Abstract. The reliable assessment of hazards or risks arising from water contamination problems and the design of efficient and effective techniques to mitigate these problems require the capability to predict the behavior of chemical contaminants in flowing water. Most attempts at quantifying contaminant transport have relied on a solution of some form of the advection-dispersion-reaction equation. In this paper, the Backward Beam Equation (BBE) method is studied and enhanced to solve the Advection-Dispersion Equation (ADE) within a contaminant source identification context. Even though the BBE has been applied successfully to parabolic problems before, it has never been applied to solving the ADE with heterogeneous parameters. The BBE employed in this work is capable of recovering the time history and spatial distribution of a groundwater contaminant plume from measurements of its current position. Using examples involving deterministic heterogeneous dispersion coefficients, we show that the method is robust enough to handle heterogeneous parameters. By altering the method, to produce a hybrid between a marching and a jury method called the Marching-Jury Backward Beam Equation (MJBBE), we were also able to make the problem practical to solve. IntroductionGroundwater amounts to about half of the U. Furthermore, by accurately identifying pollution sources, the time and cost requirements associated with the complex and lengthy process of remediation can be dramatically reduced. Most attempts at quantifying contaminant transport have relied on solving some form of a well-known governing equation referred to as advection-dispersion-reaction equation. In identifying the source of pollution we have to solve the governing equations backward in time. Modeling contaminant transport using reverse time is an ill-posed problem since the process, being dispersive, is irreversible. Because of this illposedness, the problems have discontinuous dependence on data and are sensitive to the errors in data. A problem is categorized as a well-posed problem if (1) the solution exists, (2) the solution is unique, and (3) the problem is stable [Tikhonov and Arsenin, 1977]. Problems that do not satisfy these criteria are called ill-posed. For the groundwater contamination problem the plume has to have originated from someplace, therefore, physically, the plume exists. However, in mathematically rigorous terms, the fact that we have a presentday plume concentration does not necessarily mean that we satisfy the existence criterion. The solution exists only when we have perfect and consistent model and data that satisfy extremely restrictive conditions. Meeting the stability criterion is a difficult task to accomplish since numerical schemes that are 2113
The reliable assessment of hazards or risks arising from groundwater contamination problems and the design of ecient and eective techniques to mitigate these problems require the capability to predict the behavior of chemical contaminants in¯owing water. Most attempts at quantifying contaminant transport have relied on a solution of some form of a well-known governing equation referred to as advection-dispersion-reaction equation. To choose an appropriate remediation strategy, knowledge of the contaminant release source and time release history becomes pertinent. As additional contaminated sites are being detected, it is almost impossible to perform exhaustive drilling, testing, and chemical ®ngerprint analysis every time. Moreover, chemical ®ngerprinting and site records are not sucient to allow a unique solution for the timing of source releases. The purpose of this paper is to present and review mathematical methods that have been developed during the past 15 years to identify the contaminant source location and recover the time release history.# 2001 AEHS
[1] In this paper, a comparison between the marching-jury backward beam equation (MJBBE) and the quasi-reversibility (QR) methods to perform hydrologic inversion and, more specifically, to reconstruct conservative contaminant plume spatial distributions is presented. The MJBBE, developed by Atmadja and Bagtzoglou [2000, 2001a], was used to recover contaminant spatial distributions in heterogeneous porous media, while the QR method, first applied to groundwater contamination problems by Skaggs and Kabala [1995], was modified to incorporate heterogeneity and explicitly handle the advective term of the transport equation. Spatially uncorrelated and correlated, stationary and nonstationary, heterogeneous dispersion coefficient fields were generated using the Bayesian nearest neighbor method (BNNM). Homogeneous and deterministically heterogeneous cases are also presented for comparison. In addition, contaminant plume initial data with uncertainty were also analyzed using the MJBBE and QR methods. The MJBBE is found to be robust enough to handle highly heterogeneous parameters and is able to preserve the salient features of the initial input data. On the other hand, the QR method is superior in handling cases with homogeneous parameters and with initial data that are plagued by uncertainty but it performs very poorly in cases with heterogeneous media.
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