C.J. Radke

Protein adsorption at the oil/water interface: characterization of adsorption kinetics by dynamic interfacial tension measurements.

The dynamics of protein adsorption at an oil/water interface are examined over time scales ranging from seconds to several hours. The pendant drop technique is used to determine the dynamic interfacial tension of several proteins at the heptane/aqueous buffer interface. The kinetics of adsorption of these proteins are interpreted from tension/log time plots, which often display three distinct regimes. (I) Diffusion and protein interfacial affinity determine the duration of an initial induction period of minimal tension reduction. A comparison of surface pressure profiles at the oil/water and air/water interface reveals the role of interfacial conformational changes in the early stages of adsorption. (II) Continued rearrangement defines the second regime, where the resulting number of interfacial contacts per protein molecule causes a steep tension decline. (III) The final regime occurs upon monolayer coverage, and is attributed to continued relaxation of the adsorbed layer and possible build-up of multilayers. Denaturation of proteins by urea in the bulk phase is shown to affect early regimes.

Laminar flow of a wetting liquid along the corners of a predominantly gas-occupied noncircular pore

Abstract The problem of low Reynolds number wetting liquid flow in a noncircular capillary occupied predominantly by a nonwetting gas phase is separated into individual corner flow problems and solved numerically. The solution is presented in terms of a dimensionless flow resistance, β, which is tabulated as a function of corner geometry (half angle and degree of roundedness), surface shear viscosity, and contact angle. The effect of corner geometry and contact angle is to change the cross-sectional area available for flow, while the surface shear viscosity affects the boundary condition at the gas-liquid interface. The dimensionless flow resistance is shown to increase with increasing half angle, degree of roundedness, surface shear viscosity, and contact angle. Finally, it is demonstrated htat the error resulting from the use of the hydraulic-diameter approximation for the corner flow problem is on the order of 50% or higher.

The motion of long bubbles in polygonal capillaries. Part 1. Thin films

Foam in porous media exhibits an unusually high apparent viscosity, making it useful in many industrial processes. The rheology of foam, however, is complex and not well understood. Previous pore-level models of foam are based primarily on studies of bubble flow in circular capillaries. A circular capillary, however, lacks the corners that characterize the geometry of the pores. We study the pressure–velocity relation of bubble flow in polygonal capillaries. A long bubble in a polygonal capillary acts as a leaky piston. The ‘piston’ is reluctant to move because of a large drag exerted by the capillary sidewalls. The liquid in the capillary therefore bypasses the bubble through the leaky corners at a speed an order higher than that of the bubble. Consequently, the pressure work is dissipated predominantly by the motion of the fluid and not by the motion of the bubble. This is opposite to the conclusion based on bubble flow in circular capillaries. The discovery of this new flow regime reconciles two groups of contradictory foam-flow experiments. Part 1 of this work studies the fluid films deposited on capillary walls in the limit Ca → 0 ( Ca ≡ μ U /σ, where μ is the fluid viscosity, U the bubble velocity, and σ the surface tension). Part 2 (Wong et al. 1995) uses the film profile at the back end to calculate the drag of the bubble. Since the bubble length is arbitrary, the film profile is determined here as a general function of the dimensionless downstream distance x . For 1 [Lt ] x [Lt ] Ca −1 , the film profile is frozen with a thickness of order Ca 2/3 at the centre and order Ca at the sides. For x ∼ Ca −1 , surface tension rearranges the film at the centre into a parabolic shape while the film at the sides thins to order Ca 4/3 . For x [Gt ] Ca −1 , the film is still parabolic, but the height decreases as film fluid leaks through the side constrictions. For x ∼ Ca −5/3 , the height of the parabola is order Ca 2/3 . Finally, for x [Gt ] Ca −5/3 , the height decreases as Ca 1/4 x −1/4 .

The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow

This work determines the pressure–velocity relation of bubble flow in polygonal capillaries. The liquid pressure drop needed to drive a long bubble at a given velocity U is solved by an integral method. In this method, the pressure drop is shown to balance the drag of the bubble, which is determined by the films at the two ends of the bubble. Using the liquid-film results of Part 1 (Wong, Radke & Morris 1995), we find that the drag scales as Ca 2/3 in the limit Ca → 0 ( Ca μ U /σ, where μ is the liquid viscosity and σ the surface tension). Thus, the pressure drop also scales as Ca 2/3 . The proportionality constant for six different polygonal capillaries is roughly the same and is about a third that for the circular capillary. The liquid in a polygonal capillary flows by pushing the bubble (plug flow) and by bypassing the bubble through corner channels (corner flow). The resistance to the plug flow comes mainly from the drag of the bubble. Thus, the plug flow obeys the nonlinear pressure–velocity relation of the bubble. Corner flow, however, is chiefly unidirectional because the bubble is long. The ratio of plug to corner flow varies with liquid flow rate Q (made dimensionless by σ a 2 /μ, where a is the radius of the largest inscribed sphere). The two flows are equal at a critical flow rate Q c , whose value depends strongly on capillary geometry and bubble length. For the six polygonal capillaries studied, Q c [Lt ] 10 −6 . For Q c [Lt ] Q [Lt ] 1, the plug flow dominates, and the gradient in liquid pressure varies with Q 2/3 . For Q [Lt ] Q c , the corner flow dominates, and the pressure gradient varies linearly with Q . A transition at such low flow rates is unexpected and partly explains the complex rheology of foam flow in porous media.

