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Show which the aquifer simulation is essentially at steady state. The response matrix terms need to be stored only up to this time, leading to a further reduction in memory requirements. The head H^ at any node ( i j ) at any time t may then be computed as the initial head H^ o at the node, less the total drawdown s^ due to background wells and natural recharge, and the total drawdown due to pumping at the wells considered for optimization, up to time t. The total drawdown at ( ij) due to pumping at the wells considered for optimization is determined using the principle of superposition. The computation of head Hjjt may be symbolically stated as : t w t « l W « l where H jj0 is the head at node ( i j) at time t= 0 Sjjt is the total drawdown due to all background pumping/ natural recharge upto time t ayWl- is the incremental drawdown at node ( ij) in time period t\ due to unit pumping at well w Q^ j- is the pumping rate at well w, during time period tf 3.2.4 Response Matrices for Flow Computations As was stated earlier, the maintenance of water rights and groundwater quality is achieved through control of groundwater flows. It is necessary to develop procedures that allow the computation of flows across any general boundary of interest as a response to pumping in the aquifer. This section develops such procedures in a response matrix context. The boundary across which flow is to be computed is considered to be composed of several linear segments. The flow normal to the boundary is then computed as the sum of the flows across each such segment. Procedures for computing the flow across each segment as a response to pumping are first developed with the aid of Figure 3.6. Consider that the boundary segment in the neighborhood of the node ( ij) makes an angle 9 with the x axis. The flow traversing the cell defined by the node ( i j ) may then be estimated in terms of its components Fx and Fy in the x and y directions, using a forward difference approximation as H - H F * T .. ** 1> j" i, j 8y.. ( 3.3) ox.. H - H F = T . . Uhl ' i > jSx.. ( 3.4) * yy.>. j *.. » o xxij Tyyjj cell( ij). where Txxij and T ^ j are the transmissivities in the x and y directions associated with the 45 |