 # Contents of /trunk/doc/cookbook/example11.tex

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 1 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % Copyright (c) 2003-2018 by The University of Queensland 4 5 % 6 % Primary Business: Queensland, Australia 7 % Licensed under the Apache License, version 2.0 8 9 % 10 % Development until 2012 by Earth Systems Science Computational Center (ESSCC) 11 % Development 2012-2013 by School of Earth Sciences 12 % Development from 2014 by Centre for Geoscience Computing (GeoComp) 13 % 14 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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