Английская Википедия:Buckley–Leverett equation
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:
- <math>
\frac{\partial S_w}{\partial t} + \frac{\partial}{\partial x}\left( \frac{Q}{\phi A} f_w(S_w) \right) = 0, </math>
where <math>S_w(x,t)</math> is the wetting-phase (water) saturation, <math>Q</math> is the total flow rate, <math>\phi</math> is the rock porosity, <math>A</math> is the area of the cross-section in the sample volume, and <math>f_w(S_w)</math> is the fractional flow function of the wetting phase. Typically, <math>f_w(S_w)</math> is an S-shaped, nonlinear function of the saturation <math>S_w</math>, which characterizes the relative mobilities of the two phases:
- <math>
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} }, </math>
where <math>\lambda_w</math> and <math>\lambda_n</math> denote the wetting and non-wetting phase mobilities. <math>k_{rw}(S_w)</math> and <math>k_{rn}(S_w)</math> denote the relative permeability functions of each phase and <math>\mu_w</math> and <math>\mu_n</math> 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:
- <math>U(S_w) = \frac{Q}{\phi A} \frac{\mathrm{d} f_w}{\mathrm{d} S_w}.</math>
The hyperbolic nature of the equation implies that the solution of the Buckley–Leverett equation has the form <math>S_w(x,t) = S_w(x - U t)</math>, where <math>U</math> is the characteristic velocity given above. The non-convexity of the fractional flow function <math>f_w(S_w)</math> also gives rise to the well known Buckley-Leverett profile, which consists of a shock wave immediately followed by a rarefaction wave.
See also
References
External links
