318
I n d . E n g . Chem. Res. 1987, 26, 318-325
Tarng, Y. J.; Anthony, R. G. Repot DOE/FC/10601-1; 1983 US Department of Energy (DE-AC18-83FC10601).
Villadsen, J. V.; Stewart, W. E. Chem. Eng. Sci. 1967, 22, 1483.
Received for review October 11, 1984 Revised manuscript received November 8, 1985 Accepted April 1, 1986
Villadsen, J. V.; Michelsen, M. L. Solution of Differential Equation Models by Polynomial Approximation; Prentice-Hall Englewood Cliffs, NJ, 1978: Chapter 5.
Critical Review of the Foam Rheology Literature John P. Heller* and Murty S. Kuntamukkula Petroleum Recovery Research Center, N e w Mexico Institute of Mining and Technology, Socorro. N e w Mexico 87801
T h e rheology of foam is not like that of simpler fluids which can be regarded as mathematical continua. The difference arises because the size of foam bubbles is not infinitesimal relative to the width of the flow channels and because foam (whether gas-in-liquid or liquid-in-liquid) contains a high volume fraction (in excess of 74% for a monodisperse system) of the discontinuous phase. T h e latter feature causes crowding of the bubbles so that a non-zero yield stress is required before continuous shearing motion can occur. References in the literature t o the flow of foam both in viscometers and in porous media are examined. Special emphasis is given to recent work that evaluates the critical yield stress in idealized cases and makes it possible to relate measured apparent viscosities of foams in large channels t o a usually unmeasured boundary effect. The word “foam” is used here in a broader sense than is common in much of the literature on this subject. The phrase “foamlike dispersion” is perhaps more descriptive of this composite substance and has been suggested by Lien et al. in a previous paper (Heller et al., 1982). Sebba (1984) has suggested the use of the phrases “polyaphrons” and “biliquid foams”, and Wellington (1982) has coined the word “foamulsion”. The principal distinguishing characteristic of these foamlike dispersions is the large volume fraction (generally greater than 74% in a foam of uniform cells) of the noncontinuous or “internal” fluid component. For polydisperse foams, this criterion based on the closest packing of foam cells may take on a different value. In this situation, the cells or parcels of the discontinuous phase will not be spherical because of crowding and would thus seem to be incapable of any significant independent motion. Despite this, foam can flow. It is the nature of such fluid behavior that is the subject of this paper. Although the most common foam is one in which air or some other gas constitutes the noncontinuous phase and in which an aqueous surfactant solution is the continuous fluid, there are other specialty foams in which, for instance, the continuous phase may be a hydrocarbon-based fluid or even a polymeric liquid. In a different foam that holds promise in enhanced oil recovery, the continuous phase is again an aqueous surfactant solution, while the noncontinuous fluid is dense carbon dioxide. This nonpolar fluid may be either liquid or gas, depending on the temperature being below or above the critical point of C 0 2 , 31.1 “C. At equilibrium conditions, although the surfactant solution is saturated with COz and partially dissociated carbonic acid (its pH is about 3), the quantity of C 0 2 dissolved in the aqueous phase is small compared to that which comprises the noncontinuous component of the foam. An even smaller amount of water is soluble in the dense C 0 2and thus exists as a solute in the noncontinuous phase. When a foam is generated by intimate mixing of the nonpolar fluid and liquid, it may persist for a reasonable length of time without collapsing into separate constituent phases. From a thermodynamic viewpoint, foams are unstable dispersions by their very nature and should eventually break into individual component phases in the direction of decreasing total surface free energy. Among 0888-5885/ 87 / 2626-0318$01.50/0
the mechanisms which contribute to foam decay are drainage of the continuous liquid and mass transfer across the foam lamellae. Gravitational drainage of liquid through the lamellae that surround the foam cells leads to gradual thinning and finally to rupture of foam bubbles. Especially in a polydisperse gas foam, due to the higher capillary pressure associated with smaller bubbles, there is significant spontaneous interdiffusion of gas from small bubbles i n t ~ adjacent larger bubbles through the interfacial films. This process results in shrinkage of the small bubbles and expansion of the neighboring larger bubbles, also causing the foam films to become thinner and ultimately to rupture. This decay mechanism could occur also in foamlike liquid-liquid dispersions but would require appreciable solubility of the internal phase in the external fluid. Interfacial viscosity and elasticity of lamellae may promote foam stability by retarding liquid drainage and by resisting deformations induced by ambient fluctuations in temperature or pressure, as in the case of protein-stabilized foams. For any foam to be reasonably long-lasting, the continuous phase must carry in solution one or more surfactants as foaming agents. The function of the surfactant is to stabilize the films, perhaps by populating them