ON AN AVERAGED MODEL FOR THE 2-FLUID IMMISCIBLE FLOW WITH SURFACE TENSION IN A THIN CYLINDRICAL TUBE

Abstract

We consider the two-fluid incompressible immiscible flow, with surface tension, in a thin cylindrical tube. Using the method of asymptotic expansions with respect to the thickness of the domain, effective 1D equations are derived. Supposing that the fluids are strictly separated at t=0 and that only one phase is in contact with the rigid wall, we show that the derived equations equal a nonlinear degenerate fourth order parabolic equation for the saturation. If we set the surface tension to be zero, it reduces to the Buckley–Leverett equation recently derived by Mikeli and Paoli using the same technique. In the general case we obtain the generalized Darcy's equations with extra-diagonal terms and the relative permeabilities are calculated explicitly. The capillary pressure results only from te surface tension between two fluid phases.