Buckley–Leverett equation

In fluid dynamics, the Buckley–Leverett equation is a conservation equation used to model two-phase flow in porous media.^[1] 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:

${\displaystyle {\frac {\partial S_{w}}{\partial t}}+{\frac {\partial }{\partial x}}\left({\frac {Q}{\phi A}}f_{w}(S_{w})\right)=0,}$

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

${\displaystyle f_{w}(S_{w})={\frac {\lambda _{w}}{\lambda _{w}+\lambda _{n}}}={\frac {\frac {k_{rw}}{\mu _{w}}}{{\frac {k_{rw}}{\mu _{w}}}+{\frac {k_{rn}}{\mu _{n}}}}},}$

where ${\displaystyle \lambda _{w}}$ and ${\displaystyle \lambda _{n}}$ denote the wetting and non-wetting phase mobilities. ${\displaystyle k_{rw}(S_{w})}$ and ${\displaystyle k_{rn}(S_{w})}$ denote the relative permeability functions of each phase and ${\displaystyle \mu _{w}}$ and ${\displaystyle \mu _{n}}$ 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:

${\displaystyle U(S_{w})={\frac {Q}{\phi A}}{\frac {\mathrm {d} f_{w}}{\mathrm {d} S_{w}}}.}$

The hyperbolic nature of the equation implies that the solution of the Buckley–Leverett equation has the form ${\displaystyle S_{w}(x,t)=S_{w}(x-Ut)}$, where ${\displaystyle U}$ is the characteristic velocity given above. The non-convexity of the fractional flow function ${\displaystyle f_{w}(S_{w})}$ 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