Mechanisms of Foam Flow in Porous Media: Apparent Viscosity in Smooth Capillaries

The apparent viscosity of foam flowing through smooth capillaries was measured experimentally, and a mathematical model was developed. Foam texture (a measure of bubble volume) is a key parameter in determining the following properties of foam flowing through a capillary:whether the foam exists as bulk foam or as a chain of bubbles where each pair of bubbles is separated by an individual lamella,the number of lamellae per unit length of the capillary, andthe radius of curvature of the gas-liquid interface. The apparent viscosity is the sum of three contributions:that from slugs of liquid between bubbles,the resistance to deformation of the interface of a bubble passing through a capillary, andthe surface tension gradient that results when surface active material is swept from the front of a bubble and accumulates at the back of it. The sensitivity of both measured and calculated apparent viscosity is presented as a function of bubble size, capillary radius, ratio of bubble radius to capillary radius, velocity, quality, and surface tension gradient. Introduction An early conceptual model for the relative permeability of two-phase flow was the bundle of capillary tubes model. In this model, the wetting phase flowed in the smaller capillaries and the nonwetting phase flowed in the larger capillaries. The relationship between the flow rate and pressure drop in a capillary was described by the pressure drop in a capillary was described by the Hagen-Poiseuille law. The flow of a discontinuous nonwetting phase, such as a foam, cannot be described by the Hagen-Poiseuille law. The purpose of this investigation was to determine the relationship between flow rate and pressure drop for the flow of foam through a capillary. This relationship is described by an apparent viscosity that is required to modify the Hagen-Poiseuille law for the flow of foam. Our previous observations of flow of foam lamellae in transparent porous models showed that lamellae move from pore to pore by translation. Breaking and re-forming of lamellae were rare; so was bubble coalescence. These observations suggest that the apparent viscosity of foam or lamellae in uniform, smooth capillaries is related to and, indeed, is one component of the mobility of foam in porous media. A reasonable conceptual model of a natural porous medium is a bundle of interconnected capillaries of different sizes and containing constrictions. All capillary sections, or pores, near to one another have the same capillary pressure. Thus, phase saturations may differ from pore to pore, but the radii of curvature of the gas/ liquid interfaces are equal. When flow in such an array of capillaries is modeled, resistance to flow in parallel channels of both the same and different sizes is conceived to be in parallel. Flow in smooth, uniform pore sections is in series with flow through constrictions. The component of resistance owing to smooth, uniform pore sections is approximated by resistance to flow in smooth, uniform capillaries. Measurements and theory presented here show that the most important variable affecting foam viscosity in uniform, smooth capillaries is foam texture (bubble size). Foam of finer texture has more lamellae per unit length and, as a result, greater resistance to flow. This is true both for flow of bulk foam and series of lamellae. The principal factors affecting apparent viscosity of foam in uniform capillaries are dynamic changes at gas/liquid interfaces. These are illustrated in Fig. 1.Slugs of liquid between gas bubbles resist flow.Viscous and capillary forces result in interfaces that are deformed against the restoring force of surface tension. The extent of this deformation and the resulting bubble shape partially determine apparent viscosity as a function of flow rate.Another factor determining apparent viscosity as a function of velocity is expansion of the interface at the leading end of a bubble, accompanied by compression at the trailing end. This sweeping action causes surface active material to be depleted at the front and to accumulate at the back of the bubble. The result is a surface tension gradient that resists flow. Scaling of Foam Texture and Capillary Radius Since foam texture is a measure of the average volume or equivalent radius of its bubbles, one would expect that an important scale factor is the ratio of this equivalent radius to the equivalent radius of a porous medium or the radius of a capillary. This ratio can be expressed either as the wetted perimeter per unit area of the solid or as the number of lamellae per unit length of capillary. These quantities are denoted as nL and are referred to as the number of equivalent lamellae per unit length. This concept is illustrated in Fig. 2. SPEJ P. 176