UQ and Decision Making for Groundwater Contamination: A Measure-Theoretic Approach

Abstract:

The movement of contaminant plumes in underground aquifers is highly dependent on many hydrogeological parameters. We model the transport with an advection, diffusion, reaction system requiring the specification of porosity, flow direction, flow speed, dispersivities, and effects of geochemical reactions. It is often prohibitively expensive or impossible to make accurate and reliable measurements of these parameters in the field. It is also difficult to know the position and shape of a contaminant plume at a given time or the exact details of the source of the contamination, e.g. size, location, origin time, and magnitude. If decisions are to be made regarding contaminant remediation strategies or predictions of future contaminant concentrations in and near water-supply wells, then these uncertain hydrogeological and source parameters need to be analyzed and estimated. We utilize a measure-theoretic framework to formulate and solve the physics-based stochastic inverse problem to quantify the uncertainty for these parameters.
We solve the model using both analytical and finite element solutions. We define quantities of interest (QoI) for the groundwater contaminant problem in terms of observable field measurements. We develop adjoint problems to compute accurate and reliable a posteriori error estimates of the QoIs. The adjoint solutions are also useful in the solution of the inverse problem. The measure-theoretic formulation and solution of the inverse problem and modeling framework define a solution as a probability measure on the parameter domain. In the typical case where the number of output quantities is less than the number of parameters, the inverse of the map from parameters to data defines a type of generalized contour map where the geometry plays a pivotal role in determining an optimal set of QoI. We determine and analyze solutions for geometrically distinct QoI defining reduced-dimension set-valued inverses for this measure-theoretic inverse framework.