Fall 2004
Earth Properties and Basic Theory
Reading
Today : 207-218
Next Lecture : 218-228
1

Introduction
• Link resistivity (ability of the earth to conduct an electric current) to subsurface structure.
• Useful because resistivity of earth materials varies by around 10 orders of magnitude.
• Developed by Conrad Schlumberger
(France) and Frank Wenner (United
States) in early 20th century.
• Uses: Archeology, Environmental, Mineral exploration
Electricity Basics
Voltage V - Electrical potential energy per unit charge [volts]
Current i - amount of charge per unit time [amperes] i =
1
R
V
Resistivity R is just a proportionality constant [ohms]
R relates current I to voltage I.
However, no units of length in this form of Ohm’s law.
2

Resistivity
Resistance includes length and area
We want resisitivity
ρ [ ohm/m
] because
- property of the material alone.
- no geometry included
R
= ρ
L
A
[ ohm
] ↑ length
→
↑ area
→
↑ resistance
↓ resistance
Conductivity
σ
[siemens/m] or [mhos/m]:
σ =
1
ρ
[ mhos m
]
It is the ability of the electrical charge to move through the material
Electrical Conductivity of Geomaterials
• Non-conductive minerals
Electric field Electric field
Hydrated ion mobility
Electrolyte
Soil (Archie’s law)
Soil (clays)
Equation
σ el
[ ] m
=
0
Double layer surface conduction
.
15 TDS
[ ]
L
σ soil
= a
σ el
S c r n m
σ soil
= n
σ el
+
(
1
− n
)
Θ ρ g
S s
Comments
TDS: total dissolved salts a
≈
1; m~1-2.4; c~4-5
Θ ≈
10 -9 S (for Kaolinite)
3

• Non-conductive Minerals
4

Any solutions you know for one of these flows works for the others with the analogous boundary and initial conditions.
(Wang and Anderson 1982)
Soils and rocks can be conceptually modeled as a circuit made of a resistor, capacitor, inductor and battery:
Electrodes
R
C
L
B
Resistor R: dissipator of applied energy as heat
Capacitor C: storage of energy as separation of charges
Inductor L:
Battery B: self voltage associated to electromagnetic methods electrokinetics and self-potentials
5

Current Source on Surface
Electric potential at distance away from current source on surface given as V(r)=
ρ
I/2
π r. How?
Boundary conditions:
1)As r =>
∞
, V => 0.
2) V is continuous across any boundary
3) Tangential E continuous across any boundary
4) Normal I continuous across any boundary.
5) Above leads to no vertical current crossing earth-air interface.
Current Flow in a Homogeneous and
Isotropic Medium
Point Current Source: dV
= iR shell
= i
ρ l
A
= i
ρ dr
2
π r
2
Voltage decreases as the inverse of the distance from the current source.
Shape of constant voltages are hemispheres for a single point source.
V
D
= i
ρ
2
π
= ∫
∞
D dV
−
= i
ρ
2
π
∫
∞
D dr r
2
1
D
=
= i
ρ
2
π
( ) 1 r
∞
D i
ρ
2
π
D
6

Two Current Electrodes: Source and Sink
• Why run an electrode to infinity when we can use it?
source sink r source
P r sink
Total Voltage at P:
V source
= i ρ
2 π r source
V sin k
= i
ρ
2
π r sink
V p
=
V source
−
V sin k
= i
ρ
2
π
⎛
⎜⎜
1 r source
−
1 r s ink
⎞
⎟⎟
Measurement Practicalities
Can’t measure potential at single point unless the other end of our volt meter is at infinity. This is inconvenient. It is easier to measure potential difference (
∆
V) . This lead to use of four electrode array for each measurement.
ρ
Resulting measurement given as
∆
V=V
P1
-V
P2
=
ρ
I/(2
π
)*(1/r
1
-1/r
2
-1/r
3
+1/r
4
).
Can be rewritten
∆
V=
ρ
I*G/(2
π
) where G/2
π is the Geometrical Factor of the array.
7

Current density and equipotential lines for a current dipole d i fraction total current f
=
2
π tan − 1
⎛ 2 z d ⎠ i f
=0.5 at z = d
2 i f
=0.7 at z = d
Wider spacing
→
Deeper currents
Apparent Resistivity
Previous expression can be rearranged in terms of resistivity:
ρ=(∆
V/I) (2
π
/G).
This can be done even when medium is inhomogeneous. Result is then referred to as Apparent
Resistivity.
ρ
2
ρ
Definition:Resistivity of a fictitious homogenous subsurface that would yield the same voltages as the earth over which measurements were actually made.
1
8

Geometrical Factors
Array advantages and disadvantages
Array
Wenner
Schlumberger
Dipole-Dipole
Advantages
1. Easy to calculate
ρ a field in the
2. Less demand on instrument sensivity
1. Fewer electrodes to move each sounding
2. Needs shorter potential cables
1. Cables can be shorter for deep soundings
Disadvantages
1. All electrodes moved each sounding
2. Sensitive to local shallow variations
3. Long cables for large depths
1. Can be confusing in the field
2. Requires more sensitive equipment
3. Long Current cables
1. Requires large current
2. Requires sensitive instruments
9