Phreatic seepage around a rectilinear cutoff wall: The Zhukovsky-Vedernikov-Polubarinova-Kochina dispute revisited

The problem of steady 2-D potential Darcian flow from a flat-bottomed reservoir of a coffer or other type of dam, separated from an empty tailwater by an impermeable vertical sheet piling, is one of the pillars in groundwater hydrology and soil mechanics. Analytical solutions to this problem are presented in the books by Forchheimer, Zhukovsky, Vedernikov, Polubarinova-Kochina, Harr, among others. For the case of seepage with a phreatic line, descending from the downstream face of the cutoff, the formulae presented in classical textbooks - despite the simplicity of conformal mappings of the complex potential quarter-plane onto the Zhukovsky half-strip - have lacunae and flaws. We discuss and clean the mismatch in the abovementioned formulae. The classical analytical solutions in the textbooks deal with the regime of no backwater, for which the magnitude of the specific discharge vector at infinity coincides with hydraulic conductivity of the soil. We obtain a general solution for most general flows with backwater, for which the specific discharge at infinity is zero and the free surface has an inflection point in the physical plane. The capillary fringe in the hodograph domain is imaged by a circular cut. The pressure head infinitely deep under the cutoff-wall is infinite. A circular tetragon in the hodograph domain is conformally mapped onto the quadrant in the complex potential plane via a reference plane. In the case with backwater, one extra condition is needed to fix a mathematical solution. For this purpose, the location of a point on the free surface is fixed. Mathematically, two affixes in the reference plane are found by solving a system of two nonlinear equations (in the no backwater case, only one affix is found and one equation is solved). We also examine flow with a phreatic line in a low-permeable layer (e.g. concrete apron of a dam) with an Λ-shaped topology of streamlines, bounded from the left by a cutoff. Seepage from a dam reservoir through a subjacent highly-permeable alluvium induces the pore water motion in the upper layer. A Hilbert boundary-value problem is solved for the complex potential in this cell. In the Kirkham-Brock type boundary condition along the interface between the two layers, the horizontal component of the specific discharge is constant. Phreatic surfaces rising on the downstream face of the cutoff caps Λ−shaped streamlines, which turn around a hinge point on the interface between two layers of contrasting permeability.