To define the optimization problem the user specifies: 1) a set of numerical discretization nodes to be considered as candidates for pumping or recharge in each stress period; 2) costs for the installation and unit pumping of each candidate well; and 3) requirements on the hydraulic behavior of the aquifer from among a set of possible constraints on head, streamflows and pumping rates.
Design criteria can be incorporated into the optimization procedure by selection of constraints. Minimum and maximum head constraints can be used to control excessive drawdown or mounding of the piezometric surface. Minimum head difference constraints can force groundwater to flow in a specified direction between two locations. Constraints on streamflow depletion can be imposed. Minimum and maximum pumping rates can be used to limit the amount of pumping or recharge allowed at a well. Total pumping within a stress period can have upper and lower bounds and the ratio of total extraction over total injection can be fixed by a ratio. Bounds can also be placed on the number of wells to be installed.
For groundwater problems in which the saturated thickness is fixed (confined aquifers) or the saturated thickness changes relatively little (unconfined aquifers with little drawdown) and which are at steady-state, the relationship between head and pumping is linear. By calling MODFLOW repeatedly, MODOFC determines this relationship by perturbing each well and measuring the change in head at each constraint. The head at any constraint in the model can be predicted without further need of the flow simulator.
MODOFC always seeks the pumping strategy with the lowest cost. This cost is defined by the user who designates well installation costs and unit costs for pumping and recharge at each candidate well. The total cost which is minimized is measured as the sum of the installation costs and the products of the unit costs at each well and the flow rate at the well. The use of this cost function presumes that the cost of operation of the hydraulic control system is linearly proportional to the rate of pumping.
The optimization problem defined by the user's constraints and the groundwater system's responses is written as a linear program or a mixed binary program (if well installation costs are included). A linear program is solved by the Simplex method and a mixed binary program is solved by the branch and bound algorithm. Either solution method yields the lowest cost pumping scheme.
MODOFC is also capable of solving groundwater problems with unconfined aquifers when drawdown is significant. For these problems the relationship between head and pumping is nonlinear and the perturbation method only provides a linear approximation of that relationship. Streamflow constraints also introduce a nonlinear relationship. For nonlinear problems, the linear program is solved repeatedly; at each iteration the previous solution is used as background pumping when perturbing the wells. As the iterations progress, the linear approximation converges to the true solution.