Fundamentals of foam transport in porous media

Foam in porous media is a fascinating fluid both because of its unique microstructure and because its dramatic influence on the flow of gas and liquid. A wealth of information is now compiled in the literature describing foam generation, destruction, and transport mechanisms. Yet there are conflicting views of these mechanisms and on the macroscopic results they produce. By critically reviewing how surfactant formulation and porous media topology conspire to control foam texture and flow resistance, we attempt to unify the disparate viewpoints. Evolution of texture during foam displacement is quantified by a population balance on bubble concentration, which is designed specifically for convenient incorporation into a standard reservoir simulator. Theories for the dominant bubble generation and coalescence mechanisms provide physically based rate expressions for the proposed population balance. Stone-type relative permeability functions along with the texture-sensitive and shear-thinning nature of confined foam complete the model. Quite good agreement is found between theory and new experiments for transient foam displacement in linear cores.

The role of interfacial rheology in reservoir mixed wettability

Abstract Since the early 1950s, industrial researchers have recognized that asphaltenic crude oil/water interfaces form so-called “rigid skins”. This work emphasizes the role that such oil/water interfacial microstructures play in establishing the mixed-wet state of reservoirs. We utilize a new oscillating-drop dynamic tensiometer that sinusoidally and infinitesimally expands and contracts a crude-oil droplet immersed in brine at a fixed frequency and measures the resulting dynamic interfacial stress from image analysis and axisymmetric drop-shape analysis. Linear viscoelastic theory permits evaluation of the dilatational interfacial elastic storage and viscous loss moduli. We find that for two crude oils, designated as Crude AS and Crude AH, immersed in synthetic sea water, the interface behaves primarily elastically and that the more asphaltenic the oil the stronger is the interfacial elasticity. Moreover, interfacial elasticity grows slowly in time over days and is clearly manifest even when “rigid skins” are not visible to the eye. Apparently, macroscopic, networked asphaltenic structures slowly evolve in time at the interface. Advancing and receding contact angles are also measured on smooth mica surfaces for the same crude oil/brine systems. We find that water advancing and receding contact angles when measured within hours are about equal (i.e., there is little hysteresis). However, aging of the drop over days dramatically alters the subsequent advancing and receding contact angles. Water receding angles grow somewhat in time, but the corresponding advancing angles increase over days towards 180° or towards complete pinning. Interestingly, the advancing contact angles for both crude oils do not depend on whether the drop is aged in the brine or in contact with the mica surface. Also, the measured, receding contact angles for both crude oils are much higher than those commonly assumed in the literature. Fascinatingly, aging kinetics of the contact angles correlates directly with the aging of interfacial elasticities and interfacial tensions. Based on in situ AFM studies of the asphaltene-coated mica surfaces, we explain why this happens. Upon rupture of the protective water film and adhesion of the oil droplet to the mica substrate, the surface underneath the oil droplet is pockmarked with water-wet patches in a Dalmatian microwetting pattern. To our knowledge the crucial role of oil/water interface aging in controlling wettability changes has not previously been recognized. Finally, by sketching various primary drainage and imbibition pore-level events, we emphasize the importance of the observed changes in contact angles towards the evolution of mixed-wet oil reservoirs.

An ion-binding model for ionic surfactant adsorption at aqueous-fluid interfaces

Abstract A simple ion-binding model is presented to quantify the equilibrium adsorption of ionic surfactants at aqueous-fluid interfaces. The proposed model adopts a triple layer structure for the interface: a plane of adsorbed surfactants (interface plane), a plane of partially dehydrated, contact-bound counterions (inner Helmholtz plane), and a plane of hydrated counterions (outer Helmholtz plane). An analytic expression for the surface tension is obtained as a function of the physicochemical parameters of the system. It generalizes the classical results of J.T. Davies and E.K. Rideal (Interfacial Phenomena, Academic Press, New York, 1963) as well as those, more recent, of R.P. Borwankar and D.T. Wasan (Chem. Eng. Sci., 1 (1986) 199). In the ion-binding model, the surface tension depends on the electrocapacitance in the layers closest to the interface and the distances between them, in addition to the surface charges on the planes. For the limiting case of a moderate concentration of surfactant, asymptotic formulae for the surface tension are derived. On a semilogarithmic graph of surface tension versus surfactant concentration in the presence of background electrolyte, the asymptotic slope approaches - kT (M t ), where k is Boltzmanns constant, T is temperature, and (M t ) is the surface concentration of total sites, M t , available for surfactant headgroups in the interface, the parentheses indicating concentration. In the case of no salt added, the asymptotic slope is −2 kT (M t ). The asymptotic formulae also establish the influence on the surface tension of the equilibrium constants and the lateral interaction parameter, ω, in Frumkins isotherm. The ion-binding model results are in good agreement with the surface and interfacial tension data for sodium dodecyl sulfate (SDS). Agreement with measured ξ-potentials is also found for SDS at the air-water boundary.

