A generalized equation for the shape of the water table between two base levels

Abstract

The steady-state, water table profile of unconfined aquifer systems bounded by streams is examined theoretically, and an analytical solution is obtained that is more general than heretofore encountered in the literature. The solution incorporates the Dupuit-Forchheimer assumptions as applicable to free-surface flows, and includes a non-horizontal lower boundary and unequal drainage levels as variables. A computer program is given to obtain solutions to numerical problems. Some selected solutions are presented as non-dimensional type curves for determining the location and magnitude of the highest point on the water table between the two base levels. The dimensionless ratio of recharge to hydraulic conductivity, P, is found to determine the maximum elevation of the water table (hMAX). For a symmetrical water table aquifer, this ratio is critical in determining hMAX for values of dimensionless H (ratio of the base level height to distance between the two base levels) up to 0.05. For aquifers having unequal base levels, the dimensionless ratio of A/P (A = H1² - H2²; H1 = H₁/L; H2 = H₂/L) determines the location of hMAX (or XMAX). For values of A/P less than or equal to 0.1, hMAX is essentially located halfway between the two base levels. In the case of aquifers having equal base levels, but bounded by inclined lower boundaries, for any given slope, XMAX depends on H.