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GLE 594: An introduction to applied geophysics

Electrical Resistivity Methods

Fall 2004

Theory and Measurements

Reading: Today: 210-223Next Lecture: 223-251

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• Why run an electrode to infinity when we can use it?

source sink

P

Vsource =iρ

2πrsource

rsource

sinkksin r2

iVπ

ρ=

rsink

Total Voltage at P: ⎟⎟⎠

⎞⎜⎜⎝

⎛−

πρ

=−=inkssource

ksinsourcep r1

r1

2iVVV

Two Current Electrodes: Source and Sink

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.

Measurement Practicalities

Resulting measurement given as

Can be rewritten

where G*/2π is sometimes referred as the Geometrical Factor

ρ

πρ=∆

2GIV

*

⎟⎟⎠

⎞⎜⎜⎝

⎛+−−

πρ

=−=∆4321

2P1P r1

r1

r1

r1

2IVVV

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Current density and equipotential lines for a current dipole

fraction total current

i f =2

πtan−1 2z

d

⎛ ⎝ ⎜

⎞ ⎠ ⎟

d

z =d

2if=0.5 at

if=0.7 at z = d

Wider spacing → Deeper currents

Apparent Resistivity

Previous expression can be rearranged in terms of resistivity: ρ=(∆V/I) (2π/G).

This can be done even when medium is inhomogeneous. Result is then referred to as Apparent Resistivity.

Definition:Resistivity of a fictitious homogenous subsurface that would yield the same voltages as the earth over which measurements were actually made.

ρ2

ρ1

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Geometrical Factors

Array advantages and disadvantages

1. Requires large current2. Requires sensitive instruments

1. Cables can be shorter for deep soundings

Dipole-Dipole

1. Can be confusing in the field2. Requires more sensitive

equipment3. Long Current cables

1. Fewer electrodes to move each sounding

2. Needs shorter potential cables

Schlumberger

1. All electrodes moved each sounding

2. Sensitive to local shallow variations

3. Long cables for large depths

1. Easy to calculate ρa in the field

2. Less demand on instrument sensivity

WennerDisadvantagesAdvantagesArray

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Governing Equation

0zzjy

yj

xxj

0jzzjjjy

yj

jjxxjj

zyx

zz

zyy

yxx

x

=∆∂∂

−∆∂∂

−∆∂∂

−

=⎟⎠⎞

⎜⎝⎛ −∆

∂∂

−+⎟⎟⎠

⎞⎜⎜⎝

⎛−∆

∂∂

−+⎟⎠⎞

⎜⎝⎛ −∆

∂∂

−

Continuity: What goes in must comes out

r j =

r i

A

Current Density (like hydro q):

z

xy

∆x

∆z

∆y

jy +∂jy

∂y∆y

jy

zj

xj

yyjj z

z ∆∂∂

+

yyjj x

x ∆∂∂

+

Applying Ohm’s Law: z

V1j;yV1j;

xV1j zyx ∂

∂ρ

−=∂∂

ρ−=

∂∂

ρ−=

equation sLaPlace' 0V

0zV

rV

r1

rV

0zV

yV

xV

2

2

2

2

2

2

2

2

2

2

2

⇒=∇

⎪⎪⎭

⎪⎪⎬

⎫

=∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

Governing Equation

222 ryx and , sinr = y, cosr =x

usingor

0zV1

zyV1

yxV1

x

=+θθ

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

ρ∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

ρ∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

ρ∂∂

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Governing Equation - Solution

• The Laplace’s equation is a homogeneous, partial second order differential equation

• Solution:– Exact solutions: only for simple geometries– Graphical solutions: Flow nets, master charts– Numerical solutions: finite difference and finite elements

solutions– Approximate solutions: methods of fragments– Physical analogies (electrical, hydraulic and heat flow)

Geo-electric Layering• Often the earth can be simplified within

the region of our measurement as consisting of a series of horizontal beds that are infinite in extent.

• Goal of the resistivity survey is then to determine thickness and resistivity of the layers.

