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, \(\phi\) 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.
• Flow is Linear
• Flow is Steady-State
• Formation is one Layer

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 \(S\).

See also

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