Foams.. Fundamentals and Applications in the Petroleum Industry
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Figure 9 presents one such lamella at successive times t t , and t As the lamella translates from left to right, it is squeezed upon entering the constriction at time t.
Film thickness increases to conserve liquid mass, and the disjoining pressure is correspondingly low. Jimenez and Radke 2 y v 2 y 2 ti h h Figure 9. Foam lamella translatingfromleft torightin a periodically constricted tube. Coalescence occurs at t.
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Fluid resistance in the film is inversely proportional to the local film thickness to the third power, so drainage is not instantaneous. When the lamella moves out of the pore constriction, it is stretched upon expansion into the downstream body. The film thins to conserve mass, and the disjoining pressure is now high. Thus, the thickness of the transporting lamella oscillates about the equilibrium thickness established in the stationary lamellae in a sequence of squeezing—stretching and draining-filling events.
Thus, if the film is stretched too rapidly for healing surfactant solution to flow into the film and stabilize it, rupture ensues. The proposed model of Jimenez and Radke 55 is oversimplified in that it does not account for the details of the actual curved-film breakup 74 , or for any surface tension gradients and elastic effects as the film stretches and squeezes.
Moreover, Huh et al. Lamellae coalesced in the pore bodies of the micromodel. Likewise, Chambers and Radke present photomicrographs of the identical mechanism for coalescence of N -aqueous surfactant foams in etchedglass micromodels Figure 26 of reference Aronson et al. Figure 10 summarizes the findings. Conversely, closed symbols correspond to the measured steady pressure gradients for the same solutions.crezenecex.tk
Foaming of Oils: Effect of Poly(dimethylsiloxanes) and Silica Nanoparticles
Solid lines again indicate the trends. The postulate that large rupture disjoining pressures, above 30 kPa in Figure 10, give rise to strong foam in porous media is clearly confirmed. Figure 10 then reveals that once the rupture pressure of the foam films exceeds the capillary pressure of the medium, low mobility foam emerges.
Finally, Khatib et al. Comparison of foam lamella rupture pressure and bead pack pressure gradient during steady-state foam flow. Upward directed arrows indicate that the actual rupture pressure is greater than the value indicated In Foams: Fundamentals and Applications in the Petroleum Industry; Schramm, L. The study of Aronson et al.
Gas Diffusion. The second mechanism for foam coalescence in porous media, gas diffusion, pertains primarily to the stagnant, trapped bubbles. According to the Young—Laplace equation, gas on the concave side of a curved foam film is at a higher pressure and, hence, higher chemical potential than that on the convex side. Driven by this difference in chemical potential, gas dissolves in the liquid film and escapes by diffusion from the concave to the convex side of the film.
The rate of escape is proportional to film curvature squared and, therefore, is rapid for small bubbles 16, Bulk foams coarsen with larger bubbles growing at the expense of smaller ones that eventually disappear. However, confined foam in porous media does not coarsen in a similar fashion because bubble volume is not directly related to film curvature. Rather, in porous media, lamella curvature depends on pore dimensions and on location within the pore space.
Gas diffusion still proceeds from the most highly concave bubbles forcing the lamellae to diminish their curvatures by translation toward porethroats. In the absence of an imposed pressure gradient, it is possible for gas diffusion to drive all lamellae to pore-throats to achieve an equilibrium state of zero curvature. Coalescence occurs only when two lamellae happen to reach the same pore-throat.
With steam foam, coarsening is more rapid because water can condense on one side of a lamella and evaporate from the other. A noncondensible gas may be added to retard coarsening 3. Nevertheless, coarsening of the trapped foam by gas diffusion is expected.
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As the resulting texture diminishes, a portion of the trapped bubbles may remobilize In the limiting capillary-pressure regime, the wetting-liquid saturation is sensibly constant, independent of gas and liquid velocities over a rather large range. Foam-flow behavior in the limiting capillary-pressure regime is rather remarkable 20, 36, 61, 62, 78— Varying liquid velocity and holding gas velocity constant usually yields a linearly increasing pressure drop response. Increasing both liquid and gas velocity and holding the fractional flow constant yields a linear response when pressure drop is plotted versus total flow rate.
