Analytical Solutions of Intermittent Transient Groundwater Pumping Cost
Publication: Journal of Hazardous, Toxic, and Radioactive Waste
Volume 24, Issue 4
Abstract
In this paper, we have studied ways to minimize the cost of stepwise or intermittent pumping from a system of wells under transient groundwater flow conditions. We take into account infinite confined aquifers and semi-infinite aquifers where the method of images applies Moreover, we examine optimal separation of wells in groups, in the case of alternate pumping. We have proved analytically via the method of Lagrange multipliers that, at any time, the pumping cost is minimized when the hydraulic head level drawdowns at the locations of the pumping wells are equal to each other.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The author thanks Professor K. L. Katsifarakis for his spiritual, moral, but mostly insightful support.
Notation
The following symbols are used in this paper:
- A
- constant coefficient;
- C
- pumping cost for steady state conditions;
- D
- matrix of the distances between wells;
- D.S.
- matrix of the values of different combinations;
- E.V.
- matrix of sums of D.S. matrix;
- gi
- total pumping constraint for each time step;
- H
- Hessian matrix;
- K
- pumping cost for transient conditions;
- L
- objective function;
- N
- number of wells;
- Qj
- flow rate of well j (m3/s);
- total pumping rate during period i (m3/s);
- R
- radius of influence (m);
- rmj
- distance between wells m and j (m);
- S
- storativity of aquifer;
- sj
- hydraulic head level drawdown at well j (m);
- T
- transmissivity of aquifer (m2/s);
- W(u)
- well function for infinite confined aquifer;
- x
- number of different combinations;
- δj
- distance between the initial horizontal level of the hydraulic head and the predefined reference level (m); and
- λi
- Lagrange multiplier for every constraint.
References
Ahlfeld, D. P., and M. M. Laverty. 2011. “Analytical solutions for minimization of energy use for groundwater pumping.” Water Resour. Res. 47 (6): W06508. https://doi.org/10.1029/2010WR009752.
Ahlfeld, D. P., and M. M. Laverty. 2015. “Field scale minimization of energy use for groundwater pumping.” J. Hydrol. 525: 489–495. https://doi.org/10.1016/j.jhydrol.2015.03.065.
Bayer, P., E. Duran, R. Baumann, and M. Finkel. 2009. “Optimized groundwater drawdown in a subsiding urban mining area.” J. Hydrol. 365 (1–2): 95–104. https://doi.org/10.1016/j.jhydrol.2008.11.028.
Bear, J. 1979. Hydraulics of groundwater. New York: McGraw-Hill.
Fowler, K. P., et al. 2008. “Comparison of derivative-free optimization methods for groundwater supply and hydraulic capture community problems.” Adv. Water Resour. 31 (5): 743–757. https://doi.org/10.1016/j.advwatres.2008.01.010.
Katsifarakis, K. L. 2008. “Groundwater pumping cost minimization—An analytical approach.” Water Resour. Manage. 22 (8): 1089–1099. https://doi.org/10.1007/s11269-007-9212-x.
Katsifarakis, K. L., I. A. Nikoletos, and C. Stavridis. 2018. “Minimization of transient groundwater pumping cost—Analytical and practical solutions.” Water Resour. Manage. 32 (3): 1053–1069. https://doi.org/10.1007/s11269-017-1854-8.
Katsifarakis, K. L., and K. Tselepidou. 2009. “Pumping cost minimization in aquifers with regional flow and two zones of different transmissivities.” J. Hydrol. 377 (1–2): 106–111. https://doi.org/10.1016/j.jhydrol.2009.08.010.
Khadem, M., and M. H. Afshar. 2015. “A hybridized GA with LP-LP model for the management of confined groundwater.” Groundwater 53 (2): 485–492. https://doi.org/10.1111/gwat.12234.
Lamaddalena, N., and S. Khila. 2012. “Energy saving with variable speed pumps in on-demand irrigation systems.” Irrig. Sci. 30 (2): 157–166. https://doi.org/10.1007/s00271-011-0271-7.
Latinopoulos, P. 1996. Hydraulics of groundwater. Oakdale, CA: XARIS Publication.
Mahdavi, A. 2015. “Transient-State analytical solution for groundwater recharge in anisotropic sloping aquifer.” Water Resour. Manage. 29 (10): 3735–3748. https://doi.org/10.1007/s11269-015-1026-7.
Mani, A., F. T.-C. Tsai, S.-C. Kao, B. S. Naz, M. Ashfaq, and D. Rastogi. 2016. “Conjunctive management of surface and groundwater resources under projected future climate change scenarios.” J. Hydrol. 540: 397–411. https://doi.org/10.1016/j.jhydrol.2016.06.021.
Nikoletos, I. A. 2018. “Cost minimization of intermittent transient groundwater pumping.” In Proc., 14th Int. Conf. Protection and Restoration of the Environment, 622–630, Greece: Aristotle University of Thessaloniki.
Papadopoulou, M. P., G. F. Pinder, and G. P. Karatzas. 2007. “Flexible time-varying optimization methodology for the solution of groundwater management problems.” Eur. J. Oper. Res. 180 (2): 770–785. https://doi.org/10.1016/j.ejor.2006.02.041.
Reca, J., A. Garcia-Manzano, and J. Martinez. 2014. “Optimal pumping scheduling for complex irrigation water distribution systems.” J. Water Resour. Plann. Manage. 140 (5): 630–637. https://doi.org/10.1061/%28ASCE%29WR.1943-5452.0000360.
Rodriguez-Pretelin, A., and W. Nowak. 2019. “Dynamic re-distribution of pumping rates in well fields to counter transient problems in groundwater production.” Groundwater Sustainable Dev. 8: 606–616. https://doi.org/10.1016/j.gsd.2019.02.009.
Saeedpanah, I., and R. Golmohamadi Azar. 2017. “New analytical expressions for two-dimensional aquifer adjoining with streams of varying water level.” Water Resour. Manage. 31 (1): 403–424. https://doi.org/10.1007/s11269-016-1533-1.
Shourian, M., and S. M. J. Davoudi. 2017. “Optimum pumping well placement and capacity design for a groundwater lowering system in urban areas with the minimum cost objective.” Water Resour. Manage. 31 (13): 4207–4225. https://doi.org/10.1007/s11269-017-1740-4.
Siarkos, I., D. Latinopoulos, Z. Mallios, and P. Latinopoulos. 2017. “A methodological framework to assess the environmental and economic effects of injection barriers against seawater intrusion.” J. Environ. Manage. 193: 532–540. https://doi.org/10.1016/j.jenvman.2017.02.051.
Singh, A., and S. N. Panda. 2013. “Optimization and simulation modelling for managing the problems of water resources.” Water Resour. Manage. 27 (9): 3421–3431. https://doi.org/10.1007/s11269-013-0355-7.
Theis, C. V. 1935. “The relation between lowering of the piezometric surface and the rate and duration of discharge of a well using ground water storage.” Trans. Am. Geophys. Union 16 (2): 519–524. https://doi.org/10.1029/TR016i002p00519.
Theodossiou, N. P. 2004. “Application of non-linear simulation and optimization models in groundwater aquifer management.” Water Resour. Manage. 18 (2): 125–141. https://doi.org/10.1023/B:WARM.0000024723.17916.64.
Information & Authors
Information
Published In
Copyright
© 2020 American Society of Civil Engineers.
History
Received: Jan 2, 2020
Accepted: Apr 8, 2020
Published online: Jun 19, 2020
Published in print: Oct 1, 2020
Discussion open until: Nov 19, 2020
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.