The Effect of Scaleup on Bivariate Statistics: Insight into Modeling Imperfectly Correlated Porosity and Permeability

James W. Jennings
The University of Texas at Austin, Austin, TX

Most cross plots of plug-scale porosity and permeability exhibit imperfect correlation and significant scatter. Thus, some methods for populating reservoir model grids with petrophysical properties, such as stochastic cosimulation and cloud transformation, are designed to reproduce this scatter. The statistical parameters for simulation of porosity are usually estimated from well-log data because fewer wells are cored. However, the cross statistics between porosity and permeability are almost always estimated from core data because the well logs do not include an independent measurement of permeability. Thus, three scales are involved, cores, well-logs, and grid cells, but the effects of the scale changes on the cross statistics are rarely taken into account.

This paper will present a series of numerical experiments conducted to investigate the effects of scaleup on the cross-statistics of porosity and permeability in two carbonate examples. The spatial statistics and cross statistics of porosity and permeability were estimated from a combination of outcrop and subsurface plug data. Plug-scale stochastic cosimulations were conducted using these estimated statistics, and a series of scaleup calculations were performed. The results indicate that at typical grid-cell scales the scatter is almost entirely eliminated if the averaging is limited to cells composed of a single rock-fabric class. Thus, reservoir models using grids constructed to follow rock-fabric flow units, can be accurately populated with porosity and permeability values without the use of stochastic cosimulation. It is sufficient to stochastically simulate porosity alone and to subsequently calculate permeability with a scaleup-corrected, but perfectly correlated, porosity-permeability transform.