Steady-State and Unsteady-State Flow of Non~Newtonian Fluids Through Porous Media

Abstract

Abstract Non-Newtonian fluids may be injected into a reservoir during secondary recovery operations. The non-Newtonian fluid used in this work is a power-law type of fluid; that is, the viscosity of the fluid decreases as the flow rate or rate of shear increases. This paper presents equations for steady-state, linear and radial flow of such fluids, transient behavior results from a finite difference model of a radial system, and transient behavior results from a field test. The equations that describe the flow of a non-Newtonian fluid are non-linear and are solved numerically. Finite difference solutions are presented as curves of dimensionless pressure drop at the wellbore vs dimensionless time for a constant injection rate. Solutions were obtained for 5-percent, 10-percent and 100-percent PV of a non-Newtonian fluid for injection rates of 1, 10, 100 and 1000 cc/sec and for a 5 percent PV of non-Newtonian fluid located at r = rw, 3, 10, 20, 50 and 100 ft for a flow rate of 1 cc/sec. The buildup curves do not exhibit a straight-line portion as is the case for Newtonian flow through porous media. Correlations also are shown for the productivity index vs rate for the computer model study and the field tests. INTRODUCTION During various secondary recovery operations non-Newtonian fluids are injected. Such fluids, in general, include thickened water and gelled fluids. The term "non-Newtonian" implies that the viscosity is not only dependent upon temperature and pressure, but also on the rate of shear that is applied to the fluid. For example, water, which is a Newtonian fluid, will have essentially the same viscosity no matter what rate of shear is applied. A pseudoplastic fluid exhibits a decreasing viscosity when higher rates of shear are applied; a dilated fluid has an increasing viscosity with increasing rates of shear. The objective of studies performed and described in this report is to obtain relationships and mathematical and empirical descriptions of the flow of non- Newtonian fluids through porous media. Simple mathematical relationships, computer studies that include the unsteady- state behavior of such fluids, and field tests are used. PREVIOUS LABORATORY STUDIES Laboratory studies have been performed by several investigators. The fluids used in one investigation1 were surfactant-stabilized dispersions of water in hydrocarbons. Its Fig. 6 is reproduced as our Fig. 1. This effective-viscosity vs frontal-velocity diagram shows that this fluid is of the power-law type over a fairly large range of frontal velocities. Gogarty1 developed an equation for the effective viscosity as a function of shear rate that reduces to Eq. 2 for frontal velocities greater than about 2 ft/D. The fluid characteristics used in the present study are similar to those reported by Gogarty.