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Show The response matrix method was first introduced in petroleum technology literature by Lee and Aronofsky ( 1958). Ground water management literature contains models formulated using the response matrix method under variety of guises, such as the algebraic technological function ( Maddock, 1972) and the discrete kernel generation approach ( Morel- Seytoux and Daly, 1975). Maddock ( 1972) presents the derivation of an algebraic technological function ( ATF) to determine drawdown at a well at the end of a pumping period due to pumping of the other wells in the field. The derivation is dependent on the validity of the principle of linear superposition; it requires that the drawdown at a point in a confined aquifer with multiple wells is equivalent to the sum of the drawdowns that would arise from each well pumping independently. The ATF is incorporated into the objective function of a management model aimed at minimizing pumping costs while meeting water demand. The aquifer considered in the study ( Maddock, 1972) is of uniform thickness, with negligible vertical flow. Maddock ( 1974a) develops an ATF which relates drawdown to pumping in unconfined aquifer whose flow can be modeled by Boussinesq's equation. The result is a nonlinear relationship between pumping and drawdown which is incorporated into the objective function. The constraints in the management model remain linear. Morel- Seytoux and Daly ( 1975) and Maddock ( 1974) present management models for stream- aquifer interaction. These models incorporate a response matrix, either by use of the discrete Kernel approach or by an ATF, which relates the amount of water pumped to the percent of water withdrawn from the stream and the percent mined from the aquifer. Since the principle of superposition is utilized in the response matrix approach, linearity of the governing partial differential equations of flow is necessitated. Procedures to deal with nonlinear situations, arising when unconfined, leaky, or stratified aquifers are considered are presented by Maddock ( 1974a), Yeh ( 1982), and Willis ( 1983). A finite difference or finite element discrete kernel generator is first employed to generate the drawdowns at all points of interest to unit pumping at a candidate well location. Illangasekare and Morel- Seytoux ( 1982) present several schemes for the efficient discrete kernel generators for confined, unconfined, leaky, and stratified aquifers, and for stream aquifer systems. The assemblage of the drawdown responses to unit pumping is termed the response matrix. The drawdown at any point due to pumping at a number of candidate locations may then be evaluated using Green's theorem and utilizing the principle of superposition. Since the simulation and optimization models are completely independent using this formulation, the simulation may be carried out at any desired level of discretization without increasing the size of the optimization problem. Wanakule and Mays ( 1986) state that they prefer the embedding formulation to the response matrix formulation since " it is necessary to resolve the response matrix in case the boundary or initial conditions change. " This is a rather interesting observation since the entire optimization model would need to be resolved if the boundary conditions change, regardless of whether the embedding or the response matrix formulation is used. The simulation part of the response matrix formulation remains unaffected if the initial conditions change, since the response matrices are evaluated in terms of drawdowns rather than piezometric heads. On the contrary, the entire optimization model would have to be re- solved if the embedding approach were to be used. 8 |