Three-dimensional Menisci in Polygonal Capillaries

Abstract The shapes of gravity-free, three-dimensional menisci are computed from the augmented Young-Laplace equation. Incorporation of disjoining thin-film forces in the Young-Laplace relation eliminates the contact line, thereby eliminating the free boundary from the problem. To calculate a meniscus with finite contact angles, the conjoining/disjoining pressure isotherm must also contain an attractive, sharply varying, spike function. The width of this function, w, reflects the range of the thin-film forces. In the limit of w approaching zero, a solution of the Young-Laplace equation is recovered. The proposed calculation method is demonstrated for menisci in two different types of capillaries. In the first case, the capillary is regular-polygonal in cross section with either 3, 4, or 6 sides and with contact angles Φ ranging from 0 to 45°. In the second case, the capillary is rectangular in section with aspect ratios ranging from 1.2 to 5 and with Φ = 0°, 15°, or 30°. Gas-liquid menisci inside a square glass capillary of 0.5 mm inscribed radius are measured optically for air bubbles immersed in a solution of di-n-butyl phthalate and mineral oil. This liquid mixture exhibits a zero contact angle with the wall and matches the refractive index of the glass capillary, permitting precise visual location of the interface. Excellent agreement is found with the numerical results which further demonstrates that the limiting process of the proposed method is valid. Because it avoids the issue of locating the contact line, solution of the augmented Young-Laplace equation is a simple and powerful method for the calculation of three-dimensional menisci.

An extended evolution equation for liquid film breakup in cylindrical capillaries

Abstract The dynamics of a viscous liquid film forming lenses or stable collars and the static stability of such collars are investigated in straight cylindrical capillaries for various approximations of the Young-Laplace equation. We find that the leading order approximation of the Young-Laplace equation in a thin-film expansion does not predict the known instability of wetting collars forming lenses. Hence, we propose a simple approximation that does. Numerical solution of the new extended evolution equation for a film in a tube of radius R T * gives a critical film thickness of 0.12 R T *, which demarks the transition between films that evolve into stable collars and those that break up to form lenses. New experimental measurements of the critical thickness agree well with the theoretical result, confirming the validity of the proposed evolution equation.

A mechanistic population balance model for transient and steady-state foam flow in Boise sandstone

Foam in porous media is discontinuous on a length scale that overlaps with pore dimensions. This foam-bubble microstructure determines the flow behavior of foam in porous media and, in turn, the flow of gas and liquid. Modeling of foam displacement has been frustrated because empirical extensions of the conventional continuum and Newtonian description of fluids in porous media do not reflect the coupling of foam-bubble microstructure and foam rheology. We report a mechanistic model for foam displacement in porous media that incorporates pore-level mechanisms of foam generation, coalescence, and transport in the transient flow of aqueous foams. A mean-size foam-bubble conservation equation, along with the traditional reservoir/groundwater simulation equations, provides the foundation for our mechanistic foam-displacement simulations. Since foam mobility depends heavily upon its texture, the bubble population balance is both useful and necessary, as the role of foam texture must be incorporated into any model which seeks to predict foam flow accurately. Our model employs capillary-pressure-dependent kinetic expressions for lamellae generation and coalescence, and incorporates trapping of lamellae. Additionally, the effects of surfactant chemical transport are included. All model parameters have clear physical meaning and, consequently, are independent of flow conditions. Thus, for the first time, scale up of foam-flow behavior from laboratory to field dimensions appears possible. The simulation model is verified by comparison with experiment. In situ, transient, and steady aqueous-phase liquid contents are garnered in a 1.3 μm2 Boise sandstone using scanning gamma-ray densitometry. Backpressures exceed 5 MPa, and foam quality ranges from 0.80 to 0.99. Total superficial velocities range from as little as 0.42 to 2.20 m/d. Sequential pressure taps measure flow resistance. Excellent agreement is found between experiment and theory. Further, we find that the bubble population balance is the only current means of describing all flow modes of foam self-consistently.