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Buckley–Leverett equation
Conservation law for two-phase flow in porous media From Wikipedia, the free encyclopedia
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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.
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Equation
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In a quasi-1D domain, the Buckley–Leverett equation is given by:
where is the wetting-phase (water) saturation, is the total flow rate, is the rock porosity, is the area of the cross-section in the sample volume, and is the fractional flow function of the wetting phase. Typically, is an S-shaped, nonlinear function of the saturation , which characterizes the relative mobilities of the two phases:
where and denote the wetting and non-wetting phase mobilities. and denote the relative permeability functions of each phase and and represent the phase viscosities.
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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
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The characteristic velocity of the Buckley–Leverett equation is given by:
The hyperbolic nature of the equation implies that the solution of the Buckley–Leverett equation has the form , where is the characteristic velocity given above. The non-convexity of the fractional flow function also gives rise to the well known Buckley-Leverett profile, which consists of a shock wave immediately followed by a rarefaction wave.
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