Resistivity Methods:Earth Properties and Basic TheoryReading

Today : 207-221Next Lecture : 218-228

Introduction• Link resistivity (ability of

the earth to conduct anelectric current) tosubsurface structure.

• Useful because resistivity ofearth materials varies byaround 10 orders ofmagnitude.

• Developed by ConradSchlumberger (France) andFrank Wenner (UnitedStates) in early 20thcentury.

•Uses

•Archeology

•Environmental

•Mineral exploration

Electricity BasicsVoltage V - Electrical potential energy per unit charge [volts]Current i - amount of charge per unit time [amperes]

†

i =1R

V

Resistivity R is just a proportionality constant [ohms]R relates Current to voltage.

However, no units of length in this form of Ohm’s law.

Resistivity [ohm-m]Resistance includes length and areaWe want resisitivity r because - property of the material alone.- no geometry included

↑ length Æ ↑ resistance

↑area Æ Øresistance

†

R = rlengtharea

[seiman/m] used to be mhos/m

†

s =1/R

Conductivity s

More general form of Ohm’s law

†

i =1R

V =Ar

Vl

When looking at a solid, we write ohm’s law as:

†

i =Ar

DVDl

And in 3-D we use vectors

†

I =Ar

gradV

Analogous to Heat and Fluid Flow.Any solutions you know for one of these flows works for the others with theanalogous boundary and initial conditions.

Wang and Anderson, 1982

Current Source on SurfaceElectric potential at distance away from currentsource on surface given as V(r)=rI/2pr. How?

Boundary conditions:1)As r => •, V => 0.2) V continuous acrossany boundary3)Tangential E continuousacross any boundary4) Normal I continuousacross any boundary.5) Above leads to novertical current crossingearth-air interface.

Current Flow in a Homogeneous, Isotropic Earth

†

dV = iRshell = ir lA

= ir dr2pr2

Point Current Source

†

VD = dV =ir2pD

•

Ú drr2

D

•

Ú =ir2p

-1( )1r D

•

=ir2p

-1( ) 1•

-1D

Ê

Ë Á

ˆ

¯ ˜ =

ir2pD

Voltage decreasesas the inverse ofthe distance fromthe current source.

Shape of constantvoltages arehemispheres for asingle point source

Two Current Electrodes - Source and SinkWhy run an electrode to infinity when we can use it?

source sink

P

†

Vsource =ir

2prsource

rsource

†

Vsin k =ir

2prsink

rsink

TotalVoltage at P

†

Vp =ir2p

1rsource

-1

rsink

Ê

Ë Á

ˆ

¯ ˜

Can’t measure potential at single point unless the other end ofour volt meter is at infinity. This is inconvenient. It is easierto measure potential difference (DV). This lead to use of fourelectrode array for each measurement.

Measurement Practicalities

Resulting measurement given as

DV=VP1-VP2= rI/(2p)*(1/r1-1/r2-1/r3+1/r4). Can be rewrittenDV=rI*G/(2p) where G/2p is the Geometrical Factor of the array.

r

Current Density and Equipotentiallines for a current dipole

†

fraction total current

i f =2p

tan-1 2zd

Ê

Ë Á

ˆ

¯ ˜

d

†

z =d2

if=0.5 at

if=0.7 at

†

z = d

Wider spacing Æ Deeper currents

Apparent Resistivity

r1

r2

Previous expression can berearranged in terms ofresistivity: r=(DV/I) (2p/G).

This can be done even whenmedium is inhomogenous.Result is then referred to asApparent Resistivity.

Definition:Resistivity of a fictitious homogenoussubsurface that would yield the same voltages as the earthover which measurements were actually made.

GeometricalFactors

Array advantages and disadvantages

1. Requires large current

2. Requires sensitiveinstruments

1. Cables can beshorter for deepsoundings

Dipole-Dipole

1. Can be confusing in thefield

2. Requires more sensitiveequipment

3. Long Current cables

1. Fewer electrodes tomove eachsounding

2. Needs shorterpotential cables

Schlumberger

1. All electrodes movedeach sounding

2. Sensitive to localshallow variations

3. Long cables for largedepths

1. Easy to calculate rain the field

2. Less demand oninstrument sensivity

Wenner

DisadvantagesAdvantagesArray