Skip to main content

# 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

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

4. 4

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

5. 5

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

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

7. 7

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

8. 8

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

9. 9

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

10. 10

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

11. 11

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

12. 12

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

13. 13

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

14. 14

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

15. 15

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

16. 16

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

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

19. 19

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

20. 20

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

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

22. 22

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

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

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

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

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

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

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

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

Download references

## Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.

## Author information

Authors

### Corresponding author

Correspondence to E. J. M. Veling.

## Rights and permissions

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.

Reprints and Permissions

## About this article

### Cite this article

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

Download citation

• Received:

• Accepted:

• Published:

• DOI: https://doi.org/10.1007/s10665-008-9256-9

### Keywords

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