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: 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: where \lambda_w and \lambda_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:

General solution

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, 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.