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.
In a quasi-1D domain, the Buckley–Leverett equation is given by:
∂ S w ∂ t + ∂ ∂ x ( Q ϕ A f w ( S w ) ) = 0 , where S w ( x , t ) is the wetting-phase (water) saturation, Q is the total flow rate, ϕ 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 r w μ w k r w μ w + k r n μ n , where λ w and λ n denote the wetting and non-wetting phase mobilities. k r w ( S w ) and k r n ( S w ) denote the relative permeability functions of each phase and μ w and μ n represent the phase viscosities.
The Buckley–Leverett equation is derived based on the following assumptions:
Flow is linear and horizontalBoth wetting and non-wetting phases are incompressibleImmiscible phasesNegligible capillary pressure effects (this implies that the pressures of the two phases are equal)Negligible gravitational forcesThe characteristic velocity of the Buckley–Leverett equation is given by:
U ( S w ) = Q ϕ 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.
