The Saint-Venant type isoperimetric inequalities for assessing saturated water storage in lacunary shallow perched aquifers

The Dupuit-Forchheimer (DF) approximation for unconfined groundwater flows is reduced to 2D Poisson's equation (PE), the right hand side of which involves an intensive evapotranspiration (ET) rate. A shallow water table dips inward from a constant piezometric head boundary (closed curve) such that a “dry gap,” demarcated by another closed curve (unknown front), may emerge. Apriori estimates of the volume of the saturated zone and of the “dry gap” area are important for water resources management in drylands. We use the results from the theory of linear elasticity, torsion of elastic bars, for which PE is solved for the Prandtl function. Using the Poincaré metrics with the Gaussian constant curvature (Formula presented.), isoperimetric inequalities are obtained for steady-state DF flows. Conformal moments and confromal radii of the domains and 2-D Hardy's type inequalities in domains, modeling the Saint Venant bar torsion problem, are involved in the obtained estimates. The studied boundary value problems (BVPs) are nonlinear for ET rates depending on the depth of the water table. In the DF model, the vadose zone (VZ) is considered as a “distributed sink” (similar to a standard “distributed source,” which models recharge to the water table in humid climates). The analytical VZ is collated with one obtained from a numerical solution to BVP for Richards’ equation in a 3-D saturated-unsaturated flow. BVPs for Richards’ equation in cylindrical domains, solved by HYDRUS software, give the pressure head, moisture content, Darcian velocity fields, and streamlines.