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1 ahallam 3232
2     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3     %
4     % Copyright (c) 2003-2010 by University of Queensland
5     % Earth Systems Science Computational Center (ESSCC)
6     % http://www.uq.edu.au/esscc
7     %
8     % Primary Business: Queensland, Australia
9     % Licensed under the Open Software License version 3.0
10     % http://www.opensource.org/licenses/osl-3.0.php
11     %
12     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13    
14     In this chapter we will investigate the effects of a current flow and
15     resistivity in a medium. This type of problem is related to the DC resistivity method of
16     geophysical prospecting. Currents are injected into the ground at the surface
17     and measurements of the potential are taken at various potential-dipole
18     locations along or adjacent to the survey line. From these measurements of the
19     potential it is possible to infer an approximate apparent resistivity model of
20     the subsurface.
21    
22     The following theory comes from a tutorial by \citet{Loke2004}.
23     We know from Ohm's law that the current flow in the ground is given in vector
24     form by
25     \begin{equation}
26     \vec{J}=\sigma\vec{E}
27     \end{equation}
28     where $\vec{J}$ is current density, $\vec{E}$ is the electric field intensity
29     and $\sigma$ is the conductivity. We can relate the potential to the electric
30     field intensity by
31     \begin{equation}
32     E=-\nabla\Phi
33     \end{equation}
34     where $\Phi$ is the potential. We now note that the current density is related
35     to the potential via
36     \begin{equation}
37     \vec{J}=-\sigma\nabla\Phi
38     \end{equation}
39     Geophysical surveys predominantly use current sources which individually act as
40     point poles. Considering our model will contain volumes, we can normalise the
41     input current and approximate the current density in a volume $\Delta V$ by
42     \begin{equation}
43     \nabla \vec{J} =
44 jfenwick 3308 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s})
45     \delta(y-y_{s})
46     \delta(z-z_{s})
47 ahallam 3232 \end{equation}
48    
49     \begin{equation}
50     -\nabla \cdot \left[ \sigma(x,y,z) \nabla \phi (x,y,z) \right] =
51 jfenwick 3308 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s})
52     \delta(y-y_{s})
53     \delta(z-z_{s})
54 ahallam 3232 \end{equation}
55    
56     This form is quite simple to solve in \esc.
57    
58     \section{3D Current-Dipole Potential}
59     \sslist{example11m.py; example11c.py}
60    
61     \begin{figure}[ht]
62     \centering
63     \includegraphics[width=0.95\textwidth]{figures/ex11cstreamline.png}
64     \caption{Current Density Model for layered medium.}
65     \label{fig:ex11cstream}
66     \end{figure}
67    
68     \section{Frequency Dependent Resistivity - Induced Polarisation}
69     With a more complicated resistivity model it is possible to calculate the
70     chargeability or IP effect in the model. A recent development has been the
71     Fractal model for complex resistivity \citep{Farias2010,Honig2007}.
72    
73    
74     The model is calculated over many frequencyies and transformed to the time
75     domain using a discrete fourier transform.
76    

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