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2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 % Copyright (c) 2003-2012 by University of Queensland
4 % http://www.uq.edu.au
5 %
6 % Primary Business: Queensland, Australia
7 % Licensed under the Open Software License version 3.0
8 % http://www.opensource.org/licenses/osl-3.0.php
9 %
10 % Development until 2012 by Earth Systems Science Computational Center (ESSCC)
11 % Development since 2012 by School of Earth Sciences
12 %
13 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
14
15 In this chapter we will investigate the effects of a current flow and
16 resistivity in a medium. This type of problem is related to the DC resistivity method of
17 geophysical prospecting. Currents are injected into the ground at the surface
18 and measurements of the potential are taken at various potential-dipole
19 locations along or adjacent to the survey line. From these measurements of the
20 potential it is possible to infer an approximate apparent resistivity model of
21 the subsurface.
22
23 The following theory comes from a tutorial by \citet{Loke2004}.
24 We know from Ohm's law that the current flow in the ground is given in vector
25 form by
26 \begin{equation}
27 \vec{J}=\sigma\vec{E}
28 \end{equation}
29 where $\vec{J}$ is current density, $\vec{E}$ is the electric field intensity
30 and $\sigma$ is the conductivity. We can relate the potential to the electric
31 field intensity by
32 \begin{equation}
33 E=-\nabla\Phi
34 \end{equation}
35 where $\Phi$ is the potential. We now note that the current density is related
36 to the potential via
37 \begin{equation}
38 \vec{J}=-\sigma\nabla\Phi
39 \end{equation}
40 Geophysical surveys predominantly use current sources which individually act as
41 point poles. Considering our model will contain volumes, we can normalise the
42 input current and approximate the current density in a volume $\Delta V$ by
43 \begin{equation}
44 \nabla \vec{J} =
45 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s})
46 \delta(y-y_{s})
47 \delta(z-z_{s})
48 \end{equation}
49
50 \begin{equation}
51 -\nabla \cdot \left[ \sigma(x,y,z) \nabla \phi (x,y,z) \right] =
52 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s})
53 \delta(y-y_{s})
54 \delta(z-z_{s})
55 \end{equation}
56
57 This form is quite simple to solve in \esc.
58
59 \section{3D Current-Dipole Potential}
60 \sslist{example11m.py; example11c.py}
61
62 \begin{figure}[ht]
63 \centering
64 \includegraphics[width=0.95\textwidth]{figures/ex11cstreamline.png}
65 \caption{Current Density Model for layered medium.}
66 \label{fig:ex11cstream}
67 \end{figure}
68
69 \section{Frequency Dependent Resistivity - Induced Polarisation}
70 With a more complicated resistivity model it is possible to calculate the
71 chargeability or IP effect in the model. A recent development has been the
72 Fractal model for complex resistivity \citep{Farias2010,Honig2007}.
73
74
75 The model is calculated over many frequencyies and transformed to the time
76 domain using a discrete fourier transform.
77

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