Miscible displacement in a Hele-Shaw cell (Q1198202)

A theory of viscous isothermal gravity driven flow in a Hele-Shaw cell model for two miscible incompressible each one homogeneous and isotropic liquids (such as glycerin displacing water or vice versa) is presented. The governing time and space depending partial differential equations for the laminar velocity field, pressure and volume concentration distribution of the mixture are given and discussed in detail. They are then solved under further simplifying assumptions, for given density and viscosity dependence on the concentration and appropriate initial and boundary conditions. The crucial assumption of non-solenoidal mass averaged velocity field, as it is correct from physical point of view by this problem, enables concentration gradient stresses (Korteweg's stresses) to be taken into account, too. The stability problem of this flow is also studied and solved for the case of steady miscible displacement. The numerical methods employed for solving the final partial differential equation of the problem are only mentioned. As one of the possible applications of this theory could be the problem of miscible displacement in oil-bearing sand.