What does an instrument measure? Empirical spatial weighting functions calculated from permeability data sets measured on multiple sample supports

Abstract

With the aid of linear filter theory we analyze 13,824 permeability measurements to empirically address the question, What does an instrument measure? By measure we mean the sample support or sample volume associated with an instrument, as well as how the instrument spatially weights the heterogeneities comprising that sample support. Although the theoretical aspects of linear filter analysis are well documented, physical data for testing the filtering behavior of an instrument, particularly in the context of porous media flow, are rare to nonexistent. Our exploration makes use of permeability data measured with a minipermeameter on a block of Berea sandstone. Data were collected according to a uniform grid that was resampled with tip seals of increasing size (i.e., increasing sample support). Spatial weighting (filter) functions characterizing the minipermeameter measurements were then calculated directly from the permeability data sets. In this paper we limit our presentation to one of the six rock faces, consisting of 2304 measurements, as the general results for each rock face are similar. We found that the empirical weighting functions are consistent with the basic physics of the minipermeameter measurement. They decay as a nonlinear function of radial distance from the center of the tip seal, consistent with the divergent flow geometry imposed by the minipermeameter. The magnitude of the weighting function decreases while its breadth increases with increasing tip seal size, reflecting the increasing sample support. We further demonstrate, both empirically and theoretically, that nonadditive properties like permeability are amenable to linear filter analysis under certain limiting conditions (i.e., small variances). Specifically, the weighting function is independent of the power average employed in its calculation (e.g., arithmetic versus harmonic average). Finally, we examine the implications of these results for other instruments commonly employed in hydraulic testing (e.g., slug and pump tests).