Phreatic surface flow from a near-reservoir saturated tongue

Steady and transient 2D Darcian flows in a saturated ‘tongue’ adjacent to a reservoir are studied analytically. First, a stable tongue receiving water from an inclined equipotential reservoir bed and losing moisture through a phreatic surface is considered. The hydraulic head is governed by the Laplace equation and the complex potential and complex coordinate are determined explicitly by the Polubarinova-Kochina method at an arbitrary bank slopes and evapotranspiration rates. In a particular case of a vertical slope, the tongue becomes a right-angled triangle extending into the layer for the same distance as the Dupuit-Forchheimer model predicts. Second, a saturated Dupuit-Forchheimer flow in the tongue is analyzed under the assumption of evaporation exponentially and linearly decreasing with the depth of a phreatic surface. The corresponding non-linear ordinary differential equation is integrated twice and predicts the length of the tongue as a function of the reservoir water level. Third, a transient regime is modelled by the Boussinesq equation with evaporation uniform in space, but varying cyclostationary with time. A straight-line water table translating upward–downward is found to be located always below the water table for steady regimes with an average evaporation.