GLE 594: An introduction to
applied geophysics
Electrical Resistivity Methods
Fall 2004
Theory and Measurements
Reading:
Today: 210-223
Next Lecture: 223-251
1
Two Current Electrodes: Source and Sink
• Why run an electrode to infinity when we can use it?
source
sink
rsource
P
rsink
Vsource =
Total Voltage at P:
iρ
2πrsource
Vsin k =
Vp = Vsource − Vsin k =
iρ
2 πrsink
iρ ⎛ 1
1 ⎞
⎟
⎜⎜
−
2 π ⎝ rsource rsink ⎟⎠
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 = VP1 − VP 2 =
Can be rewritten
∆V = ρI
G*
2π
ρI ⎛ 1 1 1 1 ⎞
⎜ − − + ⎟
2 π ⎜⎝ r1 r2 r3 r4 ⎟⎠
where G*/2π is sometimes referred as the Geometrical Factor
2
Current density and equipotential lines
for a current dipole
d
fraction total current
⎛ 2z ⎞
2
i f = tan−1⎜ ⎟
⎝d⎠
π
if=0.5 at
z=
d
2
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.
ρ1
ρ2
Definition:Resistivity of a fictitious homogenous subsurface
that would yield the same voltages as the earth over which
measurements were actually made.
3
Geometrical Factors
Array advantages and disadvantages
Array
Advantages
Disadvantages
Wenner
1. Easy to calculate ρa in the 1. All electrodes moved each
sounding
field
2. Sensitive to local shallow
2. Less demand on
variations
instrument sensivity
3. Long cables for large depths
Schlumberger
1. Fewer electrodes to move 1. Can be confusing in the field
each sounding
2. Requires more sensitive
2. Needs shorter potential
equipment
cables
3. Long Current cables
Dipole-Dipole
1. Cables can be shorter for
deep soundings
1. Requires large current
2. Requires sensitive instruments
4
Governing Equation
Continuity: What goes in must comes out
∂jy
⎞ ⎛
∂j
∂j
⎛
⎞ ⎛
⎞
∆y − jy ⎟⎟ + ⎜ jz − z ∆z − jz ⎟ = 0
⎜ jx − x ∆x − jx ⎟ + ⎜⎜ jy −
∂x
∂y
∂z
⎝
⎠ ⎝
⎠
⎠ ⎝
∂j
∂j
∂j
− x ∆x − y ∆y − z ∆z = 0
∂x
∂y
∂z
z
jz +
∂jz
∆y
∂y
jx
∆z
jy +
jy
jx +
x
∆y
∂jx
∆y
∂y
∆x
∂j y
∆y
∂y
Current Density
(like hydro q):
r
r i
j=
A
y
jz
Governing Equation
Applying
Ohm’s Law:
jx = −
1 ∂V
1 ∂V
1 ∂V
; jz = −
; jy = −
ρ ∂x
ρ ∂y
ρ ∂z
∂ ⎛ 1 ∂V ⎞ ∂ ⎛ 1 ∂V ⎞ ∂ ⎛ 1 ∂V ⎞
⎜
⎟+ ⎜
⎟+ ⎜
⎟=0
∂x ⎜⎝ ρ ∂x ⎟⎠ ∂y ⎜⎝ ρ ∂y ⎟⎠ ∂z ⎜⎝ ρ ∂z ⎟⎠
or using
x = r cos θ, y = r sin θ, and x 2 + y 2 = r 2
⎫
∂ 2V ∂ 2V ∂ 2V
+ 2 + 2 =0⎪
2
∂x
∂y
∂z
⎪ 2
⎬∇ V = 0 ⇒ LaPlace' s equation
2
2
∂ V 1 ∂V ∂ V
⎪
+
= 0⎪
+
∂r 2 r ∂r ∂z 2
⎭
5
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):
Transverse resistance (one layer):
Longitudinal resistivity (one layer):
Transverse resistivity (one layer):
SL=h/ρ=hσ
T=hρ
ρL=h/S
ρT=T/h
Longitudinal conductance (one layer):
Transverse resistance (one layer):
SL=Σ(hi/ρi)
T=Σ(hiρi)
6
Voltage and Flow in Layers
Tangent Law: The electrical current is bent at a boundary
i1
ρ1
a
dl1
θ1
b
ρ2
θ2
c
Relations:
Current:
Voltage:
Resistivity:
dV1
i2
i1=i2
dV1=dV2
ρ1>ρ2
ρ 2 tan θ1
=
ρ1 tan θ 2
dl2
dV2
If ρ2<ρ1 then the current lines will be refracted away from the normal
If ρ2>ρ1 then the current lines will be refracted closer to the normal
Voltage and Flow in Layers
Method of electrical image
S
Voltages at points P and Q:
r1
P
r3
ρ1
ρ2
r2
Q
VP =
Iρ1 ⎛ 1 k ⎞
⎜ + ⎟
4 π ⎜⎝ r1 r2 ⎟⎠
VQ =
Iρ 2 ⎛ 1 + k ⎞
⎜
⎟
4 π ⎜⎝ r3 ⎟⎠
S’
where k =
ρ 2 − ρ1
ρ 2 + ρ1
7
Solving the differential equation for
two layers and a source and sink
C1
Governing Equation
∂ 2 V 1 ∂V ∂ 2 V
+
+
=0
∂r 2 r ∂r ∂z 2
Boundary Conditions
a
P1
ρ1
zint = h
ρ2
1. i z = 0 z =0
No current at surface
2. V1 = V2 at z = z interface
Voltage is continuous
3.
1 ∂V1 1 ∂V2
at z = z interface
=
ρ1 ∂z ρ 2 ∂z
4. V =
(
iρ1
2π r 2 + z 2
)
1
2
at r = 0, z = 0
Normal current density is continous
Particular solution
Layer Calculations
• Can use for image theory for multiple
boundaries. For two layer case:
Vp =
=
k=
ρ 2 − ρ1
ρ 2 + ρ1
⎞
Iρ1 ⎛ 1 2k 2 k 2
2k n
⎜⎜ +
+
+ ..... +
+ .... ⎟⎟
2π ⎝ r r1
r2
rn
⎠
∞
Iρ1 ⎛ 1
kn ⎞
⎜⎜ + 2∑ ⎟⎟
2π ⎝ r
n =1 rn ⎠
where
rn = r 2 + (2 nh )
2
• It obviously gets much more difficult
with more layers.
8
Layer Calculations (cont.)
∞
Iρ
• Integral method: V p = 1 ∫ K ( λ ) J 0 (λr )dλ
2π 0
• J0 is the Bessel function of zero order.
– K(λ) given by relationship
K (λ ) =
T1 ( λ )
ρ1
• Ti(λ) solved for recursively upward from bottom layer to layer 1
using:
Ti ( λ ) =
where
[Ti +1 + ρi tanh(λhi )]
[1 + Ti +1 tanh( λhi ) / ρi ]
tanh(λh i ) =
⋅
e 2 λh i − 1
⋅
e 2 λh i + 1
and
Tn ( λ ) = ρ n ⋅
Solutions for a Wenner Array
for two layers
C1
P1
P2
C2
k=
ρ 2 − ρ1
ρ 2 + ρ1
9
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.
10
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).
Equivalence: several models produce
the same results
• 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.
11
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.
12