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 average over the derivatives of the seepage velocities of the two immiscible fluids with respect to saturation. Based on analysis of relative permeability data and computational modeling, it has been proposed that the co-moving velocity is linear when plotted against the derivative of the average seepage velocity with respect to the saturation, the flow derivative. I show here that it is enough to demand that the co-moving velocity is characterized by an additive parameter in addition to the flow derivative to be linear. This has profound consequences for relative permeability theory as it leads to a differential equation relating the two relative permeabilities describing the flow. I present this equation together with two solutions.