DNAPL
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
EngineeringTufts University, Department of Civil and Environmental
EngineeringAcknowledgements2
Alireza AghasiPhD recipient, ECE
Linda AbriolaProf. Civil and Environmental EngineeringDean Tufts
School of Engineering
Tian TangPhD Student, Tufts CEE The ProblemThree 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 ChallengesLocating and estimating extent of source
massCharacterizing mass distributionInvasive, in-source
characterization methods may mobilize contaminantsIn-source
characterization methods too costly for application at most
sites5Approach: Characterization of DNAPL source zones using
(noninvasive) electrical and (down gradient transect) hydrological
measurements Overview of Measurement Modalities
Electrical Resistance Tomography (ERT):DNAPL causes change in
electrical conductivityInject current and measure voltagesInfer
electrical conductivityHydrologySaturated DNAPL directly dissolved
by flowing groundwater Measured downstream concentrationInfer
upstream saturationhysteretic infiltration and entrapment
simulations were performed using the University of Texas Chemical
Compositional Simulator (UTCHEM 9.0)5. Natural gradient dissolution
simulations were obtained by means of a modified version of the
modular three-dimensional transport simulator (MT3D)6, which uses
the Powers et al.7 mass transfer correlation. Model hydrogeologic
parameters were based on permeability realizations from a
contaminated aquifer in Oscoda, Michigan (Bachman Road site), which
consisted of relatively homogeneous glacial outwash sands8. Matrix
properties of the permeability realizations originally developed by
Lemke et al.,9 were modified by varying the correlation length
(integral scale), to encompass a range of geologic conditions. The
ERT model is based on insertion of electrical current into a medium
and measuring the electrical potential at the periphery of the
medium to analyze the electrical conductivity distribution
throughout the medium10. Based upon the conductivity contrast
between DNAPL and the surrounding groundwater, reconstruction of
the conductivity values can be a means of characterizing the source
zone.
Mathematical Models
Electrical conductivityCurrent source distributionElectrical
potentialMass concentration of component iThe inter-phase mass
exchange of component i from one phase to otherElectrical
Resistance TomographyFlow & Mass Transport Model
7
Saturation of the corresponding phase
hysteretic infiltration and entrapment simulations were
performed using the University of Texas Chemical Compositional
Simulator (UTCHEM 9.0)5. Natural gradient dissolution simulations
were obtained by means of a modified version of the modular
three-dimensional transport simulator (MT3D)6, which uses the
Powers et al.7 mass transfer correlation. Model hydrogeologic
parameters were based on permeability realizations from a
contaminated aquifer in Oscoda, Michigan (Bachman Road site), which
consisted of relatively homogeneous glacial outwash sands8. Matrix
properties of the permeability realizations originally developed by
Lemke et al.,9 were modified by varying the correlation length
(integral scale), to encompass a range of geologic conditions. The
ERT model is based on insertion of electrical current into a medium
and measuring the electrical potential at the periphery of the
medium to analyze the electrical conductivity distribution
throughout the medium10. Based upon the conductivity contrast
between DNAPL and the surrounding groundwater, reconstruction of
the conductivity values can be a means of characterizing the source
zone.
ERT ModelingPoissons equationDiscretize using finite difference
stencilLarge sparse system of linear equationsSolved either
directly (backslash) or using iterative methodBoundary conditions
always a problemExpand grid and use a zero BCContract grid and use
a complicated absorbing BC
8
Hydrological Model9Solid Phase(Organic components)Aqueous
Phase(Organics, Water, Oxygen, Nutrients, etc) DNAPL
Phase(non-aqueous, organic components)Sorption DissolutionGas
Phase(organic components, oxygen, nitrogen, etc)Volatilization
/DissolutionVolatilization Adapted from Michigan Soil Vapor
Extraction and Remediation (MISER) Model by Ariola et. al,
EPA/600/R-97/009, Sept. 1997Solid grainSolid grainSolid
grainAdvectionAqueousNAPLHydrological ModelThe PDEs basically
enforce mass balanceAmong the phases, For the constituents within
each phase, CiIn 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
anotherLargely advection-diffusion physicsMaterial moving due to
flow of fluid (advecting)Material diffusing from regions of high to
low concentration
10
Hydrological ModelKey quantitiesSaturation (s): Percent of pore
space occupied by each phaseWe want to determine the saturation of
DNAPLConcentration (Ci): Mass per volume of component i in phase .
