Analytical solution for transient flow into a hemispherical auger hole

An explicit analytic solution for a transient groundwater flow to an auger hole of hemispherical shape is found on the base of ideas and under the same assumptions as in the original paper of Kirkham and van Bavel (1948). Theory of seepage into auger holes. Soil Sci. Soc. Am. Proc., 13, 75–82). The Legendre polynomial expansions are used to solve the Laplace equation with a combination of seepage face and equipotential conditions along the hole contour. A type curve for increasing water level in an initially empty hole is plotted, which allows for a fast determination of the hydraulic conductivity from the observation of filling of a small hole tapping a shallow water table. Time-decreasing total flow rates, velocities, particle trajectories, a capture zone of the hole are calculated using the Wolfram Mathematica software.