George P. Karatzas

Groundwater management using numerical simulation and the outer approximation method for global optimization

Groundwater quantity management problems with fixed charges have been formulated in the past as mixed integer and linear programming problems. In this paper a new methodology is presented where the fixed charges are incorporated into the objective function in an exponential form and the problem is solved as a concave minimization problem. The principal difficulty in the minimization of a concave function over a linear or nonlinear set of constraints is that the local minima which are determined by the classical minimization algorithms may not be global. In an effort to circumvent this problem the outer approximation method is introduced. This method is applicable to the global minimization of a concave function over a compact set of constraints. In the present work the outer approximation is applied to concave minimization problems over a convex compact set of constraints. Two applications of the method to groundwater management problems are presented herein, and the results are compared with an existing solution obtained using a different optimization approach.

The Solution of Groundwater Quality Management Problems with a Nonconvex Feasible Region Using a Cutting Plane Optimization Technique

In groundwater quality management problems the concentration constraints have a nonlinear behavior which may be described either as a convex or a nonconvex function. Therefore the feasible region, which is defined as the intersection of all of these constraints, can be either a convex or a nonconvex set. A review of existing optimization algorithms for the solution of the groundwater quality management problem indicates that the majority of them have an inability to determine a global optimum when nonconvexity occurs. In an earlier paper that appeared in this journal [Karatzas and Finder, 1993], the outer approximation method, a global optimization technique, was presented for the solution of groundwater management problems with convex constraints. The problem was formulated to minimize a concave objective function over a compact convex set of constraints. In the present study the same concept is applied to problems with nonconvex constraints. While the main concept of the current approach remains the same as that in our earlier study, there is a significant difference in the determination of the cutting hyperplane. The nonconvexity of the domain requires a special approach to insure that the introduction of the cutting hyperplane does not eliminate any part of the nonconvex feasible region. In this work the theory of the developed algorithm is presented and subsequently applied to two groundwater quality problems. In the first example a hypothetical aquifer is considered to illustrate the performance of the methodology. In the second example a groundwater quality management problem in Woburn, Massachusetts, is solved. Results obtained are compared with those generated by MINOS 5.1.

A methodology to determine optimal transmissivity measurement locations in groundwater quality management models with scarce field information

A new criterion to determine transmissivity measurement location for groundwater quality management models is proposed. The transmissivity is represented as a random field with known mean and covariance function, conditioned to existing measurements values. The criterion is the minimization of the coefficient of variation of the optimal remediation cost distribution. This distribution is computed numerically by means of Monte Carlo simulations in which each optimal remediation cost is evaluated by solving a deterministic management problem, where the transmissivity is a single realization of the random field. The coefficient of variation is equivalent to the relative error in the prediction of the optimal cost and is shown to be invariant with respect to the actual (unknown) measurement value according to some limiting conditions. In the more general case the coefficient of variation of the optimal cost is less sensitive than the expected optimal cost to the actual measurement value. The proposed criterion becomes a useful tool to determine the “best” transmissivity measurement location when only few measurements have been performed.

Least-Cost Groundwater Remediation Using Uncertain Hydrogeological Information

The design of groundwater remediation pump-and-treat well networks under aquifer parameter measurement uncertainty can be addressed using an optimal-design strategy based upon the concept of robust optimization. The robust-optimization approach allows for the admission of design alternatives that do not satisfy all design constraints. However in the selection process the algorithm penalizes such selections based upon the number of constraints violated. The result is a design which balances the importance of reliability with overall project cost. The robust-optimization method has been applied to the problem of groundwater plume containment and risk-based groundwater remediation design. Designs dedicated to groundwater-plume containment assure that the contaminant plume will not extend beyond a prespecified perimeter. Inwardly directed groundwater velocity must be achieved along this perimeter. The outer-approximation optimization technique in combination with a groundwater flow model ( PTC) is used to solve this optimal-design problem.

Optimal Design Of Groundwater RemediationSystems With Treatment Plant Considerations

The Outer Approximation method is extended to accomodate the solution of ground water remediation problems including treatment-plant design. The corresponding objective function is piecewise linear, monotonically increasing, and suitable for use with the Outer Approximation method. Optimal pumping well locations and corresponding pumping rates as well as optimal configuration of the treatment plant are obtained. The plant consists of units of air-stripping and granular activated carbon. The associated costs are determined using RACER, a commercially available software package for cost estimation. The proposed methodology is applied to a hypothetical case scenario.