Analytical solutions for steady-state, axisymmetric seepage to toroidal and disk-shaped drainages and tunnels: the Gauss and Weber legacy revisited

Anvar Kacimov*, Georg Houben

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Axisymmetric, steady-state, Darcian flows in homogeneous and isotropic aquifers towards a toroid or disk intake are analytically studied. Both unbounded (infinite) and bounded (by an equipotential soil surface or by an impermeable horizontal caprock-bedrock) aquifers are considered. The Gauss closed-form solution from astronomy for a gravitating circle having a uniform mass distribution and the Weber solution from electrostatics for an equipotential disk are utilized. The scalar/vector fields of piezometric head (potential)/specific discharge allow for reconstruction of stream lines, isobars, isochrones, and isotachs. An air-filled toroid drains much more water than equipotential, or–inversely–at a given flow rate, the size of an empty toroid is much smaller than that of a water-filled one. The hydraulic gradients in the vicinity of modelled wells/tunnels are very high, triggering colmation and suffusion. The functionals of dissipation and drawdown over a specified zone in the far field are evaluated.

Original languageEnglish
Pages (from-to)938-953
Number of pages16
JournalHydrological Sciences Journal
Issue number7
Publication statusPublished - May 4 2023


  • Dirichlet boundary value problems
  • dissipation and integral drawdown functionals
  • Gaussian quasi-toroidal tunnel
  • gravitational and specific discharge potentials
  • streamlines–isotachs–isochrones
  • superposition principle for harmonic function of piezometric head
  • Weber’s disk-shaped well

ASJC Scopus subject areas

  • Water Science and Technology

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