with molecules that retard flow within the interface. In films unprotected by a suitable surface-active chemical (synonyms: foamer, foamant), such surface flow causes rapid thinning and breakage of the films. Some surfactants are more effective than others for stabilizing foams and special conditions of application may require consideration of such factors as adsorption on solid surfaces and chemical, thermal, and mechanical stability of interfacial films. But the issue of surfactant screening criteria is not the concern in this paper-it is presumed here that suitable foamants are available and that their concentration in the continuous phase is sufficient to stabilize the quantity of surface film area required in a particular application. Instead, the objective here is to examine the flow of the composite fluids which have been broadly defined as foams in the previous paragraphs. The flow of a foam differs from that of conventional fluids with respect to the way the flow is affected by the size and shape of the channels that confine it. Furthermore, at least one characteristic of foam which is of particular importance to its flow-the cell size-is not easily 0 1987 American Chemical Society
Ind. Eng. Chem. Res. Vol. 26, No. 2 , 1987 319 measured and not always reported. Consequently, some published experimental results do not seem consistent with the established theories and procedures of flow measurements. Most of the general literature on foam is concerned with its structure and with the surface chemistry of the lamellae or films that separate the cells of internal-phase fluid. These subjects have been discussed in books by Bikerman (1973) and Rosen (1978). Increasing interest is also evident in the flow of foams, which has been examined from a fundamental point of view (Bretherton, 1961; Hirasaki and Lawson, 1983; Prud’homme, 1981; Princen, 1983; Kraynik, 1982) and for several specific engineering applications. These latter include the flow of foams in pipes and large annuli (Anderson, 1971; Mitchell, 1971; Krug, 1975; Wendorff and Ainley, 1981; Sanghani and Ikoku, 1983; Okpobiri and Ikoku, 1983; Lincicome, 1984) in oil-field applications for drilling, completion, fracturing and cleanup of wells. Also of great interest in the oil industry is the flow of foams or foamlike dispersions in porous media. Because the rate of flow of a foam in porous media is greatly reduced from that of either of its component phases, some investigators have considered the conditions under which a body of permeable rock could be completely sealed (Bernard and Holm, 1970; Albrecht and Marsden, 1970). This application relates to the use of foam for preventing leakage from gas-bearing formations. Much of the foam-in-porous-media work reported in the literature, however, is concerned with the use of foam as a displacement fluid. Because of the higher pressure gradient required to move foam, its use would be expected to increase the uniformity of displacement fronts. Thus, many papers (Holm, 1959; Fried, 1961; Minssieux, 1974; Chiang et al., 1980) report experiments in which oil or water is displaced from porous media by foam and in which displacement efficiencies are directly measured in the sample geometry used. In some of the porous media work, the mechanism of foam flow itself is emphasized and the effect of various parameters is studied, including the type and concentration of surfactant in the aqueous phase, the volume fraction of the nonpolar phase (i.e., foam quality), and the ”texture” (a semiquantitative measure of the average bubble size). Additional questions about foam flow in porous media have also been considered (Bernard and Holm, 1964; Bernard et al., 1980,1965;Marsden and Khan, 1966; Mast, 1972; Raza and Marsden, 1967; Raza, 1970; Hirasaki and Lawson, 1983; Patton et al., 1981, 19831, although no consensus has been reached on mechanisms. In much of the work cited above, the results of measurement and analysis have been put in terms of the traditional rheological parameters developed for use with other fluids. For instance, the frictional forces retarding the flow of foam are described in terms of an effective or apparent viscosity. While the computed coefficients are by definition suitable for the particular geometries in which they were derived, they are often not appropriate at a different scale. There is, in fact, experimental evidence in some of these papers that the apparent viscosity values, unlike those measured in simpler fluids, are geometrydependent. As the authors intend to show by consideration of the works of Kraynik, Prud’homme, and Princen, this geometry dependence is to be expected as a consequence of a flow regime that involves little or no shearing of the bulk foam itself. Background Traditionally, the macroscopic concept of viscosity has played a prominent role in the development of fluid dy-
