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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 jfenwick 3308 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s})
46     \delta(y-y_{s})
47     \delta(z-z_{s})
48 ahallam 3232 \end{equation}
49    
50     \begin{equation}
51     -\nabla \cdot \left[ \sigma(x,y,z) \nabla \phi (x,y,z) \right] =
52 jfenwick 3308 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s})
53     \delta(y-y_{s})
54     \delta(z-z_{s})
55 ahallam 3232 \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 jfenwick 4345 The model is calculated over many frequencies and transformed to the time
76 ahallam 3232 domain using a discrete fourier transform.
77    

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