Effect of Fractal Permeability Correlations on Waterflooding Performance in Carbonate Reservoirs
Authors: Babadagli T. 
Published: December 2007

Abstract
The objective of this paper is to investigate the response of correlated permeability distributions to field scale immiscible flooding and relate the output to correlation parameter. Synthetic cross section porosity maps are generated using fractional Brownian motion (fBm) statistics in vertical direction and then the porosities are converted to permeabilities using exponential correlation as data to black oil simulator. The intermittency coefficient, H that characterizes the smoothness of the distribution is varied between two extremes; 0.4 and 0.95. For different porosity and permeability distributions generated on the basis of different random number seeds, simulations are carried out to explore the effects of oil/water viscosity ratio, the number of layers and average permeabilities on the waterflooding performance. It is found that the displacement efficiency increases with increasing H of the distribution. The effect of the smoothness of the permeability field is more significant on water-oil ratio and breakthrough time than on the cumulative oil production and oil production rate. Finally, the methodology is applied to a field example. The performance of a five-spot waterflooding in a carbonate reservoir is evaluated for different distribution types namely vertical and horizontal fBm distributions as well as the dual porosity model.

Introduction
Accurate performance prediction of any field application requires realistic input of spatially distributed reservoir properties on a scale of grid block size. Due to limitations on data collection representing whole reservoir, stochastic techniques are applied to generate non-unique but representative distribution of properties.
The applicability of the stochastic methods to describe the distribution of reservoir properties has been extensively investigated in last decade. In his pioneering work, Hewett showed that the permeability distribution in sedimentary environments represents a fractal character with long range correlations. Later, conditional simulation studies of field applications in which the fractal distributions of interwell porosity and permeability fields are used as data were reported.
Some studies are available on correlating the relationship between the heterogeneity and the displacement process. Hird and Kelkar related the production performance to the reservoir connectivity parameters. Also, Aasum et al. generated fractal cross sectional distributions of a dolomitic field by fractional Brownian motion (fBm) and applied the miscible displacement process on different realizations of these distributions. They observed that the correlation parameter, H has a moderate influence on breakthrough time and saturation profiles but a negligible influence on recovery.
In summary, the studies published to date show that the porosity and permeability represent a correlation in space and the random realizations of the permeability distribution based on the fractal correlations provide an accurate estimate of performance without any history matching. Also, once the permeability distribution represents a correlation in space like fractal distributions, the performance of the displacement process can be related to the correlation parameter of the property distribution. However, more research is needed to understand and generalize the relationship between the distribution correlation parameter and the recovery performance. This has been the motivation for this study and immiscible displacement simulations were performed on synthetically generated fractal cross sections of porosity and permeability field. Then, the effect of the roughness of permeability distribution on the production performance of waterflooding was investigated. A field case was evaluated applying different fractal property distributions and dual porosity model.
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