In fluid dynamics, the Buckley–Leverett equation is a conservation equation used to model two-phase flow in porous media.[1] 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:
\( {\displaystyle {\frac {\partial S_{w}}{\partial t}}+{\frac {\partial }{\partial x}}\left({\frac {Q}{\phi A}}f_{w}(S_{w})\right)=\phi {\frac {\partial S_{w}}{\partial x}},} \)
where\( {\displaystyle S_{w}(x,t)} \) 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 \( {\displaystyle f_{w}(S_{w})} \) is the fractional flow function of the wetting phase. Typically,\( {\displaystyle f_{w}(S_{w})} \) is an 'S'-shaped, nonlinear function of the saturation \( S_{w} \) , which characterizes the relative mobilities of the two phases:
\( {\displaystyle f_{w}(S_{w})={\frac {\lambda _{w}}{\lambda _{w}+\lambda _{n}}}={\frac {\frac {k_{rw}}{\mu _{w}}}{{\frac {k_{rw}}{\mu _{w}}}+{\frac {k_{rn}}{\mu _{n}}}}},} \)
where \( {\displaystyle \lambda _{w}} \) and \( \lambda _{n} \) denote the wetting and non-wetting phase mobilities. \( {\displaystyle k_{rw}(S_{w})} \) and \( {\displaystyle k_{rn}(S_{w})} \) denote the relative permeability functions of each phase and \( {\displaystyle \mu _{w}} \) 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:
\( {\displaystyle U(S_{w})={\frac {Q}{\phi A}}{\frac {\mathrm {d} f_{w}}{\mathrm {d} S_{w}}}.} \)
The hyperbolic nature of the equation implies that the solution of the Buckley–Leverett equation has the form \( {\displaystyle S_{w}(x,t)=S_{w}(x-Ut)} \) , where U {\displaystyle U} U is the characteristic velocity given above. The non-convexity of the fractional flow function \( {\displaystyle f_{w}(S_{w})} \) also gives rise to the well known Buckley-Leverett profile, which consists of a shock wave immediately followed by a rarefaction wave.
See also
Capillary pressure
Permeability (fluid)
Relative permeability
Darcy's law
References
S.E. Buckley and M.C. Leverett (1942). "Mechanism of fluid displacements in sands". Transactions of the AIME (146): 107–116.
Hellenica World - Scientific Library
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