Mixing at the interface between two fluids in aquifer well upconing steady flow

Because of transverse pore-scale dispersion, a mixing layer develops along the interface between a lighter fluid (fresh water) that flows toward a well or sink above a heavier fluid (seawater) in a steady axisymmetric configuration. The Peclet number characterizing mixing, Pe = L/α_T, where L is the aquifer thickness and α_T is transverse dispersivity, is generally much larger than unity. The existence of a narrow transition zone calls for a boundary layer solution, which is obtained by the following steps: (1) the potential flow solution of the upper fluid above a sharp interface is assumed to be known; (2) the equations are reformulated with the potential and Stokes stream function of this flow serving as independent variables; (3) boundary layer approximate equations are formulated in terms of these variables; and (4) simple analytical solutions are obtained by von Kármán integral method. Approximate solutions are presented for a few configurations: a confined aquifer of constant thickness, a phreatic aquifer with constant recharge, and a point sink above the interface in an unbounded domain. In each case the thickness of the boundary layer and the salinity of the pumped water are determined analytically or by quadratures. The results are illustrated for a few particular cases.