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3 % Copyright (c) 2003-2018 by The University of Queensland
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7 % Licensed under the Apache License, version 2.0
8 % http://www.apache.org/licenses/LICENSE-2.0
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10 % Development until 2012 by Earth Systems Science Computational Center (ESSCC)
11 % Development 2012-2013 by School of Earth Sciences
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15
16 In this chapter we will investigate the effects of a current flow and
17 resistivity in a medium. This type of problem is related to the DC resistivity method of
18 geophysical prospecting. Currents are injected into the ground at the surface
19 and measurements of the potential are taken at various potential-dipole
20 locations along or adjacent to the survey line. From these measurements of the
21 potential it is possible to infer an approximate apparent resistivity model of
22 the subsurface.
23
24 The following theory comes from a tutorial by \citet{Loke2004}.
25 We know from Ohm's law that the current flow in the ground is given in vector
26 form by
27 \begin{equation}
28 \vec{J}=\sigma\vec{E}
29 \end{equation}
30 where $\vec{J}$ is current density, $\vec{E}$ is the electric field intensity
31 and $\sigma$ is the conductivity. We can relate the potential to the electric
32 field intensity by
33 \begin{equation}
34 E=-\nabla\Phi
35 \end{equation}
36 where $\Phi$ is the potential. We now note that the current density is related
37 to the potential via
38 \begin{equation}
39 \vec{J}=-\sigma\nabla\Phi
40 \end{equation}
41 Geophysical surveys predominantly use current sources which individually act as
42 point poles. Considering our model will contain volumes, we can normalise the
43 input current and approximate the current density in a volume $\Delta V$ by
44 \begin{equation}
45 \nabla \vec{J} =
46 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s})
47 \delta(y-y_{s})
48 \delta(z-z_{s})
49 \end{equation}
50
51 \begin{equation}
52 -\nabla \cdot \left[ \sigma(x,y,z) \nabla \phi (x,y,z) \right] =
53 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s})
54 \delta(y-y_{s})
55 \delta(z-z_{s})
56 \end{equation}
57
58 This form is quite simple to solve in \esc.
59
60 \section{3D Current-Dipole Potential}
61 \sslist{example11m.py; example11c.py}
62
63 \begin{figure}[ht]
64 \centering
65 \includegraphics[width=0.95\textwidth]{figures/ex11cstreamline.png}
66 \caption{Current Density Model for layered medium.}
67 \label{fig:ex11cstream}
68 \end{figure}
69
70 \section{Frequency Dependent Resistivity - Induced Polarisation}
71 With a more complicated resistivity model it is possible to calculate the
72 chargeability or IP effect in the model. A recent development has been the
73 Fractal model for complex resistivity \citep{Farias2010,Honig2007}.
74
75
76 The model is calculated over many frequencies and transformed to the time
77 domain using a discrete fourier transform.
78

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