Geoelectricity

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.

221

r

QQKF

Charles Augustin de Coulomb

(1736 - 1806)

Later, Ampere figured out what the units should be based on the flow of charge though parallel wires. We define a material property o called the permittivity constant:

F 1

4o

Q1Q2

r2

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 )

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:

22

1 4 r

Q

Q

FE

o

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:

E U

r

And we define U as the work per unit charge required to bring an object from infinity to r:

U Edr

r

Q

4or2 dr

r

Q

4or

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:

V IR

Georg Simon Ohm

1787-1854

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 as:

The unit of are Ohm-meters or -m:

We then write the 3D equivalent of Ohm’s law as

JEA

I

L

V

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 as 1/.

Units:R OhmsI Amperes or AmpsV Volts Ohm-meters mhos/meter or siemens/meterE Volts/meter

Electrical Conduction

1. Electronic or Ohmic: free electrons.

A property of metals. Very efficient. Ranges over ~24 orders of magnitude

Conductors < 1 Ohm-meterResistors/Insulators > 1 Ohm-meterSemi-conductors ~ 1 Ohm-meter; electrons only partially bound

Good conductors: metals, graphiteOk conductors: sulphides, arsenidesSemiconductors: most oxidesInsulators: carbonates, phosphates, nitrates (most rocks)

2. 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.

a

mSn w

where is the porosity, S is the fraction of pores filled with water, w 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.

w examples:

Meteoric Rain 30-1000 mFresh Water (Seds) 1-100 mSea Water (Ocean) 0.2 m

We use the empirical Archie’s Law for a porous medium:

Material Resistivity (Ohm-meter)

Air Infinite

Pyrite 3 x 10-1

Galena 2 x 10-3

Quartz 4 x 1010 - 2 x 1014

Calcite 1 x 1012 - 1 x 1013

Rock Salt 30 - 1 x 1013

Mica 9 x 1012 - 1 x 1014

Granite 100 - 1 x 106

Gabbro 1 x 103 - 1 x 106

Basalt 10 - 1 x 107

Limestones 50 - 1 x 107

Sandstones 1 - 1 x 108

Shales 20 - 2 x 103

Dolomite 100 - 10,000

Sand 1 - 1,000

Clay 1 - 100

Ground Water 0.5 - 300

Sea Water 0.2

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 o to as:

o

where is the dielectric constant. Note that generally is a function of frequency; (f) ~ 1/f. Here are some typical values of :

Water 80Sandstone 5-12Soil 4-30Basalt 12Gneiss 8.5

Note that in EM we define a Displacement field D as

D = E

Maxwell’s Equations

t

BE

t

DJH

Where J = E, B = H, and D = E (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:

2

22

t

E

t

EE

2

22

t

H

t

HH

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.

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.

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 separatedNernst -> 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.

Self Potential Across a Fault

Mise a la Mase

Monitoring Fluid Flow

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.

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.

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.

We measure a voltage difference, and figure that current density results in from a constriction or redirection of current.

J E

V

L

Note you can measure in perpendicular directions to get the areal direction of current and identify a resisitive body.

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):

2

22

t

H

t

HH

Let’s assume a monochromatic field:

H(x,t) = H(x)eit

Note this is like the separation of variables trick we did for heat conduction

HHiH 22

The first term is called the conduction term, the second the displacement term

The relative sizes of these terms (conductive to displacement) is 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

HiH 2

Which is the heat conductivity diffusion equation we solved before. We take the exact same steps and find

aztiazo eeHH

where 2/1

2

a

Note that for normal values of in the Earth, the attenuation term becomes

f

zze az 32/1

102exp2

exp

z is in meters, is in m, and f is in Hz. The skin depth zs is when the field is Ho/e:

fz s

500

Examples

10-4 m 102 m

f

10-2 50 m 5 x 104 m

103 0.16 m 160 m

Now, as an H field penetrates the surface it will attenuate. Maxwell says that:

t

DJH

Again, the dielectric term will be much smaller than the conductive term, so

EJH

Assuming a simple H that is oriented in the y direction (Hy component), we evaluate:

xxyyy

y

EJz

Hi

x

Hk

z

Hi

Hzyx

kji

H

00

From before

)1( iaztio

aztiazo

yx e

zeHeeH

zz

HJ

4/2/14/2 iix HeaHeJ

Thus, the current (telluric) has a /4 phase shift relative to the initial H field.

