Fractional flow 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... 

Pope, G.A., 1980. The application of fractional flow theory to enhanced oil recovery. SPEJ (June), 191-205. 

Rossen et al. [18, 19,22] applied fractional flow theory to foam processes. At the center of their application lies the concept of limiting capillary pressure , P. Experimental evidence shows that the higher the capUlary pressure the more unstable the foam. If P is surpassed rapid bubble coalescence destroys the foam as capillary suction withdraws the water out of the foam lamellae. In two phase flow (water and gas), the capillary pressure is related directly to the water saturation. For water wet porous media, as the water saturation increases the capillary pressure decreases. Therefore, P can be related to a limiting water saturation, S. Once the water saturation decreases below the limiting water saturation, existing foam should rapidly destabilize. 

Extended fractional flow theory for 1-D polymer flooding... 

Figure 8.5. Water saturation fronts in a linear polymer flood showing the nomenclature for the fractional flow theory in the text (after Pope, 1980).

The assumptions employed in applying fractional flow theory as outlined below are as follows ... 

Equations are derived below that allow calculation of the water/solvent injection ratio required for equal water/solvent velocities. The model is for ID flow and is based on fractional-flow theory. Equations and graphical solutions are presented to descaibe WAG injection for both secondaiy and tertiary recovery conditions. Walsh 4 gives a more comprehensive treatment using fractional-flow theory. 

The deterministic model with random fractional flow rates may be conceived on the basis of a deterministic transfer mechanism. In this formulation, a given replicate of the experiment is based on a particular realization of the random fractional flow rates and/or initial amounts 0. Once the realization is determined, the behavior of the system is deterministic. In principle, to obtain from the assumed distribution of 0 the distribution of common approach is to use the classical procedures for transformation of variables. When the model is expressed by a system of differential equations, the solution can be obtained through the theory of random differential equations [312-314]. 

The evolution of the polymer flood is explained by Pope in terms of the fractional flow (/ )/saturation (S ) curves for both the water and injected polymer solutions. A typical diagram showing the water and polymer solution fractional flow curves is shown in Figure 8.6. In the theory outlined below the following nomenclature is used. 

For the above assumptions and the case of continuous polymer water injection, Eqs. la and lb can be solved analytically using the method of characteristics and shock discontinuity theory. This analytical solution is illustrated here in the context of examples. Fig. 2 shows laboratory measured water-oil relative permeability curves for a 6,000-md unconsolidated, Nevada sand representative of a California viscous-oil reservoir. Fig. 3 shows die water fractional flow curves for the cases of normal water, = 1 cp, and polymer... 

However, bubble nonhomogeneous distribution exists in two-phase shear flow. As yet, the following general trends in void fraction radial profiles are being identified for bubbly upward flow (Zun, 1990) concave profiles (Serizawa et al., 1975) convex profiles (Sekoguchi et al., 1981), and intermediate profiles (Sekoguchi et al., 1981 Zun, 1988). Two theories are currently dominant ... 

The slopes of the peaks in the dynamic adsorption experiment is influenced by dispersion. The 1% acidified brine and the surfactant (dissolved in that brine) are miscible. Use of a core sample that is much longer than its diameter is intended to minimize the relative length of the transition zone produced by dispersion because excessive dispersion would make it more difficult to measure peak parameters accurately. Also, the underlying assumption of a simple theory is that adsorption occurs instantly on contact with the rock. The fraction that is classified as "permanent" in the above calculation depends on the flow rate of the experiment. It is the fraction that is not desorbed in the time available. The rest of the adsorption occurs reversibly and equilibrium is effectively maintained with the surfactant in the solution which is in contact with the pore walls. The inlet flow rate is the same as the outlet rate, since the brine and the surfactant are incompressible. Therefore, it can be clearly seen that the dynamic adsorption depends on the concentration, the flow rate, and the rock. The two parameters... 

Wu, Ruff and Faeth12491 studied the breakup of liquid jets with holography. Their measurements showed that the liquid volume fraction on the spray centerline starts to decrease from unit atZ/<70=150 for non-turbulent flows, whereas the decrease starts at aboutZ/<70=10 for fully developed turbulent flows. Their measurements of the primary breakup also showed that the classical linear wave growth theories were not effective, plausibly due to the non-linear nature of liquid breakup processes. 

---