Physics Based Forward Modeling for Inverse
Methods
Alireza Aghasi%, Tian Tang†, and Linda M. Abriola†, Eric L. Miller*
%Georgia
Tech, School of Electrical and Computer Engineering
*Tufts University, Department of Electrical and Computer Engineering
†Tufts University, Department of Civil and Environmental Engineering
Acknowledgements
Alireza Aghasi
PhD recipient, ECE
Linda Abriola
Prof. Civil and
Environmental
Engineering
Dean Tufts School of
Engineering
Tian Tang
PhD Student, Tufts CEE
2
The Problem
Three Mile Island (1979)
Love Canal (1978)
3
The Problem
Characterization of DNAPL (Dense Non-Aqueous Phase Liquid) source zones based
on electrical and hydrological measurements
The Challenges
• Locating and estimating extent of source mass
• Characterizing mass distribution
• Invasive, in-source characterization methods may mobilize
contaminants
• In-source characterization methods too costly for application
at most sites
Approach:
Characterization of DNAPL source zones using
(noninvasive) electrical and (down gradient
transect) hydrological measurements
5
Overview of Measurement Modalities
• Electrical Resistance Tomography (ERT):
– DNAPL causes change in electrical conductivity
– Inject current and measure voltages
– Infer electrical conductivity
• Hydrology
– Saturated DNAPL directly dissolved by flowing groundwater
– Measured downstream concentration
– Infer upstream saturation
Mathematical Models
Electrical Resistance Tomography
Electrical conductivity
Current source distribution
Electrical potential
Flow & Mass Transport Model
Saturation of the corresponding phase
Mass concentration of component i
The inter-phase mass exchange of
component i from one phase to other
7
ERT Modeling
Ñ is ( r ) Ñv ( r ) = i ( r )
• Poisson’s equation
• Discretize using finite
difference stencil
• Large sparse system of linear
equations
– Solved either directly
(backslash) or using iterative
method
• Boundary conditions always a
problem
– Expand grid and use a zero BC
– Contract grid and use a
complicated absorbing BC
8
Hydrological Model
Advection
Solid Phase
(Organic
components)
Solid grain
Solid grain
NAPL
Sorption
Aqueous
Dissolution
Aqueous Phase
(Organics, Water,
Oxygen,
Nutrients, etc)
Solid grain
Volatilization /
Dissolution
Gas Phase
(organic components,
oxygen, nitrogen, etc)
Volatilization
DNAPL Phase
(non-aqueous,
organic
components)
Adapted from “Michigan Soil Vapor Extraction and Remediation (MISER) Model” by Ariola et. al, EPA/600/R-97/009, Sept. 1997
9
Hydrological Model
• The PDEs basically enforce mass balance
– Among the phases, α
– For the constituents within each phase, Ciα
• In words:
Time rate of change of mass =
Divergence of mass times velocity (momentum) +
All the different ways materials can move from one phase to
another and from one component to another
• Largely advection-diffusion physics
– Material moving due to flow of fluid (advecting)
– Material diffusing from regions of high to low concentration
10
Hydrological Model
• Key quantities
– Saturation (sα):
• Percent of pore space occupied by each phase
• We want to determine the saturation of DNAPL
– Concentration (Ciα):
• Mass per volume of component i in phase α.
• Will observe NAPL concentration downstream
– Relative permeability (kra):
• Normalized measure of ability of fluid to flow in a porous
medium
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Hydrological Model
• Number of constitutive relations required for closure
– Capillary pressure nonlinearly related to aqueous saturation
– Relative permeability related to saturation
• Result is a nonlinear, coupled set of partial differential
equations
– Lots of very interesting numerics, all well beyond me
– We use a well characterized code (MT3D, roots back to the
1980’s) as a black box for this task
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Petrophysical Models
• Presence of contaminant reflected differently in
different modalities
– ERT sensitive to electrical conductivity
– Hydrology data measures contaminant concentration
• Petrophysical model used to link the two
• We use Archie’s Law
–
–
–
–
s = as wj m s n
φ = porosity
Fit a, m, and n to data.
