I_{0}}{4 π σ_{h} z} .

(19)

This cancellation constitutes the proof for the paradox of anisotropy. Any voltage measurement in a vertically aligned electrode configuration is a superposition of this Coulomb potential; the vertical conductivity

σ_{v}

has canceled out at this most fundamental level, leaving the potential only as function of the horizontal conductivity

σ_{h}

⁠.

Special case of horizontal alignment — surface surveys

Surface surveys or seafloor electric surveys align the current-injector and potential-monitor electrodes horizontally, parallel to any underlying lamination. In this case, the anisotropic distance is

r_{{\hat{σ}}^{- 1}} = \sqrt{σ_{h}^{- 1} (x^{2} + y^{2})},

(20)

and Coulomb’s potential in the laminated medium cancels the inverse horizontal conductivity

σ_{h}^{- 1}

against one

σ_{h}

factor in the normalization determinant; Coulomb’s potential simplifies to

Φ (\vec{r}) = \frac{I_{0}}{4 π \sqrt{σ_{h}^{2} σ_{v}} \sqrt{σ_{h}^{- 1} (x^{2} + y^{2})}} = \frac{I_{0}}{4 π \sqrt{σ_{h} σ_{v}} \sqrt{x^{2} + y^{2}}} .

(21)

Any surface survey will only measure the geometric mean of horizontal and vertical conductivity; it will be blind to the horizontal conductivity by itself.

Considerations for general voordinate orientation

Coulomb’s potential in an anisotropic medium 9 combines the anisotropic-conductivity tensor as Jacobian with the anisotropic distance 8. The distance vector couples to both sides of the inverse conductivity tensor. This way it projects onto one component of the inverse tensor, which in turn cancels the corresponding term in the Jacobian determinant. In the product, only the anisotropic conductivity components in the plane perpendicular to the distance vector remain.

This observation holds for a general coordinate orientation, where in general the tensor is not diagonal. If the distance vector lies along a principal axis of the tensor; i.e., the tensor is diagonal the plane perpendicular to the distance vector is defined by the two other eigenvectors; hence, the Coulomb-potential normalization will be the geometric mean of the two corresponding eigenvalues. Wasscher (1961) observed and clearly emphasized this geometric relationship.

APPLICATION TO OTHER PHYSICS DOMAINS

The field equation for Coulomb’s potential 7 reappears in similar or generalized form in various physics domains. Each such domain uses some material parameter that becomes tensor-valued in anisotropic media. Hence the close-form solution to Coulomb’s potential 9 generalizes or adapts to these physics problems as well.

Compressional elastic waves with tensor-valued bulk modulus

An elastic medium with bulk modulus

κ

and density

ρ

may be subject to some point-pressure-pulse source. A monochromatic, single-frequency source with frequency

f

(angular frequency

ω = 2 π f

⁠) generates a pressure wave

p

with the elastic-medium equation

\frac{κ}{ρ} \vec{\nabla} \cdot \vec{\nabla} p (\vec{r}, t) - \partial_{t}^{2} p (\vec{r}, t) = p_{0} δ^{3} (\vec{r}) e^{- i ω t} .

(22)

Here, the time dependence factors into the universal

e^{- i ω t}

⁠, simplifying the time derivative

\partial_{t} \equiv - i ω

⁠. In anisotropic media, the bulk modulus

\hat{κ}

is tensor valued. The field equation becomes

\vec{\nabla} \cdot \frac{\hat{κ}}{ρ} \cdot \vec{\nabla} p (\vec{r}) + ω^{2} p (\vec{r}) = p_{0} δ^{3} (\vec{r}) .

(23)

This equation has the “spherical-wave” solution

p (\vec{r}, t) = \frac{p_{0} \sqrt{ρ}}{4 π \sqrt{| \hat{κ} |}} \frac{e^{i ω \sqrt{ρ} \sqrt{\vec{r} \cdot {\hat{κ}}^{- 1} \cdot \vec{r}} - i ω t}}{\sqrt{\vec{r} \cdot {\hat{κ}}^{- 1} \cdot \vec{r}}},

(24)

where the spheres are distorted into ellipsoids by the anisotropic distance

r_{{\hat{κ}}^{- 1}} = \sqrt{\vec{r} \cdot {\hat{κ}}^{- 1} \cdot \vec{r}} .

