Accounting for heterogeneity in scale-up of apparent polymer viscosity for field scale application

Petroleum reservoirs exhibit heterogeneities over a wide range of length scales, and the impact of these heterogeneities on effective shear rate controls the flow behavior of polymer solution. Prediction of apparent polymer viscosity in field-scale models therefore requires proper scale-up of reservoir heterogeneity and the associated scale-up of effective shear rate. The focus of the paper is to demonstrate a procedure for estimating scaled-up apparent viscosity accounting for sub-scale heterogeneities based on the volume averaging approach. Volume averaging is a mathematical technique to derive continuum equations at coarse scales given representative transport equations at local or fine scales. Although treatment of transport problems with volume averaging techniques has been presented by several authors, application to real geological systems exhibiting complex heterogeneity is lacking. While volume averaging has been used in the past to derive effective rock properties, the application of the method to derive an effective fluid property (namely polymer viscosity) is novel. In this new procedure, results from a fine-scale numerical flow simulation reflecting the full physics of polymer flow transport albeit over a small sub-volume of the reservoir is integrated using the volume averaging technique to provide effective description of polymer transport at the coarse scale. The results obtained over the small sub-volume can be later extrapolated to the full volume or in the case where the effects of global boundary conditions (number and locations of wells) are important, one may divide the entire domain into various subregions and perform the volume averaging calculations within each sub-region. These sub-regions could correspond to different flow pattern (configuration of well groups), and their physical location with respect to the large-scale global boundary conditions must be tracked. We apply the volume averaging technique to single-phase flow of a shear-thinning non-Newtonian fluid and the scaling characteristics of apparent viscosity are documented for different heterogeneity correlation lengths. Our results show 1) mean of apparent viscosity decreases with length scale, and this decline becomes more drastic with increased heterogeneity; 2) at a given length scale, variability in apparent viscosity increases with heterogeneity; 3) effective shear rate increases with scale implying that the shear rate coefficient that affects polymer viscosity in a grid block of the flow simulation model also increases with scale. An important contribution of this work is to provide a framework for generating scaling relationship of apparent viscosity that accounts for reservoir heterogeneities. Our results demonstrate that shear rate coefficient, which is generally assumed to be constant regardless of the modeling scale, actually increases with scale so as to preserve the mobility of the flowing phase. Since polymer is used as a mobility control agent in all chemical flooding applications, correct scale-up of effective shear rate and apparent viscosity is crucial in accurate prediction of field-scale response and recovery factors.