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

Abstract

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-

Using data assimilation method to calibrate a heterogeneous conductivity field and improve solute transport prediction with an unknown contamination source

“…This is a unique property of the data assimilation method, not shared by conventional inverse methods that use calibration data in the batch mode. It is worth mentioning that in this study, we focus on gradually reduce influence of unknown source term on transport predictions using the EnKF, do not intend to characterize the source term as in the backward probability model of Neupauer and Wilson (2001) and Neupauer and Lin (2006).…”

Simultaneous identification of a single pollution point-source location and contamination time under known flow field conditions

“…As was formally shown by Neupauer and Wilson [16,17] and later by Cornaton and Perrochet [6,7], the forward pdf F q evaluated at a downstream location x 1 after t i of forward transport in response to a unit mass-release at an upstream location x 0 is identical to the backward pdf F Àq evaluated at the upstream location x 0 after s i of backward transport in response to a unit mass-release at x 1 . This leads to the very useful and fundamental property.…”

“…5,16,21,28]. In this context, the work of Neupauer and Wilson [16,17] is particularly relevant. They combine travel time and location probability density functions as used in the transfer function theory [11] with an adjoint method for conservative tracers.…”

“…and the backward pdf F Àq with an initial unit-mass input at a point x 1 is a solution of the transport equation with a reversed flow field [16,17] …”

Linear Programming Method for Investigating the Disposal Histories and Locations of Pollutant Sources in an Aquifer

“…Likewise, Yeh (1990a and1990b), Wilson and Liu (1996), Neupauer and Wilson (2001), Snodgrass and Kitanidis (1997), and Skaggs and Kabala (1994) used mathematiccal inverse methods to identify the contamination sources. Neupauer et al (2000) compared the relative effectiveness of the methods from studies of Skaggs and Kabala (1994) and Woodbury and Ulrych (1996).…”