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

ABSTRACT

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.

KEYWORDS:

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

Editor A. Fiori; Associate Editor G. Chiogna

Editor A. Fiori; Associate Editor G. Chiogna

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ We recall that a toroid is any 3D shape created by rotating a 2D shape about an axis outside the shape. A torus is specifically a toroid that is made by rotating a circle. Our quasi-torus is a toroid obtained by rotation of an almost circular curve.

² Other packages, e.g. Matlab or Python, can be used instead of Mathematica.

³ It is noteworthy that in applications to astronomy, the model of a “flat Earth” with a homogeneous distribution of the planet mass over an infinitely thin disk surface gives a non-constant distribution of the Newtonian potential over the disk surface (see e.g. Duboshin 1961).:

⁴ We recall that groundwater flow is assumed to be steady here. For transient seepage flows through elastic aquifers and elastic groundwater, the velocity potential obeys the diffusion (a parabolic PDE) rather than Laplace’s (an elliptic PDE) equation. In astronomy, the Newtonian potential remains a harmonic function in the space outside gravitating bodies, even if these bodies ramble, provided the velocities remain relatively small.

Additional information

Funding

This work was supported by the grants DR\RG\17, IG/VC/WRC/21/01 and IG/AGR/SWAE/22/02, Sultan Qaboos University, Oman. Helpful comments by two anonymous referees are appreciated.