Multiphase Darcy and Mobility

Each Phase Has Its Own Mobility

When two phases share the pores, Darcy's law applies to each through its mobility $\lambda_p = \dfrac{k,k_{rp}}{\mu_p}$: the rock permeability times that phase's relative permeability, divided by its viscosity.

Total Mobility and Fractional Flow

The phase mobilities add to the total mobility, and the fraction of the flow that is water is its share of that total. Water mobility climbs as water saturation rises while oil mobility falls.

The Mobility Ratio

The endpoint mobility ratio compares the water behind the front with the oil ahead of it, $M = \dfrac{\lambda_w}{\lambda_o}$. At most one the displacement is stable and piston-like; above one the more mobile water fingers through and breaks through early, leaving oil behind.