Geoelectricity

1. Geoelectricity

2. Introduction: Electrical Principles Let Q1, Q2 be electrical charges separated by a distance r. There is a force between the two charges that goes like This is called Coulomb’s law, after Charles Augustin de Coulomb who first figured this out. Charles Augustin de Coulomb (1736 - 1806)

3. Later, Ampere figured out what the units should be based on the flow of charge though parallel wires. We define a material property eo called the permittivity constant: which is approximately equal to 8.85419 x 10-12 C2N-1m-2 (C = Coulomb which is a unit of charge. One Coulomb is defined as the amount of charge that passes through a wire of 1 Ampere current flowing for 1 second). ANDRÉ-MARIE AMPÈRE ( 1775 - 1836 )

4. Note similarity to force of gravity. There are many analogues. We can define the electric field (similar to gravity acceleration field) as a force per unit charge: units of E in this form are N Q-1. We think of a field as lines along which a charge Q1 would move if were attracted by the charge Q2. Also analogous to gravity, we define an electrical potential U and relate it to the field by a negative gradient:

5. And we define U as the work per unit charge required to bring an object from infinity to r: Instead of absolute potentials we normally talk about potential differences which we call volts (V; after the Italian physicist Alessandro Volta). There is a famous relation between the voltage, current, and resistance in a wire called Ohm’s Law: Georg Simon Ohm 1787-1854

6. However, resistance is not really an intensive material property (like, say, density) and so is not appropriate for application to rocks. We define instead the resistivity r as: The unit of r are Ohm-meters or W-m:

7. We then write the 3D equivalent of Ohm’s law as where we recognize E as the potential gradient (V/L) and J = I/A is called the current density. Note that we also define the conductivity s as 1/r. Units: R Ohms I Amperes or Amps V Volts r Ohm-meters s mhos/meter or siemens/meter E Volts/meter

8. Electrical Conduction • Electronic or Ohmic: free electrons. • A property of metals. Very efficient. • Ranges over ~24 orders of magnitude • Conductors r < 1 Ohm-meter • Resistors/Insulators r > 1 Ohm-meter • Semi-conductors r ~ 1 Ohm-meter; electrons only partially bound • Good conductors: metals, graphite • Ok conductors: sulphides, arsenides • Semiconductors: most oxides • Insulators: carbonates, phosphates, nitrates (most rocks)

9. Ionic or Electrolytic: • Dissolved Ions in a fluid (water). • Very efficient but more space problems with bigger elements moving around. Thus it is not as efficient as electronic • Water is very important in this process, which makes electrical methods very good for addressing water related problems.

10. We use the empirical Archie’s Law for a porous medium: where f is the porosity, S is the fraction of pores filled with water, rw is the resistivity of the water, and m, n, and a are material constants. Generally 0.5 < a < 2.5, 1.3 < m < 2.5, and n ~ 2. Often we just assume “2” for all of them. rw examples: Meteoric Rain 30-1000 Wm Fresh Water (Seds) 1-100 Wm Sea Water (Ocean) 0.2 Wm

12. 3. Dielectric: Caused by the relative displacement of protons and electrons within their orbital shells. Of no importance at low f (to DC) but is very important at high frequency AC. The net effect is to change the permittivity eo to e as: where k is the dielectric constant. Note that k generally is a function of frequency; k (f) ~ 1/f. Here are some typical values of k: Water 80 Sandstone 5-12 Soil 4-30 Basalt 12 Gneiss 8.5 Note that in EM we define a Displacement field D as D = eE

13. Maxwell’s Equations Where J = sE, B = mH, and D = eE (and all are vectors). So in general the electric and magnetic fields are coupled. However, in the case of an isotropic, homogeneous medium they separate as: Note these are the same equation with different variables, and that they are a combination of the diffusion and wave equations. We’ll solve these in a bit when we talk about MT.

14. Electrical Methods There is an alphabet soup of electrical methods (SP, IP, MT, EM, Resistivity, GPR) which we will discuss in turn. Most are sensitive to resistivity/conductivity in some way, except for GPR (dielectric constant). As we saw before, natural materials vary in resistivity by several orders of magnitude.

15. Self Potential (SP) Measure natural potential differences in the earth Sources: Electrokinetic or streaming potentials: moving ions. Electrochemical (Nernst and diffusion) diffusion: ions with different mobilities get separated Nernst -> same electrodes, different concentrations Mineralization -> different electrodes (materials) Ore bodies always give negative potentials. Measurement with porous pots. Signals range from few mv to 1 V. 200 mV is a strong signal.

16. Self Potential Across a Fault

17. Mise a la Mase Monitoring Fluid Flow

18. The Earth’s electric field. The ground generally has negative charge, so the Earth’s E field points down into the earth. The atmosphere is generally positive, with ions produced by cosmic rays. These bombard the Earth, which neutralize the surface. However, the negative charge is replenished by lightning storms.

19. Tellurics Natural electric currents in the earth. These are cause by decaying magnetic fields in the earth. They are like large swirls that follow the sun.

