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# Buckley–Leverett equation

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.

## Assumptions for validity

The Buckley–Leverett equation is derived for a 1D sample given

## General solution

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