Buckley–Leverett equation

Short description: Conservation law for two-phase flow in porous media

In fluid dynamics, the Buckley–Leverett equation is a conservation equation used to model two-phase flow in porous media. The Buckley–Leverett equation or the Buckley–Leverett displacement describes an immiscible displacement process, such as the displacement of oil by water, in a one-dimensional or quasi-one-dimensional reservoir. This equation can be derived from the mass conservation equations of two-phase flow, under the assumptions listed below.

Equation

In a quasi-1D domain, the Buckley–Leverett equation is given by:

$\frac{\partial S_w}{\partial t}+\frac{\partial }{\partial x}(\frac{Q}{\varphi A}{f}_w(S_w))=0,$

where $S_w(x,t)$ is the wetting-phase (water) saturation, $Q$ is the total flow rate, $\varphi$ is the rock porosity, $A$ is the area of the cross-section in the sample volume, and $f_w(S_w)$ is the fractional flow function of the wetting phase. Typically, $f_w(S_w)$ is an S-shaped, nonlinear function of the saturation $S_w$, which characterizes the relative mobilities of the two phases:

$f_w(S_w)=\frac{{\lambda }_w}{{\lambda }_w+{\lambda }_n}=\frac{\frac{k_{rw}}{{\mu }_w}}{\frac{k_{rw}}{{\mu }_w}+\frac{k_{rn}}{{\mu }_n}},$

where ${\lambda }_w$ and ${\lambda }_n$ denote the wetting and non-wetting phase mobilities. $k_{rw}(S_w)$ and $k_{rn}(S_w)$ denote the relative permeability functions of each phase and ${\mu }_w$ and ${\mu }_n$ represent the phase viscosities.

Assumptions

The Buckley–Leverett equation is derived based on the following assumptions:

• Flow is linear and horizontal
• Both wetting and non-wetting phases are incompressible
• Immiscible phases
• Negligible capillary pressure effects (this implies that the pressures of the two phases are equal)
• Negligible gravitational forces

General solution

The characteristic velocity of the Buckley–Leverett equation is given by:

$U(S_w)=\frac{Q}{\varphi A}\frac{\mathrm{d}{f}_w}{\mathrm{d}{S}_w}.$

The hyperbolic nature of the equation implies that the solution of the Buckley–Leverett equation has the form $S_w(x,t)=S_w(x-Ut)$, where $U$ is the characteristic velocity given above. The non-convexity of the fractional flow function $f_w(S_w)$ also gives rise to the well known Buckley-Leverett profile, which consists of a shock wave immediately followed by a rarefaction wave.

See also

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

External links

• Buckley-Leverett Equation and Uses in Porous Media