Two optimization formulations have been developed. The first formulation minimizes the energy required to pump water from an aquifer. This formulation is non- linear, requiring the use of quadratic terms in the objective function. The second formulation minimizes the maximum lift in an aquifer and is called the MINIMAX formulation. This formulation is linear and may be an acceptable substitution for the minimize energy formulation under conditions where there are small differences in initial lifts at the wells and where the demand is sufficient to require pumping at all wells such that the lifts at all wells are equal.
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.