namics and has proven useful in engineering flow calculations. By definition, the shear viscosity, q , of a fluid is evaluated as the shear stress, r , divided by the shear rate, 9 , and it describes the frictional resistance offered by the fluid to continuous deformation when subjected to an applied shearing force. In general, liquids exhibit a strong dependence of shear viscosity on temperature and a weak dependence on pressure, except near phase transition regions. Many fluids obey Newton’s famous simplification, that the viscosity remains constant over a wide range of shear rates. However, other empirical models are needed to describe non-Newtonian behavior such as is found in concentrated dispersions and solutions containing high molecular weight solutes. Some common manifestations of non-Newtonianism include (a) nonlinearly viscous behavior (e.g., power law and Ellis fluids), (b) yield-plastic behavior (e.g., Bingham plastic, Hershel-Bulkley fluid), and (c) viscoelastic behavior that includes elastic phenomena due to normal stresses. Before examining the complications which arise in the rheological characterization of certain two-phase systems such as liquid-liquid emulsions and gas-liquid dispersions (the common foams being a part of this group), it is prudent to review two of the fundamental assumptions underlying the concept of shear viscosity (as applicable in viscometric analysis) and the resulting rheological classification of fluids. 1. A “continuum” description of the material is assumed to be valid (i.e., microscopic structural details of the fluid are deemed unimportant in evaluating the physical properties or gross-flow characteristics of the fluid). 2. Classification of fluids of interest in viscometric work is based on the assumptions of material isotropy (that there are no direction-dependent properties) and homogeneity in composition and properties. In the present discussion, the applicability of these assumptions is of primary importanee. Although no real fluids are truly “structureless continua”, the extent to which their dynamic structure may invalidate the results of continuum theory can be expected to depend on both distance and time scales. Thus, the use of “viscosity” as a space-and-time-averaged fluid property raises no difficulties in the usual consideration of low molecular weight fluids with molecular dimensions in the range 10-8-10-7 cm. Similarly, concentrated polymer solutions or polymer melts containing macromolecules of a size approaching l/lo of a micrometer (10-6 cm) can be safely considered to be continua, except perhaps in very fine or twisted channels. An emulsion or suspension of solid particles in a liquid can be treated as a “pseudo-single-phase liquid” and assigned a viscosity, if the distribution statistics are not modified by the channel boundaries and if the maximum particle size is many times smaller than the viscometer dimensions. For instance, geometry effects were evident in tube viscometry of suspensions of solids in liquids when the ratio (tube diameter)/ (maximum particle size) < 20 (Merrill, 1963). In the case of dispersions containing a gaseous phase, as in the common foams, compressibility can add further complications in viscosity measurement. Two critical requirements are therefore proposed as standards for an empirically useful rheological characterization of a material. (1)The data must be reproducible. (2) The observed rheological behavior should be unique and independent of the viscometer used or of its size. The purpose of these criteria is to test whether the scale of the fluid structure interferes with the definition and measurement of its viscosity.
320 Ind. Eng. Chem. Res. Vol. 26, No. 2, 1987
Foam Rheology. Flow of Foams. Foams, which possess more complex macroscopic structure than either single-phase non-Newtonian liquids or dilute suspensions, present serious difficulties in efforts to obtain a unique rheological description. This is primarily because foam is a heterogeneous dispersion having structural features (foam cells or bubbles) which cannot often be neglected relative to the characteristic dimension of the viscometric device used. Firstly, the very least of these difficulties would consist of statistical variations of measurements, as individual cells passed by pressure measuring ports in a tube viscometer. Further complications could arise from modification of the shear distribution within the viscometer flow channel. Secondly, many foams are compressible and can create spurious effects in tube viscometric measurements. Lastly, foam decay during the course of testing can have a profound impact on the flow characteristics. Factors which might influence the flow behavior of foams, and therefore be important in the design and interpretation of foam flow experiments, are as follows: (a) the ratio of (mean bubble size)/(flow channel size); (b) size distribution of bubbles; (c) flow-induced anisotropy of bubble distribution; (d) foamant-channel wall interactions (e.g., slip, adsorption, etc.); (e) characteristics of flow geometry and flow rate; (f) quality (or nonpolar-phase volume fraction) of foam, q ; (g) properties of the two fluid phases; (h) absolute pressure, (this can be significant with aqueous foams using air or nitrogen a t ambient conditions; the effects generally arise from a difference in compressibility of the two phases); (i) physicochemical nature and concentration of foamant; 6 ) interfacial rheological properties of foam lamellae and their variation with (aging) time (these would affict the coalescence and rupture processes of foam cells and hence the stability of foam; their role in extensional deformations of foam lamellae in the convergine-diverging regions of porous media is, however, not