⁠.

The present study does not address shear waves or the shear modulus in anisotropic media.

Thermal flow problem with tensor-valued thermal conductivity

Carslaw and Jaeger (1959) have analyzed this problem. Fourier’s law of thermal transport is a diffusion problem, relating temporal change

\partial_{t}

of a temperature distribution

T (\vec{r}) [K]

to thermal flow

\partial_{t} T (\vec{r}, t) = \frac{κ}{c ρ} \vec{\nabla} \cdot \vec{\nabla} T (\vec{r}, t) + Q_{0} (\vec{r}, t) .

(25)

Here,

Q_{0}

is an external heat source; the thermal conductivity

κ

⁠, the specific heat

c

and the material density

ρ [g / {cm}^{3}]

describe the system. In the generalized Coulomb case,

Q_{0}

may be a

δ

-pulsed point heater emitting an instant pulse at time

t = 0

\partial_{t} T (\vec{r}, t) = \frac{κ}{c ρ} \vec{\nabla} \cdot \vec{\nabla} T (\vec{r}, t) + Q_{0} δ^{3} (\vec{r}) δ (t) .

(26)

In anisotropic media, the thermal conductivity will be tensor-valued

\partial_{t} T (\vec{r}, t) = \vec{\nabla} \cdot \frac{\hat{κ}}{c ρ} \cdot \vec{\nabla} T (\vec{r}, t) + Q_{0} δ^{3} (\vec{r}) δ (t) .

(27)

Appendix B derives the Green’s function for this diffusion equation

G (\vec{r}, t) = \frac{e^{- (\vec{r} \cdot {\hat{κ}}^{- 1} \cdot \vec{r}) / 4 t}}{8 π^{3 / 2} t^{3 / 2} \sqrt{| \hat{κ} |}} .

(28)

The special case of a steady heat injection

Q_{0} / Δ T_{0}

simplifies Fourier’s equation with the point heater. This steady-state problem reduces to Coulomb’s field equation in anisotropic media

\partial_{t} T (\vec{r}) = \vec{\nabla} \cdot \frac{\hat{κ}}{c ρ} \cdot \vec{\nabla} T (\vec{r}) + \frac{Q_{0}}{Δ T_{0}} δ^{3} (\vec{r}) \Leftrightarrow \vec{\nabla} \cdot \frac{\hat{κ}}{c ρ} \cdot \vec{\nabla} T (\vec{r}) = - \frac{Q_{0}}{Δ T_{0}} δ^{3} (\vec{r}),

(29)

with the solution derived in appendix B

T (\vec{r}) = \frac{Q_{0}}{Δ T_{0}} \frac{c ρ}{4 π \sqrt{| \hat{κ} |} \sqrt{\vec{r} \cdot {\hat{κ}}^{- 1} \cdot \vec{r}}} .

(30)

Darcy flow in porous media

A porous, anisotropic medium with tensor-valued hydraulic permeability

k

may be filled with some fluid with viscosity

ν

and density

ρ

⁠. Darcy’s equation describes the pressure distribution

P (\vec{r}, t)

in this medium for a fluid-injection source

Q_{0}

A point-like source injecting an instant pressure pulse at time

t = 0

fulfills the equation

\frac{1}{ρ ν} \vec{\nabla} \cdot \hat{k} \cdot \vec{\nabla} P (\vec{r}, t) = \frac{1}{ρ} \partial_{t} P (\vec{r}, t) + Q_{0} δ^{3} (\vec{r}) δ (t) .

