Mobility Ratio and Fractional Flow

Mobility and the Mobility Ratio

When injected water pushes oil through the rock, two fluids of very different thicknesses are in a race. The mobility of a phase measures how readily it flows: the relative permeability of that phase divided by its viscosity, $\lambda = k_r/\mu$. Water is thin (low viscosity) and oil is thick (high viscosity), so water is naturally the more mobile of the two.

The mobility ratio compares the mobility of the displacing water behind the front to the mobility of the oil ahead of it, evaluated at their end-point saturations:

$M = \frac{\lambda_w}{\lambda_o} = \frac{k_{rw}^{0}/\mu_w}{k_{ro}^{0}/\mu_o}.$

When $M \le 1$ the displacement is favorable: the oil moves at least as easily as the water, so the water pushes the oil ahead of it like a piston, giving a sharp front and good sweep. When $M > 1$ it is unfavorable: the water is more mobile than the oil it is meant to push, so it fingers through and around the oil, breaks through early, and leaves large patches unswept. Lowering $M$, for example by thickening the water with polymer, is one of the central ideas of improved recovery.

Fractional Flow

To predict how a waterflood advances we need the fraction of the flowing stream that is water at any point. For one dimensional, horizontal, incompressible flow with no capillary pressure, combining the Darcy law for each phase gives the fractional flow of water:

$f_w = \frac{q_w}{q_w + q_o} = \frac{1}{,1 + \dfrac{k_{ro},\mu_w}{k_{rw},\mu_o},}.$

Because $k_{rw}$ rises and $k_{ro}$ falls as water saturation increases, $f_w$ runs from 0 at the connate-water saturation $S_{wc}$ up to 1 near the residual-oil saturation, in a characteristic S shape. Raise the oil viscosity and the curve bulges toward the upper left, the mobility ratio turns unfavorable, and the front weakens.