The Apparent Viscosity of Foams in Homogeneous Bead Packs

Abstract

Summary 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 -1/3 power when the bubble size is large cornpared with the pore size and on the -2/3 power of capillary number when the bubble size is smaller. Introduction Although foams hold promise for improving sweep efficiency in oil recovery processes (e.g., see Refs. 1 through 4), there have been relatively few fundamental studies to show how they flow through pore space. The traditional way to analyze foam flow has been to extend Darcy's law5:Equation 1 ?rj is the so-called relative mobility of Phase j and is ordinarily written as the relative permeability of the medium to Phase j divided by the phase viscosity. Eq. 1 has been used to describe the mobility of foam by treating the gas and liquid as separate phases6–8; the Newtonian viscosities of the phases are used, while their relative permeabilities are allowed to differ from those when they do not flow as a foam. Other approaches treat foam as a single phase.9–12 Regardless of how the transport of foam through porous materials is modeled, experiments have established that gas mobility depends on foam quality (fractional flow of gas) and composition of the aqueous phase as well as on the permeability and liquid saturation of the medium.13 Another important parameter is foam texture, which is the number distribution of the volumes of the discrete gas cells that constitute the foam; the average cell volume or, equivalently, the radius of a sphere of equal volume, is but one measure of foam texture. The influence of texture on the viscosity of bulk foams (i.e., foams in which the average bubble size is much smaller than the dimensions of their physical boundaries) has long been recognized14 but has been ignored by most researchers who have studied foam in porous media. Marsden et al.,13 on the other hand, realized the significance of foam texture in stating that "it appears that size, rather than quality or foamer concentration determines the mobility." Although they were unable to control the distribution of bubble sizes in foams flowing through sandpacks, Marsden et al. successfully measured texture in the effluent. More recently, Hiraski and Lawson15 showed texture to be the principal variable affecting the apparent viscosity of surfactant-stabilized foams flowing through smooth, glass capillaries of uniform diameter. To determine apparent viscosity as a function of foam texture, quality, gas velocity, capillary radius, and capillary length, they analyzed experimental data with the Hagen-Poiseuille law16:Equation 2 With a theoretical model, they also revealed three constituents of foam viscosity:the Newtonian viscosity of any liquid slugs between gas bubbles;a resistance that manifests itself as an interface deformation; anda surface traction that results from a surface tension gradient. Because flow through uniform tubes is one component of flow through porous media (at least idealized porous media), it is likely that Hirasaki and Lawson's work applies in some measure to the viscosities of foams in more complex pore structures. This work examines the flow of aqueous foams in homogeneous bead packs. Gas mobilities in foams made from a 1.0 wt% aqueous solution of sodium dodecylbenzenesulfonate were measured without oil present in glass tubes packed with either 0.23- or 0.3-cm [0.09- or 0.12-in.) -diameter glass beads. When the average foam-bubble size did not change in-situ and the relative permeability to gas could be estimated from the residence time of moving gas in the pack, the data were reduced to apparent gas viscosities. To correlate these measured viscosities, Hirasaki and Lawson's theory was extended to account forthe radii of curvature of foam lamellae being set by the capillary pressure in the porous medium andpore constrictions. The former were included through an ordinary, idealized capillary pressure function. The way pore constrictions contribute to flow resistance was determined through an analysis of the pressure gradient required to mobilize stationary lamellae and a steady-state, dynamic mechanical energy balance. To complete this description of foam in porous media, a way to predict gas-phase relative permeability must be developed. Future work, therefore, should aim at elucidating the relationships between gas relative permeability foam texture, and other variables. In this way, the usual equations of mass, momentum, and energy conservation, when coupled with balances on the densities of flowing and stationary bubbles within the foam, can be used to describe correctly the generation, destruction, and flow of foams in porous media.17 Total Apparatus and Procedures Glass Bead Packs. Bead packs were constructed from glass tubes of uniform 1D (see Fig. 1 and Table 1). To monitor the texture of the foam being injected and produced, capillary tubes were connected to the inflow and outflow ends. Foam Generators. To examine foams of controlled bubble sizes, special generators (depicted in Fig. 2) were used. A generator consists of a drawn tip that protrudes into a liquid-fiIled chamber. Gas, through the tip, and liquid, through another port, are injected into the chamber at prescribed rates. The diameter of the drawn tip and, to a lesser extent, the gas flow rate set the texture. To investigate the flow of foams of smaller average bubble size, one bead pack (Pack 1 in Table 1) was used as the generator for another (Pack 3). Controlling Gas and Liquid Flow Rates. Gas flow to the foam generator was controlled with a Matheson (Fast Rutherford, NJ) Model No. 8240 mass flow controller. Liquid rates were regulated with a syringe pump (Harvard Apparatus No. 975, South Natick, MA). Because the liquid rates were low, they were also measured by weighing the effluent from the pack. After passing through the outflow capillary, the foam traveled through a flexible, rubber connector into a glass weighing vessel that rested on a balance (Fig. 3). Monitoring Foam Texture. Bubble sizes in the foam entering and leaving a pack were observed in the inflow and outflow capillaries. Because the capillary diameters were smaller than the bubble diameters, the gas cells were cylindrical inside the capillaries. The bubble lengths were measured from photographs taken while the foam was flowing. The average bubble volume was subsequently converted to an equivalent sphere radius. The same average bubble size in the inflow and outflow capillaries for a sustained period indicated that bubble size remained uniform within the pack.