Regularization of a semivariogram

Clark, Isobel

doi:10.1016/0098-3004(77)90010-3

Cited by 41 publications

(19 citation statements)

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“…))I)) of the point-support permeability, k, (i.e., associated with an infinitely small sample volume), 0,' is the corresponding variance (0,' = Var[ln(k,(l))]), and g , is a constant accounting for the spatial anisotropy of the natural-log permeability field. Although we do not have exact measures of the correlation length scales for this medium, io (as determined from ko), we do know that io is slightly smaller than the semivariogram range values determined from the smallest, 0.15 cm, tip seal data set (Clark, 1977;Journel and Huijbregts, 1978). From the semivariograms (Figure 4) and inspection of the sampled block face it is clear that the correlation length scale parallel to stratification is much larger than that normal to stratification.…”

Section: Model Comparisonmentioning

confidence: 99%

“…In practice the local support (Le., smallest sample support for which permeability data are available) is taken to represent the point support, sometimes subject to minor modifications ( e g , Clark, 1977). However, under non-uniform flow conditions and where heterogeneity occurs on the scale of the measurement, such treatment can be dangerous.…”

Section: Discussionmentioning

confidence: 99%

“…First, the sill of the sernivariogram decreases with increasing sample support, consistent with the decreasing variance, while the general shape of the semivariogram remains unchanged (except for dampening of the hole-effect in Figure 4b). Also, the fitted range values xi increase linearly with sample support reflecting the linear increase in tip seal radius (Clark, 1977;Journel and Huijbregts, 1978). …”

Section: Spatial Permeability Measurementsmentioning

confidence: 99%

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Tidwell

Wilson

1999

Mathematical Geology

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To physically investigate permeability upscaling over 13,000 permeability values were measured with four different sample supports (Le., sample volumes) on a block of Berea Sandstone. At each sample support spatially-exhaustive permeability data sets were measured, subject to consistent flow geometry and boundary conditions, with a specially adapted minipermeameter test system.Here, we present and analyze a subset of the data consisting of 2304 permeability values collected from a single block face oriented normal to stratification. Results reveal a number of distinct and consistent trends (i.e., upscaling) relating changes in key summary statistics to an increasing sample support. Examples include the sample mean and semivariogram range that increase with increasing sample support and the sample variance that decreases. To help interpret the measured mean upscaling we compared it to theoretical models that are only available for somewhat different flow geometries. The comparison suggests that the non-uniform flow imposed by the minipermeameter coupled with permeability anisotropy at the scale of the local support (i.e., smallest sample support for which data is available) are the primary controls on the measured upscaling. This work demonstrates, experimentally, that it is not always appropriate to treat the local-support permeability as an intrinsic feature of the porous medium; that is, independent of its conditions of measurement.

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Section: Model Comparisonmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Spatial Permeability Measurementsmentioning

confidence: 99%

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Untitled

Tidwell

Wilson

1999

Mathematical Geology

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“…We begin by noting that many of the upscaling trends in the ensemble statistics are qualitatively consistent with linear volume averaging (e.g., semivariogram regularization [Clark, 1977;Journel and Huijbregts, 1978]). In porous media theory, linear volume averaging has been applied to both k(x) and its log transform, Y(x) = In k(x).…”

Section: Upscaling Of the Ensemble Statisticsmentioning

confidence: 99%

Upscaling experiments conducted on a block of volcanic tuff: Results for a bimodal permeability distribution

Tidwell¹,

Wilson²

1999

Water Resources Research

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Abstract. Permeability upscaling is physically investigated by making over 31,000 permeability measurements on a meter-scale block of volcanic tuff. The experiments are made possible by a specially adapted minipermeameter test system. Here we present and analyze 5185 permeability values, corresponding to five different sample supports (i.e., sample volumes) collected from one of the six block faces. The results show that the measured spatial permeability patterns, bimodal permeability distribution, and semivariogram structure/length scales are closely related to the strong textural contrast characterizing the tuff sample (i.e., highly porous pumice fragments embedded in a tight rock matrix). Each of the summary statistics shows distinct and consistent trends with increasing sample support (i.e., upscaling). As the sample support increases, the mean and variance decrease according to a power law relation, and the semivariogram range increases linearly, while the general structure of the semivariogram (isotropic, spherical model) remains unchanged. Interpretation of these results is pursued from two very different points of view; one addresses upscaling of the ensemble statistics, while the second examines upscaling from a local or pointwise perspective. We find the general upscaling trends exhibited by the ensemble statistics (given above) to be consistent with the basic concepts of volume averaging, albeit nonlinear volume averaging. The bimodal characteristics of the tuff sample and the nonuniform flow conditions imparted by the minipermeameter contribute to the nonlinearity. The local analysis reveals strong variability in permeability upscaling from point to point throughout the sampling domain. Specifically, the permeability upscaling exhibited by zones rich in pumice is very different from zones dominated by matrix, unless the averaging volume is significantly larger than the spatial correlation scale.

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“…Certain relationships can be obtained from previous studies and can be used to link a model range and sill of a regularized semivariogram to a deregularized one, such as the studies by (Clark, 1977;Journel & Huijbregts, 1981;Jupp et al, 1988Jupp et al, , 1989 and (Atkinson & Curran, 1995). Therefore, in general, the range a d of deregularized variogram can be computed as follows (Clark, 1977;Atkinson & Curran, 1995), where a is the range of the regularized variogram, and L is the pixel size. Moreover, the sill c d , a reverse of the equation given by (Journel & Huijbregts, 1981) for spherical models was used, where c is the sill of the regularized variogram, and a d is the range of the deregularized variogram.…”

Section: Variogram Deregularizationmentioning

confidence: 97%