Buckley–Leverett 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, $\backslash phi$ 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 $\backslash lambda\_w$ and $\backslash lambda\_n$ 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 $\backslash mu\_w$ and $\backslash mu\_n$ represent the phase viscosities.

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

The characteristic velocity of the Buckley–Leverett equation is given by:

:$U(S\_w)\; =\; \backslash frac\{Q\}\{\backslash phi\; A\}\; \backslash frac\{\backslash mathrm\{d\}\; f\_w\}\{\backslash mathrm\{d\}\; S\_w\}.$

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.

• Capillary pressure
• Permeability (fluid)
• Relative permeability
• Darcy's law

{{reflist}}

• [http://perminc.com/resources/fundamentals-of-fluid-flow-in-porous-media/chapter-4-immiscible-displacement/buckley-leverett-theory/ Buckley-Leverett Equation and Uses in Porous Media]

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Category:Conservation equations

Category:Equations of fluid dynamics

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