Very commonly used in petroleum industry
Many interesting issues
13
Joint Electrical and Hydrological Inversion
General Electrical Model
General Hydrological Model
A Petrophysical Model, Archie’s Law
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Sensitivity Calculations
• A key component of this type of inverse
problem is computing sensitivity (gradient)
information
• Either for gradient decent or quasi-Newton
type of optimization approaches
• Can be cumbersome for PDE-based models
where need e.g.,
¶v ( r )
¶s ( r )
and
¶c ( r )
¶s ( r )
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Sensitivity
• In discrete setting could try finite differences
¶v ( r ) v ( r;s i + d )
»
¶s i
d
• Requires one forward solve per pixel
• Alternative approach provided by adjoint-field
ideas
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ERT Adjoint Method
Source
at rs
Detector
at rd
¶v ( rd )
= ò ( Ñv ( r ) iÑvd ( r ))ds ( r ) dr
¶s ( r )
δσ = conductivity perturbation
Forward System
Ñ is Ñv = d ( r - rs )
Adjoint System
Ñ is Ñv = d ( r - rd )
• One forward solve per source and detector location
(more efficient)
• Derivation is messy: lots of Green’s theorem or
integration by parts
• Many related ideas (adjoint state space models, Born
approximation)
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Hydrology Adjoint Ideas
• Adjoint analysis not yet done for the full multiphase flow and transport problem
– For results in this talk, using finite differences
• Some initial results have been derived for
related problem: push-pull test
• Push aqueous tracers into formation
• Each tracer partitions differently in the
saturated contaminant
• Pull fluid from the formation
• Time history of recovered tracers reflect
saturation distribution
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Push Pull Model
State variables:
• Cw and Cn: Water/NAPL concentrations and their adjoint versions
Forward Model
k
æ
ö
¶Cw
1
+ k ç Cw Cn ÷ - Ñ i nSw Dij ÑCw - vCw
¶t
K eq ø
è
( (
)) = source
æ
ö
¶Cn
1
= k ç Cw Cn ÷
¶t
K eq ø
è
Adjoint Model
k
¶Cw
+ k Cw + Cn - Ñ inSw DijT ÑCw - nSw v ×ÑCw = adjoint source
¶(T - t )
(
)
n (1- Sw )
¶Cn
k
=
Cw + Cn
¶t
K eq
(
)
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Pixel Based vs. Level Set Method
Aghasi et al. (2011)
Pixel-Based Inversion
A level set function
Ill-posedness is an issue!
Low order texture models
Electrical Resistance Tomography
(ERT )
Level Set Method in More Detail
21
Parametric Level Set Method
22
A Flexible Parameterization
Using compactly supported functions (bumps) to
parameterize the level set function
Why use bumps?
By considering an -level set a relaxation to set operations is
achieved (a pseudo-logical property)
23
Advantages of Using the PaLS
Technique
• Low order and still highly flexible in shape representation
• No explicit need for any sort of regularization technique
• Implicitly benefiting from the smoothness of the RBFs
(regularization by parameterization)
• Offers the possibility of using high order minimization
methods such as Gauss-Newton techniques instead of
gradient descent methods
• Newton type methods are independent of variable scaling and
therefore robust against using different type of variables with
different orders of sensitivity
24
Joint Inversion: A Multi-Objective
Approach
Parameterization of the shape through the Parametric Level Set technique:
A simple approach to combining is scalarization:
25
25
Classic Newton Method
The inverse problem takes the form of a finite dimensional multi-objective
minimization problem
Classic Newton approach for minimization:
Single cost:
Determining a step at every iteration:
Desired to be
minimized
Quadratic
approximation
26
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Problem with Scalarization
Corresponding downstream
concentration
Water has limited capacity to dissolve DNAPL (saturation concentration)
saturation of a certain pixel
Representing the scalar cost as
the balance between the electrical and hydrological costs significantly alters
in the course of minimization
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Multi-objective Newton Method
Fliege et al., Newton’s Method for Multi-objective Minimization, SIAM Journal on Optimization, Vol 20,
Issue 2, pp 602-626, 2009
28
Minimization Problem to Determine
the Step
Convex Problem:
Equivalent Form:
This can be solved efficiently, facilitated by the low dimensionality of the
PaLS technique
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Examples
•
Realistic DNAPL release:
permeability fields generated
using sequential Gaussian
simulation (MVALOR3D)
•
Hydrological model: modified
MT3DMS with finite difference
approximation for PaLS
sensitivity calculations
•
ERT model: home grown 3D
finite difference code with
adjoint field method for
sensitivity
•
A parallel computing technique
used for the inversion
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Results Using a Single Level Set
Function
Initialization
Hydrology Only
ERT Only
Scalarization
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Results Using a Single Level Set
Function
Using the proposed algorithm:
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Results Using Two Level Set
Functions
Using two parametric level set functions, one for characterizing the source
zone ganglia and one for identifying the pools
Reconstruction
33
Conclusions
•
•
•
•
Considered physics-based approach for fusing highly
disparate data sets
PDE based models for both modalities as well as their
adjoint forms needed “in the loop”
Petrophysical model used to link the constitutive
properties across modalities
• Could also consider other types of prior models
For this application, value in inverting for quantities other
than pixels. Lots of fun with geometric parameterizations
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