If is small, then H penetrates to great depth, and little J is produced.

If is large,then H does not go to great depth, and big J is produced.

HiaeiaeH iaztio )1()1( )1(

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

4/2/1

4/2/1

i

yi

yx eHeHE

so

E x

Hy

1/ 2

E y

Hx

We can make a “pseudo-section” of resistivity as follows:

Ex 1

Hy

z

1

eff

Hy

zeff

so

zeff 1

eff

Hy

E x

1

eff

eff

1/ 2

1

eff

1/ 2

1

E x

Hy

T

2Ex

Hy

where T is the period and = 2/T.

Thus

1

eff

Hy

E x

eff

Hy

E x

T

2E x

Hy

or

eff T

2E x

Hy

2

Assuming a typical value for of 1.3 x 10-6 henrys/meter we can write:

eff 0.2TEx

Hy

2

zeff 1

25T

where the units are

E mV/kmH gammasT seconds mz km

So the idea is to determine as a function of frequency (for different E/H ratios) and then calculate the corresponding depth z.

MT Recording Geometry

MT Resistivity in subducting plate

MT Cross section across a fault

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):

2U 2U

dr2 2

r

U

dr0

Note that

dr

r2 U

dr

r2 2U

dr2 2r

U

drr2 2U

dr2

2

r

U

dr

so Laplace’s equation is equvalent in this case to

dr

r2 U

dr

0

or

U

dr

A

r2

U A

r

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

I 4r2J 4r2 U

r 4r2 A

r2 4A

Thus

A I

4and

U I

4r

4rU

I

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

U I

2r

2rU

I

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

UAC I

2rAC

UBC I

2rBC

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

UC UAC UBC I

2rAC

I

2rBC

I2

1

rAC

1

rBC

Similarly, the potential at another point D would be

UD I2

1

rAD

1

rDB

And so the potential difference between points C and D is VCD given by

VCD UC UD I2

1

rAC

1

rCB

1

rAD

1

rDB

2VCD

I

1

1

rAC

1

rCB

1

rAD

1

rDB

or

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

2VCD

I

1

1

rAC

1

rCB

1

rAD

1

rDB

Wenner: equal spacing between current and potential electrodes:

I

aV

aaI

V

aaaaI

V CDCDCD 2

21

21

121

21

211

12

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

rAC L x l

rBC L x l

rAD L x l

rBD L x l

2VCD

I

11

L x l 1

L x l

1

L x l 1

L x l

If L-x >> l and L+x >> l, then

1

L x l L x 1

1l

L x

1

L x 11

l

L x

1

L x

l

L x 2

and similarly for the other terms (substitute –l for l and –x for x). Plug all this in and eventually you get

VCD

2Il

L2 x 2 2

L2 x 2

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: lnlnlrAC 22)1(2

rBC 2l(n 1)

rAD 2l(n 1) 2l 2l 2l(n 1)

lnlnlrBD 22)1(2

2VCD

I

11

2ln

1

2l(n 1)

1

2l(n 1)

1

2ln

2VCDln n2 1

I

Making a resistivity pseudo section:

Measure 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).

Resistivity Imaging around a Tunnel

Resistivity Imaging in Limestone (Karst Lithology)

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):

Ex U

x

x

I2

1

r1

1

r2

we set

r1 x 2 y 2 z2

r2 L x 2 y 2 z2

2/32222/1222

1 2

21

zyx

xzyx

xrx

2/32222/1222

2 2

21

zyxL

xLzyxL

xrx

3

1

2/3222

r

xzyxx

3

1r

xL

Hence

Jx E x I

2x

r13

L x

r23

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

22221 2/ zyLrr

2/32222/3222 2/

2/

2/

2/

2 zyL

L

zyL

LIJ x

2/32222/

1

2 zyL

IL

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

2/3222

0 2/2 zyL

dydz

L

I

I zx

L

z

zL

dzL

I

I zx 2

tan2

2/1

022

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.

How about layers of resisitivity? It gets complicated fast.

Let’s first consider two halfspaces separated by in interface. The upper halfspace has a resistivity 1 and the lower halfspace has resistivity 2. We have a current electrode in the upper halfspace.

