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:

[math]\displaystyle{ \frac{\partial S_w}{\partial t} + \frac{\partial}{\partial x}\left( \frac{Q}{\phi A} f_w(S_w) \right) = 0, }[/math]

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

[math]\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} }, }[/math]

where [math]\displaystyle{ \lambda_w }[/math] and [math]\displaystyle{ \lambda_n }[/math] denote the wetting and non-wetting phase mobilities. [math]\displaystyle{ k_{rw}(S_w) }[/math] and [math]\displaystyle{ k_{rn}(S_w) }[/math] denote the relative permeability functions of each phase and [math]\displaystyle{ \mu_w }[/math] and [math]\displaystyle{ \mu_n }[/math] 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:

[math]\displaystyle{ U(S_w) = \frac{Q}{\phi A} \frac{\mathrm{d} f_w}{\mathrm{d} S_w}. }[/math]

The hyperbolic nature of the equation implies that the solution of the Buckley–Leverett equation has the form [math]\displaystyle{ S_w(x,t) = S_w(x - U t) }[/math], where [math]\displaystyle{ U }[/math] is the characteristic velocity given above. The non-convexity of the fractional flow function [math]\displaystyle{ f_w(S_w) }[/math] 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