Approximate Pore-Level Modeling for Apparent Viscosity of Polymer-Enhanced Foam in Porous Media

Abstract

Summary Foam is used in the oil industry in a variety of applications, and polymer is sometimes added to increase foam's stability and effectiveness. A variety of surfactant and polymer combinations have been employed to generate polymer-enhanced foam (PEF), typically anionic surfactants and anionic polymers, to reduce their adsorption in reservoir rock. While addition of polymer to bulk foam is known to increase its viscosity and apparent stability, polymer addition to foams for use in porous media has not been as effective. In this pore-level modeling study, we develop an apparent viscosity expression for PEF at fixed bubble size, as a preliminary step to interpret the available laboratory coreflood data. To derive the apparent viscosity, the pressure-drop calculation of Hirasaki and Lawson (1985) for gas bubbles in a circular tube is extended to include the effects of shear-thinning polymer in water, employing the Bretherton's asymptotic matching technique. For polymer rheology, the Ellis model is employed, which predicts a limiting Newtonian viscosity at the low-shear limit and the well-known power-law relation at high shear rates. While the pressure drop caused by foam can be characterized fully with only the capillary number for Newtonian liquid, the shear-thinning liquid requires one additional grouping of the Ellis-model parameters and bubble velocity. The model predicts that the apparent viscosity for PEF shows behavior more shear-thinning than that for polymer-free foam, because the polymer solution being displaced by gas bubbles in pores tends to experience a high shear rate. Foam apparent viscosity scales with gas velocity (Ug) with an exponent [−α/(α+2)], where α, the Ellis-model exponent, is greater than 1 for shear-thinning fluids. With a Newtonian fluid, for which α = 1, foam apparent viscosity is proportional to the (−⅓) power of Ug as derived by Hirasaki and Lawson. A simplified capillary-bundle model study shows that the thin-film flow around a moving foam bubble is generally in the high-shear, power-law regime. Because the flow of polymer solution in narrower, water-filled tubes is also governed by shear-thinning rheology, it affects foam mobility as revealed by plot of pressure gradient as a function of water and gas superficial velocities. The relation between the rheology of the liquid phase and that of the foam is not simple, however. The apparent rheology of the foam depends on the rheology of the liquid, the trapping and mobilization of gas as a function of pressure gradient, and capillary pressure, which affects the apparent viscosity of the flowing gas even at fixed bubble size.