Immiscible Displacement of Non-Newtonian Fluids in Communicating Stratified Reservoirs

The displacement of non-Newtonian power-law fluids in communicating stratified reservoirs with a log-normal permeability distribution is studied. Equations are derived for fractional oil recovery, water cut, injectivity ratio, and pseudorelative permeability functions, and the performance is compared with that for Newtonian fluids. Constant-injection-rate and constant-total-pressuredrop cases are studied. The effects of the following factors on performance are investigated: the flow-behavior indices, the apparent mobility ratio, the Dykstra-Parsons variation coefficient, and the flow rate. It was found that fractional oil recovery increases for nw>no and decreases for nwno. Increasing the total injection rate increases the recovery for nw>no, and the opposite is true for nw<no. It also was found that the fractional oil recovery for the displacement at constant total pressure drop is lower than that for the displacement at constant injection rate, with the effect being more significant when nw<no. Introduction Many of the fluids injected into the reservoir in enhanced-oilrecovery (EOR)/improved-oil-recovery (IOR) processes such as polymer, surfactant, and alkaline solutions may be non-Newtonian; in addition, some heavy oils exhibit non-Newtonian behavior. Flow of non-Newtonian fluids in porous media has been studied mainly for single-phase flow. Savins (1969) presented a comprehensive review of the rheological behavior of non-Newtonian fluids and their flow behavior through porous media. van Poollen and Jargon (1969) presented a finite-difference solution for transient-pressure behavior, while Odeh and Yang (1979) derived an approximate closed-form analytical solution of the problem. Chakrabarty et al. (1993) presented Laplace-space solutions for transient pressure in fractal reservoirs. For multiphase flow of non-Newtonian fluids in porous media, the problem was considered only for single-layer cases. Salman et al. (1990) presented the modifications for the Buckley-Leverett frontal-advance method and for the JBN relative permeability method for non-Newtonian power-law fluid displacing a Newtonian fluid. Wu et al. (1992) studied the displacement of a Bingham non-Newtonian fluid (oil) by a Newtonian fluid (water). Wu and Pruess (1998) introduced a numerical finite-difference solution for displacement of non-Newtonian fluids in linear systems and in a five-spot pattern. Yi (2004) developed a Buckley-Leverett model for displacement by a Newtonian fluid of a fracturing fluid having a Herschel-Bulkley rheological behavior. An iterative procedure was used to obtain a solution of the model. The methods available in the literature to predict linear waterflooding performance in stratified reservoirs are grouped into two categories depending on the assumption of communication or no communication between the different layers. In the case of noncommunicating systems, no vertical crossflow is permitted between the adjacent layers. The DykstraParsons (1950) method is the basis for performance prediction in noncommunicating stratified reservoirs. A model for communicating stratified reservoirs was presented by Hiatt (1958). This model assumes complete crossflow between layers to keep the pressure gradient the same in all layers (vertical equilibrium) at any distance. Warren and Cosgrove (1964) applied the Hiatt model to a system with log-normal permeability distribution and normal porosity distribution. El-Khatib (1999) presented a closed-form analytical solution for communicating systems with log-normal permeability distribution. Hearn (1971) used the same Hiatt model to develop expressions for pseudorelative permeabilities that can be used to reduce a 3D model to a 2D areal model with average (pseudo) functions for the vertical direction. To the best of the author’s knowledge, no analytical or numerical models are available for the displacement of non-Newtonian fluids in multilayer stratified reservoirs. In this study, an analytical model will be presented to study the performance of immiscible non-Newtonian power-law fluids in stratified reservoirs. Although the model is applicable for any permeability distribution, a stratified system with a log-normal permeability distribution is studied because its behavior is well documented in the literature (Warren and Cosgrove 1964; El-Khatib 1999) for Newtonian fluids.