Buckley-Leverett equation

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In fluid dynamics, the Buckley–Leverett equation is a transport equation used to model two phase flow in porous media. 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, $φ$ is porosity and $A$ is area of the cross-section in the sample volume.

[edit] 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
$d p c / d S = 0$ causing the pressure gradients of the two phases to be equal.

[edit] 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 $S$ .