Geostatistical solution to the inverse problem using surrogate functions for remediation of shallow aquifers

Pump-and-treat (PAT) techniques are often applied to the remediation of dissolved chemicals from shallow aquifers. A related management problem typically consists of the selection of the pumping strategy and the most appropriate treatment method, in order to minimize the total cleanup cost while meeting a set of technical, economic and social constraints. However, due to scarcity of information about the hydrogeological system, stochastic modeling approaches seem more appropriate. Of primary concern is the inherent spatial variability of hydraulic conductivity. In general, the implementation of a remediation strategy assessed based on uncertain hydrogeological parameters leads to a decision involving the risk of constraint violations. The decision-making process may then be formulated as a multiobjective optimization framework where the optimality of a pumping pattern is traded off against its reliability. Operations may be structured into a stochastic optimal control problem, in which the remediation strategy is sequentially updated based upon new measurements collected during the actual cleanup process. The procedure requires the implementation of an inverse simulation model to estimate the stochastic hydrogeological parameters based on a set of potential measurements. In this work, we follow a geostatistical conceptual model where the spatial distribution of hydraulic conductivity is represented as a realization of a log- normally distributed stationary process, characterized by an exponential covariance function. Using the maximum likelihood method, the parameter estimation problem is solved by determining the set of geostatistical parameters -- average, variance, and correlation scales. Available data may include direct measurements of hydraulic conductivity, water table elevation at a number of monitoring wells, and contaminant mass extracted from active remediation wells. A rigorous solution to this optimization problem would require a stochastic flow and transport model to be included in the optimization loop to calculate the expected values and the covariance matrix of the available measurements as functions of the decision variables. Because of the overwhelming computational effort involved, a surrogate model or response surface is introduced to approximate the objective function. The surrogate model is estimated using a multidimensional kriging interpolation over a set of data points or measurements obtained from a series of stochastic flow and transport simulations for pre-established combinations of the decision variables. Since the goodness of the solution is ultimately determined by the error of estimation of the objective function, which in turn depends on the number and the location of measurements, an optimal search procedure is used in order to optimize the pattern of data points. The method turns out to be computationally efficient and produces results that well approximate the actual geostatistical distribution of hydraulic conductivity.