What is the potential at a point P in the upper halfspace and P’ in the lower halfspace?

We define a “reflection coefficient” k and a transmission coefficient 1-k. If the point P is a distance r1 from C, then

VP I1

41

r1

k

r2

where r2 is the “ray” distance to the interface and back to P from C, following the usual reflection law (equal angles of incidence and reflection). Note that r2 can be constructed by reflecting the normal from C across the interface and drawing a straight line to P. Similarly, the potential at P’ is

VP ' I2

41 k

r3

where r3 is the distance from C to P’.

If we move the points P and P’ to the interface, (P=P’) then r1 = r2 = r3 and

I1

41

r1

k

r1

I2

41 k

r1

from which

k 2 1

2 1

Note that –1 < k < 1.

How about a layer over a half space?

As in the case of the two halfspaces, we account for the bottom interface by summing potentials from the original electode (C1) and it’s mirror across the interface (C2). BUT now we have to mirror C2 across the other interface (surface) to produce C3, and mirror C3 across the lower interface to get C4 and so on ad infinitum. Hence:

VC1 I1

41

r

VC 2 I1

4k

r1

VC 3 I1

4k ka

r1

VC 4 I1

4k ka k

r2

VC 5 I1

4k ka k ka

r2

and so on, where the reflection at the surface depends on the resistivity of the air (a) or

ka a 1

a 1

1

Because the resitivity of the air is very large.

VCi

i1

I1

21

r

2k

r1

2k 2

r2

2/1222/1221 2;2 mhrrhrr m

hence

V I1

21

r 2

km

r2 2mh 2m1

The origin of induced electrical polarization is complex and is not well understood.  This is primarily because several physio-chemical phenomena and conditions are likely responsible for its occurrence. 

Induced Polarization

Induced Polarization (IP) is the transient storage of voltage in the ground. We can produce it by turning the voltage in a resistivity array on and off.

When a metal electrode is immersed in a solution of ions of a certain concentration and valence, a potential difference is established between the metal and the solution sides of the interface. 

This difference in potential is an explicit function of the ion concentration, valence, etc. 

When an external voltage is applied across the interface, a current is caused to flow, and the potential drop across the interface changes from its initial value. 

The change in interface voltage is called the "overvoltage" or "polarization" potential of the electrode.  Overvoltages are due to an accumulation of ions on the electrolyte side of the interface waiting to be discharged.  The time constant of buildup and decay is typically several tenths of a second.

Overvoltage is therefore established whenever current is caused to flow across an interface between ionic and electronic conduction.  In normal rocks, the current that flows under the action of an applied EMF does so by ionic conduction in the electrolyte in the pores of the rock.  There are, however, certain minerals that have a measure of electronic conduction (almost all the metallic sulfides - except sphalerite - such as pyrite, graphite, some coals, magnetite, pyrolusite, native metals, some arsenides, and other minerals with a metallic lustre). 

The most important sources of nonmetallic IP in rocks are certain types of clay minerals (Vacquier 1957, Seigel 1970).  These effects are believed to be related to electrodialysis of the clay particles.  This is only one type of phenomenon that can cause "ion-sorting" or "membrane effects."  For example, the figure below shows a cation-selective membrane zone in which the mobility of the cation is increased relative to that of the anion, causing ionic concentration gradients and therefore polarization. 

In time-domain IP, several indices have been used to define the polarizability of the medium.  Seigel (1959) defined "chargeability" (in seconds) as the ratio of the area under the decay curve (in millivolt-seconds, mV‑s) to the potential difference (in mV) measured before switching the current off.  Komarov, et al., (1966) defined "polarizability" as the ratio of the potential difference after a given time from switching the current off to the potential difference before switching the current off.  Polarizability is expressed as a percentage.

M 1

Vo

V (t)dtt1

t2

Chargeabilty M

Frequency Effect

FE f F

F

MF A F f A F 1 f

1 A f F

f F

Metal Factor

A = 2 x 105

IP Example - Mapping soil and groundwater contamination. 

Cahyna, Mazac, and Vendhodova (1990) used IP to determine the slag-type material containing cyanide complexes that have contaminated groundwater in Czechoslovakia.  The figure shows contours of ηa (percent) obtained from a 10-m grid of profiles.  The largest IP anomaly (ηa = 2.44%) directly adjoined the area of the outcrop of the contaminant (labeled A).  The hatched region exhibits polarizability over 1.5% and probably represents the maximum concentration of the contaminant.  The region exhibiting polarizability of less than 0.75% was interpreted as ground free of any slag-type contaminant.

