Self-similar condensing flows in porous media

Abstract

Similarity solutions are obtained for the propagation of a condensation wave into an initially dry porous matrix which receives an inflow of saturated vapor due to a step increase in temperature and pressure at the boundary. The generalized Darcy (low Reynolds number) formulation of two-phase flow leads to hyperbolic/parabolic equations in which capillarity and heat conduction are suppressed in order to emphasize the shock-like behavior. Application of the x/√ t similarity transformation gives ordinary differential equations which are solved by shooting methods, using jump-balance (Rankine-Hugoniot) conditions to preserve discontinuities in saturation (quality), pressure gradient and sometimes temperature. The distribution of condensate (saturation) is wave-shaped, with a forward-facing shock on the leading side. For a small temperature difference, there is little condensate and it is nearly immobile; the saturation shock lies close to the boundary, and the outer region is described by a reduced system of equations. With increasing temperature difference, the shock moves forward into the flow and gains in strength until the medium is liquid-full behind the shock. Beyond this, the shock splits into a pair of back-to-back shocks separated by a subcooled liquid slug. The considered prototypic problem is representative of a broad class of two-phase flows which occur in energy-related and geologic applications.