Water Resources Management

, Volume 32, Issue 2, pp 751–767 | Cite as

A Particle Swarm Optimization Assessment for the Determination of Non-Darcian Flow Parameters in a Confined Aquifer



Being one of the preliminary in-situ testing methods, aquifer pumping tests would provide significant insights which form a basis for the aquifer characterization. The use of Darcian based flow models to describe the groundwater flow would be ineffective for the aquifer pumping tests under certain circumstances. Non-Darcian flow models could therefore construct more accurate portrayal of physical reality for the assessment of aquifer testing. The interpretation of flow parameters obtained from non-Darcian flows via classical curve matching methods seems extremely difficult to acquire a unique match since the well-defined type curves have not been developed. In this study, an evolutionary optimization based algorithm, called as Particle Swarm Optimization (PSO), was established to determine the flow parameters namely power index, storativity and the turbulent factor which serves as an apparent hydraulic conductivity. The proposed PSO based parameter estimation scheme was implemented for a number of numerical test cases and the estimation performance was evaluated by comparing with available population based algorithms. The results reveal that the PSO based estimation approach is successfully able to identify the flow parameters in an accurate and fast manner. A number of sensitivity analyses were also conducted to draw the limitations of the introduced PSO based technique. The positive findings from this study pointed the potential capability of using PSO as a viable algorithm to process the complex relations in the flow.


Confined aquifer Groundwater flow Izbash equation Non-Darcian flow Particle Swarm Optimization (PSO) 


