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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

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Physics based forward modeling for inverse methods

Physics Based Forward Modeling for Inverse Methods

AlirezaAghasi%, 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


Alireza Aghasi

PhD recipient, ECE

Linda Abriola

Prof. Civil and Environmental Engineering

Dean Tufts School of Engineering

Tian Tang

PhD Student, Tufts CEE

The problem
The Problem

Three Mile Island (1979)

Love Canal (1978)

Physics based forward modeling for inverse methods

The Problem

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

The challenges
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


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

Physics based forward modeling for inverse methods

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

Physics based forward modeling for inverse methods

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

Ert modeling
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

Hydrological model
Hydrological Model

Solid Phase

(Organic components)


Solid grain

Solid grain



Solid grain



Aqueous Phase

(Organics, Water, Oxygen, Nutrients, etc)

Volatilization /Dissolution

Gas Phase

(organic components, oxygen, nitrogen, etc)


(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

Hydrological model1
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

Hydrological model2
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 iin phase α.

      • Will observe NAPL concentration downstream

    • Relative permeability (kra):

      • Normalized measure of ability of fluid to flow in a porous medium

Hydrological model3
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

Petrophysical models
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

Physics based forward modeling for inverse methods

Joint Electrical and Hydrological Inversion

General Electrical Model

General Hydrological Model

A Petrophysical Model, Archie’s Law

Sensitivity calculations
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.,



  • In discrete setting could try finite differences

  • Requires one forward solve per pixel

  • Alternative approach provided by adjoint-field ideas

Ert adjoint method
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)

Sourceat rs

Detector at rd

δσ = conductivity perturbation

Forward System

Adjoint System

Hydrology adjoint ideas
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

  • 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

Push pull model
Push Pull Model

  • State variables:

  • Cw and Cn: Water/NAPL concentrations and their adjoint versions

Forward Model

Adjoint Model

Physics based forward modeling for inverse methods

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 )

Physics based forward modeling for inverse methods

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)

Physics based forward modeling for inverse methods

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

Physics based forward modeling for inverse methods

Joint Inversion: A Multi-Objective Approach

Parameterization of the shape through the Parametric Level Set technique:

A simple approach to combining is scalarization:


Physics based forward modeling for inverse methods

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


Physics based forward modeling for inverse methods

Problem with Scalarization

Water has limited capacity to dissolve DNAPL (saturation concentration)

Corresponding downstream 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


Physics based forward modeling for inverse methods

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

Physics based forward modeling for inverse methods

Minimization Problem to Determine the Step

Convex Problem:

Equivalent Form:

This can be solved efficiently, facilitated by the low dimensionality of the PaLS technique

Physics based forward modeling for inverse methods


  • 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

Results using a single level set function
Results Using a Single Level Set Function


ERT Only

Hydrology Only


Results using a single level set function1
Results Using a Single Level Set Function

  • Using the proposed algorithm:

Results using two level set functions
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



  • 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