(31)

Goode and Thambynayagam (1990, 1992), Kuchuk and Habashy (1995), Kuchuk (1996), and Kuchuk et al. (2010) have explained that this equation is equivalent to the thermal-flow equation with the same Green’s function

P (\vec{r}, t) = \frac{ρ ν Q_{0} e^{- ν (\vec{r} \cdot {\hat{k}}^{- 1} \cdot \vec{r}) / 4 t}}{8 π^{3 / 2} t^{3 / 2} \sqrt{| \hat{k} |}} .

(32)

CONCLUSIONS

Electric surveys in laminated formations are subject to the Paradox of Anisotropy: in vertical wells with negligible borehole through a horizontal lamination, they measure only the horizontal conductivity and are blind to the vertical conductivity. Maillet and Doll proved this theorem in 1932. Thus, in these conditions, Laterolog surveys cannot detect hydrocarbons in oil-bearing sand-shale laminae, whose horizontal conductivity is dominated by the conductive shales.

This study provides a simple proof of the paradox of anisotropy. The proof is based on Coulomb’s law in anisotropic media with a general conductivity tensor. Coulomb’s potential introduces an anisotropic distance that includes the inverse conductivity tensor in the scalar product between the two distance vectors; this way, Pythagoras’ theorem generalizes to anisotropic media. The determinant of the conductivity tensor normalizes this potential, reconciling the units of the anisotropic distance.

The paradox of anisotropy manifests itself at the level of Coulomb’s potential as cancellation of the vertical conductivity in the normalization determinant and the anisotropic distance. This case requires purely vertical alignment of source and observation point. On the other hand, a horizontal survey parallel to the laminae is sensitive to the geometric mean of horizontal and vertical conductivities.

Coulomb’s problem readily generalizes to other physics domains with their specific anisotropy. Elastic media support compressional waves, which are parameterized by the bulk modulus. This study does not address shear waves, though. Fourier’s law describes heat transport as diffusion equation with a tensor-valued thermal conductivity in anisotropic media. Darcy’s equation describes fluid flow through porous media, which is parameterized by the hydraulic permeability. In anisotropic media, the permeability becomes tensor-valued.

ACKNOWLEDGMENTS

I want to thank Constantin Bachas (Ecole Normale Supérieure) for discussions on Green’s functions in anisotropic media, which strongly influenced Appendix A. I thank Tarek M. Habashy for sharing his version of the derivation of Coulomb’s potential in anisotropic media with me. I thank Fikri Kuchuk for explaining to me his work on Darcy’s equation in Well Testing. I thank Patrice Ligneul and Laurent Mossé for their diligent reviews of the manuscript and their helpful comments. Finally, I thank Mark Everett (Texas A&M University) as referee for valuable comments about the manuscript and for providing several references.

COULOMB’S LAW IN ANISOTROPIC MEDIA

An electric current

I_{0}

is injected at the origin, described by the current density

j_{0}

⁠:

j_{0} (\vec{r}) = I_{0} δ^{3} (\vec{r})

This current density gives an electrostatic potential

Φ

⁠. This potential satisfies the field equation :

\vec{\nabla} \cdot \hat{σ} \cdot \vec{\nabla} Φ (\vec{r}) = - j_{0} (\vec{r}) = - I_{0} δ^{3} (\vec{r}) .

This field equation is readily solved in momentum space. The momentum space relates to the configuration space by a local plane-wave assumption

e^{i \vec{k} \cdot \vec{r}}

⁠, such that each gradient is replaced by the wave vector

\vec{\nabla} \Leftrightarrow i \vec{k}

⁠. In momentum space, the field equation becomes

\vec{k} \cdot \hat{σ} \cdot \vec{k} \tilde{Φ} (\vec{k}) = I_{0} .

(A-1)

This equation is solved by the scalar-valued momentum-space Green function

\tilde{G} (\vec{k})

(\vec{k} \cdot \hat{σ} \cdot \vec{k}) \tilde{G} (\vec{k}) = 1 \Leftrightarrow \tilde{G} (\vec{k}) = \frac{1}{\vec{k} \cdot \hat{σ} \cdot \vec{k}} .

