The Buckley-Leverett Displacement

How the Front Advances

Conservation of water in a slice of the reservoir gives the frontal advance equation of Buckley and Leverett (1942):

$\phi,\frac{\partial S_w}{\partial t} + q,\frac{\partial f_w}{\partial x} = 0.$

This says a given water saturation travels through the reservoir at a speed proportional to the slope of the fractional flow curve, $f_w'(S_w)$. Measuring time in pore volumes injected (PVI), a saturation $S_w$ reaches the dimensionless position $x = f_w'(S_w),\text{PVI}$.

Because $f_w'$ is largest at intermediate saturations, faster saturations would overtake slower ones, which is unphysical. The resolution is a shock, and the Welge tangent locates it: the straight line from the connate-water point $(S_{wc}, 0)$ tangent to the $f_w$ curve touches it at the shock-front saturation $S_{wf}$. The front reaches the producer, which is breakthrough, at $\text{PVI}${bt} = \frac{1}{f_w'(S{wf})}.

Press Play and watch the front sweep across. Before breakthrough the water cut is zero and recovery climbs steeply, because every volume of water displaces a volume of oil; at breakthrough the water cut jumps and the recovery curve bends over. The simulated front (solid) lags and smears relative to the analytical front (dashed): that smearing is numerical diffusion, the artifact of representing a sharp shock on a finite grid, and the biggest reason coarse grids blur fronts. Raise the oil viscosity and the mobility ratio goes unfavorable, the front weakens, and water breaks through earlier.