Statistical scale-up of reservoir properties: Concepts and applications

All petrophysical quantities are used at a scale different from the one on which they were measured. This necessitates an adjustment of the measured values before they are used to develop a reservoir model, a step referred to as scale-up. Scale-up is complicated by the properties being heterogeneously distributed in space and self- or autocorrelated. The autocorrelation means that the heterogeneity itself must be scaled up so that the adjusted measurements correctly reflect the property at the coarser scale. This paper attempts to understand the uncertainty in assigning scaled up values to finite regions of space. The region may be a formation interval or a cell thickness for numerical simulation. We use the notion of the variance of the mean of a random variable to understand the scale-up process. The behavior of the variance of the mean is used to investigate the definition of a representative elementary volume (REV), and of the behavior of lateral and vertical permeabilities with scale and the resultant impact on uncertainty distributions for reservoir properties. The paper demonstrates that the notion of the variance of a mean can be used to understand and refine the concept of representative elementary volume. The change in horizontal and vertical permeability with scale can also be explained using the variance of the mean using reasonable autocorrelation functions. The impact of scaling up on the autocorrelation structure of the simulated field is demonstrated in the paper. The results point to the importance of employing rigorous procedures to scale heterogeneity in order to derive robust estimates of uncertainty.