Analytical solution and shape optimization for groundwater flow through a leaky porous trough subjacent to an aquifer

A. R. Kacimov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

25 Citations (Scopus)


Steady two-dimensional groundwater flow in a porous low-permeable trough is studied by the method of boundary-value problems for holomorphic functions. In the overlying highly permeable aquifer the hydraulic head varies linearly, i.e. flow is unidirectional. The exchange of groundwater between aquifer and trough does not affect the flow in the aquifer. It is assumed that through a horizontal aquifer-trough interface the head is transmitted into the trough, where the bounding effect of the bed causes circulatory seepage. Triangular troughs are studied and an isosceles form with a base angle of 38° is proved to have the highest circulation rate at a given cross-sectional area. In the class of arbitrary forms, solution to this optimal shape design problem is obtained by tackling the Schwartz and Signorini singular integrals and a unique and global maximum of the rate is found. The extreme curve coincides with an optimal soil channel of minimal seepage losses having depth to width ratio of 0.371. The global maximum differs from that one for the triangular class in 3% only that corroborates stability and robustness of the optimization criterion and makes possible isoperimetric estimates of the seepage intensity through arbitrary troughs.

Original languageEnglish
Pages (from-to)1409-1423
Number of pages15
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2069
Publication statusPublished - 2006


  • Aquifer
  • Holomorphic functions
  • Optimal shape design
  • Seepage
  • Signorini formula

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)


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