The seepage exclusion problem for parabolic and paraboloidal cavities

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Abstract
The problems of water exclusion from, or entry into, parabolic‐cylindrical and paraboloidal cavities during steady downward unsaturated seepage are solved. Both the quasi‐linear and the full nonlinear solutions are found exactly and simply. The identical general mathematical character of the two types of solution, and the good agreement of numerical details, attest to the reliability and utility of the (very much simpler) quasi‐linear analysis. A remarkable result is that for both the parabolic cylinder and the paraboloid, the potential at the cavity wall is spatially uniform. These geometries thus constitute separatices between apically blunter ones (circular cylinder, sphere), where the potential at the wall decreases downstream, and sharper ones (wedge, cone, hyperbolic cylinder, hyperboloid), where it increases. The parabolic geometries are optimal in leaking nowhere or everywhere: blunter ones leak first at the apex and and sharper ones at a low point of the wall. Implications for the design of seepage‐excluding tunnels and underground repositories are indicated. Comparison of the present results with those for the circular cylinder and the sphere reveals the significance of two dimensionless parameters of cavity geometry: σ(=½ ακ−1, with α sorptive number and κ apical total curvature); and ω, apical sharpness.

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