A particularly important area of research in groundwater management modeling involves the formulation and solution of management problems that account for uncertainty associated with flow and transport simulation models. In recent years, uncertainty in model parameters has been quantified and incorporated into the groundwater management model using techniques of stochastic optimization. The stochastic management model defines the simulation model predictions as probabilistic functions of the decision variables and the parameter uncertainty, and determines the optimal strategy under consideration of the trade-offs between increasing system reliability and increasing costs.
In general, two different, but related, approaches to stochastic groundwater management have been adopted. In the first approach, uncertainty analysis is incorporated directly into the management model to define the model predictions in terms of their mean and covariance. Theoretical and applied studies using this approach are described in later sections of this paper. The second general approach to stochastic groundwater management uses a Monte Carlo-type analysis involving a series of realizations of the uncertain parameters. A robust, or reliable, groundwater management strategy is obtained by requiring the management model to simultaneously satisfy the constraints for multiple realizations of the uncertain model parameters. Early work [ Gorelick, 1987; Wagner and Gorelick, 1989] demonstrated that the multiple-realization approach can provide reliable designs, but it did not allow for prespecification of the design reliability; the reliability was determined through post-optimization Monte Carlo analyses.
A number of recent studies have attempted to develop multiple-realization management models that allow for prespecification of system reliability. Morgan et al. [1993] present a method to identify optimal pumping strategies for capture zone design in the presence of a heterogeneous and uncertain transmissivity field. The hydraulic gradient constraints of the linear multiple-realization optimization problem are formulated to include an integer `indicator' that takes a value of one if a constraint set for a particular realization is violated. A heuristic algorithm begins with N realizations of transmissivity. The model successively drops realizations to determine the optimal pumping rates for reliability levels ranging from full reliability (all constraint sets satisfied) to no reliability (all constraint sets violated). The optimal pumping strategy for reliability r (r between zero and one) is defined as the optimal design that satisfies rN of the constraint sets. The implied assumption of this approach is that if rN of the constraint sets are satisfied, it is reasonable to expect the management strategy would satisfy the same proportion r of the complete ensemble of realizations.
The study of Morgan et al. [1993] clearly demonstrates some important characteristics of the stochastic groundwater management problem. In their example, it was not possible to achieve a reliable design based on a homogeneous transmissivity field, even when the pumping rates were increased by `safety factors' ranging as high as 40. This work also shows that there is, in general, no single `worst case' realization that controls the capture zone design; therefore, the optimization problem must be solved simultaneously (and not separately) for all of the transmissivity realizations under consideration.
Chan [1994] also presents a multiple realization management model that allows for prespecification of system reliability. As in the study described above, the goal was reliable hydraulic capture zone design in a spatially variable transmissivity field. Beginning with N realizations of transmissivity and a desired system reliability r, the method progresses to a solution in which (1-r)N of the constraint sets are violated. Two heuristic methods for solving this problem are presented. The first method successively discards realizations, based on the improvement in the objective value, until (1-r)N of the constraints are violated. The second method, adds a penalty to the objective to trade off infeasibilities against improvement in the objective. Monte Carlo testing shows that the accuracy (as defined by the average and maximum deviations between actual and specified reliability) of these methods increases with the number N of realizations in the optimization model.
In a related work, Chan [1993] presents theoretical analyses (based on Bayesian principles and order statistics) to define the management design reliability r as a function of the number of realizations N included in the optimization model (note that for this study, the multiple realization model is formulated to satisfy the constrains for all N realizations). The nonparametric relations obtained through these analyses show good agreement with the average reliability levels obtained through Monte Carlo analyses. Sensitivity analyses indicate that the relationship between reliability and N is relatively insensitive to a wide range of model parameters.
In the multiple-realization approach, the number of constraints grows in direct proportion to the number of realizations N, and the computational effort is roughly proportional to N times the computational effort required to solve a single-realization optimization problem. The studies by Chan [1993, 1994] suggest that N = 50--100 realizations are needed to ensure a design has a high level of reliability. However, only those realizations with binding constraints at the optimum are critical to the solution of the optimization problem. If we could identify the `critical realizations' prior to solving the optimization problem, we could formulate the management model to include only those realizations, thereby realizing a significant savings in computational effort.
When designing hydraulic capture zones, the location, size and orientation of high and low transmissivity zones, along with the degree of variability within the transmissivity field, will have a direct influence on the level of pumping required to achieve hydraulic containment. Ranjithan et al. [1993] use an artificial neural network (ANN) as a screening tool to identify `critical realizations' for capture zone design. The ANN [ Simpson, 1990] can be thought of as a nonlinear `mapping' system that can `learn' complex patterns and associate these patterns with specific inputs or outputs. In this case, ANN is applied to `learn' the association between critical patterns of transmissivities and the associated high level of pumping required for capture zone design. The basic idea is to (1) use the neural network model to identify `critical realizations' and (2) solve the multiple-realization optimization problem for only a small number of these `critical realizations.' A comparison with the model presented by Morgan et al. [1993] demonstrates that the ANN-based approach can closely estimate the cost-reliability trade-off at higher reliabilities using fewer realizations and less total computational time.
Wagner et al. [1992] use a multiple-realization-based approach, termed stochastic programming with recourse, to identify plume containment strategies in the presence of a random hydraulic conductivity field. The goal is to identify the pumping strategy that minimizes the expected `cost' of plume containment, which includes the cost of pumping, the benefits derived from the pumped water, and a recourse `cost' that penalizes gradient constraint violations. The authors examine four different model formulations, ranging from simple deterministic optimization based on the expected values of the uncertain parameters to full stochastic programming with recourse. The optimal designs obtained using the deterministic approach are significantly more expensive; these designs incur significant recourse costs because they do not account for the heterogeneity and uncertainty of the hydraulic conductivity field. This demonstrates once again the importance of explicitly incorporating uncertainty into the management model formulation.