(A-2)

This Green’s function is Fourier-transformed back to configuration space

G (\vec{r}) = \int \frac{d^{3} k}{{(2 π)}^{3}} \tilde{G} (\vec{k}) e^{- i \vec{k} \cdot \vec{r}} = \frac{1}{(2 π)^{3}} \int d^{3} k \frac{e^{- i \vec{k} \cdot \vec{r}}}{\vec{k} \cdot \hat{σ} \cdot \vec{k}} .

(A-3)

The scalar product in the denominator includes the electric-conductivity tensor, which is equally divided between the two wave vectors

\vec{k} \cdot \hat{σ} \cdot \vec{k} = (\vec{k} \cdot {\hat{σ}}^{1 / 2}) \cdot ({\hat{σ}}^{1 / 2} \cdot \vec{k}) .

(A-4)

The wave vector is thus modified by a linear transformation

\vec{k} \to {\vec{k}}^{'} \equiv {\hat{σ}}^{1 / 2} \cdot \vec{k} .

(A-5)

The scalar product in the exponent is appropriately modified by inserting a unity matrix in the scalar product

\vec{k} \cdot \vec{r} = (\vec{k} \cdot {\hat{σ}}^{1 / 2}) \cdot ({\hat{σ}}^{- 1 / 2} \cdot \vec{r}) = {\vec{k}}^{'} \cdot ({\hat{σ}}^{- 1 / 2} \cdot \vec{r}),

(A-6)

which modifies the distance vector

\vec{r} \to {\hat{σ}}^{- 1 / 2} \cdot \vec{r}

⁠. This vector provides a new, anisotropic distance

{\hat{σ}}^{- 1 / 2} \cdot \vec{r} \to r_{{\hat{σ}}^{- 1}} \equiv \sqrt{\vec{r} \cdot {\hat{σ}}^{- 1} \cdot \vec{r}} .

(A-7)

This anisotropic distance is not even given in meters, but has mixed units.

The Jacobian of the linear transformation modifies the integration measure

d^{3} k \to d^{3} k^{'} = | {\hat{σ}}^{1 / 2} | d^{3} k \Leftrightarrow d^{3} k = | {\hat{σ}}^{- 1 / 2} | d^{3} k^{'} = d^{3} k^{'} / \sqrt{| \hat{σ} |},

(A-8)

using the identity

| {\hat{σ}}^{1 / 2} | = \sqrt{| \hat{σ} |}

⁠. The Fourier-transform of the Green’s function () becomes

G (\vec{r}) = \frac{1}{(2 π)^{3}} \int d^{3} k \frac{e^{- i \vec{k} \cdot \vec{r}}}{\vec{k} \cdot \hat{σ} \cdot \vec{k}} = \frac{1}{(2 π)^{3} \sqrt{| \hat{σ} |}} \int d^{3} k^{'} \frac{e^{- i {\vec{k}}^{'} \cdot ({\hat{σ}}^{- 1 / 2} \cdot \vec{r})}}{{\vec{k}}^{'} \cdot {\vec{k}}^{'}} .

(A-9)

This integral is the Fourier-transform of the isotropic Coulomb potential. Coulomb’s potential in an anisotropic conductive medium

\hat{σ}

then becomes

Φ (\vec{r}) = \frac{I_{0}}{4 π \sqrt{| \hat{σ} |} \sqrt{\vec{r} \cdot {\hat{σ}}^{- 1} \cdot \vec{r}}} .

The mixed units of the anisotropic distance reconcile: The Jacobian provides 3/2 powers of

σ

⁠, whereas the anisotropic distance contributes

- 1 / 2

power of

σ

to give the same units as the isotropic Coulomb potential

Φ (\vec{r}) = \frac{I_{0}}{4 π σ r} .

