WATER RESOURCES RESEARCH, VOL. 29, NO. 9, PAGES 3277-3289, SEPTEMBER 1993 Nonlinear Weighted Feedback Control of Groundwater Remediation Under Uncertainty GREGORY J. WHIFFEN AND CHRISTINE A. SHOEMAKER Schoolof Civil and Environmental Engineering, Cornell Universit)', Ithaca, New York Differential dynamic programming is used to compute optimal time-varying pumping policies for a pump and treat strategy for groundwater remediation. The feedback law generated by a constrained differential dynamic programming algorithm withpenalty functions is used asthe basis of feedback laws tested in cases where there is uncertainty in the hydraulic conductivity. Confined transient aquifer flow and transport are modeled using a two-dimensional Galerkin finite element scheme with implicit time differencing. Optimal policies are calculated using agiven or"measured" set ofhydraulic conductivities and initial conditions. The optimal policies (withandwithout feedback) are applied using the same finite element model with a second or"true" set ofconductivities. The "true" sets of conductivities are generated randomly from anautocorrelated lognormal distribution by thespectral method. The approach used here has anadvantage over other uncertainty approaches because it isnot necessary to specify precisely which parameters are considered uncertain and which are certain. Also nosingle probability distribution need beassumed foreach uncertain parameter. By adjusting the relative weight assigned each penalty function, robust feedback laws were obtained thatperform equally well under nine different assumed error distributions. Inour examples, well-designed feedback policies cost between 4% and 51% less than the cost of applying the calculated optimal policies without usinga feedbacklaw. INTRODUCTION The objective ofthis paper isto develop pumping policies for the pump and treat groundwater remediation method that perform well even when there exists substantial uncertainty inaquifer characteristics. These policies use a "weighted feedback" law that adjusts pumping ratesif the observed values ofhydraulic head and pollution concentration deviate from predicted values overtime. Groundwater contamination in the United States and in many other countries is recognized as a widespread and serious threat to drinkingwater sources. Contaminated aquifers have been identified in every state, and the rate of discovery of new polluted sites is increasing. Patrick etal. [1987] estimate that between 1% and 2% of all usable near-surface groundwater in theUnited States is contami- nated bypoint sources alone. The cost ofa single remedia- tion canrange from several thousand to'overa billion dollars. The prevalence of groundwater contamination and the high costs associated with remediation by methods such as pump and treat have led many researchers tomodel the fate of contaminants in aquifers under theinfluence of various remediation strategies. More recently, optimization methods have been used in conjunction withcontaminant transport models toidentify cost-effective remediation strategies. Lin- ear, nonlinear, and neutral network optimization methods have been used (see, forexample, Ahlfeld etal. [1988a, b], Gorelick et al. [1984], and Ranjithan andEheart [1993] respectively.) Dougherty and Marryott [1991] solve a com- binatorial remediation optimization problem using simulated annealing. Since transport models forgroundwater are non- linear innature, nonlinear optimization methods offer a more accurate toolwhen computationally practical. Copyright !993 bythe American Geophysical Union. Paper number93WR00928. 0043-1397/93/93WR-00928505.00 Nonlinear optimization methods relyonthe predictions of flowand transport models with whichthey are coupled. Since accurate modeling of anyaquifer canbe very difficult or impossible, calculated optimal policies based on deter- ministic flow and transport models will not, in fact, be optimal for therealaquifer system. There is concern that least cost policies generated by optimization analysis may not be robust against model error and wouldfail to meet cleanup goals if they areapplied to a real system. Theresearch inthis paper isan extension of workdone by the same research group on deterministic groundwater re- mediation. Chang et al. [1992] describe a method for com- puting time-varying optimal pumping policies using a modi- fied version of the optimal control algorithm differential dynamic programming. Culver and Shoernaker [1992] extend the work of Chang et al. [1992]to includemanagement periods, in which theuser chooses the number of finite element time steps to incorporate intoeach "management" period inthe optimization. This approach reduces the com- putational requirements and generates pumping policies that are more practical to implement because thepumping rates arenotallowed to change too frequently. Culver andShoe- maker [1993] use quasi-Newton approximated second deriv- atives of the transition function to reduce the number of iterations required for convergence andthe overall compu- tation time. Thework described in this paper is most closely related toChang etal. [1992], but the method can be used in conjunction with the management period and quasi-Newton approach. The development of a weighted feedback approach for constrained optimal control problems is described in this paper. The use of unweighted feedback for constrained optimal control problems has been discussed byJacobson and Mayne [1970]. The procedure described here foradjust- ing penalty function weights to obtain a robust feedback policy for constrained optimal control problems has not been previously developed. 3277