Electrical Surveying
Part I: Resistivity method
Lecture A
Dr. Laurent Marescot
laurent@tomoquest.com
1
Introduction
Electrical surveying…
• Resistivity method
• Induced polarization method (IP)
• Self-potential (SP) method
Higher frequency methods (electromagnetic surveys):
• Electromagnetic induction methods
• Ground penetrating radar (GPR)
2
Resistivity method
The resistivity method is used in the study of horizontal
and vertical discontinuities in the electrical properties
(resistivity) of the subsurface
3
Application
•
•
•
•
•
•
•
•
Exploration of bulk mineral deposit (sand, gravel)
Exploration of underground water supplies
Engineering/construction site investigation
Waste sites and pollutant investigations
Cavity, karst detection
Glaciology, permafrost
Geology
Archaeological investigations
4
Structure of the lecture
The next two lectures…
1.
2.
3.
4.
5.
Resistivity of rocks
Equations in resistivity surveying
Survey strategies and interpretation
Summary of resistivity methods: case histories
Conclusions
5
1. Resistivity of rocks
6
Resistivity and units
δL
δR = ρ
δA
δA
ρ =δR
δL
•
•
ρ
σ =1/ ρ
resistivity in ohm.m (Ωm)
conductivity in Siemens per meter (S/m)
Resistivity is the physical property which determines the aptitude of
this material to be opposed to the passage of the electrical current
7
Electronic conductibility
The current flows by
displacement of electrons.
Known as electronic
conductibility or metallic
because it is a similar
conductibility to that of metals.
This solid conductibility is really
significant only for certain
massive mineral deposits.
8
Electrolytic conductibility
The current is carried by ions. The electrical resistivity of
rocks bearing water is controlled mainly by the water which
they contain.
9
Electrolytic conductibility
The resistivity of a rock will depend :
• on the resistivity of the natural pore water and consequently
the quantity of dissolved salts in the electrolyte
1g/liter=1000 ppm
• on the quantity of electrolyte contained in the unit of rock
volume (saturation)
• on the mode of electrolyte distribution, porosity
10
Effect of temperature
ρt =
ρ18
1 + 0.025(t − 18)
A rock totally frozen is infinitely resistant and it is
impossible to implement resitivity methods (use EM
methods)
11
Archie´s Law
ρ = ρw a φ −m S −n
ρ
ρw
Φ
•
•
•
•
•
S
a
•
m
•
n
resistivity of the rock
resistivity of the fluid (water)
porosity
saturation in water
factor which depends of the lithology (varies between
0.6 and 2)
cementation factor (depends of the pores shape, of the
compaction and varies between 1.3 for unconsolidated sands to
2.2 for cimented limestone
about 2 for majority of the formations with normal porosities
containing water between 20 and 100 %.
12
Formation factor F
ρ = ρw a φ −m S −n
ρ = ρw F S −n
• For sand and sandstones: F≈ 0.62/φ2.15
• For well cemented rocks: F≈ 1/φ2
13
Permeability
There is no direct
relationship between
resistivity and
permeability.
This table shows also the
problem in identifying
rocks due to overlapping
resistivity values (no
contrast)
14
Resistivity of rocks and minerals
Air, gas or oil: infinite or very high resistivity!
Liquid materials from landfills are generally conductive (<10 ohm.m)
15
Effect of clay and graphite
• Clay has a high ionic exchange capacity, therefore the
conductivity of the pore fluid largely increases
Archie´s Law is not valid if clay is present!
• Graphite, often associated with pyrite, makes the resistivity
decrease
16
Summary…
The conductivity of a rock increases if…
•
•
•
•
The quantity of water increases
The salinity increases (quantity of ions)
The quantity of clay increases
The temperature increases
17
2. Equations in resistivity surveying
18
Maxwell equations
G
G ∂B
∇× E +
=0
Faraday induction
∂t
G
G ∂D G
∇× H −
= j
Ampère − Maxwell
∂t
G
G ∂p
G
∇ ⋅ D = p or ∇ ⋅ j =
E electrical field (V/m)
G
∂t
B magnetic induction field
G
G
∇⋅B = 0
H magnetic field (A/m)
(Wb/m 2 )
G
D displacement field (C/m 2 )
G
j current density (A/m 2 )
p charge density (C/m 3 )
19
Static approximation
G
∇× E = 0
G G
∇× H = j
G
∇⋅ j = 0
G
∇⋅B = 0
Faraday induction
Ampère − Maxwell
20
Current flow in the ground
21
Equations for DC approximation
G
G
E=ρ j
Ohm´s Law
G
∇⋅ j = 0
Divergence is null except at the
current source
G
E = −∇V
Definition of electrical field E
G
K 1
1 2
∇⋅ j = ∇⋅E = − ∇ V = 0
ρ
ρ
Laplace´s equation
22
Potential from a single electrode
In polar coordinates, Laplace´s equation rewrites:
1
∂ ⎛ 2 ∂V ⎞
∂ ⎛
∂V ⎞
1
∂ 2V
=0
⎜r
⎟+ 2
⎜ sin θ
⎟+ 2 2
2
∂θ ⎠ r sin θ ∂ψ
∂r ⎝ ∂r ⎠ r sin θ ∂θ ⎝
In polar coordinates, the current flow is symmetrical
with respect of θ and ψ directions
∂ ⎛ 2 ∂V ⎞
⎜r
⎟=0
∂r ⎝ ∂r ⎠
∂V
r
= C1
∂r
2
Direct integration can then be performed…
C1
V = − + C2
r
23
Potential from a single electrode
Determination of C1 using the definition of current I…
G
E
Remember...
