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.
