Roseanna M. Neupauer

Adjoint method for obtaining backward-in-time location and travel time probabilities of a conservative groundwater contaminant

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.

Adjoint‐derived location and travel time probabilities for a multidimensional groundwater system

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 advection-dispersion 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 two- dimensional domain.

Comparison of inverse methods for reconstructing the release history of a groundwater contamination source.

Inverse methods can be used to reconstruct the release history of a known source of groundwater contamination from concentration data describing the present-day spatial distribution of the contaminant plume. Using hypothetical release history functions and contaminant plumes, we evaluate the relative effectiveness of two proposed inverse methods, Tikhonov regularization (TR) and minimum relative entropy (MRE) inversion, in reconstructing the release history of a conservative contaminant in a one-dimensional domain [Skaggs and Kabala, 1994; Woodbury and Ulrych, 1996]. We also address issues of reproducibility of the solution and the appropriateness of models for simulating random measurement error. The results show that if error-free plume concentration data are available, both methods perform well in reconstructing a smooth source history function. With error-free data the MRE method is more robust than TR in reconstructing a nonsmooth source history function; however, the TR method is more robust if the data contain measurement error. Two error models were evaluated in this study, and we found that the particular error model does not affect the reliability of the solutions. The results for the TR method have somewhat greater reproducibility because, in some cases, its input parameters are less subjective than those of the MRE method; however, the MRE solution can identify regions where the data give little or no information about the source history function, while the TR solution cannot.

An iterative approach to multi-objective engineering design

This study contributes an iterative problem reformulation technique for multi-objective evolutionary algorithm (MOEA) decision support. Problem formulations consist of objectives, decision variables, and constraints, and directly influence the results generated by the MOEA. Typically, design problems are optimized based on a single problem formulation established a priori. In this paper, we demonstrate an approach to perform iterative optimization using problem formulations updated from analyses of results from prior rounds of optimization, which often reveal design components not initially considered. To demonstrate the approach, we consider a novel groundwater remediation technique, Engineered Injection and Extraction (EIE), which has never been optimized in the literature. Iterative problem reformulation enabled the MOEA to generate EIE solutions with better performance than the heuristically-developed solution used in prior work. We optimize a groundwater remediation strategy using multi-objective optimization.We demonstrate an iterative approach to adapt the problem formulation.We couple visualizations of objective and decision space to analyze solutions.

Chaotic advection and reaction during engineered injection and extraction in heterogeneous porous media

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.

Backward fractional advection dispersion model for contaminant source prediction

The forward Fractional Advection Dispersion Equation (FADE) provides a useful model for non-Fickian transport in heterogeneous porous media. The space FADE captures the long leading tail, skewness, and fast spreading typically seen in concentration profiles from field data. This paper develops the corresponding backward FADE model, to identify source location and release time. The backward method is developed from the theory of inverse problems, and then explained from a stochastic point of view. The resultant backward FADE differs significantly from the traditional backward Advection Dispersion Equation (ADE) because the fractional derivative is not self-adjoint and the probability density function for backward locations is highly skewed. Finally, the method is validated using tracer data from a well-known field experiment, where the peak of the backward FADE curve predicts source release time, while the median or a range of percentiles can be used to determine the most likely source location for the observed plume. The backward ADE cannot reliably identify the source in this application, since the forward ADE does not provide an adequate fit to the concentration data.

Backward probabilistic modeling to identify contaminant sources in water distribution systems.

If a chemical or biological agent is released into a water distribution system, sensors that are installed in the pipe network may detect the contamination as it travels through the system. To minimize the adverse impact of the contaminant release, the source must be characterized to determine the extent of the contamination and to remediate the contaminated area. We present a backward modeling approach that uses the data collected by the sensors to obtain probability density functions that describe the random time in the past that the observed contamination was at a particular upgradient position. These probability density functions can be used to identify the source node and release time. The approach is developed for steady flow conditions with known system demands and for a single, instantaneous source of contamination. Using a hypothetical water distribution system and release scenario, we demonstrate that the backward model is an efficient and effective approach for identifying the source node and t...

