Buckley–Leverett equation explained

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:

(\partial S _{w})/(\partial t) + (\partial)/(\partial x) \left( (Q)/(\phi A) 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)} = (λ _{w})/(λ _{w} +λ _{n}) = ((k _{rw})/(\mu _{w}))/((k _{rw})/(\mu _{w}) + (k _{rn})/(\mu _{n})) ,

where

λ _{w} and λ _{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)} = (Q)/(\phi A) (d f _{w})/(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} - 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

External links

• Buckley-Leverett Equation and Uses in Porous Media

Notes and References

1. S.E. Buckley and M.C. Leverett. Mechanism of fluid displacements in sands. Transactions of the AIME. 146. 107–116. 1942. 146 . 10.2118/942107-G . free.