20. Electromagnetic fields arise from time-varying currents in the ionosphere and tropical storms (lightning strikes). Fields propagate as plane-waves vertically into the Earth, inducing secondary currents.

21. We measure a voltage difference, and figure that current density results in from a constriction or redirection of current. Note you can measure in perpendicular directions to get the areal direction of current and identify a resisitive body.

22. Magnetotellurics Simultaneous measurement of the magnetic and electric fields in the Earth. Let’s solve Maxwell’s equation for the H field (it will be the same for the E field): Let’s assume a monochromatic field: H(x,t) = H(x)eiwt Note this is like the separation of variables trick we did for heat conduction The first term is called the conduction term, the second the displacement term

23. The relative sizes of these terms (conductive to displacement) is s/ew. So, if conductivity is large and/or frequencies are small, then the first term dominates. If conductivity is small and/or frequencies are large, the displacement term is large. For rocks and natural field frequencies, the conductive term is about 8 orders of magnitude greater than the displacement term, so for this kind of observation we have Which is the heat conductivity diffusion equation we solved before. We take the exact same steps and find where

24. Note that for normal values of m in the Earth, the attenuation term becomes z is in meters, r is in Wm, and f is in Hz. The skin depth zs is when the field is Ho/e: Examples

25. Now, as an H field penetrates the surface it will attenuate. Maxwell says that: Again, the dielectric term will be much smaller than the conductive term, so Assuming a simple H that is oriented in the y direction (Hy component), we evaluate:

26. From before Thus, the current (telluric) has a p/4 phase shift relative to the initial H field. If mws is small, then H penetrates to great depth, and little J is produced. If mws is large,then H does not go to great depth, and big J is produced.

27. The idea behind MT is to measure H and E simultaneously, and take the ratio of E in one direction to H in the perpendicular direction. From above so We can make a “pseudo-section” of resistivity as follows: so where T is the period and w = 2p/T.

28. Thus or Assuming a typical value for m of 1.3 x 10-6 henrys/meter we can write:

29. where the units are E mV/km H gammas T seconds rWm z km So the idea is to determine r as a function of frequency (for different E/H ratios) and then calculate the corresponding depth z.

30. MT Recording Geometry MT Resistivity in subducting plate

31. MT Cross section across a fault

32. Resistivity An active technique. Pump current into the ground and measure spatial variation in voltage to get a resistivity map. Let’s consider what happens if we put an electrode into the ground, and start with an infinite space. We can think of it as a charge Q with associated electric fields and potentials. Everywhere around Q, as long as there are no sources or sinks (i.e. no other charges in the volume) then the potential U satisfies Laplace’s equation (in spherical coordinates):

33. Note that so Laplace’s equation is equvalent in this case to or

34. Note that integrative constants are zero because U and gradU -> 0 as r -> infinity. The current at any radius r is related to the current density by Thus and

35. If the electrode is at the surface of a half space instead of within an infinite space, then we repeat the above but use a hemisphere instead of a sphere and find

36. Now suppose we have two electrodes at the surface at points A and B, and we want to determine the potential at an arbitrary point C. If the distances to C are rAC and rBC, then we reverse the sign on UBC because current flows of electrode one (positive Q) and into electrode two (negative Q). The total potential at point C is then

37. Similarly, the potential at another point D would be And so the potential difference between points C and D is VCD given by or

38. This is the fundamental resistivity equation. It is independent of any particular geometry, but there are some configurations which are more or less standard.

39. Wenner: equal spacing between current and potential electrodes:

40. Schlumberger: Current electrodes are a distance 2L apart, potential electrodes are a distance 2l apart, the center of the potential electrodes is a distance x from the center of the current electrodes, and L >> l and L-x >> l (we are far from the ends). In this case

41. If L-x >> l and L+x >> l, then and similarly for the other terms (substitute –l for l and –x for x). Plug all this in and eventually you get

42. Dipole-Dipole In this case we imagine that both the current and potential electrodes are separated by a distance 2l and the distance between the inner current and potential electrodes is a multiple of this distance = 2l(n-1) where n >= 1 (when n = 1, they are together). In this case:

43. Making a resistivity pseudo section: Measure r at a given separation, mark a spot half way in between (d=l(n-1)) and plot this a a depth the same distance below the surface (i.e., depth = l(n-1).

44. Resistivity Imaging around a Tunnel

45. Resistivity Imaging in Limestone (Karst Lithology)

47. Current Distribution Where is the current going, anyway? We can get an idea by examining the case of a homogeneous halfspace. Consider current electrodes a distance L apart. At a point P a distance r1 from the positive electrode and r2 from the negative electrode (and at a depth z):

48. we set

49. Hence For illustration, let’s see what happens at the midpoint between the electrodes. x = L/2, L-x = L/2, so

50. The current that flows across and element dydz is dIx = Jxdydz. Thus, the fraction of the total current I that flows between the surface and depth z is This shows that half the current crosses above a depth z = L/2, and almost 90% above z = 3L. This gives you some idea on how current distribution depends on separation of electrodes.