Contents of /branches/doubleplusgood/doc/cookbook/example11.tex

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 1 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % Copyright (c) 2003-2013 by University of Queensland 4 5 % 6 % Primary Business: Queensland, Australia 7 % Licensed under the Open Software License version 3.0 8 9 % 10 % Development until 2012 by Earth Systems Science Computational Center (ESSCC) 11 % Development since 2012 by School of Earth Sciences 12 % 13 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 14 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 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s}) 46 \delta(y-y_{s}) 47 \delta(z-z_{s}) 48 \end{equation} 49 50 \begin{equation} 51 -\nabla \cdot \left[ \sigma(x,y,z) \nabla \phi (x,y,z) \right] = 52 \left(\frac{I}{\Delta V} \right) \delta(x-x_{s}) 53 \delta(y-y_{s}) 54 \delta(z-z_{s}) 55 \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 The model is calculated over many frequencies and transformed to the time 76 domain using a discrete fourier transform. 77