(A-10)

THERMAL-TRANSPORT AND DIFFUSION EQUATIONS IN ANISOTROPIC MEDIA

Carslaw and Jaeger (1959) state the solution for the thermal-transport equation in an anisotropic medium in the eigensystem of the anisotropy tensor, i. e., with a diagonalized material tensor. This equation is an example of a diffusion processes; hence, several authors (Goode and Thambynayagam, 1990, 1992; Kuchuk and Habashy, 1995, Kuchuk, 1996, Kuchuk et al., 2010) have adapted this solution to Darcy’s problem of fluid flow in porous media, all referring back to Carslaw and Jaeger (1959).

It is not necessary to derive the solution for this thermal-transport or diffusion equation in the eigensystem of the material tensor; the tensor does not need to be diagonal. The thermal-transport equation is:

\partial_{t} T (\vec{r}, t) = \vec{\nabla} \cdot \frac{\hat{κ}}{c ρ} \cdot \vec{\nabla} T (\vec{r}, t) + Q_{0} δ^{3} (\vec{r}) δ (t) .

It describes the temperature distribution

T (\vec{r}, t)

in space and time with the tensor-valued thermal conductivity

\hat{κ}

⁠, the specific heat

c

⁠, and the material density

ρ

for a pointlike heat source that emits an instant heat pulse

Q_{0}

at time

t = 0

⁠.

Likewise, the diffusion equation describing the pressure

P (\vec{r}, t)

in space and time is

\frac{1}{ρ ν} \vec{\nabla} \cdot \hat{k} \cdot \vec{\nabla} P (\vec{r}, t) = \frac{1}{ρ} \partial_{t} P (\vec{r}, t) + Q_{0} δ^{3} (\vec{r}) δ (t) .

Here, the ratio of the tensor-valued permeability

\hat{k}

over viscosity

ν

gives the mobility

\hat{λ} \equiv \hat{k} / ν

⁠. The material density

ρ

normalizes the pressure field. The fluid-injection point source emits an instant pulse

Q_{0}

at time

t = 0

⁠.

Thermal-transport and diffusion equations are described by the Green’s function

G (\vec{r}, t)

\vec{\nabla} \cdot \hat{λ} \cdot \vec{\nabla} G (\vec{r}, t) = \partial_{t} G (\vec{r}, t) .

(B-1)

Here,

\hat{λ}

describes the generic anisotropic material tensor. The source term commonly is replaced by a boundary condition at the source point and at time

t = 0

⁠.

The separated spatial and time derivatives allow separation of variables. The spatial dependence away from the point source allows a local Fourier decomposition with the wave vector

\vec{k}

G (\vec{r}, t) \to \tilde{G} (\vec{k}, t) = \int d^{3} r G (\vec{r}, t) e^{i \vec{k} \cdot \vec{r}},

(B-2)

with the gradient

\vec{\nabla} G (\vec{r}, t) \to i \vec{k} \tilde{G} (\vec{k}, t) .

(B-3)

The diffusion equation then becomes

- \vec{k} \cdot \hat{λ} \cdot \vec{k} \tilde{G} (\vec{k}, t) = \partial_{t} \tilde{G} (\vec{k}, t) .

(B-4)

This first-order, linear differential equation has the solution

\tilde{G} (\vec{k}, t) = e^{- (\vec{k} \cdot \hat{λ} \cdot \vec{k}) t},

(B-5)

with unit normalization. The momentum-space Green function is transformed back to configuration space

G (\vec{r}, t) = \frac{1}{(2 π)^{3}} \int d^{3} k \tilde{G} (\vec{k}, t) e^{- i \vec{k} \cdot \vec{r}} = \frac{1}{{(2 π)}^{3}} \int d^{3} k e^{- (\vec{k} \cdot \hat{λ} \cdot \vec{k}) t - i \vec{k} \cdot \vec{r}} .

