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1 ahallam 3232
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15 ahallam 3232
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 jfenwick 3308 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s})
47     \delta(y-y_{s})
48     \delta(z-z_{s})
49 ahallam 3232 \end{equation}
50    
51     \begin{equation}
52     -\nabla \cdot \left[ \sigma(x,y,z) \nabla \phi (x,y,z) \right] =
53 jfenwick 3308 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s})
54     \delta(y-y_{s})
55     \delta(z-z_{s})
56 ahallam 3232 \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 caltinay 4286 The model is calculated over many frequencies and transformed to the time
77 ahallam 3232 domain using a discrete fourier transform.
78    

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