The Shin (2015) model

Introduction

The model proposed by Shin (2015) is an equivalent circuit that aims to reproduce SIP data This model predicts that the complex resistivity spectra \(\rho^*\) of a polarizable rock sample can be described by

\begin{equation} \rho^* = \sum_{i=1}^2 \frac{\rho_i}{(i\omega)^{n_i} \rho_iQ_i + 1} \end{equation}

where \(\omega\) is the measurement angular frequencies (\(\omega=2\pi f\)) and \(i\) is the imaginary unit.

Here, \(\rho^*\) depends on 3 pairs of parameters:

• \(\rho_i \in [0, \infty)\), the resistivity of the resistance element in Shin’s circuit.
• \(Q_i \in [0, \infty)\), the capacitance of the CPE.
• \(n_i \in [0, 1]\), the exponent of the CPE impedance (0 = resistor, 0.5 = warburg, 1.0 = capacitance).

In this tutorial we will perform batch inversion of all SIP data files provided with BISIP with the Shin (2015) and double Cole-Cole models, and we will compare their respective relaxation time (\(\tau\)) parameters.

Exploring the parameter space

First import the required packages.

import numpy as np
from bisip import PeltonColeCole
from bisip import Shin2015
from bisip import DataFiles

np.random.seed(42)

# Load the data file paths
data_files = DataFiles()

results = {'Shin': {},
           'Pelton': {},
          }

nsteps = 1000
for fname, fpath in data_files.items():

        if fname == 'SIP-K389175':
            model = Shin2015(fpath, nsteps=nsteps)
            model.fit()
            results['Shin'][fname] = model

#             model = PeltonColeCole(fpath, nsteps=nsteps, n_modes=2)
#             model.fit()
#             results['Pelton'][fname] = model

100%|██████████| 1000/1000 [00:01<00:00, 724.53it/s]

fig = results['Shin']['SIP-K389175'].plot_traces()

fig = results['Shin']['SIP-K389175'].plot_fit(discard=500)

import matplotlib.pyplot as plt
freq = np.logspace(-2, 100, 10000)
w = 2*np.pi*freq
theta = np.array([8.16E02, 3.12E03, np.log(1.80E-14), np.log(2.25E-06), 0.3098, 0.4584])

Z = model.forward(theta, w)
plt.plot(*Z)

[<matplotlib.lines.Line2D at 0x12336b978>]

Conclusions