Finally, steady-state saturations are independent of gas fractional flow. These observations cannot be explained by Darcy's law for multiphase flow in porous media. In particular, foam flow is not solely a function of fractional flow, but depends on individual gas and liquid velocities. Because P is constant, aqueous-phase saturation consequently remains constant and so does the relative permeability to the wetting liquid.
Persoff et al. Rossen and co-workers 30, 82 , by incorporating these observations into fractional flow theory, demonstrate how powerful the limiting capillary-pressure regime is in understanding foam flow. In fact, Persoff et al. Likewise, Ettinger and Radke 20 probedfractionalflowsbetween 0. However, the data of Khatib et al.
Abovefractionalflowsof roughly 0. More recently, Osterloh and Jante 77 probed a wide range of flow rates andfractionalflowsfor foam in a 6. At gasfractionalflowsabove 0.
At lesser gasfractionalflows,the converse was found. Pressure-gradient response was negligible with increased liquid velocity but increased with gas velocity to the 0. It was surmised that the transition between the two regimes occurs at the point where the limiting capillary pressure is attained. The variation in the range of fractional flows yielding limiting capillary-pressure behavior is likely due to the different surfactant systems employed.
Different surfactant structures and conditions such as concentration and temperature lead to different disjoining-pressure isotherms for single foam films and thus different limiting capillary-pressure characteristics. Also, the various porous media have differing capillary-pressure versus aqueous-phase saturation relationships. Although the limiting capillary-pressure regime is by no means general for all conditions of foam flow in porous media, it is an important one.
Our modeling effort to follow is directed toward predicting how this behavior emerges in a transient, one-dimensional foam displacement. Population-Balance Modeling of Foam Flow in Porous Media A variety of methods have been proposed for modeling of foamflowand displacement in porous media. These range from population-balance methods 8, 9, 20, 39, , 83 to percolation models , 47, 84 , and from semiempirical alteration of gas-phase mobilities 85—92 to applying so-calledfractionalflowtheories 30, Semiempirical models are computationally simple, but lack generality. The last method may be unsuitable for modeling of foamfloodingbecausefractionalflowtheory is approximate when applied to compressible phases 30 , severe extrapolations from available data are needed to fit model parameters 30 , and strong foam behavior is not, in general, a unique function of fractional flow 20, 36, 61, 62, 75, Constructingfractionalflowcurves for foam flow in porous media, as such, may be inappropriate as absoluteflowrates determine foam-flow behavior.
Of these four methods, only the population-balance method and network, or percolation, models arisefromfirstprinciples. Network models that allow replication of pore-level mechanisms have the decided disadvantage of requiring large amounts of computation time and providing results on a prohibitively small grid. It seems unlikely that network or percolation models can be useful in transient displacements that demand tracking of saturation, surfactant concentration, and foam on laboratory scales, let alonefieldscales.
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By analogy to balances on surfactant or other chemical species, a separate conservation equation is written for the concentration of foam bubbles. We focus on transient displacement by strong foam that at steady state achieves the limiting capillary-pressure regime. Conservation Equations. The companion mass balance for the aqueous phase is written by interchanging the subscript g, which denotes gas phase for w, which denotes the liquid phase.
Because the mobility of the foam phase is a strong function of texture 9, 18, 20, 26, 33, 48 , mechanistic prediction of foam flow in porous media s s s In Foams: Fundamentals and Applications in the Petroleum Industry; Schramm, L. Thus, n and n are, respectively, the number of foam bubbles per unit volume of flowing and stationary gas. Thefirstterm of the time derivative is the rate at whichflowingfoam texture becomes finer or coarser per unit rock volume, and the second is the net rate at which foam bubbles trap. The spatial term tracks the convection of foam bubbles.
Foam in Porous Media
The usefulness of a foam bubble population-balance, in large part, revolves around the convection of gas and aqueous phases. On the right of equation 4, the generation and coalescence rates, r and r , are expressed on a per volume of gas basis. These two terms are fundamental, for they control bubble texture.
At steady state, farfromany sources or sinks, and where rock properties are constant e. That is, the rate of bubble generation by snap-off balances the rate of bubble coalescence by capillary-pressure suction To proceed, kinetic expressions are needed for r and r. Snap-off in germination sites determines the rate expression for bubble generation following the process pictured in Figure 5.
Bubble leave-behind is neglected. The division mechanism for producing new lamellae yields a rate that is indistinguishable in form from that of coalescence and is included there.