Ground Penetrating Radar (GPR)

2E 2E

t 2

This is just like seismic waves, only in this case the reflection coefficient and wavespeed depend on dielectric constant.

2

22

t

E

t

EE

Ground-penetrating radar (GPR) uses a high-frequency (80 to 1,500 MHz) EM pulse transmitted from a radar antenna to probe the earth.  The transmitted radar pulses are reflected from various interfaces within the ground, and this return is detected by the radar receiver. Remember from Maxwell:

At high frequencies, the second (wave) term dominates, so

Reflecting interfaces may be soil horizons, the groundwater surface, soil/rock interfaces, man-made objects, or any other interface possessing a contrast in dielectric properties.  The dielectric properties of materials correlate with many of the mechanical and geologic parameters of materials.

How it works:

The radar signal is imparted to the ground by an antenna that is in close proximity to the ground.  The reflected signals can be detected by the transmitting antenna or by a second, separate receiving antenna. 

As the antenna (or antenna pair) is moved along the surface, the graphic recorder displays results in a cross-section record or radar image of the earth. 

Spatial considerations:

As GPR has short wavelengths in most earth materials, resolution of interfaces and discrete objects is very good.  However, the attenuation of the signals in earth materials is high, and depths of penetration seldom exceed 10 m.  Water and clay soils increase the attenuation, decreasing penetration.

The objective of GPR surveys is to map near-surface interfaces.  For many surveys, the location of objects such as tanks or pipes in the subsurface is the objective.  Dielectric properties of materials are not measured directly.  The method is most useful for detecting changes in the geometry of subsurface interfaces.

Signal GPR waveform

Unlike seismic waves, most of the GPR signal is

perpendicular to the antenna (straight down)

Typical Example of a GPR field trace.

GPR on a small scale.

Using GPR to locate graves, trenches, and sinkholes

Electromagnetic InductionThe way this works is you create your own magnetic field by passing current through a wire, usually a current loop. The magnetic field penetrates the ground, produces a current that induces a secondary field in an object in the ground. The receiving loop detects a combination of the original signal field and that produced by an object in the ground.

Electromagnetic Induction: Quantitative Bits

2B B

t

Recall from the MT discussion

aztiazo eeHH

where2/1

2

a

We start by generating our own magnetic field B. Use the diffusion part of Maxwell’s equation:

Some definitions:IC current in conductor

IR current in receiverIT current in transmitter

R resistance of conductorL inductance of conductor

VC voltage of conductor

Inductance produces a voltage proportional to the rate of change of current. Thus:

)( LiRIdt

dILRIV C

CCC

The current in one entity produces a voltage in another through Mutual inductance. Let

MTC Mutual inductance between transmitter and conductorMTR Mutual inductance between transmitter and receiverMCR Mutual inductance between conductor and receiver

TTCT

TCC IMidt

dIMV

Primary voltage in the receiver (due to the transmitter)

TTRT

TRP IMidt

dIMV

Secondary voltage in the receiver (due to the conductor)

CCRC

CRS IMidt

dIMV

From above

)( LiRIIMidt

dIMV CTTC

TTCC

So

)()()( 222 LiR

LR

Mi

LiR

Mi

I

I TCTC

T

C

The secondary and primary voltages at the receiver are therefore related as

)(

)(222 LR

LiRMi

M

M

IM

IM

IMi

IMi

V

V TC

TR

CR

TTR

CCR

TTR

CCR

P

S

2

2

2

222

2

222

11

i

LM

MM

R

LR

R

L

R

Li

L

R

M

MM

V

V

TR

CRTC

TR

CRTC

P

S

Where

R

L

)( 222

22

LR

LRi

M

MM

TR

CRTC

is called the response parameter.

The real part of the quotient is called the in phase component, the imaginary part is out of phase and is called the quadrature component.

i

TR

CRTC

TR

CRTC

P

S AeiLM

MMi

LM

MM

V

V

22

2

11

L

R

11 tan

1tan

By observing both the amplitude and phase of the recorded EM field, we can estimate R and L.

A variety of EM examples