3.2. Optimization Problem
The objective of the optimization model is to maximize the water
withdrawal rate from the aquifer (Total discharge, Q) and water gain of
the river from the aquifer (Leakage out, L) while having the least
drawdown at the wells in the domain. The leakage out represents the rate
of discharge of water from the aquifer to the river. The pumping rate is
assumed constant over well zones and specified time steps. The water
withdrawal rate from the well zones, leakage rate out of the aquifer
into the river, and drawdown at the wells, of the last time step, are
considered for optimization. The domain contains different zones of the
wells, based on the distance from the river and municipal zone data. To
achieve the objective, the discharge in these zones are adjusted by the
following optimization algorithm:
\begin{equation}
\text{Maximise}\left\{\begin{matrix}\sum_{i\in R}L_{i}-P\\
\sum_{i=1}^{n_{z}}{N_{i}*Q_{i}}-P\\
\end{matrix}\right.\ \nonumber \\
\end{equation}\begin{equation}
\left(Q_{i}\right)_{\text{lb}}\leq Q_{i}\leq\left(Q_{i}\right)_{\text{ub}}\nonumber \\
\end{equation}where, Li = rate of leakage out of the aquifer to the
river; Qi = rate of discharge of ithwell zone; R = set of the river grid cells with leakage out;
nz = total number of well zones; Ni =
number of wells in the ith zone; P = penalty imposed
due to drawdown constraint violation;
(Qi)lb = lower bound of the discharge
for the ith well zone;
(Qi)ub = upper bound of the discharge
for the ith well zone.
\begin{equation}
\mathbb{d}_{\text{dist}}=\sqrt{\sum_{i\in W}\left(d_{i}-d_{\text{threshold}}\right)^{2}}\nonumber \\
\end{equation}\begin{equation}
P=C_{\text{model}}*d_{\text{dist}}\nonumber \\
\end{equation}Where, di = drawdown at ith well; W =
set of wells; dthreshold = threshold value of drawdown,
taken as 2 m; ddist = distance-based metric for
drawdown; Cmodel = constant to amplify the penalty. The
value of Cmodel is calculated such that the penalty is
of the same order as that of the cost. Both the cost has values of the
order 105 and from the multiple (500) random model
evaluations, an expected value of ddist is obtained.
Consequently, the value of the Cmdoel is obtained as
105/ E(ddist).