Summary. The stability of foam lamellae is limited by capillary pressure. Consequently, as the fractional flow of gas in a foam is raised at a fixed gas velocity, the capillary pressure in a porous medium at first increases and then approaches a characteristic value, here called the "limiting capillary pressure." If the gas fractional flow is increased after the limiting capillary pressure has been attained, coalescence coarsens foam texture, the liquid saturation remains constant, and the relative gas mobility becomes proportional to the ratio of gas-to-liquid fractional flow. The limiting capillary pressure varies with the surfactant formulation, gas velocity, and permeability of the medium. Introduction The mobility of a gas flowing through a porous medium is lower when it is dispersed within a liquid as a foam. Consequently, foams can improve volumetric sweep in oil-recovery processes that use gases. To apply foams optimally, the factors that influence their transport must be understood. Previous research has shown that the mobility of a foam depends heavily on its texture, which is the distribution of bubble sizes within the dispersion. The texture of foam, in turn, is governed by mechanisms that generate and destroy bubbles inside porous media: capillary snap-off and division create bubbles, while mass transfer between bubbles resulting from condensation/evaporation or diffusion and coalescence (lamella rupture) diminish bubble density. Falls et al. recently developed a model to describe the rate of foam generation by capillary snap-off and demonstrated how it could be used to simulate foam generation and flow in a simple, one-dimensional displacement. We report here the initial steps toward achieving the same level of understanding for coalescence. On the basis of how soap films and bulk foams behave, we hypothesize in particular thatthe rate of coalescence of foam bubbles lamellae in porous media depends on capillary pressure andlamellae in porous media cannot withstand capillary pressures above a porous media cannot withstand capillary pressures above a limiting value. The way this limiting capillary pressure should influence the transport of foams through porous media is subsequently detailed. Finally, experiments in which capillary pressure and foam mobility are measured concurrently are reported; these confirm that capillary pressure does indeed play a major role in determining coalescence and phase mobilities of foams in porous media. Capillary Pressure's Role In Foam Stability Outside Porous Media The behavior of soap films is described qualitatively by Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, the classic theory of colloid stability. When gas/liquid interfaces are present, ionic surfactant molecules from an aqueous solution adsorb preferentially at the interfaces, thereby creating charged surfaces. The overlap of the electric fields from these charged layers imparts stability to a foam lamella. Long-ranged, attractive van der Waals forces tend to destabilize the system.DLVO theory considers how double-layer repulsion balances against the van der Waals forces. Films that are stable to small perturbations result only when the repulsive forces are stronger that perturbations result only when the repulsive forces are stronger that the attractive ones. Imposing capillary pressure (capillary pressure is the difference between gas- and liquid-phase pressures) on a lamella forces the charged surfaces closer to one another. The double-layer repulsion must balance the van der Waals forces plus the capillary pressure. As the capillary pressure is raised, the work required to break the film decreases. At a sufficiently high capillary pressure, this work may become so small that mechanical disturbances or even thermal fluctuations may rupture the film.DLVO theory thus predicts that capillary pressure should play a role in determining the stability of foams outside porous media. This conclusion is backed by experimental studies of how capillary pressure affects that lifetimes of soap films and bulk foams. These suggest that there is a characteristic capillary pressure above which the lifetimes of lamellae become exceedingly pressure above which the lifetimes of lamellae become exceedingly short. The value of the "critical" capillary pressure varies with the surfactant formulationi.e., the type and concentration of surfactant and electrolyte. Critical capillary pressures found by Khristov et al. for single films and bulk foams made from 0.001 M sodium dodecyl sulfate solutions are recorded in Table 1. The critical capillary pressure increases as the concentration of sodium chloride is raised. pressure increases as the concentration of sodium chloride is raised. At first glance, this would appear to go against what is expected from DLVO theory: at higher electrolyte levels, the length over which the electric field from the charged surfactant layers decay should shorten and the repulsive forces should weaken. This effect may be offset, however, because salt raises the surface concentration of surfactant at the gas/liquid interfaces. (NaCl's ability to raise the critical capillary pressure of foam films may explain why salt must be added to enable some steam-foam-forming surfactants to reduce steam Mobility.)Khristov et al. also noticed that bulk foams break at lower capillary pressures than do single films created from the same solution. They attributed this tothe films in the bulk foams having larger radii than the single to and"collective effects," where disturbances from the rupture of one film cause its neighbors to break. Concepts and Consequences of a Limiting Capillary Pressure for Foam In Porous Media The stability of foams in porous media may likewise depend on capillary pressure. In porous media, however, all lamellae do not simultaneously coalesce at some "critical" capillary pressure. Instead, because lamellae are convected or are generated in situ, the capillary pressure increases up to a limiting capillary pressure as the gas fractional flow is raised. With further increase in gas fractional flow, the capillary pressure remains at its limiting value while the foam texture becomes coarser. The limiting capillary pressure for foams in porous media should vary with more than just surfactant formulation: gas velocity and the permeability of the medium can also be expected to be important variables. 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The mobility of a foam depends heavily on its texture, which is the distribution of bubble sizes in the dispersion. To incorporate this variable in a mechanistic simulator, the usual conservation equations are coupled with balances on the densities of flowing and stationary bubbles in the foam. This approach to modeling foam flow is illustrated with a simulation of a displacement in which foam is generated in situ by capillary snap-off.

The mobilities of aqueous foams of known texture have been measured in homogeneous bead packs. To correlate the data, a theory developed recently to describe the apparent viscosity of foams in smooth capillaries is extended to account for (1) the capillary pressure imposed by the porous medium and (2) constricted flow paths. In porous media, apparent gas viscosity depends strongly on foam-bubble size; for large bubble sizes, it is proportional to the third power of the ratio of the hydraulic radius of the pack to the bubble radius. Foams of uniform texture are pseudoplastic. At low shear rates, the viscosity varies inversely with a capillary number; at higher rates, it depends on the capillary number to the -~ power when the bubble size is large compared with the pore size and on the -2fI power of capillary number when the bubble size is smaller.157

Residual stress, microstructure, and structure of tungsten thin films deposited by magnetron sputtering An integral theory and gradient theory of inhomogeneous fluid are used to predict the structure of spherical interfaces. The nonlinear integral and differential equations of the theories are solved by using state-of-the-art finite element techniques coupled with Newton's method. The numerical method, discussed at some length, is suggested as a potential tool for solving other nonlinear problems of fluid statistical mechanics. Results of integral and gradient theories are compared and found to be in qualitative agreement and, on the strength of this agreement, gradient theory is used to describe and analyze the structure and stress in microscopic spherical drops. Predictions from gradient theory indicate a breakdown of the Young-Laplace equation for drops smaller than about ten molecules wide. However, contrary to the usual estimate of the curvature dependence of surface tension, the surface tension of even very small drops (say, three or four molecules wide) is found to deviate little from the tension of a planar interface.3986

by hydrolysis-oxidation mechanisms.The particulate sulfates examined in this experiment did not appear to be isotopically coupled with the ambient water vapor in the air masses from which they were sampled. The lack of correlation between the temporal variations of P O of sulfate (Figure 2) and of water vapor (Figure 3) may indicate one or both of the following conditions: (1) the absence of a predominant secondary mechanism of formation in which the A kinetic mechanism for photochemical smog is developed to incorporate recent new information on rate constants and mechanisms. The predictions of the mechanism are compared with smog chamber data on propylene, n-butane, and propyleneln-butane systems. Areas of uncertainty are delineated, and the influence of these uncertainties on the predictions of the mechanism is discussed.