2 ∂V
r
= C1
∂r
G
2πC1
C1
I = ∫ j ⋅ ds = ∫ ds = ∫ 2 ds = −
ρ
ρr
ρ
S
s
s
24
Potential from a single electrode
C2 tends to 0, if D tends to infinity…
C1
V = − + C2
r
I =−
2π C1
ρ
ρI
V=
2πr
25
Two current electrodes
26
Potential field between
two current electrodes
A and B
A
B
27
Potential difference
Vp1 is the sum of the
potential contribution
from the current
electrodes C1 and C2
⎛1 1⎞
VP1 = Iρ / 2πr1 + (− Iρ / 2πr2 ) = (Iρ / 2π )⎜⎜ − ⎟⎟
⎝ r1 r2 ⎠
28
Two potential electrodes
ρI
VM =
2π
ρI
VN =
2π
ΔVMN
1 ⎞
⎛ 1
−
⎜
⎟
⎝ AM MB ⎠
1 ⎞
⎛ 1
−
⎟
⎜
⎝ AN NB ⎠
ρI
= VM − VN =
2π
1
1
1 ⎞
⎛ 1
−
−
+
⎜
⎟
⎝ AM MB AN NB ⎠
2πΔVMN ⎛ 1
1
1
1 ⎞
ρa =
−
−
+
⎜
⎟
I
⎝ AM MB AN NB ⎠
−1
29
Apparent resistivity
• In a heterogeneous medium, the measured resistivity is an
apparent resistivity, which is a function of the form of the
inhomogeneity and of the electrode spacing and surface
location.
• K is named the geometric factor.
ΔVMN
ρa =
K
I
30
Geometric factor
For a half-space, a general
definition for the geometric
factor can be written:
K=
4π
1
1
1
1
1
1
1
1
−
−
+
+
−
−
+
AM AN BM BN A′M A′N B′M B′N
31
Electrode spreads
32
Electrode spreads
ΔV
ρ a = 2πa
I
ΔV
ρ a = π n( n + 1)a
I
ΔV
ρ a = πn(n + 1)(n + 2)a
I
Wenner array
Schlumberger array
dipole-dipole array
33
Current penetration
⎛ 2 ⎞ −1 ⎛ 2 z ⎞
I f = ⎜ ⎟ tan ⎜
⎟
⎝ AB ⎠
⎝π ⎠
• z
• AB
• If
depth
distance between current electrodes
fraction of current penetrating between the
surface and z
34
⎛ 2 ⎞ −1 ⎛ 2 z ⎞
I f = ⎜ ⎟ tan ⎜
⎟
⎝ AB ⎠
⎝π ⎠
35
Principle of reciprocity
36
Heterogeneous Earth
37
Modified Snell´s Law
tan θ1 = Lz1 / Lx
tan θ 2 = Lz 2 / Lx
⇒ tan θ1 / tan θ 2 = Lz1 / Lz 2
V = j ρL ⇒ L = V / j ρ
⇒ Lz1 ∝ 1 / ρ1 , Lz 2 ∝ 1 / ρ 2
⇒ Lz1 / Lz 2 = tan θ1 / tan θ 2 = ρ 2 / ρ1
tan θ1 / tan θ 2 = ρ 2 / ρ1
38
tan θ1 ρ 2
=
tan θ 2 ρ1
ρ1 < ρ 2
ρ1 > ρ 2
39
Current distribution
40
Current distribution
This has an influence on the depth of investigation!
