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.