Buckley-Leverett

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 ... 

In the original Buckley-Leverett theory, gravitational, compressibility and capillarity are ignored. Devereux (36) presents the solution for the case of constant pressure, and the constant-velocity case was derived by Soo and Radke 12). The model requires a knowledge of the capillary retarding force per unit volume of the porous medium, and the relative permeabilities of the oil droplets in the emulsion and the continuous water phase. These relative permeabilities are assumed to be functions of the oil saturation in the porous medium. These must be determined before the model can be used. 

According to the Buckley-Leverett (1942) theory, the velocity of the saturation 8 2 is... 

Note that the preceding velocity is the interstitial injection velocity normalized by qt/(A( )), and that it is dimensionless. Lake (1989) and Green and Willhite (1998) used the term specific velocity for the dimensionless velocity. In this book, we follow their terminology. Corresponding to the front of the component C, we assume the water saturation is 8 3. According to the Buckley-Leverett theory (1942), the specific velocity of 8 3 is... 

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... 

When the injection starts at to = 0, the locations of saturations are given by the Buckley-Leverett theory ... 

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... 

Cores No. 2 and 3 were from the same outcrop with the same approximate permeability. We assumed that the capillary effects in the 2 cc/min run from Core No. 2 which was the longer core in practically negligible. The production and pressure data from this experiment was used in the Buckley Leverett model to estimate the relative permeability parameters. Also the 0.2 cc/min run production data from Core No. 3 were analyzed. 

A computer model was developed to estimate the oil and water relative permeability exponential functions from displacement experiments. The model consist of an optimization algorithm, a one-dimensional two phase flow simulator with capillary effects, and a simple Buckley Leverett non-capillary flow simulator. Using experimental data and the model, the effect of rate on the parameters of relative permeability functions were studied. The following conclusions can be delivered from the results of this study. 

The experimental pressure and effluent curves were history matched using an analytic simulation package, "PRIsm [30, 31], which is based on the Buckley-Leverett theory. 

In the first case above, there is an inefficient microscopic (linear) displacement efficiency for M > 1 with its associated low Buckley-Leverett front height, as shown in Figure 8.1. This promotes early breakthrough followed by a... 

The maximum possible saturation just behind the first waterfront (S ) is equal to that of a standard Buckley-Leverett waterfront, that is ... 

The resulting boundary value formulations for Siw(x,t) and S2w(x,t) lead to the so-called Buckley-Leverett problem well known to reservoir engineers. Their solutions can contain shockwaves or steep saturation discontinuities, depending on the form of the fractional flow functions and the initial conditions. The basic issues are discussed in Collins (1961) and will not be repeated here. 

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... 

Hovanessian and Payers (1961), and others, in fact, pointed out that small capillary effects can and will affect both shock structure and position. In the limit of high flow rates, computed solutions correctly gave the corresponding low-order, shock-fitted Buckley-Leverett solutions. 

IMMISCIBLE BUCKLEY-LEVERETT LINEAL FLOWS WITHOUT CAPILLARY PRESSURE... 

Allen, M.B., and Finder, G.F., The Convergence of Upstream Collocation in the Buckley-Leverett Problem, SPE Paper No. 10978, 57 Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers, New Orleans, La., Sept. 26-29, 1982. 

Immiscible Buckley-Leverett lineal flows without capillary pressure, 409 Molecular diffusion in fluid flows, 416... 

As one would expect, this improvement in displacement efficiency is reflected in a floodout test. Fig. 6 shows the results of such a test on a 1-in. diameter by 1-ft long unconsolidated core from a Louisiana field, using field fluids. According to the simplified form of the Buckley-Leverett theory, this test represents a linear model of the field on a pore volume basis. Thus, this core performance can be considered as a partial representation of the expected field performance. 

In accordance with Eqs. 2 and 3, the connate water bank will break through at / = (S - 5,)/fy, and the polymer front and associated saturation discontinuity will break through at / = (S + b)/fg. Fig. 5 shows the oil recovery curve constructed from Eqs. 2 through 4 and, for comparison, the recovery curve for a normal waterflood as calculated by the Buckley-Leverett method. 

Salman, M., Baghdikian, S.Y., Handy, L.L., and Yortsos, Y.C. 1990. Modification of Buckley-Leverett and JBN Methods for Power-Law Fluids. Paper SPE 20279 available from SPE, Richardson, Texas. 

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... 

---