Will observe NAPL concentration downstreamRelative permeability
(kra):Normalized measure of ability of fluid to flow in a porous
medium11
Hydrological ModelNumber of constitutive relations required for
closureCapillary pressure nonlinearly related to aqueous
saturationRelative permeability related to saturationResult is a
nonlinear, coupled set of partial differential equationsLots of
very interesting numerics, all well beyond meWe use a well
characterized code (MT3D, roots back to the 1980s) as a black box
for this task12
Petrophysical ModelsPresence of contaminant reflected
differently in different modalitiesERT sensitive to electrical
conductivityHydrology data measures contaminant
concentrationPetrophysical model used to link the twoWe use Archies
Law
= porosityFit a, m, and n to data. Very commonly used in
petroleum industryMany interesting issues13
General Electrical ModelA Petrophysical Model, Archies Law
Joint Electrical and Hydrological InversionGeneral Hydrological
Model
14The inverse problem associated with the hydrological modality
uses the down-gradient concentration data to invert for DNAPL
saturation values, while the inverse problem associated with the
geophysical modality inverts electrical data for the electrical
conductivity distribution in the imaging domain. To relate the two
modalities, a form of Archie's law2 is employed to link the
electrical properties of the medium to the saturation distribution.
The ultimate goal of this inversion is to extract the geometry of
the source zone region, along with a low order characterization of
the spatial variability within the region of contamination.
Sensitivity CalculationsA key component of this type of inverse
problem is computing sensitivity (gradient) informationEither for
gradient decent or quasi-Newton type of optimization approachesCan
be cumbersome for PDE-based models where need e.g.,
15
andSensitivityIn discrete setting could try finite
differences
Requires one forward solve per pixelAlternative approach
provided by adjoint-field ideas16
ERT Adjoint MethodOne forward solve per source and detector
location (more efficient)Derivation is messy: lots of Greens
theorem or integration by partsMany related ideas (adjoint state
space models, Born approximation)
17Source at rsDetector at rd
Forward SystemAdjoint System = conductivity
perturbationHydrology Adjoint IdeasAdjoint analysis not yet done
for the full multi-phase flow and transport problemFor results in
this talk, using finite differencesSome initial results have been
derived for related problem: push-pull test18
Push aqueous tracers into formationEach tracer partitions
differently in the saturated contaminantPull fluid from the
formationTime history of recovered tracers reflect saturation
distribution
Push Pull Model19Forward ModelAdjoint Model
State variables:Cw and Cn: Water/NAPL concentrations and their
adjoint versionsPixel Based vs. Level Set MethodIll-posedness is an
issue!
Pixel-Based InversionElectrical Resistance Tomography (ERT )
A level set function
Low order texture models
Aghasi et al. (2011)
The standard inversion approach to this problem would involve
the discretization of the imaging domain into a dense grid of
pixels and inversion for the saturation values associated with each
pixel. This approach results in a highly ill-posed inverse problem
that is further complicated by the need to choose suitable
regularization and associated regularization parameter(s). Here a
more geometric approach to the problem is considered, based on
level set concepts12, that is better suited both to the limited
information content in the data and the ultimate objective of the
problem; i.e., characterizing the region of the subsurface
contaminated by DNAPL. In a level set model the DNAPL saturation is
represented as where is the Heaviside step function and is a low
order texture representation for the DNAPL distribution within the
source zone. In the inverse problem, is usually replaced by a
smooth approximation. In the simplest case can be considered as a
scalar value representing the average DNAPL saturation in the
source zone. In the course of inversion, an initial function is
deformed to eventually reach a state such that its zero level set
represents the true boundary. Although this approach is
topologically flexible (i.e., requires no a priori assumption about
the number of connected components in the domain) and better poses
the problem by placing more emphasis on the geometry of the source
zone, it still requires discretizing in the whole imaging domain.
21
Level Set Method in More Detail22
Parametric Level Set MethodUsing compactly supported functions
(bumps) to parameterize the level set function By considering an
-level set a relaxation to set operations is achieved (a
pseudo-logical property)
A Flexible Parameterization
Why use bumps?
