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

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.

Automated summary

This article analytically studies axisymmetric, steady-state, Darcian flows in homogeneous and isotropic aquifers towards a toroid or disk intake. The study considers both unbounded and bounded aquifers. The results show that an air-filled toroid drains much more water than an equipotential disk, and the size of an empty toroid is much smaller than that of a water-filled one at a given flow rate. Additionally, the high hydraulic gradients in the vicinity of modeled wells/tunnels can lead to colmation and suffusion. The article also evaluates the dissipation and drawdown effects in a specified zone in the far field.