Abstract. Backward location and travel time probabilities can be used to determine the prior location of contamination in an aquifer. For a contaminant particle that was detected in an aquifer, the backward location probability is the probability of where the particle was located at some prior time. Backward travel time probability is the probability of when the particle was located at some position upgradient of the detection. These probabilities can be used to improve characterization of known sources of groundwater contamination, to identify previously unknown contamination sources, and to delineate capture zones. For simple model domains, backward probabilities can be obtained heuristically from a forward model of contaminant transport. For multidimensional problems and complex domain geometries, the heuristic approach is difficult to implement and verify. The adjoint method provides a formal approach for obtaining backward probabilities for all model domains and geometries. We formally show that the backward model probabilities are adjoint states of resident concentration. We provide a methodology for obtaining the governing equations and boundary and final conditions for these probabilities. The approach is illustrated using a one-dimensional, semi-infinite domain that mimics flow to a production well, and these results are compared to equivalent probabilities derived heuristically. IntroductionTransport of a conservative solute in groundwater is usually described by the advection-dispersion equation ( The forward ADE can also be used to solve for location probability and travel time probability. If we consider an individual solute parcel that was released from the contaminant source, then the location probability of that parcel is the probability that it is located at a given position in space at some later time IDagan, 1989; Jury and Roth, 1990; Chin and Chittaluru, 1994]. Location probability is related to resident concentration [Jury and Roth, 1990]. Resident concentration measures the mass of solute at a given location in space at a snapshot in time. If the resident concentration measurements are normalized by the total mass of solute in the system, the resulting distribution is the percentage of the total mass that is at a given location in space. Suppose we are interested in the present location of one parcel of mass that was input at the Copyright 1999 by the American Geophysical Union. Paper number 1999WR900190.0043-1397/99/1999WR900190509.00 source. The parcel is more likely to be found at a location that has a high solute concentration (or, equivalently, a high normalized solute concentration) than a location that has a low solute concentration. Thus, at any point in time, the normalized concentration distribution is equivalent to a probability density function for the location of the parcel (i.e., location probability). Note that for a unit source, the resulting resident concentration is equal to the location probability.If we again consider an individual solute parcel that was released from th...
Abstract. Backward location and travel time probabilities can be used to determine the former location of contamination in an aquifer. For a contaminant parcel that was detected in an aquifer the backward location probability describes its position at some time prior to sampling, and the backward travel time probability describes the amount of time required for it to travel to the sampling location from some upgradient position. These probabilities, which can provide information about the source of contamination, are related to adjoint states of resident concentration. The governing equations of the backward probabilities are adjoints of the forward governing equation, e.g., the advectiondispersion equation. We derive these backward governing equations and their boundary and final conditions for both location and travel time probabilities in a multidimensional system. Each governing equation contains the adjoint of the advection-dispersion operator and a load term that defines the particular adjoint state (probability). The load term depends on both the type of probability (location or travel time) and the sampling device (pumping well or monitoring well) with which the contamination was detected. The adjoint equation can also be used to efficiently determine forward location and travel time probabilities describing the future location of groundwater contamination, a feature most useful for delineating pumping well captures zones. We illustrate the use of the backward model for obtaining location and travel time probabilities in a hypothetical twodimensional domain. IntroductionWhen contamination is detected in an aquifer, the source of contamination is often unknown. To address aquifer remediation or to assign responsibility, we might want to answer questions such as the following: "Where is the contamination source?" or "When was the contamination released from the source?" A common modeling approach to answer these questions is to run one forward simulation for each potential source, modeling the movement of the contamination away from the source (forward modeling). If the number of potential sources is large, this approach can result in a significant computational burden since one simulation must be run for each potential source. Furthermore, all potential sources must be identified.Backward modeling [Wilson and Liu, 1994, 1997, available at http://www.engg.ksu.edu/HSRC/96Proceed/wilson.html] is a more efficient approach for answering these questions. With backward modeling we solve for the probability of the former position of the contamination (location probability) or for the probability of the contaminant's travel time from some upgradient location to the sampling location (travel time probability). With one backward simulation we obtain these probabilities for all locations. This reduces the computational burden of forward modeling and is not limited to the pre-identified pos-
[1] This paper proposes a new approach to the hydraulics of in situ groundwater remediation. In situ remediation promotes reactions between an injected treatment solution and the contaminated groundwater, but without a hydraulic mechanism to promote spreading, the laminar flows characteristic of porous media will keep the two fluids in approximately the same relative configuration as they travel through the aquifer, limiting the opportunity for reactions to occur. To address this fundamental limitation, this paper borrows a key result from the fluid mechanics literature : Spreading in laminar flows is optimized by chaotic advection. Previous studies have applied this result to groundwater remediation using the pulsed dipole model, but that model depends on reinjection of fluid, which presents a number of theoretical and practical limitations. Accordingly, this paper proposes a new conceptual model for plume spreading by chaotic advection, using an engineered sequence of extractions and injections of clean water at an array of wells, which generates plume spreading by stretching and folding the fluid interface between the injected treatment solution and the contaminated groundwater but does not require reinjection. The paper includes an overview of the analytical techniques-Poincaré sections, periodic points, stable and unstable manifolds, heteroclinic points, and Lyapunov exponents-used to demonstrate chaotic advection in the limiting case in which diffusion is negligible. Numerical simulations show that spreading by stretching and folding is complimentary to spreading resulting from aquifer heterogeneity.
During in situ remediation of contaminated groundwater, a treatment solution is often injected into the contaminated region to initiate reactions that degrade the contaminant. Degradation reactions only occur where the treatment solution and the contaminated groundwater are close enough that mixing will bring them together. Degradation is enhanced when the treatment solution is spread into the contaminated region, thereby increasing the spatial extent of mixing and degradation reactions. Spreading results from local velocity variations that emerge from aquifer heterogeneity and from spatial variations in the external forcings that drive flow. Certain patterns in external forcings have been shown to create chaotic advection, which is known to enhance spreading of solutes in groundwater flow and other laminar flows. This work uses numerical simulations of flow and reactive transport to investigate how aquifer heterogeneity changes the qualitative and quantitative aspects of chaotic advection in an aquifer, and the extent to which these changes enhance contaminant degradation. We generate chaotic advection using engineered injection and extraction (EIE), an approach that uses sequential injection and extraction of water in wells surrounding the contaminated region to create time-dependent flow fields that promote plume spreading. We demonstrate that as the degree of heterogeneity increases, both plume spreading and contaminant degradation increase; however, the increase in contaminant degradation is small relative to the increase in plume spreading. Our results show that the combined effects of EIE and heterogeneity produce substantially more stretching than either effect separately.
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