Buckley-Leverett equation

In the yet further specialisation of neglecting gravity effects one arrives at the celebrated Buckley-Leverett equation, [28],... 

There is not enough space to describe the properties of this equation. Suffice it to say that the Buckley-Leverett equation has shock-like solutions, where the saturation front is a wave propagating through the reservoir. This combination of an elliptic equation for the total pressure and a parabohc, but nearly hyperbolic equation for the saturation, gives rise to great mathematical interest in two-phase flow though porous media. 

Then, the equations describing reservoir fluid flow are given by the Buckley-Leverett equation... 

Here, we solve the Buckley-Leverett equation (27) with the total velocity field a given in (31) and the fractional flow specified in (30). We set the radius of the injection wells to i = 0.05 and use the initial condition... 

The front is inherently unstable, however, and this is often studied by a linear stability analysis. Infinitesimal perturbations are applied to all of the variables to simulate reservoir heterogeneities, density fluctuations, and other effects. Just as in the Buckley-Leverett solution, the perturbed variables are governed by force and mass balance equations, and they can be solved for a perturbation of any given wave number. These solutions show whether the perturbation dies out or if it grows with time. Any parameter for which the perturbation grows indicates an instability. For water flooding, the rate of growth, B, obeys the proportionality... 

The mathematical formulation of this model was made by Dever-eux 36) based on the classical Buckley-Leverett theory for two-phase flow in porous media (49) and equations developed by Scheidegger (44). They solve a set of eight equations ... 

The waterflood front is given by the classical Buckley-Leverett theory by drawing the tangent to the f versus S. curve from (S c. 0), as shown in Figure 2.16. The corresponding equation is... 

The trapping number defined by Eq. 7.103 for an arbitrary dipping angle is consistent with the conventional Buckley-Leverett fractional flow theory. In the Buckley-Leverett fractional flow equation, the gravity term is multiplied by sina (Leverett, 1941). However, Figure 7.34 shows that the trapped residual saturation predicted by Eq. 7.103 is lower than the experimental data at the same trapping number. This figure compares the relationship between the... 

This jump condition is analogous to the global mass conservation constraint enforced in the Buckley-Leverett problem (e.g., via Welge s construction )-Exact conservation laws like Equation 13-11 are just one consequence of complete models like Equation 13-9, with the exphcit form of the high-order derivative term available. Its algebraic structure controls the form of energy-like quantities that are dissipated across discontinuities. For example, multiply Equation 13-9 by u(x) throughout, so that u2 daJdx = eu This can be... 

Here, and be taken as the displaced and displacing fluids, respectively, and k and k are their relative permeabilities, which are functions of the voffime fractions (i.e., saturations) of the fluids. As shown by Buckley and Leverett, the solution to Equation 4 gives a plot of saturation versus length along the flow direction that has a discontinuity (17). (The saturation of a fluid phase is its volume divided by the sum of the fluid volumes.) This discontinuity is often referred to as a shock front, and the flood is described as a piston-like displacement. 

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