The minimize energy and MINIMAX formulations were applied to four test models: a confined, homogeneous aquifer with two wells; a confined, homogeneous aquifer with 20 wells; a confined, heterogeneous aquifer with 20 wells; and an unconfined, homogeneous aquifer. The MINIMAX formulation produced the same results as the minimize energy formulation when the non-pumping lifts were the same. As the non-pumping lifts varied, the MINIMAX formulation deviated from the minimize energy formulation.

A case study of the Lancaster subbasin of Antelope Valley, California, was used to further test the minimize energy and MINIMAX formulations. Two minimize energy formulations were examined, the first lifting water to the ground surface elevation at each well and the second lifting the water to a single reference elevation that took the value of the maximum ground surface elevation that was used in the first formulation. The MINIMAX formulation was applied to the case where the water was lifted to the reference elevation. The difference in total energy between the MINIMAX and minimize energy formulations was less than 10%, but the distribution of pumping among the wells varied greatly.

]]>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.