Variable‐density saturated flow with modified Darcy's law: The salt lake problem and circulation

Abstract

For unsteady variable‐density seepage flow, alternative solutions are obtained by taking, respectively, the curl and the divergence of a linear form of Darcy's law, and solving each problem directly, using compatible boundary conditions. This gives a vector potential formulation depending upon the horizontal density gradient, and a pressure formulation depending upon the vertical density gradient, resulting in two complementary solutions. Two velocity fields are obtained by taking the curl of the vector potential solution, and by solving Darcy's law using the gradient of the pressure solution, and corresponding vector potentials are obtained, fairly symmetrically, from these velocities. The novelty is that a linear combination of the two solutions can be made by simple addition or subtraction, with independent scalar coefficients, having broader scope than each of the alternative solutions alone. A two‐dimensional model, based on convective plumes in a Hele‐Shaw experiment with a macroscopic Rayleigh number of 3975, is treated as a benchmark salt lake problem, having a uniform evaporation layer with 1% noise along one‐third part of the upper boundary, with appropriate saline recharge. The coefficients are optimized for maximum circulation. This determines the ratio of the pressure‐based solution to the vector potential–based solution, modifying the Rayleigh number downward to an effective value of 3455. Numerical streamlines reveal secondary flow typical of Henry circulation, measured by a peak stream function equal to the circulation flux. From finger geometry, there is better agreement between the numerically calculated plumes and the experimental plumes than has been achieved previously.