On the derivation of the Buckley—Leverett model from the two fluid Navier—Stokes equations in a thin domain

We consider a two-phase incompressible immiscible flow, with neglected surface tension, in a thin domain. 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, we show that the derived equations equal the Buckley–Leverett equation for the saturation, in conjunction with the Reynolds law for each of the flowing phases. The fractional flow curve and corresponding relative permeabilities are determined explicitly and their heuristically observed properties verified.