en>Wahrmund: /* Proof of impossibility */ Missing word.

2014-02-01T15:11:25Z

Proof of impossibility: Missing word.

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In [[fluid dynamics]], the '''Buckley–Leverett equation''' is a [[Generic scalar transport equation|transport equation]] used to model [[two-phase flow]] in [[porous media]].<ref>{{cite journal|author=S.E. Buckley and M.C. Leverett|title=Mechanism of fluid displacements in sands|url=http://www.aimehq.org/library/books/Petroleum%20Technology,%201941,%20Vol.%20IV/T.P.%201337.pdf|journal=Transactions of the AIME|issue=146|pages=107–116|year=1942}}</ref> The Buckley–Leverett equation or the Buckley–Leverett ''displacement'' can be interpreted as a way of incorporating the microscopic effects due to [[capillary pressure]] in [[two-phase flow]] into [[Darcy's law]].

In a 1D sample ([[control volume]]), let <math>S(x,t)</math> be the [[water saturation]], then the Buckley–Leverett equation is

:<math>\frac{\partial S}{\partial t} = U(S)\frac{\partial S}{\partial x}</math>

where

:<math>U(S) = \frac{Q}{\phi A} \frac{\mathrm{d} f}{\mathrm{d} S}.</math>

<math>f</math> is the fractional flow rate, <math>Q</math> is the total flow, <math>\phi</math> is [[porosity]] and <math>A</math> is area of the cross-section in the sample volume.

==Assumptions for validity==
The Buckley–Leverett equation is derived for a 1D sample given
* [[Conservation of mass|mass conservation]]
* [[capillary pressure]] <math>p_c(S)</math> is a function of [[water saturation]] <math>S</math> only
* <math>\mathrm{d}p_c/\mathrm{d}S = 0</math> causing the [[pressure gradients]] of the two phases to be equal.
* Flow is Linear
* Flow is Steady-State
* Formation is one Layer

==General solution==

The solution of the Buckley–Leverett equation has the form <math>S(x,t) = S(x+U(S)t)</math> which means that <math>U(S)</math> is the [[front velocity]] of the fluids at saturation <math>S</math>.

==See also==
* [[Capillary pressure]]
* [[Permeability (fluid)]]
* [[Relative permeability]]
* [[Darcy's law]]

==References==
{{reflist}}

{{DEFAULTSORT:Buckley-Leverett equation}}
[[Category:Hyperbolic partial differential equations]]
[[Category:Equations of fluid dynamics]]

{{fluiddynamics-stub}}

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en>Wahrmund: /* Proof of impossibility */ Missing word.