Linearity of the co-moving velocity

The co-moving velocity is a new variable in the description of immiscible two-phase flow in porous media. It is the saturation-weighted derivatives of the seepage velocities of the two immiscible fluids with respect to saturation. I show that it is linear in the derivative of the average seepage velocity with respect to the saturation. This is in accordance with recent measurements. Two parameters are needed to describe it. In terms of relative permeability theory, this linear relation leads to a differential equation relating the two relative permeabilities describing the flow. I present this equation together with a solution.