General Backward Model to Identify the Source for Contaminants Undergoing Non-Fickian Diffusion in Water.

Pollutant source identification (PSI) has been conducted for four decades for tracking Fickian diffusive pollutants, while PSI for non-Fickian diffusion, well-documented in aquifers and rivers, requires novel, predictive models. To enable PSI for non-Fickian diffusive pollutants, this study derived a general backward model using the fractional-adjoint approach in sensitivity analysis for dissolved contaminants with transport governed by the spatiotemporal fractional advection-dispersion equation (fADE). The backward fADE contains a self-adjoint time-fractional term for subdiffusion and direction-dependent, non-self-adjoint space-fractional terms for superdiffusion. Field applications showed that the resultant backward location probability density function identified the point source location in all three test cases, one alluvial aquifer and two rivers. The backward model and boundary conditions derived in this study made it possible to reliably and efficiently backtrack pollutants (and may include other constituents, such as bedload) undergoing mixed sub- and superdiffusion in natural aquatic systems. The classical PSI model, however, underestimated the source location since it did not account for solute retention and preferential flow. In addition, the measured tracer snapshots (if available before PSI) can enhance the parameter predictability and improve the applicability of backward fADE PSI. Most importantly, a spatially variable dispersion coefficient is needed in the backward fADE since PSI is most likely scale dependent in natural hydrologic systems.

Evaluating backward probability model under various hydrogeologic and hydrologic conditions.

Contaminant source identification improves the understanding of contaminant source characteristics including location and release time, which can lead to more effective remediation and water resources management plans. The backward probability model can provide probabilities of source locations and release times under various contaminant properties and hydrogeologic conditions. The backward probability model has been applied to numerous synthetic and real contamination sites for locating possible contaminant sources, but it is also important to evaluate the reliability of the backward probability model through rigorous verification analyses. Here, we present a model verification framework for the backward probability model using a stepwise approach from simple to complex model settings: comparison with previous studies, transient saturated flow under various hydrogeologic conditions, and transient variably-saturated flow conditions. As a simple condition, one-dimensional homogeneous problems under steady-state and transient flow conditions were verified by comparing with previous studies. Model verifications with complex conditions were conducted by comparing forward and backward probability simulation results. The verification results demonstrate that the backward probability model performs well for homogeneous problems. For heterogeneous problems, the backward probability model results in slightly different backward travel times due to differences in solute decay and boundary conditions assigned for both forward and backward probability simulations, but the backward travel time at the maximum probability can be reproduced well.

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.

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.

Numerical implementation of a backward probabilistic model of ground water contamination.

Backward location and travel time probabilities can be used to characterize known and unknown sources or prior positions of ground water contamination. Backward location probability describes the position of the observed contamination at some time in the past; backward travel time probability describes the amount of time prior to observation that the contamination was released from its source or was at a particular upgradient location. The governing equation for backward probabilities is the adjoint of the governing equation for contaminant transport, but with new load terms. Numerical codes that have been written to solve the forward equations of contaminant transport, e.g., the advection-dispersion equation, can also be used to solve the adjoint equation for location and travel time probabilities; however, the interpretation of the results is different and some new approximations must be made for the load terms. We present the governing equations for backward location and travel time probabilities, and provide appropriate numerical approximations for these load terms using the cell-centered finite difference method, one of the most popular numerical methods in ground water hydrology. We discuss some additional numerical considerations for the backward model including boundary conditions, reversal of the flow field, and interpretation of the results. We illustrate the implementation of the backward probability model using hypothetical examples in one- and two-dimensional domains. We also present a three-dimensional application of a pump-and-treat remediation capture zone delineation at the Massachusetts Military Reservation. The illustrations are performed using MODFLOW-96 for flow simulations and MT3DMS for transport simulations.

Backward location and travel time probabilities for a decaying contaminant in an aquifer.

Backward location and travel time probabilities can be used to determine the prior position of contamination in an aquifer. These probabilities, which are related to adjoint states of concentration, can be used to improve characterization of known sources of groundwater contamination, to identify previously unknown contamination sources, and to delineate capture zones. The first contribution of this paper is to extend the adjoint model to the case of a decaying solute (first-order decay), and to describe two different interpretations of backward probabilities. The conventional interpretation accounts for the probability that a contaminant particle could decay before reaching the detection location. The other interpretation is conditioned on the fact that the detected contaminant particle actually reached the detection location, despite this possibility of decay. In either case, travel time probabilities are skewed toward earlier travel times, relative to a conservative solute. The second contribution of this paper is to verify the load term for a monitoring well observation. We provide examples using one-dimensional models and hypothetical aquifers. We employ an infinite domain in order to verify the monitoring well load. This new but simple one-dimensional adjoint solution can also be used to verify higher-dimensional numerical models of backward location and travel time probabilities. We employ a semi-infinite domain to illustrate the effect of decay on backward models of pumping well probabilistic capture zones. Decay causes the capture zones to fall closer to the well.