Longitudinal conductance (one layer): SL=h/ρ=hσTransverse resistance (one layer): T=hρLongitudinal resistivity (one layer): ρL=h/STransverse resistivity (one layer): ρT=T/h

Longitudinal conductance (one layer): SL=Σ(hi/ρi)Transverse resistance (one layer): T=Σ(hiρi)

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Voltage and Flow in Layers

Tangent Law: The electrical current is bent at a boundary

i1

i2

dl1

dl2

dV1

dV2

a

cθ1

θ2

ρ1

ρ2

Relations: Current: i1=i2Voltage: dV1=dV2Resistivity: ρ1>ρ2

b

ρ2

ρ1

=tanθ1

tanθ2

If ρ2<ρ1 then the current lines will be refracted away from the normalIf ρ2>ρ1 then the current lines will be refracted closer to the normal

Voltage and Flow in Layers

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12kρ+ρρ−ρ

=

Method of electrical image

r1S

S’

P ρ1

ρ2

Voltages at points P and Q:

⎟⎟⎠

⎞⎜⎜⎝

⎛+

πρ

=21

1P r

kr1

4IVr3

Qr2 ⎟⎟⎠

⎞⎜⎜⎝

⎛ +π

ρ=

3

2Q r

k14IV

where

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Solving the differential equation for two layers and a source and sink

Governing Equation

0zV

rV

r1

rV

2

2

2

2

=∂∂

+∂∂

+∂∂

Boundary Conditions

( )solution Particular 0=z 0,=rat

zr2

iV 4.

continous isdensity current Normalz=zat z

V1zV1 3.

continuous is Voltagez=zat VV 2.

surfaceat current No0i 1.

21

22

1

interface2

2

1

1

interface21

0zz

+π

ρ=

∂∂

ρ=

∂∂

ρ

=

==

aC1 P1

zint = hρ1

ρ2

Layer Calculations• Can use for image theory for multiple

boundaries. For two layer case:

where

• It obviously gets much more difficult with more layers.

⎟⎟⎠

⎞⎜⎜⎝

⎛+

πρ

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+++++

πρ

=

∑∞

=1n n

n1

n

n

2

2

1

1p

rk2

r1

2I

....rk2.....

rk2

rk2

r1

2IV

( )22n nh2rr +=

12

12kρ+ρρ−ρ

=

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Layer Calculations (cont.)• Integral method:• J0 is the Bessel function of zero order.

– K(λ) given by relationship

• Ti(λ) solved for recursively upward from bottom layer to layer 1 using:

where

and

∫∞

=0

01 )()(

2λλλ

πρ drJKIVp

1

1 )()(ρ

λλ TK =

[ ][ ] ⋅

++

=+

+

iii

iiii hT

hTTρλ

λρλ/)tanh(1

)tanh()(1

1

⋅+−

=λ λ

λ

1e1e)htanh(

i

i

h2

h2

i

⋅= nnT ρλ)(

Solutions for a Wenner Array for two layers

k =ρ2 − ρ1

ρ2 + ρ1

C1 C2P1 P2

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Vertical Electric Sounding

• When trying to probe how resistivity changes with depth, need multiple measurements that each give a different depth sensitivity.

• This is accomplished through resistivity sounding where greater electrode separation gives greater depth sensitivity.

VES Data Plotting Convention• Plot apparent resistivity as a

function of the log of some measure of electrode separation.• Wenner – a spacing• Schlumberger – AB/2• Dipole-Dipole – n spacing

• Asymptotes:• Short spacings << h1,

ρa=ρ1.• Long spacings >> total

thickness of overlying layers, ρa=ρn

• To get ρa=ρtrue for intermediate layers, layer must be thick relative to depth.

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Equivalence: several models produce the same results

• Ambiguity in physics of 1D interpretation such that different layered models basically yield the same response.

• Different Scenarios:• Conductive layers between two resistors, where

lateral conductance (σh) is the same. • Resistive layer between two conductors with

same transverse resistance (ρh).

• Although ER cannot determine unique parameters, can determine range of values.

• Also exists in 2D and 3D, but much more difficult to quantify. In these multidimensional cases simply referred to as non-uniqueness.

Equivalence: several models produce the same results

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Suppression

• Principle of suppression: Thin layers of small resistivity contrast with respect to background will be missed.

• Thin layers of greater resistivity contrast will be detectable, but equivalence limits resolution of boundary depths, etc.