Dupuit-Ghyben-Herzberg analysis of strip-island lenses

Abstract

Dupuit-Ghyben-Herzberg analysis follows from combination of the continuity equation and Darcy's Law with the Ghyben-Herzberg Principle and the Dupuit assumptions of horizontal flow. The analysis is used to derive the position of the water table and salt-water interface in island lenses in terms of island geometry, distribution of hydraulic conductivity (K), and distribution of recharge (R). For small islands and cases for which the salt-water head is zero, application of Dupuit-Ghyben-Herzberg analysis gives good results because the low R/K ratios of natural lenses assure that height/width ratios of natural lenses are extremely low, 1:30 to 1:100. In earlier publications, the Dupuit-Ghyben-Herzberg differential equation has been integrated with the boundary condition that the water table and interface meet at sea level at the shoreline (that is, no outflow face). The dimensions of an outflow face, however, are known from results of potential-theory analysis. Integration of the Dupuit- Ghyben-Herzberg equation with a boundary condition consistent with the potential-theory outflow face (a) places the interface indistinguishably close to its position given by potential theory and (b) allows calculation of Ghyben-Herzberg ratios (interface depth to water-table elevation), which in this case, depart from their usual value of 40 because equipotentials are curved close to the outflow face. For natural-sized lenses, however, such analysis is necessary only within a narrow strip within 1% to 5% of the island width of the shoreline. Outside this strip, the regular analysis that ignores the presence of the outflow face positions the interface indistinguishably close to that of the potential-theory solution. Analytical solutions are developed for a number of infinite-strip islands. It is shown by analysis of the homogeneous, rectangular-island case that an island can be considered an infinite strip (to 0.1% accuracy) if its length/width ratio is larger than 4.4. Asymmetric lenses occur if the island is composed of strips of different K or different R, with the greater asymmetry occurring with differences in K. A high-K basement compresses the root of the lens and thereby decreases the water table in the island. A lens perched on impermeable basement has a higher water table than would otherwise occur in the island, but the volume of the lens is less.