23
Advantages of Using the PaLS TechniqueLow order and still highly
flexible in shape representation
No explicit need for any sort of regularization
techniqueImplicitly 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
methodsNewton type methods are independent of variable scaling and
therefore robust against using different type of variables with
different orders of sensitivity2425Joint Inversion: A
Multi-Objective Approach
Parameterization of the shape through the Parametric Level Set
technique:A simple approach to combining is scalarization:
25Consider Mh to be the full hydrological model linking DNAPL
saturation values to downstream concentration observations and Me
the electrical model that relates the conductivity values to
electric potential measurements . The electrical conductivity and
saturation values are linked through the petro-physical relation .
In the PaLS approach, the DNAPL saturation is parameterized as
Sn(x,u) where u includes the PaLS parameters and probably some low
order texture parameters. Thus, the inverse problem takes the form
of a finite dimensional multi-objective minimization problem
26Classic 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 minimized26
Quadratic approximation27Problem with Scalarizationsaturation of
a certain pixelCorresponding downstream concentration Water has
limited capacity to dissolve DNAPL (saturation concentration)
Representing the scalar cost as the balance between the
electrical and hydrological costs significantly alters in the
course of minimization 27One of the main solution strategies in
multi-objective optimization problems is the scalarization
approach. In the current application, a scalarized version of the
multi-objective problem would be Ft=Fh+nFe where n is a scalar
usually chosen at the beginning of the minimization to make the two
cost terms comparable.In the context of DNAPL characterization, the
rate-limited dissolution process is governed by the equilibrium
concentration in water and is limited by the capacity of the
aqueous phase. Thus, as DNAPL saturation values increase within the
source region, the downstream concentration values eventually stop
monotonically increasing. Therefore in reconstructing relatively
high DNAPL values, the reduction rate of Fe may become more
significant than that of Fh when approaching the minima, resulting
in an unbalanced convergence. For this reason, a variant of the
multi- objective minimization technique13 is needed.Multi-objective
Newton Method
Fliege et al., Newtons Method for Multi-objective Minimization,
SIAM Journal on Optimization, Vol 20, Issue 2, pp 602-626,
200928
Minimization Problem to Determine the StepConvex Problem:
Equivalent Form: This can be solved efficiently, facilitated by the
low dimensionality of the PaLS technique29
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
30ExamplesResults Using a Single Level Set
FunctionInitializationERT Only31Hydrology OnlyScalarization
32Results Using a Single Level Set FunctionUsing the proposed
algorithm:
Results Using Two Level Set FunctionsReconstructionUsing two
parametric level set functions, one for characterizing the source
zone ganglia and one for identifying the pools
33
ConclusionsConsidered physics-based approach for fusing highly
disparate data setsPDE based models for both modalities as well as
their adjoint forms needed in the loopPetrophysical model used to
link the constitutive properties across modalitiesCould also
consider other types of prior modelsFor this application, value in
inverting for quantities other than pixels. Lots of fun with
geometric parameterizations
34The overall approach presented in this paper is based upon the
new parametric level set method, which enables rather complex
geometries and structures to be described using relatively few
parameters. Unlike conventional inversion methods, which rely on
the inversion of the desired physical property values at a dense
grid, this model-based approach significantly reduces the number of
underlying parameters in the inversion. By posing the problem in
this geometric framework, we are able to employ full 3D models as
the basis for inversion for both the ERT and the multi-phase flow
and transport. Additionally the newly developed PaLS method enables
the joint inversion approach to have a denser data set
corresponding to two different modalities (down gradient
concentration and cross-hole ERT), an inversion that is almost
impossible to conduct using traditional pixel-based
techniques.Compared to alternative approaches, such as partitioning
interwell tracer tests, the combined use of ERT and down gradient
concentration transect measurements for DNAPL source zone
characterization is both less costly and less invasive, reducing
the risk of contaminant mobilization. Prior to this research,
however, the spatial complexity and low contrast and non-uniqueness
issues associated with interpretation of these measurements greatly
hampered their successful application in source zone
characterization. The reconstruction results presented here are
very promising and demonstrate that this new method is capable of
handling inverse problems that use two different types of
modalities of data sets, which makes it a well-suited, and
computationally tractable, choice for complex inversion problems
like characterization of DNAPL sources.