(B-6)

The quadratic form in the exponent is completed by adding a suitable zero. The material tensor is factored into the square of its square root. This square root and its inverse are inserted in the Fourier term

- i \vec{k} \cdot \vec{r}

⁠:

- (\vec{k} \cdot \hat{λ} \cdot \vec{k}) t - i \vec{k} \cdot \vec{r} = - [t^{1 / 2} (\vec{k} \cdot {\hat{λ}}^{1 / 2}) \cdot ({\hat{λ}}^{1 / 2} \cdot \vec{k}) t^{1 / 2} + {it}^{1 / 2} (\vec{k} \cdot {\hat{λ}}^{1 / 2}) \cdot ({\hat{λ}}^{- 1 / 2} \cdot \vec{r}) t^{- 1 / 2}] .

(B-7)

This quadratic form is completed by adding and subtracting the term

\frac{{it}^{- 1 / 2}}{2} (\vec{r} \cdot {\hat{λ}}^{- 1 / 2}) \cdot ({\hat{λ}}^{- 1 / 2} \cdot \vec{r}) \frac{{it}^{- 1 / 2}}{2} = - \frac{1}{4 t} (\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r}) .

(B-8)

This way, the wave vector is shifted by an (imaginary) constant, and the spatial term separates out

- (\vec{k} \cdot \hat{λ} \cdot \vec{k}) t - i \vec{k} \cdot \vec{r} = - [t^{1 / 2} (\vec{k} \cdot {\hat{λ}}^{1 / 2}) + i \frac{t^{- 1 / 2}}{2} (\vec{r} \cdot {\hat{λ}}^{- 1 / 2})] \cdot [({\hat{λ}}^{1 / 2} \cdot \vec{k}) t^{1 / 2} + i ({\hat{λ}}^{- 1 / 2} \cdot \vec{r}) \frac{t^{- 1 / 2}}{2}] - \frac{1}{4 t} (\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r}) .

(B-9)

The Fourier integral B-6 becomes

G (\vec{r}, t) = \frac{e^{- (\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r}) / 4 t}}{{(2 π)}^{3}} \int d^{3} k e^{- [t^{1 / 2} (\vec{k} \cdot {\hat{λ}}^{1 / 2}) + i \frac{t^{- 1 / 2}}{2} (\vec{r} \cdot {\hat{λ}}^{- 1 / 2})] \cdot [({\hat{λ}}^{1 / 2} \cdot \vec{k}) t^{1 / 2} + i ({\hat{λ}}^{- 1 / 2} \cdot \vec{r}) \frac{t^{- 1 / 2}}{2}]} .

(B-10)

The constant shift

i (\vec{r} \cdot {\hat{λ}}^{- 1 / 2}) t^{- 1 / 2} / 2

is eliminated from the wave vector upon integration. The wave vector itself has been linearly transformed, which transforms the integral measure by the Jacobian

\vec{k} \to {\vec{k}}^{'} \equiv t^{1 / 2} {\hat{λ}}^{1 / 2} \cdot \vec{k} \Rightarrow d^{3} k \to d^{3} k^{'} = t^{3 / 2} \sqrt{| \hat{λ} |} d^{3} k .

(B-11)

The configuration-space Green function thus becomes

G (\vec{r}, t) = \frac{e^{- (\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r}) / 4 t}}{{(2 π)}^{3} t^{3 / 2} \sqrt{| \hat{λ} |}} \int d^{3} k^{'} e^{- {\vec{k}}^{'} \cdot {\vec{k}}^{'}} = \frac{e^{- (\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r}) / 4 t}}{8 π^{3 / 2} t^{3 / 2} \sqrt{| \hat{λ} |}} .

(B-12)

This close-form solution is equation 8 in chapter 10.2 of Carslaw and Jaeger (1959) without limiting the description to the eigenvalues of the material tensor

\hat{λ}

⁠.

Steady-state solution

This Green’s function assumes an instant source pulse at time $t = 0$ ⁠. Imposing a steady source from time $t = 0$ on provides the steady-state solution. For example, if a point heat source in a homogeneous, anisotropic medium is switched on at $t = 0$ it will slowly heat up the medium until it reaches and asymptotic equilibrium.

