Buckley–Leverett Equation

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:

:$\frac + \frac\left( \frac f_w(S_w) \right) = 0,$

where $S_w(x,t)$ is the wetting-phase (water) saturation, $Q$ is the total flow rate, $\phi$ 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:

:$f_w(S_w) = \frac = \frac,$

where $\lambda_w$ and $\lambda_n$ denote the wetting and non-wetting phase mobilities. $k_(S_w)$ and $k_(S_w)$ 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:

:$U(S_w) = \frac \frac.$

The hyperbolic nature of the equation implies that the solution of the Buckley–Leverett equation has the form $S_w(x,t) = S_w(x - U t)$, 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

References

External links

Buckley-Leverett Equation and Uses in Porous Media

Conservation equations
Equations of fluid dynamics

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