  We present a continuum (i.e., an effective) description of immiscible
two-phase flow in porous media characterized by two fields, the pressure and
the saturation. Gradients in these two fields are the driving forces that move
the immiscible fluids around. The fluids are characterized by two seepage
velocity fields, one for each fluid. Following Hansen et al.\ (Transport in
Porous Media, 125, 565 (2018)), we construct a two-way transformation between
the velocity couple consisting of the seepage velocity of each fluid, to a
velocity couple consisting of the average seepage velocity of both fluids and a
new velocity parameter, the co-moving velocity. The co-moving velocity is
related but not equal to velocity difference between the two immiscible fluids.
The two-way mapping, the mass conservation equation and the constitutive
equations for the average seepage velocity and the co-moving velocity form a
closed set of equations that determine the flow. There is growing experimental,
computational and theoretical evidence that constitutive equation for the
average seepage velocity has the form of a power law in the pressure gradient
over a wide range of capillary numbers. Through the transformation between the
two velocity couples, this constitutive equation may be taken directly into
account in the equations describing the flow of each fluid. This is e.g., not
possible using relative permeability theory. By reverse engineering relative
permeability data from the literature, we construct the constitutive equation
for the co-moving velocity. We also calculate the co-moving constitutive
equation using a dynamic pore network model over a wide range of parameters,
from where the flow is viscosity dominated to where the capillary and viscous
forces compete.
