Transient 2-D seepage through a porous bed of a wadi, where a small-head dam is constructed, is studied with the help of conformal mappings (the Schwarz-Christoffel formula), based on the analytical solution obtained by Pavlovsky and Polubarinova-Kochina and used by geotechnical engineers a century ago. The dam has a vertical cutoff wall, placed at an arbitrary distance between the head water and tail water. Evaporation from the reservoir horizontal water surface, as well as pumping (or injection) are contributing to the mass balance in the head water that gives an ODE, which is solved either exactly (if no pumping or a constant rate pumping that gives either a linear or the Abel ODE, solved two centuries ago), or by a computer algebra routine of numerical integration (as the Cauchy problem with a “time front” indicating a complete depletion of the reservoir). Optimal design problems are solved, aiming at smart management of the reservoir water storage that will help mini-farms or camel-goat ranches on the wadi banks, which use the reservoir water for irrigation and animal husbandry in harsh environment of Arabia (or other hyperarid regions). Numerical modeling in HYDRUS2D demonstrates kinematics of saturated-unsaturated seepage for the cases of unclogged and clogged reservoir bed. Subsurface hydrology of cascades of dams, including groundwater recharged ones, is also discussed from the perspective of smart water retention and flow in ephemeral and intermittent streams of drylands.
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