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1
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 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s})
45 \delta(y-y_{s})
46 \delta(z-z_{s})
47 \end{equation}
48
49 \begin{equation}
50 -\nabla \cdot \left[ \sigma(x,y,z) \nabla \phi (x,y,z) \right] =
51 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s})
52 \delta(y-y_{s})
53 \delta(z-z_{s})
54 \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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