Abstract
In this paper, we have investigated minimization of total cost to pump a given flow rate from any number (n) of wells up to a water tank, under steady-state flow conditions. Regarding groundwater flow, we have considered infinite or semi-infinite aquifers, to which the method of images applies. Additional regional groundwater flow can be taken into account, too. The pipe network connecting the wells to the tank can include junctions at the locations of the wells only. Moreover, all pumps have equal efficiency. We have derived a new analytical formula, which holds at the critical points of the total cost function. Based on this formula, we derived a system of n equations and n unknowns, to calculate the well flow rate combinations which correspond to the critical points of the total cost function. The n-1 equations are 2nd degree polynomials, while the remaining one is linear, expressing the constraint that the sum of well flow rates must be equal the required total flow rate. The solution of the system can be achieved using commercial solvers. Moreover, we have concluded that there is one feasible solution that minimizes the total cost. Finally, we present a tabulation process to facilitate the use of solvers and we provide and discuss two illustrative examples.
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Abbreviations
- a i,j :
-
Coefficient used to calculate the difference of interference of well j with wells i and n (for i = 1 to n-1)
- A:
-
Matrix of aij
- b :
-
Cost coefficient
- c i,j :
-
Parameter indicating whether pipe Pi carries water pumped from well Wj
- C:
-
Matrix of cij
- D i :
-
Diameter of pipe i 8
- f i :
-
Friction coefficient along pipe i
- g :
-
Gravity constant
- hf i :
-
Head loss along pipe i
- Hf sm :
-
Sum of head losses along the path of qm
- K1 tot :
-
objective function of the minimization problem
- K dr :
-
Cost factor, accounting for groundwater hydraulic head level drawdown
- K el :
-
Cost factor, accounting for the difference δj between water level at the tank and initial hydraulic head level at each well
- K tot :
-
Total pumping cost
- K trans :
-
Cost factor, accounting for hydraulic head losses at the pipe network
- L i :
-
Length of pipe i
- n :
-
Number of wells
- Q i :
-
Flow rate through pipe i
- q j :
-
Flow rate of well j
- q tot :
-
Total required flow rate
- R :
-
Radius of influence of the wells
- r kj :
-
Distance between wells j and k
- \( {\overset{\prime }{r}}_{kj} \) :
-
Distance between well j and the image of well k
- S j :
-
“Path” of well flow rate qj; it includes all pipes Pi through which water pumped from well Wj flows to the tank
- T :
-
Aquifer’s transmissivity
- δ j :
-
Difference between water level at the tank and initial hydraulic head level at each well
- ξ i :
-
Head loss coefficient of pipe i
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Acknowledgements
The authors would like to thank Professor Hara Charalambous, Department of Mathematics, Aristotle University of Thessaloniki, for her valuable comments.
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Nagkoulis, N., Katsifarakis, K. Minimization of Total Pumping Cost from an Aquifer to a Water Tank, Via a Pipe Network. Water Resour Manage 34, 4147–4162 (2020). https://doi.org/10.1007/s11269-020-02661-x
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DOI: https://doi.org/10.1007/s11269-020-02661-x