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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

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Acknowledgements 2 Alireza Aghasi PhD recipient, ECE Linda Abriola Prof. Civil and Environmental Engineering Dean Tufts School of Engineering Tian Tang PhD Student, Tufts CEE

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The Problem Three Mile Island (1979)Love Canal (1978) 3

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The Problem Characterization of DNAPL (Dense Non-Aqueous Phase Liquid) source zones based on electrical and hydrological measurements

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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 5 Approach: Characterization of DNAPL source zones using (noninvasive) electrical and (down gradient transect) hydrological measurements

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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

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Mathematical Models Electrical conductivity Current source distribution Electrical potential Mass concentration of component i The inter-phase mass exchange of component i from one phase to other Electrical Resistance Tomography Flow & Mass Transport Model 7 Saturation of the corresponding phase

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ERT Modeling 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

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Hydrological Model 9 Solid Phase (Organic components) Solid Phase (Organic components) Aqueous Phase (Organics, Water, Oxygen, Nutrients, etc) Aqueous Phase (Organics, Water, Oxygen, Nutrients, etc) DNAPL Phase (non-aqueous, organic components) DNAPL Phase (non-aqueous, organic components) Sorption Dissolution Gas Phase (organic components, oxygen, nitrogen, etc) Gas Phase (organic components, oxygen, nitrogen, etc) Volatilization / Dissolution Volatilization Adapted from “Michigan Soil Vapor Extraction and Remediation (MISER) Model” by Ariola et. al, EPA/600/R-97/009, Sept. 1997 Solid grain Advection Aqueous NAPL

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Hydrological Model The PDEs basically enforce mass balance – Among the phases, α – For the constituents within each phase, C i α 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

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Hydrological Model Key quantities – Saturation (s α ): Percent of pore space occupied by each phase We want to determine the saturation of DNAPL – Concentration (C i α ): Mass per volume of component i in phase α. Will observe NAPL concentration downstream – Relative permeability (k ra ): Normalized measure of ability of fluid to flow in a porous medium 11

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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 12

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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 – φ = porosity – Fit a, m, and n to data. – Very commonly used in petroleum industry – Many interesting issues 13

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General Electrical Model A Petrophysical Model, Archie’s Law Joint Electrical and Hydrological Inversion General Hydrological Model 14

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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., 15 and

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Sensitivity In discrete setting could try finite differences Requires one forward solve per pixel Alternative approach provided by adjoint-field ideas 16

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ERT Adjoint Method 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) 17 Source at r s Detector at r d Forward SystemAdjoint System δσ = conductivity perturbation

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Hydrology Adjoint Ideas Adjoint analysis not yet done for the full multi- phase flow and transport problem – For results in this talk, using finite differences Some initial results have been derived for related problem: push-pull test 18 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 19 Forward Model Adjoint Model State variables: C w and C n : Water/NAPL concentrations and their adjoint versions

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Pixel Based vs. Level Set Method Ill-posedness is an issue! Pixel-Based Inversion Electrical Resistance Tomography (ERT ) A level set function Low order texture models Aghasi et al. (2011)

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21 Level Set Method in More Detail

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22 Parametric Level Set Method

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Using 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

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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

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25 Joint Inversion: A Multi-Objective Approach Parameterization of the shape through the Parametric Level Set technique: A simple approach to combining is scalarization: 25

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26 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 26 Quadratic approximation

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27 Problem with Scalarization saturation of a certain pixel Corresponding 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 27

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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

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Minimization Problem to Determine the Step Convex Problem: Equivalent Form: This can be solved efficiently, facilitated by the low dimensionality of the PaLS technique 29

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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 30 Examples

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Results Using a Single Level Set Function Initialization ERT Only 31 Hydrology OnlyScalarization

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32 Results Using a Single Level Set Function Using the proposed algorithm:

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Results Using Two Level Set Functions Reconstruction Using two parametric level set functions, one for characterizing the source zone ganglia and one for identifying the pools 33

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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 34

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