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