Buckley–Leverett equation

In fluid dynamics, the Buckley–Leverett equation is a transport equation used to model two-phase flow in porous media [cite journal|author=S.E. Buckley and M.C. Leverett|title=Mechanism of fluid displacements in sands|journal=Transactions of the AIME|issue=146|pages=107–116|date=1942] . The Buckley–Leverett equation or the Buckley–Leverett "displacement" can be interpreted as a way of incorporating the microscopic effects to due capillary pressure in two-phase flow into Darcy's law.

In a 1D sample (control volume), let $S(x,t)$ be the water saturation, then the Buckley–Leverett equation is

:$frac\{partial\; S\}\{partial\; t\}\; =\; U(S)frac\{partial\; S\}\{partial\; x\}$

where

:$U(S)\; =\; frac\{Q\}\{phi\; A\}\; frac\{mathrm\{d\}\; f\}\{mathrm\{d\}\; S\}.$

$f$ is the fractional flow rate, $Q$ is the total flow, $phi$ is porosity and $A$ is area of the cross-section in the sample volume.

Assumptions for validity

The Buckley–Leverett equation is derived for a 1D sample given
* mass conservation
* capillary pressure $p\_c(S)$ is a function of water saturation $S$ only
* $mathrm\{d\}p\_c/mathrm\{d\}S\; =\; 0$ causing the pressure gradients of the two phases to be equal.

General solution

The solution of the Buckley–Leverett equation has the form $S(x,t)\; =\; S(x-U(S)t)$ which means that $U(S)$ is the front velocity of the fluids at saturation $S$.

ee also

* Capillary pressure
* Permeability (fluid)
* Relative permeability
* Darcy's law

References