## 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.

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

### Keywords

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