41
Current distribution
42
Reflection and transmission
ρ1 I ⎛ 1 ⎞ ρ1 I ⎛ k ⎞
⎜⎜ ⎟⎟ +
⎜⎜ ⎟⎟
VM =
4π ⎝ r1 ⎠ 4π ⎝ r2 ⎠
ρ2 I ⎛ 1 k ⎞
⎜⎜ − ⎟⎟
VN =
4π ⎝ r3 r3 ⎠
For r1=r2=r3
VM
(
ρ 2 − ρ1 )
= VN ⇒ k =
(ρ1 + ρ 2 )
43
Anisotropy
n
S = S1 + S 2 + ... + S n = ∑
i =1
hi
ρi
=
n
H
ρl
T = T1 + T2 + ... + Tn = ∑ hi ρ i = Hρ t
i =1
λ=
e.g.
λ ≅1 for alluvium
λ >2 for graphitic slates
• ρl
• ρt
• λ
ρt
ρl
longitudinal resistivity
transverse resistivity
44
coefficient of anisotropy
Effect of topography
Equipotential: dashed lines
45
3. Survey strategies and interpretation
46
Resistivity survey equipment
47
48
Device
• Current source: batteries in series
• Voltmeter and ammeter (resistivimeter)
• Electrodes: metallic stakes
current electrodes: stainless steel
potential electrodes: stainless steel or
nonpolarizing electrodes
Polarization occurs at the contact electrode/ground: this
creates an additional potential difference.
49
Polarization and skin depth
• Use an alternating current to avoid polarization
• Very low frequency (<10 Hz)
Skin depth:
depth δ at which the amplitude of the field
reaches 1/e of its original value a the source
δ ≈ 503
ρ
f
50
Contact resistance
dL
dL
dR = ρ
=ρ
s
2πL2
ρ ⎛1 1 ⎞
R=
⎜ − ⎟
2π ⎝ r L ⎠
L = distance to the centre of the electrode [m]
r = radius of the electrode [m]
R = resistance [ohm]
ρ = resistivity of the surrounding ground [ohm.m]
51
To decrease the contact resistance…
•
•
•
•
•
Add electrodes in parallel
Increase the current intensity
Increase the diameter of the current electrodes
Put electrode deeper into the ground
Add water (with salt) near the electrodes
About 90% of the contact resistance contribution comes
from a portion of the ground around the electrode that is
equal to 10 times the diameter of the electrode
52
Equivalent circuit
53
Origine of noise
• Telluric currents
• Man-made currents
• Metallic conductors in the ground (short-circuits)
Solutions:
• Use of alternating current
• Stacking operations
• Rejection filters (16-20 Hz, 50-60 Hz)
54
Survey strategies
• Resistivity mapping, constant separation traversing (CST):
used to determine lateral variations of resistivity. The
current and potential electrodes are maintained at a fixed
separation and moved along profiles
• Vertical electrical sounding (VES):
used in the study of near-horizontal interfaces. The
electrode spread is progressively expanded about a central
point
• Resistivity tomography (ERT):
is a mix between CST and VES. Also named electrical
imaging
55
Constant separation traversing (CST)
56
Constant separation traversing (CST)
57
Constant separation traversing (CST)
58
Constant separation traversing (CST)
59
Constant separation traversing (CST)
•
Demo during the lecture
60
Interpretation of CST
61
62
63
64
65
66
67
Pontis Nappe
SiviezMischabel
Nappe
Unstable
area
Water
infiltration
68
Small scale resistivity map (archaeology)
AB=4m
wall
fountain?
69
Mobile arrays
Source: Geocarta, Paris
100 data points/seconde
70
1 data point each 20cm
Mobile arrays
Source: Geocarta, Paris
Vineyards investigations
71
Mobile arrays
A
Current injection
M1
Resistivity measurement
(three investigation depths)
M2
M3
B
N1
N2
N372
Source: Geocarta, Paris
Mapping example with mobile array
(spacing 2m)
Surface: 140 hectares
Apparent resistivity
15 ohm.m
150 ohm.m
73
Source: Geocarta, Paris
Profile spacing 6m
Profile spacing 12m
Profile spacing 24m
Apparent resistivity
10 ohm.m
90 ohm.m
74
Source: Geocarta, Paris
Ecartement 0.5m
Ecartement 1m
Ecartement 2m
Apparent resistivity
10 ohm.m
60 ohm.m
75
Source: Geocarta, Paris
Vertical electrical sounding (VES)
76
Vertical electrical sounding (VES)
77
Vertical electrical sounding (VES)
78
Vertical electrical sounding (VES)
79
80
One layer and two layers
81
82
83
Three layers and more…
84
85
Equivalence
R = hρ
R=
h
ρ
86
Parametric sounding
A parametric sounding is a VES carried out on an outcrop
or near a borehole to precisely determine the resistivity of
a geological formation.
A precise determination of resistivity reduce the problem
of equivalence
87
Suppression
88
89
•
Demo during the lecture
90
Interpretation of VES
•
Demo during the lecture
91
Interpretation of VES
92
93
94
95
96
97
98