lheagy/anisotropic-conductivity.md

## anisotropic-conductivity.md

      
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              anisotropic-conductivity.md
            
          
    :uid: anisotropic-conductivity
:title: Anisotropic Conductivity
:description: when electrical conductivity varies with direction
:tooltip: None
:tag: conductivity,anisotropy,dc,em,physical properties
:group: simpeg
:license: CC-BY-4.0
:source: https://api.github.com/gists/56c47e9a872aa908705d
When a physical property of a material varies with direction, that
material is anisotropic (not to be confused with heterogeneity. Check out Matt Hall's discussion on anisotropy). Anisotropy in electrical conductivity can have a variety of causes.
To name a few:

alignment of cracks or pores,
bedding / lamination,
alignment of mineral grains.

Any of these, or similar phenomena can create preferential directions
for current to flow. To examine this a bit, lets look at the constitutive relation that relates the current density $J$ to the electric field $E$ through the electrical conductivity $\sigma$ (aka. Ohm's Law).
$$ \vec{J} = \sigma \vec{E}$$
In an isotropic medium, $\sigma = \sigma(x,y,z)$ is a scalar which may
be a distributed property. If you apply a constant electric field
$\vec{E}$ (which is a vector having both magnitude and direction),
that exerts a force on the little charges in the conductor which is
directly proportional to strength of the electric field. This will
cause the charges to start to move, and the resistivity ($
\sigma^{-1}$) of the medium
acts much like friction, controlling how easily the charges move. This
description of the flow of charges is described by the current density, $\vec{J}$.
In an anisotropic medium, $\sigma$ is a $3\times3$
tensor, in which case, Ohm's law looks like:
$$ \left(\begin{array}{c} J_x \ J_y \ J_z \end{array} \right)
= \left(\begin{array}{ccc}
\sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\sigma_{xy} & \sigma_{yy} & \sigma_{yz} \\
\sigma_{xz} & \sigma_{yz} & \sigma_{zz}
\end{array}\right)
\left(\begin{array}{c} E_x \ E_y \ E_z \end{array} \right)
$$
Lets break this down and consider a few examples. Since drawing
vectors and such is easier in 2D, lets consider a 2D example, where
everything is happening in the x-z plane. In the figure below, we
have a "rock" that is blue and gray.

The blue regions are conductive and the gray regions are
resistive. If we apply an electric field in the x-direction, the
charges have a choice to either go through the resistive or
conductive region, like a parallel circuit. If instead we apply
an electric field in the z-direction, the charges have no choice
which medium they can go through.
If we want to describe the conductivity of this rock, we use a $2x2$ diagonal tensor. In this case, where the principal axes of anisotropy align with our cartesian coordinate system, the conductivity tensor describing this system is diagonal i.e.
$$
\sigma =
\left(
\begin{array}{cc}
\sigma_{xx} &            \\
& \sigma_{zz}
\end{array}
\right)
$$

If we excite this with an electric field oriented in the $x$-
direction, then we see
$$
\vec{J}_{1}

$$
In a similar vein, if we use an electric field oriented in the $z$-direction, then
$$
\vec{J}_{2}

$$
So we can multiply a matrix and a vector! Thats not why I am
showing you this... lets have a look at the magnitude of
the current density in each case. For the first case, we have that
$$
|\vec{J}{1}| = \sigma{xx} E_{x}
$$
and in the second case,
$$
|\vec{J}{2}| = \sigma{zz} E_{z}
$$
For the example here, we know that \sigma_{xx} > \sigma_{zz}. So if the magnitude of the inducing field is identical in both cases, the resulting current density magnitude is different between the two, namely $|\vec{J}{1}| > |\vec{J}2|$. So if we wanted the current density to be same, we need to apply a bigger push in the $z$-direction; in particular, we would need: $E{z} = \sigma{xx}/\sigma_{zz} E_x$.
In this example, everything lined up with our coordinate system, so
the various components didn't talk to each other. Lets change that.

I started planning on writing this about the behavior of anisotropy
in cylindrically symmetric settings, but got a bit sidetracked.
Anyways, that will come at another time!
References

http://www.agilegeoscience.com/blog/2015/2/9/what-is-anisotropy