This asymptotic evolution of the medium is described by integrating the Green function from over time from

t = 0

to some later time

T

⁠, ultimately to infinity

\bar{G} (\vec{r}, T) = \int_{0}^{T} d t G (\vec{r}, t) = \frac{1}{8 π^{3 / 2} \sqrt{| \hat{λ} |}} \int_{0}^{T} d t \frac{e^{- (\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r}) / 4 t}}{t^{3 / 2}} .

(B-13)

The integration measure

d t / t^{3 / 2}

invites the substitution

u \equiv 1 / t^{1 / 2} \Rightarrow d u = - d t / 2 t^{3 / 2}

and

1 / t = u^{2}

in the exponent to give the integral

\bar{G} (\vec{r}, T) = \frac{1}{8 π^{3 / 2} \sqrt{| \hat{λ} |}} \int_{0}^{T} \frac{d t}{t^{3 / 2}} e^{- (\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r}) / 4 t} = \frac{1}{4 π^{3 / 2} \sqrt{| \hat{λ} |}} \int_{1 / \sqrt{T}}^{\infty} {due}^{- u^{2} (\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r}) / 4} .

(B-14)

This integral gives the error function

erfc (\sqrt{(\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r})} / 2 \sqrt{T})

\bar{G} (\vec{r}, T) = \frac{2}{8 π^{3 / 2} \sqrt{| \hat{λ} |}} \int_{1 / \sqrt{T}}^{\infty} {due}^{- u^{2} (\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r}) / 4} = \frac{erfc (\sqrt{(\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r})} / 2 \sqrt{T})}{4 π \sqrt{| \hat{λ} |} \sqrt{(\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r})}} .

(B-15)

The asymptotic steady-state limit

T \to \infty

reduces to Coulomb’s law in anisotropic media

\bar{\bar{G}} (\vec{r}) = \bar{G} (\vec{r}, T \to \infty) = \frac{1}{4 π \sqrt{| \hat{λ} |} \sqrt{(\vec{r} \cdot {\hat{λ}}^{- 1} \cdot \vec{r})}} .

(B-16)

The paradox of anisotropy in electric logging: A simple proof and extensions to other physics domains

ABSTRACT

INTRODUCTION

COULOMB’S LAW IN ANISOTROPIC MEDIA

ELECTRIC FIELD AND ELECTRIC-CURRENT DENSITY WITH SIMPLE EXAMPLES

PROOF OF THE PARADOX OF ANISOTROPY

Special case of vertical alignment — the proof

Special case of horizontal alignment — surface surveys

Considerations for general voordinate orientation

APPLICATION TO OTHER PHYSICS DOMAINS

Compressional elastic waves with tensor-valued bulk modulus

Thermal flow problem with tensor-valued thermal conductivity

Darcy flow in porous media

CONCLUSIONS

ACKNOWLEDGMENTS

COULOMB’S LAW IN ANISOTROPIC MEDIA

THERMAL-TRANSPORT AND DIFFUSION EQUATIONS IN ANISOTROPIC MEDIA

Steady-state solution

Data & Figures

About GSW

Resources

Explore

Connect

The paradox of anisotropy in electric logging: A simple proof and extensions to other physics domains

ABSTRACT

INTRODUCTION

COULOMB’S LAW IN ANISOTROPIC MEDIA

ELECTRIC FIELD AND ELECTRIC-CURRENT DENSITY WITH SIMPLE EXAMPLES

PROOF OF THE PARADOX OF ANISOTROPY

Special case of vertical alignment — the proof

Special case of horizontal alignment — surface surveys

Considerations for general voordinate orientation

APPLICATION TO OTHER PHYSICS DOMAINS

Compressional elastic waves with tensor-valued bulk modulus

Thermal flow problem with tensor-valued thermal conductivity

Darcy flow in porous media

CONCLUSIONS

ACKNOWLEDGMENTS

COULOMB’S LAW IN ANISOTROPIC MEDIA

THERMAL-TRANSPORT AND DIFFUSION EQUATIONS IN ANISOTROPIC MEDIA

Steady-state solution

Data & Figures

About GSW

Resources

Explore

Connect

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