Simulating Residual Saturation and Relative Permeability in Heterogeneous Formations
Authors: Paterson Lincoln , Painter Scott , Zhang Xiaodong , Pinczewski Val 
Published: December 2007

Abstract
Network models are used to investigate the effect of correlated heterogeneity on capillary dominated displacements in porous media. Residual saturations and relative permeabilities are shown to be sensitive to the degree of correlation and anisotropy but not variability. The network models reproduce the experimental observation that relative permeability is greater in the direction parallel to the bedding compared to perpendicular to the bedding. The scatter commonly observed in core measurements of residual saturation is attributed to the presence of correlated heterogeneity in actual reservoir rocks.

Introduction
A major problem in performing realistic field reservoir simulation studies is relating laboratory core measurements of relative permeability and residual oil saturation to the field scale. This is because relative permeability and residual oil saturation are strongly influenced by reservoir heterogeneity and heterogeneity occurs at all scales in the reservoir from the pore-scale to the full field-scale. In previous studies we have developed improved statistical methods for characterising reservoir heterogeneity. In the present study we investigate the effects of statistical heterogeneity on relative permeability and residual saturation in order to relate measurements from laboratory core tests to the field scale. We develop network models, based on pore-scale physics, and use these to generate relative permeability curves and residual oil saturations for different statistical descriptions of heterogeneity with capillary dominated flow. Heterogeneity correlations were generated using fractional Brownian motion, fractional Levy motion and multifractals. The choice of parameters in the correlations are based on extensive studies of outcrop data, well logs, and minipermeameter measurements on cores.
Results include a comparison of the breakthrough times, residual oil saturations and relative permeability curves for varying degrees of heterogeneity. The results show that relative permeabilities tend to be higher with correlated heterogeneity because the pore space occupied by each of the fluids is more compact. The variability in residual oil saturation is also greater in media with long-range correlations and does not show the scale dependence of the uncorrelated media. Bias in core selection can lead to incorrect estimates at the reservoir scale. This paper provides insight for interpreting residual saturation and relative permeability measurements.

Heterogeneity
Increasingly accurate statistical descriptions of heterogeneity in sedimentary rocks have become available through the analysis of borehole data and studies of outcrops. Fractional Brownian motion (fBm) was introduced as a description of reservoir heterogeneity by Hewitt. More recently a description based on the more general fractional Levy motion (fLm) has been found to have better accuracy. The fLm model is parameterised by the Levy index and the Hurst parameter H. The Levy index 0 &lt; 2 quantifies the degree of spatial variability, with increasing implying increasing variability. The situation = 2 is the special case of fBm. The Hurst parameter 0 &lt; H &lt; 1 quantifies the degree of spatial correlation; increasing H implies increasing correlation. A feature of these fractional motion statistical descriptions is that they contain long range correlations in property values. This contrasts with the usual network models and glass micromodel simulations in which adjacent properties are independent.
For our network simulations, H values were mainly chosen to be in the range 0.1 H 0.5, based on measurements at the reservoir scale. Reported measurements of H values greater than or less than 0.5 appear to depend on the choice of fractional Brownian motion or fractional Gaussian noise as the underlying model. Values appearing in the literature have been summarized by Gray et al.
The fLm model has been tested from the kilometre scale to scales as small as 0.03 m. X-ray CT scanning of laboratory core samples often reveals laminations and other heterogeneities at scales down to about 1 mm, which is the limit of resolution for most common scanners.
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