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

Abstract

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). Introduction Questions concerning what is actually measured by an in-In this way, the weighting function 13mo explicitly accounts for the influence of the measurement process on the value of k m.

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“…Results do not reveal any discernible differences in the range values fitted to the indicator semivariograms calculated for the different permeability classes. Third, the calculated spatial weighting (filter) functions in Tidwell et al (1999) did not discern significant anisotropy that would be diagnostic of preferred flow paths along the laminae.…”

“…Specifically, laminae of alternating "high" and "low" permeability are located in the same position in each of the data sets (e.g., the low-permeability laminae centered at z=7.5 cm, and at 18.5 cm). In fact, these laminae can be traced across all four block faces cut normal to the stratification (Tidwell and Wilson, 1997;Tidwell et al, 1999). Finally, distinct smoothing of the permeability field is evident as the tip seal size increases.…”

“…The sample mean increases with increasing sample support while the sample variance decreases. Inspection of the CDFs reveals that upscaling influences the distribution beyond the sample mean and variance, as To investigate the spatial correlation of the permeability data, full two-dimensional semivariograms were calculated for each of the natural-log permeability data sets using Fourier analysis (via Fast Fourier Transforms, see Bracewell, 1986;Tidwell et al, 1999). The twodimensional semivariogram calculated for the 1.27 cm data set is given in Figure 3 The sample semivariograms calculated from the four different tip seal data sets have been fit with an exponential yi model where C, is the nugget, 02 the variance of the natural-log permeability, ii = (Ax,l,Lz,l) the semivariogram range vector (cm), and S = (sx,sz) the separation vector (cm) in the principal directions (see Figure 1).…”

“…However, data beyond that provided above, can be found in other references. Tidwell and Wilson (1997) present data taken from Face 5, which is located directly opposite of the block from Face 3 (given here), while data from Face 2, orthogonal to Faces 3 and 5 and oriented normal to stratification, can be found in Tidwell et al (1999). Each sampled block face exhibits a roughly log-normal permeability distribution with a mean that increases with increasing sample support and a variance that decreases.…”

Kinematic structure of minipermeameter flow

“…Thus it may seem very natural to (r s, zs) corner of the cylindrical sample, a hemispherical support volume could be defined by this "effective radius" [Tidwell et al, 1999]. However, given the support volume geometry defined by the 10% streamline, this is misleading.…”