Buckley–Leverett equation

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:



\frac{\partial S_w}{\partial t} + \frac{\partial}{\partial x}\left( \frac{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) = \frac{\lambda_w}{\lambda_w + \lambda_n} = \frac{ \frac{k_{rw}}{\mu_w} }{ \frac{k_{rw}}{\mu_w} + \frac{k_{rn}}{\mu_n} }, $$

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:
 * 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:


 * $$U(S_w) = \frac{Q}{\phi A} \frac{\mathrm{d} f_w}{\mathrm{d} S_w}.$$

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.