Optimal design of active spreading systems to remediate sorbing groundwater contaminants in situ

The effectiveness of in situ remediation to treat contaminated aquifers is limited by the degree of contact between the injected treatment chemical and the groundwater contaminant. In this study, candidate designs that actively spread the treatment chemical into the contaminant are generated using a multi-objective evolutionary algorithm. Design parameters pertaining to the amount of treatment chemical and the duration and rate of its injection are optimized according to objectives established for the remediation - maximizing contaminant degradation while minimizing energy and material requirements. Because groundwater contaminants have different reaction and sorption properties that influence their ability to be degraded with in situ remediation, optimization was conducted for six different combinations of reaction rate coefficients and sorption rates constants to represent remediation of the common groundwater contaminants, trichloroethene, tetrachloroethene, and toluene, using the treatment chemical, permanganate. Results indicate that active spreading for contaminants with low reaction rate coefficients should be conducted by using greater amounts of treatment chemical mass and longer injection durations relative to contaminants with high reaction rate coefficients. For contaminants with slow sorption or contaminants in heterogeneous aquifers, two different design strategies are acceptable - one that injects high concentrations of treatment chemical mass over a short duration or one that injects lower concentrations of treatment chemical mass over a long duration. Thus, decision-makers can select a strategy according to their preference for material or energy use. Finally, for scenarios with high ambient groundwater velocities, the injection rate used for active spreading should be high enough for the groundwater divide to encompass the entire contaminant plume.

Bounded fractional diffusion in geological media: Definition and Lagrangian approximation

Spatiotemporal Fractional-Derivative Models (FDMs) have been increasingly used to simulate non-Fickian diffusion, but methods have not been available to define boundary conditions for FDMs in bounded domains. This study defines boundary conditions and then develops a Lagrangian solver to approximate bounded, one-dimensional fractional diffusion. Both the zero-value and non-zero-value Dirichlet, Neumann, and mixed Robin boundary conditions are defined, where the sign of Riemann-Liouville fractional derivative (capturing non-zero-value spatial-nonlocal boundary conditions with directional super-diffusion) remains consistent with the sign of the fractional-diffusive flux term in the FDMs. New Lagrangian schemes are then proposed to track solute particles moving in bounded domains, where the solutions are checked against analytical or Eularian solutions available for simplified FDMs. Numerical experiments show that the particle-tracking algorithm for non-Fickian diffusion differs from Fickian diffusion in relocating the particle position around the reflective boundary, likely due to the nonlocal and non-symmetric fractional diffusion. For a non-zero-value Neumann or Robin boundary, a source cell with a reflective face can be applied to define the release rate of random-walking particles at the specified flux boundary. Mathematical definitions of physically meaningful nonlocal boundaries combined with bounded Lagrangian solvers in this study may provide the only viable techniques at present to quantify the impact of boundaries on anomalous diffusion, expanding the applicability of FDMs from infinite domains to those with any size and boundary conditions. This article is protected by copyright. All rights reserved.

Adjoint Simulation of Stream Depletion Due to Aquifer Pumping

If an aquifer is hydraulically connected to an adjacent stream, a pumping well operating in the aquifer will draw some water from aquifer storage and some water from the stream, causing stream depletion. Several analytical, semi-analytical, and numerical approaches have been developed to estimate stream depletion due to pumping. These approaches are effective if the well location is known. If a new well is to be installed, it may be desirable to install the well at a location where stream depletion is minimal. If several possible locations are considered for the location of a new well, stream depletion would have to be estimated for all possible well locations, which can be computationally inefficient. The adjoint approach for estimating stream depletion is a more efficient alternative because with one simulation of the adjoint model, stream depletion can be estimated for pumping at a well at any location. We derive the adjoint equations for a coupled system with a confined aquifer, an overlying unconfined aquifer, and a river that is hydraulically connected to the unconfined aquifer. We assume that the stage in the river is known, and is independent of the stream depletion, consistent with the assumptions of the MODFLOW river package. We describe how the adjoint equations can be solved using MODFLOW. In an illustrative example, we show that for this scenario, the adjoint approach is as accurate as standard forward numerical simulation methods, and requires substantially less computational effort.