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Strategy for solving semi-analytically three-dimensional transient flow in a coupled N-layer aquifer system

Abstract

Efficient strategies for solving semi-analytically the transient groundwater head in a coupled N-layer aquifer system \({\phi _{i}(r,z,t)}\) , i = 1, ... , N, with radial symmetry, with full z-dependency, and partially penetrating wells are presented. Aquitards are treated as aquifers with their own horizontal and vertical permeabilities. Since the vertical direction is fully taken into account, there is no need to pose the Dupuit assumption, i.e., that the flow is mainly horizontal. To solve this problem, integral transforms will be employed: the Laplace transform for the t-variable (with transform parameter p), the Hankel transform for the r-variable (with transform parameter α) and a particular form of a generalized Fourier transform for the vertical direction z with an infinite set of eigenvalues \({\lambda _{m}^{2}}\) (with the discrete index m). It is possible to solve this problem in the form of a semi-analytical solution in the sense that an analytical expression in terms of the variables r and z, transform parameter p, and eigenvalues \({\lambda _{m}^{2}(p)}\) of the generalized Fourier transform can be given or in terms of the variables z and t, transform parameter α, and eigenvalues \({\lambda _{m}^{2}(\alpha )}\) . The calculation of the eigenvalues \({\lambda _{m}^{2}}\) and the inversion of these transformed solutions can only be done numerically. In this context the application of the generalized Fourier transform is novel. By means of this generalized Fourier transform, transient problems with horizontal symmetries other than radial can be treated as well. The notion of analytical solution versus numerical solution is discussed and a classification of analytical solutions is proposed in seven classes. The expressions found in this paper belong to Class 6, meaning that the transformed solutions are written in terms of eigenvalues which depend on one transform parameter (here p or α). Earlier solutions to the transient problem belong to Class 7, where the eigenvalues depend on two transform parameters. The theory is applied to three examples.

References

  1. 1

    Maas C (1987) Groundwater flow to a well in a layered porous medium 1. Steady flow. Water Resour Res 23(8): 1675–1681

    Article  ADS  Google Scholar 

  2. 2

    Veling EJM (1991) FLOP3N—pathlines in three-dimensional groundwater flow in a system of homogeneous anisotropic layers. Technical report Report nr. 719106001, R.I.V.M., National Institute of Public Health and Environmental Protection, Bilthoven, The Netherlands. http://www.citg.tudelft.nl/live/pagina.jsp?id=4f94d86a-e4eb-41b2-af1b-3441ed059de2&lang=en&binary=/doc/doc-f3n-t.pdf

  3. 3

    Veling EJM (1992) Three-dimensional groundwaterflow modelling for the calculation of capture zones around extraction sites. In: Hirsch Ch, Périaux J, Kordulla W (eds) Computational fluid dynamics, vol 2. In Proceedings of the first European computational fluid dynamics conference, September 7–11. Brussel, Elsevier, Amsterdam, pp 1013–1020

    Google Scholar 

  4. 4

    Neuman SP, Witherspoon PA (1969) Theory of flow in a confined two aquifer system. Water Resour Res 5(4): 803–816

    Article  ADS  Google Scholar 

  5. 5

    Hemker CJ, Maas C (1987) Unsteady flow to wells in layered and fissured aquifer systems. J Hydrol 90: 231–249

    Article  Google Scholar 

  6. 6

    Maas C (1987) Groundwater flow to a well in a layered porous medium 2. Nonsteady multiple-aquifer flow. Water Resour Res 23(8): 1683–1688

    Article  ADS  Google Scholar 

  7. 7

    Senda K, Tuzuki M (1966) Integration von anomalen linearen Anfangsrandwertaufgaben. Technol Rep Osaka Univ 16(689): 89–120

    Google Scholar 

  8. 8

    Senda K (1968) A family of integral transforms and some applications to physical problems. Technol Rep Osaka Univ 18(823): 261–286

    Google Scholar 

  9. 9

    Ölçer NY (1968) Theory of unsteady heat conduction in multicomponent finite regions. Ing Arch 36(5): 285–293

    MATH  Article  Google Scholar 

  10. 10

    Özişik MN (1980) Heat conduction. Wiley, New York

    Google Scholar 

  11. 11

    Mikhailov MD, Özişik MN (1984) Unified analysis and solutions of heat and mass diffusion. Dover Publications, Inc., New York