fully understood). Review of Experimental Work on Foam Rheology. With the above considerations in mind, it is interesting to review the many efforts which have been directed toward a fundamental understanding of the rheology of foam since the early works on froth (Sibree, 1934) and on firefighting foam (Grove et al., 1951). A variety of standard viscometric instruments and some specially fabricated, nonviscometric-type equipment were used in these and later investigations. A survey of literature on foam rheology reveals that most work dealt with aqueous foams made from surfactant solutions, the nonpolar phase being either air or nitrogen at nearly atmospheric conditions. Among the variables examined were the (1)gas phase (air, nitrogen), (2) liquid phase (foamant dissolved in water or brine), (3) foamant concentration, (4) technique of foam generation, ( 5 ) quality of foam, and (6) flow rates of gas and liquid. Since the method of preparation influences the type of foam obtained and its properties, a brief description of foam generation techniques is given before discussing the rheological results. Foam Generation and Description Filtered compressed gas and an aqueous surfactant solution have been most often used to generate foam. In one instance (Patton et al., 1981), the surfactant solution also contained a small amount of dissolved polymer, xanthan gum. In a similar and very successful effort to retard gravitational drainage (Rand, 1984), a foamant recipe was used that contained 0.2% of the acrylic polymer, Carbopol 941. The most commonly employed foam generator consists of a pack of beads, sand, or stainless steel wool through which the gas
and liquid phases are pumped simultaneously at constant flow rates. The quality of foam may be controlled by varying the ratio of flow rates of the two fluids. Foam of nearly uniform texture may be obtained by a judicious design and operation of the foam generator. Alternate designs included agitation and mixing of both phases in a chamber and forcing the mixture to pass through selected screens. In some cases, investigators have performed visualization studies and reported measurements of bubble-size distribution and mean bubble size. Often, though, the descriptions of foams used were of qualitative nature and contained insufficient information to estimate the relative effect of bubble size on the observed flow behavior in the viscometer. Again, little information other than qualitative remarks was provided in several of the published reports concerning the stability of foam during experimentation. Measurements in Rotational Viscometers. Studies on foam rheology have been conducted in a number of rotational viscometric devices, including the Brookfield viscometer (Patton et al., 1979), the Fann VG meter, and the Epprecht coaxial-cylinder viscometer (Minssieux, 1974) and modified versions of a Fann coaxial-cylinder viscometer (Fried, 1961; Marsden and Khan, 1966) and of a cone-and-plate viscometer (Wenzel et al., 1970). Sibree (1934) used a specially designed concentric cylinder “viscometer” which enabled indirect measurements of limiting values of froth “viscosity” attained at high rotational speeds. A major experimental problem perceived by some of the observers in the use of these viscometers is that a fixed amount of foam sample is contained in the instrument. This sample is subjected to shearing for a finite period of time until “steady-state” measurements of torque can be made as a function of angular velocity. Unfortunately, collapse or rearrangement of the network of foam bubbles in contact with the rotating solid surface tends to cause a partial discontinuity in the transmittal of angular motion, and therefore, the torque measured is not truly indicative of the “whole” of the foam sample undergoing shear. This is somewhat akin to the phenomenon of slippage observed with certain non-Newtonian fluids at the solid-liquid interface In addition, unless very dry foams (quality, q, exceeding 85% or 90%) are used, sufficient drainage of liquid may occur during the course of testing to change the quality of the foam being sheared (Fried, 1961). The increasing amount of liquid collecting at the bottom of the viscometer will then affect the torque measurement (end effects). Thus, the “time of shearing” will have a pronounced influence on torque readings obtained with wet foams. Consequently, reproducibility of data as well as steady-state measurements have been difficult to achieve in practice. In one study, a continuous flow of foam was provided to the viscometer (Marsden and Khan, 1966) and the stator and rotor surfaces were modified from the usual smooth cylindrical shape by installing thin, long fins to minimize drainage and slippage. Even in this case, the interpretation of the data in terms of foam viscosity was not straightforward. For instance, measurements of apparent viscosity using this modified Fann VG meter were found to lie between 50 and 500 CPfor aqueous foams having gas volume fractions in the range 0.70-0.96. Figure 1, which is reproduced from Marsden and Khan’s work, shows some of their data and indicates the shear thinning character of these foams. These values contrast with order-of-magnitude lower values (3-8 cP) obtained with static foams at high frequencies in a Bendix Ultraviscoson.
