Subsurface Characterization Using Electrical Resistivity ...
Transcript of Subsurface Characterization Using Electrical Resistivity ...
Subsurface Characterization Using Electrical Resistivity Tomography
Frank T.‐C. Tsai and Madhava PillalaLouisiana State University
Civil and Environmental Engineering(2009 World Environmental & Water Resources Congress)(7th Symposium on Groundwater Hydrology, Quality and
Management, May 18, 2009)
Outline
Motivations and objectives
ERT forward problem (MODFLOW)
ERT Inverse problem (Adjoint‐State Method)
ERT experiments
Preliminary results
Closing remarks
Motivations
Electrical Resistivity Tomography (ERT) has been successfully
used as a geophysical method to detect/monitor subsurface
environment, e.g. soil heterogeneity, saltwater intrusion, etc.
ERT is a lower cost tomographical technique and provides
qualitative and quantitative images.
ERT provides 2D (slice)or 3D (volume) dynamic images.
ERT can be applied as an non‐invasive (boundary measurement)
or invasive (borehole measurement) approach.
ERT involves an inverse technique to estimate an electrical
resistivity distribution.
ERT can provides abundant voltage data to the inverse problem.
(Source: Daily and Ramirez 2000)
(Source: USGS)
Objectives
Initiate ERT experiment.
Test if the groundwater forward and inverse
models are working
What is ERT?• A measurement technique with an inversion
technique for estimating electrical conductivity (or electrical resistivity) distribution of conductive materials.
• Multiple electrodes are arranged around the boundary of the vessel (non‐invasive) or in boreholes (invasive).
• The electrodes make electrical contact with the material with limited to no influence on the material, e.g., flow.
• Measure voltages given a pair of current source electrode and sink electrode.
• Systematically change locations of source and sink electrodes.
• Use voltage measurements and electrostatic model to estimate conductivity distribution
Electrostatic model(potential‐resistivity model)
Electrode number
source sinkFlow of Electricity
Ohm’s Law:
J‐ Current Density (Ampere/m2)
ρ‐ Electrical resistivity (ohm‐m)
φ− Electrical Potential (Volt)
σ‐ Electrical Conductivity (Siemen/m)
Electrostatic Equation:
I – Current (Ampere)
*Potential at source/sink electrode:
( ) Iσ φ∇ ∇ =i
1J σ φ ρ φ−= − ∇ = − ∇
e eV z n Uσ φ+ ∇ =i
Groundwater model(MODFLOW)
Flow of Electricity
Ohm’s Law:
J‐ Current Density (Ampere/m2)
ρ‐ Electrical resistivity (ohm‐m)
φ− Electrical Potential (Volt)
σ‐ Electrical Conductivity (Siemen/m)
Electrostatic Equation:
I – Current (Ampere)
*Potential at source/sink electrode:
Ze – contact impedance (ohm‐m2)
( ) Iσ φ∇ ∇ =i
1J σ φ ρ φ−= − ∇ = − ∇
e eV z n Uσ φ+ ∇ =i
Flow of Groundwater (MODFLOW)
Darcy’s Law
q‐ average pore velocity (m/sec)
K – hydraulic conductivity (m/sec)
h‐ ground water potential head (m)
Steady‐state groundwater flow equation:
Q – Volumetric flow rate (m3/sec)
Contact impedance in MODFLOW?
q K h= − ∇
( )K Qφ∇ ∇ =i
Representing contact impedance in MODFLOW: Horizontal Flow Barrier
1HC A1+A2+A3
ee
Az =
A1
A2
A3Ae
Contact impedance= [ohm-m^2]
I1
I2
I3
HC: hydraulic conductivityper unit width of barrier [1/T]
Source electrode, I=I1+I2+I3
Inverse ModelingEstimate conductivity at each cell
P current patternsN potential data given a current patternA total of P*N potential data
Objective function:
( )2
, ,1 1
minP N
obsp p
pE
σφ φ
= =
= −∑∑
Quasi-Newton method: BFGS
1
P
p pp
dE dd σψ φσ Ω
=
= ∇ ∇ Ω∑∫ i Depending on parameterizationscheme
Adjoint-state equation:
( ) ( ) ( ), ,1
2N
obsp p p Dσ ψ φ φ δ
=
∇ ∇ = − − −∑ x xi
Only need 2P calls on MODFLOW!
Hardware:
Source electrode
Voltage Measurement
Sink electrode
DC PowerTest cell
LabView Software: Measuring Voltage
Sampling frequency: 0.01 sec
Channel list
Measurement Block Diagram
LabView Software: Changing Current Source and Sink Electrode Locations
List of source locations
List of sink locations
Experimental Setup
Stainless steel electrode
1”
Case 1: Homogeneous Case: Estimating Contact Impedance
• a 6” by 6” box• Water depth 2”•12 electrodes (one layer)• 132 current patterns (12 by 11)• Constant current: 5 mA• tap water (homogeneous)• 0.042 S/m from conductivity meter• A total of 1452 potential data.
Results: Homogeneous CaseEstimating Contact Impedance
• 12 electrodes (one layer)• 132 current patterns (12 by 11)• tap water (homogeneous)• 0.042 S/m from conductivity meter
Inverse Model:• discretization: 37 by 37 cells• parameterization: homogeneity• 1452 potential data• 13 unknowns
Results:• K =0.045 S/m• HC value: 1.4~2.3 (1/sec)
Case 2: Synthetic CaseTrueERT result
Number of obj. function called
• 12 electrodes (one layer)• 132 current patterns (12 by 11)• tap water (homogeneous + one anomaly )
Inverse Model:• Discretization: 37 by 37 cells• Parameterization = discretization• Use identified HC values• 1452 potential data• 1369 unknowns
Borehole Tomography
• Sand + tap water• Sand of porosity: 0.4• A block of wood
Borehole Tomography
Case 3: Borehole Test: Synthetic Case
Electrical Potential Fitting Errors
0.1
1.0
10.0
100.0
1000.0
10000.0
0 50 100 150 200 250
No. of Obj Function Called
Fitti
ng E
rror
True fieldERT
TRUEK10.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.050
• 100cm by 50 cm• Water depth 2”• 32 electrodes• 512 current patterns (32 by 16)• Constant current: 2 mA• A total of 15872 potential data.
Inverse Model:• Discretization: 37 by 19 cells• Parameterization = discretization• 15872 potential data• 703 unknowns
Case 4: Real data
D50=0.45mmCu=1.8
Summary
• ERT is a potential tomographical technique to monitor
subsurface environment.
• ERT is low cost and products abundant data for the inverse
problem.
• The adjoint state method presents an efficient way to obtain
gradients.
• Need much larger number of measured data than unknowns
to obtain good quality images.
Challenges and Further Research
• High performance computing.
• Low resolution images.
• Data quality
• Contact impedance
• Optimal patterns, optimal electrode locations
• Parameterization vs. resolution
• Interpreting electrical conductivity to hydraulic property