Speed of Propagation for Some Models of Two-Phase Flow in Porous Media
Christopher E. Kees · 2004
Abstract. Flow of two immiscible, incompressible uids in a porous medium is typ-ically described by a nonlinear advection-diusion equation for one of the uid satura-tions. The diusion coecient, which represents the eect of capillary forces on the u-ids, is zero when the medium is locally saturated by either uid since in these limiting cases the eects of capillary forces tend to zero. This degeneracy in the second-order term usually gives rise to the qualitative property that perturbations in saturation propagate with nite speed through regions that are fully saturated by either uid. This qualita-tive property is physically realistic. In this work we show that, under certain choices of constitutive relations and modeling approximations, the nite speed of propagation prop-erty is lost, despite the fact that the diusion coecient is degenerate. The loss of -nite speed of propagation is due to unbounded derivatives in the closure relations as the medium becomes saturated by wetting phase. We present analytical and numerical so-lutions, compare solution dynamics that display nite and innite speed of propagation, and provide a brief account of numerical diculties related to the degenerate coecients. 1.