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

With the aid of linear filter theory we analyze 13,824 permeability measure ments 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 heterogenei ties comprising that sample support. Although the theoretical aspects of li near filter analysis are well documented, physical data for testing the fil tering behavior of an instrument, particularly in the context of porous med ia flow, are rare to nonexistent. Our exploration makes use of permeability data measured with a minipermeameter on a block of Berea sandstone. Data w ere collected according to a uniform grid that was resampled with tip seals of increasing size (i.e., increasing sample support). Spatial weighting (f ilter) 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 measurem ents, 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 functon of radia l distance from the center of the tip seal, consistent with the divergent f low geometry imposed by the minipermeameter. The magnitude of the weighting function decreases while its breadth increases with increasing tip seal si ze, reflecting the increasing sample support. We further demonstrate, both empirically and theoretically, that nonadditive properties like permeabilit y are amenable to linear filter analysis under certain limiting conditions (i.e., small variances). Specifically, the weighting function is independen t of the power average employed in its calculation (e.g., arithmetic versus harmonic average). Finally, we examine the implications of these results f or other instruments commonly employed in hydraulic testing (e.g., slug and pump tests).