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

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


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)")


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

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