  1. Abraham A, Guo H, Liu H (2006) Swarm intelligence: foundations, perspectives and applications. Stud Comput Intell 26:3–25Google Scholar
  2. Basak P, Madhav MR (1979) Analytical solutions to the problems of transient drainage through trapezoidal embankments with Darcian and non-Darcian flow. J Hydrol 41:49–57CrossRefGoogle Scholar
  3. Bolado-Lavin R, Castaings W, Tarantola S (2009) Contribution to the sample mean plot for graphical and numerical sensitivity analysis. Reliab Eng Syst Saf 94(6):1041–1049CrossRefGoogle Scholar
  4. Camacho-V RG, Vasquez-C M (1992) Comment on analytical solution incorporating nonlinear radial flow in confined aquifers by Zekai Sen. Water Resour Res 28(12):3337–3338CrossRefGoogle Scholar
  5. Choi ES, Cheema T, Islam MR (1997) A new dual-porosity/dual-permeability model with non-Darcian flow through fractures. J Petrol Sci Eng 17:331–344CrossRefGoogle Scholar
  6. Domenico PA, Schwartz FW (1997) Physical and chemical hydrogeology, 2nd edn. Wiley, New YorkGoogle Scholar
  7. Forchheimer P (1901) Wasserbewegung durch Boden. Z Ver Deutsch Ing 45:1782–1788Google Scholar
  8. Hantush MS, Jacob CE (1955) Nonsteady radial flow in an infinite leaky aquifer. Trans Am Geophys Union 36:95CrossRefGoogle Scholar
  9. Iadevaia S, Lu Y, Morales FC, Mills GB, Ram PT (2010) Identification of optimal drug combinations targeting cellular networks: integrating Phospho-proteomics and computational network analysis. Cancer Res 70:6704–6714CrossRefGoogle Scholar
  10. Izbash SV (1931) O Filtracii V Kropnozernstom Materiale. Leningrad, USSR (in Russian)Google Scholar
  11. Kabala Z (2001) Sensitivity analysis of a pumping test on a well with wellbore storage and skin. Adv Water Resour 24(5):483–504CrossRefGoogle Scholar
  12. Kennedy J, Eberhart R (1995) Particle swarm optimization. Proc. IEEE Int’l. Conf. on Neural Networks (Perth, Australia), IEEE Service Center, vol IV. Piscataway, pp 1942–2948Google Scholar
  13. Lee T-C (1999) Applied mathematics in hydrology. Lewis PublishersGoogle Scholar
  14. Mathias S, Butler A, Zhan HB (2008) Approximate solutions for Forchheimer flow to a well. J Hydraul Eng 134(9):1318–1325CrossRefGoogle Scholar
  15. Neuman SP (1972) Theory of flow in unconfined aquifers considering delay response of the water table. Water Resour Res 8(4):1031–1045. CrossRefGoogle Scholar
  16. Papadopulos IS, Cooper HH (1967) Drawdown in a well of large diameter. Water Resour Res 3(1):241–244CrossRefGoogle Scholar
  17. Plischke E (2012) An adaptive correlation ratio method using the cumulative sum of the reordered output. Reliab Eng Syst Saf 107:149–156CrossRefGoogle Scholar
  18. Poli R (2007) An analysis of publications on particle swarm optimization applications. Tech. Rep. CSM-469, Department of Computing and Electronic Systems, University of Essex, Colchester, May–November 2007Google Scholar
  19. Poli R, Kennedy J, Blackwell T (2007) Particle swarm optimisation: an overview. Swarm Intell J 1(1):33–57.
  20. Polubariava-Kochina P (1962) Theory of ground water movement. Princeton University Press, Princeton, New Jersey, USA (translated by J.M. De Wiest)Google Scholar
  21. Qian J, Zhan H, Zhao W, Sun F (2005) Experimental study of turbulent unconfined groundwater flow in a single fracture. J Hydrol 311:134–142CrossRefGoogle Scholar
  22. Roose T, Fowler AC, Darrah PR (2001) A mathematical model of plant nutrient uptake. J Math Biol 42:347–360CrossRefGoogle Scholar
  23. Sen Z (1988) Analytical solution incorporating nonlinear radial flow in confined aquifers. Water Resour Res 24(4):601–606CrossRefGoogle Scholar
  24. Sen Z (1989) Nonlinear flow toward wells. J Hydraul Eng 115(2):193–209CrossRefGoogle Scholar
  25. Stehfest H (1970) Algorithm 368 numerical inversion of Laplace transforms. Commun ACM 13(1):47–49. CrossRefGoogle Scholar
  26. Storn R, Price K (1997) Differential evolution: a simple and efficient heuristic for global optimization over continuous spaces. J Global Optimization 11:341–359CrossRefGoogle Scholar
  27. Tarantola S, Kopustinskas V, Bolado-Lavin R, Kaliatka A, Uspuras E, Vaisnoras M (2011) Sensitivity analysis using contribution to sample variance plot: application to a water hammer model. Reliab Eng Syst Saf 99:62–73CrossRefGoogle Scholar
  28. Teh CI, Nie X (2002) Coupled consolidation theory with non-Darcian flow. Comput Geotech 29:169–209CrossRefGoogle Scholar
  29. Theis CV (1935) The relation between the lowering of the piezometric surface and the rate and duration of discharge of well using groundwater storage. Trans Amer Geophys Union 2:519–524CrossRefGoogle Scholar
  30. Venkataraman P, Rao PRM (2000) Validation of Forchheimer’s law for flow through porous media with converging boundaries. J Hydraul Eng-ASCE 126(1):63–71CrossRefGoogle Scholar
  31. Wang F, He X-S, Wang Y, Yang SM (2012) Markov model and convergence analysis based on cuckoo search algorithm. Comput Eng 38(11):180–185Google Scholar
  32. Wattenbarger RA, Ramey HJ Jr (1969) Well test interpretation of vertically fractures gas wells. J Petrol Tech AIME 246:625–632CrossRefGoogle Scholar
  33. Wen Z, Huang GH, Zhan HB (2008) An analytical solution for non-Darcian flow in a confined aquifer using the power law function. Adv Water Res 31(1):44–55CrossRefGoogle Scholar
  34. Wen Z, Huang GH, Zhan HB (2009) A numerical solution for non-Darcian flow to a well in a confined aquifer using the power law function. J Hydrol 364(1–2):99–106CrossRefGoogle Scholar
  35. Wen Z, Liu K, Zhan H (2014) Non-Darcian flow toward a larger-diameter partially penetrating well in a confined aquifer. Environ Earth Sci 72:4617. CrossRefGoogle Scholar
  36. Yang XS (2014) Nature-inspired optimization algorithms, 1st edn. Elsevier Science Publishers B. V, AmsterdamGoogle Scholar
  37. Yang XS, Deb S (2009) Cuckoo search via Lévy flights. In: Proceedings of world congress on nature & biologically inspired computing (NaBIC 2009). IEEE Publications, USA, pp 210–214CrossRefGoogle Scholar
  38. Zaharie D (2009) Influence of crossover on the behavior of the differential evolution algorithm. Appl Soft Comput 9(3):1126–1138CrossRefGoogle Scholar
  39. Zhan ZH, Zhang J, Li Y, Chung HS (2009) Adaptive particle swarm optimization. IEEE Trans Syst Man Cybern B Cybern 39(6):1362–1381. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Civil EngineeringHacettepe UniversityAnkaraTurkey

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