theory and measurements - richard...
TRANSCRIPT
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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.