Ind. Eng. Chem. Res. Vol. 26, No. 2, 1987 321 500
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( R e f nirasaki 8 Lawson. S P E J - 2 5 April 1985, p 186 1
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Observations of slip for foams in contact with a smooth surface were reported by Wenzel et al. (1970) in the case of a coaxial-cylinder viscometer whose inner cylinder wall was roughened with vanes and the outer cylinder (rotor) wall was smooth. Data were also obtained in a modified cone-plate viscometer in which both surfaces were vaned to minimize slip effects. Values of a small but finite yield stress below which no flow occurred were given for a variety of dry foams of different texture and quality (q > 0.97) by extrapolating data of wall shear stress vs. rotational velocity. Capillary-Tube Viscometry. Continuous-flow-tube viscometry had also been utilized by several investigators (Fried, 1961; Wenzel et al., 1967; Raza and Marsden, 1967; David and Marsden, 1969; Patton et al., 1981, 1983) to evaluate the rheology of foam. In most studies the externally generated aqueous foam passed through capillary tubes and exited into the atmosphere on the downstream side. An apparent motivation for the use of small diameter tubes is the potential applicability of such results to interpret foam flow data in porous materials. A few papers dealing with the application of foams in drilling (Mitchell, 1971; Beyer et al., 1972; Blauer and Holcomb, 1975; Sanghani and Ikoku, 1983) present data on flow experiments in larger tubes, pipes and annuli a t high flow rates. In flow through long tubes or channels, the different compressibilities of the component fluids in foam would be a significant factor whenever the foam density varied appreciably between the entrance and exit. An indirect estimate of this effect is given by examining whether (Pin - Po,)/Pinis small compared to unity. In many reported experiments (Raza and Marsden, 1967; David and Marsden, 1969; Patton et al., 1983), this measure is of the order of 1, indicating there was significant expansion of foam bubbles as they proceeded along the length of the tube. Three consequences of this expansion are as follows. (1)The bubble-size distribution of foam will change with the distance along the tube, resulting in a larger mean bubble size and tending to make the geometry effect, if any, more pronounced. (2) The quality of foam will be higher at the exit compared to entrance conditions. Such a change in quality itself can apparently have a substantial impact on foam flow. (3) When the compressibility of foam cannot be ignored under the experimental conditions, another complication
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is encountered in applying conventional viscometric analysis. That is, the pressure gradient along the channel length is not a constant and the decrease in pressure, P , in the flow direction, z , will be nonlinear. In this case it can no longer be assumed that
AB a result, the equations to calculate wall shear stress, wall shear rate, etc., require corrections. The accepted rheological procedure by which slip effects at the tube wall are considered (Mooney, 1931) cannot be rigorously applied in data interpretation for a compressible foam, or when the (bubble size)/(tube diameter) is nonnegligible. In these instances, the use of viscometric equations will not yield a unique foam-viscosity function. Additionally, the stability of foam during its transit through the tube is important in maintaining the integrity of foam. Liquid drainage from the interfacial films and coalescence and rupture of bubbles can alter the size distribution of bubbles and, thus, the rheology of foam. With these phenomena having a marked influence on the flow mechanism of foam, it is not surprising to see the reported "viscosity" values of foams being highly dependent upon the dimensions of the tubes used. Efforts to explain the variation of apparent foam viscosity with the tube radius on the basis of slippage a t the tube wall were not entirely successful (David and Marsden, 1969), since the slip-corrected viscosity still retained a dependence on tube diameter. Similar geometry dependence of "apparent foam viscosity" on the tube radius was observed by others (Fried, 1961; Raza and Marsden, 1967; Blauer et al., 1974; Patton et al., 1981,1983;Hirasaki and Lawson, 1983). It is evident from the theoretical and experimental work of Hirasaki and Lawson that foam texture has an important influence on foam rheology in small-diameter capillary tubes. As an example, the data in Figure 2 (reproduced from their paper) illustrate that the apparent viscosity varied over a wide range (1000-0.2 cP) when the ratio of mean bubble radius to tube radius was varied from 0.1 to 30 for an aqueous foam of q = 0.83 moving a t a steady velocity of 1cm/s. These experimental conditions covered lamellae motion as well as bulk foam flow. Pat-
322 Ind. Eng. Chem. Res. Vol. 26, No. 2, 1987
ton’s work exhibited an apparent viscosity that was also a function of tube length for tubes of the same radius for a foam of given quality, but this observation has not been reported elsewhere. Wenzel et al. (1967) performed steady-state flow experiments through two round tubes (one smooth and one rough) and three rectangular channels for foams of different texture. The observed relationship between AP/L and the average flow velocity was nonlinear for foams characterized by different mean bubble diameters of
