#### Title

Extension of MF2005-GWM (Ground-Water Management Process) to Solve Management Formulations which Optimize Hydraulic Head and Solve Quadratic Programming Problems

#### Publication Date

9-2008

#### Abstract

Groundwater management problems usually have many possible alternative solutions. Groundwater simulation models can be coupled with mathematical optimization techniques to determine the best alternative given an objective and set of constraints. Ground-Water Management (MF2005-GWM 1.1.1) is a process for the U.S. Geological Survey modular three-dimensional groundwater flow model, MODFLOW-2005. The GWM Process uses the response matrix approach coupled with mathematical programming to solve groundwater management problems for systems which can be simulated using MODFLOW. The current version of GWM is MF2005-GWM 1.1.1 and allows the formulation of management problems by defining decision variables (primarily flow-rate variables for wells) and to optimize the value of these decision variables subject to constraints on hydraulic head, streamflow and/or summations of decision variables, among other capabilities.

This work introduces a new type of variable to GWM, called a state variable, which is defined as either a hydraulic head or streamflow type variable. The value of the state variables are entirely dependent on the flow-rate variables and are calculated using the response matrix approach. State variables may be included in the objective function and summation constraints in this new MF2005-GWM version 2.0. The mathematical formulation, implementation, and sample problems are included to describe the theory and use of state variables.

Additional functionality is added to GWM in a Beta version for the capability to solve quadratic programming problems. In version 1.1.1 of GWM, the objective function must be a linear sum of decision variables. The addition of the state variable package in version 2.0 expands the definition of the objective to allow linear sums of state variables. The expanded capability to quadratic terms in the objective is motivated by the desire to represent energy costs, which are a function of the product of hydraulic head (a type of state variable) and flow-rate variables. This report includes the motivation, mathematical development, implementation, verification and a sample problem for a version of GWM including quadratic objective terms.