In this paper we study minimization of pumping cost of a given total flow rate from any number and layout of wells. Building on previously published results, we have considered steady state flow in aquifers with two zones of different transmissivities, to which the method of images applies. Moreover, we have taken into account additional regional flow, which results in different hydraulic head values at the location of the wells, independent of their operation. We prove analytically that in this general case, pumping cost is minimized, when final differences between hydraulic head values at the locations of the wells, resulting from superposition of the regional flow and the operation of the system of the wells, are equal to the half of those, which are due to the regional flow only. Finally we present the analytical calculation procedure of the optimal distribution of the required total flow rate to the individual wells and we provide illustrative examples.
Pumping cost minimization is one of the most common problems in groundwater resources management, e.g. Sidiropoulos and Tolikas (2004). Its complexity depends on the respective constraints, such as limits on well flow rates due to pump capacities or aquifer features and limits on hydraulic head drawdown in parts of the aquifer (e.g. Bayer et al., 2009). In other cases pumping cost is examined together with other cost items, such as amortization of well or pipe network construction cost, e.g. Wang and Zheng, 1998 and Cunha, 1999. Water quality considerations, from seawater intrusion to nitrate pollution, may also enter the optimization process, e.g. Katsifarakis et al., 1999, Park and Aral, 2004, Katsifarakis and Petala, 2006 and Minciardi et al., 2007. In many cases, pumping cost is the main item in aquifer restoration problems, e.g. Shieh and Peralta, 2005, Matott et al., 2006 and Papadopoulou et al., 2007.
Due to the importance of proper development of groundwater resources, the respective problems have been tackled over the years by many optimization methods, such as linear and non-linear programming (e.g. Bear, 1979, Rastogi, 1989 and Theodossiou, 2004), genetic algorithms and other evolutionary techniques (e.g. Ouazar and Cheng, 2000, Mantoglou et al., 2004, Kalwij and Peralta, 2008 and He et al., 2008), the outer approximation method (e.g. Spiliotopoulos et al., 2004), etc. Critical evaluations of different optimization methods have been presented by Mayer et al., 2002 and Fowler et al., 2008.
For steady flow in confined infinite aquifers, as well as in semi-infinite ones to which the method of images applies, the following proposition has been proved by Katsifarakis (2008): the cost to pump a given total flow rate QT from any number and layout of wells is minimized, when hydraulic head levels at all wells are equal to each other, as long as flow is due to that system of wells only.
In this paper we extend the aforementioned work to steady flows in aquifers with two zones of different transmissivities, to which the method of images applies. We prove that pumping cost is minimized when hydraulic head levels at all wells are equal to each other, as long as flow is due to that system of wells only. Moreover we outline the analytical calculation of the optimal distribution of QT to the individual wells and we provide an illustrative example.
Then we take into account regional flow, independent of the operation of the wells. We prove analytically that in this general case, pumping cost is minimized, when final differences between hydraulic head values at the locations of the wells, resulting from superposition of the regional flow and the operation of the system of the wells, are equal to the half of those, which are due to the regional flow only. Finally we present the analytical calculation procedure of the optimal distribution of QT to the individual wells and we provide an illustrative example.
In this paper we have studied minimization of pumping cost in confined aquifers under steady state flow conditions. We have considered aquifers with two zones of different transmissivities, to which the method of images applies. We have proved that pumping cost is minimized when hydraulic head levels at all wells are equal to each other, as long as flow is due to that system of wells only.
Then, we have taken into account regional flow, independent of the operation of the wells. We have proved analytically that in this general case, pumping cost is minimized, when final differences between hydraulic head values at the locations of the wells, resulting from superposition of the regional flow and the operation of the system of the wells, are equal to the half of those, which are due to the regional flow only. In other words, if there are initial differences in hydraulic head level between wells, minimization of pumping cost requires that these differences are drastically reduced, but not completely eliminated.
The minimum pumping cost for a given total well flow rate QT and layout of wells can be easily calculated analytically, by solving a linear system of N equations and N unknowns, as described in “Aquifers with 2 zones of different transmissivities” and “Additional regional flow”.
The aforementioned conclusions, regarding the hydraulic head level values at the wells, which correspond to the minimum cost, can be used as quality criteria to evaluate solutions obtained by optimization techniques for more complex problems. Let’s examine, for instance, optimization of both the layout of the wells and the distribution of a given total flow rate to them. If differences between final hI values at the locations of the wells are not equal to the half of the initial ones (which are due to the additional flow only), the global optimum has not been found. When discrepancies are small, local fine-tuning of the obtained suboptimal solution could lead to better results, although without guarantying that the global optimum is reached. In other cases, discrepancies might be justified by reductions in other cost items (e.g. network amortization cost or water treatment cost) or by restriction observance (e.g. maximum permissible flow rate per well or minimum permissible hydraulic head level) and should be evaluated accordingly.
