Spatial geostatistics, lateral variability, and scale invariance of variogram properties within dolomites

Previous research into the lateral variability of dolostones' attributes at outcrop and subsurface interwell scale suggest that variogram properties such as the correlation range and hole-effect wavelength may follow a power-law function with respect to the scale at which sampling is conducted. The research question posed in this thesis is, does a power-law relationship exist between variogram attributes (i.e., correlation range and hole-effect wavelength) and sampling scale across greater magnitudes of scale (i.e. from thin-section to interwell scale). This question was addressed by analyzing the spatial variability of various properties of dolomites at sub-meter sampling scales (thin-section, outcrop slab, and outcrop photomosaics). Porosity and permeability were analyzed on five outcrop slabs and were shown to have either long-range trends or apparent correlation ranges of 4-20 cm and hole-effects with wavelengths of 20-30 cm. Thin-sections were collected from each of these slabs and analyzed for porosity. These sample were shown to have either long-range trends or apparent correlation ranges of 0.2-0.5 cm and hole-effect wavelengths of 2.8-4.6 cm. Outcrop photomosaics were shown to have either long-range trends or apparent correlation ranges of 5-230 cm and hole effects with wavelengths of 70-560 cm.;The data collected herein and previously collected literature data do define a power-law relationship for both correlation range and hole-effect wavelength over several magnitudes of sampling scale. Three possible hypotheses were considered as explanations of this relationship. (1) Variogram attributes are related to the sample spacing. (2) Variogram attributes are related to the volume of investigation of each sample. (3) Variogram attributes are illusory because short transect lengths (i.e., slabs and thin-sections) are not long enough to exhibit statistical stationarity. Hypotheses 2 and 3 were shown to be the most likely explanations for the presence of the power-law function. Hypothesis 2 likely accounts for the long correlation ranges seen in subsurface data because each sampling point captures heterogeneity at a much larger scale than sampling for outcrop studies. Hypothesis 3 is valid because short transects (slabs and thin-sections ) do not exhibit stationarity relative to the longer outcrop studies. Therefore the variograms based on slab and thin-section data have characteristics that should not be interpreted as ranges and hole-effects. While it is possible that a power-law relationship could exist for correlation ranges and hole-effect wavelengths from micro to megascales, the data collected herein does not demonstrate its existence.