Two-phase mixing in heterogeneous porous media

For a two-phase immiscible flow through a heterogeneous porous medium a macroscale model of first order is derived by a two-scale homogenization method while capturing the effects of fluid mixing. The capillary pressure is taken in consideration. An asymptotic two-scale homogenization method is applied which derives homogenization equations as a two-scale limit of the system when the medium heterogeneity tends to zero. The obtained macroscale flow equation has revealed that the mixing is manifested in the form of a nonlinear hydrodynamic dispersion and a transport velocity shift ("velocity renormalization"). The dispersion tensor is shown to be a nonlinear function of saturation. In the case of flow without gravity and without capillarity this function is proportional to the fractional flow derivative and depends on the viscosity ratio. The capillary forces change the structure of the dispersion tensor and the qualitative dependence on saturation. The case of fractured medium is also considered in the form of a periodic anisotropic network. In the case of asymptotically thin fractures the limit solution to the cell problem is shown to become non-unique due to a physical effect of stream configuration collapse in the nodes of fracture intersections. For a 2D periodic network, all the probable stream configurations are determined. The solution to the regularized problem and to the dispersion tensor is obtained in an analytical form. The longitudinal dispersion is the linear function of heterogeneity degree while the transverse dispersion is bounded. In the behaviour of the dispersion tensor singular regimes are revealed which are characterized by an infinite growth of dispersion. These regimes correspond to the trapping of a phase.