Kouli_etal_2008_Groundwater modelling_BOOK.pdf - Pantelis ...

Flow Simulation and Optimal Management of Groundwater Resources… 299
The chromosome evaluation function VB for both cases has the following simple form:
N
VB = ∑Q
i=
1
i
−
PEN
(1)
where N is the number of the wells, Q i is the flow rate of well i and PEN is a penalty,
imposed if the pumping scheme induces seawater intrusion to the wells. But, in order to check
whether this happens, namely in order to calculate the term PEN, one has to use the
groundwater flow simulation code.
The form of the penalty function, that has been finally chosen for both cases, is:
= −70
⋅ k − 7 ⋅ T1 ⋅∑ qi
⋅ li
PEN (2)
where T 1 is the transmissivity of the coastal zone 1 and l i is the length of coastal boundary
element i, while summation extends only to the k boundary elements with positive q i values.
It can be seen that PEN depends both on the number of violated constraints and on the
magnitude of each violation.
Due to the comparatively low computational volume of the flow simulation model, it was
possible to set the population size PS = 50 and to use values up to 300, for the number of
generations NG. These values were adequate to lead the optimization process to the optimum
solution of the first problem, namely the best combination of safe flow rates Q 1 and Q 2 , for all
test runs. This has been proved by means of independent runs of the simulation code.
Increase of either Q 1 or Q 2 by even 1 l/s leads to seawater intrusion. Near optimal solutions
have been achieved for the second problem, too. It has been concluded then, that the proposed
approach does not compromise the optimization process. But, for each practical application,
the optimum solution of the simplified problem should be checked by means of a more
detailed flow simulation model.
5.2. Use of an Approximate Form of the Method of Images
The method of images is extensively presented in a number of textbooks on groundwater
hydraulics (e.g. Bear, 1979). It offers analytical solutions for homogeneous semi-infinite
aquifers bounded by one rectilinear (impermeable or constant head) boundary. It can also be
used in flow fields with two rectilinear boundaries, provided that the angle θ formed by the
two boundaries is an integer submultiple of 180 0 (for boundaries of the same kind) or of 90 0
(for boundaries of different kinds). Only for these cases, the number N W of imaginary wells is
finite, all of them lie outside the real flow field and the sign of their flow rate (pumped or
injected) is uniquely defined. N W is a function of θ and is given as:
N W
360
−1
θ
= (3)