# Effective Medium Theory Mapping¶

This example uses Self Consistent Effective Medium Theory to estimate the
electrical conductivity of a mixture of two phases of materials. Given
the electrical conductivity of each of the phases (\(\sigma_0\),
\(\sigma_1\)), the `SimPEG.Maps.SelfConsistentEffectiveMedium`

map takes the concentration of phase-1 (\(\phi_1\)) and maps this to an
electrical conductivity.

This mapping is used in chapter 2 of:

Heagy, Lindsey J.(2018, in prep) *Electromagnetic methods for imaging
subsurface injections.* University of British Columbia

author: | @lheagy |
---|

```
import numpy as np
import matplotlib.pyplot as plt
from SimPEG import Maps
from matplotlib import rcParams
rcParams['font.size'] = 12
```

## Conductivities¶

Here we consider a mixture composed of fluid (3 S/m) and conductive particles which we will vary the conductivity of.

```
sigma_fluid = 3
sigma1 = np.logspace(1, 5, 5) # look at a range of particle conductivities
phi = np.linspace(0.0, 1, 1000) # vary the volume of particles
```

## Construct the Mapping¶

We set the conductivity of the phase-0 material to the conductivity of the fluid. The mapping will then take a concentration (by volume), of phase-1 material and compute the effective conductivity

```
scemt = Maps.SelfConsistentEffectiveMedium(sigma0=sigma_fluid, sigma1=1)
```

## Loop over a range of particle conductivities¶

We loop over the values defined as sigma1 and compute the effective conductivity of the mixture for each concentration in the phi vector

```
sige = np.zeros([phi.size, sigma1.size])
for i, s in enumerate(sigma1):
scemt.sigma1 = s
sige[:, i] = scemt * phi
```

## Plot the effective conductivity¶

The plot shows the effective conductivity of 5 difference mixtures. In all cases, the conductivity of the fluid, \(\sigma_0\), is 3 S/m. The conductivity of the particles is indicated in the legend

```
fig, ax = plt.subplots(1, 1, figsize=(7, 4), dpi=350)
ax.semilogy(phi, sige)
ax.grid(which="both", alpha=0.4)
ax.legend(["{:1.0e} S/m".format(s) for s in sigma1])
ax.set_xlabel("Volume fraction of proppant $\phi$")
ax.set_ylabel("Effective conductivity (S/m)")
plt.tight_layout()
```

**Total running time of the script:** ( 0 minutes 0.288 seconds)