The Analytical Solution of the Boussinesq Equation for Flow Induced by a Step Change of the Water Table Elevation Revisited

Abstract

The classical problem of flow induced by a sudden change of the piezometic head in a semi-infinite aquifer is re-examined. A new analytical solution is derived, by combining an expression describing the water table elevation upstream, obtained by the Adomian’s decomposition approach, to an existing polynomial expression (Tolikas et al. in Water Resour Res 20:24–28, 1984), adequate for the downstream region; the parameters of both approximations are computed by matching the two solutions at the inflection point of the water table. Although several analytical solutions are available in the literature, we demonstrate that the expression we have developed in this issue is the most accurate for strong or moderate non-linear flows, where the degree of non-linearity is defined as the ratio of the piezometric head elevation at the origin to the initial water table elevation. For this type of flows the perturbation-series solution of Polubarinova-Kochina, characterized by previous studies as the best available analytical solution provides physically unacceptable results, while the analytical solution of Lockington (J Irrig Drain Eng 123:24–27, 1997), used to check the accuracy of numerical schemes, underestimates the penetration distance of the recharging front.