Self-similar radial two-phase flows

Abstract

A general mathematical formulation is developed, appropriate to both single- and two-phase conditions in a porous medium. A new similarity solution, generalizing the well known Theis solution, is derived for radial flow to a well in a region initially containing a two-phase mixture of steam and water, in which either steam or water is immobile. This generalized Theis solution follows from a new Ricatti equation for mass flow, which includes an additional nonlinear effect resulting from quadratic pressure gradient terms. Existing results for the saturation profile are extended by inclusion of nonlinear contributions, which are shown to be necessary for accurate descriptions of the saturation profile. A boundary-layer analysis is developed for flow about the well, where both mass flux and flowing enthaply are almost constant, which enables both the pressure and saturation profiles to be determined analytically. An analysis of two-phase self-similar shocks is given, together with the associated entropy conditions constraining the existence of shocks. Finally, numerical examples are discussed showing the agreement between theory and numerical simulations.