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.

is the wetting-phase (water) saturation,

is the total flow rate,

is the area of the cross-section in the sample volume, and

is the fractional flow function of the wetting phase.

is an S-shaped, nonlinear function of the saturation

, which characterizes the relative mobilities of the two phases: where

denote the wetting and non-wetting phase mobilities.

denote the relative permeability functions of each phase and

represent the phase viscosities.

The Buckley–Leverett equation is derived based on the following assumptions: The characteristic velocity of the Buckley–Leverett equation is given by: The hyperbolic nature of the equation implies that the solution of the Buckley–Leverett equation has the form

The non-convexity of the fractional flow function

also gives rise to the well known Buckley-Leverett profile, which consists of a shock wave immediately followed by a rarefaction wave.

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