    Google Scholar 

  12. 12

    Maas C (1986) The use of matrix differential calculus in problems of multiple-aquifer flow. J Hydrol 88: 43–67

    Article  Google Scholar 

  13. 13

    Hemker CJ (1984) Steady groundwater flow in leaky multiple-aquifer systems. J Hydrol 72: 355–374

    Article  Google Scholar 

  14. 14

    Hemker CJ (1985) Transient well flow in multiple-aquifer systems. J Hydrol 81: 111–126

    Article  Google Scholar 

  15. 15

    Hemker CJ (1999a) Transient flow in vertically heterogeneous aquifers. J Hydrol 225: 1–18

    Article  Google Scholar 

  16. 16

    Hemker CJ (1999b) Transient well flow in layered aquifer systems: the uniform well-face drawdown solution. J Hydrol 225: 19–44

    Article  Google Scholar 

  17. 17

    Bruggeman GA (1999) Analytical solutions of geohydrological problems. Developments in Water Science, nr. 46. Elsevier, Amsterdam

  18. 18

    Sneddon IN (1972) The use of integral transforms. McGraw-Hill, New York

    MATH  Google Scholar 

  19. 19

    Mikhailov MD, Vulchanov NL (1983) Computational Procedure for Sturm–Liouville problems. J Comput Phys 50: 323–336

    MATH  Article  MathSciNet  ADS  Google Scholar 

  20. 20

    Zettl A (2005) Sturm–Liouville theory. Mathematical surveys and monographs, vol 121. American Mathematical Society, Providence

    Google Scholar 

  21. 21

    Wittrick WH, Williams FW (1971) A general algorithm for computing natural frequencies of elastic structures. Q J Mech Appl Math 24(1): 263–284

    MATH  Article  MathSciNet  Google Scholar 

  22. 22

    Stehfest H (1970) Algorithm 368, numerical inversion of Laplace transforms. Commun ACM 13(1,10): 47–49 and 624

    Article  Google Scholar 

  23. 23

    de Hoog FR, Knight JH, Stokes AN (1982) An improved method for numerical inversion of Laplace transforms. SIAM J Sci Stat Comput 3(3): 357–366

    MATH  Article  Google Scholar 

  24. 24

    Strack ODL (1989) Groundwater mechanics. Prentice Hall, Englewood Cliffs. Out of print, currently available by Strack Consulting Inc., St. Paul

  25. 25

    Furman A, Neuman SP (2003) Laplace-transform analytic element solution of transient flow in porous media. Adv Water Resour 26: 1229–1237

    Article  ADS  Google Scholar 

  26. 26

    Lenoach B, Ramakrishnan TS, Thambynayagam RKM (2004) Transient flow of a compressible fluid in a connected layered permeable medium. Trans Porous Media 57: 153–169

    Article  Google Scholar 

  27. 27

    Malama B, Kuhlman KL, Barrash W (2007) Semi-analytical solution for flow in leaky unconfined aquifer-aquitard systems. J Hydrol 346: 59–68. doi:101016/jjhydrol200708018

    Article  Google Scholar 

  28. 28

    Bruggeman GA, Veling EJM (2006) Nonmonotonic trajectories to a partially penetrating well in a semiconfined aquifer. Water Resour Res 42(2): W02501. doi:101029/2005WR003951

    Article  Google Scholar 

  29. 29

    Neuman SP (1972) Theory of flow in unconfined aquifers considering delayed response of the water table. Water Resour Res 8(4): 1031–1045

    Article  MathSciNet  ADS  Google Scholar 

  30. 30

    Abramowitz M, Stegun IA (eds) (1964) Handbook of mathematical functions. National Bureau of Standards, Washington, DC. http://www.math.hkbu.edu.hk/support/aands/tochtm

  31. 31

    Chaudhry AM, Zubair SM (2002) On a class of incomplete gamma functions with applications. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Veling, E.J.M., Maas, C. Strategy for solving semi-analytically three-dimensional transient flow in a coupled N-layer aquifer system. J Eng Math 64, 145 (2009). https://doi.org/10.1007/s10665-008-9256-9

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Keywords

  • Analytical solution
  • Coupled aquifers
  • Integral transformation
  • Transient flow