Buckley-Leverett equation

An equation for fluid flow in a porous medium, proposed in [a1], that models the displacement of oil by water in a one-dimensional porous medium (e.g., sand). If $ S _ {w} = S _ {w} ( x,t ) $ denotes the saturation of water, i.e., the fraction of pore volume occupied by water, and $ \lambda _ {w} ( S _ {w} ) $ and $ \lambda _ {o} ( S _ {w} ) $ are the mobilities of water and oil, respectively, i.e., the ratio of relative permeability and viscosity, then the Buckley–Leverett equation reads:

$$ { \frac{\partial S _ {w} }{\partial t } } + { \frac \partial {\partial x } } \left ( { \frac{\lambda _ {w} ( S _ {w} ) }{\lambda _ {w} ( S _ {w} ) + \lambda _ {o} ( S _ {w} ) } } \right ) = 0, $$

$$ S _ {w} ( x,0 ) = S _ {0} ( x ) . $$

Under standard conditions, the flux function $ \lambda _ {o} ( S _ {w} ) / ( \lambda _ {w} ( S _ {w} ) + \lambda _ {o} ( S _ {w} ) ) $ has an "S" shape with one inflection point. The equation is a quasi-linear hyperbolic partial differential equation (cf. Quasi-linear hyperbolic equations and systems and Hyperbolic partial differential equation). In the case of displacement of oil by water, the initial data are such that they give rise to the Buckley–Leverett profile, consisting of a shock (cf. Shock waves, mathematical theory of) followed by a rarefaction wave. The solution method was simplified by H.J. Welge [a2], who showed that one could get the shock speed by constructing a particular tangent to the flux function. This technique is equivalent to the Rankine–Hugoniot condition.

The Buckley–Leverett equation is derived under the assumptions of incompressible fluids and immiscible flow. Capillary pressure and gravitation are neglected, and the permeable medium is assumed to obey Darcy's law, stating that the velocity of each phase is proportional to the gradient of its pressure. Let $ \phi $ denote the porosity of the medium, and $ \phi _ \alpha = \phi S _ \alpha $, where $ \alpha = o, w $ denotes the phase. Conservation of mass of each phase gives that

$$ { \frac \partial {\partial t } } ( \phi _ \alpha \rho _ \alpha ) + \nabla \cdot ( \phi _ \alpha \rho _ \alpha v _ \alpha ) = 0, $$

where $ v _ \alpha $ and $ \rho _ \alpha $ denote the velocity and density of each phase, respectively. Darcy's law reads:

$$ v _ \alpha = - K { \frac{\lambda _ \alpha }{\phi _ \alpha } } \nabla p _ \alpha , $$

where $ K $ is the (absolute) permeability and $ p _ \alpha $ the pressure of phase $ \alpha $. The pressures are identical in the absence of capillary forces. In one space dimension one can eliminate the pressure, and subsequently scale the time variable, making the total velocity constant, thereby obtaining the Buckley–Leverett equation.

References

[a1]	S.E. Buckley, M.C. Leverett, "Mechanism of fluid displacement in sands" Trans. AIME , 146 (1942) pp. 187–196
[a2]	H.J. Welge, "A simplified method for computing oil recovery by gas or water drive" Trans. AIME , 195 (1952) pp. 97–108